National Aeronautics and

Space Administration

IS04

George C. Marshall Space Flight Center

Marshall Space Flight Center, Alabama

35812
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NASA/TM—2005–213848

A “Kane’s Dynamics” Model for the 

Active Rack Isolation System

J.K. Rupert

Dynetics, Inc., Huntsville, Alabama

R.D. Hampton

United States Military Academy, West Point, New York

G.S. Beech

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

February 2005

National Aeronautics and

Space Administration

Marshall Space Flight Center • MSFC, Alabama 35812

Part Three: Addition of Umbilicals to the Nonlinear Model
ii

Available from:

NASA Center for AeroSpace Information National Technical Information Service

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076–1320 Springfield, VA 22161

301–621–0390 703–487–4650

TRADEMARKS

Trade names and trademarks are used in this report for identification only. This usage does not constitute an official 

endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
iii

TABLE OF CONTENTS

1. INTRODUCTION ........................................................................................................................ 1

2. COORDINATE SYSTEMS .......................................................................................................... 3

3. GENERALIZED COORDINATES AND GENERALIZED SPEEDS ........................................ 5

4. ANGULAR VELOCITIES OF REFERENCE FRAMES AND RIGID BODIES ....................... 6

5. GENERALIZED ACTIVE FORCE CONTRIBUTIONS DUE TO THE UMBILICALS .......... 8

5.1 Generalized Active Force Equations ..................................................................................... 8

5.2 Partial Velocities S

r

F

v U j 1 ........................................................................................................ 9

5.3 Partial Angular Velocities S r 1 1 F ω ......................................................................................... 11

5.4 General Form for Umbilical Forces and Moments ................................................................ 11

5.5 Umbilical Elongations and Elongation Rates ........................................................................ 12

5.6 Umbilical Angles of Twist and Twist Rates ........................................................................... 14

6. IMPLEMENTATION IN AUTOLEV .......................................................................................... 17

7. MODEL VERIFICATION ............................................................................................................ 18

7.1 Onboard Impulsive-Disturbance Force, No Damping ........................................................... 19

7.2 Onboard Sinusoidal-Disturbance Force, With Damping ....................................................... 19

7.3 Onboard Impulsive-Disturbance Moment, No Damping ...................................................... 21

7.4 Onboard Disturbance Moment, With Damping ..................................................................... 22

7.5 Off-Board Translational Disturbance .................................................................................... 24

7.6 Off-Board Rotational Disturbance ......................................................................................... 27

7.7 Comparisons Using Nondiagonal Stiffness and Damping Matrices ..................................... 29

8. CONCLUSION ............................................................................................................................ 30

REFERENCES .................................................................................................................................. 31
iv

LIST OF FIGURES

1. Detailed diagram of the umbilical assembly ...................................................................... 1

2. Kinematic diagram, including the ith actuator assembly and the umbilical ...................... 3

3. CAD-based technique for determining umbilical attachment locations ............................ 17

4. Translational single-degree-of-freedom truth model ......................................................... 18

5. Translational displacement due to onboard impulsive-disturbance force .......................... 19

6. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 1 1 and ........................................................................................................ 20

7. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 5 1 and ........................................................................................................ 20

8. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 1 0 and .25 ................................................................................................... 21

9. Rotational displacement due to onboard impulsive-disturbance moment ......................... 22

10. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 1 1 and ........................................................................................................ 23

11. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 5 1 and ........................................................................................................ 23

12. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 1 0 and .25 ................................................................................................... 24

13. Translational single-degree-of-freedom truth model, with base motion ............................ 25
LIST OF FIGURES (Continued)

14. Translational displacement due to off-board sinusoidal position disturbance 

with ω

ω

ζ

n

= = 0 1. . and 0 707 ............................................................................................. 25

15. Translational displacement due to off-board sinusoidal position disturbance 

with 

ω

ω

ζ

n

= = 1 0 and .25 ................................................................................................... 26

16. Translational displacement due to off-board sinusoidal position disturbance 

with 

ω

ω

ζ

n

= = 10 and . 0 707 .............................................................................................. 26

17. Rotational displacement due to off-board sinusoidal rotation disturbance 

with ω

ω

ζ

n

= = 10 and . 0 707 .............................................................................................. 27

18. Rotational displacement due to off-board sinusoidal rotation disturbance 

with ω

ω

ζ

n

= = 0 1. . and 0 707 ............................................................................................. 28

19. Rotational displacement due to off-board sinusoidal rotation disturbance 

with ω

ω

ζ

n

= = 1 0 and .25 ................................................................................................... 28
vi

LIST OF ACRONYMS

ARIS active rack isolation system

CAD computer-aided design

ISPR international standard payload rack

ISS	 International Space Station

TM Technical Memorandum
vii

NOMENCLATURE

x vector of arbitrary quantity x

x first time derivative of arbitrary quantity x

xˆ unit length of arbitrary quantity x



x arbitrary reference frame x

x arbitrary dynamical system x

l x linearized arbitrary quantity x

x* arbitrary rigid-body x center of mass

x

T transpose of matrix x

d

d t ordinary (or total) derivative with respect to time t

∂

∂u partial derivative with respect to scalar u

tr A	 trace of arbitrary matrix A

Scalars

Uppercase

Ai

1 intersection point on ith actuator arm (fig. 2)

Ai

2 point locating ith upper stinger (fig. 2)

Ai

3 point locating ith Lorentz coil (fig. 2)

C system damping matrix: lbf⋅s/ft 

E	 system disturbance input matrix
viii

NOMENCLATURE (Continued)

FCi magnitude of ith Lorentz coil force vector FCi : lbf

Fi point locating ith cross-flexure (fig. 2)

Fr system holonomic generalized active force, for the rth generalized speed: ft⋅lbf

Fr

* system holonomic generalized inertia force, for the rth generalized speed: ft⋅lbf

F

r system nonholonomic generalized active force, for the rth generalized speed: ft⋅lbf

F

r

* system nonholonomic generalized inertia force, for the rth generalized speed: ft⋅lbf

Fu umbilical attachment point at the flotor end

Fuh stator-fixed reference position of Fu

I	 identity matrix

I jk

A A/ *

central inertia scalar of rigid body A, for body-fixed coordinate directions aˆ j

and aˆk : ft⋅lbf⋅s2

K system stiffness matrix: lbf/ft

M system mass matrix: slug, lbf⋅s2/ft

O	 zero matrix

Q rotation matrix from the 

Si

1 coordinate system to the 

Fi

1 coordinate system

Qij ijth element of rotation matrix Q

Qr

B contribution to the system holonomic generalized active force for the rth general-

ized speed, due to rigid body B: ft⋅lbf

Qr

B * ( ) contribution to the system holonomic generalized inertia force for the rth general-

ized speed, due to rigid body B: ft⋅lbf

Si point locating ith lower stinger (fig. 2)

Su umbilical attachment point at the stator end
ix

NOMENCLATURE (Continued)



S total dynamical system (stator, flotor, actuators, umbilical)

