APPROACHES AND POSSIBLE 

IMPROVEMENTS IN THE AREA 

OF MULTIBODY DYNAMICS MODELING 

[tlASA-CR-l’79227) APPBOACHE S AND POSSIBLE M88-14067 

IHPROVE?lEITS IN THE AREA OF MULTIBODY 

DIHAMICS MODELING (Honeyuell) 45 pCSCL OOA 

Unclas 

G3/02 0103654 

Principal Investigator : 

Dr. K. W. Lips 

Space Systems Engineering 

(SSFCS) 

Prepared for NASA Marshall Space Flight Center 

Under Contract NAS8-34588 by Honeywell Space 

and Strategic Avionics Divison, Clearwater, Florida 

Date: 30 October 1987 
Acknowledgement 

Subcontracting support was 

provided by DYNACS Engineering 

of Clearwater, Florida and is 

reflected primarily in 

Sections 4.4, 5.1 and 5.3. 

Dr. R. Singh served as 

principal investigator. 
ABSTRACT 

This study takes a wide ranging look at issues involved 

in the dynamic modeling of complex, multibodied 

orbiting space systems. Capabilities and limitations of 

two major codes (DISCOS, TREETOPS) are assessed 

and possible extensions to the CONTOPS software are 

outlined. In addition recommendations are made 

concerning the direction future development should take 

in order to achieve higher fidelity, more computationally 

efficient multibody software solutions. 

-1- 
TABLE OF CONTENTS 

1 .o 

2.0 

3.0 

4.0 

5.0 

6.0 

Abst rad 1 

INTRODUCTION 4 

MU LTI BODY DY N AM IC SI MU LATl ON 5 

2.1 Role of Simulation, Analysis Tools 5 

2.2 Nature of the Multibody Problem 5 

2.3 Derivation of Equations 5 

2.5 Evaluation of Representative MuIti body Software 7 

2.6 Symbolic Manipulation, Parallel Processing a 

2.4 Status of Multibody Code Development 6 

ADEQUACY OF STRUCTURAL MODELS 9 

3.1 Basic Mathematical Descriptions for Flexibility 9 

3.2 Specialized Considerations 10 

3.2.1 Energy Dissipation 10 

3.2.2 Mass Flow Deployment 10 

Foreshortening Effect 11 

3.2.3 Reduced Dimension Models and the 

3.2.4 Joint Idealization 13 

OTHER FACTORS IMPACTING ON FIDELITY OF MULTIBODY 

S I MU LATlO N 14 

4.1 Fluid Interactions 14 

4.2 Environmental Loads 15 

4.3 Requirement for Controlled Maneuvers 15 

4.4 Numerical Considerations in the Solution of Mixed 

Differential Algebraic Equations 15 

EXTENSIONS TO THE CONTOPS ALGORITHM 17 

5.1 Generalization of the CONTOPS Hinge 17 

5.2 Implications of Deployment and Flex Foreshortening 21 

5.2.1 Existing CONTOPS Kinematics 22 

5.2.2 Modifications due to Changing Mass 23 

5.2.3 Incorporation of Flex Foreshortening 24 

5.3 A Model Reduction Approach for CONTOPS 24 

DISCUSSION AND CONCLUSIONS 27 
7.0 

TABLE OF CONTENTS (cont'd) 

RECOMMENDATIONS 30 

REFERENCES 32 

APPENDIX: Previous Assessments of Multibody 

For mu lat io ns 

To po I og y : A Co m put e r 0 ri e n t ed Approach " 

ATTACHMENT: "Dynamics of Flexible Bodies in Tree fl&wZ/#uY 

R~ZV 0vk-B 

-3- 
1 .o INTRODUCTION 

Planned space systems (large space antennas, Space-Based-Radar, 

directed energy weapons) may be called upon to provide a precision in 

maneuverability and in pointing which is orders of magnitude beyond that 

possible to date.' While the performance requirements are becoming more 

stringent, the spacecraft themselves are becoming more complex, larger and 

more flexible and consequently are more difficult to model. 

As well, spacecraft configurations are characteristically quite diverse from 

one another. It is for this reason that "generic" multibody codes have been 

developed 2,s to model dynamic behavior and for use in control design. 

Reference 4 provides a good general overview, from a controls viewpoint, of 

the difficulties involved in the analysis of large flexible space structures 

(LFSS). The emphasis here, however, is on fidelity of the multibody 

modeling process and on computational efficiency. The spirit of the 

investigation is to initiate a fresh look at the major codes and to not be 

dedicated to any one methodology a priori. 

The open literature is reviewed to establish the state-of-the-art in formulation 

of multibody dynamic equations of motion and in multibody solution 

methodologies. Fundamental issues relate to the system topology are 

examined. Pros and cons of two existing codes (DISCOS, TREETOPS) are 

compared and the suitability of present structural modeling are looked at. 

Effects of other influences such as fluid motions, environmental loads and 

implications of maneuvering are also discussed briefly. As a consequence of 

the literature review, several useful extensions to CONTOPS are evident 

("free-free" flex modes, deployment, foreshortening) and an outline is 

provided for their implementation. To complete the study, some general 

conclusions are reached and suggestions are made on what might be done 

to generate more efficient, higher fidelity models in the future. 

-4- 
2.0 MULTIBODY DYNAMIC SIMULATION 

2.1 Role of Simulation, Analysis Tools 

Closed form analytic solutions for dynamic response exist only for the 

simplest of systems: mass particles, translating finite masses, rotations of 

symmetric (or at least partly symmetric) rigid bodies, etc. As system 

complexity grows due to number of bodies, types of bodies (fluid, rigid, 

flex,...), geometric irregularity, range of motions (linear, nonlinear), the only 

recourse one has is a numerical solution. The specific list of assumptions 

grows longer as one strives for an evermore generic representation and can 

seem endless with the growth in demand for higher accuracies and the 

ensuing need to more fully capture all interaction effects. 

Ultimately situations can arise for which it is not possible to generate 

simulations capable of providing the desired accuracies. Nevertheless the 

analytic exercise remains an important one. Best-available-engineering 

models can be constructed and used to gain an understanding of important 

aspects of the problem. The modeling process itself develops insight into 

much of the basic physics involved and thus, in addition, is valuable training 

for the sorting out and interpreting of any measured data. From the viewpoint 

of controls design, this expertise forms an important building block in the 

pursuit of a workable control methodology. 

2.2 Nature of the Multibody Problem 

A multibody system is made up of any collection of interconnected bodies. In 

general, these bodies will have finite dimensions and will have mass 

distributed throughout the spatial domain. The motion dynamics tends to 

become quite complex because of kinematicAoad coupling introduced via the 

interconnections. Major analytic efforts to date have been concentrated on 

determining a systematic manner for computing and keeping track of the 

forces of inertia associated with the accelerating masses. Comparatively, 

much less work has gone into establishing consistent phenomenological 

force models (elastic strain energy, dissipative mechanisms, etc.) and 

representations of environmental loads as .applied to multibody 

configurations. 

2.3 Derivation of Equations 

All derivation procedures rely on some combination of DAlembert, Newton, 

Euler, Hamilton or Lagrange principles for establishing conditions for 

dynamic equilibrium. It is taken as prerequisite that the individual procedures 

result in equations which are equivalent if not identical in mathematical form. 

Consequently this topic is not dealt with further in this study. Discussion and 

evaluations of derivation techniques can be found, for example, in 

References 5-1 4. 

-5- 
2.4 Status of Multibody Code Development 

Earlier efforts in development of multibody formalisms and computer codes 

have already been reviewed or commented on in References 5-7, 15, 16. 

Reference 17 reviews developments from a user perspective. Overall 

assessments by Jerkovsky (1978) 15 and by Covington, et. al. (1984)16 are 

summarized in the Appendix (Tables 1 and 2, respectively). What emerges is 

evidence of an evolutionary path being followed in the search of ever more 

realistic and hence more useful formulations. The configurations modeled 

range from the cluster and chain categories shown in Figures 1 (a), 1 (b) 

through to the open and closed tree topologies of Figures 1 (c), 1 (d). Some 

important factors addressed include: Degree of rigidity (1 00% rigid? 1,2, .... DOF vibrations?), type of state variable (displacement? velocity? momenta?), 

choice of reference coordinates (relative? inertial?), choice of reference 

points (center of mass? fixed hinge?) and character of the interconnection 

(point? finite domain? #DOF? constraints?). 

Q. 

6 

Figure 1 Classes of multibody configurations: (a) Cluster 

(b) Chain 

(c) Open-loop tree 

(d) Closed-loop 

-6- 
The question of which formalism to adopt from the basic laws of mechanics is 

also an important one, but the exact method does not appear to be a critical 

consideration since all methodologies end up providing equivalent sets of 

equations. 5912 Suffice it to say that for multibody applications it simplifies 

the derivation process considerably to not include nonworking forces of 

constraint. 

Another key issue in the realm of multibody modeling is centered around 

methods for dealing with the "system" aspects of the topology. For example, 

is it always advantageous to work through the barycenter of augmented 

bodies18 or is the direct path approach as advanced by H07 more beneficial 

all round? Hughes6 states that, while the use of augmented bodies does 

help uncouple the system translational motion from rotations, the price paid is 

a more complicated form for the rotational dynamics. Furthermore, Reference 

6 points out that although the translationshotations are coupled in the direct 

path procedure, the constraint forces are eliminated at the outset by choosing 

interbody hinge points as reference points and more straightforward 

interpretations are possible for quantities such as the inertia coefficients. 

All in all, a consensus (References 6,8) appears to be building in favor of the 

direct path method. This, by the way, is consistent with the methodology 

employed in the TREETOPS algorithm.3 Still other attempts at organizing 

the system kinematic, dynamic topology is the "vector-networking" of 

Reference 19 and the "bond-graph" strategy followed in Reference 20. Such 

procedures are well suited to efficient implementation on a digital computer. 

