The computation of constrined dynamical systems: matching physical ...

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THE COMPUTATION OF CONSTRAINED DYNAMICAL SYSTEMS: MATCHING PHYSICAL MODELING WITH NUMERICAL METHODS Researchers have investigated modeling and computation of constrained dynamical systems, but scientists and engineers sometimes overlook consistent matching of mathematical modeling and numerical methods. The authors use multibody system software to study offhighway vehicles, automobile and android crash simulation, and planetary ephemerides.

M

odeling and computing constrained dynamical systems—a time-varying system consisting of bodies connected by joints that can be represented by constraint equations—involves modeling real-world problems and constructing differential algebraic equations and numerical integration. The purpose is to obtain generalized coordinates, their time derivatives, and body forces for dynamic analysis and simulation. Researchers have developed general approaches and software packages for multibody kinematic and dynamic analysis,1–4 yet specific examination of systems from first principles offer differing perspectives of old and new problems and reveal features that might have initially passed unnoticed. For example, an incompatibility between physical modeling and numerical methods might lead to inaccurate conclusions about system design. Authors such as Edward Haug, 5 Parviz Nikravesh,2 and Ahmed Shabana3,4 have substan-

1521-9615/01/$10.00 © 2001 IEEE

BUD FOX, LESLIE JENNINGS, AND ALBERT Y. ZOMAYA University of Western Australia

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tially pursued planar and spatial multibody modeling and constrained variational dynamics. The multibody system equations that result from the modeling are in fact differential-algebraic equations (DAEs), because the system equations are augmented with algebraic constraints that define the connectivity of the system’s components. (For information about additional DAE research, see the sidebar.) In this article, we use our own planar and spatial constrained variational system software, Multibody System,6–8 to compute offhighway vehicle dynamics, crash simulations, and planetary ephemerides and to record the integration method’s accuracy. Studying force modeling in multibody systems by investigating progressively smoother body interaction forces should encourage scientists and engineers to develop new methods to detect numerical problems in system computation. These methods might involve replacing the forces, in part with the equivalent constraint equations, or using stiffness detection techniques to employ further physical modeling near the time interval concerned. Multibody system equations The following derivation follows the work of Shabana and shows how we can formulate the general equations of dynamic equilibrium for

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Related work Multibody system equations are in fact differential-algebraic equations (DAEs), because the system equations are augmented with algebraic constraints defining the connectivity of the system’s components. Bernd Simeon and his colleagues1 provide a summary of the theory, computation, and applications of DAEs. Kathryn Brenan and her colleagues introduce the basic types of DAEs, constrained variational problems, the theory, solvability and index concept, linear and nonlinear systems, numerical methods, algorithms, applications and DAE examples.2 Uri Ascher and Linda Petzold have carefully organized the study of ODEs and DAEs and present methods for their computation.3 Ahmed Shabana has studied track vehicle dynamics,4–7 his work pursuing a different method of dynamic equations computation. The investigation of constraint compliance reveals the difficulty encountered in numerical computation of the track vehicle equations. Contact force modeling rather than the differentiation of the constraint equations appears to be the cause of computational difficulty, and new methods and algorithms might be required to detect numerical problems in advance, to facilitate integration. (For further details of track vehicle dynamics see the article by Bud Fox and his colleagues.8) We have also investigated automobile crash simulation involving two planar android multibody system models. The first is a simplified android system, and the second is a 46-body-part android muscle-skeletal and vertebral system. The soft android tissue contacts result in smoother contact force models and, hence, are easier to compute. For further details of automobile and android modeling and computation, see the article by Fox and his colleagues.9 Planetary ephemerides computed by JPL are available in the Astronomical Almanac.10 We inves-

multibody systems using generalized Cartesian coordinates and the virtual work principle.4 This principle states that the virtual work due to the inertia forces is equal to the sum of the virtual work due to the externally applied forces and constraint forces. That is,

δWI,i = δWe,i + δWc,i.