XFu, YFu, ZFu geometric lengths: ft

Lowercase

aj

i geometric length for ith actuator assembly: ft 

ci umbilical damping in the ˆ

ξ i

 direction: lbf⋅s/ft 

cj

i cosine of angle qj

 for ith actuator assembly

fj

i geometric length for ith actuator assembly: ft 

fjk

i rotation matrix element 

gi direction cosines for nˆ

φ in the 

F1 coordinate system

ki umbilical stiffness in the ˆ

ξ i

 direction: lbf/ft

lj

i geometric length for ith actuator assembly: ft 

mA mass of arbitrary rigid body A: slug, lbf⋅s2/ft

pi

2 geometric length for ith actuator assembly: ft 

qj

i jth generalized coordinate for ith actuator: rad

rjk rotation matrix element 

sj

i sine of angle qj

 for ith actuator assembly

u j

i jth generalized speed for ith actuator: rad/s

vj

i geometric length for ith actuator assembly: ft 

x0, y0, z0 geometric lengths: ft 
NOMENCLATURE (Continued)

xi umbilical elongation in the ˆ

ξ i

 direction: ft 

xFu, yFu, zFu geometric lengths: ft 

xSu, ySu, zSu geometric lengths: ft 

κi torsional umbilical stiffness about the ˆ

ξ i

 axis: ft⋅lbf/rad

φ flotor angle of twist relative to stator: rad

φi angle-of-twist component in the ˆ

ξ i

 direction: rad

γi torsional umbilical damping about the ˆ

ξ i

 axis: ft⋅lbf⋅s/rad

Vectors

Uppercase

Fb umbilical bias force in the reference position: lbf

FCi force exerted on the ith Lorentz coil by the flotor: lbf

FD unknown disturbance force acting directly on the flotor: lbf

FFi force exerted on the flotor by the ith actuator arm: lbf

FU force exerted on the flotor by the umbilical: lbf

HA/A* angular momentum of arbitrary rigid body A with respect to its mass center A*: 

ft⋅lbf⋅s

M Ai moment exerted on the ith actuator arm through the upper stinger, due to the ith 

pushrod: ft⋅lbf

Mb umbilical bias moment in the reference position: ft⋅lbf

M Ci moment exerted on the ith Lorentz coil by the flotor, with FCi assumed to act 

at the ith cross flexure: ft⋅lbf
xi

NOMENCLATURE (Continued)

MD unknown disturbance moment acting directly on the flotor, with FD assumed to act 

at the flotor mass center: ft⋅lbf

M Fi moment exerted on the flotor by the ith actuator, with FFi assumed to act at the ith 

cross-flexure: ft⋅lbf

Lowercase

AaB acceleration of arbitrary point B, with arbitrary reference frame A assumed 

fixed: ft/s2

aI translational acceleration of stator (due to g-jitter): ft/s2

d system disturbance vector: lbf and ft⋅lbf elements 

i system control current vector: amp

nˆ

φ unit vector in direction of rotation axis for stator-to-flotor rotation φ φ nˆ

q Vector of generalized coordinates: rad

rAB position vector from arbitrary point A to arbitrary point B: ft

u vector of generalized speeds: rad/s

uI vector of independent generalized speeds: rad/s

AvB velocity of arbitrary point B, with arbitrary reference frame A assumed fixed: ft/s

AαB angular acceleration of arbitrary reference frame B with respect to arbitrary refer-

 ence frame A: rad/s2

AωB angular velocity of arbitrary reference frame B with respect to arbitrary reference 

 frame A: rad/s
xii
TECHNICAL MEMORANDUM

A “KANE’S DYNAMICS” MODEL 

FOR THE ACTIVE RACK ISOLATION SYSTEM

PART THREE: ADDITION OF UMBILICALS 

TO THE NONLINEAR MODEL

1. INTRODUCTION

As the only active rack-level isolation system for the International Space Station (ISS), the active 

rack isolation system (ARIS) is the central component of the stationwide strategy to meet ISS isolation requirements.1 It serves to attenuate the various disturbances that unavoidably accompany manned 

space flight. Umbilicals, as shown in figure 1, are to be used in support of many experiments planned 

for space, providing services such as cooling, power, vacuum, and data transmission. ARIS is nominally 

equipped with thirteen umbilicals.1

Vacuum Resource

Safing Power Gaseous Nitrogen

Main Power

High Rate Date

Fire Detection/Maintenance

1553 Bus A

1553 Bus B

Vacuum Exhaust

Moderate Temperature

Cooling Supply

Moderate Temperature

Cooling Return

Video 

Figure 1. Detailed diagram of the umbilical assembly.

Unfortunately, the ARIS umbilicals are not only the primary transmitters of indirect; i.e., offboard, disturbances from ISS to the international standard payload rack (ISPR), they are also typically 

nonlinear, hysteretic, and poorly characterized.2 Since the respective umbilicals are attached at different 

locations to ISS at the bottom plate and to the base of the ISPR at the z-panel, it is desirable to include 

this attachment-point information as part of the ARIS model to improve model fidelity and reduce the 

amount of uncertainty for which the controller must compensate.
In part one of this series, NASA/TM—2001–211063, a high-fidelity, linearized, analytical model 

of ARIS was derived using traditional, hand-calculation methods.3 The model was developed directly, 

using Thomas Kane’s method, without intermediate development of the full nonlinear model.4 Part one 

briefly outlined an approach for incorporating an ARIS umbilical into the model, using diagonalized 

stiffness and damping matrices.

Part two, NASA/TM—2004–213552, presented four computer-based, numerical models of 

ARIS, one of which was purely kinematical, and the remaining three, dynamical.5 These numerical 

models were used collectively to verify and simplify the linearized analytical model developed in 

part one. 

This Technical Memorandum (TM) completes the nonlinear ARIS model of part two, by adding 

multiple Hookean umbilicals with full stiffness and damping matrices. Comparisons against simple 

single-degree-of-freedom truth models indicate that the completed nonlinear model has input responses 

that follow those expected given the laws of physics.

This TM (1) briefly summarizes the existing Kane’s model, (2) describes the addition of an 

arbitrary number of massless umbilicals with specified arbitrary attachment points and arbitrary (parallel) stiffness and dampings, to that model, (3) provides a brief description of AutolevTM in which the 

underlying untethered model was developed, and (4) presents the process by which verification of the 

enhanced model was accomplished along with some verification results. 

To develop the enhanced model, it was assumed that the umbilicals were to be included individually in the model, rather than as a combined, or effective, umbilical. For the sake of simplicity and since 

the range of motion is small, each umbilical is assumed to be Hookean and massless since an algebraic 

state-space model, one that is useful for controller design using state-space-based methods, is desired. 

Also, each is modeled as a three-dimensional (six-by-six, constant) stiffness matrix in parallel with a 

three-dimensional (six-by-six, constant) damping matrix. 

