Survey of Numerical Methods for Trajectory Optimization

Theoretical and numerical analysis of differential-algebraic equations

Preface. Introduction. 1. Theory of differential equations: an introduction. 1.1 General solvability theory. 1.2 Stability of the initial value problem. 1.3 Direction fields. Problems. 2. Euler's method. 2.1 Euler's method. 2.2 Error analysis of Euler's method. 2.3 Asymptotic error analysis. 2.3.1 Richardson extrapolation. 2.4 Numerical stability. 2.4.1 Rounding error accumulation. Problems. 3. Systems of differential equations. 3.1 Higher order differential equations. 3.2 Numerical methods for systems. Problems. 4. The backward Euler method and the trapezoidal method. 4.1 The backward Euler method. 4.2 The trapezoidal method. Problems. 5. Taylor and Runge-Kutta methods. 5.1 Taylor methods. 5.2 Runge-Kutta methods. 5.3 Convergence, stability, and asymptotic error. 5.4 Runge-Kutta-Fehlberg methods. 5.5 Matlab codes. 5.6 Implicit Runge-Kutta methods. Problems. 6. Multistep methods. 6.1 Adams-Bashforth methods. 6.2 Adams-Moulton methods. 6.3 Computer codes. Problems. 7. General error analysis for multistep methods. 7.1 Truncation error. 7.2 Convergence. 7.3 A general error analysis. Problems. 8. Stiff differential equations. 8.1 The method of lines for a parabolic equation. 8.2 Backward differentiation formulas. 8.3 Stability regions for multistep methods. 8.4 Additional sources of difficulty. 8.5 Solving the finite difference method. 8.6 Computer codes. Problems. 9. Implicit RK methods for stiff differential equations. 9.1 Families of implicit Runge-Kutta methods. 9.2 Stability of Runge-Kutta methods. 9.3 Order reduction. 9.4 Runge-Kutta methods for stiff equations in practice. Problems. 10. Differential algebraic equations. 10.1 Initial conditions and drift. 10.2 DAEs as stiff differential equations. 10.3 Numerical issues: higher index problems. 10.4 Backward differentiation methods for DAEs. 10.5 Runge-Kutta methods for DAEs. 10.6 Index three problems from mechanics. 10.7 Higher index DAEs. Problems. 11. Two-point boundary value problems. 11.1 A finite difference method. 11.2 Nonlinear two-point boundary value problems. Problems. 12. Volterra integral equations. 12.1 Solvability theory. 12.2 Numerical methods. 12.3 Numerical methods - Theory. Problems. Appendix A. Taylor's theorem. Appendix B. Polynomial interpolation. Bibliography. Index.

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Numerical Solution of Ordinary Differential Equations

The paper presents theoretical and implementation aspects related to a numerical integrator used for the simulation of large mechanical systems with flexible bodies and contact/impact. The proposed algorithm is based on the Hilber-Hughes-Taylor (HHT) implicit method and is tailored to answer the challenges posed by the numerical solution of index 3 differential-algebraic equations that govern the time evolution of a multibody system. One of the salient attributes of the algorithm is the good conditioning of the Jacobian matrix associated with the implicit integrator. Error estimation, integration step-size control, and nonlinear system stopping criteria are discussed in detail. Simulations using the proposed algorithm of an engine model, a model with contacts, and a model with flexible bodies indicate a 2 to 3 speedup factor when compared against benchmark MSC.ADAMS runs. The proposed HHT-based algorithm has been released in the 2005 version of the MSC.ADAMS/Solver.

http://homepages.cae.wisc.edu/~negrut/PDFpapers/hhtJCND.pdf

On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096)

The paper deals with controlled mechanical systems in which the number of control inputs is equal to the number of desired system outputs, and is smaller than the number of degrees of freedom of the system. The determination of control input strategy that force the underactuated system to complete the partly specified motion is a challenging problem. In the present formulation, the outputs (performance goals), expressed in terms of system states, are treated as constraints on the system—called control or program constraints as distinct from contact constraints in the classical sense, and a mathematical resemblance of the inverse control problem to the constrained system dynamics is exploited. However, while the reactions of contact constraints act in the directions orthogonal to the respective constraint manifold, the available control reactions may have arbitrary directions with respect to the program constraint manifold, and in the extreme may be tangent. A specific methodology must then be developed to find the solution of such “singular” problems, related to a class of control tracking problems such as position control of elastic joint robots, control of cranes, and aircraft control in prescribed trajectory flight. The governing equations of the problem arise as a set of differential-algebraic equations (DAEs), and an effective method for solving the DAEs, based on backward Euler method, is proposed. The open-loop control formulation obtained this way is then extended by a closed-loop control law to provide stable tracking of the required reference trajectories in the presence of perturbations. Some examples of applications of the theory and results of numerical simulations are reported.

A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework

Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

/pdf/backward-differentiation-approximations-of-nonlinear-4lnwsb26hy.pdf

Backward differentiation approximations of nonlinear differential/algebraic systems

This paper describes the design of a “sparse” nonlinear programming algorithm for a “large scale” flight optimization and sizing code. This program is designed to solve trajectory optimal control problems with path constraints. The direct transcription technique is applied to discretize an optimal control problem into a sparse, large scale parameter optimization problem for subsequent numerical solution by a nonlinear programming algorithm. This paper focuses on the design of a generalized reduced gradient algorithm to exploit the sparsity structure of the collocation equations.

A Generalized Reduced Gradient Algorithm for Large-scale Trajectory Optimization Problems

As we did for the index-two system, we must prove there always exists a set of consistent initial values (c(tk..1), b(tk.1), e(tk.1)) for the linear, index-three DAE system (2.11)-(2.13) corresponding to any set of initial values (vk-1, wk.1, uk.1) satisfying (2.3),(2.4). From (2.13) it is clear we should choose b(tk,.l) E AK(A32(tk.l)). The consistency condition (2.3) implies we can write wA.-i = w(tk_.) + hk+lcl,k..l and vk._1 = v(tAk_1) + hk+1c2,A,.l for some bounded vectors clk-l and C2,kl1. Since we want our initial values to be consistent with the asymptotic error expansions (2.7),(2.8), we select for t = tk._1 and i = k 1,

Supplement to Backward Differentiation Approximations of Nonlinear Differential/Algebraic Systems

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Backward differentiation approximations of nonlinear differential/algebraic systems

A Generalized Reduced Gradient Algorithm for Large-scale Trajectory Optimization Problems

Supplement to Backward Differentiation Approximations of Nonlinear Differential/Algebraic Systems