NUMERICAL INTEGRATION OF CONSTRAINED MULTI-BODY DYNAMICAL SYSTEMS USING 5 T H ORDER EXACT ANALYTIC CONTINUATION ALGORITHM

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1 (Preprint) AAS NUMERICAL INTEGRATION OF CONSTRAINED MULTI-BODY DYNAMICAL SYSTEMS USING 5 T H ORDER EXACT ANALYTIC CONTINUATION ALGORITHM Ahmad Bani Younes, and James Turner Many numerical integration methods have been developed for predicting the evolution of the response of dynamical systems. Standard algorithms approach approximate the solution at a future time by introducing using a truncated power series representation that attempts to recover an n-th order Taylor series approximation, while only numerically sampling a single derivative model. An exact fifthorder analytic continuation method is presented for integrating constrained multibody vector-valued systems of equations, where the Jacobi form of the Routh-Voss equations of motion simultaneously generates the acceleration and Lagrange multiplier solution. The constraint drift problem is addressed by introducing an analytic continuation method that rigorously enforces the kinematic constraints through five time derivatives. The proposed approach is expected to be particularly useful for stiff dynamical systems, as well as systems where implicit integration formulations are introduced. Numerical examples are presented that demonstrate the effectiveness of the proposed methodology. INTRODUCTION Analytic Continuation Method An exact fifth-order analytic continuation method is presented for integrating vector-valued constrained multi-body dynamical systems of equations. The major innovation of this paper is the introduction of computational differentiation (CD) techniques which enable the development of a rigorous n-th order Taylor series analytic continuation model for vector systems of differential equations. Classical numerical methods sample a single ordinary differential equation motion several times to develop a weighted approximation for the integrated solution. The proposed analytic continuation method computes exact derivatives for all of the terms retained in the series approximation. Two classes of dynamical systems are considered: (1) unconstrained differential equations, and (2) constrained multi-body dynamical systems. Special attention is given to handling matrix-vector equations were the matrix is time-varying and is too large to be symbolically inverted. Both constant and time-varying kinematic constraints are considered for multi-body systems. Numerical results are presented that demonstrate the proposed analytical/numerical modeling approach. For general applications in science and engineering, the system equations of motion are described PhD Candidate, Aerospace Engineering, Texas A&M University, College Station, Texas, olalahmad@gmail.com. Research Professor, Aerospace Engineering, Texas A&M University, College Station, Texas, turner@aero.tamu. edu. 1

2 by the following first-order vector differential equation: ẋ = f(t, x); given x(t 0 ) = a (1) where x R n denotes the state vector, t denotes time, and ẋ = dx dt. A rigorous Taylor series expansion is developed for Eq.(1), that defines the future response of the system by summing contributions form n-time derivatives of f( ), where time is evaluated at the point t = a. Analytically continuing x(t) leads to the following Taylor series expansion x(t) = x(a) + x (a) 1! (t a) + x (a) 2! (t a) 2 + x (a) (t a) 3 + (2) 3! Theoretically, the solution is complete once a numerical method is specified for evaluating the time derivatives for: dx dt, d2 x, d3 x, unfortunately these time derivatives are generally not available because of the analyst time required for deriving and coding and verifying the derivative calculations. dt 2 dt 3 Classical methods overcome this problem by a repeated sampling and averaging strategy. The main contribution of this paper is to demonstrate that CD tools can be brought to bear on the problem of numerically integrating constrained-complex nonlinear vector systems of ordinary differential equations by returning to the fundamental Taylor expansion that defines the evolution of the system response. To this end, CD is used to numerically evaluate derivative approximations for: dx dt, d2 x dt 2, d3 x dt 3, which are then used in the truncated series x(t+h) x(t)+ 5 1 d i x i! h i. This ex- dt i trapolation equation replaces the weighted averaging approach of conventional solution algorithms. Numerical time derivative values are computed from the following cascade of tensor contraction operations: ẋ = f ẍ = f.ẋ x (3) = 2 f.ẋ.ẋ + f.ẍ x (4) = 3 f.ẋ.ẋ.ẋ f.ẍ.ẋ + 2 f.ẋ.ẍ + f.x (3) (3) x (5) = 4 f.ẋ.ẋ.ẋ.ẋ f.ẍ.ẋ.ẋ f.ẋ.ẍ.ẋ f.x (3).ẋ + 3 f.ẋ.ẋ.ẍ f.ẍ.ẍ + 2 f.ẋ.x (3) + f.x (4) These equations assume that time does not appear explicitly in the differential equation. Higherorder time derivative terms are easily generated. Turner s computational differentiation-based Object- Oriented Coordinate Embedding Algorithm (OCEA) provides numerical values the n-th order Jacobians. The analyst only codes f(x, t); OCEA uses hidden operator-overloaded software-based tools for automatically deriving, coding, and evaluating the Jacobian tensors required in Eq.(2). Without the availability of CD Eq.(3) only represents a theoretical solution for an exact series approximation. COMPUTATIONAL DIFFERENTIATION Computational differentiation is a specialized topic in applied mathematics and computer science for developing and fielding software tools for numerically evaluating partial derivative models. 1, 2 This paper makes use of Turner s Object-Oriented Coordinate Embedding Algorithm (OCEA) program for computing 1 st -4 th -order mixed partial derivative models. 3, 4, 5, 6 OCEA is a CD tool for computing arbitrarily complex partial derivative models. At compile time, OCEA uses the programmer s math model as a template for deriving, coding, and generating an executable for simultaneously compiling the simulation and sensitivity models. No symbolic or finite difference tools 2

