February 2011 

NASA/CR–2011-217051 

4D Dynamic Required Navigation Performance 

Final Report 

Daniel M. Finkelsztein and James L. Sturdy 

Raytheon Company, Hampton, Virginia 

Omeed Alaverdi 

SAIC, Lexington Park, Maryland 

Joachim K. Hochwarth 

GE Aviation, Grand Rapids, Michigan 
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February 2011 

NASA/CR–2011-217051 

4D Dynamic Required Navigation Performance 

Final Report 

Daniel M. Finkelsztein and James L. Sturdy 

Raytheon Company, Hampton, Virginia 

Omeed Alaverdi 

SAIC, Lexington Park, Maryland 

Joachim K. Hochwarth 

GE Aviation, Grand Rapids, Michigan 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1 Summary 

New advanced four dimensional trajectory (4DT) procedures under consideration for the Next 

Generation Air Transportation System (NextGen) require an aircraft to precisely navigate 

relative to a moving reference such as another aircraft. Examples are Self-Separation for enroute operations and Interval Management for in-trail and merging operations. The current 

construct of Required Navigation Performance (RNP), defined for fixed-reference-frame 

navigation, is not sufficiently specified to be applicable to defining performance levels of such 

air-to-air procedures. An extension of RNP to air-to-air navigation would enable these advanced 

procedures to be implemented with a specified level of performance. The objective of this 

research effort was to propose new 4D Dynamic RNP constructs that account for the dynamic 

spatial and temporal nature of Interval Management and Self-Separation, develop mathematical 

models of the Dynamic RNP constructs, “Required Self-Separation Performance” and “Required 

Interval Management Performance,” and to analyze the performance characteristics of these airto-air procedures using the newly developed models. This final report summarizes the activities 

led by Raytheon, in collaboration with GE Aviation and SAIC, and presents the results from this 

research effort to expand the RNP concept to a dynamic 4D frame of reference. 

5
6 

TABLE OF CONTENTS 

1 Summary 5

2 Introduction 14

3 Technical Approach 15

4 RNP State-of-the-Art Assessment 15

5 4D Dynamic RNP Overview 16

5.1 Introduction 16

6 Required Self Separation Performance Conceptual Framework 17

6.1 Introduction 17

6.1.1 Separation Background 17

6.1.2 Enabling Technologies: RNP and ADS-B 17

6.2 System Overview 18

6.2.1 ADS-B Transceiver 18

6.2.2 Surveillance 19

6.2.3 State Estimation 19

6.2.4 Navigation 19

6.2.5 ASAS 19

6.2.6 Display and System Alerting 19

6.3 Required Self-Separation Performance 19

6.3.1 Separation System Categories 20

6.3.2 RSSP Concept 20

6.3.3 Application of RSSP 21

6.3.4 RSSP Types 21

6.4 Operational Goals and Applications 21

6.4.1 Overview 22

6.4.2 Concept of Operation 22

6.4.3 Procedure Description 24

6.4.4 Errors and Uncertainties 25

6.5 Assumptions 27

6.5.1 Assumptions Applicable to All Classes 27

6.5.2 Assumptions Applicable to Class A Systems 28

6.6 Definitions 28

6.6.1 Required Self-Separation Performance 28

6.6.2 Trajectory 30

6.6.3 Error Terms 30

6.6.4 Separation Containment Concept 31

6.7 System Performance Requirements 33

6.7.1 Accuracy 33

6.7.2 State Data Accuracy Metrics 33

6.7.3 Additional Metrics 34

6.8 Integrity 34

6.9 Continuity 35

6.10 Availability 35

7 Required Interval Management Performance Conceptual Framework 36

7.1 Introduction 36
7 

7.2 System Overview 36

7.2.1 State Estimation 39

7.2.2 4D Path Definition 40

7.2.3 Speed Control 41

7.2.4 User Interface 42

7.3 Required Interval Management Performance 42

7.3.1 Time Navigation 42

7.3.2 RIMP Concept 42

7.3.3 Application of RIMP 43

7.3.4 RIMP Types 43

7.4 Assumptions 43

7.4.1 Reference Aircraft State 43

7.4.2 Ownship State 44

7.4.3 Reference Aircraft Intent 44

7.4.4 Ownship Intent 44

7.5 Definitions 44

7.5.1 Required Interval Management Performance 44

7.5.2 Trajectory 45

7.5.3 Error Terms 46

7.6 System Performance Requirements 51

7.6.1 Accuracy 51

7.6.2 Integrity 51

7.6.3 Continuity 51

7.6.4 Availability 51

8 Required Self Separation Performance Model 52

8.1 Introduction 52

8.2 Section Overview 52

8.3 TSE Model Derivation 52

8.3.1 Expected Separation Computation 52

8.3.2 Analysis of Sensitivity of Expected Minimum Separation to State Variable 

Variations 54

8.3.3 Analysis of Error Distribution of Minimum Separation 58

8.3.4 Extension to Other Uncertainty/Error Terms 59

8.4 MATLAB Linearized Sensitivity Model Description 59

8.4.1 Scenario Inputs 59

8.4.2 Sensitivity Analysis Modeling 60

8.4.3 Distribution Analysis Modeling 61

8.5 MATLAB Model Operation 63

8.5.1 Overview 63

8.5.2 System Requirements 63

8.5.3 Installation & Start 63

8.5.4 Scenario Definition 63

8.5.5 Sensitivity Analysis 65

8.5.6 Distribution Analysis 70

8.5.7 Class B Systems 77

9 Required Interval Management Performance Model 82
8 

9.1.1 Introduction 82

9.1.2 Section Overview 82

9.2 RIMP Operation Model 83

9.2.1 Modeling Approach 84

9.2.2 Aircraft Model Configuration 87

9.2.3 Elements of Uncertainty and Operational Limits 91

9.3 Aircraft and Environment Model Derivation 92

9.3.1 Point-Mass Equations of Motion 92

9.3.2 Aircraft Outputs and Measured Parameters 94

9.3.3 Navigation Calculations 97

9.3.4 Guidance and Control Calculations 105

9.4 TTG Estimation 107

9.4.1 Velocity Profile and Associated Assumptions 108

9.4.2 Distance and Time-to-Go Calculations 111

9.4.3 TTG Algorithms 112

9.5 RIMP Speed Control 112

9.5.1 Minimum Required Speed Change 113

9.5.2 Nominal Speed Change 114

9.5.3 Speed Change Logic 114

9.6 Model Setup and Execution Procedures 115

9.6.1 Model Setup 115

9.6.2 Individual Simulation Runs 119

9.6.3 Monte Carlo Simulation Runs 121

10 Required Self Separation Performance Study 122

10.1 Objectives and Motivation 122

10.2 Background 122

10.3 Simulation Platform and Scenarios Description 123

10.3.1 Basic Separation Distribution Model 123

10.3.2 Computation of Required Target RCPA 125

10.3.3 Outer Framework 125

10.4 Study Design 125

10.4.1 Plan and Approach 125

10.4.2 Metrics 126

10.4.3 Test Plan 126

10.5 Study Results 127

10.5.1 Class A Systems 127

10.5.2 Class B Systems 131

10.5.3 Comparison of Class A and Class B RSSP Systems 135

10.6 RSSP Study Conclusion 135

11 Required interval Management Performance Study 136

11.1 Objectives and Motivation 136

11.2 Background 137

11.3 Simulation Platform and Scenarios Description 137

11.3.1 Aircraft Flight Dynamics Model 138

11.3.2 Air Traffic Simulation 141

11.3.3 Monte Carlo Simulation 144
9 

11.4 RIMP Study Design 144

11.4.1 RIMP Study Plan and Approach 144

11.4.2 Metrics 147

11.4.3 RIMP Test Plan 148

11.5 RIMP Study Results 149

11.5.1 Baseline Test 149

11.5.2 Single Pair IM 159

11.5.3 Large IM Stream 170

11.6 RIMP Study Conclusion 181

12 Conclusions 182

12.1 Observations 182

12.2 Next Steps 183

Acknowledgement 185

References 185

Acronyms 186
10 

LIST OF FIGURES 

Figure 1 - Separation System Functional Block Diagram ............................................................ 18

Figure 2 - Original Estimated Geometry ...................................................................................... 23

Figure 3 - Estimated Geometry after Ownship Slows .................................................................. 23

Figure 4 - Estimated Geometry after Ownship Turns ................................................................... 24

Figure 5 - Estimated Geometry with Uncertainties and Errors ..................................................... 26

Figure 6 - Overview of APS-RIMP Operation ............................................................................. 37

Figure 7 - APS System Block Diagram ........................................................................................ 39

Figure 8 - Representative Layout of Paired Airplanes in Precision Spacing Mode ..................... 40

Figure 9 - Along Track Components of Navigation Error Terms ................................................ 46

Figure 10 - Reference Time Profile with Uncertainty .................................................................. 48

Figure 11 - Along-Track and Cross-Track Containment Region ................................................. 49

Figure 12 – RSSPSimulator Control Panel ................................................................................... 63

Figure 13 – Mode Control Panel Configuration for Scenario Inputs ............................................ 64

Figure 14 – Scenario Display Example ........................................................................................ 65

Figure 15 – SPS Control Panel Configuration to Vary Crossing Angle (). ............................... 66

Figure 16 – SPS Sensitivity to Position Error as a Function of Crossing Angle (). ................... 67

Figure 17 – SPS Sensitivity to Velocity Error as a Function of Crossing Angle (). .................. 68

Figure 18 – TPS Control Panel Configuration Varying SO and SR. .............................................. 69

Figure 19 – TPS Sensitivity to Position Error as a Function of SO and SR. .................................. 69

Figure 20 – TPS Sensitivity to Velocity Error as a Function of SO and SR. .................................. 70

Figure 21 – Distribution Analysis Controls. ................................................................................. 71

Figure 22 – Histogram Limit Warning Dialog. ............................................................................ 71

Figure 23 – Class A Error Distribution Inputs. ............................................................................. 72

Figure 24 – Class A Single Distribution Control Panel Setup. ..................................................... 72

Figure 25 – Class A Single Distribution Result. ........................................................................... 73

Figure 26 – Class A SPS Distribution Analysis Control Panel Setup. ......................................... 74

Figure 27 – Class A SPS Distribution Analysis Distribution Result. ........................................... 74

Figure 28 – Class A SPS Distribution Analysis Statistics Result. ................................................ 75

Figure 29 – Class A TPS Distribution Analysis Control Panel Setup. ......................................... 76

Figure 30 – Class A TPS Distribution Analysis Statistics Result. ................................................ 76

Figure 31 – Class B Error Distribution Inputs. ............................................................................. 77

Figure 32 – Class B Single Distribution Control Panel Setup. ..................................................... 78

Figure 33 – Class B Single Distribution Result. ........................................................................... 79

Figure 34 – Class B SPS Distribution Analysis Control Panel Setup........................................... 80

Figure 35 – Class B SPS Distribution Analysis Distribution Result. ........................................... 80

Figure 36 – Class B SPS Distribution Analysis Statistics Result. ................................................ 81

Figure 37 – RIMP Operation Flow Diagram ................................................................................ 83

Figure 38 – RIMP Operation Model ............................................................................................. 85

Figure 39 – RIMP model data flow .............................................................................................. 86

Figure 40 – RIMP Aircraft Model ................................................................................................ 88

Figure 41 – Flight Segment and Navigation Modes ..................................................................... 98

Figure 42 – TRACK Mode Parameters ........................................................................................ 98
11 

Figure 43 – Required Distance for Turn to New Course ............................................................ 100

Figure 44 – Waypoint Capture in TURN Mode ......................................................................... 101

Figure 45 – Corrected Segment Length ...................................................................................... 102

Figure 46 – True Airspeed Control System ................................................................................ 106

Figure 47 – Vertical Flight Path Control System ........................................................................ 106

Figure 48 – Lateral Control System in TRACK mode ............................................................... 107

Figure 49 – Assumed velocity profile for level segments .......................................................... 108

Figure 50 – Assumed velocity profile for descent segments ...................................................... 109

Figure 51 – Speed Change Logic ................................................................................................ 114

Figure 52 - Original Estimated Geometry .................................................................................. 123

Figure 53 -Scenarios A-1—A-4: Required Buffer versus Crossing Angle and Normalized Speed 

Difference. .................................................................................................................................. 129

Figure 54 - Scenarios A-1—A-4: Required Buffer versus Crossing Angle, Normalized Speed 

Difference in the Z-Axis. ............................................................................................................ 130

Figure 55 - Scenarios A-1—A-4: Required Buffer versus Normalized Speed Difference, 

Crossing Angle in the Z-Axis. .................................................................................................... 131

Figure 56 - Scenarios B-1 – B-6: Required Buffer versus Crossing Angle, Normalized Speed 

Difference in the Z-Axis. ............................................................................................................ 134

Figure 57 - Scenarios B-1 – B-6: Required Buffer versus Normalized Speed Difference, Crossing 

Angle in the Z-Axis. ................................................................................................................... 135

Figure 58 Aircraft Flight Dynamics Model in Simulink ............................................................ 139

Figure 59 Sensor Model Parameters ........................................................................................... 140

Figure 60 Aircraft Indicated Airspeed Limits ............................................................................. 140

Figure 61 Air Traffic Model in Simulink ................................................................................... 142

Figure 62 Aircraft Setup File for IM Operations ........................................................................ 143

Figure 63 Global Simulation Constants ...................................................................................... 143

Figure 64 Flight Plan File for BGEST Arrival ........................................................................... 145

Figure 65 Flight Plan Surface Map ............................................................................................. 146

Figure 66 Wind Temperature and Pressure Data ........................................................................ 147

Figure 67 Altitude Profile ........................................................................................................... 151

Figure 68 Flight Segment Completion Histories ........................................................................ 151

Figure 69 Indicated Airspeed Profile .......................................................................................... 152

Figure 70 Mach Number Profile ................................................................................................. 152

Figure 71 True Airspeed Profile ................................................................................................. 153

Figure 72 Measured Wind Magnitudes and Direction ................................................................ 153

Figure 73 Resulting Ground Speed ............................................................................................. 154

Figure 74 Rate of Climb Profile.................................................................................................. 154

Figure 75 Along-Track Distance to Go for Current Segment ..................................................... 155

Figure 76 Computed TTG for Nominal Case ............................................................................. 156

Figure 77 Distribution of Final Spacing with Wind Uncertainty and no IM .............................. 157

Figure 78 Predicted Spacing Signal ............................................................................................ 158

Figure 79 Final Spacing with Wind and Initial Spacing Uncertainty and no IM ....................... 159

Figure 80 Achieved Spacing Histogram ..................................................................................... 161

Figure 81 IM Frequency Histogram Total Operation ................................................................. 161

Figure 82 IM Frequency Histogram below 10K ......................................................................... 162

Figure 83 IM Frequency Histogram per Segment ...................................................................... 162
12 

Figure 84 Estimated Spacing and IM Command History ........................................................... 163

Figure 85 Achieved Spacing Histogram. .................................................................................... 165

Figure 86 IM Frequency Histogram Total Operation ................................................................. 165

Figure 87 IM Frequency Histogram below 10K ......................................................................... 166

Figure 88 IM Frequency Histogram per Segment ...................................................................... 166

Figure 89 Achieved Spacing Histogram. .................................................................................... 168

Figure 90 IM Frequency Histogram Total Operation ................................................................. 168

Figure 91 IM Frequency Histogram below 10K ......................................................................... 169

Figure 92 IM Frequency Histogram per Segment ...................................................................... 169

Figure 93 Achieved Spacing Histogram. .................................................................................... 171

Figure 94 IM Frequency Histogram Total Operation ................................................................. 171

Figure 95 IM Frequency Histogram below 10K ......................................................................... 172

Figure 96 IM Frequency Histogram per Segment AC#2 ............................................................ 172

Figure 97 IM Frequency Histogram per Segment AC#6 ............................................................ 173

Figure 98 Achieved Spacing Histogram. .................................................................................... 175

Figure 99 IM Frequency Histogram Total Operation ................................................................. 175

Figure 100 IM Frequency Histogram below 10K ....................................................................... 176

Figure 101 IM Frequency Histogram per Segment AC#2 .......................................................... 176

Figure 102 IM Frequency Histogram per Segment AC#6 .......................................................... 177

Figure 103 Achieved Spacing Histogram. .................................................................................. 179

Figure 104 IM Frequency Histogram Total Operation ............................................................... 179

Figure 105 IM Frequency Histogram below 10K ....................................................................... 180

Figure 106 IM Frequency Histogram per Segment AC#2 .......................................................... 180

Figure 107 IM Frequency Histogram per Segment AC#6 .......................................................... 181
13 

LIST OF TABLES 

Table 1: Scenario geometry parameters 126

Table 2: Class A system condition parameters 127

Table 3: Class B system condition parameters 127

Table 4: Class A system statistical results 128

Table 5: Class B system statistical results 132

Table 6: Class B system analysis of uncertainties 133

Table 7: Statistics for baseline test case with wind uncertainty 157

Table 8: Statistics for baseline test case with wind and initial spacing uncertainty 159

Table 9: Statistics for single pair IM test case without uncertainty 160

Table 10: Statistics for single pair IM test case with wind uncertainty 164

Table 11: Statistics for single pair IM test case with wind and initial spacing uncertainty 167

Table 12: Statistics for multi pair IM test case with wind uncertainty 174

Table 13: Statistics for multi pair IM test case with wind and initial spacing uncertainty 178
2 Introduction 

The Joint Program Development Office (JPDO) Concept of Operations for the Next Generation 

Air Transportation System (NextGen) considers Four Dimensional Trajectory (4DT) procedures 

a key enabler to Trajectory Based Operations (TBO). The JPDO defines 4DT as “a precise 

description of an aircraft path in space and time”. While NextGen assumes that this path is 

defined within an Earth-reference frame, many 4DT procedure implementations will require an 

aircraft to precisely navigate relative to a moving reference such as another aircraft to form 

aggregate flows or a weather cell to allow for flows to shift. Current methods of implementing 

routes and flight paths rely on aircraft meeting a Required Navigation Performance (RNP) 

specification and being equipped with a monitoring and alerting capability to annunciate when 

the aircraft system is unable to meet the performance specification required for the operation. 

Since all aircraft today operate within the NAS relative to fixed reference points, the current 

RNP definition is deemed satisfactory. However, it is not well understood how the current RNP 

construct will support NextGen 4DT procedures where aircraft operate relative to each other or 

to other dynamic frames of reference. The objective of this research effort is to analyze candidate 

4DT procedures from both an Air Navigation Service Provider (ANSP) and aircraft perspective, 

to identify their specific navigational requirements, assess the shortcomings of the current RNP 

construct to meet these requirements, to propose an extended 4D “dynamic” RNP construct that 

accounts for the dynamic spatial and temporal nature of the selected 4DT procedures, and finally, 

to model and analyze the Dynamic RNP characteristics of these procedures. 

This Final Report summarizes the activities led by Raytheon, in collaboration with GE Aviation 

and SAIC, and presents the results from this research effort to expand the RNP concept to four 

dimensions relative to a dynamic frame of reference. 

A comprehensive assessment of the state-of-the-art international implementation of current RNP 

was completed and presented in the Contractor Report “RNP State-of-the-Art Assessment, 

Version 4, 17 December 2008”. The team defined in detail two 4DT operations, Airborne 

Precision Spacing and Self-Separation, that are ideally suited to be supported by 4D Dynamic 

RNP and developed their respective conceptual frameworks, “Required Interval Management 

Performance (RIMP) Version 1.1, 13 April 2009” and “Required Self Separation Performance 

(RSSP) Version 1.1, 13 April 2009”. 

Once the conceptual framework for the RSSP and RIMP operation was established, the team 

developed respective mathematical models and implemented them in simulation tools. The 

RSSP simulator , developed using MATLAB, allows a researcher to assess the impact that 

various error sources and uncertainties have on an aircraft’s ability to accurately predict (and 

hence control) its minimum separation from a reference aircraft. The model includes 

computation of the encounter geometry, linearized analysis of the impact of error sources on the 

predicted separation, and a Monte Carlo based analysis of the impact of error sources on the 

distribution of separation as a function of encounter geometry in the context of key error terms 

for RSSP systems. The RSSP model and its simulation tool was documented in “Required Self 

Separation Performance Modeling, Version 2.2, 31 March 2010”. For RIMP operations, the 

team developed a framework to analyze RIMP operations in the presence of various uncertainties 

and operational limitations. An accompanying MATLAB/Simulink model provided a means for 

14
a time-based simulation of paired approaches using the RIMP concept. The RIMP model and its 

simulation tool is documented in “Required Interval Management Performance Modeling, 

Version 3.1, 30 May 2010”. 

Finally, the team performed a RSSP and a RIMP study using these simulation tools. The RSSP 

study explored the relationship between various lateral separation geometry parameters (speeds, 

crossing angle, and time-to-go) and system accuracy metrics (navigation accuracy and 

surveillance accuracy) and the statistical distribution of expected actual separation distances. 

The RIMP study assessed the feasibility of a long stream of aircraft to perform RIMP operation 

in the presence of various uncertainties and aims at quantifying the performance of airborne IM 

and define the limits of safe and stable operation. The results of the RSSP and RIMP studies 

were respectively documented in “Required Self Separation Performance Study Report, Version 

1.1, 30 July 2010” and “Required Interval Management Performance Study Report, Version 1.1, 

30 July 2010”. 

3 Technical Approach 

The Raytheon team brings a broad and comprehensive perspective on RNP and Trajectory Based 

Operations including operational support to the FAA for the implementation of Area Navigation 

(RNAV) and RNP, implementation of advanced 4D capable avionics for airframe manufacturers 

and operational support to operators, modeling and simulation development of advanced CNS 

capabilities into ATOS, and support of various experiments for the NextGen Airspace Project. 

While our technical approach attempts to use and build upon the content and document structure 

of the Minimum Aviation System Performance Standards (MASPS, DO-236B) and Minimum 

Operational Performance Specifications (MOPS, DO-283A) developed by RTCA, we have 

leveraged the team’s breadth of perspective and expertise to determine the practicability of 

extending this construct to 4D RNP relative to a dynamic frame of reference. 

The key elements of our technical approach were: 

1. Assessment of international state-of-the-art in 2D, 3D and 4D RNP capability and use of 

RNP procedures and performance levels 

2. Selection of candidate NextGen procedures and applications, and definition of the 

relevant 4D Dynamic RNP requirements 

3. Develop a mathematical construct and a model for 4D Dynamic RNP in support of the 

selected procedures 

4. Implement these models into simulation tools that enable the of operation specific 

performance metrics as a function of errors and uncertainties 

5. Perform one study for each selected procedures using the developed simulation tools 

4 RNP State-of-the-Art Assessment 

The “RNP State-of-the-Art Assessment, v4, 17 December 2008” retraces the concept of 

Performance Based Navigation (PBN) from its origin in the early nineties and the development 

of satellite based technologies to the latest implementations throughout the world. The report 

documents the conceptual and operational perspectives of lateral, vertical, and time Required 

Navigation Performance. It also provides a directory of the documents related to 

PBN/RNAV/RNP, the RNAV and RNP Procedures and a status of international RNP 

implementation. 

15
5 4D Dynamic RNP Overview 

5.1 Introduction 

RTCA document DO 236B contains MASPS for RNAV systems operating in an RNP 

environment. This report will start with DO-236B requirements and apply them to developing 

performance requirements for aircraft self-separation and self-spacing applications. 

The Next Generation Air Transportation System (NextGen) is envisioned as a revolutionary 

transformation of the U.S. airspace to a performance-based, scalable, network enabled system 

that will be flexible enough to meet future air traffic needs. One of the major transformations is 

the use of TBO as the main mechanism for managing traffic at high density or in highly-complex 

airspace. These TBOs will be specified between the user and the air navigation service provider 

(ANSP) and agreed in a “contract”, using advanced automation. Overall, preferences for all users 

are accommodated to the greatest extent possible, and trajectories constrained only to the extent 

required to accommodate congestion, or for security, safety or environmental reasons. Changes 

to that “contract” will be made collaboratively, balancing the user preferences with the ANSP 

constraints. 

A major element of TBO is trajectory-based separation management (SM), which uses 

automation and shared trajectories to better manage separation among aircraft and airspace and 

hazards such as weather and terrain. TBO provides a means for maintaining a target level of 

safety (TLS) while increasing traffic densities well beyond what is possible today given the 

workload, uncertainty, and execution delays inherent in current ground-based air traffic 

management. 

16
6 Required Self Separation Performance Conceptual 

Framework 

6.1 Introduction 

This section examines what would be required in order to ensure a TLS for a scenario, in which 

aircraft have primary responsibility for providing separation from other aircraft in contrast to the 

current system where that responsibility lies with the ground-based ANSP. It describes the 

system components, identifies key system parameters, and derives an RNP-like construct for 

self-separation operations. 

6.1.1 Separation Background 

Separation is the term used to describe the act of keeping aircraft at such distances from each 

other that the risk of their colliding with each other is below a TLS. Such separation distances 

are specified as horizontal or vertical standards. Separation in the horizontal plane can be 

applied either longitudinally, spacing aircraft behind each other; laterally, spacing aircraft side by 

side; or a composite of the two, providing separation for aircraft whose paths cross. When not 

horizontally separated, vertical separation is achieved by aircraft operating at different altitudes 

(flight levels). The required separation is usually expressed in terms of minimum distances in 

each dimension, which should not be infringed. In the case of horizontal separation, the 

minimum distance can be expressed in either nautical miles, degrees of angular displacement 

(e.g.: on departure) or, in the longitudinal dimension, as either time-based or distance-based 

minima. 

The separation minima used by Air Traffic Services (ATS) today in radar-controlled airspace 

takes into account that any decisions are based on a picture of the airspace derived from radar 

surveillance. The separation minima used must therefore ensure that even in the worst case 

surveillance conditions the positive separation of aircraft can be maintained. The 

implementation of Automatic Dependent Surveillance (ADS) and digital data link 

communications technologies into the NAS will provide significant improvements beyond 

current procedural ground-based control. This is due to the increased frequency and accuracy 

of position updates as well as information on the future intent of the aircraft. The technology 

should enable significant reductions in separation and spacing minima. 

6.1.2 Enabling Technologies: RNP and ADS-B 

Figure 1-1, “Navigation System Block Diagram”, from DO-236B provides a general description 

of the functions and describes the relationship between the various elements of the navigation 

system. One of the elements critical to precision separation is the path definition function since 

it computes the defined path to be flown in relation to the vertical, horizontal, and time 

dimensions. The RNP concept provides the means for quantifying lateral containment integrity 

and containment continuity, both of which are needed in order to demonstrate a TLS for TBO. 

Unfortunately, DO-236B only discusses, in detail, lateral path definition whereas NextGen 

envisions TBOs based on full 4DTs. Two dimensional trajectory (2DT) operations are defined 

by longitudinal and lateral positions and define a ground track. Three dimensional trajectories 

17
(3DT) add the vertical position so that altitude is defined anywhere along the ground track. 4DT 

adds the time element to trajectory-based operations. 

Full 4D TBO will require updated equipment standards and aircraft capabilities: 

a. Navigation and Guidance - more precise definition of vertical profiles and better 

definition and precision for along track/time-control, 

b. Communication – ability to describe trajectory windows and performance in all 

dimensions and to access available data and negotiate trajectories, 

c. Surveillance – ability to conduct relative navigation (spacing) or accept delegated 

separation, and finally 

d. Flight Crew Displays and Decision Support – flight crew awareness and management of 

TBO. 

This report assumes the use of a surveillance technology with the full capabilities defined for 

Automatic Dependent Surveillance – Broadcast (ADS-B) systems defined in DO-242A to 

provide aircraft with position, velocity, and identification data about the aircraft operating in 

their vicinity, and in some instances, more complete intent information than is currently defined 

in the published ADS-B standards. 

6.2 System Overview 

A key new technology required to support self-separation is an Airborne Separation Assistance 

System (ASAS). This system is needed to monitor the state and intent of the ownship aircraft in 

the context of the state and intent of reference aircraft as determined by the onboard surveillance 

function. The ASAS is expected to not only alert the flight crew to predicted losses of separation 

(conflicts), but also to supply one or more maneuvers that, if implemented, should resolve such 

conflict(s). The relationship of the ASAS to other aircraft systems is depicted in Figure 1. 

Figure 1 - Separation System Functional Block Diagram 

6.2.1 ADS-B Transceiver 

The ADS-B transceiver transmits and receives ADS-B signals. It collects outgoing data from the 

position estimation and navigation functions. Received ADS-B messages are passed to the 

Surveillance function for processing. 

18
6.2.2 Surveillance 

The Surveillance function assembles ADS-B messages and maintains target and track 

information for each aircraft within ADS-B range. The Surveillance function is responsible for 

track initiation and maintenance and ensures that a consistent set of intent data for cooperating 

targets is available for the other aircraft functions. 

6.2.3 State Estimation 

The state estimation function maintains the estimate of the aircraft’s position, velocity, and the 

current time. It provides estimates of the quality of the position and velocity data. This function 

also includes the real-time estimation of the local wind vector based on the difference between 

ground-referenced velocity and air-referenced velocity. 

6.2.4 Navigation 

The navigation function generates the expected 4D trajectory of the ownship aircraft (ownship 

intent). This trajectory is updated in accordance with changes generated by the pilot or when the 

aircraft state deviates from the trajectory in dimensions that are not being controlled (e.g. time 

drift when there is not RTA constraint). The navigation function also provides the guidance 

signals to the flight control system and to the flight crew. 

6.2.5 ASAS 

The ASAS function compares the aircraft state and trajectory to the estimated state and trajectory 

of reference aircraft to monitor predicted separation, detect potential conflicts, and provide 

conflict resolution guidance in order to ensure that safe separation will be maintained. The 

estimated trajectory of reference aircraft is based either on broadcast intent information (Class A 

systems) or projection of state information in the absence of reference intent. When a loss of 

separation is projected within a specified time horizon, the ASAS function presents an indication 

to the flight crew and computes one or more resolution maneuvers for pilot selection and 

execution. 

6.2.6 Display and System Alerting 

This function encompasses the interfaces between the aircraft systems and the flight crew. 

6.3 Required Self-Separation Performance 

Analogous to the different performance levels used to define RNP capabilities and airspace user 

requirements, a suitable performance characterization, Required Self-Separation Performance 

(RSSP), is needed for ASAS equipped aircraft. The expectation is that an RSSP performance 

level will define a sufficient set of system performance attributes to permit determination of the 

separation buffer required to attain a specified TLS. Unlike RNP, the RSSP will need to be 

based on a multi-dimensional set of metrics in order to encompass the main interdependent 

parameters that affect achieving the target level of safety1 (TLS) for a given traffic 

 

1

 TLS = the requisite Target Level of Safety for the airspace domain, e.g. Where, “fatal accidents per flight hour” is 

considered to be an appropriate metric, a TLS of 5x10-9 fatal accidents per flight hour per dimension for the enroute domain. NOTE: Where fatal accidents per flight hour is not considered to be an appropriate metric, 

justifiable alternative metrics and methods of assessment providing an acceptable TLS may be established. 

19
density/complexity (or, conversely, the traffic density/complexity for which a given TLS can be 

sustained) 2. These interdependent parameters include: 

a) Aircraft navigation and guidance performance (4D RNP); 

b) Surveillance performance (range, accuracy, update rate/latency, update reliability, level 

of intent detail, and level of intent accuracy which has not yet been defined); 

c) Conflict resolution algorithm capabilities (how boxed in does the ownship need to get 

before no resolution can be found); and 

d) Look ahead time horizon (how far ahead are conflicts typically detected and resolution 

maneuvers reflected in intent broadcasts). 

The expectation is that controlling these parameters will permit progressively reducing 

separation standards by reducing uncertainty and unpredictability in the overall system, thereby 

permitting higher-performing aircraft to operate in higher density airspace. These parameters are 

interdependent in that an increase or decrease in any single parameter may result in a 

corresponding increase or decrease in some or all of the others. RSSP is a consolidation of RNP, 

Required Communications Performance (RCP) of the surveillance datalink, and Required 

Surveillance Performance (RSP)3. RSSP is effectively a Communications, Navigation, and 

Surveillance (CNS)/ATM matrix where the outcome is directly related to a level of capability 

that is required to operate in a particular airspace. 

6.3.1 Separation System Categories 

This document recognizes two distinct classes of self-separation system. Class B systems use 

only position and velocity surveillance data (state data) from the reference aircraft in order to 

detect conflicts and determine separation maneuvers. Class A systems use not only the state data 

of class B systems, but also provide and receive intent data as part of the surveillance data 

exchange. Having this intent data greatly reduces the errors and uncertainties associated with 

linearly projecting state data into the future and permits Class A systems to make more accurate 

predictions further into the future than Class B systems and to account for planned maneuvers of 

other aircraft. The two classes are distinguished because the presence or absence of intent data 

has a significant impact on the analysis of separation errors. 

6.3.2 RSSP Concept 

The accuracy, integrity, and continuity concepts for longitudinal/time performance (and vertical 

performance) has not been addressed by ICAO or FAA to the level of detail that RNP (lateral) 

has, and the MASPS address accuracy requirements of Estimated Time of arrival (ETA), Time of 

Arrival Control (TOAC), and Vertical Navigation (VNAV) only at a very high level. Currently, 

the standards only address performance relative to a fixed, earth-referenced path and do not in 

any way address performance measurement in a dynamic frame of reference, such as flight 

 United States TERPS criteria provide an example for obstacle clearance at 10-7 per approach or 10-9 instantaneous 

collision from an obstacle. 2

 The numeric determination of a level of safety will depend upon the traffic density/complexity because the traffic 

density/complexity is a primary driver of conflict probability and the likelihood of finding conflict-free airspace 

to use for a conflict resolution. 3

 Definitions and standards for RCP and RSP are in development. The concepts of RCP and RSP have been 

discussed at ICAO. The FAA Associate Administrator for Safety talked about RCP, RSP and Required Total 

System Performance (RTSP) at the New technologies workshop in January 2007. Both are discussed in the 

NextGen ConOps. 

20
relative to another aircraft. Because of the need to be able to quantify a level of safety associated 

with self-separation operations, this document develops a proposed RSSP concept and associated 

requirements. 

For consistency with the ICAO definition of RNP, the document defines RSSP as: A statement 

of the self-separation performance accuracy, integrity, continuity, and availability necessary for 

conducting self-separation operations within self-separation permissible airspace4. 

6.3.3 Application of RSSP 

The term RSSP is applied as a descriptor for airspace where self-separation is permitted. This 

could encompass a large area, such as one or more en route or oceanic sectors, or to a more 

constrained volume such as a flow corridor as defined in the NextGen Concept of Operations 

(ConOps). 

6.3.4 RSSP Types 

The term RSSP -x-y is introduced to denote both an area of self-separation permitted airspace 

and, simultaneously, the minimum aircraft system performance required to perform selfseparation within that airspace. 

The ‘x’ indicates the class of RSSP system. Currently, two classes are defined: 

 Class A systems provide and use both state and detailed intent information. 

 Class B systems need only provide and use state information. 

The ‘y’ indicates the category of required system performance. Each such category is associated 

with performance metrics that, taken together, define an aircraft’s ability to identify and resolve 

conflicts to a particular standard. The set of metrics is enumerated in Section 6.4.4. The 

definition of the specific categories and the metric thresholds associated with them, however, is 

beyond the scope of this document. 

