Driving simulator with double-wishbone suspension using efficient block-triangularized kinematic equations

When modeled with ideal joints, many vehicle suspensions contain closed kinematic chains, or kinematic loops, and are most conveniently modeled 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 for a double-wishbone suspension 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. Symbolic computation is then used to triangularize a subset of the kinematic constraint equations, thereby producing a recursively solvable system for calculating a subset of the dependent generalized coordinates. Thus, the kinematic equations are reduced to a block-triangular form, which results in a more computationally efficient solution strategy than that obtained by iterating over the original constraint equations. The efficiency of this block-triangular kinematic solution is exploited in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardwareand operator-in-the-loop driving simulator.