In the verification stage, the model of the tethered ARIS system is compared with two baseline, 

or truth, models. One truth model consists of a translational, one-dimensional, second-order springmass-damper system, and the other consists of a corresponding rotational system.6 The verification 

checks indicate that the proposed Kane’s model, when properly calibrated against experimental data, 

will provide a high-fidelity representation of ARIS for simulation and controller-design purposes. 
2. COORDINATE SYSTEMS

The coordinate systems, generalized coordinates, generalized speeds, and angular velocities used 

for the Kane’s model were presented previously in part one.3,7,8 They are reviewed in this section and in 

sections 3 and 4 for the reader’s convenience. 

With the ISPR (flotor) in the reference position; i.e., centered in its ISS-fixed rattle space, fix 

eight right-handed, orthogonal coordinate systems in the ISPR, one at each of the cross-flexure centers (fig. 2). Let the ith coordinate system have origin Fi

, located at the center of the ith cross flexure, 

(i=1,...,8), with axis directions determined by an orthonormal set of unit vectors, ˆ

f j

i

 (j=1,2,3). The 

overhat indicates unit length, the index, i, corresponds to the ith actuator assembly, and the index, j, 

distinguishes the three vectors. Orient the unit vectors such that ˆ

f

i

2 is along the ith arm, toward the ith 

voice coil; ˆ

f

i

1 is directed parallel to the other segment of the ith arm and toward the upper stinger, which 

is located at Ai

2 ; and ˆ

f

i

3 is in the direction ˆ ˆ f f i i1 2

× along the intersection of the two crosspieces of the 

ith cross flexure. 

Coil

Arm

Flotor

Umbilical

Pushrod

Stator

Rattlespace

Ai

3

Ai

2 Ai

1

Fi

Si

SUj

FUj P*i

i

4

l

Lower Stinger

(angles qi, qi, qi) 4 5 6

i

2

l

i

3

l

i

1

l

Upper Stinger

(angles qi

, qi) 2 3

Cross flexure

(angle qi ) 1

Figure 2. Kinematic diagram, including the ith actuator assembly and the umbilical.
Fix a similar right-handed coordinate system, αˆ j

i

(j=1,2,3), in the arm of each actuator. Locate 

each system, αˆ j

i

, such that it is coincident with the corresponding flotor-fixed coordinate system, ˆ

f j

i

, 

when the flotor is in the reference position. 

Locate eight more arm-fixed coordinate systems, described by aˆ j

i , at the respective upper stingers, points Ai

2 . At the respective lower stingers, points Si

, place eight pushrod-fixed coordinate systems 

pˆ

j

i , and eight stator-fixed (ISS-fixed) coordinate systems, sˆji . Orient these 24 coordinate systems such 

that when the stingers are relaxed; i.e., with the flotor in the reference position, the coordinate directions, 

aˆ j

i , pˆ

j

i , and sˆji , are coaligned for the ith actuator, with pˆi

2

 (along with aˆi

2 and sˆi2 , in the reference 

position) directed from Si towards Ai

2 . Directions pˆi

1

 and pˆi

3

 (in the reference position) are determined 

by fixed rotation matrices relating the aˆ j

i to the αˆ j

i

 coordinate systems.

Finally, define a primary, central, flotor-fixed, reference coordinate system with coordinate directions, ˆ

f j . All other flotor-fixed coordinate systems, ˆ

f j

i

, are assumed capable of being referenced; e.g., 

by known direction cosine angles, to this system.

Define a stator-fixed coordinate system, ˆξ i

j

, that is associated with the jth umbilical. Define the 

rotation matrix from the stator-fixed coordinate system, sˆ

i

1 , to the new coordinate system, ˆ

ξ i

j

, by 

ˆ

ˆ

ˆ

ˆ

ˆ

ξ

ξ

ξ

1

2

3

1

1

j

j

j

j

s

s

































= R

  2

1

3

1

ˆ

 ,

s





























(1)

where

R 

j =

j j j

j j j

j j j

r r r

r r r

r r r

11 12 13

21 22 23

31 32 33

























 . (2)

Note that R j is a constant matrix, and it could differ for different umbilical circumstances. 
3. GENERALIZED COORDINATES AND GENERALIZED SPEEDS

Let the αˆ j

i

 coordinate system rotate, relative to the ˆ

f j

i

 coordinate system, through the positive 

angle, qi

1, about the ˆ

f

i

3 axis against the cross-flexure stiffness, ki

1. Similarly, let the orientation of the 

aˆ j

i

 coordinate system, relative to the pˆ

j

i coordinate system, be described by consecutive positive rotations—qi

2 about the pˆi

1

 axis and qi

3 about the moved 3 axis. Also, let the orientation of the pˆ

j

i coordinate system, relative to the sˆj

i coordinate system, be described by consecutive positive rotations—qi

4

about the sˆi

3 axis, qi

5 about the moved 2 axis, and qi6 about the moved 1 axis. The six generalized 

speeds are defined as the time rates of change of the respective generalized coordinates:

u q i j j

i

j

i = =  ( 1 8 , ,   ; , = 1 6, ) . (3)

Finally, let c s j

i

j

i and represent the cosines and sines of the respective angles, qji .
4. ANGULAR VELOCITIES OF REFERENCE FRAMES AND RIGID BODIES

Designate the reference frames corresponding to the stator, the ith pushrod, the ith arm, and the 

flotor, as the symbols     S P, i i , A F , and , respectively. Let  S F i i and represent, respectively, the coordinate 

systems in  S F and , respectively defined by s s ˆ ˆ sˆ i i i T

1 2 3 

  and ˆ ˆ ˆ f f f i i i T

1 2 3



  . Two intermediate 

reference frames were introduced previously to permit describing the angular velocity of each pushrod 

relative to the stator. Designate those intermediate frames corresponding to the ith actuator assembly as  

R Q i i and . Another intermediate reference frame was previously introduced between frames  P A i i and ;

designate that frame as Ti . Finally, designate the stator-fixed reference frame corresponding to the jth 

umbilical as Ξj .

Let each intermediate reference frame have a frame-fixed, dextral set of unit vectors. Indicate 

the unit vectors for each of these frame-fixed coordinate systems by using the corresponding lowercase 

letter; i.e., rˆj

i corresponding to 

Ri, etc. The following gives the expressions for the angular velocities 

of the various reference frames and rigid bodies of S :

F A i i

iω i = u f 1 3

ˆ , (4)

P T i i i i ω = u p 2 1

ˆ , (5)

T A i i

i i ω = u t 3 3

ˆ , (6)

S R i i i i ω = u s 4 3

ˆ , (7)

R Q i i i i ω = u r 5 2

ˆ , (8)

and

Q P i i i i ω = u q6 1

ˆ . (9)

Using the addition theorem for angular velocities, the angular velocities of the rigid bodies 

of S are 

S A i i i i i i i i i i i ω = + u p 2 u t + + u s u r + u q 1 3 3 4 3 5 2 6 ˆ ˆ ˆ ˆ ˆ

1

i

 , (10)
S P i i i i i i i i ω = + u s 4 u r + u q 3 5 2 6 1 ˆ ˆ ˆ , (11)

and

S F i i i i i i i i ω = + u s 4 u r + u q 3 5 2 6 1 ˆ ˆ ˆ . (12)
5. GENERALIZED ACTIVE FORCE CONTRIBUTIONS DUE TO THE UMBILICALS

5.1 Generalized Active Force Equations

Equations of motion developed using Kane’s method consist of contributions called generalized 

active forces (GAFs), related to system forces and moments, and contributions called generalized inertia 

forces (GIFs), related to the time derivatives of linear and angular momenta.4 For ARIS, the development of the GAF and GIF contributions due to the actuator arms and pushrods, the flotor, and a single 

umbilical with diagonal stiffness and damping matrices is detailed in part one.3 Because the umbilicals 

are assumed to be massless, they make no contributions to the GIFs, and thus, they contribute only to 

the GAFs.