JANUARY/FEBRUARY 2001

(1)

tigated the numerical accuracy of integrating smooth astrophysical systems by computing the motion of an artificial satellite and ephemerides of the planets of the solar system. For further details of astrophysical systems see the article by Fox and his colleagues.11

References 1. B. Simeon, C. Fuhrer, and R. Rentrop, “Differential-Algebraic Equations in Vehicle System Dynamics,” Surveys on Mathematics for Industry, 1991, pp. 1–37. 2. K.E. Brenan, S.L. Campbell, and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier Science, New York, 1989. 3. U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, Penn., 1998. 4. T. Nakanishi and A.A. Shabana, “Contact Forces in the Nonlinear Dynamic Analysis of Tracked Vehicles,” Int’l J. Numerical Computing Methods in Eng., Vol. 37, 1994, pp. 1251–1275. 5. T. Nakanishi and A.A. Shabana, “On the Numerical Solution of Tracked Vehicle Dynamic Equations,” Nonlinear Dynamics, Vol. 6, No. 6, Dec. 1994, pp. 391–417. 6. M.K. Sarwar, T. Nakanishi, and A.A. Shabana, “Chain Link Deformation in the Nonlinear Dynamics of Tracked Vehicles,” J. Vibration and Control, Vol. 1, No. 2, May 1995, pp. 201–224. 7. A.A. Shabana, Computational Dynamics, John Wiley & Sons, New York, 1994. 8. B. Fox, L.S. Jennings, and A.Y. Zomaya, “Numerical Computation of Differential Algebraic Equations for Non-linear Dynamics of Multibody Systems involving Contact Forces,” to be published in ASME J. Mechanical Design, 2000. 9. B. Fox, L.S. Jennings, and A.Y. Zomaya, “Numerical Computation of Differential-Algebraic Equations for Non-linear Dynamics of Multibody Android Systems in Automobile Crash Simulation,” IEEE Trans. Biomedical Engineering, Vol. 46, No. 10, 1999, pp. 1199–1206. 10. The 1998 Astronomical Almanac, US Government Printing Office, Washington, DC, 1997. 11. B. Fox, L.S. Jennings, and A.Y. Zomaya, “Numerical Computation of Differential Algebraic Equations for the Approximation of Artificial Satellite Trajectories and Planetary Ephemerides,” ASME J. Applied Mechanics, Vol. 67, No. 3, Sept. 2000, pp. 574–580.

The virtual work of the externally applied forces acting on the rigid body i is

δWe,i = QeT,iδqi

(2)

where Qe,i is the vector of generalized externally applied forces corresponding to the vector of generalized planar coordinates,

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qi = [Rx,i, Ry,i, θi]T.

(3)

The virtual work of the joint constraint forces acting on the rigid body is

δWc,i = QcT,iδqi

(4)

where Qc,i is the vector of generalized constraint forces corresponding to the generalized coordinates qi. Finally, the virtual work due to the inertia forces is given as

∫

δWI ,i = ρi r˙˙iTδri dVi

(5)

Vi

where ρi and Vi are the density and volume of body i, respectively. The position of an arbitrary point on the body i with respect to the global coordinate system is ri = Ri + Aiui

(6)

dA ⇒ r˙i = R˙ i + i uiθ˙i dθi

(7)

and the rotation matrix of the body i with angle of rotation θi is cos θi − sin θi  Ai =  .  sin θi cos θi 

(8)

We can express the virtual work of the inertia forces of body i as

δWI ,i = [ Mi q˙˙i − Qv,i ]T δqi

C(q, t) = 0

(12)

and differentiating twice with respect to time gives Cq q˙˙ = Qd = −(Cq q˙ )q q˙ − 2Cqt q˙ − Ctt .

(13)

Augmenting Equation 11 with Equation 13, we obtain the complete system of 3NB + NC equations in 3NB + NC unknowns,  M CqT   q˙˙  Qe  C 0  λ  = Q   q    d 

(14)

For spatial systems, Euler parameters Θi = (θ 0 ,θ1,θ 2 ,θ3 )iT

(15)

rather than Euler angles are used, resulting in the generalized coordinates qi = [ Rx Ry Rz θ 0 θ1 θ 2 θ3 ]iT .