6.4 Operational Goals and Applications 

Under a performance-based system, as envisioned by NextGen, excess separation resulting from 

today’s control imprecision (a product of available data and controller workload) and lack of 

predictability are minimized which enables reduced separation among aircraft. Also, separation 

management responsibility may be delegated or transferred to aircraft having the capability to 

perform that function. Self separating aircraft, as envisioned by NextGen, are required to 

maintain separation from all other aircraft, and obstacles and hazards, in the airspace. Aircraft 

follow the proper separation procedures and avoid any maneuvers that generate immediate 

conflicts with any other aircraft. Self separation procedures are conducted only in self separation 

airspace. Eventually, self separating aircraft will have 4 dimensional trajectories (4DT). Either 

the separation maneuver will leave the aircraft within the constraints of its 4DT, or the flight 

crew will need to negotiate a new 4DT. This document does not attempt to define a complete 

 

4

 While self-separation is only expected to take place in RSSP designated airspace, this statement does not imply 

that such airspace may not also contain aircraft for whom separation assurance is being provided for by the 

ANSP. This report does not address such ‘mixed’ operations or the related equipage requirements for non-RSSP 

aircraft that would result if RSSP aircraft are required to maintain separation from the non-RSSP aircraft. 

21
concept of operations for self-separation, and in particular, does not address issues such as the 

unambiguous handoff of separation responsibility when transitioning into and out of selfseparation airspace. 

6.4.1 Overview 

The self-separation scenario looks at the situation when one aircraft (ownship) must pass no 

closer than a specified separation distance (or more generally, an exclusion zone)5

 from another 

aircraft (reference)6

. In order to perform this maneuver reliably, ownship must target a 

separation that includes some extra buffer in addition to the required separation to account for 

uncertainties in ownship and reference aircraft state estimation and performance. Because this 

additional buffer represents inefficiency in the operation (use of airspace, excess ownship 

maneuvering, etc.), it is desirable to keep it to the practical minimum. The necessary size of this 

additional buffer, however, is a complex function of the uncertainties inherent in the 

measurement of the ownship and reference aircraft state, knowledge of reference aircraft intent 

(future maneuvers before reaching the point of closest approach), crossing geometry, and 

predictions of future aircraft positions based on current measurements. The necessary buffer size 

is also a function of the ownship ability to execute the required maneuvers accurately and in a 

timely fashion. The goal of this analysis is to be able to specify a mathematical construct, 

analogous to RNP in the cross-track direction, that can be used for determining the size of buffer 

required to produce a given level of assurance that ownship will not pass within the exclusion 

zone of reference (or cause reference to pass through ownship’s exclusion zone) in the lateral 

and/or vertical dimensions7

, though for the purposes of this analysis, the focus will be on lateral 

separation. 

6.4.2 Concept of Operation 

The scenario assumes that at some point in time T0, ownship determines (via ATC instruction, 

automation, or pilot) that it must maneuver to miss reference (i.e. be outside of the exclusion 

zone at the point of closest approach (PCA) to the boundary of the exclusion zone) by at least a 

specified distance, Figure 2.8 The determination of which aircraft must maneuver is expected to 

be based on a set of rules, analogous to the International Regulations for Preventing Collisions 

at Sea and applied without the need for explicit coordination between aircraft. 

 

5

 This separation distance or exclusion zone must at a minimum account for the physical extent of each aircraft plus 

the wake vortex avoidance area behind each (not only must ownship not pass through reference’s wake before it 

has decayed sufficiently, ownship must also not make reference pass through ownship’s wake). If mitigation 

strategies are relied upon to account for excess uncertainty, mitigation will add to the separation distance. 6

 Or, in the general case, a non-stationary constraint such as a convective weather cell. 7

 This level of assurance is an element, along with alerting and mitigation strategies, of the analysis required to 

demonstrate a TLS. 8

 While many of the errors and uncertainties considered in this analysis pertain to the accuracy of this 

determination, the focus will be on the post-maneuver portion of the case in order to include the contribution of 

maneuver uncertainty to the total. 

22
Figure 2 - Original Estimated Geometry 

At T0, ownship has an estimate of the position and velocity of both ownship and reference. 

Ownship may also have an estimate of the future trajectory of ownship and reference (intent 

information). In the horizontal plane, the maneuver could consist of a change of speed, as shown 

in Figure 3, or direction, as shown in Figure 4. Of course, a combination of these, or a change of 

altitude are also possible resolution maneuvers. 

Figure 3 - Estimated Geometry after Ownship Slows 

VO0 

VR0 

LatR0 , LonR0 

PCA

LatO0 , LonO0 

Exclusion 

Zone

Ownship 

slows 

VO0 

VR0 

LatR0 , LonR0 

PCA

LatO0 , LonO0 

Exclusion 

Zone

23
Figure 4 - Estimated Geometry after Ownship Turns 

The scenario assumes that ownship has measurements of relevant state parameters and their 

accuracy and integrity from the onboard navigation system. It also assumes that ownship has 

some measurement of reference’s state parameters and their accuracy and integrity via ADS-B 

transmissions from reference received by ownship’s ADS-B receiver. For Class A systems, the 

scenario also assumes that some level of reference aircraft intent information is available, for 

example by ADS-B transmission of one or more Trajectory Change Points (TCPs), but the extent 

of that information will be kept as a parameter in the mathematical construct. The extent to 

which reference is permitted to maneuver without broadcasting intent adds directly to the 

uncertainty of the predicted separation and is outside the scope of this document. The analysis 

will also assume that ownship has measurements of wind and other relevant atmospheric 

parameters as well as knowledge of the accuracy and integrity of those measurements. 

6.4.3 Procedure Description 

How responsibility for maintaining the minimum separation is transferred from ATC to ownship 

is immaterial to this analysis and will not be considered as part of the procedure. It is sufficient 

to begin with the premise that ownship has this responsibility. Furthermore, barring the 

introduction of new constraints, this procedure assumes that ownship will execute the 

maneuver(s) determined after T0 without further alteration to compensate for error accumulation, 

so the maneuver(s) must include an adequate separation buffer9

. 

6.4.3.1 Timeline/Decomposition of Events 

T0 Ownship systems determine that a maneuver will be required to assure maintenance 

of the specified separation and inform the pilot. 

 

9

 This assumption/constraint is introduced in recognition of the cockpit workload implications of a strategy that 

continuously updates the maneuver based on the evolving state of the two aircraft. 

VO0 

VR0 

LatR0 , LonR0 

PCA

LatO0 , LonO0 

Exclusion 

Zone

Ownship 

turns 

24
T1 Ownship automation determines one or more candidate maneuvers that will assure 

maintenance of the specified separation and present them to the pilot.10

T2 Ownship pilot selects which maneuver candidate will be executed. 

T3 Ownship pilot configures onboard navigation system to perform the selected 

maneuver sequence. 

T4 Ownship ADS-B begins broadcasting new intent information (Class A systems). 

T5 Ownship executes planned maneuver(s). 

T6 Ownship passes Point of Closest Approach (PCA) to reference, which, if the buffer 

used was adequate, meets or exceeds the specified separation distance. 

6.4.3.2 Pilot Procedures 

The ownship pilot is assumed to have already accepted responsibility for maintaining separation 

from reference before the scenario begins. Within the scenario, the pilot is responsible for: 

1. Selecting the preferred separation maneuver(s), 

2. Configuring the navigation system to perform the selected maneuver(s), 

3. Ensuring that ownship performs the selected maneuver(s), and 

4. Monitoring to ensure that the maneuver provides at least the specified separation (this 

monitoring is aided by the onboard tools). 

6.4.4 Errors and Uncertainties 

The construct for determining the required separation performance needs to account for: 

1. Accuracy and integrity of ownship state data. 

2. Accuracy of ownship planned trajectory (both current and planned maneuver), which 

depends on: 

a. Control method accuracy (manual vs. automatic guidance) 

b. Accuracy of modeling method (e.g. 6 DOF vs. kinematic) and modeling 

parameters (thrust, drag, etc). 

c. Accuracy of predicted wind and temperature data. 

d. Accuracy of earth model. 

e. Correspondence between planned/predicted maneuver and actual maneuver 

executed (time of execution, acceleration/deceleration magnitude and transition, 

roll-in/roll-out transitions, turn-rate/bank angle used). This should directly 

correspond to 4D RNP performance level. 

3. Accuracy and integrity of reference state data. 

4. Accuracy and completeness of reference intent data (which depends on the same factors 

as the ownship data). 

5. Encounter geometry (e.g. crossing angle). 

6. ETE to point of closest approach in the resolution maneuver. 

7. Anticipated ownship FTE bounds near point of closest approach. 

The effects of these errors are represented in Figure 5. Note that it is expected that the total 

uncertainty will increase the further into the future that the aircraft state is predicted. For Class 

 

10 We assume that the tools leave sufficient time before the assumed initiation of each maneuver candidate to allow 

pilot selection and implementation to take place before the initiation time. 

25
A systems, this increase will be capped by the 4D RNP accuracy and containment associated 

with the ownship trajectory and reference aircraft intent. For Class B systems, this growth is a 

result of integrating velocity over time in order to make the predictions of future position. In 

practice, this growth may be exponential for the prediction of the reference aircraft’s state 

because of the possibility of unannounced maneuvers by the reference aircraft.11 For the 

purposes of performance evaluation, however, any such maneuvering would be considered to 

create a new encounter situation. 

Figure 5 - Estimated Geometry with Uncertainties and Errors 

The overall separation performance can be quantified by defining containment boundaries in the 

lateral and vertical dimension. These boundaries correspond to the region within which the 

actual separation will fall 95% of the time. They can also be used to define a further region that 

the actual separation will not exceed without alert with extremely high probability. Unlike in the 

case of RNP-RNAV, however, definition of the containment boundaries is not, in itself, adequate 

to be able to quantify a TLS. For RNP-RNAV, considerable freedom is left to the designer to 

make performance tradeoffs among the various system characteristics that contribute to overall 

lateral performance. For RSSP, however, there is a need to place additional constraints on this 

trade-space because the separation performance depends on the combined performance 

characteristics of two independent vehicles. 

In order to ensure that any pair of vehicles conforming to a given RSSP performance level will 

yield the desired overall separation performance, additional vehicle error, uncertainty, and 

system characteristics must be constrained for a given RSSP performance level. These errors, 

uncertainties, and system characteristics are captured using the following metrics which will be 

further defined in Section 6.7.1: 

 

11 The expectation is that such maneuvers would be away from ownship, except in the case of near-simultaneous 

maneuvering. 

VO0 

VR0

LatR0 , LonR0 

PCA

LatO0 , LonO0 

Estimation 

Uncertainty 

Exclusion 

Zone 

26
a) 4D RNP performance level12; 

b) Accuracy, integrity and continuity of surveillance data from reference aircraft, including 

intent data if present; 

c) How quickly the aircraft can respond (the relative time between recognizing that a 

conflict requires a resolution maneuver and completion of that resolution maneuver); 

d) How long before closest point of approach conflict resolution maneuvers can be selected; 

e) How long before initiation of non-emergency maneuvers new intent information is 

broadcast; 

f) Surveillance system range at X% message reception probability and other surveillance 

datalink performance characteristics; and 

g) Degrees of freedom employed for resolution maneuvers (lateral, longitudinal, vertical, 

and combinations thereof); 

6.5 Assumptions 

This document makes several assumptions applicable to all classes of system, and the remainder 

applies to Class A systems only. 

6.5.1 Assumptions Applicable to All Classes 

6.5.1.1 Uniformity of Equipage 

Because the RSSP capability is expected to be applied to a volume of airspace, it is assumed that 

all aircraft operating in that airspace will at least meet the minimum RSSP standards required by 

the airspace. 

6.5.1.2 Reference Aircraft State 

The RSSP concept assumes that all aircraft in the airspace continuously transmit and receive 

ADS-B data in compliance with RTCA DO-242A including the following parameters: 

 Target identity 

 Latitude and longitude 

 Pressure altitude 

 Position accuracy and integrity 

 North and east velocity 

 Vertical speed 

 Velocity accuracy and integrity 

 Time of applicability (reference time of the other state parameters) 

It is further assumed that ownship and reference are either using the same transmission format 

for ADS-B (e.g. 1090ES or UAT) or the airspace is served by an Automatic Dependent 

Surveillance – Rebroadcast (ADS-R) service that reliably echoes messages between the two 

transmission formats with negligible delay. 

 

12 The 4D RNP performance level defines the accuracy, integrity, and continuity performance of each aircraft in the 

lateral (RNP-RNAV), longitudinal/time (TOAC), and vertical (VNAV) dimensions. 

27
6.5.1.3 Ownship State 

The ASAS function has access to at least the set of ownship state parameters for that is available 

via ADS-B for the reference aircraft. 

6.5.2 Assumptions Applicable to Class A Systems 

6.5.2.1 Reference Aircraft Intent 

The primary differentiator between Class A and Class B systems is the availability of intent 

information for the reference aircraft. The intent information, which is not yet fully defined in 

the relevant ADS-B standards, will at least describe the intended motion of reference as a 

sequence of segments within which intermediate 4D states may be computed without significant 

accuracy degradation using the assumption of uniform (possibly zero) acceleration along the 

three primary aircraft axes. This intent information should cover at least the minimum time 

horizon required by the applicable RSSP performance level for conflict detection. The intent 

information will be updated whenever the reference aircraft’s current intent differs significantly 

relative to the accuracy requirements of the intent broadcast (which will in turn be specified 

either by the ADS-B standard or by the RSSP performance level). For RSSP performance levels 

intended for use in dense and complex airspace, there is a requirement that the reference aircraft 

broadcast a change in intent at least a minimum notification time in advance of performing a 

maneuver13 that had not been included in prior intent broadcasts. This requirement is needed to 

reduce the likelihood of simultaneous conflicting near-term maneuvers. 

6.5.2.2 Ownship Intent 

The RSSP concept assumes that a Class A system has access to detailed and accurate ownship 

intent information and associated accuracy metrics from the onboard Flight Management System 

(FMS) or equivalent. Ownship is assumed to be operating at a specified 4D RNP level (a 

successor to the current standard that defines containment in the longitudinal/time and vertical 

dimensions in addition to the lateral dimension) such that guidance is provided to maintain 

conformance to broadcast intent in all dimensions. Alternatively, ownship intent will be updated 

before significant deviations accumulate in the longitudinal/time or vertical dimensions. 

6.6 Definitions 

The definition of key terms used in this section is collected below for convenience. Italics are 

used to denote terms in these definitions that may be found in this section. 

6.6.1 Required Self-Separation Performance 

SELF-SEPARATION 

The process by which one aircraft assumes responsibility for managing the distance between 

itself and one or more reference aircraft without direct involvement of the ANSP. 

 

13 Obviously, this requirement would not apply to the non-normal case of a maneuver required for collision 

avoidance or other safety-related exigency. 

28
REQUIRED SELF-SEPARATION PERFORMANCE (RSSP)

A statement of the self-separation performance necessary for operation within a defined 

airspace. 

CLASS 

The type or category of RSSP compliant system. 

CLASS A 

An RSSP compliant system that uses intent information to provide more robust conflict detection 

and avoidance capabilities in airspace where aircraft can reasonably be expected to be 

maneuvering (e.g. terminal or high density airspace). 

CLASS B 

An RSSP compliant system that needs only position and velocity information to perform conflict 

detection and resolution. 

EXCLUSION ZONE 

The volume around an aircraft that no other aircraft may be permitted to transgress, either due to 

risk of physical contact or risk of significant wake vortex encounter. 

POINT OF CLOSEST APPROACH (PCA) 

The 3D position where the lateral distance of one aircraft to the other aircraft’s exclusion zone is 

at a minimum, or, in the vertical dimension, the 3D position where the vertical distance of one 

aircraft to the other aircraft is at a minimum while either aircraft is within the lateral extent of the 

other aircraft’s exclusion zone14. 

DESIRED SEPARATION 

The required lateral (or vertical) distance, between Ownship and Reference Aircraft at the PCA. 

DEFINED SEPARATION 

The lateral (or vertical) distance, between Ownship and Reference Aircraft at the PCA that is 

input to the ASAS function. 

ACTUAL SEPARATION 

The true lateral (or vertical) distance, between Ownship and Reference Aircraft when they are at 

the true PCA. 

ESTIMATED SEPARATION 

The output of the separation computation function, equal to the computed lateral (or vertical) 

separation at the computed PCA. This does not take into account potential future maneuvers that 

are not reflected in the ownship predicted trajectory or the reference aircraft intent. 

 

14 The concept of exclusion zone is introduced to distinguish between pure metal-on-metal contact-based standards 

versus future standards that need to account for such things as wake vortices because of their reduced dimensions. 

29
6.6.2 Trajectory 

3D POSITION 

A position in space consisting of latitude, longitude, and altitude. 

TIME REFERENCE 

The reference time source for the trajectory; for example, Universal Coordinated Time (UTC). 

CURRENT TIME 

The present time according to the time reference. 

PREDICTED TRAJECTORY 

The predicted 3D position of an aircraft as a function of time. 

INTENT 

The representation of an aircraft’s predicted trajectory that is encoded in ADS-B messages. 

ESTIMATED TIME 

The time input to the trajectory prediction function, synchronized to the time reference. 

ESTIMATED 3D POSITION 

The 3D position input to the trajectory prediction function. This position is determined by a 

Navigation function based on sensor inputs. 

ESTIMATED TIME OF ARRIVAL (ETA) 

The time at which an aircraft will arrive at a defined point on the predicted trajectory.

TIME TO GO (TTG) 

The duration from the current time until the aircraft reaches a defined point, typically in this 

discussion, the PCA. 

TRAJECTORY CHANGE POINT (TCP) 

A discrete point on the predicted trajectory containing the latitude, longitude, altitude, time, and 

speed (either CAS or Mach). This is a basic set of variables that may be standardized and/or 

expanded in the future. 

6.6.3 Error Terms 

SEPARATION ERROR 

The difference between the planned separation at the PCA when a clearance maneuver was 

formulated and the separation at the actual PCA. The tails of the statistical distribution of this 

parameter are effectively bounded for each combination of RSSP class and performance level 

and conflict encounter geometry (crossing angle and speeds). 

LATERAL SEPARATION ERROR 

The separation error measured in the horizontal plane. 

30
SEPARATION DEFINITION ERROR (SDE) 

The difference between the desired separation and the defined separation, for example due to 

approximations used in the separation function to define the exclusion zone. 

ESTIMATED TIME ERROR (ETE) 

The difference between the current time and the estimated time, synchronized to the time 

reference. 

ESTIMATED POSITION ERROR (EPE) 

The difference between the true 3D position and the estimated 3D position of the aircraft. 

COMPUTED SEPARATION ERROR (CSE) 

The difference between the estimated separation and the defined separation. 

SEPARATION ESTIMATION ERROR (SEE) 

The error in the computation of estimated separation.

SEPARATION CONTROL AUTHORITY (SCA) 

The ability of ownship to maneuver in order to compensate for computed separation error and

separation estimation error.

TOTAL SEPARATION ERROR (TSE) 

The difference between the actual separation and the desired separation. It is equal to the sum 

of the separation definition error, computed separation error, and separation estimation error 

less the separation control authority. The separation control authority is subtracted because it 

represents a controllable portion of the error. The uncontrollable portion of the error is 

analogous to the FTE in RNP. 

6.6.4 Separation Containment Concept 

The concept of separation containment is based on the concept of lateral (or cross-track) 

containment as defined in RTCA DO-236B, but the manner in which it is measured differs. For 

RSSP, the “RNP value” used in RNP-RNAV is replaced with a pair of measures, Separation 

Boundary Lateral (SBL) and Separation Boundary Vertical (SBV). This pair of measures 

defines limits, in the lateral and vertical dimensions respectively, around the desired separation

within which the actual separation will fall for at least 95% of encounters requiring some form 

of resolution maneuver. 

Separation containment integrity requirements limit the probability of the Total Separation 

Error exceeding the SBL/SBV requirements with no annunciation at least some specified time T 

before reaching the PCA. 

Separation containment continuity requirements limit the probability of a loss of function. In 

this context, function is defined as the ability to meet the separation containment requirement 

(i.e. to be within the desired SBL/SBV limits at and after time T). 

There are five possible system states relative to the SBL/SBV limits. These states are: 

31
1) S1: TSE > SBL/SBV, TTG > T, no alert (Integrity) 

2) S2: TTG > T, alert (Continuity) 

3) S3: TSE > SBL/SBV, TTG < T, no alert (Integrity) 

4) S4: TTG < T, alert (Continuity) 

5) S5: TSE < SBL/SBV, no alert (Normal Operations) 

Note: P(S1) + P(S2) + P(S3) + P(S4) + P(S5) = 1 where P(x) is the probability of the system 

being in State x. 

T is the minimum amount of time prior to reaching PCA that the alert must be made if the 

aircraft cannot achieve the desired separation. 

States 1 and 3 represent the set of events associated with a loss of separation performance with 

no annunciation (alert), and are thus the integrity requirements. The uncertainties associated 

with separation predictions generally increase with time or distance from the current position 

(primarily due to forecast wind uncertainties). Thus, the probability of State 1 will be greater 

than the probability of State 3 (P(S1) >> P(S3)) due to TTG being > T. However, it also needs to 

be considered that the control authority that ownship has to correct for these uncertainties will 

also generally decrease as the aircraft approaches the PCA. This needs to be considered when 

determining a prior time T beyond which it is critical that an annunciation be received. Thus, the 

probability of S1 must be less than XX15 per flight hour, while the probability of S3 must be less 

than YY per operation. 

The continuity requirement applies to states S2 and S4 where the loss of function is detected 

and annunciated. When current time is still far before T, the available control authority will be 

larger, and thus the probability of state S2 must be less than WW per flight hour. When the 

aircraft is close to time T the control authority is greatly reduced, and thus the probability of S4 

must be less than ZZ per flight hour. S5 represents the normal operation where the error is less 

than the SBL/SBV. 

In addition to the containment requirements, there is an uncertainty parameter associated with 

the predicted separation. This uncertainty must be bounded to ensure appropriate control 

authority is available. 

CONTAINMENT 

A set of interrelated parameters used to define the performance of a separation control system. 

These parameters are containment integrity and containment continuity. 

CONTAINMENT INTEGRITY 

A measure of confidence in the estimate, expressed as the probability that the system will detect 

and annunciate the condition where the TSE is greater than the SBL/SBV limit and the condition 

has not been detected. 

 

15 This and the following probability parameters will need to be determined based on a TLS and any mitigations that 

are identified as part of the procedures defined by the ANSP. In practice, they will likely be tailored to specific 

RSSP performance categories. 

32
CONTAINMENT CONTINUITY 

The capability of the total system to satisfy the performance limit without nonscheduled 

interruptions during the intended operation. Nonscheduled interruption is defined to be either 1) 

total loss of time prediction capability; 2) a failure of the system which is annunciated as a loss 

of time performance capability; or 3) a false annunciation of loss of time performance capability. 

CONTAINMENT REGION 

A region in both the along-track and cross-track dimensions, centered on the desired along track 

position on the desired path at a desired time T, to which the containment integrity and 

containment continuity are referenced. 

SEPARATION BOUNDARY LATERAL (SBL) 

The limit in the lateral dimension around the desired separation within which the actual 

separation will fall for at least 95% of encounters requiring some form of resolution maneuver. 

SEPARATION BOUNDARY VERTICAL (SBV) 

The limit in the vertical dimension around the desired separation within which the actual 

separation will fall for at least 95% of encounters requiring some form of resolution maneuver. 

6.7 System Performance Requirements 

6.7.1 Accuracy 

Accuracy is a measure of the difference between an aircraft‘s intended separation from a 

reference aircraft at the PCA (relative to the exclusion zone boundary) and the true separation at 

the true PCA. It is given by a value associated with a confidence interval. For example; within 

airspace requiring a particular RSSP performance level, the separation error will not be greater 

than 0.5 nautical miles for 95% of the conflicts requiring a resolution maneuver. 

6.7.2 State Data Accuracy Metrics 

6.7.2.1 The first two metrics deal with the accuracy of navigation data 

available to ownship and of the surveillance data provided by 

reference (and by inference, provided by ownship). Ownship 

Navigation Accuracy 

If ownship is operating according to 4D RNP standards, then those standards define the lateral 

and longitudinal accuracy and containment parameters for ownship state estimation both at the 

present position and along ownship’s 4D trajectory. This is the expected norm for Class A 

systems. For Class B systems, it will be necessary to break this metric into individual lateral 

position, longitudinal position, and velocity sub-metrics in order to specify the ownship 

navigation accuracy. 

6.7.2.2 Reference Surveillance Accuracy 

The position and velocity accuracy and integrity for the reference accuracy are reported in ADSB messages as the navigation accuracy for position and velocity plus the surveillance integrity 

level. When intent data is broadcast, it will need similar reported metrics. For a given RSSP 

33
performance level, a set of minimum acceptable values for each of these quality metrics will 

constitute the reference surveillance accuracy metric. 

6.7.3 Additional Metrics 

In order to characterize the ability of an aircraft to perform self-separation, especially in more 

congested airspace, additional metrics also need to be specified. 

6.7.3.1 Response Time 

This metric measures how quickly the aircraft can respond to the need to resolve a conflict. This 

applies whether the time to go to an anticipated conflict has just reached a threshold value or the 

reference aircraft has either maneuvered or broadcast new intent. It is measured as the relative 

time between recognizing that a conflict requires a resolution maneuver and completion of that 

resolution maneuver. 

6.7.3.2 Look Ahead 

Look ahead measures how long before the closest point of approach a conflict resolution 

maneuvers can be selected. It is primarily dictated by the accuracy of separation prediction as a 

function of time to go to PCA and the characteristics of the surveillance datalink (reception 

probability versus range). 

6.7.3.3 Intended Maneuver Anticipation Time 

For Class A systems, it is expected that the ASAS function will plan the resolution maneuver to 

commence some time in the future in order to provide notice of the change in intent to 

neighboring aircraft before the maneuver begins. This metric captures the amount of time before 

the initiation of non-emergency maneuvers that new intent information is broadcast. 

6.7.3.4 Surveillance Datalink Performance 

In order to function effectively, the ASAS function needs to have surveillance data at a range 

that exceeds its required look ahead time. Given assumptions of maximum closing speeds 

expected in a particular class of airspace, this can be translated into a surveillance range 

requirement defined as the range at which there is at least an X% (X >= 95) probability of 

receiving each transmission from reference. 

6.7.3.5 Resolution Flexibility 

Lower performance category ASAS systems might only provide simple lateral maneuvers, such 

as turns or sidesteps, to resolve conflicts. Higher performance systems would take advantage of 

lateral, vertical, and speed controls to find resolutions. The highest performing systems might be 

able to plan complex combinations of the three and be able to find resolution paths in the 

presence of multiple potentially conflicting aircraft. This metric seeks to enumerate or quantify 

these differing performance levels. Looked at another way, this metric captures the robustness 

with which the ASAS can find a path through multiple constraints. 

6.8 Integrity 

Integrity is a measure of the probability that information provided by a system will not be 

hazardously misleading. It is generally expressed as a probability of occurrence per operating 

34
hour, which expresses likelihood that a system will send the user bad/incorrect information that 

could cause a potentially dangerous situation without a timely warning. For example the 

likelihood that the separation error exceeds a maximum of 1.0 nautical miles, without having 

alerted the flight crew at least 90 seconds in advance of the exceedance, shall be less than 10-5

per flight hour. 

6.9 Continuity 

Continuity is the capability of a total system to perform its intended function without a nonscheduled interruption during the intended operation assuming the system was available when 

the procedure was initiated. Continuity is expressed per unit of time. For example; the 

calculations of a single thread transponder mean time to failure of 5,000 hours. 

6.10 Availability 

Availability is the probability that a system will perform its required function at the initiation of 

the intended operation. For example, the FAA target for availability of the Wide Area 

Augmentation System is 0.999. 

35
7 Required Interval Management Performance Conceptual 

Framework 

7.1 Introduction 

Airborne Precision Spacing (APS) has been developed by NASA as an element of pair-wise 

interval management to improve the merging and spacing performance during approach and 

landing for high-density operations. APS provides speed control guidance to the flight crew with 

an assigned spacing interval in the time domain at the runway threshold or a common waypoint 

with respect to a paired leading airplane (reference aircraft). Speed control guidance is derived 

from both the reference aircraft and ownship’s flight profiles, current states of the reference 

aircraft and the ownship, and speed and altitude constraints at intermediate points along the 

approach routes. The projected benefits of APS include providing the flight crew and air traffic 

controllers an automated speed control tool and advisories in delivering airplanes at the runway 

threshold at precise time intervals. This precise performance can optimize the flow of traffic 

approaching the runway, and may increase the runway throughput [1]. The concept of APS is 

also in full conformance with two key NextGen environment components as defined by Joint 

Planning and Development Office (JPDO) [2], which are Performance-Based Operations and 

Services and Aircraft Trajectory-Based Operations (TBO). The extension of Required 

Navigation Performance (RNP) in the time domain based on APS is a natural step towards the 

objectives of the NextGen air traffic environment. The Required Interval Management 

Performance (RIMP) will provide a means to assure the interval management accuracy, 

availability, continuity, and integrity during terminal and approach phases of flight. This can 

lead to a time-based interval management concept of operations. One of the benefits of a timebased RIMP is to allow enroute and terminal air traffic controllers to manage the spacing via 

time in seconds instead of closely monitoring merging traffic’s speed and distance for 

maintaining safe separation and compressing the traffic flow from en-route minima [3]. With the 

combination of APS and RIMP (APS-RIMP), workload of Air Navigation Service Provider 

(ANSP) controller can be potentially reduced. RIMP compliant airplanes will also ensure the 

performance and accuracy in delivering airplanes at managed spacing intervals. This document 

will present the concept of applying RIMP in APS, describe the system components, identify key 

system parameters, and discuss the derivation of RIMP requirements, procedures, assumptions, 

and RIMP dependency of uncertainties in airplane’s flying performance and equipment 

performance. 

7.2 System Overview 

The air traffic operations outlined in this document provide an APS-RIMP concept for aircraft 

converging to a common destination such as a runway or an intermediate navigation fix. APS is 

a potential speed advisory element of future high-density arrival and departure operations 

envisioned in the NextGen air traffic system. The focus of this APS-RIMP concept is the 

terminal and final approach portions of the flight. The precision spacing is accomplished by 

sequencing the arriving aircraft early in the initial descent phase. Each aircraft is assigned a 

reference aircraft to follow as shown in Figure 6. The paired aircraft may share a portion of the 

final assigned route, but this is not required. To achieve significant arrival density increase, this 

concept relies on precise spatial and temporal navigation that must be achieved for safe flight 

36
operation. This is accomplished by extending the RNP methodology to include an interval 

management time performance with associated uncertainty in seconds. Similar to the lateral 

RNP rating, an interval management time rating will define a maximum uncertainty and a 

containment time which is defined in section 1.4.4 of this document. 

All aircraft engaged in this APS operation must meet a certain level of RIMP. Different levels of 

RIMP are described in section 1.4.4 of this document. This applies to both the reference aircraft 

and the ownship. This concept applies to sequences of paired aircraft where the ownship itself 

becomes the reference aircraft for the subsequent following aircraft. 

Figure 6 - Overview of APS-RIMP Operation 

The APS-RIMP precision spacing concept outlined here uses many aspects of the NextGen 

technology to perform the required navigation and guidance. This includes integrated 

surveillance capabilities provided by Automated Dependent Surveillance – Broadcast (ADS-B), 

Traffic Information Service – Broadcast (TIS-B), Flight information Service – Broadcast (FIS-B) 

and associated ground radars, and precision navigation and timing provided through Global 

Navigation Surveillance System (GNSS). It also relies on digital data link communication with 

the ANSP facilities and personnel. The techniques used to accomplish the precision spacing 

include automatic 4 Dimension (4D) trajectory generation to the desired destination or 

37
navigation fix, real-time adjustments to the prescribed velocity profile to maintain the desired 

separation, and 4D navigation performance monitoring to ensure the RNP and RIMP for the 

precision spacing operation is available and attainable. 

An integrated navigation system as addressed here includes one or more navigation sensors (such 

as Inertial Navigation System (INS)/Global Positioning System (GPS)), one or more air data 

sensors, and other relevant aircraft sensors (such as angle of attack, propulsion system sensors, 

etc.) to assess overall system performance, surveillance sensors (such as ADS-B), digital data 

link with the air traffic control authority (ANSP) and extensive processing capability to define 

4D trajectories and perform the required 4D guidance (Figure 7). The navigation and guidance 

algorithms developed in support of this concept perform position estimation, path definition, path 

steering, and calculation of Time-To-Go (TTG) to the destination (navigation fix or runway 

threshold) for the aircraft as well as for the lead aircraft to which the ownship has been paired to 

follow. 

The APS system will provide a precise adjustment to the nominal speed command to achieve the 

spacing constraint, and provide the flight crew with situation indications and alert them if the 

required spacing can not be achieved. 

38
Displays & 

System 

Alerting 

Aircraft Flight 

Control System 

Speed 

Control 

4D Path 

Definition 

(ownship & 

reference 

A/C) 

State 

Estimation 

(ownship & 

reference 

A/C) 

Navigation 

Database 

Surveillance and 

Ground/Air 

Data Link 

Communication 

Navigation, 

Timing, and 

Air data 

Sensors 

Navigation 

System 

Controls 

User Interface 

State 

Estimation 

Path Definition 

Speed Control 

Navaid 

Position, 

elevation, 

and TOA or 

required 

interval 

constraint 

Sensor Data Position, 

velocity, time 

data & sensors 

in use 

Est. Position 

& velocity 

Target speed 

Speed Cmd 

Flight Plan 

Database ID, RNP-(x) 

RNAV, Fixes, Navaid IDs, 

etc. 

Ownship data 

Reference aircraft data 

Figure 7 - APS System Block Diagram 

7.2.1 State Estimation 

Position estimation requirements are outlined in RTCA DO-236B (section 1.2.1) [3]. 

Additionally, APS-RIMP operation will require timing, surveillance, wind and air temperature 

information. Precise time measurement using a common time reference and an accurate estimate 

of time uncertainty associated with that measurement are needed to arrive at the ownship timeto-go estimates. The current position and speed of reference aircraft and its intended 4D 

trajectory are needed for the reference aircraft’s time-to-go calculations. Finally, the predicted or 

measured wind field along the terminal operation and approach path is needed. 

The method by which the precise timing, surveillance data, and wind and air temperature 

estimates are acquired is not provided here. However, this document provides overall 

requirements for system performance, accuracy, integrity, continuity, and availability of the data. 

39
7.2.2 4D Path Definition 

The path definition function computes the defined 3D path and associated speeds to be flown. A 

complete 4D path will consist of a number of straight or arc segments in the horizontal plane 

plus altitude and speed specifications as shown in Figure 8. Each segment ends at a trajectory 

change point (TCP). The TCPs are assigned by a 4D trajectory generation algorithm which 

attempts to optimize the spacing and merging of the ownship with the assigned reference aircraft. 

A TCP may overlap an existing navigation fix (Area Navigation (RNAV) waypoint, airway fix, 

etc.) but this is not required. It is, however, anticipated that the trajectory endpoint will coincide 

with a known navigation fix such as the runway threshold or final approach fix (FAF). Each 

generated trajectory segment or leg will include complete horizontal and vertical navigation 

information as well as a nominal target speed (CAS or Mach) to fly. 

Figure 8 - Representative Layout of Paired Airplanes in Precision Spacing Mode 

For each leg of the trajectory, the lateral aspects of path definition determine a geographically 

fixed ground track from origin to destination. This defines a 2D component of the flight path. 