It has been commonly assumed that to solve the governing multibody 

equations one must ultimately face the task of numerically inverting a mass 

matrix. There is some recent development which introduces recursive 

algebra early in the formulation to avoid this situation (References 21 -23). 

2.5 Evaluation of Representative Multi body Software 

Rather than duplicate any assessments carried out to date, the focus here will 

be on comparing and contrasting essential characteristics of two of the more 

contemporary multibody codes resident in the public domain. Both 

DISCOS2 and TREETOPS3 capture the fully coupled, large angle, 6DOF 

translational rotational dynamics of a contiguous collection of rigid, flexible 

bodies. Vibrational displacements are handled via an Assumed-Mode type 

of expansion. In DISCOS, bodies are modeled independently and 

constrained through the kinematics and Lagrange multipliers, which also 

allows for loop closure. The net result is a fairly high order system which can 

be computationally sensitive if there are a large number of constraint forces. 

Reference 24 points out a specific problem of the closed loop formulation of 

DISCOS is the requirement that at least one member in the loop be modeled 

as elastic. TREETOPS makes use of a direct path methodology, thus 

minimizing presence of constraint forces and, in addition, the order of the 

system is adjusted according to the number of degrees of freedom specified 

- 7- 
2.6 

by the user. To encompass closed loop topologies, the TREETOPS has 

been modified to include specific classes of equality (e.g. prescribed 

velocities) and inequality (e.g. hardstops) constraints. This version is referred 

to as CONTOPS25 and makes use of Singular Value Decompostion26 to 

further minimize system degrees of freedom in the presence of constraints. 

There are other differences between these two codes. DISCOS allows for 

local body spin or locally stored momentum which TREETOPS does not. In 

DISCOS, one of the bodies can be made to follow a prescribed orbital 

trajectory whereas it is not clear that TREETOPS has this capacity. Neither 

does TREETOPS embody the aerodynamic, gravitational or thermal 

environmental loading, although this development is understood to be 

underway. Also, in DISCOS, the reference points are taken to be at the 

center of mass or at "hard" points, whereas in TREETOPS only the inner 

hinge point need be fixed. Common to both algorithms is the lack of 

provision for component mass flow (which would be needed in study of tether 

problems) or provision for coupling of axial load effects with transverse 

vibrations (see Kane, et. ai? or Lips28) and, in both cases, body 

interconnections are assumed to be point-like. More recently some effort has 

been made to modify DISCOS so that it might include foreshortening.29,30 

A significant extension to this class of multibody model is work of Reference 

31 which allows for relative translation along not a rigid, straight path, but 

rather along a flexible, curved path. 

This discussion shows how far multibody simulations have progressed. On 

the European scene, the MEDYNA32 software is also of this calibre. As well, 

a fairly sophisticated but computationally intensive code IATDYN has been 

developed by NASA Langley33 to deal specifically with the case of 

deployment, unfolding of the joint-dominated structures. 

Symbolic Manipulation, Parallel Processing 

Generating equations governing multibody dynamics and the development of 

the associated computer software tends to be labor intensive and error 

prone. This situation has spawned a new approach in which the computer is 

harnessed to do both tasks using symbolic manipulation. In the United 

States, for example, SD/EXACT34 is a symbolic code dedicated to multibody 

dynamics whereas in Europe NEWEUL35 and MESA VERDE36 have been 

applied to multibody spacecraft dynamics. 

Another attempt aimed at improving computational efficiencies of a multibody 

simulation is the processing of dynamic events in paralle1.37 

-8- 
3.0 ADEQUACY OF STRUCTURAL MODELS 

The question of exactly how to best represent the structural flexibility of an 

individual body is indeed a very large subject in its own right. It is 

undergoing constant and considerable revision and we will not attempt here 

to deal with all aspects in detail. Rather the intent is to leave the reader with 

a sense of what some of the main issues are at present and to indicate 

current approaches to solutions. 

3.1 Basic Mathematical Descriptions for Flexibility 

The most common manner of representing flexibility is to express the 

displacement field as superposition of an infinite number of assumed-modes. 

These modes are normally derivable from an eigenvalue description of the 

mass, elastic properties and amounts to a standing wave model. The 

approach ultimately leads to discrete equations which can be truncated, thus 

reducing system order and rendering a transient dynamic analysis tractable. 

Truncation criteria used to be a matter of selecting the first few lower 

frequency, higher energy modes (this also eliminates high frequency modes 

which may or may not be desirable). Currently, however, the literature 

contains more sophisticated approaches which take into account, for 

example, the contributions of a given mode to rigid/flex coupling (References 

38-41). These modal cost assignment techniques have been used to arrive 

at reduced models for entire systems (see References 42-44). This approach 

can be particularly significant to a control designer faced with a high order 

finite element system model. 

The question of exactly which modal set to adopt is not clear in a multibodied 

setting because of the dynamic coupling and because of the complicated and 

not always well defined boundary condition. Ideally, the modes chosen 

would be the eigenfunctions, but they are usually not easily arrived at so that 

simplified sets of dynamic situations (e.g. constant spin) are imposed as 

reviewed in Section 1.2 of Reference 45. Perhaps it is not essential to work 

with the closest approximation to the eigenfunctions. Meirovitch, et. a1.46 

conclude that for any linear gyroscopic system it is sufficient to use any set of 

admissible functions provided the set is complete. For a function to be 

considered admissible, it must satisfy geometric boundary conditions and be 

differentiable to order P for a system of order 2P. 

Finite Element Models (FEM) (e.g. NASTRAN) are commonly used to 

generate modal data for cases having nonuniform geometry and nonuniform 

mass, stiffness distributions. The way in which this data is used in a 

multibody code such as DISCOS has been referred to as inconsistent 

according to References 47,48. Other subtleties embodied in FEM are 

discussed in Reference 49. When dealing with very large structures,the 

order of a FEM approach can become quite large. References 50-52 discuss 

the feasibility of adopting equivalent lower order configuration models for 

those situations when a structure is built up from a series of repetitive 

-9- 
3.2 

3.2.1 

3.2.2 

subelements. Truncated discrete standing wave solution is not the only 

modeling option. No truncation is necessary if one works in the frequency 

domain as suggested by Poelart 45, 54, but this is not possible for a 

nonlinear system. A travelling wave approach is discussed in References 55, 

56. Alternately, given the uncertainty surrounding structural characteristics, 

are stochastic models (as described in Reference 57) really more 

appropriate? 

Specialized Considerations 

Energy Dissipation 

Historically there has been, and continues to be, considerable seat-of-thepants judgment exercised when arriving at a value to be assigned the 

damping factor of a given mode. This parameter is vital to response stability 

and settling times (particularly at high frequency) and cannot be left to 

chance if high accuracy control is mandatory. Fortunately research of a 

fundamental nature is underway as outlined in Reference 58. The objective 

is to measure damping of a small material piece of a structure, relate it, via a 

viscoelastic modeling process and time domain realization, to the full sized 

structure. Evidence of the continuing effort to rethink and experimentally 

quantify damping parameters is contained in the work of References 51 39- 

62, for example. Some success has been had in measuring damping in-orbit 

as cited in References 63,64 for systems having solar arrays 

(Communications Technology Satellite and Orbiter-Based-SAFE experiment) 

but for some modes the damping is much different than the predicted levels. 

Mass Flow Deployment 

By now we have gained some appreciation as to the complexities associated 

with configuration geometries and vehicle elasticity. Analysis becomes even 

more involved during extension or retraction of rigid and/or flexible 

components. This type of deployment introduces a variable mass distribution 

(and hence variable moments of inertia) together with additional relative 

velocities and accelerations, all factors that affect transient dynamic behavior. 

Perhaps because of its inherent complexity the problem has received 

relatively little attention. In general, available investigations tend to be more 

limited in scope than those dealing with nondeploying flexible structures as 

reviewed in Section 1.2.5 of Reference 21. One of the more significant 

studies (Reference 65) considers deployment of a single beam. Reference 

66 investigates the characteristics of a spinning, deploying beam in-orbit 

under the influence of the gravity gradient. One of the more general 

multibody software developments to consider individual body mass flow is 

the cluster configuration investigated by Lips (Reference 21) where it is 

shown that deployment-related Coriolis loads can be significant. More 

recently, Keat, et. al (References 16,67) are developing a multibody code 

applicable to spacecraft with unfolding and with mass flow and, as 

mentioned, Housner, et. al. (References 33,68) have developed such a code 

for joint-dominant structures. 

-10- 
3.2.3 

Y 

Later in the report a look is taken at what the implications are of extending a 

code such as TREETOPS/CONTOPS to include deployment. 

Reduced Dimension Models and the Foreshortening Effect 

Reference 27 asserts that a fundamental flaw exists in multibody codes such 

as DISCOS and TREETOPS (which is not the case for the clustered 

appendage configuration of Reference 21 or the model in Reference 66). 

Concern appears to be centered around the absence of the flexforeshortening-induced dynamic coupling between the axial loading and 

transverse vibration. Concern over this issue was raised earlier by Lips 

(Reference 28). In all cases the problem is described for a one-dimensional 

beam-like structure - a simplified model sometimes used to represent space 

trusses or astromast types of booms. 

Here, the literature in this area is reviewed and an attempt is made to 

determine the relative significance of this type of coupling. An indication of 

the main steps needed to introduce such an effect into the TREETOPS 

formulation follows later in the report. 

As shown in Figure 2(a), axial foreshortening refers to the translational 

displacement which a mass element experiences along the axial direction (x) 

due to elastic deformation (v) normal to that direction. Differential line 

element theory can be used to relate the deformed element dimension ds to 

its projected undeformed length dx as seen from Figure 2(b). This concept 

dates back to at least 1964.69 Over the years it has been introduced directly 

as a working displacement, 69970 indirectly during evaluation of spatial 

integrals,71 172 explicitly as an added component of translation,21,66,73,74 

and implicitly as a nonlinear strain displacement effect.75 

Y 

Figure 2 Concept of axial foreshortening showing: a) deformation (v) 

along a transverse direction together with the constrained 

translation (ufs) along the axial direction; and (b) length of 

deformed element (ds) and its shortened length (dx) as projected 

along the undeformed x axis. 

