Large space structures (LSSs) and other dynamical systems of current interest are often extremely complex assemblies of rigid and flexible bodies subjected to kinematical constraints. A formulation is presented for the governing equations of constrained multibody systems via the application of singular value decomposition (SVD). The resulting equations of motion are shown to be of minimum dimension

Likins, P. W.

Singh, R. P.

English

NASA Technical Reports Server

N89-24667 
C O N S T R A I N T  E L I M I N A T I O N  
I N  O Y N A M I C A L  SYSTEMS 
R. P. Singh and P. W. Likins* 
DYNACS Engineering Company, Inc 
Clearwater, FL 
T H E M E  
Large Space Structures (LSS) and other dynamical systems of current interest 
are often extremely complex assemblies of rigid and flexible bodies subjected 
to kinematical constraints. This paper presents a formulation of the govern- 
ing equations of constrained multibody systems via the application of singular 
value decomposition (SVD). The resulting equations of motion are shown to be 
of minimum dimension. 
The motivation for this work was the development of a generic computer program 
for simulating space structures an? similar electromechanical systems amenable 
to mathematical representation as a set of flexible bodies interconnected in a 
topological configuration. This representation may include closed loops of 
bodies, prescribed motion, or other constraints that may qualify as simple 
monholonomic. The equations of motion appropriate for a set of flexible bodies 
in an open loop configuration appear in Refs. 1, 2 .  A computer program 
(TREETOPS) developed to simulate the dynamic response of flexible structures 
in a topological tree configuration is described in Ref. 3 .  The SVD technique 
of the present paper is being incorporated in an extension of the TREETOPS 
program that permits application to constrained systems. This extension 
permits direct use of the dynamical equations for the less constrained system 
in Refs. 1, 2 ,  with augmentation by kinematical constraint equations and re- 
duction of dimension by SVD. 
Basically, there are two conceptual approaches to solving the equations of 
motion of such systems. (1) One can introduce unknown forces and torques at 
the interfaces between constrained bodies (often accomplishing this symbolical- 
ly with Lagrange multipliers), and then solve the dynamical equations simul- 
taneously with the constraint equations to determine the constraint forces and 
torques as well as the kinematical variables, Ref. 4 .  ( 2 )  Alternatively, one 
can use the constraint equations to reduce the dimension of the system of 
dynamical equations to be solved by partitioning generalized coordinates, Refs. 
5, 6. Techniques presented in Refs. 4 ,  5, 6 may encounter numerical singular- 
ities. Also, systems undergoing large motion may present problems of 
inconsistency in the constraints such as three-dimensional loops during the 
system motion becoming two dimensional or one-dimensional loops. In what 
follows, the SVD method will be shown to avoid mathematical singularities. 
*Lehigh University, Bethlehem, PA. 
537 
https://ntrs.nasa.gov/search.jsp?R=19890015296 2020-03-20T02:57:39+00:00Z
. .  . , 
... . . .  
C O N T E N T S  
Singular Value Decomposition: Orthogonal decomposition of an mxn matrix L by 
singular value decomposition is closely related to the eigenvalue-eigenvector 
decomposition of the symmetric positive semidefinite matrices L L and LL . 
Let rsm be the rank of L. Then there are orthogonal matrices U and V of order 
mxm and nxn rexpectively such that 
T T 
UTLV = r" Ol (1) 
where C = diag (A1, X2,....,Xr) and X L X  > .lXr>O. 1 2-" 
The diagonal elements of the decomposition are called the singular values of 
the matrix L. The singular values are unique, although U and V are not. 
It is easy to verify that 
T T  2 V L LV = diag ( C  , 0) ( 2 )  
2 2 T 
Thus (A1, ...., A 
descending order and the requirement that A. be nonnegative completely 
determines the A 
has a multiple eigenvalue X >O, the corresponding columns of V may be chosen 
as an orthonormal basis for the space spanned by the eigenvectors corresponding 
to X * 
) must be the nonzero eigenvalues of L L arranged in the r 
T1 The eigenvectors of L L are the columns of V. If LTL 
2 
i '  
2 
From eq. (1) 
L = usv T 
Now with proper partitioning of U and V eq. ( 3 )  can be expressed as 
From the above one obtains 
-1 u1 = LV1 c 
( 3 )  
(51 
Thus once V is chosen U is obtained by eq. (5). The matrices U and V may 1 1 2 2 
be any matrices with orthonormal columns spanning the null spaces of LT and L, 
respectively. It is worthwhile to mention that the null space of L is the 
space of all vectors x such that 
538 
Lx = 0 (6) 
+ 
With the orthogonal decomposition given by eq. ( 3 ) ,  an nxm matrix L , called 
the pseudoinverse of L, is defined by 
+ 
L is uniquely defined by L; it does not depend on the particular orthogonal 