In previous treatments (part one of this series and a paper by Rupert and Hampton), the umbilical 

stiffness and damping matrices were assumed to be diagonal with no coupling between translation and 

rotation, or between axes.3,8 However, coupling will always exist between translation and rotation and 

typically between axes as well.2,9 In other words, for an arbitrary set of orthonormal axes, one expects 

that the six-by-six umbilical stiffness and damping matrices will be full. Consequently, full matrices are 

assumed in the following treatment.

Designate the rth partial velocity of the jth umbilical’s flotor-attachment point (with reference 

frame S assumed to be fixed, for purposes of differentiation) as S r

F

v1 Uj and the rth partial angular 

velocity of the flotor relative to the stator as S r

F ω . Designate the force and moment that the jth umbilical exerts on the flotor as FUj

 and MUj, respectively, where the force is assumed to act on the flotor at 

the jth umbilical attachment point FUj. Let all direct disturbance forces and moments that are exerted 

on the flotor—such as those due to air currents, to direct contact by astronauts, or to moving parts on 

flotor-mounted experiments—be designated collectively by the respective symbols FD and MD. Finally, 

let –FCi and –MCi represent, respectively, the force and moment exerted by the ith Lorentz coil, located 

at Ai

3, on the flotor, where

F F a C C i i = i ˆ , 1 (13) 

assumed to act at point Fi and

M r F F l l a C F A C C i i i i i

i

= ×3 i = − i 1 4 + 3 ( )ˆ . (14)

Then, assuming a nominal complement of 13 umbilicals, the GAF contribution due to the flotor, for 

the rth generalized speed, can be written as
Q v r F M F S r

F U S r

F U

j

S r U F j j j := ⋅ + ⋅ 



 



 + ⋅

=

∑ 1

1

13

ω ω M k q f

v F v

D i i i

i

S rF D S rF

+

















+ ⋅ −

=

∑ 1 1 3

1

8

1 1

ˆ

* i i F M C S i r

F C

i

( ⋅ + ⋅ ) =

∑ 1

1

8

ω . (15)

The jth umbilical affects only the first two terms, with the contribution

j rF S r

F U S r

F U Q v Uj F M j j := ⋅ + ⋅ . 1 ω (16) 

These two terms will be treated in sections 5.2 and 5.3.

5.2 Partial Velocities S

r

F

v U j 1

Represent by rAB the position vector from arbitrary point A to arbitrary point B, and define 

the following position vectors using the indicated scalars:

r l a l a

F A i i i i i

i

2

1 2 2 1 = + ˆ ˆ , (17)

r l p

S A i i i

i

2

3 2 = ˆ , (18)

r p p

S P i i i i*

= ˆ , 2 2 (19)

r a a a a

A A i i i i i

2 i

1 1 2 2

*

= + ˆ ˆ , (20) 

and

r f f f f f f F F i i i i i i i * ˆ ˆ ˆ = + + . 1 1 2 2 3 3 (21)

Represent the position vector from the flotor center of mass to the jth umbilical’s flotor-attachment 

point as 

r X f Y f Z f F F

F

j

F

j

F

j U j

U U U

* ˆ ˆ ˆ = + + 1

1

2

1

3

1 (22)

for appropriately defined measure numbers. Then, referring to figure 2, the position vector from 

the first-lower-stinger attachment point to the flotor center of mass is seen to be 

r l p l a l a f f f f S F1 3

1

2

1

1

1

2

1

2

1

1

1

1

1

1

1

2

1 *

ˆ ˆ ˆ ˆ ˆ = − − + + 2

1

3

1

3

1

+ f fˆ . (23)
10

Since

r r r S FU S F F F 1 j = + 1 U j

* *

 , (24)

the rth partial velocity of the flotor-attachment point FUjis6

S r

F S r

F

r

S F F F v v

u

r 1 U j = + 1 1 1 U j

∂

∂

× ( ) 



 





* *

ω . (25)

The position vectors rS F1 ∗ and rF FU j

∗

, and the angular velocity vector S1ωF1, can be expressed 

in terms of the coordinates, qj

1 (j=1,...,6), and the associated generalized speeds of actuator number 1. 

This means that the velocity of each attachment point, FUj, can be expressed in terms of these same 

coordinates and generalized speeds so that the partial velocities defined in equation (25) are nonzero 

for only those six generalized speeds; i.e., for r=1,...,6. 

Expanding equation (25) in terms of the jth stator-fixed coordinate system, ˆξ i

j

, (associated with 

the jth umbilical) and collecting terms, the rth partial velocity for FUj can be expressed as

S

r

F

ri

U

i

i

j

v v U j 1 j

1

3

=

=

∑ ˆ

ξ , (26)

where the scalars, vri

Uj

, are simply the measure numbers in analytical (algebraic) form. In matrix form, 

the six nonzero partial velocities for the jth umbilical’s flotor-attachment point are 

S F

S F

S F

S F

S F

S F

v

v

v

v

v

v

U j

U j

U j

U j

U j

U j

1

1

1

1

1

1

1

2

3

4

5

6





















































=

=

∑v

v

i

U

i

j

i

j

1

1

3

2

ˆ

ξ

i

U

i

j

i

i

U

i

j

i

i

U

i

j

i

j

j

j

v

v

v

ˆ

ˆ

ˆ

ξ

ξ

ξ

=

=

=

∑

∑

∑

1

3

3

1

3

4

1

3

5i

U

i

j

i

i

U

i

j

i

j

j v

ˆ

ˆ

ξ

ξ

=

=

∑

∑

































1

3

6

1

3





























































=

v v v

U U j j

11 12 13

21 22 23

31 32 33

41 42

U

U U U

U U U

U

j

j j j

j j j

j

v v v

v v v

v v

U U

U U U

U U U

j j

j j j

j j j

v

v v v

v v v

43

51 52 53

61 62 63





































































ˆ

ˆ

ˆ

ξ

ξ

ξ

1

2

3

j

j

j

















 . (27)
11

5.3 Partial Angular Velocities S r 1 1 F ω

From equation (12), the angular velocity of the flotor relative to the stator can be expressed as

