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Optimization of holonomic as well as non-holonomic multi body systems is presented as a nonlinear programming problem that can be solved with general-purpose optimization codes. The adjoint variable approach is used for calculating design derivatives of a rather general integral type performance measure with respect to design parameters. The resulting equations are solved by numerical integration backward in time. A multistep integration algorithm with order and step-size control is adapted for this application by including an interpolation scheme. Numerical experiments and a comparison to the common approach of approximating the gradient of the performance measure by finite differences show that high efficiency, accuracy, and reliability are achievable. 'Communicated by E. J. Haug.

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Abstract. Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variablestepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

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