Using Quasi-linearization for Real Time Dynamic Simulation of a Quarter Vehicle Suspension 2007-01-0833

Real time dynamic simulation of mechanisms with kinematically closed loops requires solving systems of nonlinear differential algebraic equations (DAE). Examples of such mechanisms are racing car suspensions and certain robotic arms. Simulating such systems in real time requires significant computational power. This paper explores an alternative approach in an attempt to minimize computational effort. A hybrid, two-step approach is employed of first applying symbolic math methods followed by numerical simulation. Explored is the possibility of optimizing formulae and precalculating coefficients to speed up real time simulation. Quasi-linearization is suggested as a method of solving and simulating nonlinear DAE in real time. The equations of motion are linearized in every point of the state space. The result is a system of linear ordinary differential equations with varying coefficients. These coefficients are functions of the simulated system state space (positions and their first derivatives). A symbolic math package - Mathematica™ - is used to derive and simplify the varying coefficients. The derived coefficient functions and ordinary differential equations are simulated numerically by a Simulink™ model. Real time simulation is performed with C code generated from the Simulink™ model. To achieve maximum simulation speed, the paper explores precalculating the varying coefficients in lookup tables. The outlined process was applied to the simulation of a quarter car double wishbone suspension. The quasi-linearization approach provided numerically exact solutions for both small and large signal amplitudes. A vital requirement for successful simulation proved to be the selection of states to be actual physical values (positions of parts, and their derivatives). The validity of solutions was checked against iterative solution as well as standard linearized model.