(16)

We can show that the rotation matrix is

θ Ai = I + sin θiV˜i + 2(V˜i )2 sin 2  i   2

(17)

and the theoretical constraint equation for body i is ΘiT Θi − 1 = 0

(18)

(9) The kinetic energy of body i is

where Qv,i is the vector of centrifugal inertia forces. By substituting Equations 2, 4, and 9 into Equation 1, we obtain [ Mi q˙˙i − Qv,i − Qe,i − Qc,i ]T δqi = 0.

(10)

We can show that the system equations are Mq˙˙ + CqT λ = Qe

1 ρi r˙iT r˙i dVi 2

∫

(19)

Vi

and we can show that Ti =

1 ˙ 1 ˙ ˙ Ri mRR,i R˙ i + Θ i mΘΘ,iΘi . 2 2

(20)

(11)

which represents a system of 3NB equations in 3NB + NC unknowns. NB is the total number of bodies in the planar system, and NC is the total number of independent constraint equations. Because there are more unknowns than equations, the constraint equations are adjoined to the system equations. We can write the constraint equations denoting body connectivity as

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Ti =

By substituting this expression into the Lagrange equations of motion d  ∂Ti  ∂Ti = QiT  − dt  ∂q˙i  ∂qi

(21)

we arrive at the equations of motion for body i, which are

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Body i

Yi

Xi

r θi

Yj up up

j

Xj Ri Rj

i

pj

Y

rp

rp

ij

j

rp

Surface body j

pi

i

X Global coordinate system

Figure 1. Body surface contact geometry indicating vector information required to complete the body’s spring-damper contact force model.

˙˙   QR   0  mRR,i 0   R i = i  Q  + −2G˙ T I ω  . (22)  0 mΘΘ,i  Θ ˙˙ i i ΘΘ , i    i   Θi   

Augmenting these equations with the constraint equations yields the spatial augmented system equations for body i ˙˙  mRR,i 0   0   R i   0 m  2ΘT  Θ ˙˙ ΘΘ,i   i    i   0   λi   [0 2Θi ]  Qe, R   0  i =  Qe,Θi  + Qv,i  ˙ T Θ   0  −2Θ i i

˙ x, t ) = 0 F( x,

(23)

from which we can construct the complete system of equations, repeated here as  M CqT   q˙˙  Qe  C 0  λ  = Q .  q    d 

(24)

Differential algebraic equations DAEs are differential equations augmented

JANUARY/FEBRUARY 2001

with algebraic constraints. Linda Petzold and her colleagues have published material on the solvability and computation of DAEs; their book gives an introduction to DAEs and references therein concerning the earlier work on the theory and computation.9 Uri Ascher and Petzold5 present many important numerical methods for the computation of ordinary differential equations (ODEs) including initial-value problems and boundary-value problems, and DAEs. A DAE might be of the form (25)

The DAE of the multibody system considered here is of the form  I 0 0  q˙  v     ∂C T    ˙ = v f t M = , 0 q ( )  Qe    ∂q  µ˙     C(q, t ) 0 0 0 

(26)

where µ˙ = λ and v˙ = q˙˙. However, the matrix on the left-hand side is singular. Differentiating the third equation of the matrix–vector system given by Equation 26, with respect to time twice,

31

(a)

(b)

Figure 2. (a) Simple android model whose generalized body parts represent clusters of organs and is used in automobile accident simulation. (b) Vertebral android model that shows all vertebrae in the spinal column and is used to model android dynamics in both forward and rear automobile accident simulation.

tional and numerical difficulties that can arise as a result of differentiating the constraint equations. The constraint equations might not be satisfied as the integration progresses, and they do not recommend excessive differentiation of the constraints.

yields  I  0   0 

0 M

∂C ∂q

 0  T  ∂C       ∂q   0  

 q˙   v˙  = f (q,q˙, t ) = f (q, v, t ) ,  ˙ µ  (27)

which has a nonsingular leading matrix providing ∂C/∂q is of full rank for all time. The following definition classifies a DAE with respect to differentiation of the system equations given by Equation 25. The minimum number of times that all or part ˙ x, t ) = 0 must be differentiated of the DAE F( x, with respect to t to determine x˙ as a continuous function of x and t, for t in some interval, is the index of the DAE.9