-11- 
Kane, et. al. (1 987)27 introduce effects of foreshortening explicitly as an 

added translational velocity. However, in addition to the subtleties 

associated with foreshortening, one must deal with the terminology and 

methodology unique to the Kane approach. To the uninitiated this tends to 

obscure somewhat the handling of foreshortening and the coupling induced 

through it. The problem is not helped by the absence of any reference to the 

term foreshortening or to differential line element theory. The handling of this 

problem is a new application for the Kane methodology but, at least as far as 

beam foreshortening is concerned, it does not qualify as a "new" theory. 

Nevertheless, Reference 27 is correct in pointing out that the foreshortening - 

related effects have not been incorporated into the existing multibody codes. 

The question of how significant the coupling terms are is addressed, in part, 

in Reference 76, where an approximate result is given relating the 

foreshortening - corrected frequency of a spinning beam with that of a nonspinning beam. The results are plotted in Figure 3 where it is evident that 

foreshortening-induced effects can be expected to become significant if spin 

rates are comparable or much higher than that frequency characterizing the 

non-spinning structure. 

3 

NONDIhENSIONAL 

FREQUENCYOF 

dl = "DAMENTAL MODE FREQUENCY 

OF NONSPINNTNG BEAM 

0 / 2 

(WW 

NONDIMENSIONAL SPIN RATE 

Figure 3 Foreshortening-induced spin stiffening effect on a 

uniform beam 

-12- 
3.2.4 Joint Idealization 

Large flexible frame structures in space will be characterized by large 

numbers of joints each of which can present the analyst with a host of 

unknowns. Although idealized as points, joints in fact involve finite physical 

domains (a factor also in the setup of actuators, sensors). There can be 

sloppiness in joints, different friction regimes, considerable nonlinear 

behavior and flexibility. Such features could make the cost of computing the 

response prohibitive. Only recently has attention been focused on this area. 

Perhaps the most advanced effort to date is the Large Angle Transient 

Dynamics (LATDYN) software under development at NASA Langley 

(References 33,68). Independent work of Reference 77 suggests that the 

presence of joints may not alter modal frequencies significantly provided the 

overall structure is not too short. Some work is also being done to identify 

energy dissipation, taking place at the joint (see References 78-80). 

-13- 
4.0 OTHER FACTORS IMPACTING FIDELITY OF MULTIBODY 

SIMULATION 

4.1 FI uid Interact ion 

Just as elastic systems make it difficult to quantify response, the presence of 

fluids can present an even more vexing modeling problem being particularly 

sensitive to temperature, pressure. With the advent of liquid apogee motors, 

the liquidkolid ratio can now be as high as 55%, and hence the problem is 

commanding much more serious attention both in Europe and in the U.S.A., 

not to mention SDI applications. An overview of the state-of-the-art can be 

had by reviewing the papers given at the colloquium dedicated to this 

problem and cited in Reference 81. 

A number of presentations concentrated on development of fundamental 

models applicable to rotating, free-surface, fluid behavior. However, due to 

the complex nature of the problem only restricted approximate solutions, in 

the form of one homogeneous eigenmode for example81a are forthcoming. 

As well as analysis, a significant amount of the work is dedicated to 

experimental testing of ground- based gyro-f luid systems (e. g . Van yo , Reference 81 b). While the test data appears to be reliable, it is not certain 

that fluid character observed at ground level will be the same in orbit. To 

address this question, ESA has observed fluid behavior in orbit on the 

Shuttle SPACELAB 1 mission8lc. 

An experimental method which appears to give good agreement (better than 

20%) with in orbit measurement, is the ground-based "Drop-Test" B1d . A 

dynamically scaled model of the satellite/fluid system is allowed to free-fall. 

Nutation data is transmitted continuously throughout the duration of the drop. 

In addition to generating analytic fluid models and measuring fluid/system 

behavior, serious efforts have been made to construct overall system models 

81a, 81e. Mclntyre of Hughes Aircraft outlines an approach that is analogous 

to that used when dealing with flexibility effects. Attitude equations are 

formulated for the spacecraft which couple directly to fluid motions which, in 

turn, are characterized in terms of discrete coordinates (derived from a 

truncated Assumed-Mode expansion of the fluid displacement). Required is 

a set of representative fluid mode shapes which can be arrived at either 

analytically, experimentally or in combination. Such a method, when 

generalized, can also model fluid/flexibility interactions (platform asymmetry, 

etc.). Another effect which can become large as fluid mass ratios increase, 

and which can effect the system dynamics, is a shift in center of mass 

location. 

Attempts to come to grips with this complex phenomenon are continuing at 

both the experimental and theoretical levels as evident from References 82, 

83 which deal with coning cylindrical fluids and with clusters of gaseous 

bubbles contained within an oscillating column of liquid, respectively. 

-14- 
4.2 Environmental Loads 

An analyst may go to great lengths to generate a high fidelity model of the 

inertia-related forces of a vibrating gyroscopic system only to find that 

external forces applied by the environment are known only very 

approximately in some instances. References 84, 85 are among the more 

recent efforts aimed at improving atmospheric density modeling, whereas 

Reference 86 is one of the latest attempts to come to grips with solarradiation-pressure-induced loads. To develop the' best possible 

environmental modeling, one should also consult with the comprehensive 

bibliography on the subject contained in Reference 87. To date, generic 

multibody environmental models are not available since the description used 

is linked closely to the kinematic gendre adopted within a given formalism. 

4.3 Requirement for Controlled Maneuvers 

It is expected that upcoming space systems will be required to exhibit a 

higher degree of maneuverability. For example, the entire spacecraft (or part 

of it) may have to be re-oriented rapidly or it may have to be spun up or spun 

down in the shortest time while keeping vibrations to a minimum. Reference 

88 discusses open loop strategies for vibration control of an Orbiter-based 

experiment having long (up to 150m), highly flexible booms (Note that in this 

case flex foreshortening is included at the outset.). References 89-92 also 

point out strategies for slewing flexible systems so as to minimize the postmaneuver oscillation. References 93-94, on the other hand, deal with 

optimal attitude control for large angle maneuvering of rigid vehicles. None 

of these studies yet rely on any of the major multibody codes. 

4.4 Numerical Considerations in the Solution of Mixed DifferentialAlgebraic Equations 

Several methods have been developed to solve sets of mixed DifferentialAlgebraic Equations, hereafter abbreviated DAE. Gear 9596 has shown 

that his stiff numerical integration algorithm can be used to solve certain 

kinds of DAE. Certain other DAE can be converted into equivalent forms that 

are solvable by Gear's method. There exist, however, systems of DAE that 

cannot be solved by this method. In References 97-100, Gear's idea has 

been successfully employed for analysis and optimal design of mechanical 

systems and electrical networks. Mathematical models of dynamic 

mechanical systems, using Gear's method of solving DAE, make the 

equations of motion stiff, thus involving considerable computational cost and 

time. Furthermore, this formulation requires several problem dependent 

derivative expressions that make automatic computer generation of the 

equations of motion difficult 98,99 In fact, Reference 101 shows that DAE 

systems are not ordinary differential equations at all. 

A second approach, due to Baumgarte, 102 uses ideas from feedback control 

theory to construct a modifed differential equation that implicitly accounts for 

the constraint equations. A difficulty, however, is the selection of factors in 

-15- 
this method that greatly influence accuracy of solutions. Proper values of 

these factors are not known and experts are in disagreement over suitable 

values. Furthermore, there is no way to impose positive error control on 

constraint violations for this method. Work is in progress to develop methods 

to determine these factors, based on constraint violations that ensure that the 

constraint violations at the next time step are minimized. The major 

drawback of these methods is that all n generalized coordinates must be 

integrated, whereas only n-m of them are independent. Furthermore, m is 

often large and n-m is small. Thus, it would be attractive to integrate for only 

n-m independent generalized coordinates. Solution for the remaining m 

dependent generalized coordinates from constraint equations can then 

ensure positive error control on constraint violations. The n-m generalized 

coordinates that are computed by integrating the equations of motion are 

called independent generalized coordinates and the remaining generalized 

coordinates are called dependent generalized coordinates. 

Independent generalized coordinates can be picked from the set of n 

generalized coordinates that describe the system configuration. Wehage 

and Haug 103,104 developed an algorithm to pick an acceptable choice of 

independent coordinates, using LU decomposition of the constraint Jacobian 

matrix to identify independent and dependent coordinates. This algorithm 

performs satisfactorily, but occasionally leads to poorly conditioned matrices. 

It is then required that LU factorization be repeated and a new set of 

independent generalized coordinates be selected. The result is an increase 

in computing time and greater propagation of integration error than desired. 

In Reference 26 a method based on singular value decomposition is used to 

solve the equations of motion of dynamic systems. This method, however, 

does not provide physical insight into the solution process and is not 

convenient for automatic generation of the equations of motion for general 

mechanical systems in a general purpose computer program. 

A broader class of vectors p of independent generalized coordinates can be 

P= VI xq (n-m)xl (n-m)xn nxl 

where VI is a transformation matrix. In the algorithm of References 103,104 , 

matrix VI is a Boolean matrix. Therefore, only individual generalized 

coordinates are chosen as independent. Use of a Boolean transformation 

matrix to pick independent generalized coordinates from q is rather 

restrictive. It is desirable to enlarge the set of possible independent 

generalized coordinates by allowing matrix VI to be a more general, possibly 

nonsparse matrix. Such a choice of VI can allow linear combinations of 

physical generalized coordinates to be selected as independent. If the rows 

of VI are mutually orthogonal, the resulting linear combinations of individual 

generalized coordinates will be mutually independent. Specific forms of 

such a transformation can be chosen, based on properties of the DAE under 

consideration and desired properties of the transformed coordinates. 