The sub-functions involved are: 

1. TCP location definition in latitude and longitude, or in the case of TCP definitions that do 

not have exact location information, an estimate of the TCP position based on the known 

or predicted state of the aircraft; 

2. Leg type definition as provided in RTCA DO-236B and ARINC 424; 

40
3. Leg transition definition which specifies turning to the next leg prior to intercepting the 

waypoint or flying through the waypoint prior to turning, etc.; 

For each leg of the trajectory, the vertical aspect of the path definition determines an elevation 

change from the TCP at the beginning of the leg to the TCP at the end of the leg. This adds the 

3rd dimension to the defined flight path. The vertical aspect sub-functions are: 

1. Altitude or flight level constraints associated with waypoint definition; (Flight Level, 

corrected barometric altitude, temperature compensated altitude, or height above ellipsoid 

for GPS based precision approach) 

2. Optional vertical angle or altitude change associated with the leg definition and vertical 

transition definition which specifies how the leg should be flown. For example, climb 

and hold or follow a glideslope; 

3. Optional speed constraint; 

The temporal aspects of the path definition determine the method used to satisfy time constraints 

associated with either the precision separation requirement or time of arrival constraint at the 

destination waypoint. The time constraints must take into account the lateral and vertical 

trajectory, the flight environment, and real-time parameters provided by the ANSP and the 

assigned reference aircraft. This is accomplished through speed control which controls the 

fourth dimension (time) for the flight path. The temporal aspect sub-functions are: 

1. Nominal interval management time; 

2. Optional speed constraint; 

7.2.3 Speed Control 

The speed control function uses the planned 4D trajectories of both the ownship and the lead 

aircraft along with current measured positions and velocities of both the ownship and the 

reference aircraft, latest wind estimates, and other associated parameters to determine the alongpath range to the spacing point, speed variation, and target speed command. These parameters 

are used to correct errors and interval variations relative to the reference aircraft. The lateral and 

vertical steering functions are defined in RTCA DO-236B [3] and are omitted from this 

document. This document instead focuses on the speed control function. 

To achieve precise arrival spacing, the navigation system must compute a number of parameters 

for both the ownship and the reference aircraft. These parameters include the distance along the 

trajectory from the current estimated position to the interval management point (such as the 

runway threshold or a common merge point), estimated ground speed for each segment of the 

trajectory, and a total TTG to arrive at the interval management point. These calculations require 

knowledge of the current and predicted states of the ownship and lead aircraft, as well as external 

factors such as the forecast wind and temperatures along the trajectory. Prior to the start of this 

paired approach, the trajectory generation algorithms generate an optimum profile for both the 

ownship and the reference aircraft. When executed, these 4D trajectories will ensure achieving 

the desired interval at the interval management point, and will provide an efficient terminal area 

operation (with constraints such as fuel burn, noise abatement, etc.). Once the aircraft are paired, 

the precision spacing function computes a change in planned speed for the ownship as needed to 

41
attain the precise spacing interval upon arrival at the interval management point. The target 

speed will be the sum of the planned trajectory speed at the current location on the trajectory plus 

the required adjustment to account for variations of aircraft states of both the ownship and the 

reference aircraft. The target speed may be periodically updated as new navigation data and/or 

new constraints become available and as the estimates of the atmospheric parameters change. 

This target speed will be displayed to the flight crew and could also be supplied to the flight 

control system for possible automated speed control operations. 

The accuracy of TTG prediction for interval control purposes will depend upon the accuracy of 

the parameters input or calculated, the control system accuracy, external factors such as winds, 

and the aircraft performance (including operating limits). As the reference aircraft approaches 

the interval management point, the accuracy of the TTG prediction will increase. The aircraft 

performance limitations such as speed and angle of attack limits will become the primary 

constraint on achieving the required spacing time without altering the flight path. 

7.2.4 User Interface 

7.2.4.1 Controls, Displays, and System Alerting 

The user interface is achieved through system controls, displays, and alerting functions. These 

functions provide the means for system initialization, 4D trajectory generation and progress 

monitoring, active speed control and presentation of spacing time data for flight crew situational 

awareness. Refer to RTCA DO-236B (section 1.2.4.1) for additional information on user 

interfaces. 

7.2.4.2 Reporting 

The aircraft’s instantaneous position and velocity must be reported via a digital data link (ADSB). Depending on the required level of RIMP, other critical parameters may be required in the 

data link such as the planned 4D trajectory currently being flown by the aircraft. 

7.3 Required Interval Management Performance 

7.3.1 Time Navigation 

The NextGen TBO environment is expected to include precise time-based operations where 

aircraft will be required to arrive at a point at a specified time. This time may be a fixed time of 

arrival or a time relative to the arrival time of the reference aircraft. The first type of time 

operation is referred to as Time of Arrival Control (TOAC), while the second is referred to as 

relative time spacing. 

In addition to time computation and control accuracy, there is also a need to provide a level of 

confidence in which aircraft can perform time operations at the geographic fixes in a TBO 

environment. 

7.3.2 RIMP Concept 

The concept of time performance has not been defined by the ICAO in the same manner as RNP, 

and the Minimum Aviation System Performance Specification (MASPS) addresses accuracy 

42
requirements of ETA and TOAC at only a very high level. Because of the perceived importance 

of time-based operations in the NextGen Air Traffic Management (ATM) environment, this 

document develops a proposed RIMP construct and associated requirements. 

For consistency with the ICAO definition of RNP, this document defines RIMP as: A statement 

of the time navigation performance accuracy, integrity, continuity and availability necessary for 

time interval management operations within a defined airspace. 

7.3.3 Application of RIMP 

The term RIMP is applied as a descriptor for operations that can be part of airspaces, routes, and 

procedures. Unlike RNP, the descriptor is only applied to each instance where RIMP is required 

and not the entire airspace, since interval management is done on an operation-by-operation 

basis. This concept, however, could eventually be applied to the entire airspace. 

7.3.4 RIMP Types 

The term RIMP-x-y is introduced to denote RIMP operations. 

The x indicates the type of RIMP operation 

A – Absolute Time 

R – Relative Time (ASAS Spacing) 

O – Open Loop (Time prediction only, no time control). 

Note: The open loop is a place holder in anticipation of future changes to DO-236B [2]. 

Additional types of operations may be defined in the future as well. One such operation could be 

a combination of absolute time at a geographic fix with relative spacing prior to the fix. 

The y indicates the limit of RIMP accuracy in seconds (e.g. RIMP-A-10 indicates absolute time 

control with 10 seconds required accuracy). There is a correlation between the type and 

accuracy where the Open Loop type can only achieve a low accuracy. The RIMP accuracy for 

open loop is expressed in % of TTG as it is expected to only be used for pairings where ownship 

and the reference aircraft are further out. 

7.4 Assumptions 

The APS concept of operations relies on the premise that by knowing both the ownship and the 

reference aircraft’s 4D trajectories and the aircraft position along that 4D trajectory, it is possible 

to determine the ownship and the reference aircraft’s Time-To-Go to a given point on their 

individual trajectories. Therefore, it is necessary to define a means of acquiring the aircraft 

planned trajectory and position on that trajectory, and define an assumptions-based predictive 

model for the computation of a 4D trajectory and Time-To-Go. 

7.4.1 Reference Aircraft State 

The RIMP concept assumes that all aircraft in the airspace continuously transmit and receive 

ADS-B data in compliance with RTCA DO-242A including the following parameters: 

 Target identity 

 Latitude and longitude 

43
 Pressure altitude 

 Position accuracy and integrity 

 North and east velocity 

 Vertical speed 

 Velocity accuracy and integrity 

 Time of applicability (reference time of the other state parameters) 

It is further assumed that ownship and reference are either using the same transmission format 

for ADS-B (e.g. 1090ES or UAT) or the airspace is served by an Automatic Dependent 

Surveillance – Rebroadcast (ADS-R) service that reliably echoes messages between the two 

transmission formats with negligible delay. 

7.4.2 Ownship State 

The spacing function has access to at least the set of ownship state parameters that is available 

via ADS-B for the reference aircraft. 

7.4.3 Reference Aircraft Intent 

The APS concept assumes that the aircraft has knowledge of the reference aircraft lateral, 

vertical, and speed profile. There are many possible mechanisms for acquiring an aircraft 

trajectory ranging from assuming that the reference aircraft will adhere to a published augmented 

Standard Terminal Arrival Route (STAR) with the augmentation in the form of altitude or speed 

crossing restrictions at waypoints on the route, to the acquisition in real time of a Flight 

Management System (FMS) profile being broadcast by the reference aircraft. The method used 

is a determinant factor in quantifying the level of accuracy and integrity of the trajectory. 

Regardless of the method used, the RIMP level of performance will be a function of: 

1. The errors and uncertainty in the reference aircraft actual lateral, vertical and speed 

profile flown versus its published profile. The Reference aircraft is assumed to be RNP 

compliant in that its lateral and vertical deviation from the published profile is bounded 

by the profile’s published RNP accuracy and integrity figures of merit. 

2. The errors and uncertainties in the true wind field and temperature profile versus the 

forecast. It is assumed that ownship has access to a forecast wind field and temperature 

profile applicable to both the ownship and the reference aircraft. 

7.4.4 Ownship Intent 

The RIMP concept assumes access to detailed and accurate ownship intent information and 

associated accuracy metrics from the onboard FMS or an equivalent system. Ownship is 

assumed to be operating at a specified RNP level. 

7.5 Definitions 

This section defines key terms that are used throughout this document. 

7.5.1 Required Interval Management Performance 

REQUIRED INTERVAL MANAGEMENT PERFORMANCE (RIMP) 

A statement of the performance accuracy of an in-trail paired interval management operation 

between two aircraft. 

44
PAIRED INTERVAL MANAGEMENT 

An operation where a trailing aircraft is required to arrive at a waypoint a specified time period 

after a specified reference aircraft arrives at that same waypoint. 

OWNSHIP 

The trailing aircraft which is controlling the interval in a paired interval management operation. 

REFERENCE AIRCRAFT 

The leading aircraft which Ownship is required to space relative to. 

7.5.2 Trajectory 

3D POSITION 

A position in space consisting of latitude, longitude, and altitude. 

WAYPOINT 

A predetermined geographical position used for route definition and/or progress reporting 

purposes that is defined by latitude and longitude [2]. 

TIME REFERENCE

The reference time source for the trajectory; for example, Universal Coordinated Time (UTC). 

CURRENT TIME 

The present time according to the time reference. 

PREDICTED TRAJECTORY 

The predicted 3D position of an aircraft as a function of time. 

ESTIMATED TIME 

The time input to the trajectory prediction function, synchronized to the time reference. 

ESTIMATED TIME OF ARRIVAL (ETA) 

The time at which an aircraft will arrive at a defined point on the predicted trajectory.

ESTIMATED TIME TO GO (ETTG) 

The predicted duration from the current time until the aircraft reaches a defined point. 

TIME TO GO (TTG) 

The duration from the current time until the aircraft reaches a defined point, typically in this 

discussion, the Interval Management Point. 

TRAJECTORY CHANGE POINT (TCP) 

A discrete point on the predicted trajectory containing the latitude, longitude, altitude, time, and 

speed (either CAS or Mach). This is a basic set of variables that may be standardized and/or 

expanded in the future. 

45
INTERVAL MANAGEMENT POINT 

A defined waypoint on the predicted trajectory at which the specified interval between ownship 

and the reference aircraft is defined. 

7.5.3 Error Terms 

7.5.3.1 Along-Track 

This document considers only along-track error. Cross-Track and Vertical errors are considered 

only as they contribute to the along track error. Figure 9 below summarizes these terms. 

Figure 9 - Along Track Components of Navigation Error Terms 

ESTIMATED TIME ERROR (ETE) 

The difference between the current time and the estimated time, synchronized to the time 

reference. 

ESTIMATED POSITION ERROR (EPE) 

The arrival time error due to difference between the true position and the estimated position

along the current heading of the aircraft, expressed in seconds. 

TIME TO GO ERROR (TTGE) 

The difference between the true time to go and the estimated time to go. 

ESTIMATED TIME OF ARRIVAL ERROR (ETAE) 

The difference between the true time of arrival and the estimated time of arrival at a point. This 

is equal to the vector sum of the estimated time error, estimated position error, and time to go 

error.

7.5.3.2 Interval Management Performance 

There are errors associated with a paired interval management operation which must be 

considered in the Required Interval Management Performance. 

Estimated 

Time 

Estimated Time To Go 

Estimated 

Time Error 

Estimated 

Position 

Error 

True Time 

To Go 

Estimated Time of 

Arrival Error 

Estimated 

Time of 

Arrival 

True Time 

of Arrival 

46
DESIRED SPACING 

The required time interval, in seconds, between Ownship and Reference Aircraft at the Interval 

Management Point as might be contained in an instruction issued by the ANSP or defined in a 

procedure. 

DEFINED SPACING 

The time interval, in seconds, between Ownship and Reference Aircraft at the Interval 

Management Point that is input to the spacing computation function. 

ESTIMATED SPACING 

The output of the spacing computation function, equal to the difference between Reference 

Aircraft’s estimated time of arrival and ownship’s estimated time of arrival at the interval 

management point. This does not take into account potential future speed changes prior to the 

interval management point that are not reflected in the predicted trajectory. 

ESTIMATED TIME BIAS (ETB) 

The portion of the estimated time of arrival error that will be identical between ownship and 

reference aircraft (for example due to a shared wind error along a common path segment). 

SPACING DEFINITION ERROR (SDE) 

The difference between the desired spacing and the defined spacing, for example due to a 

difference in resolution between the spacing input and the necessary spacing.16

COMPUTED SPACING ERROR (CSE) 

The difference between the estimated spacing and the defined spacing. 

SPACING ESTIMATION ERROR (SEE) 

The error in the computation of estimated spacing. This is equal to the sum of the estimated time 

of arrival error for both ownship and reference aircraft minus any estimated time bias. 

SPACING CONTROL AUTHORITY (SCA) 

The ability of ownship to maneuver in order to compensate for computed spacing error and

spacing estimate error.

TOTAL SPACING ERROR (TSE) 

The difference between the true spacing and the desired spacing. Until ownship has reached the 

interval management point, TSE can only be estimated based on a prediction of owship time to 

go, either reference time to go or measurement of reference IMP crossing time, and estimated 

spacing control authority. It is equal to the sum of the spacing definition error, computed 

spacing error, and spacing estimation error less the spacing control authority. The spacing 

control authority is subtracted because it represents a controllable portion of the error. The 

uncontrollable portion of the error is analogous to the Flight Technical Error (FTE) in RNP. 

 

16 For example, if the desired spacing specified by ATC is in seconds but the defined spacing that is input into the 

system can only be in tenths of minutes, there could be up to a 3 second spacing definition error. 

47
Since the spacing estimation error has to rely on a probabilistic prediction, there may be a need 

for future control actions to correct for “true” disturbances and errors that were not part of the 

prediction. Figure 10 also shows how the uncertainty of the predicted time grows the farther 

from ownship the spacing point is located. 

Figure 10 - Reference Time Profile with Uncertainty 

7.5.3.3 Along Track Containment Concept 

The concept of along-track containment must be coupled with the concept of lateral (or crosstrack) containment as defined in RTCA DO-236B. If a desired lateral path exists, the CrossTrack Containment Limit combined with the Along-Track Containment Limit defines a region 

around the desired location of the aircraft at a defined future time T. The probability that the 

aircraft will be within that region in both the cross-track and along-track domains at time T, can 

be bounded. This time T may be a fixed time, or it may be an offset time relative to another 

aircraft’s ETA. This concept is depicted in Figure 11 below, and the definitions of these terms 

follow. 

ETA

ETA+ 2σ 

ETA ‐ 2σ

Current Time Uncertainty

Predicted Time Uncertainty (95%)

Distance

Time of Arrival Offset

48
Figure 11 - Along-Track and Cross-Track Containment Region 

Along-Track containment integrity requirements limit the probability of the Total Spacing 

Error exceeding the Along-Track Performance requirements with no annunciation at least some 

specified time before the defined time T. 

Along-Track containment continuity requirements limit the probability of a loss of function. In 

this context, function is defined as the ability of the APS system to meet the along track 

containment requirement (i.e. to be within the desired containment limit at time T). 

There are five possible system states relative to along-track containment limit, C. These states 

are: 

6) S1: TSE Error > C, TTG > ∆T, no alert (Integrity) 

7) S2: TSE Error > C, TTG > ∆T, alert (Continuity) 

8) S3: TSE Error > C, TTG < ∆T, no alert (Integrity) 

9) S4: TSE Error > C, TTG < ∆T, alert (Continuity) 

10) S5: TSE Error < C, no alert (Normal Operations) 

Note: P(S1) + P(S2) + P(S3) + P(S4) + P(S5) = 1 where P(x) is the probability of the system 

being in State x. 

∆T is the minimum amount of time prior to reaching the Interval Management Point that the alert 

must be made if the aircraft cannot achieve the required spacing time. 

States 1 and 3 represent the set of events associated with a loss of along-track performance with 

no annunciation (alert), and are thus the integrity requirements. The uncertainties associated 

with time predictions generally increase with distance from the current trajectory point (primarily 

due to forecast wind uncertainties). Therefore, the probability that the along-track performance 

cannot be achieved will generally decrease with distance to the Interval Management Point. 

Thus, the probability of State 1 will be greater than the probability of State 3 (P(S1) >> P(S3)) 

due to TTG being > ∆T. However, it also needs to be considered that the control authority that 

ownship has to correct for these uncertainties will also generally decrease as the aircraft 

RNP 

Value 

Cross-Track 

Containment Limit 

= 2 x RNP value 

Desired 

Path 

Desired AlongTrack Position 

at time T 

Along-Track 

Containment Limit 

Containment 

Region 

49
approaches the Interval Management Point. This needs to be considered when determining a 

time T beyond which it is critical that an annunciation be received. Thus, the probability of S1 

must be less than XX17 per flight hour, while the probability of S3 must be less than YY per 

operation. 

The continuity requirement applies to states S2 and S4 where the loss of function is detected 

and annunciated. When current time is still far before T (TTG > ∆T), the available control 

authority will be larger, and thus the probability of state S2 must be less than WW per flight 

hour. When the aircraft is close to time T the control authority is greatly reduced, and thus the 

probability of S4 must be less than ZZ per flight hour. S5 represents the normal operation where 

the error is less than the containment limit. 

In addition to the containment requirements, there is an uncertainty parameter associated with 

the predicted ETA at various points on the predicted trajectory. This uncertainty must be 

bounded to ensure appropriate control authority is available. 

CONTAINMENT 

A set of interrelated parameters used to define the performance of a time control system. These 

parameters are containment integrity, containment continuity, and containment region. 

CONTAINMENT INTEGRITY 

A measure of confidence in the estimate, expressed as the probability that the system will detect 

and annunciate the condition where the TSE is greater than the along-track containment limit and 

the condition has not been detected. 

CONTAINMENT CONTINUITY 

The capability of the total system to satisfy the performance limit without nonscheduled 

interruptions during the intended operation. Nonscheduled interruption is defined to be either 1) 

total loss of time prediction capability; 2) a failure of the system which is annunciated as a loss 

of time performance capability; or 3) a false annunciation of loss of time performance capability. 

CONTAINMENT REGION 

A region in both the along-track and cross-track dimensions, centered on the desired along track 

position on the desired path at a desired time T, to which the containment integrity and 

containment continuity are referenced. 

CROSS-TRACK CONTAINMENT LIMIT 

As defined in DO-236B MASPS. 

ALONG-TRACK CONTAINMENT LIMIT 

A time that defines the one-dimensional containment limit in the along-track dimension. The 

resulting containment region is centered on desired along-track position at desired time T and is 

bounded by ± the along-track containment limit in the along-track dimension. 

 

17 This and the following probability parameters will need to be determined based on a TLS and any mitigations that 

are identified as part of the procedures defined by the ANSP. In practice, they will likely be tailored to specific 

RIMP performance categories. 

50
ESTIMATED TIME OF ARRIVAL UNCERTAINTY 

A measure based on a defined scale in seconds which conveys the performance of the estimated 

time of arrival at the center of that containment region. 

This measure is required because a type of integrity performance measure is needed. It is based 

on a defined scale (e.g. 95%). 

The containment region is bound around ETA where the actual time of arrival will be within the 

ETA uncertainty in seconds. 

The RNP-equivalent for the ETA uncertainty is the Estimated Position Uncertainty. 

7.6 System Performance Requirements 

7.6.1 Accuracy 

A measure of the difference between an aircraft‘s reported planned interval, i.e. ADS-B report, 

as compared to its achieved interval. It is given by a number bounded by a confidence value. 

For example; within the RIMP-R-10 interval management operation the Total Spacing Error 

cannot be greater than 10.0 seconds 95% of the time. 

7.6.2 Integrity 

The probability that information provided by a system will not be hazardously misleading. It is 

generally a number, per operating hour, which expresses likelihood that a system will send the 

user bad/incorrect information that could cause a potentially dangerous situation without a timely 

warning. For example; the likelihood that a Total Spacing Error exceeds a maximum of 10.0 

seconds containment, without detection, shall be less than 10-5 per flight hour. 

7.6.3 Continuity 

The capability of a total system to perform its intended function without a non-scheduled 

interruption during the intended operation assuming the system was available when the 

procedure was initiated. Continuity is expressed per unit of time. For example; the calculations 

of a single thread transponder mean time to failure of 5,000 hours. 

7.6.4 Availability 

The probability that a system will perform its required function at the initiation of the intended 

operation. For example; the FAA target for availability of the Wide Area Augmentation System 

is 0.999. 

51
52 

8 Required Self Separation Performance Model 

8.1 Introduction 

The document, Required Self Separation Performance, Version 1.1, examined what would be 

required in order to ensure a Target Level of Safety (TLS) for a scenario, such as TrajectoryBased Airspace and Operations (TBO), in which properly equipped aircraft have primary 

responsibility for avoiding other aircraft. This is in contrast to the current system where that 

responsibility lies with the Air Navigation Service Provider (ANSP). It described the system 

components, identified key system parameters, and derived a Required Navigation Performance 

(RNP)-like construct for self-separation operations. This section, and the accompanying 

MATLAB model, expands upon the Total System Error (TSE) analysis included as Appendix B 

in that document to provide a means of assessing the impact that various error sources and 

uncertainties have on an aircraft’s ability to accurately predict (and hence control) its minimum 

separation from a reference aircraft. The model includes computation of the encounter 

geometry, linearized analysis of the impact of error sources on the predicted separation, and a 

Monte Carlo based analysis of the impact of error sources on the distribution of separation as a 

function of encounter geometry in the context of key error terms for Class A and Class B RSSP 

systems. This analysis is currently confined to separation in the horizontal plane. 

8.2 Section Overview 

Derivation of the basic mathematical model for separation prediction is presented in Section 8.3. 

Section 8.4 describes the translation of the mathematical model into a MATLAB tool 

framework. Information on the operation of the MATLAB tool is provided in Section 8.5. 

Error! Reference source not found. provides a table of acronyms and a list of symbols used 

throughout this section and the model. 

8.3 TSE Model Derivation 

The first step in developing a mathematical construct for characterizing the errors in computing 

and achieving lateral separation from another vehicle is to be able to calculate the expected 

separation at closest approach for two targets in unaccelerated (at least in the horizontal plane) 

flight. With such an solution in hand, it is then possible to assess the linearized sensitivity of the 

computed separation to various present or future state estimation uncertainties by taking partial 

derivatives of the solution equation with respect to the various uncertainty terms, such as 

current/initial position, groundspeed, and ground track angle or to assess the impact of 

perturbations of input parameters about a nominal condition. The solution can also be used to 

perform Monte Carlo analysis to predict the distribution of actual separations about a nominal 

condition based on error distributions for input terms. 

8.3.1 Expected Separation Computation 

The expected minimum separation can be relatively simply expressed given the assumptions of 

unaccelerated (i.e. linear with time) motion in a flat plane18. The derivation is accomplished by 

expressing the separation between the two vehicles (ownship, O, and reference aircraft, R) as 

 

18 This is a reasonably good assumption given the relatively small distances of interest here. 
53 

functions of time, finding the time for which the derivative of the separation with respect to time 

is zero (corresponding to the minimum of the function), and then substituting this time back into 

the distance function. 

The coordinates of ownship as a function of time can be expressed as the time-dependent 

coordinates 

P  t x v t y v t O  0O  xO 0O  yO , (1) 

Likewise, the position of the reference aircraft can be expressed as 

P  t x v t y v t R  0R  xR 0R  yR , (2) 

The distance R between the two at any time t can be expressed as 

        2 2 R t x (t) x t y t y t  R  O  R  O (3) 

which can be expanded to 

           

2

0 0

2

0 0 R t x x v v t y y v v t  R  O  xR  xO  R  O  yR  yO (4) 

Making the following substitutions to express the differences in initial position and speed along 

each axis 

  R O x x x 0  0  0 ,   R O y y y 0  0  0 and (5), (6) 

  x xR xO v  v  v ,   y yR yO v  v  v (7), (8) 

the range equation simplifies to 

   2

0

2

0 R t x v t ( y v t)   x   y (9) 

which can be expanded and regrouped (using 2

0

2

0

2

0 R  x  y to express the range at time 0) 

      2 2 2

0 0

2

0 R t R 2 x v y v t v v t x y x y        (10) 
54 

The point of closest approach occurs when the derivative of range with respect to time is zero, 

which is equivalent to when the derivative of range squared with respect to time is zero 

    x v y v   v v t

t

R t x y x y        2 2

0 0

2

0 2 2 (11) 

So, the time of closest approach, tCPA, is given by 

 

  2 2

0 0

x y

x y

CPA

v v

x v y v

t



   (12) 

Note that a negative tCPA indicates that the two aircraft are already beyond their point of closest 

approach and that tCPA is defined in all cases except when the aircraft are traveling at the same 

speed along parallel tracks. 

The range at closest approach can be found by substituting the expression for tCPA back into the 

range equation and simplifying 

 

  2 2

2

2 0 0

0

x y

x y

CPA

v v

x v y v

R R



   (13) 

8.3.2 Analysis of Sensitivity of Expected Minimum Separation to State 

Variable Variations 

The first analysis performed by the model computes the sensitivity of the RCPA prediction to 

errors in position and velocity inputs. The sensitivity of RCPA to variations in the initial position 

or velocity of either aircraft can be found by taking the partial derivative of RCPA with respect to 

the parameter of interest. As a first step, U will be defined as 

 

  2 2

2

2 0 0

x y

x y

o

v v

x v y v

U R



   so RCPA  U (14), (15) 

By the chain rule, the derivative of RCPA with respect to any parameter  is 

 

  

 U

U

RCPA

2

1 (16) 
55 

Noting that U  RCPA , this can also be written 

 

  

 U

R

R

CPA

CPA

2

1 (17) 

Further, substituting 

 

  2 2

2

0 0

x y

x y

v v

x v y v

V



  (18) 

the partial derivative becomes 



















  

  

  

R V

R

R

CPA

CPA

2

0

2

1 (19) 

Expanding V by letting B

A V  where  

2

0 x 0 y A  x v  y v and   2 2x y B  v  v and using the 

quotient rule, 2 B

B A A B V  





  

 

 , the general form of the partial derivative is 



























  

 

  

2

2

0

2

1

B

B A A B R

R

R

CPA

CPA  

 

 or (20) 

























  

  

   

B

B

A A

B

R

R

R

CPA

CPA

2

2

0 1

2

1

 (21) 

8.3.2.1 Position Uncertainty 

For this first case, consider the sensitivity of RCPA to variation in the x coordinate of ownship or 

reference aircraft. Using the above expressions and noting that 0

0

2

0 2x

x

R

R

 

 , 1

0

0  



R x

x , and 

0

0

 



R

x

x

v it can be shown that 
56 

 

  

















   



2 2

0

2

0

0

0

1

x y

x x y

R CPA

CPA

v v

x v y v v

x

x R

R

 and (22) 

 

  

















    



2 2

0

2

0

0

0

1

x y

x x y

O CPA

CPA

v v

x v y v v

x

x R

R

 (23) 

Likewise, 

 

  

















   



2 2

0

2

0

0

0

1

x y

y x y

R CPA

CPA

v v

y v x v v

y

y R

R

 and (24) 

 

  

















    



2 2

0

2

0

0

0

1

x y

y x y

O CPA

CPA

v v

y v x v v

y

y R

R

 (25) 

8.3.2.2 Velocity Uncertainty 

Since R0 is not a function of any velocity term, its derivative with respect to velocity is zero. 

Therefore, the impact of velocity uncertainty reduces to just the second term 





 







   



v

V

v R

R

CPA

CPA

2

1 (26) 

Or, using the same expansion of the inner derivative as before, 





 







  

  



v

A

v B

B

B

A

v R

R

CPA

CPA 1

2

1

2 (27) 

Using the partial derivatives 

  x o y

x

x v x y v

v

A

0

2  2 0 



 ,   y o x

y

y v x y v

v

A

0

2  2 0 



 (28), (29) 

x

x

v

v

B  2



 , and y

x

v

vy

B  2



 (30), (31) 
57 

yields 

 

 

 

  

















 



  



2 2

0

2

0

2 2 2

2

1 0 0

x y

x o y

x

x y

x y

x CPA

CPA

v v

x v x y v

v

v v

x v y v

v R

R

 and (32) 

 

 

 

  

















 



  



2 2

0

2

0

2 2 2

2

1 0 0

x y

y o x

y

x y

x y

y CPA

CPA

v v

y v x y v

v

v v

x v y v

v R

R

 (33) 

Sensitivity to the individual aircraft velocities can then be found by observing 

 1





xR

x

v

v ,  1





xO

x

v

v ,  1





yR

y

v

v

, and  1





yO

y

v

v

 (34), (35), (36), (37) 

so that 

x

CPA

xR

CPA

v

R

v

R



  

 , 

x

CPA

xO

CPA

v

R

v

R



   



, (38), (39) 

y

CPA

yR

CPA

v

R

v

R



  

 , and 

y

CPA

yO

CPA

v

R

v

R



   



. (40), (41) 

8.3.2.3 Uncertainty with respect to Groundspeed and Ground Track Angle 

If the velocity components are expressed in terms of groundspeed, S, and ground track angle, 

(measured clockwise from North), so that the speed components may be written 

vx  S sin and vy  S cos (42), (43) 

The sensitivity to these measures of velocity can also be determined from 

R O

y

y

CPA

R O

x

x

CPA

R O

CPA

S

v

v

R

S

v

v

R

S

R

, , , 















  



 and (44) 

R O

y

y

CPA

R O

x

x

CPA

R O

CPA v

v

v R

v

R R

 ,  ,  , 















  

 (45) 
58 

Using the partial derivatives yields 

R

R

x

S

v  sin 

 , R

R

y

S

v  cos 



, (46), (47) 

O

O

x

S

v  sin 

 , O

O

y

S

v  cos 



, (48), (49) 

R R

R

x S v



  cos

 , R R

R

y S

v



   sin



, (50), (51) 

O O

O

x S v



   cos



, and O O

O

y S

v



  sin



 (52), (53) 

8.3.3 Analysis of Error Distribution of Minimum Separation 

The second analysis performed by the model generates a probabilistic distribution of achieved 

RCPA values as a function of geometry and the error distributions associated with the input 

geometry parameters. This analysis will capture the full non-linear contribution of the various 

error sources to errors in the prediction of the separation distance at closest approach. 

The approach for this analysis is to start with a nominal geometry as input as well as 

specifications for the distribution functions of each measurement variable. For Class A 

systems19, the distribution functions are for the lateral uncertainty (traditional RNP) and 

longitudinal deviation (along-track position uncertainty) for each aircraft appropriate for the 

intent trajectory segments containing the nominal Closest Point of Approach (CPA). This 

assumes that the prediction is for a time far enough in the future that the error terms associated 

with the intent information dominate the errors associated with current position and velocity 

measurement. For Class B systems20, the distribution functions are for position uncertainty and 

velocity uncertainty (both uncertainties assumed to be uniformly distributed in azimuth). Once 

the nominal conditions and measurement variable distributions have been defined, a Monte Carlo 

analysis approach is used to determine the distribution of RCPA for comparison with the nominal 

value. For cases where the errors are significant relative to the nominal RCPA, it is expected that 

crossing order given a particular set of perturbed parameters may be the opposite of the nominal 

case crossing order. When this occurs, the computed RCPA for that case should be considered a 

negative quantity. 

 

19 Class A systems transmit and receive intent information as part of their ADS-B message set. 20 Class B systems need only transmit and receive position and velocity information as part of their ADS-B message 

set. 
59 

8.3.4 Extension to Other Uncertainty/Error Terms 

To the extent that other uncertainties and errors can be modeled as the above uncertainties in the 

initial conditions, their impact can also be understood by extension of the above analyses. 

For example, the impact of modeling and execution errors in making a speed or track adjustment 

can be understood if the errors and uncertainties can be translated to errors and uncertainties in 

the vehicle position and velocity at the end of the maneuver. The end of maneuver state can then 

be used to compute the new expected minimum separation and the associated sensitivity to the 

(perhaps larger) initial condition uncertainties. 

8.4 MATLAB Linearized Sensitivity Model Description 

8.4.1 Scenario Inputs 

The fundamental inputs to the mathematical model are: 

1. Groundspeed of ownship (SO) 

2. Groundspeed of reference (SR) 

3. Crossing angle (difference in ground track angle) ()

21

4. Order of crossing (ownship crosses in front of reference or behind) 

5. Time to CPA (tCPA) 

6. Separation at CPA (RCPA) 

All parameters are for the nominal condition before the introduction of uncertainties. 

Assuming a coordinate frame aligned such that ownship is nominally traveling in the direction of 

the positive x-axis, the relative speeds of the two aircraft along each axis are given by: 

x SRCos SO v  ( )  (54) 

v S Sin( ) y  R (55) 

Given the scenario input parameters and using (54) and (55) to compute vx and vy, the initial 

position of reference with respect to ownship, (x0,y0), can be found by solving the two 

simultaneous equations using equations (12) and (13), which, by expanding 2

0

2

0

2

0 R  x  y , can 

be rewritten explicitly in terms of these variables and the input parameters as shown below: 

   

  2 2

2

2 0 0

0

2

0

x y

x y

CPA

v v

x v y v

R x y



    (56) 

 

21  is positive if reference is crossing from ownship’s right to left (mathematically,  is measured from ownship’s 

speed vector (x-axis) towards reference’s speed vector counter-clockwise positive). 
60 

The solution is given by the following equations: 

     

  x y x

CPA x y y CPA x x y y

x

CPA x y

v v v

t v v v R v v v v

v

t v v

x

2 2

2 2 2 3 2 2

0



  

    and (57) 

 

  2 2

2 3 2 2

0

x y

CPA x y y CPA x x y

v v

t v v v R v v v

y



     (58) 

The ambiguity in whether to take the positive or negative square root is determined from the 

specified crossing order. The two solutions for (x0, y0), one corresponding to using positive 

square roots, the other corresponding to using negative square roots, represent the two crossing 

orders. The solution with the smaller value of x0 represents the case where ownship crosses 

behind reference. The solution with the larger value of x0 represents the case where ownship 

passes in front of reference. 

The above solution for x0 also contains a singularity for scenarios that result in a vx of 0 (DIV/0). 

In that case, the solution for x0 equals RCPA with the sign, again, depending on the crossing 

order. 

8.4.2 Sensitivity Analysis Modeling 

The linearized implementation of the model uses partial derivatives to provide an initial look at 

the sensitivity of separation distance prediction to errors in some of the input parameters for a 

nominal initial geometry. 

Partial differential equations derived in Section 8.3.2 reflect the fundamental sensitivities of the 

minimum separation to state estimation uncertainties22, but it is not certain that the nonlinearities can be neglected in the general case, and they likely cannot be neglected in the case 

where the uncertainty is not a small fraction of the nominal separation at CPA23. 

For the linear case, the relation between input parameter uncertainty and RCPA uncertainty is a 

simple scaling factor. The model can generate a plot of the scaling factor as a function of one 

input (2-D plot) or two inputs (3-D plot) with the remaining inputs held at a specified value. 

For positional errors (either ownship or intent-based reference), positional uncertainty may be 

asymmetrically expressed in the along-track and cross-track dimensions. Using the coordinate 

system introduced above, ownship along-track uncertainty corresponds to uncertainty along the 

x-axis, and cross-track uncertainty coincides with the y-axis. For reference, along-track and 

cross-track uncertainties are mapped to the x- and y-axes using the crossing angle. 

