Radu Serban and Jeffrey S. Freeman .... of finding b * from Eq. 6 with V(b) given by Eq. 8 is a nonlinear least square problem (Hiebert, 1981, Dennis and.
PARAMETER IDENTIFICATION FOR MULTIBODY DYNAMIC SYSTEMS Radu Serban and Jeffrey S. Freeman Department of Mechanical Engineering The University of Iowa Iowa City, Iowa 52242–1000, USA
ABSTRACT This paper presents a parameter identification technique for multibody dynamic systems, based on a nonlinear least–square optimization procedure. The procedure identifies unknown parameters in the differential–algebraic multibody system model by matching the acceleration time history of a point of interest with given data. Derivative information for the optimization process is obtained through dynamic sensitivity analysis. Direct differentiation methods are used to perform the sensitivity analysis. Examples of the procedure are presented, applying the technique both to perfect data; i.e. data produced by the assumed model with the optimal choice of parameters, and to experimental data; i.e. data measured on the real system and thus subject to noise and modelling imperfections.
INTRODUCTION The objective of parameter estimation is to fit a given, fixed mathematical model, , to experimentally measured data. A clear distinction should be drawn between the problem of parameter estimation and the problem of system identification. When performing system identification, one has the freedom to select both the model which describes the physical phenomena and the model parameters which minimize the differences between the model and the data. When performing parameter estimation, the model is predetermined whereas the parameters are the only free variables which can be used to minimize differences between the model and the data. The goal of this work is to apply the general theory of parameter identification to the particular type of multibody system models, taking advantage of their special structure.
ANALYSIS The multibody dynamic systems under consideration involve constrained rigid body motion under the action of time varying loads. The differential– algebraic equations of motion for these systems can be expressed in the form ..
.
M(q)q Tq(q) Q(t, q, q) (1)
(q) 0
1
.
..
where q, q, and q are n
1 vectors representing the generalized position, velocity, and acceleration, M(q) is the n
.
1 generalized force vector, is the nc
matrix, Q(t, q, q) is the n
1 vector of Lagrange multipliers, and F q(q) +
n generalized mass ēF(q) is the nc ēq
n
Jacobian matrix (Nikravesh, 1988; Haug, 1989). In the context of this study, we assume that the kinematic constraints are holonomic and the constraint Jacobian matrix has full row rank.
Parameter Identification Typically in the parameter identification process, experimental data in the form {(t i, y i) | i + 1AAAn m} has been collected, and the design parameters b Ů R p for the given model M(t, b) must be selected such that y m(t i) [ y i,, where y m(t) + M(t, b)
(2)
Generally, there are far more data points than parameters, that is n m ơ p . In Eq. 2, the mathematical model, M(t, b), is an arbitrary nonlinear model. When performing parameter estimation for multibody dynamic systems defined by Eq. 1, the system will include a vector of p design parameters, b Ů R p. These parameters represent physical properties of the rigid body mechanism, such as dimensions, masses, inertias, spring constants, damping coefficients. Note that any of the terms in Eq. 1 can be a function of the unknown parameters b. Therefore, the model, M(t, b), for constrained multibody systems is of the form .
..
