Triangularizing kinematic constraint equations using Gröbner bases for real-time dynamic simulation

Abstract

Real-time simulation is an essential component of hardware- and operator-in-the-loop applications, such as driving simulators, and can greatly facilitate the design, implementation, and testing of dynamic controllers. Such applications may involve multibody systems containing closed kinematic chains, which are most readily modeled using a set of redundant generalized coordinates. The governing dynamic equations for such systems are differential-algebraic in nature—that is, they consist of a set of ordinary differential equations coupled with a set of nonlinear algebraic constraint equations—and can be difficult to solve in real time. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Grobner bases is then used to triangularize the kinematic constraint equations, thereby producing a recursively solvable system for calculating the dependent generalized coordinates given values of the independent coordinates. The proposed approach can be used to generate computationally efficient simulation code that avoids the use of iteration, which makes it particularly suitable for real-time applications.