Studies On State-space Based Methods For Differential-algebraic Equations Of Multibody System Dynamics

In this dissertation,state-space based methods for differential-algebraic equations of multibody system dynamics are studied.Several new state-space based methods with LU decomposition used in the coordinate partitioning process are proposed to solve four kinds of differential-algebraic equations of multibody system dynamics with different types of constraints:with only holonomic constraints,with only linear nonholonomic constraints,with mixed holonomic and linear nonholonomic constraints,and with mixed holonomic and nonlinear nonholonomic constraints.For the differential-algebraic equations with only holonomic constraints,the two-loop structures of the state-space based methods with implicit integration methods are studied.A two-loop implicit state-space based method with position and velocity as basic unknowns is proposed.To use the implicit Runge-Kutta method as the integration method in state-space based methods,the fixed-point iterative method for implicit Runge-Kutta methods is proposed.The iteration of nonlinear position constraint equations and the solving of linear velocity constraint equations are embedded into the iteration of the implicit integration method with position and velocity as basic unknowns,so that the two-loop structure is constructed.This two-loop structure makes sure that the dependent coordinates in the differential-algebraic equations are updated together with the independent coordinates in the iterative process of implicit integration methods.Thus the accuracy and stability of state-space based methods are improved,and the numerical results satisfy the constraint equations in a strict way.Backward differentiation formulas can also be used in this two-loop implicit state-space based method as the integration method.Classical state-space based methods can not be used to solve the differential-algebraic equations with nonholonomic constraints.New state-space based methods are proposed to solve the differential-algebraic equations with nonholonomic constraints.In the proposed methods,the ODEs based on the minimal set of variables are replaced by the ODEs obtained from the index-1 form of differential-algebraic equations,and coordinate partitioning procedures based on LU decomposition are applied respectively for velocity constraints and position constraints to identify the independent velocities and independent positions of nonholonomic systems.Three state-space based methods are proposed to solve differential-algebraic equations of system with only nonholonomic constraints,system with mixed holonomic and linear nonholonomic constraints,and system with mixed holonomic and nonlinear nonholonomic constraints.These three different state-space based methods are put together under a unified framework to construct variable step-size algorithms.Explicit and implicit Runge-Kutta methods are used to construct variable step size algorithms.In the module of integration,both independent variables and dependent variables are obtained,and then the position constraint equations and the velocity constraint equations are solved to eliminate the violation of constraints.The numerical results of examples of mobile robots and control systems show that the proposed state-space based methods can be used to solve differential-algebraic equations with nonholonomic constraints effectively.The relationship of constraint equations at different levels can be analyzed by applying Taylor's series,and it can be proved that the violation of the position constraint equations in the differential-algebraic equations is in a limited range if the velocity constraint equations has been strictly satisfied.The procedure which makes sure the position constraint equations are satisfied is not absolutely necessary in the numerical methods.Based on this fact,the modified state-space based methods are proposed to improve the generality of state-space based methods.In the modified state-space based methods,the solving of position constraint equations is replaced by the integration of ODEs,and the velocity constraint equations are solved to eliminate the violation of velocity constraint equations.The linear velocity constraint equations can be solved by using numerical methods for linear algebraic equations,and nonlinear velocity constraint equations are solved by using Newton's method.Linear velocity constraint equations can also be solved by Newton's method,The four DAEs studied in this dissertation can all be solved by a same modified state-space based methods.It is shown in this dissertation that,different differential-algebraic equations can be effectively solved by using state-space based methods.The state-space based methods with the implicit integration methods for ODEs are good in accuracy,generality,stability and efficiency.