M(t, b) + f (t, q, q, q, l, b)
(3)
Performance Measure The general approach to the solution of the parameter identification problem is to minimize a cost function of the model residuals, r i(b), which are defined as follows: r i(b) + y i * y m(t i)
(4)
The cost function, V(b), is given as the quadratic form
ȍ nm
V(b) + 1 r 2(b) 2 i+1 i
(5)
The estimate of the design parameters, b, based on n m data points, is then defined as b *+ min[V(b)]
(6)
b
By assembling the model residuals into a vector R(b), given as R(b) + NJ r 1(b), AAA, r nm(b) Nj
T
(7)
2
we wish to choose the design parameters b so that the residual vector R(b) is as small as possible, in some sense. The quadratic performance function V(b) of Eq. 5 becomes V(b) + 1 R T(b) @ R(b) 2
(8)
Optimization Procedure The problem of finding b * from Eq. 6 with V(b) given by Eq. 8 is a nonlinear least square problem (Hiebert, 1981, Dennis and Schnabel, 1989). If the vector of residuals, R(b), is continuous, and both first and second derivatives are available, then the nonlinear least square problem can be solved by standard unconstrained optimization methods. Otherwise, a method that requires only the first derivatives of R(b) must be used (Dennis et al., 1981). Since second derivatives of the constrained multibody system model of Eq. 3 are difficult to obtain, only methods using first–order derivative information have been developed in this study. The first derivative, G(b), of Eq. 8 is defined as
ȍ ʼnr (b)r (b) + J (b) @ R(b) nm
G(b) +
T
i
(9)
i
i+1
where J(b) Ů R nm
p
is the Jacobian matrix of R(b) with respect to the design parameters. The second derivative, H(b), of Eq. 8 is defined
as H(b) + ȍƪʼnr (b)ʼnr (b) nm
i
T
i
) ʼn 2r i(b)r i(b)ƫ
(10)
i+1
+ J T(b) @ J(b) ) S(b) where S(b) Ů R p
p
is the part of H(b) that is function of second derivatives of R(b). Thus, the knowledge of J(b) supplies G(b) and the
part of H(b) dependent on first–order derivative information, but not the second order part S(b). Both the Gauss–Newton and Levenberg–Marquardt methods simply omit S(b) and base their step selection on the model given by m(b ) d) + V(b) ) G T(b)d ) 1 dJ T(b)J(b)d 2
(11)
Provided the experimental data is fit well by the optimal model M(t, b *), then this assumption is reasonable, since in this case R(b *) and S(b *) will be almost zero. Eq. 11 leads to the following optimization procedure: b (k)1) + b (k) ) kd (k)
(12)
J T(b (k))J(b (k)) @ d (k) + * G(b (k))
(13)
k + minƪ V(b (k) ) d (k)) ƫ
(14)
with
a
3
Note that, for J(b (k)) non–singular, J T(b (k))J(b (k)) is positive definite, and thus d (k) given by Eq. 13 represents a descent direction.
Sensitivity Analysis In general, the objective of parameter identification for multibody dynamic systems is to match the acceleration time history of a point of interest with given data. This acceleration can be expressed in terms of the states, derivatives of the states and Lagrange multipliers. Since the procedure developed by Eqs. 11 through 14 uses the Jacobian of the residual vector, sensitivities of the generalized coordinates, derivatives of the generalized coordinates, and Lagrange multipliers with respect to the design parameters are required. Obtaining these derivatives is the main issue in design sensitivity of multibody dynamic systems (Krishnaswami et al., 1983, Haug, 1987, Bestle and Seybold, 1992). The direct differentiation method for dynamic sensitivity analysis is used. The adjoint variable method, although faster than the direct differentiation method has the drawback of not yielding explicit derivatives for accelerations. The following set of DAE’s yields both the solution of Eq. 1 and the solution of the sensitivity equations (Serban and Freeman, 1996). ^
^ ..
Mr ) P Tr m + Q
(15)
P+0 where
NJ
r + q T, q Tb , q Tb , ..., q Tbp 1
2
Nj
NJ
T
m + l Tb , l Tb , @, l Tbp, l T
ȱM ) P M 0 @ 0 ȳ ȧM @) P @0 M@ @ @0 ȧ M +ȧ ȧ ȧM ) P 0 0 @ Mȧ Ȳ M 0 0 @ 0ȴ ȱ(F q ) F) (F ) F0 00 ȧ P +ȧ(F q ) ) (F ) 0 F @ @ @ ȧ (F q ) ) (F ) 0 0 Ȳ
1
2
Nj
T
1
b1
2
b2
^
ȍ ēm ēq n
with the components of P h defined as p hi,j +
k+1
p
i,j k
(q k) bh
bp
q
r
q b q 1
b1 q
q b q 2
b2 q
q bp q
bp q
q
q
ȳ @ 0ȧ ȧ @ȧ @ F ȴ @ 0 @ 0
q
ȡQ ȧQ Q+ȥ ȧQ Ȣ ^
.
b1
) Q q. q b ) Q qq b .