-16- 
5.0 EXTENSIONS TO THE CONTOPS ALGORITHM 

5.1 

This section discusses revisions that, if fully implemented, could increase 

both range of applicability and efficiency of the CONTOPS code. 

Generalization of the CONTOPS Hinge 

Currently, there are two restrictions imposed in CONTOPS. 

(i) 

center, 

The reference point hk (Figure 4) of the reference body is its mass 

*c (kl 

Figure 4 Geometry of deformable body j. 

(ii) Modal displacements of reference point hk and modal rotations of frame 

bk (Figure 4) are assumed to be zero for all bodies. For the reference body 

the second restriction is naturally satisfied if the free-free mode shapes are 

utilized in defining its elastic deformations. For all other bodies in the system 

the second restriction is naturally satisfied by the fixed-free modes of the 

individual body (fixed at point hk in Figure 4). CONTOPS results may not 

yield desired accuracy if the 'modeshapes are obtained with boundary 

conditions other than those described above. 

Because the mass center is taken as the reference point for the reference 

body two vector quantities hki and lk defined by equations 18,22 of 

Reference 3 (see attached copy) have been assumed null (for free-free 

modes of the reference body this is a valid assumption). With this exception 

the form of the equations is same for all bodies in the system. The first 

restriction may be removed by inserting non-null vectors for hki and lk for the 

reference body Bk . 

-1 7- 
Removal of the second restriction is not an easy matter. It is pointed out in 

Reference 3 that the elastic deformation, & , is measured with respect to 

frame bk . Furthermore, it is required that the elastic deformations remain 

small. Thus the frame bk and point hk cannot be arbitrarily defined. In what 

follows the formulation of Reference 3 will be modified which will allow a 

CONTOPS user to: 

(i) 

independent of the definition of hinge. 

define reference point and reference frame for individual bodies 

(ii) generalize the existing hinge definition to accommodate a wider class of 

boundary conditions for the mode descriptions of the substructures. 

Figure 5 describes the kinematics of body Bk and the kth hinge. Frames pk 

and qk and vector yk fully define the kth hinge. The reference point ck for Bk 

is arbitrarily chosen. 

Figure 5 kth hinge and kth body kinematics 

-18- 
Following the notations of Reference 3 from Figure 5. 

the inertial derivative of equation (1 ) yields 

+ 

-19- 
Equations'3,4,5 are sufficient for identifying the coefficients of generalized 

speed (equations 26,27,28 of Reference 3). 

Figure 6 here would replace Figure 4 of Reference 3. 

Figure 6 Vectors defining coefficients of generalized speed. 

Replace equation 26 of Reference 3 by 

-20- 
Replace equation 27 of Reference 3 by 

Replace equation 28 of Reference 3 by 

Equations 14,15 of Reference 3 remain intact except superscript hk is 

replaced by ck . 

Implications of Deployment and Flex Foreshortening 

As currently formulated, the CONTOPS multibody algorithm does not allow 

for mass deployment or retraction of individual members nor does it take into 

account the geometric foreshortening that accompanies flex deformation. In 

light of this, the existing recursive equations are reviewed once again. This 

report discusses some basic considerations involved in modifying CONTOPS 

to incorporate such effects. 

-21- 
5.2.1 Existing CONTOPS Kinematics (Reference 3) 

The kth body (Bk), shown earlier in Figure 4, is in the deformed state. 

Elemental mass dm in Bk is located relative to an inertial reference by the 

vector (using Equation 1): 

hk (9) 

- R 

- rk is fixed in B 

locates the hinge point relative to inertial 

k 

k - u represents elastic deformation 

As already implied, in order to arrive at a solution in terms of a set of ordinary 

differential equations (and to pave the way for truncation to an approximate 

1 

Overall inertial velocity of dm then is given by 

- @=$k+ 2 k x(+ k +II)+~ k ok 

where 

( ) = inertial time derivative. 

0 

( ) = differentiation in Bk - fixed frame. 

-22- 
5.2.2 Modifications due to Changing Mass 

It is common in satellite design to store elements in as compact a manner as 

possible prior to flights and then to unfurl the device once in orbit, allowing it 

to extend to full size. Such Storable Extendable Members (STEM) 

complicate dynamic studies by introducing a time variable mass feature into 

the analysis which becomes even more complex when components are 

flexible. To allow for this in the CONTOPS type of multibody code implies the 

following (Reference 21,28): 

1. A structural mass element can be viewed asbeing convected by a 

prescribed rate of deployment (ID ) in a manner analogous to a fluid 

continuum model. Then any local time derivative must be augmented 

so that 

body no deployment 

2. When differentiating the discrete expansion given by Equation 10, it 

must be recognized that G!! is not zero but now is implicitly time 

dependent as a result of the convective motions discussed above. 

3. Mass must be conserved at all times. This leads to another equation 

which governs the motion - that is, the equation of continuity. 

4. Mass flow changes system properties such as moment of inertia, 

momentum. 

Implementation of these considerations is simplified if, rdll ,er than a general 

continuum, one analyzes a one-dimensional beam-like substructure, for 

example. If it is uniform and if only linear vibrations are involved, then 

deployment velocity is identically equal to rate of change in length as 

measured along the undeformed axial direction. 

-23- 
5.2.3 Incorporating Flex Foreshortening L 

To include foreshortening (ufs) in a CONTOPS type of algorithm, it can be 

added explicitly to the displacement field 21 so that Equation (9) becomes 

Rk=R hk +r 1.: +g k +gfs k 

- - 

To date ufs has been approximated only for beam-like, plate-like structures. 

If we take, 

where unit vector 81 lies along the undeformed axial or 'long' direction of the 

structure, then 

k 

-fs - 

+ higher order terms 

where ( )' = a( ) /axk 

When differentiating to arrive at velocity, acceleration it must be kept in mind 

that the approximated foreshortening displacement is time dependent. 

Alternatively, since the TREETOPS approach makes use of the partial 

velocity description in the kinematics, one can adopt the procedure given by 

Kane et. al 27. In this case the foreshortening is embedded in the velocity 

field. 

5.3 A Model Reduction Approach for CONTOPS 

The CONTOPS formulation is based on the method of substructures. The 

elastic deformations are expanded in terms of the natural modes of individual 

flexible substructures. The number and type of substructure modes required 

to accurately describe the elastic behavior has not always been addressed 

properly. Unwise choice of mode shapes will result in modeling error 

producing poor quality solutions. Also, retaining the substructure modes with 

insignificant contribution to the solution results in unnecessary computational 

penalty. 

In what follows we outline an approach based on the modal cost analysis 

(MCA) approach, a well proven technique for model reduction. The MCA 

approach to model reduction is widely used in the control science discipline. 

This approach was first proposed by Dr. R.E. Skelton of Purdue University 

(See Ref. 38, for example). In this approach a system is defined to be 

comprised of many components. The central idea of the modal reduction 

-24- 
method of the MCA is to exploit the precise statement of the optimal control 

problem in order to predict which system components will make the largest 

contributions in the total quadratic performance criterion. These components 

are retained and the balance are discarded. Mathematically, the concepts of 

this model reduction process are explained as follows. Let a state space 

description of a time invariant system be given as 

x(t) = Ax(t) + Dw(t) 

where the n states are represented by x(t), the k outputs are y(t) and the q 

process noise are w(t). w(t) is assumed to be zero white noise with unit 

intensity. It is assumed that the model is controllable, observable and 

asymptotically stable. It is also assumed that there are k independent 

outputs, i.e., rank (C) = k and q independent inputs, i.e., rank (D) = q. 

Let a quadratic cost function associated with (16) be given as 

where E is the expectation operator and Q is a symmetric positive definite 

weighting matrix. 

Now, partitioning the state vector x(t) = (xi (t), ~2(t), ...., xn(t))T, the component 

costs Vi associated with each component Xi is defined by 

vi 5 m)Xil, lim E[( i = 1, 2, ..., n 

1 

t-)- % 

and are calculated according to the following component cost formula 

where (.)ii denotes the (i,i) element of (.) and where X, the steady state 

covariance of the states, is defined by 

satisfies 

-25- 
uT t AX t DD~ = o 

Clearly, since V = Tr(CTQCX) the component costs satisfy the cost 

decomposition property 

The component costs are ranked in the following manner 

vz 2 e.. vn 

and the state vector is reordered correspondingly. The most critical 

component of the system is x1 having component cost VI and the least 

critical component of the system is Xn having component cost Vn. 

Based on this component cost ordering, components having lower 

component cost are truncated. 

The MCA approach outlined above may be applied to CONTOPS for 

selecting the appropriate modes from a given set of modes for individual 

substructures. 

For CONTOPS application the state vector x(t) may be arranged as follows 

ath, (a+l)tb 

entry entry 

whereqif t )is the 1 th modal coordinate of the kth substructure (body Bk). The 

component cost corresponding to the lth mode of Bk is 

where X satisfies equation (21). In the technical discussion above the 

component cost computation is based on a scalar, yTQy. This concept may 

be expanded to account for multiple objective problem. This can be 

accomplished by computing the component costs for each of 

(26) 

2 

yin i P 10 ..., k 

The component costs for each y21 should be ranked and the resulting 

dominant modes based on each y21 should be retained. The MCA algorithm 

can easily be integrated into CONTOPS. 

-26- 
6.0 DISCUSSION AND CONCLUSIONS 

A wide range of issues related to the modeling of the dynamics of multibodied systems has been reviewed In this report. While it has not been 

possible to evaluate all areas fully, it is hoped that bringing so many of these 

areas together in one report has served to demonstrate the scope of the 

problem. 

Casting a multibody dynamic system into an understandable, manageable 

framework is a challenging task because of the large numbers of bodies 

involved and because of the high degree of coupling between them. 