3 are used for any of the gradient tensor calculations; all results are immediately processed to produce numerical results for all partial derivative orders. Each time a program is complied OCEA derives, assemblies, and codes the partial derivative model in the program executable, yielding numerical results that are accurate to the working level of precision for the machine. This approach benefits the user in four ways: (1) only the basic math model is programmed and checked out; (2) no analyst effort is required to either derive or code sensitivity models; (3) the user recaptures the development time normally required for developing sensitivity math models, coding, and validating nonlinear and high-order models; and (4) most importantly, math model changes are automatically handled each time the code is complied. Hand derived models are very vulnerable to model changes that can potentially force a new derivation and coding for the sensitivity model. OCEA transforms all math and intrinsic functions to embed multiple levels of the chain rule of calculus for building partial derivative models. OCEA consists of a suite of programs/modules that process the user supplied software math models as a template for generating high-order sensitivity models. Minimal modifications of the User s existing software models are required for enabling derivative-enhanced calculations. Users identify independent variables for the derivative calculations and define the variables for which partial derivatives are required (e.g., typically one is required to convert real variable(s) to an OCEA variable TYPE(EB), which instructs the complier that partial derivatives are computed for the TYPE(EB) variable(s). The software accepts user math models coded in standard FORTRAN 95/2003 language constructs. At compile time the partial derivative solution is assembled by identifying variable data types, intrinsic/lib functions, and inserting Function/Subroutine calls in the complier generated executable for computing the associated partial derivatives. From the user perspective, OCEA behaves as a language extension for FORTRAN 95/2003: its hidden operations exploit operator overloading and user-defined data types for handling all memory management details and derivative enhanced operations. It should not be surprising that OCEA introduces a computational overhead when compared to highly optimized hand coded partial derivative models. The trade off is this: if the basic math model requires X man-months to develop, then the derivation, coding, and validation of the sensitivity model can add 5X-10X man-months to the project development effort. Since the computer time required for deriving and compiling the sensitivity solution is measured in seconds vs. 5X-10X manmonths for an analyst, OCEA s impact is both clear and unambiguous for developing and solving real-world projects subject to man-power resource constraints. OCEA is particularly valuable in the normally fluid engineering design environment, where frequent design changes and -what ifexperiments must be carried out to fully explore the opportunities available in the notional system design space. The analyst always has the most up to date model. This is in stark contrast the case of a hand-derived model where even seemingly simple model changes can devastate all previous derivation efforts; thereby, forcing a restart for the derivation and coding effort from scratch for each new model is a daunting unwelcome task. OCEA s derivative enhanced variables are defined as abstract compound data objects, where objects such as the variable F, defined below, consist of the following list of concatenated data in computer memory: F := { F F 2 F n F } where m F denotes the m th order tensor gradient operation. The only variable visible to the analyst is F. Numerical values for the sub-object component values of the tensor gradient operators are obtained by using structure constructor designators (e.g., m F = F %T m, where T m denotes the tensor order). A detailed description of all of OCEA s capabilities are found in the software user manual. 7 3