1
ȣ ȧ Ȧ ȧ Ȥ
1
) Q q. q b ) Q qq b 2 2 @ . . b p ) Q qq b p ) Q qq b p b2
Q
An important issue is that of initial values for the sensitivity variables. As noted by Haug (1987), if initial values of the coordinates and velocities are given in explicit form in terms of design parameters, then the initial sensitivity values can be obtained by simply differentiating these expressions with respect to b. However, it is much more common to define the initial configuration in an implicit form. In the case of vehicle systems for example, all simulations are desired to start from a static equilibrium configuration. Such an equilibrium configuration is defined by the following conditions: F Tq l + Q
(16)
F+0
4
Note that it is usually difficult to solve the nonlinear equations, Eqs. 16, for the static equilibrium configuration since an initial estimate is not easily available. This configuration is thus obtained by simulating the multibody system until accelerations and velocities converge to zero. Taking the derivative of Eq. 16 with respect to a design parameter, b, the following expressions are obtained: F Tq l b ) ǒF Tq lǓ qq b ) F Tqbl + Q qq b ) Q b
(17)
F qq b ) F b + 0 Once the stable equilibrium configuration is known, Eqs. 17 can be rewritten as
ƪ
ǒF Tq lǓ * Q q q
F Tq
Fq
0
ƫƪ ƫ ƪ qb lb +
Q b * F Tqbl Fb
ƫ
(18)
and solved for the initial sensitivity values q b, and the equilibrium constraint forces l b.
SOLUTION EXAMPLES In order to illustrate the above technique we give the example of an actual vehicle suspension in two different cases. First, perfect data identification was performed. In the second case, experimentally measured data is used in the parameter identification procedure. The data for this example includes an unknown noise component. The left rear suspension of the HMMWV (High Mobility Multipurpose Wheeled Vehicle) was modeled as a planar mechanism with two degrees of freedom, these being the vertical displacement of the chassis center of gravity, and the vertical displacement of the wheel center of gravity. The model is graphically presented in Figs. 1 and 2 and the parameters are listed in Table 1. The mechanism consists of four bodies (chassis, upper control arm, lower control arm, and wheel assembly) connected by four revolute joints (B,C,D, and E), as shown in Fig. 2. A spring–damper with nonlinear spring characteristics connects points P on body 1 and Q on body 3. Due to its design, the HMMWV rear suspension can be modeled as a planar mechanism. The suspension is a short–long arm (SLA) type with 0% anti–squat and no caster angle changes during vertical motion. This is due to the fact that both the inboard revolute joints of the suspension control arms, denoted by points B and E in Fig. 2, lie in on a common plane, and axes of the joints are parallel. Thus the motion of the wheel center point is restricted to a vertical plane which is orthogonal to the axes of the inboard revolute joints. For this mechanism, the terms involved in the equations of motion, Eq. 1, are q + {y 1 , q 2 , q 3 , x 4 , y 4 , q 4} m )m ȱmm)cos(q )l ȧ m cos(q )l M +ȧ ȧ 0 ȧ 0 Ȳ 0 1
2
2
2
B
3
3
E
3
T
m 2cos(q 2)l B m 3cos(q 3)l E J 2 ) m 2l 2B 0 0 J 3 ) m 3l 2E 0 0 0 0 0 0
(19) 0 0 0 m4 0 0
0 0 0 0 m4 0
ȳ ȧ ȧ ȧ ȧ ȴ
0 0 0 0 0 J4
(20)
5
xȀ ) (l ) l )cosq * x * xȀ cos q ) yȀ sin q ȡy ) ȣ yȀ ) (l ) l )sinq * y * xȀ sin q * yȀ cos q ȧ ȧ + ȥ xȀ ) (l ) l )cosq * x * xȀ cos q ) yȀ sin q Ȧ ȧ Ȣy ) yȀ ) (l ) l )sinq * y * xȀ sin q * yȀ cos q ȧ Ȥ B
B
D
) sin q ȱ01 *(l (l))l l) cos q +ȧ0 0 0 Ȳ1 B
C
C
3
D
4
D
4
4
C
4
D
4
C
4
C
4
3
E
B
2
D
E
C
4
C
E
1
2
B
E
q
C
B
1
4
(21)
4
D
4
4
ȳ ȧ ȴ
0 xȀ C sin q 6 ) yȀ C cos q 6 0 *1 0 0 * 1 * xȀ C cos q 6 ) yȀ C sin q 6 0 xȀ D sin q 6 ) yȀ D cos q 6 * (l E ) l D) sin q 3 * 1 0 * 1 * xȀ D cos q 6 ) yȀ D sin q 6 (l E ) l D) cos q 3
2
2
ȡ(m ) m ) m )g ) m l q sin q ) m l q sin q ȣ m gl cos q ȧ ȧ då ) m gl cos q Q+ȥ Ȧ 0 ȧ m g ) k (y * y * l ) ) c (y * y ) ȧ 0 Ȣ Ȥ .2
1
2
.2
2 B 2
3
B
2
3
3 e 3
2
T
ground
4
3
2
E
3
.