decomposition of L. 
Application of SVD to Dynamical System with Constraints: 
comprise a set of generalized coordinates that fully defines the configuration 
of the dynamical system. The equations of motion of the system can be written 
as 
Let q = ql, ....,qn 
where the elements of nxn matrix M are functions of q's and the inertia pro- 
perties of the system; the elements of nxl column vector F are functions of q's, 
their time derivatives 4 ' s  and applied forces (moments) on the systems. If 
the generalized coordinates are related by constraint equations then they are 
not independent and the right hand side of eq. (8a) will also include the non- 
working forces of constraints. Let the unknown constraint forces be denoted 
Fc. 
the following form 
Now for the general case of constrained dynamical system, eq. (Ea) takes 
M{ = F + FC 
Suppose however that the constraint equations can be written as 
A 4  = B (9)  
where A is of dimension mxn (m<n) and B is an mxl column vector. 
Holonomic constraint equations can always be placed in the f o r m  of eq. (9) and 
nonholonomic constraints in the class called Pfaffian or simple have this 
structure also. 
If the rank of matrix A is rlm then r of the kinematical variables in q are 
related by eq. (9) and there are only n-r independent generalized coordinates. 
In other words the dynamical system possesses n-r degrees of freedom. 
The SVD of the mxn matrix A provides 
T 
A = USV (10) 
The orthogonal matrices U and V (of dimension mxm and nxn, respectively) are 
partitioned as 
539 
L J 
where U and V are respectively mxr and nxr matrices; U and V are respec- 
tively mx(m-r) and nx(n-r) matrices. Note that r is the rank of A. 
1 1 2 2 
Because AV = 0, eq. (9)  is satisfied by 2 
4 = A+B + v 2 2 (13) 
for any vector i, A+ is the pseudoinverse of A. 
reduced set of (n-r) coordinates. 
We shall refer to z as the 
Differentiation of eq. (9)  with respect to time yields 
.. .. . 
Aq = -Aq+B 
or, A{ = B' (14) 
Following eq. (13) express 6 in terms of 2 as 
.. 
(15) 
Note from eq. (13) or eq. (15) that V maps the n kinematic variables 4 (or 4) 
.. + q = A B' + V2z 
2 
to n-r variables t (or z ) .  
is given as 
Thus a consistent set of equations of motion in E 
T T T c  T +  V M V ; = V F + V  F - V  MAB' 2 2  2 2 2 (16) 
The coefficient of E is a symmetric positive definite matrix with the charac- 
teristic of an "inertia matrix" for the reduce? set of coordinates z .  
With the Lagrange multiplier method, F is established via (see Ref. 4) C 
T F ~ = A ~  
where (I is the column vector of Lagrange 
Premultiply eq. (17) by V2 to obtain the T 
T T  VTFC = V2A a 
(17) 
multipliers. 
following 
T = (AV2) a 
= o  
540 
! 
Thus it is seen that the nonworking constraint forces make no contribution to 
the equations of motion (eq. (16)) and need not be recorded. 
Employing the transformations given by eqs. (13) and (151, the minimum 
dimension governing differential equations of motion are given by 
T T T +  V M V i = V F - V  MAB' 2 2  2 2 
and 
+ ; = A B + V ;  2 
(19) 
(20) 
This method eliminates the forces of constraints which when included serve not 
only to enlarge the dimension of the dynamical system but also quite often 
introduce computational problems. 
A C K N O W L E D G M E N T  
An excellent treatment of the computational efficiency of the SVD is given 
in Ref. 7. 
R E F E R E N C E S  
1. Singh, R. P. and Likins, P. W., "Manipulator Interactive Design with 
Interconnected Flexible Elements", Proceedings of the 1983 Automatic 
Control Conference, San Francisco, CA., June 1983, pp. 505-512. 
2. Singh, R. P., VanderVoort, R. J. and Likins, P. W., "Dynamics of Flexible 
Bodies in Tree Topology - A Computer Oriented Approach", Proceedings of 
the 25th SDM and AIAA Dynamics Specialist Conference, Palm Springs, CA., 
May 1984, Part 2, pp. 327-337. 
3. VanderVoort, R. J. and Singh, R. P., "A Pointing Control System Simulation 
and Generic Model Generator for Multi-Body Flexible Structures" Proceedings 
of the Ninth DARPA Strategic and Space Symposium, Monterey, CA., October 
1983. 
4. Bodley, C., Devers, A., Park A. and Frisch, H., "A Digital Computer 
Program for Dynamic Interaction Simulation of Controls and Structures 
(DISCOS)", NASA Technical Paper 1219, May 1978. 
5. Kane, T. R,. and Wang, C. F., "On-the Derivation of Equations of Motion", 
J. SIAM, Vol. 13, 1965, pp. 487 - 492. 
6. Wehage, R. A. and Haug, E. J., "Generalized Coordinate Partitioning for 
Dimension Reduction in Analysis of Constrained Dynamic Systems", ASME 
Journal of Mechanical Design, Vol. 104, January 1982, pp. 247-255. 
7. Juang, Jer-Nan and Pinson, Larry D., "Application of Singular Value 
Decomposition to Structural Dynamics Systems with Constraints," AIAA 
Paper No. 85-0687-CP, April 1985. 
541 


Constraint elimination in dynamical systems

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