Real-time Dynamic Simulation of Constrained Multibody Systems using Symbolic Computation

Abstract

The main objective of this research is the development of a framework for the automatic generation of systems of kinematic and dynamic equations that are suitable for real-time applications. In particular, the efficient simulation of constrained multibody systems is addressed. When modelled with ideal joints, many mechanical systems of practical interest contain closed kinematic chains, or kinematic loops, and are most conveniently modelled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques.

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 Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing recursively solvable systems for calculating the dependent generalized coordinates given values of the independent coordinates. For systems that can be fully triangularized, the kinematic constraints are always satisfied exactly and in a fixed amount of time. Where full triangularization is not possible, a block-triangular form can be obtained that still results in more efficient simulations than existing iterative and constraint stabilization techniques.

The proposed approach is applied to the kinematic and dynamic simulation of several mechanical systems, including six-bar mechanisms, parallel robots, and two vehicle suspensions: a five-link and a double-wishbone. The efficient kinematic solution generated for the latter is used in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator. The Gröbner basis approach is particularly suitable for situations requiring very efficient simulations of multibody systems whose parameters are constant, such as the plant models in model-predictive control strategies and the vehicle models in driving simulators.