4
(22)
T0
T
(23)
.
ground
4
where . c d + k 1(h * l 0) 2 ) k 2(h * l 0) * cq 3(l E * l Q) h
c å + h (l E * l Q) h + ƪ(xȀ E * xȀ P) 2 ) (yȀ E * yȀ P) 2 ) (l E * l Q) 2 ) 2(l E * l Q)zƫ
1ń2
c + (xȀ E * xȀ P) sin q 3 * (yȀ E * yȀ P) cos q 3 z + (xȀ E * xȀ P) cos q 3 ) (yȀ E * yȀ P) sin q 3 In this example we perform the parameter identification process to estimate the spring constants k 1 and k 2, and the damping coefficient c. The vehicle was run at constant forward speed over the approximately sinusoidal road profile shown in Fig. 3. This roadway surface excites only the bounce and pitch modes of the vehicle, and does not induce roll motion. The optimization criteria is based on the vertical acceleration of the upper control arm center of gravity position, given in its local coordinate frame. Once the solution of the equations of motion is known, the local vertical acceleration of the upper control arm center of gravity position is obtained as ..
..
..
yȀ 2 + l Bq 2 ) y 1 cos q 2
(24)
Note that the initial static configuration of the suspension mechanism depends upon the value of the unknown design parameters. Since an analytical relationship for the static configuration is difficult to obtain, for each estimate of the spring constant, the mechanism is first simulated until the equilibrium configuration is reached. With the values of the states at this stabilized configuration as initial conditions and with initial sensitivity conditions obtained from Eq. 18 , the simulation is then carried out to get the time history of both the dynamic and sensitivity variables, using Eq. 15.
6
Perfect Data Identification In this case the data to be matched was obtained by running the model with the following choice of design parameters: k 1 + 1, 000, 000.0 Nńm 2, k 2 + 238, 000.0 Nńm, and c + 22, 000.0 Nsńm. In this idealized case, the unknown parameters can be identified to within any given tolerance and the optimal behavior can be perfectly matched. At the optimal configuration, the second order part in the Hessian of the cost function, S(b *), becomes zero. Starting with an initial guess of b + [100, 000.0 ; 100, 000.0 ; 15, 000.0], the correct solution was identified within the prescribed tolerance of 10 *5 after seven iterations. The evolution of the cost function during the optimization process is presented in Fig. 4.
Experimental Data Identification Experimental data was collected from an instrumented HMMWV at the U.S. Army’s Aberdeen Test Center. Instrumentation for this test consisted of accelerometers mounted at the vehicle’s center of gravity and at a convenient location on the upper control arm of the suspension. Examples of raw and filtered experimental data from the control arm accelerometer are shown in Fig. 5. This data was recorded at 100 Hz. using a 40 Hz. anti–aliasing filter, while the filtered data has been additionally processed using a phaseless filter at 5 Hz. Starting with an initial guess k 1 + 1, 000, 000.0 Nńm 2, k 2 + 160, 000.0 Nńm, and c + 16, 000.0 Nsńm, the values of the unknown parameters are identified as k 1 + 29.963 Nńm 2 , k 2 + 231, 481.0 Nńm, and c + 17, 575.5 Nsńm in 11 iterations. Intermediate steps in the optimization process are shown in Fig. 6. ..
Figs. 7 and 8 present the acceleration yȀ 2 corresponding to the initial guess and to the final values of the design parameters as compared to the unfiltered experimental data. Unlike the first example, which used perfect data, the output of the optimal model does not perfectly match the measured data when using experimental data. The reason for this difference is twofold. First, the experimental data was subject to measurement noise. And second, the assumed model is only an approximation of the real multibody system. However, even for such a simplified model, the output of the optimal model is close to that of the real system.