The original system has been differentiated twice, and the substitution of µ˙ = λ can be considered as an additional differentiation. This results in the ODE in Equation 27 and hence the original system Equation 26 is a DAE of index three. Note that here

µ(t ) =

t

∫0 λ(τ )dτ

is computed by the ODE software. To find λ, µ(t) must be differentiated—this is, an unstable process. Ascher and Petzold10 and Kathryn Brenan and her colleagues9 discuss the computa-

32

Figure 3. Track vehicle model showing the sprocket, idler, roller, and track links used for offhighway vehicle modeling.

Multibody models and contact forces Investigating simple models reveals dynamical information useful for modeling larger systems. Figure 1 represents a body-surface contact model. We can observe oscillatory motion through this form of contact and change the spring and damping coefficients to simulate differing material compounds. The model employs a spring-damper element between points pi and pj, the former residing on the most penetrated part of the body and the latter on the surface. The magnitude of this force is Fcij = kδ ij + cδ˙ij

(28)

where k and c are the spring and damping coefficients, respectively, and

δij = Ry,i – Ry,j – r.

(29)

Shabana4 states that the virtual work due to a force with magnitude Fcij is

δW = Fcij

rpij  ∂rpij ∂rpij  δqi     rpij  ∂qi ∂q j  δq j 

= QiTδqi + QTj δq j

(30)

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4. A front impact of 15000 N is applied to a vertebral model of an automobile occupant restrained as shown. Images are captured at 0.025-second intervals.

where  I  rp Q  Qi =  R, i  = Fcij u T AT  ij  Qθ , i   pi θ , i  rp ij

Fcij = kδ ij + cδ ijδ˙ij (31a)

and  I  rp Q  Q j =  R, j  = − Fcij u T AT  ij  Qθ , j   p j θ , j  rpij

(31b)

The position vector written in the global coordinate system is rpij = rpi − rp j = Ri + Ai u pi − R j − Aj u p j

(32)

where   3π  r cos 2 − θ i   u pi =    r sin 3π − θ i    2   

(33a)

and  R − R j, x  u p j =  i, x  . 0 

(33b)

We can use other contact models such as the progressive pseudo-damping contact force7

JANUARY/FEBRUARY 2001

(34)

to model other physical systems such as those in Figure 2. Track vehicle modeling (see Figure 3) uses the contact force models we showed earlier, and changing the spring and damping coefficients to simulate soft body contacts can approximate the behavior of harder bodies. Employing specialized constraints in the contact region can alleviate numerical computation dealing with “stiffer systems.” Results Off-highway vehicle dynamics, crash simulation, and astrophysical modeling all involve sparse systems of equations and require Gaussian elimination to be performed at each time step in the computation and integration. Spatial systems may require Euler parameters as their orientation variables rather than Euler angles, because according to Shabana,3 the body-rotation matrix might not be well defined for certain body orientations. Also, for certain values of time and angular velocity ω, the submatrix mθθ,i may be singular and hence the mass matrix Mi alone may be singular, although the leading coefficient matrix is still nonsingular. The following matrix in-

33

version technique for the computation of the system equations should not be used, because it requires that the upper left block A be invertible (for all time steps).  A B C D

−1

 A−1 + E∆−1 F − E∆−1  = −1 ∆−1   −∆ F

(35)

where ∆ = D – CA–1B, E = A–1B, and F = CA–1. We should also be aware that there are four Euler parameters—that is, four angular generalized coordinates per body. There should be no more angular constraint equations per body than there are angular coordinates per We used the virtual body to avoid a dimensionally inconsistent system of equawork principle in tions. Figure 4 shows a typical conjunction with the simulation slide-set of a multibody system. We can augmentation visibly detect the fluidity of method to derive a set motion and perform the following constraint compliance test to determine the accuracy of planar DAEs. of the results; h is the timestep size used in the integration scheme, and C(q,t) are the constraint equations: C(q, t ) − C(q, 0) < h .

