Stability and convergence results for difference approximations of overdetermined differential-algebraic equations with application to implicit integration of constrained mechanical system dynamics

Methods for forming and solving differential-algebraic equations (DAE) of motion for constrained mechanical system dynamics have advanced to the point that real-time simulation is possible using parallel processing and recursive formulations. However, stability of the simulation is still a major issue, suggesting that implicit numerical integration methods be used to take advantage of their larger stability regions. The governing DAE are first written as overdetermined differential-algebraic equations (ODAE) by twice differentiating the kinematic constraint equations. The ODAE is then treated as an ordinary differential equation on the constraint manifold. Using a backward differentiation formula (BDF) to discretize the ODAE, a system of nonlinear algebraic equations is obtained and solved by Newton-like methods that require the construction of a Jacobian matrix of derivatives that cannot be evaluated using elementary calculus. Instead of using finite difference approximation in evaluating this Jacobian matrix, recursive differentiation is used. This method is shown to be theoretically valid and numerical experiments show that it is numerically efficient. In this thesis, finite difference approximations of the ODAE are analyzed. Convergence and stability results for a k-step backward differentiation formula are proved for certain subclasses of DAEs, when initial values satisfy certain constraints. Both sufficient and necessary conditions for convergence with explicit and implicit numerical integration schemes are given. With the introduction of two linear operators, the stability results derived for both explicit and implicit numerical integration schemes are proved to be true in a small neighborhood of the solution.