 

22 It would be equally valid to use small perturbations to approximate the derivatives rather than using the partial 

derivative formulae. 23 Or, in the case of velocity or time uncertainty, does not translate to a position uncertainty at the time of CPA that 

is a significant proportion of the nominal separation. 
61 

Time of arrival uncertainty (ownship or reference) may be mapped to along-track position 

uncertainty using the appropriate nominal groundspeed. 

8.4.3 Distribution Analysis Modeling 

The distribution analysis is performed using a Monte Carlo approach and then examining the set 

of outcomes. Unlike the sensitivity analysis, this analysis does account for the non-linearities 

inherent in determining the distance at closest approach. 

The fundamental inputs to establish the nominal condition before the introduction of 

uncertainties for this analysis are those described in Section 8.4.1. 

Once the nominal conditions have been determined, perturbations are applied to the positions 

(Class A systems) or the positions and velocities (Class B systems) according to the input 

uncertainties. Equation (56) for RCPA is then re-solved using the perturbed values to determine 

RCPA for the perturbed condition, with care taken to use a negative result if the perturbation 

results in a reversal of the crossing order. Sufficient numbers of perturbed condition samples are 

examined to enable generation of a resulting distribution plot for the variation in RCPA. 

The distribution function inputs will depend on whether a Class A pair of systems or a Class B 

pair of systems is being evaluated. For Class A systems, it is assumed that the aircraft are both 

operating according to 4D flight plans and are broadcasting Trajectory Change Points (TCPs) 

with known lateral and longitudinal accuracy characteristics. The input parameters will therefore 

be taken to represent the standard deviation for lateral and longitudinal uncertainties (a Gaussian 

distribution is assumed). These uncertainties are meant to reflect a time approximately tCPA in 

the future. For Class B systems, it is assumed that the aircraft only rely on current position and 

velocity measurements, and that the accuracy characteristics of these measurements are Gaussian 

and non-directional (errors are uniformly distributed in azimuth). 

The following equations are used to apply the errors to the initial positions of both ownship and 

the reference aircraft for a Class A system: 

( ) 0O 0O NormDist ATO x x Perturbed    (59) 

( ) 0O 0O NormDist CTO y y Perturbed    (60) 

cos( ) ( ) sin( ) ( ) 0R 0R NormDist ATR NormDist CTR x x Perturbed        (61) 

sin( ) ( ) cos( ) ( ) 0R 0R NormDist ATR NormDist CTR y y Perturbed        (62) 

Where: 

Norm Dist() is a Gaussian distribution with standard deviation .
62 

In the case of a Class B system, the errors are applied to the initial positions and speeds of both 

ownship and the reference aircraft24: 

( ) cos (0 2 ) x0O/R x0O/R NormDist  pO/R UniformDist   Perturbed   (63) 

( ) sin (0 2 ) y0O/R y0O/R NormDist  pO/R UniformDist   Perturbed   (64) 

v S NormDist( ) cos  UniformDist(02 ) xOPerturbed  O  vO (65) 

v NormDist( ) sin  UniformDist(02 ) yOPerturbed  vO (66) 

v S cos( ) NormDist( ) cosUniformDist(02 ) xRPerturbed  R   vR (67) 

v S sin( ) NormDist( ) sinUniformDist(02 ) yRPerturbed  R   vR (68) 

For single distribution and SPS analyses, the probability mass function (fX(RCPA)) will be 

determined. In addition, the resulting mean () and standard deviation () of the samples are 

calculated as follows: 





NB

i

CPA

S

Samples Samples i R i

N 1

( ) ( ) 1 (69) 

   

 

NB

i

CPA Samples

S

Samples Samples i R i

N 1

2 ( ) ( ) 1

  (70) 

Where: 

NB Number of Bins 

NS Total Number of Samples 

RCPA(i) RCPA of Bin i

Samples(i) Number of Samples in Bin i

The calculated mean and standard deviation are used to calculate a normal distribution that is 

plotted over the probability mass function for comparison purposes in single distribution 

analyses. 

Additionally, the probability distribution function is “integrated” to find the value of RCPA that 

contains 5% of the samples (RCPA(5%)). RCPA(5%) is the separation distance for which there is a 

 

24 Note that the same sample from UniformDist(02 ) is used for each pair of equations (63)-(64), (65)-(66), 

and (67)-(68). 
63 

probability of 5% that the RCPA could be less than RCPA(5%) given the specified geometry and 

errors, and conversely, there is a 95% probability that RCPA will be greater than RCPA(5%). 

8.5 MATLAB Model Operation 

8.5.1 Overview 

The RSSPSimulator allows for the simulation and visualization of certain aspects of the 

mathematical model derived in the previous Section. 

8.5.2 System Requirements 

The RSSPSimulator has been developed and tested with the following versions of MATLAB 

(Help > About MATLAB): 

 R2008a (Version 7.6) 

 R2009a (Version 7.8) 

8.5.3 Installation & Start 

The RSSPSimulator.fig and RSSPSimulator.m files need to be copied into a directory 

to which the current directory of MATLAB has to be set. The RSSPSimulator can be started by 

typing RSSPSimulator in the MATLAB Command Window (i.e. >> RSSPSimulator ). 

MATLAB will then display the control panel shown in Figure 12 below. 

Figure 12 – RSSPSimulator Control Panel 

8.5.4 Scenario Definition 

The scenario parameters are defined using the left side of the control panel as shown in Figure 13 

The initial values for the modifiable variables are set to a typical scenario. 
64 

Figure 13 – Mode Control Panel Configuration for Scenario Inputs 

The Order determines whether ownship passes Ahead or Behind of the reference aircraft. 

The following ranges for of these variables were chosen to allow the analysis of reasonable 

scenarios: 

Symbol Parameter Minimum Maximum Units 

SO/SR Groundspeed of ownship/reference aircraft 200 600 knots 

 Crossing angle 10 170 degrees 

tCPA Time to CPA 2 20 minutes 

RCPA Separation at CPA 0.5 10 nmi 

Pressing the “Scenario” button will open a MATLAB figure titled “Figure 1: Scenario” that 

shows the geometry of the scenario for the specified values as shown below. 
65 

Figure 14 – Scenario Display Example 

The “Scenario” button is available as long as none of the “Plot” checkboxes are selected. 

8.5.5 Sensitivity Analysis 

The Sensitivity Analysis implements the analyses described in Section 8.4.2. 

8.5.5.1 Single-Parameter Sweep Mode 

The Single-Parameter Sweep (SPS) mode of the sensitivity analysis allows for a set of 

computations where a single selected variable is varied (“swept”) through a pre-defined range25. 

The other variables are kept constant and the results of the sensitivity analysis of the separation 

at the closest point of approach (RCPA) with respect to position and speed uncertainties are 

visualized. 

Figure 15 shows how the crossing angle () was chosen for the analysis by selecting its “Plot” 

checkbox. Pressing the “Sensitivity Analysis” button will open two MATLAB figures that 

visualize the sensitivity of RCPA to position and speed errors. Examples of these can be seen in 

 

25 The ranges are specified by the entry range of each variable. 
66 

Figure 16 and Figure 17 respectively. The positions and speeds are shown in along-track and 

cross-track components with regard to the respective aircraft (e.g. ownship or reference). The 

SPS sensitivity analysis may be performed whenever a single “Plot” checkbox is selected. 

Figure 15 – SPS Control Panel Configuration to Vary Crossing Angle (). 

The following two equations are used to transform the x/y components of the reference aircraft 

position and speed to along- and cross-track components: 

Along-TrackR = cos() x + sin() y (71) 

Cross-TrackR = -sin() x + cos() y (72) 

The step-sizes for the sweep were determined for best computational performance and 

visualization: 

SO/SR 20 [kts] 

 5 [deg] 

tCPA 1 [min] 

RCPA 0.5 [nmi] 
67 

Figure 16 – SPS Sensitivity to Position Error as a Function of Crossing Angle (). 

The interpretation of this figure is that the prediction of RCPA is most sensitive to along-track 

position errors when the crossing angle is small (tail-chase) and is most sensitive to cross-track 

position errors when the crossing angle is large (near head-on). Because of the linearized nature 

of this analysis, the magnitude of the sensitivity to position errors will never exceed unity. 
68 

Figure 17 – SPS Sensitivity to Velocity Error as a Function of Crossing Angle (). 

The interpretation of this figure is that the prediction of RCPA is most sensitive to along-track 

velocity errors when the crossing angle is small (tail-chase) and is most sensitive to cross-track 

velocity errors when the crossing angle is large (near head-on). 

8.5.5.2 Two-Parameter Sweep Mode 

The Two-Parameter Sweep (TPS) mode of sensitivity analysis allows for a set of computations 

where two selected variable are varied (“swept”) through a pre-defined range. The other 

variables are kept constant and the results of the sensitivity analysis of the separation at the 

closest point of approach (RCPA) with regard to position and speed uncertainties are visualized in 

3D MATLAB figures. 

This is similar to the SPS mode with the only difference that two of the “Plot” checkboxes have 

to be checked as demonstrated in Figure 18. Pressing the “Sensitivity Analysis” button will then 

open two MATLAB figures that visualize the sensitivity of RCPA to position and speed errors. 

Examples of these can be seen in Figure 19 and Figure 20 respectively. 
69 

Figure 18 – TPS Control Panel Configuration Varying SO and SR. 

Figure 19 – TPS Sensitivity to Position Error as a Function of SO and SR. 

The interpretation of this figure is that, for this nominal geometry, the prediction of RCPA is most 

sensitive to along-track position errors when the two groundspeeds are equal and is most 

sensitive to cross-track position errors when the two groundspeeds are quite different. Again, 
70 

because of the linearized nature of this analysis, the magnitude of the sensitivity to position 

errors will never exceed unity. 

Figure 20 – TPS Sensitivity to Velocity Error as a Function of SO and SR. 

The interpretation of this figure is that, for this nominal geometry, the prediction of RCPA is most 

sensitive to along-track velocity errors when the two groundspeeds are equal and is most 

sensitive to cross-track velocity errors when the two groundspeeds are quite different. 

8.5.6 Distribution Analysis 

The Distribution Analysis implements the analyses described in Section 8.4.3. These analyses 

require more input parameters than the Sensitivity Analysis as discussed in the following. 

8.5.6.1 Additional Configuration Inputs 

The additional configuration inputs applicable to both Class A and Class B system distribution 

analyses are contained in the middle portion of the RSSPSimulator control panel as depicted in 

Figure 21. 
71 

Figure 21 – Distribution Analysis Controls. 

Here, the user can specify whether the analysis is to be performed for a Class A or Class B 

system and control the parameters used to generate the sample distributions and the histograms 

used to plot the results. 

The random number generator can be set to use either a fixed value (“Fixed”) or a seed 

randomized using the system clock (“Time”). 

There is no hard-coded upper limit to the value of Samples but performance will slow down 

significantly with large values. This is in particular the case when performing swept-parameter 

analyses. 

The “Bins” parameters control the width and granularity of the generated RCPA histograms. If the 

Minimum or Maximum limits are exceeded by some samples, a dialog, such as shown in Figure 

22, will appear in addition to the histogram plot. Examination of the histogram will typically 

reveal which limit needs to be relaxed. 

Figure 22 – Histogram Limit Warning Dialog. 

8.5.6.2 Class A Systems 

Having selected the “Class A” radio button in the upper center of the control panel, the user can 

set the corresponding error distribution inputs in the upper right portion of the panel as shown in 

Figure 23 below. 
72 

Figure 23 – Class A Error Distribution Inputs. 

This set of inputs permits entry of standard deviations for the along-track and cross-track error 

distributions of ownship and the reference aircraft. The drop-down menus to the right of each 

entry permit selection of the standard deviation by manual entry (“Manual”), RNP equivalent 

(“RNP-*”), or no error (“Disable”). For consistency with the definitions used in the RNP 

standards, the RNP-* selections will represent a 95% (= 2) confidence boundary for an 

assumed Gaussian distribution (Norm Dist()) and the RNP values will be converted into the 

standard deviation using the following equation: 

2 (95%) 1  erf

RNP

 (73) 

Where: 1 erf is the Inverse Error Function. 

8.5.6.3 Single Distribution 

With none of the “Plot” checkboxes selected, pressing the “Distribution Analysis” button will 

generate a single distribution plot. While MATLAB is busy processing, a “Running...” progress 

bar will appear to the left of the version number label. For the example control panel shown in 

Figure 24, the resultant distribution is depicted in Figure 25. 

Figure 24 – Class A Single Distribution Control Panel Setup. 
73 

Figure 25 – Class A Single Distribution Result. 

The blue line in this figure represents the normalized histogram of outputs for a random 

sampling of input conditions about the nominal condition given the statistics of the position 

errors. The mean and standard deviation of the distribution are displayed in the legend at the 

right. The green points represent a normal distribution with an equivalent mean and standard 

deviation for reference. The 5% figure, as well as the red shaded area on the plot, represent the 

value for RCPA below which 5% of the samples lie (or, conversely, above which 95% of the 

samples lie). It is common for the mean to be slightly lower than the target value for RCPA

because of the non-linearities involved. 

8.5.6.4 Single-Parameter Sweep Distribution Analysis 

With one of the “Plot” checkboxes selected, pressing the “Distribution Analysis” button will 

generate a set of distribution plots corresponding to a sweep of values for the checked parameter. 

In addition to the 3D plot showing the distribution as a function of the swept parameter, a second 

set of plots will appear to show the variation in the mean, standard deviation, and 5% point as a 

function of the swept parameter. For the example control panel shown in Figure 26, the resultant 

plots are depicted in Figure 27 and Figure 28. 
74 

Figure 26 – Class A SPS Distribution Analysis Control Panel Setup. 

Figure 27 – Class A SPS Distribution Analysis Distribution Result. 

The 3D figure represents the normalized histogram of outputs for a random sampling of input 

conditions about the nominal condition given the statistics of the position errors for each value of 
75 

the swept parameter (here, crossing angle, ). This example shows a broadening of the 

distribution for crossing angles around 60 degrees. 

Figure 28 – Class A SPS Distribution Analysis Statistics Result. 

The mean and standard deviation of the distributions are plotted in the second set of graphs along 

with the 5% figure that represent the value for RCPA below which 5% of the samples lie (or, 

conversely, above which 95% of the samples lie) for each distribution. 

8.5.6.5 Two-Parameter Sweep Distribution Analysis 

With two of the “Plot” checkboxes selected, pressing the “Distribution Analysis” button will 

generate a set of statistical value plots corresponding to a sweep of values for the checked 

parameters. A 3D set of plots will appear to show the variation in the mean, standard deviation, 

and 5% point as a function of the swept parameters. For the example control panel shown in 

Figure 29, the resultant plots are depicted in Figure 30. These plots are analogous to the plots in 

Figure 28, but now represented across the two parameters chosen for the sweep. 
76 

Figure 29 – Class A TPS Distribution Analysis Control Panel Setup. 

Figure 30 – Class A TPS Distribution Analysis Statistics Result. 
77 

8.5.7 Class B Systems 

Having selected the “Class B” radio button in the upper center of the control panel, the user can 

set the corresponding error distribution inputs in the lower right portion of the panel as shown in 

Figure 31 below. 

Figure 31 – Class B Error Distribution Inputs. 

This set of inputs permits entry of standard deviations for the position and velocity error 

distributions of ownship and the reference aircraft. These errors are assumed to be uniformly 

distributed in azimuth. The drop-down menus to the right of each entry permit selection of the 

standard deviation by manual entry (“Manual”), NACP/NACV equivalent (“NACp/v-*”), or no 

error (“Disable”). For consistency with the definitions used for NACP and NACV in the ADS-B 

standards, the NACp/v-* selections represent a specific accuracy category, and the standard 

deviation will be computed based on the corresponding 95% confidence boundary assuming a 

Gaussian distribution (Norm Dist()) for error magnitude that is uniformly distributed (Uniform 

Dist(Range)) in azimuth. The pre-defined NACp and NACv values (shown with their 

corresponding 95% confidence boundaries) are: 

NACp-1 10 [nmi] 

NACp-2 4 [nmi] 

NACp-3 2 [nmi] 

NACp-4 1 [nmi] 

NACp-5 0.5 [nmi] 

NACp-6 0.3 [nmi] 

NACp-7 0.1 [nmi] 

NACp-8 0.05 [nmi] 

 

NACv-1 10 [m/s] 

NACp-2 3 [m/s] 

NACp-3 1 [m/s] 

NACp-4 0.3 [m/s] 

These values will be converted into standard deviations using Equation (73). 
78 

8.5.7.1 Single Distribution 

With none of the “Plot” checkboxes selected, pressing the “Distribution Analysis” button will 

generate a single distribution plot. For the example control panel shown in Figure 32, the 

resultant distribution is depicted in Figure 33. 

Figure 32 – Class B Single Distribution Control Panel Setup. 
79 

Figure 33 – Class B Single Distribution Result. 

The blue line in this figure represents the normalized histogram of outputs for a random 

sampling of input conditions about the nominal condition given the statistics of the position and 

velocity errors. The mean and standard deviation of the distribution are displayed in the legend 

at the right. The green points represent a normal distribution with an equivalent mean and 

standard deviation for reference. The 5% figure, as well as the red shaded area on the plot, 

represent the value for RCPA below which 5% of the samples lie (or, conversely, above which 

95% of the samples lie). 

8.5.7.2 Single-Parameter Sweep Distribution Analysis 

As was shown above for Class A system distribution analysis, with one of the “Plot” checkboxes 

selected, pressing the “Distribution Analysis” button will generate a set of distribution plots 

corresponding to a sweep of values for the checked parameter. In addition to the 3D plot 

showing the distribution as a function of the swept parameter, a second set of plots will appear to 

show the variation in the mean, standard deviation, and 5% point as a function of the swept 

parameter. For the example control panel shown in Figure 34, the resultant plots are depicted in 

Figure 35 and Figure 36. 
80 

Figure 34 – Class B SPS Distribution Analysis Control Panel Setup. 

Figure 35 – Class B SPS Distribution Analysis Distribution Result. 

The 3D figure represents the normalized histogram of outputs for a random sampling of input 

conditions about the nominal condition given the statistics of the position and velocity errors for 
81 

each value of the swept parameter (here, tCPA). This example shows a broadening of the 

distribution as tCPA increases. 

Figure 36 – Class B SPS Distribution Analysis Statistics Result. 

The mean and standard deviation of the distributions are plotted in the second set of graphs along 

with the 5% figure that represent the value for RCPA below which 5% of the samples lie (or, 

conversely, above which 95% of the samples lie) for each distribution. 

8.5.7.3 Two-Parameter Sweep Distribution Analysis 

Likewise, Two-Parameter Sweeps can be performed for Class B systems in the same manner as 

demonstrated in Section 8.5.6.5 for Class A systems. 
82 

9 Required Interval Management Performance Model 

9.1.1 Introduction 

The concept of Required Interval Management Performance (RIMP) was presented in an earlier 

report titled “Required Interval Management Performance, Version 1.1”. It is a metric to 

describe the performance of an interval management system which provides guidance for an 

aircraft (ownship) to arrive at an “Interval Management Point” at a precise time interval after 

another aircraft (reference aircraft). RIMP operation relies on the NextGen concepts of 

Trajectory-Based Airspace and Operations (TBO), airborne surveillance technologies such as 

ADS-B, precise navigation technologies including the RNAV/RNP operations, and various data 

link technologies. 

Aircraft engaged in TBO fly a flight plan agreed-upon by the operator and the Air Navigation 

Service Provider (ANSP). That planned trajectory is available to other aircraft within the 

airspace. The planned trajectory provides 4-dimensional intent information that includes the 

horizontal route (ground track) of the flight, altitude constraints and vertical profile, and the 

planned speed for each leg (segment) of the planned trajectory plan. The planned speed is 

presented here as the aircraft true airspeed. In addition, it is assumed that all participating 

aircraft have access to weather information containing detailed wind and altitude data for the 

areas of interest. Access to this information is needed to perform the required calculations to 

estimate the enroute time or “Time to Go” (TTG) to the spacing point for both the ownship as 

well as the reference aircraft. This will be necessary as it is not assumed that the TTG (plus an 

associated uncertainty) will be broadcast. It is also expected that excessive deviation from the 

agreed-upon flight plan will necessitate the negotiation of a revised flight plan between the 

operator and the ANSP. Many parameters can influence the planned operations and impact the 

behavior of the system. This section provides a framework to analyze RIMP operations in the 

presence of various uncertainties and operational limitations. An accompanying 

MATLAB/Simulink model provides a means for a time-based simulation of paired approaches 

using the RIMP concept. The results of this study will provide a benchmark to measure Interval 

Management (IM) performance and provide a way to determine the RIMP in order to achieve a 

spacing interval that meets appropriate levels of safety, predictability, and reliability. 

9.1.2 Section Overview 

Section 9.2.1 provides an overview of the simulation model, a description of RIMP operations, 

and the parameters that insert error and uncertainty into this operation. 

Section 9.3 describes the aircraft equations of motion, environmental variables such as wind and 

temperature, and sensor model. 

Section 9.4 describes the Time-to-Go (TTG) calculations. 

Section 9.5 outlines a simplified RIMP Speed Control concept implemented within the 

simulation model. 
83 

9.2 RIMP Operation Model 

An operation flow diagram for RIMP is provided in Figure 37. The process is initiated when the 

ANSP assigns ownship a reference aircraft to follow. 

Figure 37 – RIMP Operation Flow Diagram 

There are two primary processes involved in this operation. First, a 4-D planned trajectory is 

developed and optimized based on the current estimates of wind and temperature at specific 

points throughout the flight plan. This process includes information regarding position, altitude, 

time, reference aircraft intent information, ownship navigation and intent information, and 

ownship expected performance. 

The output of this step is a 4-D trajectory that specifies a series of 3-D legs and an expected true 

airspeed for each leg. Once the flight crew accepts the proposed solution and engages the paired 

operation, this trajectory information is broadcast to other traffic as well as Air Navigation 

Service Provider (ANSP). 

The second process (highlighted) is the execution of interval management. This includes 

monitoring ownship progress relative to the reference aircraft and providing interval 

management guidance commands (speed changes). It also includes monitoring the required 

performance associated with RIMP. It may be necessary to revise the original flight plan in 
84 

order to maintain the desired level of RIMP. In that case, a cycle through the first process will 

be necessary. 

The current version of the RIMP analysis model includes the ability to operate a 4-D trajectory. 

The trajectory consists of Trajectory Change Points (TCP) defined by their latitude, longitude, 

altitude, and a true airspeed to fly. The IM utilizes speed change commands for a finite length of 

time. Additional provisions are made to enable implementation and testing of the “IM Turn26” in 

the planned trajectory. It is assumed that significant changes to the planned trajectory including 

prolonged speed change will require a rebroadcast and possibly re-negotiation with other aircraft 

engaged in paired operation and/or ANSP. 

The focus of this mathematical modeling effort is the execution of the paired IM operation. The 

results of this analysis may also be used in the future to further develop the 4-D planned 

trajectory generation and optimization process. The intent of developing a representative 

mathematical model for IM operations is to quantify the effects of real-life variations and 

limitations on its “behavior”. 

The current aircraft spacing standards in Instrument Flight Rules (IFR) conditions are based on 

various limitations imposed on the IM operations such as the Air Traffic Control (ATC) 

workload, surveillance system accuracy and update rate, communication limitations, and speed 

control limitations (e.g. aerodynamic limits). A mathematical representation of the system 

should be able to shed some light on the impact of various limitations and allow to verify the 

validity of current operations as well as to demonstrate how the spacing time can be safely 

reduced when the appropriate navigation, surveillance, and automated procedures are used. 

9.2.1 Modeling Approach 

A simplified non-linear simulation of an aircraft executing the RIMP operation was developed 

using the MATLAB/Simulink environment. This model can be used for a Monte Carlo 

simulation which will collect statistics that are not analytically obtainable due to the lack of a 

closed-form solution. A Simulink concept called “Reference Model27” is used to represent each 

of the aircraft engaged in the RIMP operation. The aircraft models are identical in their format, 

but can be provided with different internal parameters to make them represent different aircraft 

performance and limitations. 

Figure 38 shows the Simulink top level view of the RIMP model. An implementation of 6-

aircraft RIMP pairings is depicted. 

Each aircraft within the RIMP operation model is assigned a unique identification number (e.g. 

#4). This enables different aircraft to be associated with a specific flight plan and a specific 

initial condition file. In addition, a unique aircraft performance data file is associated with each 

aircraft. Two other data files are used by the RIMP model. These are common among all 

aircraft and specify general simulation constants as well as sensor characteristics. 

 

26 The “IM Turn” is a form of path length modification used to make larger adjustments to the time of arrival 

estimate than can be achieved by speed control alone. 27 A Simulink Reference Model allows the use of a single model in multiple instances for easier maintainability. 
85 

Figure 38 – RIMP Operation Model 
86 

Figure 39 provides an overview of the RIMP operation model data flow. At the start of the 

simulation, the flight plan, aircraft limits, and aircraft setup files are read and the appropriate 

variables within the MATLAB simulation are populated. The aircraft setup file points to the 

appropriate flight plan, aircraft limit, and pre-processed wind data file. The setup file also 

includes data needed to initialize and configure the specific aircraft operation within the interval 

management stream. The execution of the model also includes two common files that describe 

the sensor properties (common among all aircraft) as well as other constant parameters used for 

the execution of RIMP. 

The model generates a time series of aircraft states at a sampling rate of once per second for each 

of the aircraft. In addition, the processed flight plan information for each aircraft is available. 

Finally, a set of statistics for each aircraft RIMP execution will be stored. 

 Figure 39 – RIMP model data flow 

The RIMP operation simulation model consists of a number of aircraft models executing interval 

management operations. The communication between the aircraft is currently accommodated by 

a simple message structure where each aircraft writes its own states and from it reads the states 

of all other aircraft from. At present, a single frame delay is used in this communication and it is 

assumed that the total system latency is accounted for in the sensor model for each aircraft. 

However, the mechanism is in place to add additional delay frames for the communication 

system. It should be noted that the nominal system update is currently done at 100 Hz (10 

millisecond update rate). 

RIMP 

Simulator 

RUC 

data 

Constants 

Sensor 

Param. 

A/C states and

parameters (time)

Processed 

flight plan 

Engagement 

statistics 

Flight Plan ”b” 

Flight Plan ”a” 

A/C Perf ”757” 

A/C Perf ”767” 

AtmosData 002 

AC#2 

Setup 

& IC 

AC#1 

Setup 

& IC 

AtmosData 001 

Data Library
87 

9.2.2 Aircraft Model Configuration 

Figure 40 provides an overview of the RIMP aircraft model. The primary input to each aircraft 

is a unique aircraft number. Upon the start of the simulation, each aircraft model will access the 

base MATLAB workspace and retrieve the parameters associated with the specified aircraft 

number. This includes aircraft operational and performance parameters as well as the assigned 

flight plan. In addition, a dynamic dataset called “ADS_Data” is read by each aircraft at each 

execution frame (ADSB_in). This dataset provides “current” information on all the vehicles 

involved in the RIMP operations. The aircraft model generates two output vectors. One is the 

“ADSB_Out” vector which is shared by all the other aircraft during operation. The other is the 

relevant aircraft data including the measurement of dynamic states as well as other desired 

simulation parameters. 
88 

Figure 40 – RIMP Aircraft Model 
89 

Each aircraft model contains all the necessary components to complete the 4-D flight plan and to 

perform RIMP operations. At the core of the aircraft model is the “A/C Dynamics” block that 

contains the equations of motion, initialization, and wind environment components. Other 

system components include the “measured output” module which contains various sensor 

properties, “navigation” module, “guidance and control” module, “time-to-go estimator”, “ADSB input”, “ADS-B output”, and “RIMP control” modules. 

9.2.2.1 Aircraft Dynamics Model 

The aircraft dynamics model provides a simplified set of equations of motion for propagation of 

a point mass in three-dimensions plus the east and north wind velocities. The output of the 

aircraft dynamics model is considered as the truth model and is fed to the sensor model. The 

inertial velocities are defined using the total inertial velocity, plus a ground track and a vertical 

flight path angle. The position states include the Easting, Northing, and altitude relative to sea 

level and a navigation origin point defined in the “constants” data file. 

The wind states include east and north components. Wind states model a spatially varying wind 

defined at each waypoint and linearly varying to the adjoining waypoints. Each wind “node” 

includes forecast wind plus a random wind component which is different for each aircraft. 

Essentially, this represents a slowly time-varying component of the wind in addition to the 

forecast wind since aircraft are separated by approximately one minute. 

9.2.2.2 Measurement Model 

The measurement model receives the truth aircraft states from the aircraft model which it uses to 

compute several derived parameters such as groundspeed, vertical speed, and airspeed. The 

sensor model adds various elements of errors and uncertainties associated with each specific 

sensor to “perturb” the data and simulate typical sensor output. The output of the sensor model 

is then used by various components of the simulation to generate a realistic system response. 

9.2.2.3 Navigation Module 

The flight plan is preprocessed for each aircraft to generate a sequence of segments for the 

aircraft to fly. Each segment includes an origin waypoint and a destination waypoint with 

associated altitudes. This information is translated to a required ground track, a required flight 

path angle and a planned airspeed to fly. 

During the execution of the simulation, the navigation module compares the current 

measurement of aircraft states against the desired navigation information and calculates such 

navigation parameters as cross-track error, glideslope deviation, desired turn rate, desired climb 

rate, etc. A critical function of the navigation module is to keep track of the progress of the 

flight and automatically sequence through the specified waypoints. As a part of this function, the 

navigation module computes when the aircraft must start turning towards the next segment 

course and when to declare a waypoint sequenced. 

The navigation module operates under two normal navigation modes; “Turn Mode” and “Track 

Mode”. “Turn Mode” provides navigation commands to intercept the next segment course at the 

proper moment using a constant radius turn. As the aircraft inertial speed is changed due to the 
90 

wind environment, the rate of turn is adjusted to maintain a constant radius to complete the 

intercept turn correctly. When in Track mode, the navigation system provides the appropriate 

parameters to enable the aircraft to track the desired course and null any detected cross-track 

error. In either mode, the navigation system provides the appropriate desired flight path angle 

and true airspeed to fly the current segment. 

9.2.2.4 Guidance and Control 

The guidance model attempts to fly the aircraft as directed by the navigation module and RIMP 

control module. In essence, the guidance and control module replaces the function of the pilot or 

autopilot. The linear acceleration command is generated and adjusted such that the commanded 

true airspeed is achieved and maintained. Similarly, turn rate and rate of change of vertical flight 

path are adjusted in order to either achieve the desired navigation commands or to null the crosstrack or glideslope error predicted by the navigation module. 

The guidance and control module also monitors the specified aircraft envelope and limits the 

commands to preserve the required flight envelope. For example, if the RIMP controller 

requests an airspeed reduction beyond the aircraft specified envelope, the guidance and control 

module will limit the airspeed command. 

9.2.2.5 TTG Estimator 

The “time-to-go” or TTG estimator computes an estimate of time required to fly to the RIMP 

interval management point. It uses the current position and velocity information from the sensor 

model, as well as current and future planned trajectory parameters from the flight plan. The TTG 

estimator uses the planned airspeed and average forecast winds to compute an “expected” 

groundspeed for each segment. The current segment calculations use the measured winds and 

current airspeed, but the system assumes that the airspeed will be adjusted to the planned value. 

Since the actual winds are statistically different than the forecast values and the simulation model 

linearly changes the winds while enroute, the wind error terms will directly affect the TTG 

estimates. 

The current version of the simulation assumes that each aircraft broadcasts its time-to-go 

estimate. This is a simplification in the current model. However, since all aircraft have access to 

the flight plan of their reference aircraft, a duplicate copy of the TTG estimator can be used to 

estimate TTG for the reference aircraft. 

9.2.2.6 Surveillance Model 

The surveillance model acts as a conduit to supply the reference aircraft position, speed, and 

other parameters to the ownship. In practice, a combination of ADS-B messages and other data 

link information provide the surveillance function. In the simulation model, this component adds 

the appropriate time delays to the affected parameters as needed. In the initial version of the 

simulation, this model is essentially a pass-through model providing the TTG broadcast by the 

reference aircraft. 
91 

9.2.2.7 RIMP Speed Control 

The current version of RIMP speed control monitors the TTG for ownship and the reference 

aircraft and generates a positive or negative increment to the speed guidance commands. This is 

effectively an increment to the planned true airspeed of aircraft. 

9.2.3 Elements of Uncertainty and Operational Limits 

The interval management operation relies on ownship aircraft navigation, extensive surveillance 

information received from other participating aircraft, and a number of assumptions concerning 

the intent and procedures executed by all participating aircraft. One key element of this 

procedure is the estimate of TTG to the interval management point for both ownship and the 

reference aircraft. The desired or commanded interval time is written as the difference between 

the time-to-go of ownship and the reference aircraft. To model the real-life uncertainties in 

navigation, flight planning, wind variation from the forecast values and many other terms 

affecting the RIMP operations, the following modeling terms have been added: 

1- Wind uncertainty (relative to predicted local winds) 

2- Speed measurement uncertainty 

3- 3-D Position/Distance-to-go uncertainty 

4- Altitude measurement uncertainty 

5- Integration error 

6- Input time uncertainty 

7- Speed guidance accuracy 

8- Temperature and pressure uncertainty impacts the calculation of speed of sound and 

air density. This affects the calculation of Mach number and indicated airspeed. In 

the current implementation, the temperature variation is included but its uncertainty is 

not modeled. 

9- Performance data accuracy (for predictions/trajectory integration) affects the ability 

of the aircraft to meet the planned trajectory. The airspeed limitations are modeled as 

a function of altitude. However, propulsion system limitation and aerodynamic drag 

uncertainties are not directly introduced in the current version of the simulation. 

In addition to the above, a number of “pure time delay” terms are added to the interval time 

calculations. These are: 

10- Surveillance time delay 

11- Onboard calculation time delay 

12- Communication time delay 

13- Equivalent human performance delay and/or autopilot time delays are indirectly 

modeled. These terms are pure time delays associated with the onset of responses to 

events. 

The interval-time equations include the following basic parameters that may be adjusted, and 

need to be maintained (controlled) within a given limit: 

14- Planned acceleration or deceleration 
92 

15- Planned Mach or indicated airspeed is maintained. True airspeed is calculated. The 

TTG for the current segment is based on the current true airspeed and current 

measured winds. The time-to-go for future segments is based on forecast wind values 

along with the planned true airspeed. 

16- Planned 3-D flight path 

Effective resolution of signals or parameters used to control the RIMP operation introduces an 

equivalent dead-band28. For example, airspeed change is customarily commanded by air traffic 

control in increments of 10 Knots. The indicated airspeed is typically displayed with a resolution 

of 1 Knot. Several “dead-band” components are included: 

17- Airspeed control dead-band 

18- Interval management dead-band 

19- Position and altitude control dead-band 

Several limits are used to restrict the available solution-space. In most cases, reaching a limit 

could affect the method of control or may require a revision of the 4-D flight plan. 

20- Indicated or Equivalent airspeed upper and lower limits as a function of altitude 

21- Acceleration and deceleration limits 

22- Rate of turn limits 

23- Rate of change of flight path angle limits 

24- Available power for airspeed control is not currently implemented. 

The interval time equations essentially yield the dynamic response of the system in the presence 

of the above terms. Since Interval Management is essentially a feedback control system, the 

intent of this analysis is to assess the performance and stability properties of this system by 

exercising these equations. 

9.3 Aircraft and Environment Model Derivation 

This section defines the simplified point-mass equations of motion, sensor model, and various 

uncertainty and variability components added to the basic model. 