(36)

The track vehicle dynamics study involving contact forces explores integration of the system equations using different integration schemes. Studying constraint compliance reveals that nonlinear, nondifferentiable contact forces of high magnitude (arising from the body-interaction modeling method) present numerical difficulties for software that expects almost analytically smooth data. The twice-differentiated constraint equations might produce some numerical inaccuracy, but we regard the nondifferentiable contact force model as the numerical problem’s real cause. The automobile crash simulation involves numerically difficult computation due to the contact and constraint forces. The contact force modeling uses a progressive damping force, which is a function of both the body penetration distance and its time derivative (see Equation 34), and is not differentiable everywhere in time. The simplified model largely predicts the body dynamics behavior; however, a more involved vertebral model reveals greater constituent

34

body-part detail, including the whiplash effect on the cervical vertebrae.7 The third system we investigated involves the spatial modeling of a planetary system using Euler parameters. This system has data that is analytically smooth, because there is no contact between any of the bodies. System equation integration is commenced using available initial heliocentric rectangular equatorial generalized coordinates and their time derivatives and continues over a time interval to generate the planetary ephemerides and artificial satellite trajectories. We investigated constraint compliance to establish the computation’s accuracy within a range of three to five significant figures of the data provided by the Astronomical Almanac after simulation times of 200 and 360 days. The constraint equations are the theoretical constraints introduced by the Euler parameter method we described earlier. Lessons learned In multibody modeling involving writing code for numerical computation of equations, researchers should pay careful attention to mathematical modeling, numerical methods, and software. We found that if the mathematical modeling and numerical methods are matched in a way that caters to the dynamics of the problem being studied, the resulting software development will provide a useful reusable tool for problem investigation in the future. Mathematical modeling

Coordinate selection must be carefully made, usually between generalized Cartesian coordinates and relative coordinates. Inevitably, the sparsity of the system equations obtained using the generalized coordinates is lost if we decide on the relative description; however, this leads to a smaller set of equations we must solve. A decision between Euler angles and Euler parameters may be required; invertibility of system matrices should be investigated by those working on the problem to prevent the inadvertent computation of a singular system. A general multibody and computational dynamics method might allow easier generalization of a system than a more specific or particular approach, such as the Denavit–Hartenberg method commonly found in robotics work. Researchers should carefully pursue force modeling with an emphasis on continuity and differentiability of the appropriate functions so that the associated numerical methods of the cor-

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rect order are selected. Higher-order schemes can make use of the available higher-order derivatives and lead to more accurate computation. Numerical methods

Researchers need to consider the stability of the problem being computed and the stability of the numerical methods employed to compute the system of equations. The researcher should investigate the nature of the eigenvalues of the leading system matrix and define a stability constant to aid the discussion of the results. (See the Ascher and Petzold book for more details.10) The method’s efficiency might be at the programmers’ discretion. If they use industrial software (for example, the Livermore Solver of Ordinary Differential Equations [LSODAR] line of integrators or the differential algebraic software Differential Algebraic Equations System Solver [DASSL]), then automatic method switching will match the appropriate method to the nature of the problem. The constraint compliance error in system computation should be quoted. For example, in the current work, the constraint compliance given by Equation 36 is made for each time step, and the computation is terminated should this difference be too large. In certain cases, researchers should investigate the local and global errors and note theorem 3.1 in the Ascher and Petzold book,10 which relates consistency, stability, and convergence. Requiring too little accuracy of LSODAR lets the software skip over the regions where the data might not be able to be integrated accurately. The resultant trajectories are then inaccurate, but the user does not know this. Asking for high accuracy means that LSODAR “knows” when it cannot integrate over a region, so it stops with an indicated error condition. The user then does not have trajectories for the full time interval but knows that what has been computed is accurate.

27. We used LSODAR to numerically integrate the system. However, it might be more suitable to directly use DASSL and, in so doing, avoid the time consuming analytic differentiation of the constraint equations. Researchers should also carefully consider selecting the appropriate software for the existing hardware with a view to porting the resulting project code to other platforms in advance.