CONCLUSIONS A parameter identification procedure for multibody dynamic systems has been formulated and its feasibility has been verified through its application on various examples. The procedure identifies unknown model parameters in a multibody system by matching the acceleration time history of a point of interest on a body with experimental data. The formulation presented in this study employs differential–algebraic equations to express the multibody system, and employs a nonlinear least–square optimization algorithm to identify the unknown model parameters. Dynamic design sensitivity is accomplished using a direct differentiation technique which directly yields the first–order derivative information with respect to the design parameters.
7
The examples presented as part of the study highlight the ability of the procedure to identify multiple parameters. Testing with perfect data showed the expected result of identification of the optimal solution. Testing with experimental data, using a vehicle suspension model, also resulted in the identification of an optimal model. Unlike the example using perfect data, the optimal solution identified from the experimental data example did not exactly match the original data. Reasons for this discrepancy were identified as noise in the original data set, and imperfections in modeling the physical system. In all, the identification procedure developed in this study enables the recovery of unknown model parameters from experimental test data. As such, dynamic multibody system models can be better correlated to physical systems.
ACKNOWLEDGMENTS Financial support for this study was provided by the U.S. Army Tank–Automotive Command (TACOM) through the University of Michigan Automotive Research Center, a U.S. Army Center of Excellence for Automotive Research, under contract number DAAE07–94–C–R094. Additional support for vehicle experimental testing was provided under the DARPA and National Science Foundation contract number CDR–87–15397.
REFERENCES Bestle, D. and Eberhard, P., 1992, “Analyzing and Optimizing Multibody Systems,” Mech. Struct. & Mach., Vol 20(1), pp. 67–92 Bestle, D. and Seybold, J., 1992, “Sensitivity Analysis of Constrained Multibody Systems,” Archive of Applied Mechanics, Vol. 62, pp. 181–190 Chang, C.O. and Nikravesh, P.E., 1985, “Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method,” ASME J. of Mechanisms, Transmissions, and Automation in Design, Vol. 107, pp. 493–498 Dennis, J.E. JR., Gay, D.M., and Welsch, R.E., 1981, “An Addaptive Nonlinear Least–Squares Algorithm,” ACM Transactions on Mathematical Software, Vol. 7, No. 3, pp. 348–360 Dennis, J.E. JR. and Schnabel, R.B., 1989, “A View of Unconstrained Optimization,” Handbooks in OR & MS, G.L. Nemhauser et al., Ed., Elsevier Science Publishers B.V. (North–Holland) Haug, E.J. and Ehle, P.E., 1982, “Second–Order Design Sensitivity Analysis of Mechanical System Dynamics,” Intl. J. for Numerical Methods in Engineering, Vol. 18, pp. 1699–1717 Haug, E.J., 1987, “Design Sensitivity Analysis of Dynamic Systems,” Computer–Aided Design: Structural and Mechanical Systems, C.A. Mota– Soares, Ed., Springer–Verlag, Berlin Haug, E.J., 1989, Computer–Aided Kinematics and Dynamics of Mechanical Systems, Volume I: Basic Methods, Allyn and Bacon, Needham Heights, Massachusetts
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Hiebert, K.L., 1981, “An Evaluation of Mathematical Software that Solves Nonlinear Least Squares Problems,” ACM Transactions on Mathematical Software, Vol. 7, No. 1, pp. 1–16 Krishnaswami, P., Wehage, R.A. and Haug, E.J., 1983, “Design Sensitivity Analysis of Constrained Dynamic Systems by Direct Differentiation,” Technical Report No. 83–5, Center for Computer– Aided Design, The University of Iowa, Iowa City, Iowa Nikravesh, P.E., 1988, Computer–Aided Analysis of Mechanical Systems, Prentice–Hall, Englewood Cliffs, New Jersey Serban, R. and Freeman, J.S., 1996, “Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems,” to appear in Journal of Mechanical Design “Vehicle Test Facilities at Aberdeen Proving Ground,” US Army Test and Evaluation Command, Test Operations Procedure, Report #TOP 1–1–011, 57 pages.
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Figure 1 Left Rear HMMWV Suspension
10
Figure 2 Planar Model of the Left Rear HMMWV Suspension
11
Figure 3 Aberdeen Test Center Roadway Profile
12
Figure 4 Evolution of the Cost Function for Perfect Data Identification
13
14