9.3.1 Point-Mass Equations of Motion 

9.3.1.1 Inertial States 

For this analysis it is assumed that the aircraft is strictly a point mass. Therefore, orientation 

concepts such as heading and pitch angles are undefined. However, the aircraft inertial velocity 

vector is used to define a total velocity, a ground track, and a vertical flight path angle. The time 

derivative of the inertial velocity vector is the input to the equations of motion. The equations of 

motion propagate the following relations. 

 

28 A dead-band is an area of signal range or band where no action occurs. 
93 





















 











































 





















 

Altitude

Northing

Easting

v

v

a V v

U

N

E











 (74) 

Since the aerodynamic forces such as lift and drag depend on the true airspeed and air density 

experienced by the aircraft, it is necessary to compute the instantaneous true airspeed 

experienced at the aircraft position. This is accomplished by computing an instantaneous wind 

vector independent of the aircraft’s inertial states described above. 

The acceleration and angular rates are computed by the guidance and control module. The total 

acceleration and angular rates used by the equations of motion include the commanded terms 

plus random external components that can be used to mimic external disturbances such as 

atmospheric turbulence or other factors altering the intended flight path. 

Total Command ExternalDisturbance

a a a























































































 (75) 

The east, north, and up components of the velocity vector are computed using the integrated 

acceleration and angular rate terms above. 

  

  

  





















 

 

























 

 

sin

cos cos

cos sin

V

V

V

v

v

v

U

N

E

 (76) 

The position and altitude states are computed by direct integration of east, north, and up 

velocities. 

9.3.1.2 Wind States 

The horizontal wind magnitude and direction is pre-defined at each waypoint location and 

altitude. These are the forecast wind parameters. For a better representation of a real-world 

wind field, a random component is added to the forecast wind value for each waypoint and each 

aircraft. Therefore, subsequent aircraft flying over the same waypoint at the same altitude will 

experience somewhat different wind field. The waypoint winds are defined as: 

U Forecast

N

E

dir Forecast

Speed

w

w

w

W

W























 



 





0

 (77) 
94 

Random

N

E

U Forecast

N

E

U Total

N

E

w

w

w

w

w

w

w

w







































































 0 0 0

 (78) 

The random wind components do not change during the simulation execution. However, the 

implementation of the wind variation for different aircraft mimics a slowly varying random wind 

field affecting the flow of traffic. To ensure a continuously varying wind field from one 

waypoint to the next, a set of wind rate of change terms are computed. These rates are integrated 

to compute the instantaneous east and north wind components. 





















 























0 0

N

E

N

E

w

w

w

w





 (79) 

When traveling towards waypoint i, the instantaneous rate of change of east and north wind 

velocities are computed using the following relationship relating the current east and north winds 

and total wind velocity defined at the waypoint. 

AlongTrack

AlongTrack

N N

N

AlongTrack

AlongTrack

E E

E

V

d

w w

w

V

d

w w

w

i current

i current

  

  





 (80) 

The terms d AlongTrack and VAlongTrack are the instantaneous along-track distance-to-go and alongtrack inertial velocity of the ownship to the current destination waypoint. 

9.3.2 Aircraft Outputs and Measured Parameters 

The integrated aircraft states corrected for initial conditions constitute the output vector. The 

aircraft output vector is the input to the sensor model. The sensor model output vector is the 

aircraft output parameters adjusted for sensor error, uncertainty, and dynamics. The measured 

parameter vector is based on the sensor output vector with additional derived parameters, such as 

indicated and true airspeeds, Mach number, density, and speed of sound. 

9.3.2.1 Sensor Model Error and Uncertainty 

The modeled sensor outputs include errors and uncertainties that affect the basic navigation and 

guidance of the aircraft as well as RIMP TTG estimate and speed control functions. The model 
95 

includes a complete list of parameter uncertainties and has the ability to vary them 

deterministically or randomly to accommodate different types of analyses. 

A general sensor formulation includes the following components: 

1- Scale factor correction (nominal = 1) 

2- Bias correction (nominal = 0) 

3- Measurement noise (nominal mean = 0, appropriate variance) 

4- Measurement dynamics. This is essentially the sensor noise attenuation ability. For 

simplicity, two first-order lag filters with appropriate time constants are placed in 

series and filter the added noise signal. 

5- Upper and lower measurement limits (sensor-dependent) 

6- Signal quantization, defining the resolution of digital signals and effective resolution 

of analog signals or gauges. 

7- A pure time delay is added to the signal to account for processing delays associated 

with multiple digital avionics components that are integrated in a typical aircraft. 

Additionally, this term can be used to represent an overall lag associated with control 

actuation and aircraft response. 

8- For certain signals such as GPS-based position estimates, the typical long term drifts 

can be modeled using a properly configure random walk signal. 

The overall form for each sensor is represented as: 

v   t delaydiscrete  bias SF v t noise random walk up it

m low it _ lim     lim  (81) 

The sensor output vector consists of the following parameters: 

1- Measured inertial velocity in Meters per second 

2- Measured ground track in Radians 

3- Measured vertical flight path angle in Radians 

4- Measured East wind in Meters per second 

5- Measured North wind in Meters per second 

6- Measured outside air temperature in degrees Kelvin 

7- Measured static pressure in Millibars 

8- Measured East distance from reference coordinates in Meters 

9- Measured North distance from reference coordinates in Meters 

10- Measured altitude (MSL) in Meters 

9.3.2.2 Temperature and Pressure Prediction Uncertainty 

The forecast pressure and temperature as a function of position, altitude, and time is provided in 

the RUC atmospheric data. This data is used to develop the flight plan which is then executed by 

the RIMP model. Since flight operation is conducted with constant Mach number or constant 

equivalent (or indicated) airspeed, variation between the predicted and actual local air 
96 

temperature and pressure will result in a difference between the predicted and actual true 

airspeed and thus the predicted ground speed of the aircraft. The temperature and pressure 

uncertainty terms are not currently implemented in the simulation model. The aircraft is exposed 

to continually varying temperature and pressure based on the forecast values. However, small 

changes from the forecast will be observed in measured temperature and pressure values due 

sensor uncertainties described above. 

9.3.2.3 Airspeed and Mach Calculations 

The true airspeed signal is computed from the total inertial velocity and the instantaneous wind 

vector using the following equation. 

       2 2 2 ( ) VE WE VN WN VU TAS t      (82) 

As described in the equations of motion section (9.3.1.1), the instantaneous wind vector is 

computed by integrating wind acceleration terms. The current formulation essentially provides a 

linearly varying wind from one waypoint to the next. At each waypoint, the total wind vector 

consists of the forecast wind plus a random wind uncertainty or error term that is constant and 

determined at the beginning of the simulation run. 

The local speed of sound in meters per second is computed based on the measured outside air 

temperature. 

C(t)    RT (83) 

Assuming dry adiabatic air properties, =1.4, R=287.05 (N.m)/(Kg.K), and T is temperature in 

degrees Kelvin. The local density in kilogram per cubic meter is computed based on the 

measured outside air temperature and pressure 

T  K

Kg K

N m R

m

N P

t

 















 







 







2

( ) (84) 

The atmospheric pressure data is provided in millibars and the above equation is multiplied by 

100 to account for that unit. The aircraft Mach number is calculate using the following equation. 

C

TAS M  (85) 

The indicated or equivalent airspeed is computed as: 
97 

  IAS  TAS   TAS  1.2250 (86) 

9.3.3 Navigation Calculations 

The navigation module monitors the progress of the aircraft along its flight plan, sequences the 

flight plan waypoints, and directs the aircraft to follow a straight path or turn to a specified new 

direction. An extended description of the navigation module is provided here. 

A special case for the navigation model is given when the aircraft arrives at the runway. Upon 

arriving over the runway touchdown point, the navigation system commands the appropriate 

deceleration and rotates the aircraft to a zero vertical flight path angle. This special case is 

described later in this section. The normal flight operation is described first. 

The navigation module treats the flight plan to be executed as a sequence of flight segments. 

Each segment is defined with an origin waypoint and a destination waypoint. These waypoints 

or “trajectory change points” (TCP) include a latitude, longitude, altitude, and speed. The speed 

to fly the segment is the true airspeed defined at the destination waypoint. It is assumed that the 

aircraft will decelerate or accelerate at a pre-defined level to achieve and maintain the specified 

true airspeed. The line connecting the origin and destination waypoint defines the course 

heading, glideslope, and absolute segment length. In general, the aircraft will attempt to fly as 

close as possible to this line. 

The navigation module monitors the progress of the flight and retains a memory of which 

segment of flight is completed and which waypoints are sequenced. Therefore, the current 

segment number is a “state” parameter for the navigation module and this memory is used to 

select the appropriate flight plan information during execution of the flight. 

During normal operation (when the aircraft has not arrived at the touchdown point on the 

runway), the navigation mode operates in one of two modes: TRACK or TURN. The module 

retains a memory of in which operating mode it is and this mode is only changed when the 

appropriate set of requirements are met. Figure 41 provides a pictorial example of the origin and 

destination waypoints defining a segment. The blue segments indicate that the navigation mode 

is TURN, and the orange line indicates where the navigation mode is in TRACK. 
98 

Figure 41 – Flight Segment and Navigation Modes 

9.3.3.1 TRACK Mode 

When operating in TRACK mode, the “desired” turn rate is set to zero and the “desired” track 

angle is set to the segment course. This combination is used by the guidance and control module 

to attempt to zero the cross-track error computed by the navigation module. Figure 42 depicts 

the segment length (Dsegment), cross-track error (CTE), and along-track distance-to-go (ATD2G) 

geometries. 

Figure 42 – TRACK Mode Parameters 

     2 2 Dseg  Edest  Eorig  Ndest  Norig (87) 

       

seg

dest A C dest orig dest A C dest orig

D

N N N N E E E E

ATD G         / / 2 (88) 

       

seg

dest A C dest orig dest A C dest orig

D

E E N N N N E E

CTE         / / (89) 

Destination 

Waypoint 

Origin 

Waypoint 

Dsegment

ATD2G 

CTE 

TURN 

TRACK 

TURN 

Destination 

Waypoint 

Origin 

Waypoint 
99 

The cross-track error is positive when the course path is on the right side of the aircraft and a 

right turn is required to correct for it. 

When the aircraft is operating in the TRACK mode, the navigation module computes the 

required distance before reaching the destination waypoint at which point a turn to a new course 

is necessary. This calculation depends on the current groundspeed of aircraft, the nominal rate of 

turn specified in the “constants” file, and the total change of course required. The nominal turn 

radius is computed as: 

 nominal 

GS

turn

V R  (90) 

The required turn angle is the rectified difference between the next segment course and the 

current aircraft track angle. The rectified difference provides a result between - and +. It 

should be noted that both the segment course and aircraft ground track are rectified between 0 

and 2 (0 to 360 degrees) as measured from true North. The magnitude of the turn angle is 

limited to 150 degrees. This is consistent with airline flight operations. Figure 43 provides a 

pictorial presentation of the required distance before the waypoint to start the turn. 

  seg A C rect    1   / (91) 

 

  



 

   1 cossin turn

before

R

d (92) 

When the aircraft reaches the required turn before the waypoint, the navigation system will set 

the mode state to TURN. At this point, the reference groundspeed is set to the current 

groundspeed. This parameter is needed in the memory in order to maintain the turn radius 

constant as groundspeed changes due to varying wind impact on the aircraft. In addition, the 

“Start Capture” flag is set to TRUE, the “desired” turn rate is set to the nominal value, and the 

“desired” final track angle is set to the next segment course. 
100 

Figure 43 – Required Distance for Turn to New Course 

9.3.3.2 TURN Mode 

When the navigation module is in TURN mode, the along-track distance-to-go or cross-track 

error parameters are not used for navigation or guidance. Instead, a turn radius is computed 

based on the reference groundspeed set prior to switching to TURN mode. This procedure 

ensures that the system maintains a constant radius of turn through the course capture procedure. 

The desired turn rate is then scaled based on the current groundspeed measurements as time 

progresses and the aircraft changes its orientation. 

desired

GS current

no al

GS reference

turn

V V

R

 

_

min

_   (93) 

Continually varying the desired turn rate to maintain the turn radius ensures accurate intercept 

and merging with the next segment course. When the magnitude of the difference between the 

desired final track angle and current aircraft track angle is less than a tolerance value set in the 

“Constants” file, the navigation mode is switched to TRACK and the desired track is set to the 

current segment course. 

9.3.3.3 Waypoint Sequencing Computations 

While the aircraft executes segment i of the flight plan, it is traveling from waypoint i-1 to 

waypoint i. The waypoint capture criteria depend on the mode of the navigation system. 

When in TRACK mode, if the direct distance to the waypoint is less than a set value (defined in 

the “constants” file) the waypoint is flagged as “captured”. Otherwise, a check is made to see if 

the missed waypoint conditions are met. If the waypoint is missed, the waypoint is considered 

complete and the segment number is sequenced. Otherwise, the navigation module continues 

along the current segment. The missed waypoint logic compares the current aircraft bearing with 

the segment course. If the rectified difference between these angles exceeds 85 degrees and the 

waypoint has not been captured yet, it flags it as a missed waypoint. 

dbefore

abs()

Rturn
101 

     



















  

_ int 85

tan

/

1 /

missed waypo rectify Bearing

N N

E E

Bearing

seg

dest A C

dest A C



 (94) 

When in TURN mode, a waypoint is considered captured when the aircraft track reaches halfway 

through the total planned turn. For example, if the aircraft was turning from a ground track of 

220 to 300 degrees, the waypoint is considered captured once the ground track reaches 260 

degrees. Figure 44 depicts the capture geometry in TURN mode. 

Figure 44 – Waypoint Capture in TURN Mode 

The logic is configured such that the segment sequencing is completed once during over-flight of 

a waypoint or during turn to a new direction. Therefore, addition of noise in the position and 

velocity signals will not affect the waypoint sequencing logic. 

9.3.3.4 Corrected Segment Length 

The lateral navigation parameters described above rely on a segment length defined by the actual 

horizontal distance between adjoining waypoints. In the general case where the aircraft starts a 

turn prior to reaching each waypoint, the actual distance traveled by the aircraft as it covers the 

planned trajectory is slightly less than the full distance between fly-by waypoints. This 

difference in distance covered affects the vertical navigation calculations and time-to-go 

estimation which will be covered later in this section. Both of these issues can be addressed 

through the introduction of a corrected segment length concept. 

Figure 45 provides a description of the geometry being considered. The flight plan switches to 

segment i halfway during the turn into the current segment and will switch to the next segment 

halfway through the turn to the next segment. 

Waypoint 

capture 
102 

Figure 45 – Corrected Segment Length 

The corrected segment length consists of the arc lengths at the start and the end of each segment 

plus the linear distance in-between. The total length is always slightly less than the linear 

distance between fly-by waypoints. To perform this calculation a nominal rate of turn is 

computed based on planned average groundspeed of the aircraft as it covers the entire segment. 

It is understood that uncertainty in actual wind will introduce an error into this equation. 

However, this allows providing a first-order approximation to the segment length problem. 

Since the planned ground speed may be very different between the start and end turns on each 

segment, two separate turn radii are defined. 

nominal

_ 2

2

nominal

_ 1

1









GS plan

turn

GS plan

turn

V

R

V

R





(95) 

The planned groundspeed is based on the planned Mach number or indicated airspeed, forecast 

wind speed and forecast wind direction at the closest waypoints. Using the forecast air 

temperature and pressure at each waypoint an estimate of true airspeed can be calculated. The 

true airspeed is then used to arrive at the planned ground speed. The turn angle and distance 

before (and after) the waypoints and associated arc lengths are computed below. 

 

  seg seg

seg seg

rect

rect

  

  

  

  





2 1

1 1

 (96) 

 

 

 

  2

2 2

2

1

1 1

1

1 cos

sin

1 cos

sin









 

  

 

  

turn

before

turn

before

R

d

R

d

 (97) 

2 2 2

1 1 1

0.5

0.5





   

   

arc turn

arc turn

d R

d R

 (98) 

arc1 d

arc 2 d

before2 d

before1 d
103 

The corrected segment length is defined as: 

Dseg _ corr  Dseg  dbefore1  dbefore2  darc1  darc2 (99) 

Similarly, a corrected along-track distance needs to be computed to go based on the simple linear 

ATD2G computed above. This can be approximated by comparing the linear ATD2G value with 

the segment lengths computed above and taking into account the current ground track offset from 

the segment course. 

If ATD2G  Dseg  dbefore1 , the aircraft is still turning to intercept the segment course. The 

difference between current track and segment course is used to compute the appropriate arc 

length. 

  seg A C rect      / (100) 

darc  Rturn1   (101) 

1 2 2 2 corr seg before before arc arc ATD G  D  d  d  d  d (102) 

If ATD2G  Dseg  dbefore1  dbefore2 , the aircraft is on the straight section of the segment and has 

not yet started its turn to the next track. The corrected along-track distance-to-go is computed as: 

2 2 2 2 corr before arc ATD G  ATD G  d  d (103) 

If ATD2G  dbefore2 , the aircraft has started its turn to the next track. The corrected along-track 

distance-to-go is computed as: 

  A C seg   rect  /   (104) 

darc  Rturn2   (105) 

corr arc arc ATD G  d  d 2 2 (106) 

9.3.3.5 Vertical Navigation 

The vertical navigation parameters include the vertical flight path angle and a glide slope 

deviation. The vertical flight path angle is the angle in the vertical plane for a line connecting 
104 

the origin waypoint to the destination waypoint. Assuming a straight-in flight profile between 

the origin and destination waypoints, the desired flight path can be written as: 















   

seg

dest orig

seg D

Alt Alt 1  tan (107) 

The instantaneous glideslope deviation is a measure of vertical distance above or below the 

nominal vertical flight path line described above. Given the current along-track position of the 

aircraft (along-track distance-to-go on the current segment) and the current altitude, the 

following equation computes the glideslope error. If the current altitude of the aircraft is below 

the glideslope, the error will be positive and a positive increase in the aircraft flight path angle 

will be required. 















     

seg

dest orig

dest A C D

Alt Alt

GSE Alt Alt / ATD2G (108) 

The dependency of this calculation on the assumed segment length and along-track distance-togo is evident. As described above, when the aircraft turns to intercept a new flight plan segment, 

it effectively shortens the segment length and alters the along-track distance-to-go. Therefore, an 

apparent glideslope error (GSE) will be experienced during such turn maneuvers. The current 

version of the RIMP simulation model does not fully correct for this apparent error. This error is 

small when total ground track angle changes less than 90 degrees. 

9.3.3.6 Landing Mode Operations 

Once the aircraft arrives over the touchdown waypoint, a special set of modified navigation 

equations are used to command the aircraft to stop on the runway. 

In the lateral axis, the desired track is set equal to the segment course. This course is the same as 

the previous segment course. The desired turn rate is set to zero and the cross-track error is also 

set to zero. 

In the vertical axis, the desired vertical path is set to zero and the glideslope error is also set to 

zero. It should be noted that typically, the final approach segment has a vertical glideslope angle 

of -3 degrees. Therefore, the navigation system will command the aircraft flight path angle to be 

increased to zero. 

The desired segment airspeed is set to zero for the landing operation, and the along-track speed is 

set to the current aircraft speed. Within the guidance and control module, the speed specification 

during landing is interpreted as slowing the aircraft to zero groundspeed. Therefore, in the 

presence of non-zero surface wind, the aircraft will retain a non-zero airspeed. 
105 

9.3.4 Guidance and Control Calculations 

The guidance and control module compares the current measured aircraft states and navigation 

module outputs to derive acceleration and angular rate commands. The module uses several 

guidance and control gains that are specified in the “constants” data file. 

9.3.4.1 Airspeed Control 

Depending on the mode of operation the aircraft will be commanded to maintain a constant 

Mach number or maintain a constant indicated airspeed. In either case, the aircraft must stay 

within a specified indicated airspeed (or dynamic pressure) range. As a result, during the early 

stages of descent, the aircraft will maintain a Mach number in the descent until the indicated 

airspeed reaches the nominal high value. The aircraft will then maintain this nominal value until 

reaching the destination waypoint of the current segment. 

The above behavior will result in a continually increasing true airspeed until the desired dynamic 

pressure is reached. After this, the true airspeed will be continually decreasing as the aircraft 

descends. Since the RIMP operation model computes the instantaneous pressure, temperature, 

density, and speed of sound, the commanded Mach number or commanded indicated airspeed 

can be related to an instantaneous commanded true airspeed. The total airspeed command 

includes a RIMP airspeed increment. The total airspeed is limited within the indicated airspeed 

upper and lower limits set for each aircraft type as a function of aircraft altitude (or Flight Level). 

If the speed command is less than 1.1, it is assumed that a constant Mach number is commanded. 

IASplan  Mcmd  c   (109) 

Otherwise, it is assumed that a constant indicated airspeed is commanded. The RIMP speed 

increment is then added to the resulting indicated airspeed command, and the result is limited 

before converting back to an instantaneous true airspeed signal. Figure 46 depicts the airspeed 

control system. 

 

 

 

/ 

max

min

max

min

cmd cmd

IAS

RIMP IAS cmd plan

TAS IAS

IAS IAS IAS

KIAS f Alt

KIAS f Alt



  





 (110) 
106 

Figure 46 – True Airspeed Control System 

Additional logic within the acceleration limiter block ensures that the system inhibits zerocrossing within an update frame when the airspeed error signal is very small. 

9.3.4.2 Vertical Flight Path Control 

The total vertical flight path command consists of the planned vertical flight path angle plus 

corrections needed to zero the computed glideslope error (GSE). To mimic typical aircraft 

operations, the flight path correction angle is limited to ±2.8 degrees of the nominal value. 

Figure 47 depicts the vertical flight path control system. 

Figure 47 – Vertical Flight Path Control System 

Additional logic within the flight path rate limiter block ensures that the system inhibits zerocrossing within an update frame when the flight path error signal is very small. 

9.3.4.3 Lateral Path Control 

When the navigation module operates in TRACK mode, a desired track angle and a cross-track 

correction angle are provided to the guidance and control module. The controller attempts to fly 

the aircraft to zero the cross-track error (CTE). A track correction angle corr is computed 

using the following equation. 

sign  CTE

R

CTE

turn

corr 





















   

1.0

0.0

1

2  sin (111) 

Gamma-dot 

command 

Reference Gamma-dot 

(Constants data file) 

Gamma 

command 

Measured 

Gamma

Aircraft 

point-mass 

EOM 

+ 

- 

+ 

+ 

Kgam

GSE 

± 2.8 

degrees 

Linear 

acceleration 

command 

Nominal acceleration 

(Constants data file) 

TAS command 

Measured TAS 

Aircraft 

point-mass 

EOM 

+ 

- 
107 

The turn radius in the above equation is computed based on the current ground speed and a 

maximum bank angle of 10 degrees (Appendix A). The controller attempts to zero the error 

between the measured ground track and the sum of desired track and the track correction angle. 

Figure 48 depicts the lateral control system when operating in TRACK mode. To ensure correct 

operation, the sum of desired track angle and track correction angle must be rectified between 0 

and 360 degrees as measured from true North (0 to 2). In addition, the subtraction of measured 

and total commanded angles must be put through an “error rectifier” algorithm to properly 

rectify the difference between - and +. 

Figure 48 – Lateral Control System in TRACK mode 

The limited turn rate signal shown is the smaller of the reference turn rate and a turn rate that 

would result in a 10 degrees bank angle at the current groundspeed. Additional logic within the 

turn rate limiter block ensures that the system inhibits zero-crossing within an update frame 

when the track error signal is very small. 

When the navigation module operates in TURN mode, a desired turn rate and a desired final 

track angle are provided. The difference between the desired track angle and current measured 

ground track is put through an error rectifier circuit to define the proper direction of turn. The 

sign of this signal is then applied to the desired turn rate supplied by the navigation module to 

create the turn rate command sent to the equation of motion. 

9.3.4.4 Operation while Landing 

During landing operation, the controller operates in a special mode. The speed control switches 

to using groundspeed and attempts to zero the groundspeed. The maximum deceleration limit is 

increased to 2.0 meters per second or 0.2 g’s. The vertical flight path controller attempts to zero 

the vertical flight path angle at an increased rate of 2.8 degrees per second. Finally, the turn rate 

command is set to zero and no attempt is made to “stay on the runway”. Since the intent of the 

RIMP simulation model is the flight portion, this landing arrangement provides and adequate 

termination condition and not necessarily an accurate landing model for the air vehicles. 

9.4 TTG Estimation 

The TTG estimation function is required for the RIMP operation. TTG calculations can be 

performed for both the ownship and the reference aircraft. 

Turn rate 

command 

turn rate limit 

(Constants data file) 

desired 

track 

Measured 

ground track 

Aircraft 

point-mass 

EOM 

+ 

- 

+ 

+ 

 corr

Rectifier Error 

rectifier 
108 

9.4.1 Velocity Profile and Associated Assumptions 

For calculating estimated time of travel or time to go on a segment an estimate of the ground 

speed is needed. As described in the earlier sections, the typical speed command will be either a 

constant Mach number or a constant indicated airspeed. A typical segment is 15 nautical miles 

or less and wind variation within a level altitude segment is considered relatively small. It is 

therefore reasobale to assume that the velocity profile for such a level segment follows an 

“accelerate and hold” behavior. This means that while executing the current segment, the 

aircraft will accelerate or decelerate from the velocity specified for previous segment i-1 using a 

known constant acceleration to the velocity specified for the current segment i. This is depicted 

in Figure 49 – Assumed velocity profile for level segments. In addition, it is assumed that the 

algorithms used to generate the flight plan provide adequate segment length so that the required 

acceleration/deceleration can be accommodated in the current segment. 

Figure 49 – Assumed velocity profile for level segments 

However, in the case of descending flight, a constant Mach number will result in a continuously 

increasing true airspeed and a constant indicated airspeed will result in a continuously decreasing 

true airspeed. In addition, it is normal for experience significant wind variation as a function of 

altitude. Therefore, during descents it is typical to experience an almost linearly varying true 

airspeed and ground speed as the aircraft proceeds to the current destination waypoint. 

This is a very different velocity profile from the one described for level segments. For 

simplification it is necessary to assume that the resulting inertial velocity is linearly changing 

from one waypoint (position and altitude) to the next. This velocity profile is depicted in Figure 

50. 

t0 t td 1

Vi

Vi-1
109 

Figure 50 – Assumed velocity profile for descent segments 

Based on historical and empirical data as well as aeronautical standards for passenger comfort 

the following acceleration/deceleration levels during maneuvering are anticipated: 

Comfortable level: <0.315 [m/s2

] or <0.032 [g] 

Slightly uncomfortable: 0.315 – 0.63 [m/s2

] or 0.032 – 0.064 [g] 

Maintaining a constant acceleration both in the vertical (z) and horizontal (x, y) axes during 

flight operation is generally considered necessary for passenger comfort. For the purpose of this 

study, a constant acceleration for each speed change maneuver is assumed, and it is assumed that 

this acceleration is a known or pre-defined value. The general case would associate a constant 

acceleration with a given flight plan segment. An additional simplifying assumption is made for 

this version of study is to fix the acceleration to a given value. So, all speed changes are 

performed using a constant known acceleration referred to as ref a . Possible variations in 

acceleration are considered as uncertainty in the model. 

The along-track acceleration and inertial velocity change occur as the aircraft changes heading 

(into and away from the wind) or descends or climbs into different wind. The guidance 

algorithms attempt to limit the commanded linear acceleration at the aref level. However, no 

attempt is made to limit the combined acceleration from the wind and the airspeed change. In 

the general case, a net acceleration slightly larger than the magnitude of the reference 

acceleration may be experienced. For the purpose of this analysis tool, the differences between 

commanded and achieved acceleration terms are considered as modeling uncertainty. 

For the segment i of the trajectory, the velocity profile can be written as: 

      

t

t i i v t V a t dt0 1 (112) 

      

1

0 1

t

t i i V V a t dt (113) 

Vi

Vi-1

t0 t1
110 

Vi1 is the previous desired inertial speed. The time t0 is the initial time for the integration. It is 

either the start time for the segment, or as will be discussed, it could be the current time of the 

calculations. The time t1 is the arrival-time at TCP number i. 

Based on the previously discussed constant acceleration profile, 

  



  seg a

a t

0

(114) 

The term aseg is equal to the reference acceleration during level segments and is set equal to the 

required speed change during the segment for descending segments as described earlier. A 

default maximum value for this acceleration will be ±0.05 [g]. With the above acceleration 

profile, the required time to accelerate or decelerate to the new commanded speed in the level 

segment can be written as: 

    0 1

1   i  i

ref

d V V

a

t t (115) 

In order to achieve the required or specified velocity Vi while still on the current segment of the 

trajectory, the following condition must hold: 

   1 0 0 t t t t   d  (116) 

It is assumed that the planned trajectory being executed by this simulation correctly 

accommodates for time and distance to decelerate to the commanded airspeed. For the 

descending segments, the required acceleration is computed using the following relation. 

 

  1 0

1

t t

V V

a i i

seg 

   (117) 

Note that the inertial velocity at each waypoint can be derived by subtracting the wind 

components from the planned true airspeed values. However, since it is assumed that the wind 

velocity varies linearly between the waypoints, the assumption of equation 36 will not apply if a 

large change in wind velocity or direction is introduced in a short trajectory segment. In 

addition, we assume that an average wind value for the entire segment provides sufficient fidelity 

to use in these simplified equations. 
111 

9.4.2 Distance and Time-to-Go Calculations 

The segment length or distance to travel given the above velocity profile is written as: 

     1

0

t

t i i R v t dt (118) 

Substituting the above velocity profile, the following closed-form solution can be derived 

   2

1 0 1 2

1       i i

seg

i i V V

a

R V t t (119) 

Define a trajectory segment time as the difference between the initial and final on the segment. 

All “time” parameters are in seconds. 

  1 0 T t t i   (120) 

The time of travel for each segment can therefore be written as: 

  













    

2

1 2

1 1

i i

seg

i

i

i V V

a

R

V

T (121) 

Note that the velocity terms here are inertial velocities in the horizontal plane (groundspeed). 

The remaining time on segment k is written as: 

  













    2

2

1 1

k m

seg

current

k

current V v

a

R

V

T (122) 

The total time to go (TTG) to the spacing point is calculated as: 

 

 

n

i k

TTG Tcurrent Ti

1

 (123) 

The spacing time is computed as: 

Spacing  TTGownship TTGrefA/C (124) 
112 

9.4.3 TTG Algorithms 

9.4.3.1 Current Segment Calculations 

The current segment index is denoted as k. The current inertial velocity in the along-track 

direction is AT v and is provided by the navigation module. The current velocity command is 

Vcmd , which is derived from the planned true airspeed and average forecast winds along the 

current segment. It is also resolved in the along-track direction. 

The navigation module computes the remaining distance on the current segment. To account for 

the turns at the start and end of each segment, a corrected along-track distance to go parameter is 

computed. This corrected value is used here. 

Rcurrent ATD Gcorr  2 (125) 

The time of travel remaining on the current segment is computed using the following equation. 

  













    2

2

1 1

k AT

seg

current

k

current V v

a

R

V

T (126) 

9.4.3.2 Subsequent Segment Calculations 

The next segment index is denoted as k+1. The velocity and time of travel calculations for the 

next segment is comparable to the current segment. The remaining distance on the segment is 

equal to the total length of the segment. Again, the corrected segment length is used for this 

calculation. Vk1 is the flight plan groundspeed based on specified true airspeed and forecast 

wind components. 

RK1  Dseg _ corr (127) 

  













    





2

_ 1

1

1 2

1 1

k k

seg

seg corr

k

k V V

a

D

V

T (128) 

9.5 RIMP Speed Control 

Although the design of a RIMP control system is not the intent of this effort, at least a simplified 

functioning control system is required to manipulate the RIMP environment and assess the 

sensitivity of various components to uncertainties, errors, and perturbations. A simple RIMP 

control described below consists of a minimum speed change calculation used as a scaling 

parameter followed by calculation of the required speed change and associated logic. 
113 

9.5.1 Minimum Required Speed Change 

The minimum required speed change is a calculated speed change based on the current estimate 

of the spacing error and current time-to-go of the reference aircraft. It defines a speed change 

which if started instantly and applied up to the interval management point, will adjust the spacing 

interval by the time the reference aircraft arrives at the interval management point. This provides 

for the longest duration allowed for the speed change to adjust the interval time. As expected, 

when the aircraft is far from the spacing point, the calculated minimum speed change will be 

very small. As the aircraft approaches the reference point, the minimum speed change value will 

increase. Therefore, it can be seen that the minimum speed change is akin to a scaling parameter 

for the RIMP operation. Instead of using the distance to the reference point, the simplified 

equations use the TTG of the reference aircraft for this measure. 

The along-track distance to go must be flown at either a nominal speed or at the nominal speed 

plus a speed increment. For this calculation it is assumed that the current aircraft ground speed 

is the nominal value to fly the rest of the plan. 

ATD G V TTG   V V TTG TTG 2  measured  ownnominal  measured   min  ownnominal   (129) 

Assuming a general speed trend of faster speed to slower speed during RIMP operation, a 

simplified but conservative solution can be derived based on the current speed of ownship, timeto-go of the reference aircraft, and the spacing error. This solution is conservative since the 

current speed used in the above equation is equal or faster than the speed used in the planned 

trajectory. Therefore, the estimated TTG adjustment is equal or less than what is achievable in 

the course of the flight. 

The simplified equations do not work at the limit when the reference aircraft is very close to the 

RIMP reference point. However, assuming that   TTGrefA/C TIM >> 1 second, the following 

minimum required speed change can be computed. 

  

  refA C IM

measured IM own ref

TTG T

V T TTG TTG

V



     

/

min (130) 

The sign of the required speed change is negative for deceleration and positive for acceleration. 

This value is significant since it has the longest time to impact the RIMP operation. If no action 

is taken as the paired aircraft proceed on their flight plans, the magnitude of the minim required 

speed change will increase. Operationally, once a reasonable threshold value is reached, the 

minimum required speed change must become the speed change command, and should be used 

to re-negotiate the flight plan. 
114 

9.5.2 Nominal Speed Change 

In the simplified RIMP control model, the nominal speed change is computed as the speed 

change to resolve the current spacing error in 600 seconds. 

  

600

measured IM own ref

nom

V T TTG TTG

V       (131) 

The required speed change is the larger of the nominal and minimum speed change values. 

  min VIM  sign(Vnom ) *max Vnom , V (132) 

The resulting speed change is then passed through a speed change logic (described below) to 

quantize the speed change events. The result is then multiplied by the speed change factor to 

generate the commanded change in speed, and in this particular case, change in true airspeed of 

the aircraft. This commanded speed change does not take into account if the aircraft is operating 

at the specified airspeed limit. The guidance module limits the sum of planned and RIMP 

commanded airspeed. 

9.5.3 Speed Change Logic 

To ensure that the RIMP speed command is engaged in a pseudo-discrete sense and to minimize 

the nuisance occurrence of speed change events the following simplified logic has been 

implemented. A RIMP speed logic state is defined in the memory. It is currently setup to 

include two speed-up steps, three slow-down steps and a null step. Figure 51 provides a logic 

flow diagram for the speed change algorithm. 

Figure 51 – Speed Change Logic 

null 

-1 

-2 

-3 

+1 

+2 

V<-1.5 

V>+1.5 

V>+12.5 

V>+5.0 

V<-5.0 

V<-12.5 

V<-22.5 
115 

When the logic state is in the null mode, the speed change command is set to zero. If the “IM 

speed change” described above becomes greater than 5.0, the logic mode is switched to “+1 

Speed-Up”. If the speed change drop below -5.0, the logic mode is switched to “-1 SlowDown”. 

When the logic state is in “+1 Speed-Up” mode if the IM speed change drops below -1.5, the 

mode is switched to null. If the IM speed change increases past +12.5, the mode is switched to 

+2 Speed-Up. 

When the logic state is in +2 Speed-Up mode if the IM speed change drops below -1.5, the mode 

is switched to null. 