4 CONSTRAINED MULTI-BODY DYNAMICAL SYSTEMS A problem of long standing in the science and engineering community is concerned with providing accurate integrations of math models that consist of both differential and algebraic equations. All numerical techniques produce drift in the constraint values that must be addressed for the simulations to remain physically meaningful. Many strategies have been proposed for addressing this problem, ranging from differential correction to defining artificial constraint differential equations. 8, 9, 10 Our goal is to exactly model the constraint through several time derivatives so that the constraint drift is minimized. Other researchers have considered developing CD-based integration methods; however, these efforts have not dealt with the time-varying system matrix derivative handled in this formulation. The new capability is critically important for enabling CD methods to handle real-world applications in engineering and science. To this end, the following constrained multi-body dynamics system is considered M q = f + [B] T λ subject to [B] q + b = 0 (4) where M denotes the mass matrix, q is the generalized coordinates of the system, ( q = dq dt, q = d 2 q ), f denotes all forces acting on the system, [B] denotes the velocity constraint matrix, λ dt 2 ( R Nc 1 ) denotes the constraint Lagrange multiplier, N c is the number of constraints. This system of equations is re-cast in the Jacobi form of the Routh-Voss equations as [ M B T B 0 ] [ ] [ ] q f = λ ḃ Ḃ q MẎ = F (5) Equation (5) completely describes the constrained system dynamics and recovers both the system acceleration and the negative of the constraint Lagrange multiplier. Equation (5) is known as the mass descriptor form of the constrained equations of motion. An exact series expansion for the system position and velocity is obtained by repeatedly differentiating Eq.(5) w.r.t. time, yielding the cascade of equations M Ẏ = F Ẏ = M 1 F MŸ + MẎ = F M Y... Ÿ = M 1 ( F MẎ) + 2 MŸ + MẎ = F... Y = M 1 ( F 2 MŸ MẎ) (6).. where the time derivatives of the mass descriptor matrix M and the force vector F are calculated using this equation Ψ = Ψ(q, q) Ψ = Ψ = N q N q ( Ψ q i ( Ψ q i + Ψ ) q i q i q i + Ψ... q i q i N q q i + j=1 + 2 Ψ q i q i q q j + 2 Ψ )) q i q j j q i q j ( 2 Ψ q i q i q j q j + 2 Ψ q i q j q i q j (7) 4

5 where Ψ can be replaced by M or F, and N q denotes to the number of generalized coordinates. The numerical solutions account for the curvature of the constraints through four derivatives. These equations permit the following exact Taylor expansions for the constrained response ( 1 R(t + h) R(t) + V(t)h + S Yh Yh 5) 2!Ẏh2 3!Ÿh3 4! 5! ( 1 V(t + h) V(t) + S 1!Ẏh Yh Yh 4) (8) 2!Ÿh2 3! 4! S = [I 0] Selecting Operator Yielding 6 th order prediction for the position and 5 th order prediction for the velocity. Several examples are presented for both unconstrained and constrained systems. Constraint drift during integration is expected to be minimized during the simulation of complex mechanical systems. Numerical experiments will be performed to determine the impact of the proposed high-order constraint tracking algorithm. NUMERICAL EXAMPLES Acceleration Constraints This section considers presents one of the common constrained problems in engineering. Figure (1) represents slider crank mechanism connected to a block mass that is allowed to slide on the horizontal plane. Therefore, we have one constraint equation that forces the M 3 mass moves horizontally. The system equations of motion can be derived using the Lagrangian approach assuming the bars are too thin and the mass center locates at the middle point of the body. The generalized coordinates of the system are given by (q 1 = θ, q 2 = θ 2 ). To start the derivation, one needs to write the position vector for each mass: Y M 1, L 1 M 2, L 2 M3 θ 1 θ 2 X Figure 1. Slider Crank Mechanism. r 1 = l 1 2 ( ) cos(q1 ) ; r sin(q 1 ) 2 = 2r 1 + l ( ) ( ) 2 cos(q2 ) cos(q2 ) ; r 2 sin(q 2 ) 3 = 2r 1 + l 2 sin(q 2 ) (9) The Lagrangian equations of motion can be written in the form: d ( L ) L dt q q = Q nc + [B] T λ (10) 5