M

atching physical properties with numerical methods when modeling constrained dynamical systems lets scientists and engineers perform accurate analyses of system behavior. We used the virtual work principle in conjunction with the augmentation method to derive a set of planar DAEs. We derived the spatial system of DAEs by substituting an expression for body kinetic energy into the Lagrange equations of motion. The DAEs were cast as ODEs through differentiation of the constraint equations. Contact force modeling and integration schemes must be carefully chosen and constraint compliance should be investigated to record the computational accuracy. The difficulty of integrating over forces such as these in the right-hand side of the ODE can only be resolved if we employ new contact models or nondifferentiability detection methods. We used the numerical integrator LSODAR to integrate ODEs, whereas DASSL directly computes DAEs. In this work, we differentiated the constraint equations to cast the DAE as the underlying ODE. Brenan and her colleagues do not recommend this procedure.9 It might result in reduced accuracy, but we have shown that the nature of the right-hand side, more so than the differentiated constraint equations, is the cause of numerical problems, especially in systems involving impacts.

Software

Software tools such as Maple, Mathematica, and Matlab are common packages that we can use alone or, in the case of Matlab, with the C and Fortran programming languages. These tools might provide insight into the problem under investigation. Project selection and evolution often centers around the availability of quality and informative software that is suitable for large, ongoing projects. Here, we converted the DAE given by Equation 26 to the underlying ODE, Equation

JANUARY/FEBRUARY 2001

References 1. R.R. Ryan, “Adams—Multibody System Analysis Software,” Multibody Systems Handbook, Springer–Verlag, Berlin, 1990. 2. P.E. Nikravesh, Computer Aided Analysis of Mechanical Systems, Prentice-Hall, Upper Saddle River, N.J., 1988. 3. A.A. Shabana, Dynamics of Multibody Systems, John Wiley & Sons, New York, 1988. 4. A.A. Shabana, Computational Dynamics, John Wiley & Sons, New York, 1994. 5. E.J. Haug, Computer Aided Analysis and Optimization of Mechanical System Dynamics, NATO ASI Series, Vol. F9, Springer-Verlag, Berlin, 1984.

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9. K.E. Brenan, S.L. Campbell, and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier Science, New York, 1989. 10. U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, Penn., 1998.

Bud Fox is a research associate in the Parallel Computing Research Laboratory in the Department of Electrical and Electronic Engineering at the University of Western Australia. He received his BSc and PhD from the University of Western Australia. His research interests are in multibody dynamics, computational science, and mathematical modeling. Contact him at the Parallel Computing Research Laboratory, Dept. of Electrical & Electronic Eng., Univ. of Western Australia, Nedlands, Perth Western Australia 6907; [email protected]. edu.au. Leslie Jennings is an associate professor in the Department of Mathematics at the University of Western Australia. Currently, he is working on the interface of optimal control, numerical analysis, and software engineering. He received a BSc from the University of Adelaide, Adelaide, South Australia and a PhD in numerical analysis from the Australian National University, Canberra, Australia. His research interests lie in numerical analysis and in the application of optimal control to human movement modeling, multibody systems, chemical engineering, and filter design. Contact him at the Dept. of Mathematics, Univ. of Western Australia, WA, 6907, Australia; [email protected]. Albert Y. Zomaya is a professor and Deputy-Head in the Department of Electrical and Electronic Engineering at the University of Western Australia, where he also leads the Parallel Computing Research Laboratory. He is an associate editor for the IEEE Transactions on Parallel and Distributed Systems and IEEE Transactions on Systems, Man, and Cybernetics. He is the founding editor-in-chief of the Wiley book series on Parallel and Distributed Computing. In September 2000, he was awarded the IEEE Computer Society’s Meritorious Service Award. His research interests are in high-performance computing, parallel and distributed algorithms, computational machine learning, scientific computing, adaptive computing systems, mobile computing, data mining, cluster computing, megacomputing, and wireless networks. He received his PhD from the Department of Automatic Control and Systems Engineering, Sheffield University, United Kingdom. Contact him at the Parallel Computing Research Laboratory, Dept. of Electrical and Electronic Eng., Univ. of Western Australia, Nedlands, Perth, Western Australia 6907; zomaya@ee. uwa.edu.au; www.ee.uwa.edu.au/staff/zomaya.a.html.

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