When the logic state is in -1 Slow-Down mode if the IM speed change drops below -12.5, the 

mode is switched to “-2 Slow-Down”. If the IM speed change increases past +1.5, the mode is 

switched to null. 

When the logic state is in “-2 Slow-Down” mode if the IM speed change drops below -22.5, the 

mode is switched to “-3 Slow-Down”. If the IM speed change increases past +1.5, the mode is 

switched to null. 

When the logic state is in “-3 Slow-Down” mode if the IM speed change increases past +1.5, the 

mode is switched to null. 

The final RIMP speed command is set equal to the above mode number multiplied by the IM 

spacing multiplier specified in the “Constants” data file. Nominally, this multiplier is set to 

equal to 10 Knots. RIMP speed control is disabled when commanded spacing is set to zero, and 

upon reaching the reference waypoint. 

9.6 Model Setup and Execution Procedures 

This section describes the steps in preparing the input and configuration data files, and details the 

execution of the RIMP simulation. The RIMP simulation model is coded in MATLAB/Simulink 

environment and is configured for specific analysis tasks. The description provided below is for 

a multi-aircraft scenario where continuous-descent arrivals are performed into Louisville, 

Kentucky. 

9.6.1 Model Setup 

9.6.1.1 Flight Plan Specification 

The “Flight Plan” is used to define the aircraft operation during the simulation. Multiple aircraft 

may use the same flight plan. This file is read by a MATLAB script and all text to the right of % 

sign is comment. 
116 

% Louisville BGEST CDA1

%

% Flight Plan Data File Version 3.0

% Waypoint ID number

% Waypoint type (1=enroute (default), 2=turn back, 3=land)

% Waypoint Latitude (degrees decimal)

% Waypoint Longitude (degrees decimal)

% Altitude at waypoint (feet, MSL)

% IAS/Mach at waypoint (Knots/Mach)

% Waypoint name

%

%---------------------------------------------------------------

% #, Type, Latitude, Longitude, Altitude, IAS or M, Name

%---------------------------------------------------------------

FP = [ 

 0 1 38.53117000 -91.63363900 37000 0.8 cellstr('start') 

 1 1 38.41999860 -89.15899780 37000 0.8 cellstr(' ENL') 

 2 1 38.38000000 -88.40000000 37000 0.8 cellstr('xxTOD') 

 3 1 38.37723889 -88.20648056 36000 0.8 cellstr('ZARDA') 

 4 1 38.32920278 -87.61643611 28000 0.8 cellstr('PRINC') 

 5 1 38.12002778 -86.40452778 11000 240 cellstr('BGEST') 

 6 1 38.01983055 -85.85327778 4600 190 cellstr('PTINO') 

 7 1 37.98333333 -85.75000000 4000 190 cellstr('BASKT') 

 8 1 38.02111111 -85.69444444 3000 180 cellstr('DNKIT') 

 9 1 38.06438889 -85.70835000 2400 170 cellstr('CRDNL') 

10 3 38.16081600 -85.74017300 513 140 cellstr('RW35L')]; 

9.6.1.2 Atmospheric Data 

The “Wind-Temperature-Pressure” file is pre-processed from the desired RUC data for the 

specific flight plan to be flown. It includes the East and North winds, outside air temperature, 

and static pressure at each waypoint location and altitude. 

% 

% Wind pressure and temperature data 

% First column is waypoint number 

% Second column is East wind in meters per second 

% Third column is North wind in meters per second 

% Fourth column is static pressure in millibars 

% Fifth column is OAT in degrees Kelvin 

% 

 0.0 -6.3341029e+001 -3.2106923e+001 2.1730000e+002 2.1923351e+002 

 1.0 -6.8598146e+001 -3.3449491e+001 2.1730000e+002 2.1831580e+002 

 2.0 -7.0793884e+001 -3.1034921e+001 2.1730000e+002 2.1809949e+002 

 3.0 -7.1651931e+001 -3.2510421e+001 2.2800000e+002 2.2015837e+002 

 4.0 -5.7738696e+001 -3.2149985e+001 3.2990000e+002 2.3670349e+002 

 5.0 -1.9763707e+001 -8.0108517e+000 6.6704391e+002 2.6876573e+002 

 6.0 -5.4049351e+000 -8.5042564e+000 8.5055624e+002 2.7791416e+002 

 7.0 -3.8262500e+000 -6.1950000e+000 8.6968250e+002 2.7832625e+002 

 8.0 -3.0130864e-001 -3.1112469e+000 9.0268727e+002 2.7755280e+002 

 9.0 3.7290236e+000 -2.1433883e+000 9.2352290e+002 2.7520866e+002 

 1.0 4.1435416e+000 8.3606477e-001 9.8653519e+002 2.7357225e+002 

% 
117 

9.6.1.3 Aircraft Performance Models 

The “Aircraft Limits” file is used to specify maximum and minimum airspeeds (indicated or 

equivalent airspeeds) in Knots as a function of altitude in Flight Level. These limits are used by 

the aircraft guidance and control algorithms to limit the commanded speeds when necessary. An 

additional “nominal” equivalent airspeed specified is used as a limiting speed during constant 

Mach descents. So, during such operation, when the equivalent airspeed reaches this nominal 

value, the speed is limited to maintain it. In this case, additional speed increase may still be 

commanded by the RIMP controller. 

% AircraftLimits_767.m 

% Airspeed limits for 767-200 

% 

% Altitude [ft], VMin [kts], VMax [kts] 

FltLvl = [ 0, 50, 100, 120, 180, 240, 300, 340, 400]; 

MaxIAS = [ 200, 240, 250, 310, 330, 330, 330, 330, 330]; 

NomIAS = [ 150, 240, 250, 290, 310, 310, 310, 310, 310]; 

MinIAS = [ 110, 160, 170, 180, 190, 210, 210, 210, 210]; 

% 

9.6.1.4 Aircraft Setup and Initialization Data 

The following “Setup File” is used to specify the initial conditions, flight plan, aircraft limits, 

and wind information for each aircraft within the RIMP simulation. Each aircraft must have a 

unique copy of the Setup file. The numbers specified below are for example only and are 

adjusted for each specific application. This file is read by a MATLAB script and all text to the 

right of % sign is comment. 
118 

% AC_Setup_5.m 

% Aircraft number 5. 

% 

FPfilename = 'FlightPlan_BGEST_CDA1'; 

WindTempFi = 'BGEST_CDA1_001.txt'; 

ALfilename = 'AircraftLimits_767'; 

Temp1 = [ 

 0. % Initial true airspeed 

error (TAS) [kts] 

 0. % Initial altitude error 

[Ft] 

 0. % Initial east position 

error [Nautical miles] 

 0. % Initial north position 

error [Nautical miles] 

 200. % Start time offset [Seconds] 

 5 % Aircraft number (ATC 

Sequence) 

 4 % Assigned reference 

aircraft (ATC Sequence) 

 10 % Assigned reference waypoint number 

 50. ]; % Nominal spacing time [Seconds] 

% 

9.6.1.5 Sensor Model Setup 

The following sensor model parameters are specified in the Sensor Parameter data file. The 

numbers used in this example case are representative values. They do not represent a particular 

system implementation and are not aircraft specific. 

In the current simulation setup all aircraft share the same sensor parameters. Future development 

of this model may change this to have sensor variations between different aircrafts. 

% Sensor Number 

% Scale Factor (SF) 

% Bias 

% Noise Signal Mean (Mean) 

% Noise Signal Variance (Var) 

% Filter Denominator coeff (change based on Tau and dt) 

% Lower Limit (LLim) 

% Upper Limit (ULim) 

% Quantization Interval (Quant) 

% Time Delay (tDel) [s] 

% 

% SF | Bias | Mean | Var | Tau | LLim | ULim | Quant | tDel 

1, 0.0, 0.0, 0.0, 0.05, 0.990, 0.0, 400.0, 0.1, 0.10 % Inertial Speed [m/s] 

2, 0.0, 0.0, 0.0, 0.0001, 0.990, -15.0, 15.0, 0.001, 0.01 % Ground track [rad] 

3, 0.0, 0.0, 0.0, 0.0001, 0.990, -2.0, 2.0, 0.001, 0.01 % Gamma [rad] 

4, 0.0, 0.0, 0.0, 0.05, 0.990, -250.0, 250.0, 0.01, 0.10 % Wind East [m/s] 

5, 0.0, 0.0, 0.0, 0.05, 0.990, -250.0, 250.0, 0.01, 0.10 % Wind North [m/s] 

6, 0.0, 0.0, 0.0, 0.05, 0.990, 160.0, 328.0, 0.1, 0.10 % OAT [degK] 

7, 0.0, 0.0, 0.0, 0.1, 0.990, 50.0, 1090.0, 0.1, 0.01 % P static [mBar] 

8, 0.0, 0.0, 0.0, 0.1, 0.990, -800000.0, 800000.0, 0.1, 0.10 % East Position [m] 

9, 0.0, 0.0, 0.0, 0.1, 0.990, -800000.0, 800000.0, 0.1, 0.10 % North Position [m] 

10,0.0, 0.0, 0.0, 0.1, 0.990, -1000.0, 20000.0, 0.1, 0.10 % Altitude [m] 

% 
119 

9.6.1.6 Model Constants 

The following information is specified in the Constants data file. The numbers specified below 

are for example only and are adjusted for each specific application as required. This file is read 

by a MATLAB script and all text to the right of % sign is comment. 

 0.05 % 1- Reference linear acceleration [g] 

 1.5 % 2- Reference rate of turn [deg/sec] 

 0.5 % 3- Reference rate of vertical flight path [deg/sec] 

100.0 % 4- Waypoint capture tolerance (meters) 

 1.0 % 5- Course track capture tolerance (deg) 

10.0 % 6- Reference RIMP Delta Velocity (Delta V Reference) [kts] 

 0.0 % 7- Reserved 

 0.0 % 8- Reserved 

 0.01 % 9- Flight path correction constant (deg/m) 

 0.0 % 10- Reserved 

 0.0 % 11- Reserved 

 0.0 % 12- Reserved 

-1 % 13- Random Sequence Initial Seed (-1 = CLOCK) 

 0.0 % 14- Wind Error Mean [m/s] 

 5.0 % 15- Wind Error Standard Deviation (m/s) ------------ 

 5.0 % 16- Start time offset standard deviation (sec) ------ 

 0.0 % 17- Reserved 

 38.16081600 % 18- Zero Reference Latitude, degrees 

-85.74017300 % 19- Zero Reference Longitude, degrees 

 0.0 % 20- Zero Reference Altitude, Meters 

%--------------------------------- 

9.6.2 Individual Simulation Runs 

To perform an individual RIMP simulation run, the associated setup files, flight plan files, wind 

and temperature files, and other data files described above must be configured and appropriately 

referenced in the setup file or placed in the working directory of the MATLAB session. 

9.6.2.1 Executing the Model 

To execute a single simulation run from the MATLAB command line the following MATLAB 

commands are used. The first command initializes the simulation parameters. The second 

command executes the Simulink model. 

>> run RIMP_Configure_6

>> sim('RIMPSimulator_v310_6') 

9.6.2.2 Output Parameters and Plotting 

The following parameters are saved during each simulation run. These values are store once 

every second. Parameters are stored for all the aircraft in the simulation. 
120 

%--------------------------------------------------------------- 

% "AC_Data" vector: 

% 

% 1- Destination waypoint number 

% 2- Flight Status (0 = in progress, 1= arrived, 2= frozen) 

% 3- 

% 4- Along-Track Distance to Go "ATDG" (Nm) 

% 5- Along-Track inertial speed (Kts) 

% 6- Required distance before WP to start turn (m) 

% 7- Desired track (rad) 

% 8- Cross track error (m) 

% 9- Desired vert. flight path (rad) 

% 10- Glideslope (vert.) error (m) 

% 11- Desired turn rate (deg/sec) 

% 12- Current speed command (M or KIAS) 

% 

% 13- Ground Speed (Kts) 

% 14- Vertical Speed (Kts) 

% 15- Ground track (deg) (rectified 0-360) 

% 16- Vertical flight path angle (deg) 

% 17- Wind Speed (Kts) 

% 18- Wind Direction (deg) 

% 19– East Position (nautical miles) 

% 20- North Position (nautical miles) 

% 21- Altitude (ft) 

% 22- True Airspeed (Kts) 

% 23- Indicated Airspeed (Kts) 

% 24- Mach number 

% 25- OAT (degK) 

% 26- Static Pressure (mBar) 

% 27- Air Density (Kg/m3) 

% 28- Speed of sound (m/s) 

% 

% 29- RIMP speed (Knots) 

%--------------------------------- 

In addition, the time-vector “ADS_Data” holds the “Time to Go” information for each aircraft as 

a function of simulation time. Other simulation and flight plan parameters are also available for 

examination and plotting. 

To extract the pre-defined statistics related to IM frequency and achieved spacing parameters, the 

following MATLAB scripts can be executed. 

>> run RIMP_statistics

There are several scripts for plotting the flight history and various performance parameters. The 

following commands can be used to plot lead aircraft only, an aircraft pair performing IM, and 

three aircraft (2 pairs) performing IM respectively. 

>> run RIMP_plotData_base

>> run RIMP_plotData2

>> run RIMP_plotData
121 

9.6.3 Monte Carlo Simulation Runs 

The RIMP simulation model is designed to address the random components of the simulation 

using the MATLAB pseudo-random number stream with a specified seed. If the random seed is 

set to “-1” the seed is initialized using the computer time. This setting allows for repeated 

simulation runs with random components as would be needed in a Monte Carlo analysis. 

9.6.3.1 Model Setup and Execution 

A MATLAB script file is used to execute the RIMP simulation model repeatedly and to extract 

the required statistics from each run. Prior to executing this script, the analyst must examine the 

script and all associated simulation setup files to ensure that proper parameters are specified. 

The desired number of simulation runs is specified at the top of the Monte Carlo script. This will 

executes the entire Monte Carlo simulation analysis. 

>> run RIMP_MonteCarlo

9.6.3.2 Model Statistics Collected 

The following statistics are collected for each simulation run: 

 IM frequency per flight plan segment is the number of IM speed changes (increase or 

decrease) during the segment divided by the duration of the segment. 

 IM frequency below flight level 100 is the number of IM speed changes (increase or 

decrease) during that part of flight below 10,000 feet divided by the duration of the flight 

below 10,000 feet. 

 IM frequency total flight is the number of IM speed changes (increase or decrease) during 

that entire flight operation from the “merge” point to the reference point divided by the 

duration of this operation. 

 IM achieved spacing is the difference in time of arrival of the reference aircraft and 

ownship at the IM reference waypoint. 

 IM achieved spacing error is the difference between the commanded spacing and actual 

or achieved spacing at the reference waypoint. 

9.6.3.3 Model Output and Plotting 

At the completion of a Monte Carlo simulation run, the histogram of the statistics and associated 

mean, standard deviation, maximum, and minimum values can be plotted and displayed by 

executing the following MATLAB script file. 

>> run RIMP_plotData_stats

In addition, the time history data from the last simulation run will also be available to plot as an 

example of simulation parameters.
122 

10 Required Self Separation Performance Study 

The Next Generation Air Transportation System (NextGen) is envisioned as a revolutionary 

transformation of the U.S. airspace to a performance-based, scalable, network-enabled system 

that will be flexible enough to meet future air traffic needs. One of the major transformations is 

the use of Trajectory-Based Operations (TBO) as the main mechanism for managing traffic at 

high density or in highly-complex airspace. These TBOs will be specified between the user and 

the Air Navigation Service Provider (ANSP) and agreed in a “contract”, using advanced 

automation. Overall, preferences for all users are accommodated to the greatest extent possible, 

and trajectories constrained only to the extent required to accommodate congestion, or for 

security, safety, or environmental reasons. Changes to that “contract” will be made 

collaboratively, balancing the user preferences with the ANSP constraints. 

A major element of TBO is trajectory-based Separation Management (SM), which uses 

automation and shared trajectories to better manage separation among aircraft, airspace, and 

hazards such as weather and terrain. TBO provides a means for maintaining a Target Level of 

Safety (TLS) while increasing traffic densities well beyond what is possible today given the 

workload, uncertainty, and execution delays inherent in current ground-based air traffic 

management. Concepts for moving the SM function into the cockpit have been explored [6], and 

the need to characterize the performance of such functions in a manner that supports quantitative 

analysis of the overall safety of the ATM system has been identified [7]. An analogous construct 

for the characterization of system performance has already been defined and adopted by the 

aviation community for lateral navigation. This construct has been termed Required Navigation 

Performance (RNP) [8]. 

The document, Required Self Separation Performance, Version 1.1 [9], defined an RNP-like 

construct for defining the performance of systems capable of performing airborne self-separation 

operations. A key metric for this construct was overall separation prediction accuracy and the 

assumption that this prediction accuracy could be determined based on parameters (state and 

system accuracy) known at the time of prediction. 

10.1 Objectives and Motivation 

This study explored the relationship between various lateral separation geometry parameters 

(speeds, crossing angle, and time-to-go) and system accuracy metrics (navigation accuracy and 

surveillance accuracy) and the statistical distribution of expected actual separation distances. 

With an eye to demonstrating separation containment, this study determined the minimum 

separation distance that should be targeted in order to ensure with 95% confidence that an 

encounter will result in an actual separation of at least a certain target distance. 

10.2 Background 

Reference [9] examined what would be required in order to ensure a TLS for a scenario, such as 

TBO, in which properly equipped aircraft have primary responsibility for avoiding other aircraft. 

This concept contrasts with the current system where that responsibility lies with the ANSP. It 

described the system components, identified key system parameters, and derived a RNP-like 

construct for self-separation operations. An accompanying MATLAB model, the 
123 

RSSPSimulator, expanded upon the Total System Error (TSE) analysis included as Appendix B 

in that document to provide a means of assessing the impact that various error sources and 

uncertainties have on an aircraft’s ability to accurately predict (and hence control) its minimum 

separation from a reference aircraft. The model included computation of the encounter 

geometry, linearized analysis of the impact of error sources on the predicted separation, and a 

Monte Carlo based analysis of the impact of error sources on the distribution of separation as a 

function of encounter geometry in the context of key error terms for Class A and Class B 

Required Self-Separation Performance (RSSP) systems in the horizontal plane. The two classes 

of RSSP capable systems are examined independently because of their different characteristics 

and input error terms. As defined in [9], RSSP Class A systems generate and use intent 

information whose spatial error characteristics are numerically bounded. RSSP Class B systems, 

on the other hand, need only produce and make use of present position and velocity inputs for the 

computation of conflicts and resolutions. 

10.3 Simulation Platform and Scenarios Description 

10.3.1 Basic Separation Distribution Model 

The simulation platform used for this study is based on the distribution analysis core of the 

RSSPSimulator V2.1.1 [10]. The distribution analysis was performed using a Monte Carlo 

approach. The inputs used to establish the nominal geometry and state before the introduction of 

uncertainties for this analysis are the scenario geometry parameters (ownship groundspeed (SO), 

reference aircraft groundspeed (SR), crossing angle (), nominal time to CPA (tCPA), target range 

at CPA (RCPA), and crossing order (ownship passes ahead or behind reference) as represented in 

Figure 52. These inputs are used to compute the additional dependent geometry parameters 

which define the initial relative position of the two aircraft (x0 and y0). 

Figure 52 - Original Estimated Geometry 

Once the nominal geometry and state have been determined, perturbations are applied to the 

positions (Class A systems) or the positions and velocities (Class B systems) according to the 

SR 

SO 

Ownship

Reference



0 y

0 x
124 

input uncertainties. The equation for RCPA, which was used to determine x0 and y0, is then solved 

using the perturbed values to determine the sample outcome RCPA resulting from the perturbed 

geometry and state, with care taken to use a negative result if the perturbation results in a 

reversal of the crossing order. Sufficient numbers of perturbed condition samples are examined 

to enable generation of a resulting distribution of statistics for the resulting actual RCPA for each 

perturbed condition. 

The distribution function inputs used to perturb the nominal geometry and state will depend on 

whether a Class A pair of systems or a Class B pair of systems is being evaluated. Classification 

as a Class A RSSP system, implies that the aircraft are both operating according to four 

dimensional (4D) flight plans and are broadcasting TCPs with known lateral and longitudinal 

accuracy characteristics. The input parameters will therefore be taken to represent the standard 

deviation for lateral and longitudinal uncertainties (a Gaussian distribution is assumed). These 

uncertainties are meant to reflect a time approximately tCPA in the future. Classification as a 

Class B RSSP system, on the other hand, implies that the aircraft only rely on current position 

and velocity measurements, and that the accuracy characteristics of these measurements are 

Gaussian and non-directional (errors are uniformly distributed in azimuth). The formulae used to 

compute perturbed parameters based on the system class and uncertainties may be found in 10. 

The distribution statistics computed for each set of uncertainties and nominal geometry include 

the resulting 5% confidence boundary (RCPA(5%)), mean ( RCPA  ) and standard deviation ( RCPA  ) 

of the “actual” separation at the point of closest approach. Once the Monte Carlo results have 

been organized into bins to form a histogram, the values for RCPA  and RCPA  are calculated as 

follows: 



 B

CPA

N

i

CPA

S

R Samples i R i

N 1

( ) ( ) 1 (133) 

   

  B

CPA CPA

N

i

CPA R

S

R Samples i R i

N 1

2 ( ) ( ) 1

  (134) 

Where: 

NB Number of Bins 

NS Total Number of Samples 

RCPA(i) RCPA of Bin i

Samples(i) Number of Samples in Bin i

Finally, the probability distribution function is “integrated” to find the value of RCPA that 

contains 5% of the samples (RCPA(5%)). RCPA(5%) is the separation distance for which there is a 

probability of 5% that the actual RCPA could be less than RCPA(5%) given the specified geometry 

and errors. That is, RCPA(5%) represents the larger separation distance that must be targeted to 

ensure that the smaller minimum separation distance will be achieved at least 95% of the time. 
125 

10.3.2 Computation of Required Target RCPA

A MATLAB function, RSSPExperiment, was written around the above core to iteratively find 

the required target RCPA that yields an RCPA(5%) within 0.005 nmi of a specified minimum 

separation over a set of speed and crossing angles. This MATLAB function takes scenario 

parameters of system class and associated uncertainty parameters, tCPA, and minimum separation 

as input. It then sets the target RCPA and uses the core separation distribution model to determine 

RCPA (5%) for each combination of ownship groundspeed (SO), reference aircraft groundspeed 

(SR), and crossing angle (). If this is not within tolerance of the specified minimum separation, 

the target RCPA is adjusted and the process is repeated until the solution converges. Then, the 

relevant geometry sweep parameters and output variables (RCPA(5%), RCPA, RCPA  , and RCPA  ) are 

stored in the results file. 

10.3.3 Outer Framework 

An additional MATLAB script, RSSPExperimentLauncher, was used to invoke 

RSSPExperiment for each of the test conditions. This script permits the unattended execution of 

multiple study scenarios. 

10.4 Study Design 

10.4.1 Plan and Approach 

In order to explore the relationship between various lateral separation geometry parameters and 

system accuracy metrics and the lateral separation target required to ensure (with 95% 

confidence) that the actual separation will be above a required minimum value (required target 

separation)29, the RSSPSimulator was used to perform Monte Carlo studies over a range of 

encounter geometry and system accuracy parameters while adjusting the target separation for 

each parametric run to ensure that only 5% of cases fall below a specified minimum separation 

value. The analysis was conducted for both Class A (intent based) and Class B (state only) 

RSSP types of systems. 

This study was designed to verify the following expected behavior: 

1. For a given minimum separation value, the required target separation will decrease as 

navigation and surveillance accuracy increases. 

2. The required target separation will be a function of encounter geometry in addition to 

navigation and surveillance accuracy. 

3. The average range at closest approach will always be less than the targeted range at 

closest approach by an amount that is a function of both geometry and accuracy. 

4. The standard deviation of actual range at closest approach can be predicted using the 

sensitivity analysis to scale the navigation and surveillance accuracy as a function of 

encounter geometry. 

 

29 The required target separation is equivalent to the minimum separation plus the additional buffer required to 

account for the uncertainties. 
126 

10.4.2 Metrics 

The primary metric for this study were the target lateral separation required to ensure that the 

actual separation will be greater than a specified minimum with 95% confidence. In order to 

facilitate comparisons across different separation minima, this will typically be reported as a 

buffer rather than the full target separation distance. Additional metrics include the mean and 

standard deviation of actual separation values (i.e., RCPA) achieved for a given geometry and 

across geometries for a given system type and set of uncertainties. These metrics can be 

summarized as: 

Required Buffer (Buffer) The target RCPA (needed to attain the 95% confidence) minus the 

specified minimum separation 

Average RCPA Offset 

(  RCPA  RCPA ) 

The average actual RCPA for perturbations of each input geometry 

within a scenario minus the nominal RCPA for the scenario. 

RCPA Standard Deviation 

( RCPA  ) 

The standard deviation of the actual RCPA for perturbations of each 

input geometry within a scenario. 

10.4.3 Test Plan 

For each scenario of specified minimum lateral separation, RSSP class, and system accuracy 

indicated in the tables below, the necessary target separation was determined for geometry 

parameters generated according to Table 1. 

Table 1: Scenario geometry parameters 

Parameter Min Max Step Units 

Ownship Ground Speed 200 600 50 kts 

Reference Aircraft Ground Speed 200 600 50 kts 

Crossing Angle 10 170 10 degrees

For Class A systems, the analysis was conducted for the following input parameter scenarios 

indicated in Table 2. 
127 

Table 2: Class A system condition parameters 

Scenario Ownship 

cross-track 

(nmi)

Reference 

cross-track 

(nmi)

Ownship 

along-track 

(nmi)

Reference 

along-track 

(nmi)

Minimum 

Separation 

(nmi)

A-1 0.1531 

(RNP-0.3) 

0.1531 

(RNP-0.3) 

0.1531 

(RNP-0.3) 

0.1531 

(RNP-0.3) 1.0 

A-2 0.1531 

(RNP-0.3) 

0.1531 

(RNP-0.3) 

0.5102 

(RNP-1) 

0.5102 

(RNP-1) 1.0 

A-3 0.1531 

(RNP-0.3) 

0.1531 

(RNP-0.3) 

0.5102 

(RNP-1) 

0.5102 

(RNP-1) 5.0 

A-4 0.5102 

(RNP-1) 

0.5102 

(RNP-1) 

0.5102 

(RNP-1) 

0.5102 

(RNP-1) 5.0 

For Class B systems, the analysis was conducted for the following input parameter scenarios 

indicated in Table 3. The values used for input uncertainties were selected to be consistent with 

the Navigation Accuracy Category for position (NACp) and Velocity (NACv) limits specified in 

11. 

Table 3: Class B system condition parameters 

Scenario Ownship 

Position 

 (nmi)

Reference 

Position 

 (nmi)

Ownship 

Velocity 

 (kts) 

Reference 

Velocity 

 (kts) 

Time to 

CPA 

(min) 

Minimum 

Separation 

(nmi) 

B-1 0.51 

(NACp-4) 

0.51 

 (NACp-4) 

0.99 

(NACv-3) 

0.99 

(NACv-3) 2 5.0 

B-2 0.51 

(NACp-4) 

0.51 

 (NACp-4) 

0.99 

(NACv-3) 

0.99 

(NACv-3) 10 5.0 

B-3 0.51 

(NACp-4) 

0.51 

 (NACp-4) 

0.99 

(NACv-3) 

0.99 

(NACv-3) 20 5.0 

B-4 2.04 

(NACp-2) 

2.04 

 (NACp-2) 

9.9 

(NACv-1) 

9.9 

(NACv-1) 2 5.0 

B-5 2.04 

(NACp-2) 

2.04 

(NACp-2) 

9.9 

(NACv-1) 

9.9 

(NACv-1) 20 5.0 

B-6 0.51 

(NACp-4) 

2.04 

(NACp-2) 

0.99 

(NACv-3) 

9.9 

(NACv-1) 20 5.0 

10.5 Study Results 

Because Class A RSSP systems and Class B RSSP systems have fundamentally different error 

sources and characteristics the results for each class are examined separately. 

10.5.1 Class A Systems 

Table 4 summarizes the statistics of the metrics for the Class A scenarios defined in Table 2 

computed across the geometry parameter sweeps specified in Table 1. 

None of the Class A system results showed any sensitivity to crossing order. 
128 

Table 4: Class A system statistical results 

Scenario Metric Mean StdDev Min Max Median

A-1 

Buffer 0.359 0.000 0.359 0.359 0.359 

 RCPA  RCPA -0.050 0.000 -0.051 -0.049 -0.050 

RCPA  0.218 0.000 0.218 0.219 0.218 

A-2 

Buffer 0.768 0.234 0.371 1.189 0.813 

 RCPA  RCPA -0.050 0.001 -0.052 -0.048 -0.050 

RCPA  0.466 0.142 0.226 0.721 0.494 

A-3 

Buffer 0.768 0.234 0.371 1.188 0.813 

 RCPA  RCPA -0.050 0.001 -0.052 -0.048 -0.050 

RCPA  0.466 0.142 0.226 0.722 0.494 

A-4 

Buffer 1.187 0.002 1.181 1.194 1.187 

 RCPA  RCPA -0.050 0.001 -0.054 -0.047 -0.050 

RCPA  0.722 0.001 0.720 0.725 0.722 

For conditions A-1 and A-4, the standard deviation of all three metrics was zero or near zero. 

This result indicates that the Buffer metrics were all insensitive (within the 0.005 nmi 

convergence threshold) to the swept scenario geometry parameters (Ownship Ground Speed, 

Reference Aircraft Ground Speed, and Crossing Angle). For both of these symmetric scenarios, 

the standard deviation of RCPA, RCPA  , is approximately the √2 times the standard deviation of 

each aircraft’s overall position uncertainty30. It is interesting to note that in both scenarios, the 

mean Buffer is almost exactly 1.645 RCPA  . This is consistent with the value used for a 90% 

confidence interval about the mean of a Gaussian distribution31, but does not seem to include the 

slight negative bias of the mean separation. 

The results for scenarios A-2 and A-3, the two scenarios with different lateral and longitudinal 

uncertainties, showed considerable sensitivity to encounter geometry. That their normalized 

statistics are effectively identical indicates a lack of correlation to minimum separation (the only 

control variable that differs between the two scenarios). 

In order to facilitate visualization of the output metrics as a function of the three geometry 

variables, a means was sought to combine the two input groundspeeds into a single 

normalization parameter to collapse the 4-dimensional space into 3 dimensions for easier 

representation in 2 dimensions. As a first attempt, the two speed dimensions, SO and SR, were 

represented in a single dimension using the difference in speeds (SO – SR), but this normalization 

 

30 Since the two components of position uncertainty for a Class A system are independent, orthogonal, and equal in 

magnitude, the standard deviation of the overall position uncertainty is the same as the single-axis standard 

deviation. When the position uncertainties of the two aircraft are combined, the relative position uncertainty is 

then √2 times the individual position uncertainty. 31 It is appropriate to use the 90% interval rather than the 95% interval here because the concern in this case is only 

with the half of the outliers in one tail of the distribution. 
129 

showed evidence of sensitivity to overall speed. When the two speeds were instead represented 

as the difference in speed divided by the average speed, no normalization artifacts were found 

across the entire speed profile. Therefore, all plots below will use normalized speed difference, 

(SO – SR) / [(SO + SR)/2], and crossing angle for the geometry axes. 

Figure 53, Figure 24, and Figure 55 show two perspectives of the 3D plot of the average Buffer 

metric versus geometry parameters for scenarios A-1, A-2, A-3, and A-4. Scenario A-4 is top 

plane. Scenario A-1 is the bottom plane. Scenarios A-2 and A-3 were indistinguishable in this 

representation. 

Figure 53 -Scenarios A-1—A-4: Required Buffer versus Crossing Angle and Normalized Speed Difference. 
130 

Figure 54 - Scenarios A-1—A-4: Required Buffer versus Crossing Angle, Normalized Speed Difference in the 

Z-Axis. 
131 

Figure 55 - Scenarios A-1—A-4: Required Buffer versus Normalized Speed Difference, Crossing Angle in the 

Z-Axis. 

These figures demonstrate the strong dependence of required buffer to encounter geometry 

(speeds and crossing angle). It is also interesting to note that all of the data points lie between 

the required buffers found in scenarios A-1 and A-4. This is as expected given that the 

uncertainties in scenarios A-2 and A-3 are a hybrid of the A-1 and A-4 uncertainties. These 

figures also show that the larger lateral uncertainty plays the largest role in geometries in which 

the speeds are similar and the crossing angle is small (tail-chase) and much less of a role in near 

head-on geometries regardless of relative speed. 

10.5.2 Class B Systems 

Table 5 summarizes the statistics of the metrics for the Class B scenarios defined in Table 3 

computed across the geometry parameter sweeps specified in Table 1 as was done for the Class 

A systems in the previous section. Again, none of the Class B system results showed any 

significant sensitivity to crossing order. 
132 

Table 5: Class B system statistical results 

Scenario Parameter Mean StdDev Min Max Median

B-1 

Buffer 0.849 0.002 0.844 0.856 0.849 

 RCPA  RCPA -0.050 0.001 -0.052 -0.048 -0.050 

RCPA  0.512 0.001 0.510 0.514 0.512 

B-2 

Buffer 0.888 0.002 0.882 0.894 0.887 

 RCPA  RCPA -0.050 0.001 -0.052 -0.048 -0.050 

RCPA  0.537 0.001 0.535 0.539 0.537 

B-3 

Buffer 1.002 0.002 0.996 1.009 1.002 

 RCPA  RCPA -0.050 0.001 -0.053 -0.048 -0.050 

RCPA  0.609 0.001 0.607 0.611 0.609 

B-4 

Buffer 3.435 0.025 3.411 4.085 3.432 

 RCPA  RCPA -0.055 0.017 -0.457 -0.043 -0.052 

RCPA  2.069 0.010 2.060 2.194 2.068 

B-5 

Buffer 6.418 0.056 6.374 7.720 6.409 

 RCPA  RCPA -0.057 0.025 -0.618 -0.035 -0.053 

RCPA  3.888 0.017 3.871 4.038 3.886 

B-6 

Buffer 4.626 0.024 4.587 5.147 4.623 

 RCPA  RCPA -0.053 0.010 -0.267 -0.040 -0.052 

RCPA  2.781 0.003 2.771 2.814 2.781 

These data show the expected increase in required buffer as uncertainties increase, either as the 

result of greater look-ahead times as seen comparing scenarios B-1, B-2, and B-3, or greater data 

uncertainty as seen by comparing scenarios B-1 and B-4 or B-3 and B-5. As was seen for 

symmetric Class A scenarios, the required buffer is approximately 1.65 times the measured RCPA  , 

which is consistent with a normal distribution. A comparison of the measured RCPA  to an 

approximation for each aircraft’s position uncertainty at the nominal time of CPA32 reveals that 

the RMS of the standard deviation of each aircraft’s position uncertainty is a good predictor of 

RCPA  as seen in Table 6. 

 

32 The standard deviation of the position uncertainty of each aircraft at TCPA is taken as the RMS of the standard 

deviation of the initial position uncertainty and the standard deviation of the velocity uncertainty times TCPA. 
133 

Table 6: Class B system analysis of uncertainties 

Scenario 

Ownship 

Position 

 (nmi) 

at TCPA

Ownship 

Position 

 (nmi) 

at TCPA

RMS of 

Position 

Uncertainties RMS * √2 RCPA 

B-1 0.361 0.361 0.361 0.511 0.512 

B-2 0.379 0.379 0.379 0.536 0.537 

B-3 0.430 0.430 0.430 0.607 0.609 

B-4 1.461 1.461 1.461 2.067 2.069 

B-5 2.743 2.743 2.743 3.880 3.888 

B-6 0.430 2.743 1.963 2.777 2.781 

A comparison of scenario B-6 (a hybrid of B-3 and B-5) to scenarios B-5 and B-3 shows that the 

required buffer is much more than found for B-3, but less than found for B-5. This implies that 

buffers may be reduced somewhat if ownship uncertainties are less than reference aircraft 

uncertainties, though the effects of the greater uncertainty will likely dominate. 