Nevertheless, the exact procedure adopted for derivation of the equation set 

does not appear to be a concern since all aproaches will, if competent, 

produce equivalent mathematical representations. For instance, equations 

can be put into a 1st order velocity format using either a Newton-Euler 

approach or using the procedure favoured by Kane and Levinson 10. The 

degree of effort and preference for a method will, in large measure, always 

be dictated by the analysts experience. Choice of coordinates and frames of 

reference may leave more room for debate for specialized applications but 

for a very general model this is not so much the case (6DOF is 6DOF!). What 

does stand out as a predominant issue facing multibody experts is the choice 

of a configuration topology and of methods for dealing with it. In this regard it 

would appear that the direct path method (which is conducive to use of 

recursive interbody equations and makes it straight forward to eliminate 

nonworking constrained forces) is gaining acceptance over the barycentric 

approach. One of the more versatile topologies is the tree analogy with open 

or closed branch loops. The bookkeeping aspects of the kinematics might 

also be helped through vector-networking or through bond-graph models. 

Deriving equations need not be done manually. According to the literature, 

a digital computer can be used as a symbol manipulator to generate 

equations based on whatever rules of mechanics one chooses and, if 

desired, to translate these equations directly into computer code (e.g. 

FORTRAN) for solution. This approach can be expected to yield more and 

more efficient computational algorithms and to be less error prone as a 

problem grows in complexity. A decided advantage here is the capacity to 

generate equations for the specific problem at hand rather than to generate 

one algorithm which attempts to cover all possible situations. Numerical 

efficiency can be further enhanced by carrying out computations in parallel 

whenever possible rather than in a serial manner as has been more the 

custom to date. This strategy would appear to complement the order n 

recursive modeling which avoids the need to directly invert a system order 

mass matrix and hence should augment further the speed at which the 

simulation can be executed. 

Obviously the order of a model has a strong influence on computational 

performance. As alluded to already, the order n approach combined with 

symbolic manipulation can provide a much reduced model for the system. 

Two other factors are important in determining system size - flexibility and 

-2 7- 
constraints. Equivalent continuum modeling together with modal cost 

analysis or simple truncation can be used to minimize the flex contribution. 

Singular Value Decomposition can be applied to minimize degrees of 

freedom for constrained systems. Model reduction can then be applied to the 

resulting system. This combined reduction may yield the most meaningful 

and efficient definition of the system dynamics depending on the 

computational penalty paid to achieve each stage of the reduction. 

The algorithm adopted to carry out a numerical integration also plays an 

important role in how accurately and how quickly a particular piece of 

software can respond. While this has not been addressed in great detail 

here, it is implicit throughout. Evaluations should be made to etermine 

relative merits of Runge-Kutta versus Predictor-Corrector methods. The Gear 

approach can be vital to the success of a 'stiff' multibody situation in which 

characteritic times associated with the motions are highly divergent (this can 

occur easily for cases involving extension of a substructure from a very small 

length say out to a very large value). 

It has been pointed out that it is not the fundamental principles of mechanics 

which stand in the way of achieving a high level of modeling fidelity. From 

this review it is concluded that, in a very general sense, what does restrict the 

modeling, simulation process is: 

(i) level of geometric detail used to describe a finite dimensioned physical 

system; 

(11) adequacy of the phenomenological models. {This ranges from 

uncertainties present in the basic structural model (including, in 

particular, energy dissipation) through to questions related to fluid 

behaviour and including effectiveness in accurately capturing the effect 

of environmental influences.} 

(iii) efficiency of the solution algorithm; and 

(iv) numerical techniques (integrators, parallel vs serial processing, ...) 

It is clear from this investigation that the state-of-the-art has matured to the 

point where a contemporary generic multibody code can be assumed to 

include the following features: 

arbitrary number of point-connected bodies 

allows large angle relative interbody rotations (3DOF) 

allows large relative interbody translations (3DOF) 

allows for linear vibrations (3DOF translation) 

some form of loop closure exists 

-28- 
While DISCOS and TREETOPS/CONTOPS fall into this category, they do not 

routinely account for deployment, for example, TREETOPS is superior from a 

numerical viewpoint in that, as much as possible, constraint forces are 

eliminated. The DISCOS Lagrange multiplier feedback approach, which 

includes a large number of constraint forces, can be expected to provide for a 

much more rapid build up of the numerical error. Further, the size of system 

matrices in TREETOPS is not fixed by the number of hinge points and bodies 

(as in the case of DISCOS) but is determined only by the number of degrees 

of freedom input to the program for a specific application. DISCOS, on the 

other hand, has the advantage of permitting prescribed motion (including 

orbital) and body spin (or locally stored momentum). This demonstrates that 

the search for a completely unrestricted 'generic' code is as allusive as ever. 

One area neglected in both of the codes discussed above, and in other 

known general purpose software (except for the development in Ref. 21), is 

the effect of nonlinearity in the strain displacement relationships of a flexible 

body which gives rise to the foreshortening in the displacement field as 

discussed in Section 3.2.3. The significance of this effect is summarized 

qualitatively here by paraphrasing Ref. 28.: "The generality of a formulation 

is compromised if one is interested in simulating response for space systems 

which are large physically, which undergo large attitude excursions (in terms 

of both displacement and rate) and which have vibrating appendages that 

engage in significant local translations if the effect of interaction of axial 

loading with the transverse vibrations is not accounted for. It is well known, 

for example, that centrifugal loads associated with rotation of a beam-like 

appendage affect terms which are linear in the vibration variables." In this 

report, it has been shown quantitatively that the spin-stiffening becomes 

important as spin rates approach or exceed the fundamental frequency 

characterizing the nonspinning structure. Note that this is the same effect 

focused on most recently by Kane et al.27 Their claim that it has not been 

included in the major multibody codes is a valid one, however, the 

phenomenon itself is not new and has already been dealt with extensively in 

the general literature. Fortunately, existing codes can be modified to 

accommodate this effect without completely rederiving equations or 

completely reprogramming the existing codes. 

-29- 
7.0 RECOMMENDATIONS 

This investigation has given insight into the level of complexity and the level 

of uncertainty surrounding multibody simulation dynamics. A series of 

recommendations is discussed here aimed at providing for improved 

dynamic models or improved computational efficiency of either present-day 

or of next-generation software. 

The foreshortening effect forms a valid part of the kinematics and should be a 

part of existing multibody codes and should be allowed for in any future 

modeling developments. It can require considerable additional calculations 

even though it is not significant in every application. For this reason, it should 

be retained in the form of an option. Note that CONTOPS can likely be 

amended more easily through the generalized speed than can DISCOS 

which relies on kinetic energy. 

The fact that the major codes are amenable to the type of revision alluded to 

above is encouraging, since it implies that significant enhancements can be 

achieved without generating an entirely new program. Every effort, therefore, 

should be made to capitalize fully on existing codes before contemplating 

construction of any new package. On the other hand, any single piece of 

software should not be relied on to provide the best solution for all 

applications. Rather it is more realistic to expect each code to apply to a 

certain class of problems and configurations. What is important is that the 

code description spell out clearly the known capabilities as well as the 

known limitations of a given formulation. 

Regardless of which code is being worked with or whether a new model is 

being generated, in order to arrive at higher fidelity synthesized models it is 

crucial that the best possible phenomenological models (impedance, 

standing wave, traveling wave, stochastic, viscoelastic ,...) of substructures 

and components be first determined and then implemented. Of course, 

because of the potential for dynamic interactions, all known subsystems 

should be modeled as well Another very basic factor that should not be 

overlooked is the level of geometric detail used in describing the physical 

domains and interbody connections. The more complete the representation, 

the more realistic the model and the less.likely it is that important coupling 

effects will be lost. This can become particularly important when dealing with 

very large frame, truss space structures. 

An area that will always be of interest is the manner in which the fundamental 

principles of classical mechanics are interpreted and applied in order to 

derive the governing equations. At this point in time this represents a fairly 

mature part of the technology. Consequently it is recommended that the 

emphasis now be shifted toward topological considerations and to questions 

relating to the efficiency of the solution algorithm itself. For example, this 

study indicates that use of a direct path method of formulation which 

eliminates constraint forces whenever possible is preferable for that based 

on use of barycenters and augmented bodies. Graph theory should be relied 

-30- 
on to provide a method of accounting for interbody connectivity and related 

kinematics. A newer approach that merits attention for any new development 

is the recursive, order n algorithm which avoids the need for a direct 

inversion of the system mass matrix. 

Another tool that should be developed further to aid the analysis is symbolic 

manipulation. This technique harnesses the power of the computer to 

provide more rapid, less error prone sets of equations for complex 

dynamically coup led systems. 

From a numerical viewpoint there are many challenging issues. In order to 

bring the system dimension down to a tractable order, model reduction 

should be applied at both the subsystem and at the system level. The order 

of constrained systems can also be minimized through techniques such as 

singular value decomposition. As with model reduction, however, there are 

computational costs which have to be factored in to see if real savings are 

achieved overall. 

The accuracy and efficiency of numerical integrators (Runge-Kutta, predictorcorrector, ...) needs further evaluation for space dynamic applications. How 

do the methods compare in general? What are the real benefits of having a 

variable step size in the presence of controllers? A major issue is how 

effectively can the "stiff" dynamics be dealt with. Is accuracy degraded 

seriously in presence of constraint feedback? 

Computational efficiency can be enhanced by carrying out calculations in 

parallel whenever possible rather than following the step by step serial 

approach so common in the past. In addition to parallel processing, one 

should take advantage of symbolic manipulation to write the necessary 

computer code. This promises to be faster and should definitely cut down on 

debugging costs. It is likely that this can be used when extending existing 

codes. 