S F 1 1 u p 2 u t u s u r u q

1

1

1

3

1

3

1

4

1

3

1

5

1

2

1

6

1 ω = + ˆ ˆ + + ˆ ˆ + ˆ

1

1

1

1

3

1 − u fˆ . (28)

Following the procedure and using a notation analogous to that of the previous subsection, the nonzero 

partial angular velocities can be expressed in the matrix form,

S F

S F

S F

S F

S F

S F

1

1

1

1

1

1

1

2

3

4

5

6

ω

ω

ω

ω

ω

ω













































=

=

=

∑

∑

ω ξ

ω ξ

ω

1

1

3

2

1

3

3

i

U

i

j

i

i

U

i

j

i

i

U

j

j

ˆ

ˆ

j

j

j

i

j

i

i

U

i

j

i

i

U

i

j

i

i

U

ˆ

ˆ

ˆ

ξ

ω ξ

ω ξ

ω

=

=

=

∑

∑

∑

1

3

4

1

3

5

1

3

6

j

i

j

i

ˆ

ξ

=

∑

































































1

3





























=

ω ω ω

ω ω ω

11 12 13

21 22 2

U U U

U U

j j j

j j

3

31 32 33

41 42 43

51 52

U

U U U

U U U

U U

j

j j j

j j j

j

ω ω ω

ω ω ω

ω ω j j

j j j

U

U U U

ω

ω ω ω

53

61 62 63





















































































ˆ

ˆ

ˆ

.

ξ

ξ

ξ

1

2

3

j

j

j

(29)

5.4 General Form for Umbilical Forces and Moments

With full stiffness and damping matrices, the jth-umbilical force, FUj , and moment, MU j, will 

each be expressed in terms of umbilical-elongation-, elongation-rate-, angle-of-twist-, and twist-ratecomponent measure numbers. In vector form, the umbilical elongation is rF F Uhj U j, where FUhj

 designates the stator-fixed location of the umbilical attachment point, FUj

, in the reference position. Let φ φ nˆ

represent, in vector form, the rotation of the flotor, relative to the stator, from the reference position. 

The term nˆ

φ is in the positive direction of the rotation axis, and φ is the positive angle of twist about 

that axis. Then, for the jth umbilical, the elongations; i.e., the elongation measure numbers, are

x r i

j F F

i

j = ⋅ Uhj U j ˆ

ξ , (30)

the elongation rates are x

i

j , the angles of twist are
12

φ φ ξ i φ

j

i

j = ⋅ nˆ ˆ , (31)

and the twist rates are 

φi

j

.

Let the bias force and moment exerted on the flotor by the jth umbilical be F M b

U

b

j Uj and ,

respectively. Next, for i=1,2,3, define

F F bi

U

b

U

i

j j j = ⋅ ˆ

ξ , (32)

M M bi

U

b

U

i

j j j = ⋅ ˆ

ξ , (33)

F F i

U U

i

j j = ⋅ j ˆ

ξ , (34)

and

M M i

U U

i

j j = ⋅ j ˆ

ξ . (35)

Then, in matrix form, the combined force and moment equations are as follows:

F

F

F

M

M

M

U

U

U

U

U

U

j

j

j

j

j

j

1

2

3

1

2

3





















































=

















 k k

k k

U U

U U

j j

j j

11 16

61 66



  























K

x

x

x

U

U

U

U

U

U

j

j

j

j

j

j

   

1

2

3

1

2

3

φ

φ

φ





































+

c c

c c

U U

U

j j

j

11 16

61 6



  

 6

1

2

3

U 1

U

U

U

U

j

j

j

j

C

x

x

x

























   









φ j

j

j

U

U

b

U

F





φ

φ

2

3

1





















































+

j

j

j

j

j

j

F

F

M

M

M

b

U

b

U

b

U

b

U

b

U

2

3

1

2

3





















































 , (36)

where the stiffness and damping elements, obtained by measurements or other means, and the stiffness 

and damping matrices are indicated by the designations k, c, K, and C, respectively. The stiffness and 

damping element superscripts and subscripts have their obvious meanings. To use the left-hand elements 

of equation (36) in equation (16), these elongations and angles of twist, along with their rates, must now 

be reexpressed in terms of the system coordinates and generalized speeds.

5.5 Umbilical Elongations and Elongation Rates

Recall equation (30), for i=1,2,3; but

r r r r r F F Uh S F F F S S S F j U j = + 1 1 1 U j − − 1 U j U j Uhj , (37)
13

where the right-hand-side terms can be expressed by 

r l p l a l a S F1 1 3

1

2

1

2

1

1

1

1

1

2

1 = − ˆ ˆ − ˆ , (38)

r x f y f z f F F

F

j

F

j

F

j U j

U U U

1

1

1

2

1

3

1

= + + ˆ ˆ ˆ , (39)

r x s y s z s S S

S

j

S

j

S

j U j

U U U

1

1

1

2

1

3

1 = + ˆ ˆ + ˆ , (40)

and

r x s y s z s S F j j j U j Uhj = + + 0 1

1

0 2

1

0 3

1 ˆ ˆ ˆ (41)

for appropriately defined coefficients. 

Using equation (1), equation (37) can now be written in terms of the ˆξ i

j

 coordinate system in linearized form

r x x x F F j j j j j j Uhj U j = + + 1 1 2 2 3 3

ˆ ˆ ˆ ξ ξ ξ , (42)

where

x

x

x

j C y q

j

j

j

j

F

j

1 U

2

3

1 1

1 



























= R

 ⋅

+ − q q z q l q l q q

C z

F

j

j

U 3

1

4

1

5

1

3

1

4

1

1

1

3

1

4

1

2

( ) − + − + + ( )

− F

j

F

j

U U q q 2 x q q q l q q

1

6

1

1

1

3

1

4

1

2

1

3

1

4

1 ( ) + − − ( ) − − + ( )

C l + − ( ) x q + − ( ) y l ( ) q q + + l j

F

j

F

j

U U 3 2

1

5

1

1

1

2

1

6

1

3

1

q6

1





























(43)

for

C x x l x j

F

j

S

j j

U U 1 2

1

0 = − − − , (44)

C y y l l y j

F

j

S

j j

U U 2 3

1

1

1

0 = − + − − , (45)

and

C z z z j

F

j

S

j j

U U 3 0 = − − . (46)

Differentiating equation (43),6
14







x

x

x

y u u j

j

j

j

F

j

1 U

2

3

1

1 



























= R

 ⋅

− 3

1

4

1

5

1

3

1

4

1

1

1

3

1

4

1 ( ) − + − + + ( )

−

u z u l u l u u

z u

F

j

F

j

U

U 2

1

6

1

1

1

3

1

4

1

2

1

3

1

4

1

2

1

( ) + u x − − ( ) u u − u l − + ( ) u u

l

F

j

U

( ) − + − ( )( ) + +









 x u y l u u l u F

j

F

j

U U 5

1

1

1

2

1

6

1

3

1

6

1



















 . (47)

Equations (43) and (47) give the umbilical elongations and elongation rates in terms of the six independent generalized speeds. It is now required to do the same for the umbilical angles and angles of twist.