Given that all uncertainties for Class B systems are uniform with azimuth, the results from 

scenarios A-1 and A-4, which also had uncertainties that were uniform in azimuth and showed 

no variation with geometry, might suggest a lack of dependence on geometry for Class B 

systems, and that largely seems to be the case, especially for scenarios B-1, B-2, and B-3, which 

show near zero standard deviation and little difference between minima and maxima across the 

tested encounter geometries. While the standard deviation of the parameters for scenarios B-4, 

B-5, and B-6 are also relatively small, the data does show some significant differences between 

minima and maxima for each parameter. The location of these extremes can be seen in Figure 56 

and Figure 57 which depict two views of the results in like manner to Figure 24 and Figure 55. 

These figures show the flatness of the results for the first three scenarios, similar in form to 

surfaces for scenarios A-1 and A-4, but show significant peaks for the latter three scenarios at 

small crossing angles and small to slightly negative relative speeds (SR > SO). There are also 

slight peaks (which are difficult to see in the figures) at the 170 degree crossing angle. 
134 

Figure 56 - Scenarios B-1 – B-6: Required Buffer versus Crossing Angle, Normalized Speed Difference in the 

Z-Axis. 
135 

Figure 57 - Scenarios B-1 – B-6: Required Buffer versus Normalized Speed Difference, Crossing Angle in the 

Z-Axis. 

A probable source of these peaks is the larger magnitude of the velocity uncertainty in scenarios 

B-4, B-5, and B-6. At low speeds and shallow crossing angles (e.g. 10 degrees and 170 degrees), 

the lateral closing velocities are quite small. The standard deviation for a velocity of 9.9 knots 

(NACv-1) used in these scenarios will result in significant changes in the actual crossing angle, 

and when the nominal crossing angle is quite small, will sometimes result in actual crossing 

angles near the singularity points of 0 and 180 degrees. 

10.5.3 Comparison of Class A and Class B RSSP Systems 

For all scenarios in which uncertainties were uniformly distributed in azimuth, there was almost 

no dependence of the required buffer with the encounter geometry parameters (speeds and 

crossing angle). The exception to this was for Class B RSSP systems at very narrow crossing 

angles (near 0 degrees or near 180 degrees). In general, the Class A systems required smaller 

buffers partly because smaller uncertainties were used and partly because the overall uncertainty 

does not increase with time as it does for a Class B RSSP system. As demonstrated by scenarios 

A-2 and A-3, Class A RSSP systems open up the opportunity to reduce buffer sizes when 

favorable encounter geometries exist and lateral RNP performance is better than longitudinal 

RNP performance (which is typical). 

10.6 RSSP Study Conclusion 

A mathematical model of Required Self-Separation Performance was exercised in a Monte Carlo 

study to provide insight into the separation buffers required to achieve a minimum separation 

with a given degree of confidence. The results indicated that, for a given minimum separation 

value, the required target separation will decrease as navigation and surveillance accuracy 

increases. While the data did show that the average range at closest approach will always be less 

than the targeted range at closest approach, the difference is relatively unaffected by geometry or 

the magnitude of uncertainties except in extreme cases where lateral uncertainties (especially 

velocity) result in the possibility of a 0/180 degree crossing angle. While the required target 

separation was found to be largely independent of encounter geometry for cases with 

directionally uniform uncertainties, it was found to be a complex function of encounter geometry 

in the case where lateral and longitudinal uncertainties differed significantly. 
136 

11 Required interval Management Performance Study 

The Next Generation Air Transportation System (NextGen) is envisioned as a revolutionary 

transformation of the U.S. airspace to a performance-based, scalable, network-enabled system 

that will be flexible enough to meet future air traffic needs. One of the major transformations is 

the use of Trajectory-Based Operations (TBO) as the main mechanism for managing traffic at 

high density or in highly-complex airspace. These TBOs will be specified between the user and 

the Air Navigation Service Provider (ANSP) and agreed in a “contract”, using advanced 

automation. Overall, preferences for all users are accommodated to the greatest extent possible, 

and trajectories constrained only to the extent required to accommodate congestion, or for 

security, safety, or environmental reasons. Changes to that “contract” will be made 

collaboratively, balancing the user preferences with the ANSP constraints. 

A component of 4-D RNP operation is onboard Interval Management (IM) performed by 

appropriately equipped aircraft. At present, interval management is performed by ground 

controllers and through the use of air traffic control radar and with little or no automation. It is 

expected that insertion of modern ADS-B surveillance technologies, precision navigation and 

guidance technologies, airborne automation tools, and other NextGen concepts, will enable a 

more precise airborne interval management operation that significantly increases the system 

capacity without adversely impacting the safety and reliability of air traffic operations. 

A complete set of operational requirements for automated IM has not yet been fully defined. A 

number of studies are being conducted to evaluate various uncertainties and operational 

parameters that affect IM. The document Required Interval Management Performance Version 

1.1, submitted under contract #NNL08AA15B, NASA Task Order #NNL08AB13T, defined a 

Required Navigation Performance (RNP)-like construct for defining the performance of systems 

capable of performing airborne interval management operations. Several significant uncertainty 

elements were identified and a number of parameters affecting the IM performance were 

considered. The Required Interval Management Performance (RIMP) must ultimately address 

the ability of each participating aircraft to achieve an agreed-upon spacing at the defined IM 

point behind the reference (lead) aircraft. The accuracy of this operation in the presence of 

various uncertainties is the topic of this study. 

11.1 Objectives and Motivation 

This study assessed the feasibility of a long stream of aircraft to perform RIMP operation in the 

presence of various uncertainties. The resulting data is intended to quantify the performance of 

airborne IM and define the limits of safe and stable operation. 

Successful IM achievement depends on many factors including accuracy of forecast winds, 

surveillance capabilities of aircraft engaged in IM, precision of navigation system, and the 

specific capabilities of the IM control system itself. One specific objective of this study is to 

develop metrics to assess how a given IM system performs compared to other comparable IM 

control systems. This type of quantification will allow further development of RIMP standards 

to ensure reliable achievement of the Target Level of Safety (TLS). 
137 

11.2 Background 

Earlier investigation identified a number of key elements of navigation and surveillance systems 

used in IM operation that would contribute to achievement of an overall required TLS. The goal 

of this effort was to identify a set of performance-based parameters that can relate the 

capabilities of an equipped aircraft to safely execute IM operation within a given performance 

envelope. This is comparable to the Required Navigation Performance (RNP) operation, but its 

scope extends to both trailing and leading (reference) aircraft equipage and capabilities and 

includes an estimate of relative enroute time as a key element of the operational performance. 

To evaluate the proposed RIMP standards, a simplified RIMP operation model was developed in 

the MATLAB/Simulink environment. The model included along-track motion only. Additional 

simplifications and assumptions included constant acceleration in the along-track direction. A 

Time-To-Go (TTG) estimator based on constant forecast winds for each segment of the flight 

plan was implemented and a simple IM control system was developed to adjust aircraft speed as 

needed to maintain the desired interval. The initial model included a reference aircraft and one 

trailing aircraft performing airborne IM operations. 

The current RIMP model included a four dimensional (4-D) point-mass model of multiple 

aircraft performing RIMP operations. The model also incorporated spatially varying wind states 

to provide a more realistic atmospheric environment. The flight operations were based on a 

commanded true airspeed profile specified in the flight plan. A comprehensive navigation and 

guidance model was used to realistically operate the aircraft through its flight plan. The model 

included random elements and uncertainty terms in the wind field, starting conditions, sensor 

outputs, and overall aircraft motion. This model used a very simple IM control scheme with 

limited ability to tune for specific performance goals. 

NASA research efforts have looked at various aspects of airborne IM operations and have 

attempted to identify capacity increase and fuel saving metrics [14]. IM accuracy and reliability, 

as well as stability of a stream of aircraft performing airborne IM, have also been studied [15]. 

However, variation in assumptions concerning the IM operation itself, along with different IM 

control strategies selected by different investigators, can introduce differing results. Therefore, 

this effort produced a series of sensitivity and robustness studies to show how different IM 

control schemes or different assumptions on the IM operation itself can impact the performance 

of airborne IM. 

11.3 Simulation Platform and Scenarios Description 

The simulation platform is a MATLAB/Simulink model utilizing simplified aircraft motion 

models. The modeling consists of three structural levels. The lowest level contains the 

individual aircraft flight dynamics models. The second level modeled the stream of aircraft 

performing the interval management operation. Individual aircraft models are utilized to 

generate this stream. The last level provided the Monte Carlo engine which repeatedly executes 

the air traffic stream models and collects the relevant statistics. 
138 

11.3.1 Aircraft Flight Dynamics Model 

The general layout of the aircraft flight dynamics model is depicted in Figure 58. The model 

contains several subsystems and embedded MATLAB scripts. A complete description of the 

model is provided in reference [12]. Specific model parameters are setup using data files. This 

provides the flexibility to adjust the model to fit the study requirements. 

The flight dynamics equations solve a point-mass model flying in 3-dimensional space with 

varying linear acceleration and rotation rates. The primary aircraft state derivatives are the rate 

of change of inertial speed, rate of change of vertical flight path angle, and rate of change of 

ground track. The state vector includes the inertial speed, vertical flight path angle, ground 

track, East relative position, North relative position, and altitude. Additional states being 

integrated are the instantaneous east and north wind components, as well as instantaneous 

atmospheric temperature and pressure. The air-relative velocities including indicated and true 

airspeeds and Mach number are computed based on local atmospheric properties, wind 

velocities, and aircraft inertial speed. 

The wind, temperature, and pressure field is specified in a Rapid Update Cycle (RUC) data file. 

The complete RUC file is parsed for the expected position and altitudes specified in the desired 

flight plan. This reduced data is used during the execution of the simulation model to provide 

the “forecast” wind, temperature, and pressure values at each specified waypoint in the flight 

plan. It is assumed that the wind and temperature states vary linearly (in position and altitude) 

between the waypoints. To represent the wind uncertainty, a set of random wind increments are 

added at each waypoint for each simulation run. Since different aircraft performing the flight are 

spatially and temporally separated, different wind increments are applied for each aircraft, 

although they share the same forecast wind field. 

The aircraft simulation includes a sensor model that introduces error, noise and uncertainty, and 

time delay into the measured parameters used for guidance and navigation. Currently, all aircraft 

utilize the same set of sensor parameters outlined in Figure 59. 
139 

Figure 58 Aircraft Flight Dynamics Model in Simulink 
140 

% Sensor Number 

% Scale Factor (SF) 

% Bias 

% Noise Signal Mean (Mean) 

% Noise Signal Variance (Var) 

% Filter Denominator coeff (change based on Tau and dt) 

% Lower Limit (LLim) 

% Upper Limit (ULim) 

% Quantization Interval (Quant) 

% Time Delay (tDel) [s] 

% 

% SF | Bias | Mean | Var | Tau | LLim | ULim | Quant | tDel 

1, 0.0, 0.0, 0.0, 0.05, 0.990, 0.0, 400.0, 0.1, 0.10 % Inertial Speed [m/s] 

2, 0.0, 0.0, 0.0, 0.0001, 0.990, -15.0, 15.0, 0.001, 0.01 % Ground track [rad] 

3, 0.0, 0.0, 0.0, 0.0001, 0.990, -2.0, 2.0, 0.001, 0.01 % Gamma [rad] 

4, 0.0, 0.0, 0.0, 0.05, 0.990, -250.0, 250.0, 0.01, 0.10 % Wind East [m/s] 

5, 0.0, 0.0, 0.0, 0.05, 0.990, -250.0, 250.0, 0.01, 0.10 % Wind North [m/s] 

6, 0.0, 0.0, 0.0, 0.05, 0.990, 160.0, 328.0, 0.1, 0.10 % OAT [degK] 

7, 0.0, 0.0, 0.0, 0.1, 0.990, 50.0, 1090.0, 0.1, 0.01 % P static [mBar] 

8, 0.0, 0.0, 0.0, 0.1, 0.990, -800000.0, 800000.0, 0.1, 0.10 % East Position [m] 

9, 0.0, 0.0, 0.0, 0.1, 0.990, -800000.0, 800000.0, 0.1, 0.10 % North Position [m] 

10,0.0, 0.0, 0.0, 0.1, 0.990, -1000.0, 20000.0, 0.1, 0.10 % Altitude [m] 

% 

Figure 59 Sensor Model Parameters 

The guidance and control module within each aircraft model attempts to achieve and maintain 

the specified flight profile. In the vertical axis, it adjusts the flight path angle to achieve the 

desired instantaneous altitude and to zero the glideslope error, if any. In the horizontal plane, it 

changes the aircraft heading (ground track) to zero any cross track error and maintain the desired 

ground track. In the longitudinal axis, the speed is adjusted in order to achieve and maintain the 

desired Mach number or the desired indicated (equivalent) airspeed (IAS). The desired Mach or 

IAS values are specified in the flight plan. The IM guidance is added to the desired speed 

profile. The resulting total airspeed command is limited using maximum and minimum IAS 

tables. These tables are presented as a function of altitude and are aircraft-dependent. An 

example is provided in Figure 60. 

% AircraftLimits_767.m 

% Airspeed limits for 767-200 

% 

% Altitude [ft], VMin [kts], VMax [kts] 

FltLvl = [ 0, 50, 100, 120, 180, 240, 300, 340, 400]; 

MaxIAS = [ 200, 240, 250, 310, 330, 330, 330, 330, 330]; 

NomIAS = [ 150, 240, 250, 290, 310, 310, 310, 310, 310]; 

MinIAS = [ 110, 160, 170, 180, 190, 210, 210, 210, 210]; 

% 

Figure 60 Aircraft Indicated Airspeed Limits 

During typical Continuous Descent Approach (CDA) operation, a constant Mach number is 

maintained during the initial descent until the IAS reaches the value specified in the “NomIAS” 

table. At this point, the “NomIAS” table values are maintained as the aircraft descends. The 

flight plan specifies a crossover altitude at which point speed command is switched to a specific 

IAS. During this operation, the IM speed change command is added to the speed command and 

the resulting total speed command is limited using the MaxIAS and MinIAS curves specified for 

the aircraft. 

Two other important components of each aircraft model are the Total TTG estimator and the IM 

controller. The TTG estimator assumes linear speed variation from the current measured speed 
141 

to the next waypoint speed specified in the flight plan. It is assumed that the wind will also vary 

linearly from the current measured value to the forecast value at the waypoint. For all the flight 

plan segments ahead of the current segment, the TTG estimator assumes that the forecast winds 

and specified airspeeds hold true. The IM controller uses the TTG value from the TTG estimator 

for both the ownship and the reference aircraft to derive an instantaneous interval time. This 

time is compared with the specified interval and appropriate speed-up or slow-down commands 

are issued to zero the error between the commanded and measured (estimated) intervals. 

11.3.2 Air Traffic Simulation 

The next layer of the simulation modeling is to combine the operation of multiple aircraft 

performing a stream of arrivals into a common destination. This is done using the 

MATLAB/Simulink environment. Figure 61 depicts a six-aircraft configuration of this stream. 

Each aircraft is implemented as a Simulink model. Each aircraft publishes its own TTG estimate 

and receives TTG estimates from all other aircraft. This setup is intended to simplify the 

modeling process needed for IM operation. 

Figure 62 provides an example of an aircraft “Setup” file. Shown is the setup file for aircraft 

number 5 in the IM stream. Each aircraft is assigned a flight plan filename, a wind and 

temperature data filename, and an appropriate aircraft limits filename. The start time offset (in 

this case 200 seconds) specifies the number of seconds after the start of the overall simulation 

when the aircraft “leaves” the initial waypoint. For the current simulation setup, the initial 

waypoint is effectively the “merge” point and the start of the IM operations. The ownship 

assigned number (in this case 5) and the reference aircraft number (in this case 4) are also 

specified. The assigned reference waypoint (in this case 10) is the common waypoint where the 

TTG calculations are based at. Time of arrival at the reference waypoint is used to compute the 

achieved spacing for each IM pair. Finally, the nominal spacing time is specified (in this case 50 

seconds) for each aircraft in the stream. 
142 

Figure 61 Air Traffic Model in Simulink 
143 

% AC_Setup_5.m 

% Aircraft number 5. 

% 

FPfilename = 'FlightPlan_BGEST_CDA1'; 

WindTempFi = 'BGEST_CDA1_001.txt'; 

ALfilename = 'AircraftLimits_767'; 

Temp1 = [ 

0. % Initial true airspeed error 

(TAS) [kts] 

0. % Initial altitude error [Ft] 

0. % Initial east position error 

[Nautical miles] 

0. % Initial north position error 

[Nautical miles] 

 200. % Start time offset [Seconds] 

 5 % Aircraft number (ATC Sequence) 

4 % Assigned reference aircraft 

(ATC Sequence) 

 10 % Assigned reference waypoint number 

 50. ]; % Nominal spacing time [Seconds] 

% 

Figure 62 Aircraft Setup File for IM Operations 

The “Constants” file specifies several global simulation parameters shared by all aircraft within 

the stream. Figure 63 provides an example as used in the current simulation study. 

0.05 % 1- Reference linear acceleration [g] 

 1.5 % 2- Reference rate of turn [deg/sec] 

 0.5 % 3- Referencce rate of vertical flight path [deg/sec] 

100.0 % 4- Waypoint capture tolerance (meters) 

 1.0 % 5- Course track capture tolerance (deg) 

10.0 % 6- Reference RIMP Delta Velocity (Delta V Reference) [kts] 

 0.0 % 7- Reserved 

 0.0 % 8- Reserved 

 0.01 % 9- Flight path correction constant (deg/m) 

 0.0 % 10- Reserved 

 0.0 % 11- Reserved 

 0.0 % 12- Reserved 

-1 % 13- Random Sequence Initial Seed (-1 = CLOCK) 

 0.0 % 14- Wind Error Mean [m/s] 

 5.0 % 15- Wind Error Standard Deviation (m/s) ------------ 

 5.0 % 16- Start time offset standard deviation (sec) ------ 

 0.0 % 17- Reserved 

 38.16081600 % 18- Zero Reference Latitude, degrees 

-85.74017300 % 19- Zero Reference Longitude, degrees 

 0.0 % 20- Zero Reference Altitude, Meters 

%--------------------------------- 

% 

Figure 63 Global Simulation Constants 

The MATLAB random number generator is used to generate a pseudo-random sequence used for 

all of the signals that have uncertainty associated with them. The random sequence is initialized 

at the start of each simulation run. When the random seed is set to “-1” (Constant array element 

13) the random sequence is initialized using the computer time. This is the default setting for all 

the uncertainty terms in the RIMP simulator. The wind uncertainty mean and standard deviation 

terms as well as the standard deviation for the start time uncertainty are set in the Constant array 

(Figure 63) as well. The mean and standard deviation values for sensor noise are set in the 

“Sensor Parameter” data file (Figure 59). 
144 

11.3.3 Monte Carlo Simulation 

The MATLAB script file “RIMP_MonteCarlo.m” is used to execute a Monte Carlo analysis 

using the air traffic control simulation model. The simulation model is setup for a pre 

determined run duration (3600 seconds in this case) and is repeatedly executed by the Monte 

Carlo script file. At the end of each simulation run, the desired statistics are computed and 

stored. The random sequence is initialized based on the current computer time and thus each run 

will have a different initial seed. 

11.4 RIMP Study Design 

11.4.1 RIMP Study Plan and Approach 

This study evaluates the RIMP operations while performing a CDA procedure into Louisville, 

Kentucky (SDF). The arrival procedure is based on the “BGEST ONE” arrival. This is a 

continuous descent and approach to runway 35L at SDF developed by UPS. 

The study will be performed in three stages. The first stage is the baseline configuration of the 

air traffic movement without IM operation. The impact of wind and initial start time offset in the 

spacing is investigated. The second stage will include one reference aircraft and one trailing 

aircraft performing IM operations. This simple setup will be used to evaluate the sensitivity of 

key metrics to initial spacing and wind uncertainty given the IM controller developed for this 

study. A Monte Carlo simulation setup will be used to collect the required statistical data 

samples. 

The third stage of the study will utilize a stream of 6 aircraft performing IM operations. The 

intent of this part of the study is to demonstrate the stability of IM operations in a complex 

stream. The same metrics used for the single pair operation will be collected for each of the 

engaged aircraft pairings. The study will attempt to quantify the behavioral differences as the 

size of the aircraft stream grows. Again, Monte Carlo simulation setup will be used to collect the 

desired samples. 

The hypothesis of this study is that airborne IM can accurately deliver a stream of intervalmanaged aircraft using only moderate speed variations from nominal speed profiles under 

conditions of X and Y uncertainty. 

11.4.1.1 Flight Plan 

The flight plan used for this study is a CDA procedure into Louisville, Kentucky (SDF) runway 

35L. It utilizes the BGEST waypoint and hence is referred to as the “BGEST ONE” arrival. The 

merging point in this case is also the “start” waypoint, so each aircraft in the air traffic simulation 

arrives at the “start” waypoint at the prescribed initial start time. The flight plan is depicted in 

Figure 64. 
145 

% Louisville BGEST CDA1

%

% Flight Plan Data File Version 3.0

%

% Waypoint ID number

% Waypoint type (1=enroute (default), 2=turn back, 3=land)

% Waypoint Latitude (degrees decimal)

% Waypoint Longitude (degrees decimal)

% Altitude at waypoint (feet, MSL)

% IAS/Mach at waypoint (Knots/Mach)

%

% Waypoint name

%

%---------------------------------------------------------------

% #, Type, Latitude, Longitude, Altitude, IAS or M, Name

%---------------------------------------------------------------

FP = [ 

 0 1 38.53117000 -91.63363900 37000 0.8 cellstr('start') 

 1 1 38.41999860 -89.15899780 37000 0.8 cellstr(' ENL') 

 2 1 38.38000000 -88.40000000 37000 0.8 cellstr('xxTOD') 

 3 1 38.37723889 -88.20648056 36000 0.8 cellstr('ZARDA') 

 4 1 38.32920278 -87.61643611 28000 0.8 cellstr('PRINC') 

 5 1 38.12002778 -86.40452778 11000 240 cellstr('BGEST') 

 6 1 38.01983055 -85.85327778 4600 190 cellstr('PTINO') 

 7 1 37.98333333 -85.75000000 4000 190 cellstr('BASKT') 

 8 1 38.02111111 -85.69444444 3000 180 cellstr('DNKIT') 

 9 1 38.06438889 -85.70835000 2400 170 cellstr('CRDNL') 

10 3 38.16081600 -85.74017300 513 140 cellstr('RW35L') 

]; 

Figure 64 Flight Plan File for BGEST Arrival 

Figure 65 depicts the shape of the flight profile in an East-North map where the East and North 

axes have different scales. The IM reference waypoint is CRDNL. This defines segment 

number 9 of the flight plan and the 10th waypoint specified. The achieved spacing values are 

measured at this waypoint. The final waypoint is RWY35L. Once the aircraft arrives at this 

waypoint, it is commanded to maintain runway heading, rotate to a zero degree vertical flight 

path angle and decelerate to zero ground speed. 

As shown in Figure 64, airspeed values specified as less than 1.0 are interpreted as commanded 

Mach numbers. Values greater than 1.0 are interpreted as IAS (or equivalent airspeed) in Knots. 
146 

Figure 65 Flight Plan Surface Map 
147 

11.4.1.2 Forecast Wind and Temperature 

For this analysis, the RUC data for 0800Z January 6, 2009 is used. To simply the simulation 

process, a pre-processor is used to extract wind, temperature, and pressure data from the RUC 

data file for the specific waypoint positions, altitudes, and times as applicable. An example of a 

reduced data file is provided in Figure 66. 

% 

% Wind pressure and temperature data 

% First column is waypoint number 

% Second column is East wind in meters per second 

% Third column is North wind in meters per second 

% Fourth column is static pressure in millibars 

% Fifth column is OAT in degrees Kelvin 

% 

 0.0000000e+000 -6.3341029e+001 -3.2106923e+001 2.1730000e+002 2.1923351e+002 

 1.0000000e+000 -6.8598146e+001 -3.3449491e+001 2.1730000e+002 2.1831580e+002 

 2.0000000e+000 -7.0793884e+001 -3.1034921e+001 2.1730000e+002 2.1809949e+002 

 3.0000000e+000 -7.1651931e+001 -3.2510421e+001 2.2800000e+002 2.2015837e+002 

 4.0000000e+000 -5.7738696e+001 -3.2149985e+001 3.2990000e+002 2.3670349e+002 

 5.0000000e+000 -1.9763707e+001 -8.0108517e+000 6.6704391e+002 2.6876573e+002 

 6.0000000e+000 -5.4049351e+000 -8.5042564e+000 8.5055624e+002 2.7791416e+002 

 7.0000000e+000 -3.8262500e+000 -6.1950000e+000 8.6968250e+002 2.7832625e+002 

 8.0000000e+000 -3.0130864e-001 -3.1112469e+000 9.0268727e+002 2.7755280e+002 

 9.0000000e+000 3.7290236e+000 -2.1433883e+000 9.2352290e+002 2.7520866e+002 

 1.0000000e+001 4.1435416e+000 8.3606477e-001 9.8653519e+002 2.7357225e+002 

% 

Figure 66 Wind Temperature and Pressure Data 

11.4.2 Metrics 

Several key metrics are relevant to interval management performance. These are provided in the 

order of their importance below. 

11.4.2.1 Achieved Spacing Error 

The achieved spacing error is defined as the actual time of arrival of the trailing aircraft at the 

reference IM point less the actual time of arrival of the lead aircraft at the same location. In the 

presence of various uncertainties, it is expected to have a statistically relevant sample of 

achieved spacing error using Monte Carlo simulation setup. Therefore the following statistics 

will be computed: mean, standard deviation, lower 5% tail, upper 5% tail, and absolute 95% 

(absolute of ± error) statistics. 

11.4.2.2 Frequency of IM Speed Commands 

The number of speed commands per minute is a key measure of pilot workload. Several speed 

command frequency metrics will be considered. 

 Per flight plan segment 

o Maximum 

o Average 

o 95% 

 Pre-merge (Not used in this analysis) 

o Maximum 

o Average 
148 

o 95% 

 During CDA 

o Maximum 

o Average 

o 95% 

 Below 10,000 feet 

o Maximum 

o Average 

o 95% 

 Overall 

o Maximum 

o Average 

o 95% 

11.4.2.3 Cumulative Speed Changes 

The cumulative speed change is a measure of total true airspeed or indicated airspeed change 

from the planned value. It is used to measure excessive speed changes required by the IM 

control system. 

11.4.2.4 Predicted Spacing Error 

The predicted spacing error is defined as the difference between the estimated TTG of the 

trailing and reference aircraft. A history of the predicted spacing error as the flight progresses, 

enables the evaluation of the validity of assumptions and simplifications made in the modeling of 

the IM operation. 

11.4.3 RIMP Test Plan 

The RIMP simulation software was used to perform this study. Although the simulation can be 

set up with an arbitrary number of aircraft in a stream, with different aircraft performance 

parameters, different flight plans, and different IM control parameters, this study utilized only 

one type of aircraft performance and a common flight plan. Furthermore, only one forecast wind 

field was used for all the tests. 

11.4.3.1 Baseline Test 

To generate a measure of basic operational parameters, a pair of aircraft flying the specified 

flight plan in a varied wind field and varied initial spacing is tested. The resulting data provide 

the necessary datum for the flight plan and simulation configurations used in this study. Also, 

the basic performance characteristics of the TTG estimator will be assessed. This test is 

completed in three steps: 

 Zero wind uncertainty and unbiased initial spacing. The aircraft will have variation in 

sensor noise and thus there will be slight differences in flight profile. 

 Include wind uncertainty but have unbiased initial spacing. 

 Include both wind uncertainty and initial spacing error. 
149 

The initial spacing error is computed based on the difference between the trailing aircraft’s TTG 

and the reference aircraft TTG when the trailing aircraft starts its simulation run. However, no 

IM speed change is performed during this test scenario. 

11.4.3.2 Single Pair IM 

For this test, the reference aircraft executes the assigned flight plan and the trailing aircraft 

performs IM. This test is also completed in three steps: 

 Zero wind uncertainty and unbiased initial spacing. Each aircraft has variation in sensor 

noise and thus there will be slight differences in flight profile. 

 Include wind uncertainty but have unbiased initial spacing. 

 Include wind uncertainty and initial spacing error; set interval time to 50 seconds. 

Due to schedule constraints, the evaluation of IM operations at various interval times was not 

completed. 

11.4.3.3 Large IM Stream 

This test simulates a large stream of aircraft performing IM. This includes one reference aircraft 

with a fixed flight plan and 5 trailing aircraft performing IM. The intent of the test is to evaluate 

the overall operations including accuracy and stability of the stream. Additionally, this test is 

desired to assess when re-negotiation of flight plans becomes necessary. This test is completed 

in three steps as well: 

 Zero wind uncertainty and unbiased initial spacing. The aircraft will have variation in 

sensor noise and thus there will be slight differences in its flight profile. 

 Include wind uncertainty but have unbiased initial spacing. 

 Include wind uncertainty and initial spacing error; set interval time to 50 seconds. 

Due to schedule constraints, the evaluation of IM operations at various interval times was not 

completed. 

11.5 RIMP Study Results 

11.5.1 Baseline Test 

11.5.1.1 Baseline Test - No Wind or Initial Spacing Error 

For these tests, the wind and start time (initial spacing) standard deviations were set to zero 

(“Constants” file) to effectively turn off the random terms. In addition the commanded spacing 

time (“Setup” file) was set to zero to disable IM operations. The resulting nominal operation is 

depicted in Figure 67Figure 67 through Figure 76. 

The altitude profile is shown in Figure 67. The aircraft maintains an altitude of 37,000 feet until 

arriving at the Top of Descent (TOD). Figure 68 shows the progression of the flight as 

subsequent segments are completed. For this particular aircraft example, the TOD occurs at 

approximately 1000 seconds into the simulation run and the aircraft arrives at the reference 

waypoint at approximately 2300 seconds into the simulation run. 
150 

Figure 69 and Figure 70 show the indicated airspeed and Mach number, flown by the aircraft, 

respectively. Based on the flight plan described earlier, the aircraft must maintain M=0.8 until 

arriving at the BGEST waypoint. However, during descent, the indicated airspeed increases. 

Upon reaching a nominal airspeed value specified in the aircraft limit file (in this case 310 

Knots), the flight control system limits the commanded speed to maintain the specified indicated 

airspeed. This behavior is shown in the indicated airspeed (Figure 69) and Mach number (Figure 

70) plots. The constant Mach number is maintained beyond the TOD (approximately 950 

seconds into the run) while the IAS increases. Shortly after reaching the nominal maximum IAS 

of 310 Knots (approximately 1200 seconds into the run), the flight plan calls for a reduction of 

IAS to 240 Knots. The guidance and control systems decelerate the aircraft and maintain the 

commanded IAS. The acceleration and deceleration are performed at the specified rate of 0.05 

g’s referenced to the inertial speed of the aircraft. 

The true airspeed achieved using the specified Mach/IAS speed profile is shown in Figure 71. 

As expected, during descents, the true airspeed linearly varies between waypoints. The true 

airspeed signal along with measured wind magnitude and direction, as shown in Figure 72, are 

used to compute the ground speed for the aircraft. The wind profile shown does not include any 

added uncertainty. It is the forecast winds at each waypoint location and altitude, and is linearly 

interpolated between waypoints. The ground speed signal is depicted in Figure 73, showing the 

nearly linear variation of ground speed from one waypoint to the next as the aircraft progresses 

through the flight plan. 

Although the ground speed generally varies linearly between waypoints, during certain portions 

of the descent, there are significant differences between the planned and actual velocity profile 

due to IAS limiting. The TTG estimator currently implemented in the simulation model 

accounts for the linearly varying ground speed, but it does not fully account for the possible large 

speed differences. During certain long segments of the flight plan, a difference between the 

planned speed profile and actual speed profile will result in substantial Estimate Time Enroute 

(ETE) error for that segment. This introduces significant problems in the interval management 

since the total TTG is based on the summation of individual segment ETE. These problems will 

be addressed later in this section. 

The rate of climb profile is shown in Figure 74. Although the change in altitude and resulting 

total energy change is fairly smooth (as shown in Figure 67), the change in air-relative and 

inertial velocities during descent requires fairly noticeable change in the rate of climb while 

performing this continuous descent. 
151 

Figure 67 Altitude Profile 

Figure 68 Flight Segment Completion Histories 

TOD 

Landing 

Arrival at BGEST

Arrival at CRDNL 

(reference point) 

TOD 

Landing 
152 

Figure 69 Indicated Airspeed Profile 

Figure 70 Mach Number Profile 

TOD 

Landing 

TOD 

Landing 

Arrival at BGEST
153 

Figure 71 True Airspeed Profile 

Figure 72 Measured Wind Magnitudes and Direction 

TOD 

Landing 

Arrival at BGEST
154 

Figure 73 Resulting Ground Speed 

Figure 74 Rate of Climb Profile 

The navigation module within each aircraft model computed the instantaneous along-track 

distance to the waypoint for each segment of the flight plan. This output is shown in Figure 75. 

This shows the distance flown for each waypoint as a function of simulation run time. The 

TOD 

Landing 

TOD 

Landing 
155 

along-track distance is adjusted to account for initial and final turns from the previous segment 

track and to the next segment track. 

Figure 75 Along-Track Distance to Go for Current Segment 

The estimated TTG is depicted in Figure 76. Ideally, the TTG signal is smooth with a slope of -

45 degrees. However, as shown in this figure, any difference between the planned ground speed 

and actual ground speed will result in “kinks” in the TTG curve. The TTG from the current 

waypoint k to the reference waypoint n is computed using the ETE for the current segment and 

all the future segments. 

 (135) 

In the current implementation of the TTG estimator, ETEs for all future segments are computed 

based on planned speeds and forecast winds. The current segment ETE uses the current 

measured ground speed of the vehicle and assumes that the aircraft will linearly decelerate to the 

planned speed at the waypoint. As a result of this assumption, variations in the achieved speed at 

the waypoint will result in a vertical shift in the TTG curve as shown in Figure 76 at 1200 and 

1700 seconds into the run. The significant jumps shown at the point of crossing the PRINC and 

BGEST waypoints are due to larger differences between the planned and actual ground speeds 

and the length of segments associated with these speeds. 

TOD 

Landing 
156 

Figure 76 Computed TTG for Nominal Case 

The achieved spacing for the baseline case with no wind uncertainty and no start time error is 

50±1 seconds, which is the nominal start time offset. The resolution of the simulation data is 1 

second. 

11.5.1.2 Baseline Test - With Wind Uncertainty 

For these tests, the random wind components were activated with a mean of zero and a standard 

deviation of 5 meters per second for both the East and North winds. The standard deviation for 

the start time (initial spacing) uncertainty was set to zero. In addition, the commanded spacing 

time (“Setup” file) was set to zero to disable IM operations. The initial spacing was fixed at 50 

seconds. In this setup, neither aircraft is performing IM nor adjusting speed to achieve a 

Required Arrival Time (RTA). In this case, the difference between the time of arrival of the lead 

aircraft and the trailing aircraft at the reference waypoint is referred to as the “equivalent” 

spacing. 

A total of 500 simulation runs were conducted. Figure 77 shows a histogram of the equivalent 

spacing at the reference waypoint with wind uncertainty and with IM operations disabled. Due 

to limited sample size, this histogram only roughly represents a normal distribution about the 

nominal 50 seconds of separation at the reference waypoint. A list of statistics for this case is 

provided in Table 7: Statistics for baseline test case with wind uncertainty. The collected timebased data has a sample rate of 1 second. Therefore, resolution of the statistics presented here is 

±1 second. 