What is clear from this study, is that considerable uncertainty persists in the 

modeling of multibodied dynamic systems. Ideally one would like to verify 

the working of simulation software by experimental measurement. This is not 

yet practicable for the full scale in-orbit environment. Hence, it is 

recommended here that a set of space-related baseline simulation test cases 

be established for use in evaluating performance of a given code. The test 

cases should include simplified, idealized configurations (e.9. rigid dumbell) 

through to actual mission con fig u ratio ns (representative co m mu n icat io n s 

satellite, Orbiter and payload, spinner;...). This process does of course raise 

questions of its own. Nevertheless, it is viewed here as a very important step 

forward in providing a framework for validating complex multibody codes and 

for establishing their range of application. 

-31- 
REFERENCES 

1. Tolivar A.F. and Dahlgren, J.B., "NASA Large Space Systems Missions," 

presented at Workshop on Identification and Control of Flexible Space 

Structures, San Diego, California, 4-1 6 June, 1984. 

2. Bodley, Carl S., Devers, Darrel A., Park, A. Colton, Frisch, Harold P., 

"A Digital Computer Program for the Dynamic Interaction Simulation of 

Controls and Structure (DISCOS)." Vol. I, NAS A Tec hnical Pmer No, 

1219, 1978. 

3. Singh, Ramendra P., VanderVoort, Richard J., Likins, Peter W., 

"Dynamics of Flexible Bodies in Tree Topology - A Computer-Oriented 

Approach," J. Gu idance. Co ntrol. and Dv .namics, Vol. 8, No. 5, Sept. - Oct. 

1985, pp. 584-590. 

4. Balas, M.J., "Trends in Large Space Structure Control Theory: Fondest 

Hopes, Wildest Dreams, I' IEEE Transactions on Automatic Controls, Vol. 

27, 1982, pp. 522-535. 

5. Hughes, P.C., "Dynamics of a Chain of Flexible Bodies, " JL Astronauticd 

Sciences, Vol. XXVII, No. 4, Oct. - Dec. 1979, pp. 359-380. 

6. Hughes, P.C., "Spacecraft Attitude Dynamics," Wilev, 1986. 

7. Ho, J. Y. L., "The Direct Path Method of Deriving the Dynamic Equations 

of Motion of a Multibody Flexible Spacecraft with Topological Tree 

Configuration," Paper No. 74-786, AlAA Mechanics and Co ntrol of Fliaht 

Conf., Anaheim, California, 5-9 August 1974. 

8. Ho, J. Y. L., "Direct Path Method for Flexible Multibodv Spacecraft 

Dynamics, " J. Swecraft and RockeB, Vol. 14, No. 2, February 1977, pp. 

102-1 10. 

9. Ho, J. Y.L. and Herber, D.R., "Development of Dynamics and Control 

Simulation of Large Flexible Space Systems." J. Guidance. Co ntrol. and 

namicL Vol. 8, No. 3, May-June 1985, pp. 374-383. 

10. . Kane, Thomas R. and Levinson, David A., "Formulation of Equations of 

Motion for Complex Spacecraft," J. Guidance a nd Co ntrol, Vol. 3, No. 2, 

March-April 1980, pp. 99-1 12. 

1 1 . Likins, P.W., "Analytical Dynamics and Nonrigid Spacecraft Simulation," 

BASA, Report No. 32-1593,1974. 

12. Likins, P.W., "Quasicoordinate Equations for Flexible Spacecraft," AlAA 

Journal, Vol. 13, No. 4, April 1975, pp. 524-526. 

13. Meirovitch, L., "Methods of Analytical Dynamics," McG raw-Hill. 1970. 

-32- 
14. Desloge, Edward A., "Relationship Between Kane's Equations and the 

Gibbs - Appell Equations," J. Gu idance. Co ntrol. and Dv namicg, Vol. 10, 

No. 1, January-February 1987, pp. 120-122. 

Jerkovsky, W., "The Structure of Multibody Equations," J. Guidance a nd 

Control, Vol. 1, No. 3, May-June 1978, pp. 173-1 82. 

15. 

16. Covington, A., Harcrow, H., Turner, J., and Keat, J., "The Next Generation 

Multi-Body Codes Modeling and Computational Requirements," AlAA 

Pvnamics Spec ialist Co nf., Paper No. 84-1 01 6-CP, Palm Springs, 

California, 17-1 8 May 1984. 

17. Fleischer, G.E., "Multibody System Applications -- and Simulations at the .. Jet Propulsion Laboratory, " proc. Svmp. Dv namics of Multlbodv 

Systems, Munich, August 29-September 3, 1977, Ed. K. Magnus, pp. 36- 

47. 

18. Wittenburg, Jens, "Dynamics of Systems of Rigid Bodies," Teubner, 

Stuttgart, 1977. 

19. Andrews, G.C. and Kesavan, H.K., "Simulation of Multibody Systems 

Using the Vector-Network Model," Proc. Sv mo. . Dv nam ics of Multi bodv 

Svstems, Munich, August 29-September 3, 1977, Ed. K. Magnus, pp. 1 - 

13. 

20. Margolis, D. L. and Karnapp, D.C., "Bond Graphs for Flexible Multibody 

Systems," proc. Svmp. Dvnamics of Mult iw Svstems, Munich, August 

29-September 3, 1977, Ed. K. Magnus, pp. 208-21 9. 

21. Lips, K.W., "Dynamics of a Large Class of Satellites with Deploying 

Flexible Appendages," Ph.D. Dissertation, Universitv of British Colombia, 

Vancouver, Canada, August 1980. 

22. Rosenthal, Dan, "Order N Formulation for Equations of Motion of 

Multibody Systems" presented at SDIO/NASA Workshoo on Multibodv 

Simulation, Arcadia, California, 1-3 September 1987. 

23. Schwertassek, Richard, "Recursive Generation of Multibody System 

Equations," presented at SDIO/NASA Wo rkshoD on Multiboa 

Simulation, Arcadia, California, 1-3 September; 1987. 

24. Wie, B., Furumoto, N., Banerjee, A.K. and Barba, P.M., "Modeling and 

Simulation of Spacecraft Solar Array Deployment," J. Guidance. C ontrol, 

md nvnaw, Vol. 9, No. 6, September-October 1986, pp. 593-598. 

25. Waites, Henry and Singh, Ramen, "Dynamics of Flexible Bodies: 

CONTOPS - A Computer Oriented Approach," presented at SDIO/NASA 

Workshop on Multibodv - Simulation, Arcadia, California, 1-3 September 

1987. 

-33- 
26. Singh, R. P. and Likins, P.W., "Singular Value Decomposition for 

Constrained Dynamical Systems," Jour nal of AD . plied Mechan ics, Vol. 52, 

December 1985, pp. 943-948. 

27. Kane, T.R., Ryan, R. R. and Banerjee, A.K., "Dynamics of a Cantilever 

Beam Attached to a Moving Base," ,L Guidance. Co ntrol and Dv namics, 

Vol. 10, No. 2, March-April 1987, pp. 139-151. 

28. Lips, K.W., "Comment on A General Formulation for Librational Dynamics 

of Spacecraft with Deploying Appendages," J. Guidance. Co ntrol. and 

Dvnamics, Vol. 9, No. 1 , Jan-Feb. 1986, pp. 125-1 26. 

29. Soosaar, Keto, Turner, James and Larkin, Laura, "Multibody Dynamics 

Modeling Issues for SDI and NASA Missions," presented at Solo/ NASA 

Workshot) on Multibodv - Simulation, Arcadia, California, 1-3 September, 

1987. 

30. Eke, Fidelis and Laskin, Robert A., "On the Inadequacies of Current MultiFlexible Body Simulation Codes," Paper No. 87-2248, AIAA Guidance, 

aviaion and Control Conf.. Monterey, California, August 17-1 9, 1987. .. 

31. Li, Dechang and Likins, Peter W., "Dynamics of a Multibody System with 

Relative Translation on Curved, Flexible Tracks," J. Guidance. Co ntrol, 

md Dvnarnics, Vol. 10, No. 3, May-June 1987, pp. 299-306. 

32. Schwertassek, Richards, "MEDYNA and its Capabilities," presented at 

SDIO/NASA WorkshoD on Multibodv Simulation, Arcadia, California, 1-3 

September 1987. 

33. Pinson, Larry and Housner, Jerrold, "A Finite Element Approach to 

Multibody Dynamics and Related Topics," presented at SDIOINASA 

2 

.. 

* 1 ' Arcadia, California, 1-3 September 

1987. 

34. Rosenthal, Dan E. and Sherman, Michael A., "Symbolic Multibody 

Equations via Kanes Method," presented at AASIAIAA Astrodvna m ics 

wialist Conf, , Paper No. 83-303, Lake Placid, New York: August 22- 

25, 1983. 

. 

35. Kreuzer, E. J. and Schiehlen, W. O., "Computerized Generation of 

Symbolic Equations of Motion for Spacecraft," J. Guidance. C ontrol. and 

pvnarnics, Vol. 8, No. 2, March-April 1985, pp. 284-287. 

36. Wittenburg, Jens and Wok, Udo, "MESA VERDE: A Symbolic Program 

for Nonlinear Articulated-Rigid-Body Dynamics," Paper No. 85-DET-151 , 

ASME Des ian Enaineerina Division Conference and Exhibit on 

Mechanical Vibration and Noise, Cincinnati, Ohio, 10-1 3 September 

1985. 

-34- 
37. 

38. 

39. 

40. 

41. 

42. 

43. 

44. 

45. 

46. 

47. 

48. 

Gluck, R. and Sridhar, S., "A Demonstrator for a Custom Architectured 

Parallel Processing Flexible Spacecraft Simulator," PIAA Dv namics 

Spec ialists Co nf., Palm Springs, California, 17-1 8 May 1984. 

Hughes, Peter C. and Skelton, Robert E., "Modal Truncation for Flexible 

Spacecraft," J. Gu idance and Co ntrol, Vol. 4, No. 3, May-June 1981, pp. 

291 -297. 