5.6 Umbilical Angles of Twist and Twist Rates

Express nˆ

φ as 

n g ˆ ˆs g s g ˆ ˆs . φ = + 1 + 1

1

2 2

1

3 3

1 (48)

Define rotation matrix Q by 

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

f

f

f

Q

s

s

s

1

1

2

1

3

1

1

1

2

1

3

1





























= [ ]





























, (49)

and then the linearized three-by-three rotation matrix, lQ, has elements, lQij, defined as

ˆ

ˆ

ˆ

f

f

f

1 q q q q

1

2

1

3

1

1

1

3

1

4

1 1 5





























=

− + + − 1

1

1

3

1

4

1

2

1

6

1

5

1

2

1

6

1

1

1

q q q q q

q q q

− − +

− −





















































ˆ

ˆ

ˆ

 .

s

s

s

1

1

2

1

3

1

(50)

Let the postsuperscript T indicate matrix transposition and let tr lQ represent the trace of lQ. For small 

angles it can be shown that10

φ

0

0

0 1

3 2

3 1

2 1

g g

g g

g g

Q Q

Q

l l

T

l

−

−

−





















= −

( + tr )

1 2/ . (51)
15

Substitution for lQ, as defined in equation (50), into equation (51) and simplification yield

g q 1 2 q

1

6

1 1 = ⋅( ) +

φ , (52)

g q 2 5

1 1 = ⋅ φ , (53)

and

g q 3 1 q q

1

3

1

4

1 1 = − ⋅ − ( ) − φ . (54)

Substituting from equations (52)–(54) into equation (48) and transforming into the sˆ

i

1 coordinate systems by use of l

Q yield the following linearized expression for the flotor rotation:

φ φ n q ˆ = + q sˆ q sˆ q q q sˆ ( ) 2 + − ( ) − − 1

6

1

1

1

5

1

2

1

1

1

3

1

4

1

3

1 . (55)

Use of equations (31) and (48) leads at last to the linearized forms for angular position and rotation rate:

φ

φ

φ

1

2

3

2

1

6

1

5

1

1

j

j

j

j

q q

q

q





























= R

 

+

− 1

3

1

4

1 ( ) − −



























q q 

(56)

and







φ

φ

φ

1

2

3

2

1

6

1

5

1

j

j

j

j

u u

u





























= R

 

+

− − ( ) −



























u u 1 u  1

3

1

4

1

 . (57)

The jth umbilical portion of the GAF contribution due to the flotor for the rth generalized speed 

can now be expressed in terms of the coordinates and independent generalized speeds: 
16

j rF S r

F U S r

F U

r

U

r

r

U

r

Q v F M

v

v

U j j j

j

j

= ⋅ + ⋅ =

=

=

∑

1

1

1

6

2

ω

1

6

3

1

6

1

1

6

2

1

6

3

1

6

∑

∑

∑

∑

=

=

=

=

vr

U

r

r

U

r

r

U

r

r

U

r

j

j

j

j

ω

ω

∑ω





















































































































T

U

U

U

U

U

U

F

F

F

M

M

M

j

j

j

j

j

j

1

2

3

1

2

3 



























 , (58)

where

F

F

F

M

M

M

U

U

U

U

U

U

j

j

j

j

j

j

1

2

3

1

2

3





















































=

















 k k

k k

U U

U U

j j

j j

11 16

61 66



  







































x

x

x

U

U

U

U

U

U

j

j

j

j

j

j

1

2

3

1

2

3

φ

φ

φ























+











c c

c c

U U

U U

j j

j j

11 16

61 66



  

 



































x

x

x

U

U

U

U

U

U

j

j

j

j

j

j

1

2

3

1

2

3

φ

φ

φ











































+

F

F

F

M

b

U

b

U

b

U

b

U

j

j

j

j

1

2

3

1

M

M

b

U

b

U

j

j

2

3





















































 , (59)

and the elongations, elongation rates, angles of twist, and twist rates are as defined in equations (43), 

(47), (56), and (57). The model is now ready for calibration and computer implementation. 
17

6. IMPLEMENTATION IN AUTOLEV

A nonlinear, rigid-body model for the untethered ARIS; i.e., without umbilicals, with full 

actuator dynamics (masses and inertias), was previously developed using OnLine Dynamics’ Autolev, 

a DOS-based interpreter specifically designed to solve dynamics problems using Kane’s method.11

The umbilical contributions developed above (eqs. (58) and (59)) were incorporated into that model 

by appropriately adding those contributions to the untethered-model GAFs. In Autolev, this required 

entering the locations of the umbilical attachment points, the orientations of the stator-fixed coordinate 

systems associated with the umbilicals, and the umbilical forces and moments determined above.

The geometric data required to include the umbilicals were determined from CAD models, such 

as in figure 3. Since umbilical stiffness and damping values were not available, representative estimates 

were entered and varied as needed for the process of model verification (described in section 7). 

Figure 3. CAD-based technique for determining umbilical attachment locations.

After adding necessary terms to the Autolev model of ARIS, the code was compiled in the 

Autolev workspace. Autolev then auto-generated a C-program file containing the equations of motion 

and a numerical integration routine for their solution. Results of executing the C-program were plotted 

using Microsoft Excel.
18

7. MODEL VERIFICATION

Numerous simulations were run to verify the model for special cases. The basic procedure was 

to reduce the actuator stiffness to very small levels and apply selected test forces (moments) through 

(about) the flotor center of mass. Corresponding simulations were compared between the full nonlinear 

model and appropriate (translational or torsional) single-degree-of-freedom truth models. Representative 

comparison checks are described in sections 7.1–7.8. 

For each of the following comparisons, a single umbilical is assumed to be attached between the 

stator and the flotor center of mass. Umbilical biases are assumed to be zero. The umbilical coordinate 

system, ˆξ i, is assumed to be aligned with the flotor-fixed coordinate system, ˆ

f

i , when the flotor is in the 

reference position. Stiffness and damping terms that couple translation and rotation are all assumed to be 

zero. For forces (moments) applied through (about) the flotor mass center and for small-enough actuator 

stiffness, the ARIS system dynamics should approximate those of a simple second-order spring-massdamper system (fig. 4). 

c

k

m

m

F

F

x

kx cx•

k

c 

m

F

x

Translational Stiffness

Translational (Viscous) Damping

Mass

Applied Force

Translational Displacement

Figure 4. Translational single-degree-of-freedom truth model.