Arrival at PRINC 

Arrival at BGEST 
157 

Table 7: Statistics for baseline test case with wind uncertainty 

Mean Value for equivalent spacing 49.4 seconds 

Standard Deviation 35.2 seconds 

Maximum equivalent spacing 160 seconds 

Minimum equivalent spacing -54 seconds 

Figure 78 provides an example of the predicted spacing signal between the lead aircraft and the 

ownship given the wind uncertainty. The large spikes in predicted spacing signal are due to the 

nature of the TTG estimated signal described in section 11.5.1. 

Figure 77 Distribution of Final Spacing with Wind Uncertainty and no IM 
158 

Figure 78 Predicted Spacing Signal 

11.5.1.3 Baseline Test - With Wind and Initial Spacing Uncertainty 

For these tests, the random wind components are activated with a mean of zero and a standard 

deviation of 5 meters per second for both the East and North winds. The standard deviation for 

the start time (initial spacing) uncertainty is set to 5 seconds. In addition, the commanded 

spacing time (“Setup” file) is set to zero to disable IM operations. The initial spacing was fixed 

at 50 seconds. As with previous test described in section 11.5.2, neither aircraft is performing 

IM nor adjusting speed to achieve a RTA. The difference between the time of arrival of the lead 

aircraft and the trailing aircraft at the reference waypoint is referred to as the “equivalent” 

spacing. 

A total of 500 simulation runs were conducted to arrive at the results. Figure 79 shows a 

histogram of the equivalent spacing at the reference waypoint with wind and initial uncertainty, 

and with IM operations disabled. The “rough” shape of the distribution is due to limited sample 

size. The expected nominal shape of the histogram is a normal distribution about the nominal 50 

seconds of separation at the reference waypoint. A list of statistics for this case is provided in 

Table 8. The resolution of the statistics presented here in this table is ±1 second. Qualitatively, 

the distribution of equivalent spacing at the reference waypoint is comparable to the distribution 

for the case with wind uncertainty only. 

Arrival at PRINC 

Arrival at BGEST 
159 

Table 8: Statistics for baseline test case with wind and initial spacing uncertainty 

Mean Value for achieved spacing 48.4 seconds 

Standard Deviation 36.7 seconds 

Maximum achieved spacing 143 seconds 

Minimum achieved spacing -45 seconds 

Figure 79 Final Spacing with Wind and Initial Spacing Uncertainty and no IM 

11.5.2 Single Pair IM 

11.5.2.1 Single Pair IM - No Wind or Initial Spacing Uncertainty 

For these tests, the random wind components and the initial spacing uncertainty are deactivated 

by setting the standard deviations to 0. The commanded spacing time (“Setup” file) is set to 50 

seconds for the in-trail aircraft (aircraft number 2) in order to activate the IM operation. 

A total of 250 simulation runs were conducted to arrive at the results. Zero wind or initial 

spacing uncertainty was included and the timing signal was collected at a rate of once per 

second. The simulation includes random noise and variation in the sensor model, however, the 

guidance and control system for each aircraft compensates for this variation within the once per 

second output signal resolution. As a result, the IM system response is strictly deterministic. 

Figure 80 shows a histogram of the achieved spacing at the reference waypoint. Since the 

response is deterministic, all the achieved spacing values fall within the 52 second frame. Figure 

81 provides a histogram of the IM frequency (number of IM speed commands per minute) 

averaged over the total duration of the operation. This indicates an average IM speed command 

frequency of less than 1 per minute. Figure 82 provides a histogram of the IM frequency (per 
160 

minute) for that portion of operation below an altitude of 10,000 feet. The results indicate an IM 

frequency of less than 2 per minute below 10,000 ft. Figure 83 shows a histogram of IM 

frequency (per minute) for each segment of the flight plan. It is evident that all the “high 

activity” IM actions with a frequency of 2 per minute or higher occur during the final three 

segments of the operation. 

The basic statistics associated with the IM operation are listed in Table 9. 

Table 9: Statistics for single pair IM test case without uncertainty 

Mean Value for achieved spacing 52 seconds 

Standard Deviation 0 seconds 

Maximum achieved spacing 52 seconds 

Minimum achieved spacing 52 seconds 

Mean Value for IM frequency, total 0.5337 per minute 

Standard Deviation 0 

Maximum 0.5337 

Minimum 0.5337 

Mean Value for IM frequency, below 10K 1.59 per minute 

Standard Deviation 0 

Maximum achieved spacing 1.59 

Minimum achieved spacing 1.59 

Mean Value for IM frequency, Seg. #1 0 per minute 

Mean Value for IM frequency, Seg. #2 0 

Mean Value for IM frequency, Seg. #3 1.09 

Mean Value for IM frequency, Seg. #4 0.70 

Mean Value for IM frequency, Seg. #5 0.35 

Mean Value for IM frequency, Seg. #6 0.43 

Mean Value for IM frequency, Seg. #7 2.73 

Mean Value for IM frequency, Seg. #8 3.33 

Mean Value for IM frequency, Seg. #9 6.25 
161 

Figure 80 Achieved Spacing Histogram 

Figure 81 IM Frequency Histogram Total Operation 
162 

Figure 82 IM Frequency Histogram below 10K 

Figure 83 IM Frequency Histogram per Segment 

Figure 84 provides an example of the estimated interval and the resulting IM command signal. 

The estimated interval is computed as the difference between the ownship TTG and reference 

aircraft TTG. The current implementation does not include any form of filtering for this signal 
163 

and as described before, ground speed variation between the planned and actual flight operations 

introduce large enough differences to adversely impact the IM operation. As shown in this 

figure, an apparent error of 200 seconds in interval time is estimated as the reference aircraft 

passes the PRINC waypoint at approximately 1200 seconds into the simulation. Once the 

trailing aircraft passes through the same waypoint, the estimated interval returns to a more 

accurate level. This error in interval time is caused by the assumptions in the TTG estimator. 

The IM controller attempts to correct the 200 seconds excess in interval time by commanding a 

higher speed. Once the trailing aircraft passes the PRINC waypoint, the TTG estimate for the 

ownship is similarly reduced and the interval time used by the IM controller is abruptly reduced 

by more than 200 seconds. 

Therefore, it can be seen that modifications to the TTG estimator are needed before any IM 

control optimization can be attempted. An estimation scheme that avoids the observed jumps in 

the TTG calculation is required for this operation. 

Additionally, no attempt is made in this analysis to abort the IM operation if an excessive IM 

error is encountered. For future studies, a method for aborting IM operation and reengaging such 

operation must also be developed and implemented. 

Figure 84 Estimated Spacing and IM Command History 

11.5.2.2 Single Pair IM - With Wind Uncertainty 

For these tests, the random wind components are activated with a mean of zero and a standard 

deviation of 5 meters per second for both the East and North winds. The standard deviation for 
164 

the start time (initial spacing) uncertainty is set to zero. In addition, the commanded spacing 

time (“Setup” file) is set to 50 seconds for the in-trail aircraft (aircraft number 2). 

A total of 250 simulation runs were conducted to arrive at the results. Figure 85 shows a 

histogram of the achieved spacing at the reference waypoint with wind uncertainty and with the 

second aircraft performing IM operations. Although the available sample size is limited, the 

achieved spacing appears to have a normal distribution about approximately 51 seconds. This is 

consistent with the spacing signal resolution of 1.0 second. 

The IM speed change frequency (per minute) averaged over the total duration of the operation is 

shown in Figure 86. This histogram depicts an average IM frequency of 0.5 speed changes per 

minute. Figure 87 provides a histogram of the IM frequency (per minute) for that portion of 

operation below an altitude of 10,000 feet. Approximately 1.5 to 2 speed change events per 

minute are depicted as average below 10,000 ft. Figure 88 shows a histogram of IM frequency 

(per minute) for each segment of the flight plan. As expected, all the “high activity” IM actions 

with a frequency of 2 per minute or higher occur during the final three segments of the operation 

(segments 7, 8, and 9). This is in part due to the simple design of the IM controller and can be 

significantly improved upon using a more optimized IM control scheme. 

The basic statistics associated with the IM operation are listed in Table 10 below. 

Table 10: Statistics for single pair IM test case with wind uncertainty 

Mean Value for achieved spacing 51.3 seconds 

Standard Deviation 5.9 seconds 

Maximum achieved spacing 74 seconds 

Minimum achieved spacing 11 seconds 

Mean Value for IM frequency, total 0.529 per minute 

Standard Deviation 0.101 

Maximum 0.752 

Minimum 0.225 

Mean Value for IM frequency, below 10K 1.579 per minute 

Standard Deviation 0.414 

Maximum achieved spacing 2.584 

Minimum achieved spacing 0.204 

Mean Value for IM frequency, Seg. #1 0.078 per minute 

Mean Value for IM frequency, Seg. #2 0.076 

Mean Value for IM frequency, Seg. #3 0.443 

Mean Value for IM frequency, Seg. #4 0.506 

Mean Value for IM frequency, Seg. #5 0.324 

Mean Value for IM frequency, Seg. #6 0.473 

Mean Value for IM frequency, Seg. #7 2.589 

Mean Value for IM frequency, Seg. #8 4.365 

Mean Value for IM frequency, Seg. #9 4.124 
165 

Figure 85 Achieved Spacing Histogram. 

Figure 86 IM Frequency Histogram Total Operation 
166 

Figure 87 IM Frequency Histogram below 10K 

Figure 88 IM Frequency Histogram per Segment 
167 

11.5.2.3 Single Pair IM - With Wind and Initial Spacing Uncertainty 

For these tests, the random wind components are activated with a mean of zero and a standard 

deviation of 5 meters per second for both the East and North winds. The standard deviation for 

the start time (initial spacing) uncertainty is set to 5 seconds. In addition the commanded 

spacing time (“Setup” file) is set to 50 seconds for the in-trail aircraft (aircraft number 2). 

A total of 173 simulation runs were conducted to arrive at the results. Figure 89 shows a 

histogram of the achieved spacing at the reference waypoint with wind and initial spacing 

uncertainty and with the second aircraft performing IM operations. Figure 90 provides a 

histogram of the IM frequency (per minute) averaged over the total duration of the operation. 

Figure 91 provides a histogram of the IM frequency (per minute) for that portion of operation 

below an altitude of 10,000 feet. Figure 92 shows a histogram of IM frequency (per minute) for 

each segment of the flight plan. 

Normal distributions were expected for all of the histograms presented in this section. However, 

due to the limited number of test runs, the distributions appear irregular. The basic statistics 

associated with the IM operation are listed in Table 11. 

Table 11: Statistics for single pair IM test case with wind and initial spacing uncertainty 

Mean Value for achieved spacing 51.7 seconds 

Standard Deviation 5.3 seconds 

Maximum achieved spacing 71 seconds 

Minimum achieved spacing 34 seconds 

Mean Value for IM frequency, total 0.553 per minute 

Standard Deviation 0.10 

Maximum 0.763 

Minimum 0.278 

Mean Value for IM frequency, below 10K 1.627 per minute 

Standard Deviation 0.383 

Maximum achieved spacing 2.491 

Minimum achieved spacing 0.587 

Mean Value for IM frequency, Seg. #1 0.124 per minute 

Mean Value for IM frequency, Seg. #2 0.069 

Mean Value for IM frequency, Seg. #3 0.418 

Mean Value for IM frequency, Seg. #4 0.525 

Mean Value for IM frequency, Seg. #5 0.324 

Mean Value for IM frequency, Seg. #6 0.465 

Mean Value for IM frequency, Seg. #7 2.784 

Mean Value for IM frequency, Seg. #8 4.518 

Mean Value for IM frequency, Seg. #9 4.239 
168 

Figure 89 Achieved Spacing Histogram. 

Figure 90 IM Frequency Histogram Total Operation 
169 

Figure 91 IM Frequency Histogram below 10K 

Figure 92 IM Frequency Histogram per Segment 
170 

11.5.3 Large IM Stream 

In this section, an IM stream of 6 aircraft is analyzed. The first aircraft follows the flight planspecified speeds. The following aircraft perform interval management relative to the aircraft 

directly ahead of them. 

11.5.3.1 Multi Pair IM - No Wind or Initial Spacing Uncertainty 

For these tests, the random wind components and the initial spacing uncertainty are deactivated 

by setting the standard deviations to 0. The commanded spacing time (“Setup” file) is set to 50 

seconds for the in-trail aircraft (aircraft number 2 through 6) in order to activate the IM 

operation. Each aircraft model contains random noise and variability in its sensor model, 

however, the guidance and control system for each aircraft compensates for this variation. As a 

result, the IM system response is strictly deterministic within the once per second output data. 

A total of 250 simulation runs were conducted to arrive at the results. Figure 93 shows a 

histogram of the achieved spacing at the reference waypoint. The response is deterministic; 

however different aircraft in the IM stream achieve slightly different spacing. The second 

aircraft achieves a spacing of 52 seconds. Aircraft number 4 achieves a spacing of 59 seconds 

while aircraft number 3 consistently achieves a spacing of less than 47 seconds. 

Figure 94 provides a histogram of the IM frequency averaged over the total duration of the 

operation. This indicates an average IM speed command frequency of around 0.5 for all aircraft. 

However, aircraft number 5 and number 6 are shown to have a slightly lower average IM 

frequency. Figure 95 provides a histogram of the IM frequency (per minute) for that portion of 

operation below an altitude of 10,000 feet. The results indicate an IM frequency of around 1.5 

per minute below 10,000 feet for aircraft number 2, 3, and 4. Aircraft number 5 has an IM 

frequency of 1.1, while aircraft number 6 has an IM frequency of less than 1 per minute. Figure 

96 shows a histogram of IM frequency (per minute) for aircraft number 2 at each segment of the 

flight plan. Figure 97 shows a histogram of IM frequency (per minute) for aircraft number 6 at 

each segment of the flight plan. 

The mechanism for the variation in achieved spacing and IM activity among the participating 

aircraft is not fully investigated. This variation changes with changes to the flight plan 

parameters, as well as by changing the guidance, control, and IM control parameters within the 

simulation model. A main contributor is the significant difference between actual and estimated 

time enroute for certain long flight segments. 
171 

Figure 93 Achieved Spacing Histogram. 

Figure 94 IM Frequency Histogram Total Operation 
172 

Figure 95 IM Frequency Histogram below 10K 

Figure 96 IM Frequency Histogram per Segment AC#2 
173 

Figure 97 IM Frequency Histogram per Segment AC#6 

11.5.3.2 Multi Pair IM - With Wind Uncertainty 

The random wind components are activated with a mean of zero and a standard deviation of 5 

meters per second for both the East and North winds. The standard deviation for the start time 

(initial spacing) uncertainty is set to zero. All aircraft arrive at the merge point (starting 

waypoint in the simulation) exactly 50 seconds apart. In addition, the commanded spacing time 

(“Setup” file) is set to 50 seconds for aircraft numbers 2 through 6. 

A total of 250 simulation runs were conducted to arrive at the results. Figure 98 shows a 

combined histogram of the achieved spacing at the reference waypoint with wind uncertainty for 

all aircraft engaged in IM. Aircraft number 1 follows a specified flight plan with no additional 

speed changes. Aircraft numbers 2 through 6 perform IM operations. This indicates a normal 

distribution with a mean of around 51 seconds. There is no obvious difference in achieved 

spacing statistics for aircraft 5 and 6 at the end of the IM stream. 

Figure 99 provides a histogram of the IM speed command frequency (per minute) averaged over 

the total duration of the operation. Figure 100 provides a histogram of the IM speed command 

frequency (per minute) for that portion of operation below an altitude of 10,000 feet. In both 

cases, aircraft number 5 and 6 appear to have fewer IM speed commands than the aircraft in front 

of the stream. Although the number of samples is very limited, the results indicate a normal 

distribution. Figure 101 and Figure 102 show histograms of IM frequency (per minute) for each 

segment of the flight plan for aircraft number 2 and aircraft number 6 respectively. As with the 

earlier results, all the “high activity” IM actions with a frequency of 2 per minute or higher occur 

during the final three segments of the operation. This is in part due to the simple design of the 
174 

IM controller and can be significantly improved upon using a more optimized IM control 

scheme. 

The basic statistics associated with the IM operation are listed in Table 12 below. 

Table 12: Statistics for multi pair IM test case with wind uncertainty 

Achieved spacing, seconds mean  max min 

AC #2 51.3 5.9 74 11 

AC #3 51.2 7.6 79 21 

AC #4 51.2 7.6 71 15 

AC #5 49.9 7.4 69 25 

AC #6 50.2 7.4 66 24 

IM frequency, total (per minute) mean  max min 

AC #2 0.529 0.10 0.752 0.225 

AC #3 0.481 0.10 0.778 0.245 

AC #4 0.463 0.10 0.731 0.197 

AC #5 0.442 0.10 0.674 0.142 

AC #6 0.427 0.10 0.706 0.210 

IM frequency, below 10K (per minute) mean  max min 

AC #2 1.579 0.41 2.584 0.204 

AC #3 1.325 0.37 2.300 0.475 

AC #4 1.193 0.34 2.095 0.183 

AC #5 1.076 0.30 1.752 0.168 

AC #6 0.982 0.28 1.823 0.155 

Although only very limited data has been collected, it is evident that there is no indication of 

instability due to the size of the stream in the presence of random wind uncertainty with a 

standard deviation of 5 m/s (or approximately 10 Knots) in both East and North directions. The 

tests indicated that even with the observed shortcomings of the TTG estimator and the IM 

controller implemented in this model, the last IM aircraft in the stream performs equally well as 

the first IM aircraft. This implies that the IM stream is stable. 
175 

Figure 98 Achieved Spacing Histogram. 

Figure 99 IM Frequency Histogram Total Operation 
176 

Figure 100 IM Frequency Histogram below 10K 

Figure 101 IM Frequency Histogram per Segment AC#2 
177 

Figure 102 IM Frequency Histogram per Segment AC#6 

11.5.3.3 Multi Pair IM - With Wind and Initial Spacing Uncertainty 

For these tests, the random wind components are activated with a mean of zero and a standard 

deviation of 5 meters per second for both the East and North winds. All aircraft arrive at the 

merge point (starting waypoint in the simulation) at a mean of 50 seconds apart and with a 

standard deviation of 5 seconds. In addition, the commanded spacing time (“Setup” file) is set to 

50 seconds for aircraft numbers 2 through 6. 

A total of 173 simulation runs were conducted to arrive at the results. This is not a sufficient 

sample. However, limited schedule and computational limitations prevented a larger sample. 

A combined histogram of the achieved spacing at the reference waypoint with wind and initial 

spacing uncertainties is shown in Figure 103. The lead aircraft is flown to the specified flight 

plan with no additional speed changes. The second through sixth aircraft perform IM speed 

change operations. This is consistent with a normal distribution with a mean of between 50 and 

51 seconds. Considering the resolution of sample time resolution of one second, it can be 

deduced that the average achieved spacing is nearly the same for all aircraft in the IM stream. 

Figure 104 provides a histogram of the IM speed change frequency (per minute) averaged over 

the total duration of the operation. Figure 105 provides a histogram of the IM speed change 

frequency (per minute) for that portion of operation below an altitude of 10,000. These results 

are consistent with the wind-only uncertainty case. The average IM speed change frequency for 

the aircraft in front of the IM stream appears to be slightly higher than that for the aircraft later in 
178 

the stream. Figure 106 and Figure 107 show histograms of IM frequency (per minute) for each 

segment of the flight plan for aircraft number 2 and aircraft number 6 respectively. 

The basic statistics associated with the IM operation are listed in Table 13. 

Table 13: Statistics for multi pair IM test case with wind and initial spacing uncertainty 

Achieved spacing, seconds mean  max min 

AC #2 51.7 5.3 71 34 

AC #3 51.5 8.1 75 13 

AC #4 51.2 7.1 65 22 

AC #5 49.3 8.1 66 13 

AC #6 49.9 7.8 74 19 

IM frequency, total (per minute) mean  max min 

AC #2 0.553 0.10 0.763 0.278 

AC #3 0.494 0.10 0.761 0.150 

AC #4 0.477 0.10 0.767 0.191 

AC #5 0.456 0.10 0.750 0.190 

AC #6 0.447 0.10 0.684 0.162 

IM frequency, below 10K (per minute) mean  max min 

AC #2 1.627 0.38 2.491 0.587 

AC #3 1.328 0.39 2.256 0.192 

AC #4 1.188 0.34 1.960 0.176 

AC #5 1.060 0.33 1.966 0.174 

AC #6 0.991 0.27 1.577 0.158 
179 

Figure 103 Achieved Spacing Histogram. 

Figure 104 IM Frequency Histogram Total Operation 
180 

Figure 105 IM Frequency Histogram below 10K 

Figure 106 IM Frequency Histogram per Segment AC#2 
181 

Figure 107 IM Frequency Histogram per Segment AC#6 

11.6 RIMP Study Conclusion 

A simulation of a multi-aircraft stream performing interval management operations was 

developed and used to analyze the sensitivity of the operation to various uncertainties, in 

particular forecast wind and initial spacing uncertainty. The simulation was developed in the 

MATLAB/Simulink environment and was set up to run in a Monte Carlo simulation 

environment. Each simulation run consisted of six 767-like aircraft flying a CDA procedure into 

Louisville, KY. The simulation started at a predefine point at the cruise altitude and proceeded 

to landing of all aircraft. The simulation execution length was 3600 seconds. The aircraft flight 

dynamic states were updated every 0.01 seconds, and data sampling from the model was taken 

once per second. 

Each aircraft in the simulation computed an estimate of its TTG to the reference waypoint. This 

estimated time was transmitted to all aircraft in the stream. The aircraft performing IM 

compared their ownship TTG to their respective reference aircraft TTG to compute an 

instantaneous interval time. The IM control algorithm in each aircraft used this interval time to 

command a speed increase or a decrease as necessary. 

A series of assumptions on the aircraft speed profile and how to predict the ground speed during 

the CDA were needed in order to develop the TTG estimator. The study results presented here 

show several shortcomings with the current scheme for this TTG estimation. During longer 

CDA segments where the aircraft speed deviated from a linear profile between the start and end 

waypoints, a large change to the estimated time enroute introduce a significant error. This error 

manifested itself as a step change in TTG when the lead aircraft crosses a waypoint, followed by 
182 

another step change back to the original timing when the trailing aircraft reaches the same 

waypoint. Since the size of this step change in estimated time is as big as 200 seconds, it 

overwhelms the IM control algorithm and introduces an incorrect system response. It is 

therefore necessary to address this shortcoming before optimization or redesign of the IM 

controller is attempted. 

Another anticipated problem with the IM operation implemented here is that the IM command 

frequency increases as the aircraft comes closer to the reference waypoint. Additional filtering 

of the TTG estimates as well as optimization of the IM controller will alleviate this problem. 

However, more detailed Concepts of Operations (ConOps) need to be developed in order to 

proceed with such a design. 

No procedure for aborting the IM operation was installed. Such a step is required before 

proceeding with additional testing. Once the “Required” IM performance is violated, the aircraft 

must exit the IM stream. 

Due to computational shortcomings and limited schedule, a relatively small set of simulation 

runs were completed and thus many of the relevant statistics are sparse. However, the trends 

indicate that the IM operation can accurately deliver a stream of aircraft using moderate speed 

variations. The results indicated that over 90% of the aircraft arrived within ± 10 seconds of the 

commanded spacing of 50 seconds. 

The simulation setup provides a suitable research platform to evaluate many aspects of IM 

operations. Aside from evaluating the sensitivity of the operation to various uncertainties, the 

tool has shown the complexities associated with different time of travel estimates and their 

impact on performing IM. 

12 Conclusions 

12.1 Observations 

From the outset, this effort planned to use the existing RNP concept as a baseline for extending 

the construct to encompass several operations involving constraints that are not fixed relative to 

the Earth’s surface (‘static’ RNP) but rather are relative to the position of other aircraft 

(‘dynamic’ RNP). Basing these dynamic RNP constructs on the existing RNP formulation 

provided a useful starting point and encouraged casting the problems in generally recognized 

terms, such as TLS and containment. 

An attempt was made to follow the RNP MASPS as a template to document these concepts. The 

MASPS document outline was adequate for capturing, explaining, and organizing requirements, 

but it was not well suited to the task of providing a clear description and analysis of new and 

unfamiliar air traffic management concepts. Initially, the study team sought a unified construct 

capable of encompassing a wide range of multi-vehicle operations, but the team was unable to 

formulate such a generic construct for defining containment and integrity without defining the 

nature of the operation. Therefore, the study was refocused on two specific types of two-vehicle 

operation: RSSP and RIMP. 
183 

The extension of the static RNP construct to the 4D dynamic operations described in the 

preceeding sections uncovered a number of complications. The first difficulty was the lack of 

well developed existing constructs for vertical and longitudinal RNP. Though the existing RNP 

standards include vertical and longitudinal performance measures, they do not contain fully 

developed constructs for containment and integrity as has been developed for the lateral case. At 

a minimum, the longitudinal construct was required for both operations examined in this work. 

To complete the RSSP construct, a detailed vertical construct will be required. While extending 

the lateral containment and integrity constructs to the longitudinal dimension was difficult, the 

issues associated with defining a performance standard based on the interaction of two 

independent vehicles further compounded the problem. Finally, extending the analysis beyond 

the present state of the vehicle to encompass key metrics that exist only as predicted future 

quantities added another layer of complexity. Trying to properly account for external (not 

intrinsic to the aircraft system) error sources, such as the impact of unquantified wind 

prediction/estimation errors on trajectory predictions, proved particularly elusive. 

Trying to define generic constructs not coupled to a specific set of procedures and algorithms led 

to unresolved issues for both constructs examined. Some aspects of the self-separation 

performance were found to depend on how and when trajectory adjustments are made (e.g. a few 

discrete maneuvers vs. continuous adjustment). Similarly for the spacing problem, the spacing 

performance was found to likely depend on the tuning of the speed adjustment algorithm (e.g., 

whether predicted errors are nulled early or late; whether speed changes are continuous or 

discrete). Issues also arose as to how mixed-equipage operations would be addressed in a 

performance standard, as well as transitions into and out of self-separation and intervalmanagement operations. 

12.2 Next Steps 

Before the level of safety can be analyzed for either type of operation, each construct needs to 

have a detailed ConOps developed. The ConOps would include procedures, operational 

constraints, and mitigation strategies for non-normal conditions. 

The complete RSSP concept needs to address the vertical dimension. The containment construct 

for RSSP should capture the nonlinearities of the problem and address non-circular exclusion 

zones. The safety case for RSSP will require a much deeper understanding of the impact of 

traffic density/complexity and trajectory predictability and stability on the achieved level of 

safety. Extensive Monte Carlo testing will be required to generate a quantatative understanding 

of these factors. Extension of RSSP concepts to constraints other than air vehicles (e.g. weather 

cells, prohibited airspace, volcanic ash clouds, etc.) is expected to be relatively straightforward. 

For the RIMP construct, one or more representative interval management algorithms (including 

both ETA prediction and speed profile management) need to be employed to permit quantatative 

analysis of performance. The authors have determined that such analysis can only be performed 

using simulation-based techniques. A basic capability has been constructed using a simple 

interval control algorithm. 
184 

The following recommendations for future work aim at improving the performance and the 

capabilities of th RIMP simulator to support expanded research: 

1- Update Matlab/Simulink code to support faster simulation and more robust operation 

needed for large sample Monte Carlo analysis. The modification will enable the 

optimized model to run without the need for re-compiling. 

2- Upgrade the calculation of Estimated Time Enroute and Total Time-to-Go to include the 

speed changes performed by the ownship RIMP controller. The current algorithms 

assume that the aircraft will continually try to achieve the planned speed. Thus, during 

RIMP operation, the RIMP speed change introduces a momentary error in calculation of 

ETE, which is fed back to the RIMP controller. 

3- Upgrade the calculation of ETE and TTG to compensate for constant Mach and 

maximum dynamic pressure descent profiles. The current algorithms ignore this specific 

behavior. During longer segments of the flight plan, this deficiency can result in an 

apparent spacing error in excess of 100 seconds. 

4- Add appropriate logic so that each aircraft can monitor its own RIMP progress and when 

necessary to abort the RIMP operation. This feature addresses the required performance 

construct associated with RIMP and will add considerable fidelity to the IM stream 

simulation. It will be used to assess the behavior of IM streams in the extreme cases as 

well in mixed operations where self-spacing aircraft operate along with conventionally 

controlled aircraft. 

5- Repeat Monte Carlo simulation runs with a large sample of at least 10,000 runs or until 

the histogram plots indicate appropriate distribution. 

6- Implement alternate ETE calculations and Interval Management control schemes. The 

current model is a simplified approach developed by this team for this effort. Other IM 

concepts have addressed various aspects of this operation and may prove to be more 

effective or have other desirable characteristics. This will enable comparative analysis. 

7- Implement IM turn capability and add associated control logic. The IM-turn uses the 

ability to lengthen or shorten a specific leg of the flight plan in order to accommodate 

precise spacing of the aircraft. This approach is expected to significantly enhance the 

speed-change-only IM operation. In particular during the final segments of IM operation 

prior to arriving at the reference waypoint, the IM turn may completely replace speed 

change. Further analysis is needed to refine the procedures, algorithms, and performance 

metrics for this operation. 
185 

Acknowledgement 

The team wishes to thank David Wing, the NASA Technical Monitor, for sponsoring this 

research effort as well as the many subject matter experts who provide valuable feedback and 

guidance: Terrance Abbott, Bryan Barmore, Brian Baxley, Joel Klooster, and Mike Cramer, 

References 

1. Barmore, Bryan E., Abbott, T.S., Capron, W.R., and Baxley, B.T.: “Results for Airborne 

Precision Spacing along Continuous Descent Arrivals,” AIAA ATIO 2008. 

2. Joint Planning and Development Office: “Concept of Operation for the Next Generation Air 

Transportation System”, Version 2.0, 13 June 2007 

3. RTCA DO-236B: “Minimum Aviation System Performance Standards: Required Navigation 

Performance for Area Navigation” RTCA, Incorporated 

4. Hull, James, Barmore, B., and Abbott, T.: “Technology-Enabled Airborne Spacing and 

Merging,” IEEE 2004. 

5. Abbott, T.S.: “A Trajectory Algorithm to Support En Route and Terminal Area Self-Spacing 

Concepts,” NASA/CR-2007-214899. 

6. Wing D.: A Potentially Useful Role for Airborne Separation in 4D-Trajectory ATM 

Operations. 5th AIAA Aviation Technology, Integration and Operations Conference, 

Arlington, VA, AIAA-2005-7336, 2005. 

7. Wing D.: Performance Basis For Airborne Separation. 26th International Congress of the 

Aeronautical Sciences, Anchorage, Alaska, 14-19 September 2008. 

8. RTCA: Minimum Aviation System Performance Standards: Required Navigation 

Performance for Area Navigation. DO-236B, Washington, DC, October 2003. 

9. Required Self Separation Performance Version 1.1, report submitted under Contract 

#NNL08AA15B, NASA Task Order #NNL08AB13T on April 13th 2009. 

10. Required Self Separation Performance Accuracy Modeling Version 2.1.1, report submitted 

under Contract #NNL08AA15B, NASA Task Order #NNL08AB13T on May 28th 2010. 

11. RTCA: Minimum Aviation System Performance Standards For Automatic Dependent 

Surveillance Broadcast (ADS-B). DO-242A, Washington, DC, June 25, 2002. 

12. “Required Interval Management Performance Modeling, Version 3.1” May 2010 

13. “Required Interval Management Performance Study Plan, Version 1.0” March 2010 

14. Barmore, Bryan; Abbott, Terence; Krishnamurthy, Karthik, “Airborne-Managed Spacing in 

Multiple Arrival Streams”, 2004, NASA 20040110959 

15. Barmore, B, “Airborne Precision Spacing: A Trajectory-based Approach to Improve 

Terminal Area Operations”, 2006, NASA 20080013631 
186 

Acronyms 

4D 4 Dimensional 

A/C Aircraft 

ADS-B Automatic Dependent Surveillance - Broadcast 

ADS-R Automatic Dependent Surveillance - Rebroadcast 

ANSP Air Navigation Service Provider 

APS Airborne Precision Spacing 

ASAS Airborne Separation Asssurance/Assistance System 

ATM Air Traffic Management 

ATS Air Traffic Services 

CAS Calibrated Air Speed 

CNS Communication, Navigation, Surveillance 

ConOps Concept of Operations 

CSE Computed Spacing Error 

DOF Degrees Of Freedom 

EPE Estimated Position Error 

ETA Estimated Time of Arrival 

ETAE Estimated Time Of Arrival Error 

ETB Estimated Time Bias 

ETE Estimated Time Error 

ETTG Estimated Time To Go 

FAA Federal Aviation Administration 

FAF Final Approach Fix 

FIS-B Flight Information Service - Broadcast 

FMS Flight Management System 

FTE Flight Technical Error 

GNSS Global Navigation Surveillance System 

GPS Global Positioning System 

ICAO International Civil Aviation Organization 

INS Inertial Navigation System 

JPDO Joint Planning and Development Office 

MASPS Minimum Aviation System Performance Standards 

NAS National Airspace System 

NASA National Aeronautics and Space Administration 

PCA Point of Closest Approach 

RIMP Required Interval Management Performance 

RNAV Area Navigation 

RNP Required Navigation Performance 

RSSP Required Self Separation Performance 

RTCA Radio Technical Commission for Aeronautics 

SBL Separation Boundary Lateral 

SBV Separation Boundary Vertical 

SCA Spacing Control Authority 

SDE Spacing Definition Error 

SEE Spacing Estimation Error 
187 

SM Separation Management 

STAR Standard Terminal Arrival Route 

TBO Trajectory Based Operation 

TCP Trajectory Change Point 

TIS-B Traffic Information Service - Broadcast 

TLS Target Level of Safety 

TOAC Time Of Arrival Control 

TSE Total Spacing Error 

TTG Time To Go 

TTGE Time To Go Error 

UAT Universal Access Transceiver 

UTC Universal Coordinated Time 

VNAV Vertical Navigation 
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Contractor Report

 4. TITLE AND SUBTITLE

4D Dynamic Required Navigation Performance Final Report

5a. CONTRACT NUMBER

NNL08AA15B

 6. AUTHOR(S)

Finkelsztein, Daniel M.; Sturdy, James L.; Alaverdi, Omeed; Hochwarth, 

Joachim K.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

Washington, DC 20546-0001

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NASA

13. SUPPLEMENTARY NOTES

Langley Technical Monitor: David J. Wing

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Unclassified - Unlimited

Subject Category 03

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

New advanced four dimensional trajectory (4DT) procedures under consideration for the Next Generation Air Transportation System (NextGen) require an 

aircraft to precisely navigate relative to a moving reference such as another aircraft. Examples are Self-Separation for enroute operations and Interval 

Management for in-trail and merging operations. The current construct of Required Navigation Performance (RNP), defined for fixed-reference-frame 

navigation, is not sufficiently specified to be applicable to defining performance levels of such air-to-air procedures. An extension of RNP to air-to-air 

navigation would enable these advanced procedures to be implemented with a specified level of performance. The objective of this research effort was to 

propose new 4D Dynamic RNP constructs that account for the dynamic spatial and temporal nature of Interval Management and Self-Separation, develop 

mathematical models of the Dynamic RNP constructs, "Required Self-Separation Performance" and "Required Interval Management Performance," and to 

analyze the performance characteristics of these air-to-air procedures using the newly developed models. This final report summarizes the activities led by 

Raytheon, in collaboration with GE Aviation and SAIC, and presents the results from this research effort to expand the RNP concept to a dynamic 4D frame of 

reference.

15. SUBJECT TERMS

Interval management; Performance based operations; RNP; Self-separation

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188

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