Hablani, Hari B., "Constrained and Unconstrained Modes: Some 

Modeling Aspects of Flexible Spacecraft," J. Guidance. Co ntrol. and 

Dvnamics, Vol. 5, No. 2, March-April 1982, pp. 164-173. 

Likins, P., et al., "Appendage Modal Coordinate.Truncation Criteria in 

Hybrid Coordinate Dynamic Analysis," J. Spacec raft and Roc kets, Vol. 

13, NO. 10, 1976, pp. 61 1-61 7. 

Hughes, P.C., "Space Structure Vibration Modes: How Many Exist? 

Which Ones are Important?." UTIAS Tech. Note No. 252, Toronto, 

Canada, April 1984. 

Skelton, Robert E., Hughes, Peter C. and Hablani, Hari B., "Order 

Reduction for Models of Space Structures Using Modal Cost Analysis," L 

Guidance a nd Control, Vol. 5, No. 4, July-August 1982, pp. 351-357. 

West-Vukovich, G., "Development of System Order Reduction 

Communications Report No. DOC -CR-84-044 Contractor: DY NACON), 

Ottawa, Canada, October 1984. 

Techniques Applicable to Third Generation Spacecraft," Wartment o f 

Skelton, R.E., "Model Truncation Using Controllability, Observability 

Measures," Proc. Sv mD. . Dv namics of Mu ltibodv Svstems, Munich, August 29-September 3, 1977, Ed. K. Magnus, pp. 331-344. 

Poelart, 0. , "Exact Modal Analysis for Spinning Flexible Spacecraft," 

Proc. ESA SvmDosium on Dvnamics and Co ntrol of Non-Riaid 

Sgacecraft, ESA SP 11 7, 1976, pp. 31 -38. 

Meirovitch, L., et.al., "Distributed Control of Spinning Flexible 

Spacecraft," J. Guidance a nd Control, Vol. 2, No. 5, 1979, pp. 407-41 5. 

Gates, Stephen, "DISCOS Method for Incorporation of Finite-Element 

Model," Dr;rw Interlab Memo DYN-82-1, January, 1982. 

Storch, Joel, "Dynamic Equations for Arbitrary Motion of a Flexible BodyFinite Element Idealization," Draper Lab. Memo No. DYN-82-4, to 

Stephen Gates, January 1982. 

-35- 
49. 

50. 

51. 

52. 

53. 

54. 

55. 

56. 

57. 

58. 

59. 

Strang, Gilbert and Fix, George J., "An Analysis of the Finite Element 

Method," Prentice -Hall. InL 1973. 

Dow, John O., Bodley, Carl S. and Feng, Chuan C., "An Equivalent 

Representation of Structures Composed of Repeated Elements," Paper 

No. 83-1 007, AIAA/ASME/ASCE/AHS 24th Structures St ructural 

pvnmics and Mater ials Conf., Lake Tahoe, Nevada, 2-4 May 1983. 

Sun, C.T. and Juang, J. N., "Modeling Global Structural Damping in 

Trusses Using Simple Continuum Models," Paper No. 83-1 008, 

A I AA/ASM E/ASC E/A H S 24t h St ruct u res. S t ructural Dvnamics and 

Materials Conf., Lake Tahoe, Nevada, 2-4 May 1983. 

Renton, John D., "The Beam-Like Behavior of Space Trusses," AlAA J,, 

Vol. 22, No. 2, February 1984, pp. 273-280. 

Lust, R.V. and Schmit, L.A., "Alternative Approximation Concepts for 

Space Frame Synthesis," AIAA J., Vol. 24, No. 10, October 1986, pp. 

1676-1 684. 

Poelart, Daniel, 'Continuum Methods for Multibody Systems, 'I presented 

at SDIO/NASA Workshop on Multibodv S imulation, Arcadia, California, 

1-3 September 1987. 

Von Flotow, A.H., "Traveling Wave Control for Large Spacecraft 

Structures," J. Gu idance. Control. and Ovnam icq, Vol. 9, No. 9, JulyAugust 1986, pp. 462-468. 

Von Flotow, A.H., and Schaier, B., "Wave-Absorbing Controllers for a 

Flexible Beam," J. Guidance. Cont rol. and Dv namic5, Vol. 9, No. 6, NovDec, 1986, pp. 673-680. 

Elishakoff, Isaac, "Probabilistic Methods in the Theory of Structures," 

John Wilev and So ns, 1983. 

Hughes, P.C., McTavish, D. J., Lips, K.W., and Vigneron, F.R., "A New 

Approach to Modeling Linear Viscoelastic Damping for Space 

Structures," presented at Vibration Dampina Workshop 11, sponsored by 

Flight Dynamics Lab of Air Force and Wright Aeronautical Labs, Paper 

No. ED, Las Vegas, Nevada: 5-7 March, 1986. 

Crawley, Edward F., Sarrer, George L. and Mohr, David G., 

"Experimental Measurement of Passive Material and Structural Damping 

for Flexible Space Structures," Acta Astronauticq, Vol. 10, No. 5-6, 1983, 

pp. 381-393. 

-36- 
60. Bagley, Ronald L. and Torvik, Peter J., "Fractional Calculus in the 

Transient Analysis of Viscoelastically Damped Structures," 

A I AA/ASMF/ ASCF/A HS 34th Structu res. St ructural nvnami 'cs and 

Materials Conf., Lake Tahoe, Nevada, 2-4 May 1983. 

61 . Johnson, C.D., "Design Analyses and Tests of Viscoelastically Damped 

Generic Space Trusses," Paper No. 85-DET-39, ASME Des ia q 

atneer ina Techn ical Conf, , Cincinnati, Ohio, 10-1 3 September 1985. 

62. Edberg, Donald L., "Material Damping of Simple Structures in a 

Simulated Space Environment," J. Spacec raft and Rockets, Vol. 23, No. 

3, May-June 1986, pp. 288-296. 

63. Lips, K.W. and Vigneron, F. R., "Damping Synthesis for a Spacecraft 

Using Substructure and Component Data," J. SDacecraft and Roc k ets, I' 

Vol. 23, No. 2, March-April 1986, pp. 158-164. 

64. Shock, Richard W., "Solar Array Flight Experiment," AAS Guidance and 

Control Conf,, Keystone, Colorado, February 1-5, 1986, Paper No. AAS 

86-050. 

65. Taborrok, et.al., "On the Dynamics of an Axially Moving Beam," A 

Franklin Institute, Vol. 297, No. 3, 1974, pp. 201-220. 

66. Lips, K.W., and Modi, V.J., "Dynamical Characteristics Associated with 

Deploying, Orbiting, Beam-type Appendages," presented at AIAA/AAS 

Astrodvnamics Conference, Palo Alto, Calif., August 7-9, 1978, Paper No. 

78-1 399. 

67. Keat, J. and Turner, J., "Large Space Structure Deployment Dynamics," 

Drape[ Labs, Report No. CSDL-R-1843, March, 1986. 

68. Housner, Jerrold, "Convected Transient Analysis for Large Space 

Structures Maneuver and Deployment ," presented at 

AIAA/ASME/ASCE/ AHS 25t h Structures. Structu ral Dvnamics and 

Materials Conf., Paper No. 84-1 023, Palm Springs, California: May 17- 

18,, 1984, pp. 616-629. 

69. Hurty, W.C., and Rubinstein, M.F., "Dynamics of Structures," PrenticeHall, 1964. 

70. Meirovitch, L., "Analytical Methods in Vibrations," MacMillan Co., 1967. 

71. Vigneron, F. R., "Stability of a Freely Spinning Satellite of CrossedDipole Configuration," C.A.S.I. Transact ions, Vol. 3, No. 1 , March 1970. 

72. Hughes, P.C., and Fung, J.C., "Liapunov Stability of Spinning Satellites 

with Long Flexible Appendages," J. Celestial Mechanics, Vol. 4, 1971, 

pp. 295-308. 

-37- 
73. 

74. 

75. 

76. 

77. 

78. 

79. 

80. 

81. 

Budynas, R.G., "The Dynamics of Orbiting Elastic Bodies," Ph.D. 

Dissertation, University of Massachusetts. Mechanical Ena ineeri ng, 

1970. 

Vigneron, F.R., "Comment on 'Mathematical Modeling of Spinning Elastic 

Bodies for Modal Analysis," AlAA A, Vol 13, No. 1, 1975, pp. 126-1 27. 

Kaza, K.R.V. and Kvaternik, R.G., "Nonlinear Flap-Lag-Axial Equations'of 

a Rotating Beam," AlAA J., Vol 15, No. 6, June 1977, pp. 871 -874. 

Lips, K.W., "The Foreshortening Effect," internal memo to C.C. Browning, 

Space Systems Operations (SSFCS), n~, Clearwater, 

Florida, 12 June 1987. 

Hutton, David V., "Modal Analysis of a DeDlovable Truss Using the Finite .* 

Element Method," J. Spacec rafi and Rockets, Vol. 21, No. 5, Slept. - Oct. 

1984, pp. 468-472. 

Hertz, Terrence J., "Displacement Dependent Friction in Space Structural 

Joints," pIAA J,, Vol. 23, No. 12, December 1985, pp. 1998 - 2000. 

Prucz, Jack, "Analysis of Design Tradeoffs for Passively Damped 

Structural Joi nt s ,I1 ,I . Spacecraft and Rockets, Vol. 23, No. 6, Nov.-Dec., 

1986, pp. 576-584. 

Prucz, Jack and Trudell, R.W. , "Experimental Characterization of 

Passively Damped Joints for Space Structures," J. Spacec raft a nd 

Rockets, Vol. 23, No. 6, Nov.-Dec. 1986, pp. 568-575. 

Proceedings of the Ul SAT/FSA SDonsored Colloauium on: 

"Dynamic Effect of Liquids on Spacecraft Attitude Control," Washington, 

D.C.: April 25-26, 1984. 