The flotor mass is assumed to be 55.6 slugs (1,790 lbm, 811.94 kg), and the central principal 

moments of inertia, 166.8 slug-ft2 (5,371 lbm-ft2, 226.34 kg-m2). The umbilical stiffness matrix is 

assumed to be diagonal with each diagonal element set at 30 (lbf/ft for translational elements and 

lbf-ft/rad for rotational elements). The damping matrix is also assumed diagonal with all diagonal 

terms numerically equal but varied as needed to control the damping ratios, depending on the particular simulation. 
19

7.1 Onboard Impulsive-Disturbance Force, No Damping

For the first test of the ARIS model, an impulsive disturbance force was applied directly to 

the flotor mass center along the ˆξ 2 direction, with zero damping. The disturbance was also applied 

to a simple spring-mass truth model with the same mass and stiffness. The impulse was approximated 

by a rapidly decaying exponential function: 

F Ae dt = − ˆξ , 2 (60)

where d=100 s–1 and A=100 lbf (444.82 N). The two systems had identical sinusoidal responses (fig. 5) 

at the expected natural frequency of 0.7346 rad/s (0.1169 Hz). 

0

– 0.03

– 0.02

– 0.01

0

0.01

0.02

0.03

5 10 15 20 25 30

Time (s)

Position (ft)

ARIS and Truth Model Results

Figure 5. Translational displacement due to onboard impulsive-disturbance force.

7.2 Onboard Sinusoidal-Disturbance Force, With Damping

In a second test, a sinusoidal disturbance force with an amplitude of 3 lbf (13.34 N) and a frequency of ω was applied to the flotor mass center, again in the ˆ

ξ 2

 direction, using various frequency 

ratios, ω/ωn, and damping factors, ζ. As before, the disturbance was also applied to a corresponding 

truth model. The two system responses were identical (figs. 6–8).
20

0

– 0.06

– 0.04

– 0.02

0

0.02

0.04

0.06

5 10 15 20 25

Time (s)

Displacement (ft)

ARIS and Truth Model Results

Figure 6. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 1 1 and .

0

– 0.004

– 0.002

0

0.002

0.004

0.008

0.006

0.01

0.012

5 10 15 20

Time (s)

Displacement (ft)

ARIS and Truth Model Results

Figure 7. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 5 1 and .
21

0

– 0.2

– 0.15

– 0.05

– 0.1

0

0.05

0.15

0.1

0.2

5 10 15 20 25

Time (s)

Displacement (ft)

ARIS and Truth Model Results

Figure 8. Translational displacement due to onboard sinusoidal-disturbance force 

with ω

ω

ζ

n

= = 1 0 and .25.

7.3 Onboard Impulsive-Disturbance Moment, No Damping

To check the ARIS model rotational response, an impulsive disturbance moment was applied 

directly to the flotor at its mass center about the ˆ

ξ 2

 direction with zero damping. The disturbance was 

also applied to a simple rotational spring-mass truth model, with the same inertia and rotational stiffness. 

Again, the impulse was approximated by a rapidly decaying exponential function: 

M Ae dt = − ˆξ , 2 (61)

where d=100 s–1 and A=100 lbf-ft/s (135.58 N-m/s). The two systems had identical sinusoidal responses 

(fig. 9) at the expected natural frequency of 0.4241 rad/s (0.0675 Hz). 
22

0

– 0.02

– 0.005

– 0.015

– 0.01

0

0.005

0.015

0.01

0.02

20 40 60 80 100 120

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 9. Rotational displacement due to onboard impulsive-disturbance moment.

7.4 Onboard Disturbance Moment, With Damping

To test the damped rotational response, a sinusoidal disturbance moment with amplitude 

0.09 lbf-ft/s (0.122 N-m/s) and frequency, ω, was applied at the flotor mass center, about the ˆ

ξ 2 direction, using various frequency ratios, ω/ωn, and damping factors, ζ. The disturbance was also applied 

to a corresponding truth model. Again, the two system responses were identical (figs. 10–12).
23

0

– 0.002

– 0.0005

– 0.0015

– 0.001

0

0.0005

0.0015

0.001

0.002

5 10 15 20 25 30 35 40

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 10. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 1 1 and .

0

– 0.0002

0

– 0.0001

0.0001

0.0003

0.0002

0.0004

5 10 15 20 25 30

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 11. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 5 1 and .
24

0

– 0.008

– 0.002

– 0.004

– 0.006

0

0.006

0.004

0.002

0.008

10 20 30 40 50

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 12. Rotational displacement due to onboard sinusoidal-disturbance moment 

with ω

ω

ζ

n

= = 1 0 and .25.

7.5 Off-Board Translational Disturbance

In normal operation, the flotor will also be subject to off-board disturbances, which are transmitted via the ARIS umbilicals and actuators. With negligible actuator stiffness, the system response should 

match that of a corresponding truth model (fig. 13). A translational sinusoidal disturbance of amplitude 

0.25 in (0.635 cm) was applied to the stator (base) in the ˆξ 2 direction with the ARIS umbilical oriented 

so that its stator attachment point was collinear with the flotor mass center, along the ˆξ 2 direction. (The 

flotor attachment point was assumed to be at the mass center, as before.) Various frequency ratios, ω/ωn, 

and damping factors, ζ, were used with the flotor-to-stator relative displacements compared to those of 

the truth model. The responses of the two systems were identical (figs. 14–16). 
25

M

k

2

k

2 x

Base

C

y

Figure 13. Translational single-degree-of-freedom truth model, with base motion.

0

– 0.00005

0.00005

0

0.0001

0.0002

0.00015

0.00025

5 10 15 20 25 30

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 14. Translational displacement due to off-board sinusoidal position disturbance 

with ω

ω

ζ

n

= = 0 1. . and 0 707.
26

0

– 0.05

– 0.02

– 0.01

– 0.03

– 0.04

0

0.01

0.04

0.02

0.03

0.05

5 10 15 20 25 30

Time (s)

Displacement (ft)

ARIS and Truth Model Results

Figure 15. Translational displacement due to off-board sinusoidal position disturbance 

with ω

ω

ζ

n

= = 1 0 and .25.

0

– 0.04

– 0.02

0.02

0

0.04

0.1

0.08

0.06

0.12

2 4 6 8 10 12 14

Time (s)

Displacement (ft)

ARIS and Truth Model Results

Figure 16. Translational displacement due to off-board sinusoidal position disturbance 

with ω

ω

ζ

n

= = 10 and . 0 707.
27

7.6 Off-Board Rotational Disturbance

A rotational sinusoidal disturbance, of amplitude 0.001 rad (0.0573o) was applied to the stator 

(base) about the ˆξ 2 direction with the ARIS umbilical attached and oriented as in the previous section. 

Various frequency ratios, ω/ωn, and damping factors, ζ, were used with the flotor-to-stator relative rotations compared to those of the truth model. The responses of the two systems were identical (figs. 17–19). 

0

– 0.002

– 0.001

0.001

0

0.002

0.006

0.004

0.003

0.005

5 10 15 20 25

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 17. Rotational displacement due to off-board sinusoidal rotation disturbance

with ω

ω

ζ

n

= = 10 and . 0 707.
28

0

– 0.000015

– 0.00001

– 0.000005

0

0.000015

0.000005

0.00001

10 20 30 40 50 60 70 80 90 100

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 18. Rotational displacement due to off-board sinusoidal rotation disturbance

with ω

ω

ζ

n

= = 0 1. . and 0 707.