6 where L(q, q, t) = T (q, q, t) V (q, t). Q nc is the non-conservative force vector; assumed zero. T and V are the kinetic and potential energy of the system, respectively. T = M i (ṙ i.ṙ i ); V = g M i l i sin(q i ) + M 3 g(l 1 sin(q 1 ) l 2 sin(q 2 )) (11) where g is the earth gravity constant (= 9.81m/s 2 ). And the system Pfaffian constraint is given by: ( ) q1 [B] q + b = 0 [l 1 cos(q 1 ) l 2 cos(q 2 )] = 0 (12) q 2 Following a few mathematical manipulations, the mass coefficient matrix and the force vector can be written as: (0.25M 1 + M 2 + M 3 )l 2 1 (0.5M 2 + M 3 )l 1 l 2 cos(q 1 + q 2 ) l 1 cos(q 1 ) M = (0.5M 2 + M 3 )l 1 l 2 cos(q 1 + q 2 ) (0.25M 2 + M 3 )l2 2 l 2 cos(q 2 ) l 1 cos(q 1 ) l 2 cos(q 2 ) 0 (0.5M 2 + M 3 )l 1 l 2 q 2 2 F sin(q 1 + q 2 ) (0.5M 1 + M 3 )l 1 gcos(q 1 ) = (0.5M 2 + M 3 )l 1 l 2 q 1 2sin(q 1 + q 2 ) (0.5M 2 M 3 )l 2 gcos(q 2 ) (13) l 1 q 1 2sin(q 1) l 2 q 2 2sin(q 2) We do not claim that this is the best way to integrate the multi-body dynamics. However, the advantages are noticeable if it is compared with the most common numerical integrator; Runge- Kutta (RK4). The solution is compared with Runge-Kutta (RK4) to demonstrate these advantages. Figure 2 shows the numerical integration results obtained by the two cases. Figure (a) shows the time histories of the state variables as integrated by the Analytic Integration Method. Figure (b) shows the % error between the solution obtained by Runge-Kutta and the analytic integration methods. Figures (c) and (d) show xy-trajectories as integrated by the Runge-Kutta method and Analytic Integration Method, respectively. One can notice the drift is less in the Analytic Integration case. Velocity and Acceleration Constraints The previous developments only constrain the solution at the acceleration level. The solution accuracy is further improved by constraining both the velocity and acceleration simultaneously. This is accomplished by generalizing the previous developments. Assembling the equation of motion and both the velocity constraint, and acceleration constraint, one obtains: M q B T λ = F M 0 B T q F B q = b B q + Ḃ q = ḃ 0 B 0 Λ = b (14) B Ḃ 0 λ ḃ The mass matrix is thus extended to include the velocity constraints coefficients. This produces a non-square non-invertable matrix. To fix this problem, however, recall the velocity correction term q + = q + M 1 B T Λ, which when introduced above yields M q B T λ = F M 0 B T q F B( q + M 1 B T Λ) = b B q + Ḃ( q + M 1 B T Λ) = ḃ 0 BM 1 B T 0 Λ = b B q B ḂM 1 B T 0 λ ḃ Ḃ q (15) 6

7 (a) (b) (c) (d) Figure 2. Acceleration Constraints Results. This system of equations is implemented and compared with the previous results, as well as with Runge-Kutta method. The Numerical Results are presented in Figure 3; (a) shows the constrained force and constrained momentum histories with time. (b) shows the % error between the solution obtained by Runge-Kutta and the analytic integration methods (both methods with velocity constraints). (e) and (f) show xy-trajectories as integrated by the Runge-Kutta method and Analytic Integration Method (both methods with velocity constraints), respectively. One can notice the drift is slightly reduced in the Analytic Integration case, as it is compared with the solution without velocity constraints. Adding the constrained velocity constraint is seen to further improve the solution accuracy. SUMMERY AND CONCLUSION A computational differentiation method is presented for analytically computing a Taylor series representation of the response nonlinear vector systems. Formulations are presented for constrained multi-body systems. Planer slider crank mechanism problem is presented that demonstrates the proposed analytic Taylor series integration of the constrained nonlinear ordinary vector differential equations produce very accurate results. Further research is required for establishing the utility of these new approaches for addressing engineering level of fidelity simulations. Constrained multibody systems showed very rigorous effect on enforcing the kinematic constraints and decrease the 7

8 (a) (b) (c) (d) Figure 3. Velocity and Acceleration Constraints Results. constraint drift level. REFERENCES [1] C. H. Bischof, A. Carle, P. D. Hovland, P. Khademi, and A. Mauer, ADIFOR 2.0 User s Guide (Revision D), tech. rep., Mathematics and Computer Science Division Technical Memorandum no. 192 and Center for Research on Parallel Computation Technical Report CRPC S, [2] A. Griewank, On Automatic Differentiation, Mathematical Programming (M. Iri and K. Tanabe, eds.), pp , Dordrecht: Kluwer Academic Publishers, [3] J. L. Junkins, J. D. Turner, and M. Majji, Generalizations and Applications of the Lagrange Implicit Function Theorem, Special Issue: The F. Landis Markley Astronautics Symposium, The Journal of the Astronautical Sciences, Vol. 57, January-June 2009, pp [4] Macsyma, Inc, Macsyma, Symbolic/numeric/graphical mathematics software: Mathematics and System Reference Manual, 16th ed., [5] J. Turner, Automated Generation of High-Order Partial Derivative Models, Vol. 41, AIAA, August 2003, pp [6] J. D. Turner, M. Majji, and J. L. Junkins, Keynote Paper: Fifth-Order Exact Analytic Continuation Numerical Integration Algorithm, Nanjing, China, Presented to International Conference on Computational and Experimental Engineering and Sciences, April [7] J. Turner, OCEA User Manual. Amdyn System, [8] E. Routh, Dynamics of a System of Rigid Bodies, Advanced Part. London:Macmillan/St. Martin: Reprinted 1955 by Dovr, New York, 6th ed., [9] E. Routh, Dynamics of a System of Rigid Bodies, Elementry Part. London:Macmillan/St. Martin: Reprinted 1960 by Dovr, New York, 7th ed.,

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