(Reference a-e were all presented at the INTELSAT/ESA Colloquium) 

a. Pfeiffer, F., "On Contained Rotating Fluids in Satellite 

D y n am ics" . 

b. Vanyo, James I., "Fluid Velocities and Energy Dissipation: 

Experiments". 

c. Vreeberg, J.P.B., "Dynamics of Weightless Liquids in 

Closed Contain e rs". 

d. Harrison, J., "A Review of the Scale Model Test Method for 

Predicting Spacecraft Nutation Divergence". 

-38- 
82. 

83. 

84. 

85. 

86. 

87. 

88. 

89. 

90. 

91. 

92. 

e. Mclntyre, J.E., "Spacecraft Fluids Interaction Program at 

Hughes Aircraft Company". 

Murphy, Charles H., "Side Moment Exerted by a Two-Component Liquid 

Payload on a Spinning Projectile," J. Guidance. Co ntrol. and Dvnamics, 

Vol. 10, No. 1, Jan.- Feb. 1987, pp. 20-26. 

Hashimoto, Hiroyoki and Sudo, Seiichi, "Frequency Characterisitics of a 

Vol. 22, No. 6, Nov. - Dec., 1985, pp. 649-655. 

Bubble Cluster in a Vibrated Liquid Column, I' J. Spacec raft a nd Roc kets, 

deLafontaine, J. and Hughes, P.C., "A Semi-Analytic Satellite Theory for 

Orbital Decay Predictions," Paper No. AAS 85-31 9, AAS/AIAA 

PIst rod! n am ics spec ialist Conf, , Vail, Colorado, 12-15 August 1985. 

Menees, Gene P., Park, Chul, Wilson, John F. and Brown, Kevin G., 

"Determining Atmosphere Density Using a Space Launched Projectile," 

J. SDacecraft and Rockets, Vol. 23, No. 3, May - June 1986, pp. 273-280. 

Sincarsin, G. 6. and Hughes, P.C., "Torques from Solar Radiation 

Pressure Gradient During Eclipse," J. Gu idance. Cont rol and Dynamics, 

Vol. 6, No. 6, November - December 1983, pp. 51 1-51 7. 

Shrivastava, S. K. and Modi, V. J., "Satellite Attitude Dynamics and 

Control in the Presence of Environmental Torques - A Brief Summary," 

nce. Control and DvnamiG, Vol. 6, No. 6, November - December 

1983, pp. 461-471. 

Lips, K.W., Graham, W.B., Vigneron, F.R., Hunter, D. G., "Dynamics and 

Control Characteristics for the WISP 300m Dipole AntenndShuttle 

Configuration," /IAS/AIAA Astrodvnam' ICS SDec ialist Conf.. Paper No. 

AAS 85-365, Vail, Colorado: August 12-1 5, 1985. 

Joshi, S.M., Garrett, L. B., and Andersen, G.C., "Maneuvering of Large 

Flexible Space Antennas Using Optimal Discrete-Time Controllers," 

Paper No. A84-36833, 27th Midwest Sv mDosium on Circuits and 

-, Morgantown, West Virginia, 1 1-1 2 June 1984. 

Hale, Arthur L. and Lisowski, Ronald J., "Characteristic Elastic Systems of 

Time-Limited Optimal Maneuvers," J. Guidance. Co ntrol and Dv namics, 

Vol. 8, No. 5, Sept. - Oct. 1985, pp. 628-636. 

Skaar, S.B., Tang, L. and Yalda-Mooshabad, I., "On-off Attitude Control of 

Flexible Satellites," J. Guidance. Contro. a nd Dvn amics.Vol. 9, No. 4, 

July - August 1986, pp. 507-51 0. 

Juang, J. N., Horta, L. G. and Robertshaw, H.H., "A Slewing Control 

Experiment for Flexible Structures," J. Guidance. Control. and Dvnamics, 

Vol. 9, No. 5, September - October 1985, pp. 599-607. 

-39- 
93. 

94. 

95. 

96. 

97. 

98. 

99. 

100. 

101. 

102. 

103. 

104. 

Vadali, S. R. Kraige, L. G. and Junkins, J. L., "New Results on the Optimal 

Spacecraft Attitude Maneuver Problem," J. Guidance. Control. and 

Dvnamics, Vol. 7, No. 3, May - June 1984, pp. 378-380. 

Dwyer 111, Thomas A.W., "Exact Nonlinear Control of Spacecraft Slewing 

Maneuvers with Internal Momentun Transfer," J. Guidance. Contro 1. and 

Dvnamics, Vol. 9, No. 2, March - April 1986, pp. 240-247. 

Gear, C. W., "Diff erential-Algebraic Equations," Computer Aided Analvsis 

and OD timization of Mechanical Svste rn Dvnamics (ed. E. J. Haug), 

Springer-Verlag, Heidelberg, 1984. 

Gear, C. W., "Simultaneous Numerical Solution of Differential-Algebraic 

Equations," Trans. on Circuit Theory , VOI. CT-18, NO. 1 , 1971. .. 

Orlandea, N., Chace, M.A., and Calahan, D.A., "A Sparcity Oriented 

Approach to Dynamic Analysis and Design of Mechanical systems, Parts 

I and 11," ASMF J. of E naineerina for lndu- , Ser. B, Vol. 99, 1977, pp. 

773-784. 

Barman, N.C., "Design Sensitivity Analysis and Optimization of 

Constrained Dynamic Systems," Ph.D. Thesis, Universitv of Iowa, 1979. 

Haug, E. J., Wehage, R.A., and Barman, N.C., "Design Sensitivity 

Analysis of Planar Mechanism and Machine Dynamics, J. of Mechan id 

Trans of ASME, Vol. 103, 1981 , pp. 560-570. 

Hachtel, G. D., Brayton, R.K., and Gustavson, F.G., "Sparse Tableau 

Approach to Network Analysis and Design," IEEE Transactions on Circuit 

Theory, Vol. CT-18, No. 1 , 1971. 

Petzold, L., "DifferentiaVAlgebraic Equations Are Not ODE'S," SlAM J. of 

Sci. Stat . Cornput, Vol. 3, No. 3, 1982. 

Baumgarte, J., "Stabilization of Constraints and Integrals of Motion in 

Dynamic Systems," Computer Methods in ADD lied Mechanics in 

Engrneering, North Holland Publication Co., 1972, pp. 1-1 6. 

.. 

Wehage, R.A., "Generalized Coordinate Partitioning in Dynamic Analysis 

of Mechanical Systems," Ph.D. Thesis, Universitv of Iowa, 1980. 

Wehage, R.A. and Haug, E.J., "Generalized Coordinate Partitioning for 

Dimension Reduction in Analysis of Constrained Dynamic Systems," LGaf 

ch. Desian. - Trans. ASE Vol. 104, No. 1, 1982, pp. 247-255. 

-40- 
APPENDIX 

PREVIOUS ASSESSMENTS 

OF MULTIBODY FORMULATIONS 
TABLE 1 

Jerkovsky = Reference 15 
Table 1 Comparison of 10 n-body dynamics formulations 

n-bodv confirruration Dynamics state variables Comments Date.Authors 

' Tree of rigid bodies; Inertial angular velocities Uses barycenters and '65 Hooker, Margulies 

no relative translation. of each body; inertial augmented bodies; '67 Roberson, Wittenbu pg 

linear velocity of composite. obtained via Lagrange constraint torques 

mu1 tipliers. 

Tree of rigid bodies; no Relative angular veloci- Use of relative angular '67 Velman 

relative translation (ex- ties between bodies; velocities result sin forcept for point masses). inertial linear velocity of 

composite. constraint torques 

mation of composites; 

removed by use of 

symmetric projectors. 

' Cluster of rigid bodies; Relative gimbal angle Constraint torques do '67 Palmer 

no relative translation. rates between bodies; not appear. inertial linear velocity of 

composite. 

Tree of rigid bodies; no Free components of Constraint torques do '69 Russell 

relative translation. angular momentum of not appear, mass matrix 

outward composites. is not computed 

explicitly. 

Tree of rigid bodies23; Relative gimbal angle Equations are obtained '69 Farrenkopf 

terminal bodies may be rates between bodies; inductively; constraint '7 1 Ness, Farrenkopf 

flexible;% hinge points inertial linear velocity torques do not appear; 

may be time-dependent; 

no relative translation. Body 1. ''mass matrix." 

of a material point of uses a nonsymmetric 

Tree of rigid bodies25 ; Relative gimbal angle Uses barycenters and '70 Hooker 

terminal bodies may be rates between bodies; augmented bodies; con- '73 Likins 

flexible26 no relative inertial linear velocity of straint torques do 

translation. composite. not appear. 

Tree of flexible bodies; Relative gimbal angle rates; Uses barycenters and '72 Roberson 

closed loops treated via relative displacement rates; augmented bodies; '74 Wittenburg Lagrange multipliers30 ; inertial linear velocity of constraint torques do '74, '75 Boland, Samin, 

relative translation al- composite. not appear except with Willems 

lowed. closed loops. 

Tree of rigid bodies; Relative gimbal angle rates; Does not use bary- '74, '75 Frisch 

terminal bodies may be inertial linear velocity of centers and augmented '74, '77 Ho 

flexible; no relative a material point of Body 1. bodies; constraint '75 Hooker 

translation. torques do not appear. 

Chain of flexible bodies; Relative gimbal angle rates; Uses quasistatic modes '74 Ho, Hooker, 

no relative translation. inertial linear velocity of plus vibration modes; Margulies, Winarske 

a material pint of Body 1. constraint forces and 

torques do not appear. 

Arbitrary configuration Inertial angular velocities Free body equations are '75 Bodley, Devers, 

of flexible bodies; closed of each body; inertial linear written for each body; Park 

loops allowed; relative velocity of material point all constraint forces and 

translation allowed, pre- of each body. torques are obtained via 

scribed motion allowed. Lagrange multipliers.