0

– 0.0025

– 0.002

– 0.0015

– 0.001

– 0.0005

0

0.0005

0.0025

0.001

0.002

0.0015

10 20 30 40 50 60

Time (s)

Angle (rad)

ARIS and Truth Model Results

Figure 19. Rotational displacement due to off-board sinusoidal rotation disturbance 

with ω

ω

ζ

n

= = 1 0 and .25.
29

7.7 Comparisons Using Nondiagonal Stiffness and Damping Matrices

Suppose now that the stiffness and damping matrices, [ ] K and [ ] C , respectively, used for the 

above simulations are changed by selecting a new coordinate system, ˆξ i

'

, rotated through R 



' from 

the previous coordinate system, ˆ

ξ i

:

ˆ

ˆ

ˆ

ˆ

ˆ

ξ

ξ

ξ

ξ

ξ

1

'

2

'

3

'

'

































= R

 

1

2

ˆ

ξ 3

1

1

2

1

3

1





































= R

 

R 

 '

s

s

s





























 . (62)

Then the upper left and lower right block diagonal three-by-three submatrices, [ ] K11 and [ ] K22 , of the 

(initially diagonal) stiffness matrix, [ ] K , become, respectively,

K K

T

11 11 ' 

  = R 



 

R 

 ' ' (63)

and

K K

T

22 22 '  .  

 = R 



 

R 

 ' ' (64)

Similar relationships hold for the damping matrix, [ ] C , and its corresponding submatrices. If the 

same disturbance force or moment is now applied as in any of the preceding simulations, then the resulting system motion should remain unmodified, which was shown to be the case.12 This indicates that the 

ARIS model correctly handles cross-coupling among axes; i.e., full stiffness and damping submatrices, 

[ ] K11 , [ ] K22 , [ ] C11 , and [ ] C22 . 
30

8. CONCLUSION

This TM has shown how multiple ARIS umbilicals, modeled as massless three-dimensional 

Hookean springs in parallel with three-dimensional viscous dampers, are to be included in the nonlinear 

Kane’s model of ARIS. The umbilical force, moment, partial-velocity, and partial-angular-velocity terms 

were determined for the flotor contributions to the GAFs in Kane’s dynamical equations.

Next, these modifications were incorporated into the ARIS Autolev simulation code. Simplifying assumptions were made to permit the comparison of the Autolev model’s input responses to those 

of simple second-order truth models for selected test inputs. Numerous simulations, involving onboard 

and off-board force and moment disturbances, showed that the ARIS-model responses match those of 

the truth models. This indicates that the proposed model, when properly calibrated against experimental 

data, will provide a high-fidelity representation of ARIS for simulation and controller-design purposes. 
31

REFERENCES

1. Bushnell, G.S.; Anderson, T.M.; Becraft, M.D.; and Jacot, A.D.: “Active Rack Isolation System 

Development for the International Space Station,” AIAA 1997–1203, April 1997.

2. Bushnell, G.S.; and Becraft, M.D.: “Microgravity Performance Flight Characteristics of an International Space Station Active Rack Isolation Prototype System,” IEEE–0–7803–5276–9, September 

1999. 

3. Hampton, R.D.; and Beech, G.S.: “A ‘Kane’s Dynamics’ Model for the Active Rack Isolation 

System,” NASA/TM—2001–2110063, 36 pp., Marshall Space Flight Center, AL, June 2001.

4. Kane, T.R.; and Levinson, D.A.: Dynamics: Theory and Applications, 379 pp., McGraw-Hill, Inc., 

New York, 1985.

5. Beech, G.S.; Hampton, R.D.; and Rupert, J.K.: “A ‘Kane’s Dynamics’ Model for the Active Rack 

Isolation System: Part Two: Nonlinear Model Development Verification and Simplification,” 

NASA/TM—2004–213552, 32 pp., Marshall Space Flight Center, AL, November 2004.

7. Beech, G.S.: A High Fidelity Model for the Active Rack Isolation System, Master’s Thesis, 

69 pp., The University of Alabama in Huntsville, Huntsville, AL, December 2000.

6. Thomson, W.T.: Mechanical Vibrations, 2nd Ed., Prentice-Hall, Inc., New York, 1953.

8. Rupert, J.K.; and Hampton, R.D.: “An Improved Umbilical Model for The Active Rack Isolation 

System,” AIAA 2000–0573, January 2000.

9. Johnson, T.L.; and Tolson, H.R.: “Development of a Simulation Capability for the Space Station 

Active Rack Isolation System,” Specification NASA/CR—1998–206942, Langley Research Center, 

March 1998. 

10. Greenwood, D.T.: Principles of Dynamics, 2nd Ed., Prentice-Hall, Inc., New Jersey, 1988.

11. Kane, T.R.; and Levinson, D.A.: Dynamics Online: Theory and Implementation with AutolevTM, 

408 pp., Sunnyvale, CA, 1996.

12. Rupert, J.K.: An Umbilical Model for the Active Rack Isolation System, Masters Thesis, The University of Alabama in Huntsville, 2001.
32

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Prescribed by ANSI Std. 239-18

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Unclassified Unclassified Unclassified Unlimited

A “Kane’s Dynamics” Model for the Active Rack Isolation System

Part Three: Addition of Umbilicals to the Nonlinear Model

J.K. Rupert,* R.D. Hampton,** and G.S. Beech

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Prepared by the Engineering Systems Department, Engineering Directorate

*Dynetics, Inc., **United States Military Academy

Unclassified-Unlimited

Subject Category 88

Availability: NASA CASI 301–621–0390

In the late 1980s, microgravity researchers began to voice their concern that umbilical-transmitted energy could significantly 

degrade the acceleration environment of microgravity space science experiments onboard manned spacecraft. Since umbilicals are 

necessary for many experiments, control designers began to seek ways to compensate for these “indirect” disturbances.

Hampton, et al., used the Kane’s method to develop a model of the active rack isolation system (ARIS) that includes (1) actuator 

control forces, (2) direct disturbance forces, and (3) indirect, actuator-transmitted disturbances. Their model does not, however, 

include the indirect, umbilical-transmitted disturbances. Since the umbilical stiffnesses are not negligible, these indirect disturbances must be included in the model. Until the umbilicals have been appropriately included, the model will be incomplete.

This Technical Memorandum presents a nonlinear model of ARIS with umbilicals included. Model verification was achieved by 

utilizing two commercial-off-the-shelf software tools. Various forces and moments were applied to the model to yield simulated 

responses of the system. Plots of the simulation results show how various critical points on an ARIS-outfitted international standard payload rack behave under the application of direct disturbances, indirect disturbances, and control forces. Simulations also 

show system response to a variety of initial conditions. 

44

M–1138

February 2005 Technical Memorandum

NASA/TM—2005–213848

ARIS, umbilicals, dynamics, control, math model
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