Mathematical Problems in Engineering

Volume 2018, Article ID 8945301, 10 pages

https://doi.org/10.1155/2018/8945301

## Determination of Stability Correction Parameters for Dynamic Equations of Constrained Multibody Systems

^{1}Robotics Institute, School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China^{2}School of Mechanical and Electrical Engineering, Heze University, Heze 274015, China

Correspondence should be addressed to Lyu Guizhi; moc.621@ihziugvl

Received 11 February 2018; Revised 26 March 2018; Accepted 28 March 2018; Published 6 May 2018

Academic Editor: Giovanni Falsone

Copyright © 2018 Lyu Guizhi and Liu Rong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

When analyzing mechanical systems with numerical simulation by the Udwadia and Kalaba method, numerical integral results of dynamic equations will gradually deviate from requirements of constraint equations and eventually lead to constraint violation. It is a common method to solve the constraint violation by using constraint stability to modify the constraint equation. Selection of stability parameters is critical in the particular form of the corrected equation. In this paper, the method of selecting and determining of stability parameters is given, and these parameters will be used to correct the Udwadia-Kalaba basic equation by the Baumgarte constraint stability method. The selection domain of stability parameters is further reduced in view of the singularity of the constraint matrix during the integration procedure based on the selection domain which is obtained by the system stability analysis method. Errors of velocity violation and position violation are defined in the workspace, so as to determine the parameter values. Finally, the 3-link spatial manipulator is used to verify stability parameters of the proposed method. Numerical simulation results verify the effectiveness of the proposed method.

#### 1. Introduction

The theory, which was proposed for modeling and analysis of the constrained system dynamics by Udwadia and Kalaba, has been applied more and more in mechanical systems [1–5]. However, when the mechanical system is analyzed and simulated with Udwadia and Kalaba method, the results of numerical integration of system dynamics equation will deviate from requirements of constraint equations over time and eventually lead to constraint violation. In order to solve the constraint violation problem, Udwadia has proposed a tracking control method for nonlinear systems based on the Lyapunov stability principle [6]. The asymptotic stability control of the system is realized by modifying the constraint equations. The basic idea of the method is essentially consistent with the Baumgarte stabilization method for constraint violations [7] and can be widely used in various working environment [8–15].

The stability parameters are important for the Baumgarte stabilization method to modify constraint violations, and they are often selected by the experience method. Although it has been mentioned that different values of stability parameters should be chosen for different constraint equations [16], few people further research on this. A variety of methods have been studied in order to select appropriate parameters to keep the system stable after constraint equations modified. The Taylor expansion comparison method can directly calculate values of stability parameters [17, 18]. However, the choice of stability parameters is strictly related to the integral time step. Stability parameters will be too large if the time step is too small, which will make dynamic equations distorted and the system unstable. The method of system stability analysis is used to select stable parameters, in which different numerical integration methods [19–21] and different integration time step [22, 23] would be considered to affect the selection area of stability parameters. But parameters, selected from the area of stability parameters with the method of system stability analysis, could not guarantee the continuous stability of the system in the numerical integration process. This paper focuses on further reducing the area of stability parameters on the basis of system stability analysis through the singularity determination for the constraint matrix and selecting specific stability parameters according to engineering requirements.

The outline of the paper is presented as follows: in Section 2 the form of modified Udwadia-Kalaba dynamic equation with constraint stability is presented. In Section 3 the selectable region of stability parameters and the determination method of stability parameters are given. In Section 4, the proposed method is verified and discussed; after the circular trajectory is defined as constraint, a dynamic model of the spatial 3-link manipulators is established and the numerical simulation is completed. Finally, conclusions are provided in Section 5.

#### 2. Udwadia-Kalaba Equation with Violation Stability

Basing on the modeling idea of multibody dynamics system, the equation of motion for a constrained mechanical system can be described by where is the mass matrix, is the generalized force for unconstrained system, and is the generalized force needed to be applied in the system in order to satisfy a given constraint.

Each part of the mechanical system needs to move along a specific trajectory, which can be regarded as a constraint in order to accomplish a given task. If the multibody system is composed of generalized coordinates, and there are independent movements at the position level, the constraint equation can be written as where is the generalized coordinate matrix. Formula (2) is differentiated twice with respect to time; the constraint equation at acceleration level is where is the Jacobian matrix and is a vector.

According to the Udwadia-Kalaba equation, if the initial condition of the system satisfies the constraint Eq. (2), then the closed solution of the generalized constraint force on the system can be obtained from [24]in which “+” represents the Moore–Penrose generalized inverse.

Eq. (4) indicates the control force that the system needs to be applied, when the unconstrained mechanical system was required to move along the given constraint trajectory by (2). Substitute the expression item of (4) into (1) and rewrite the formulation in a more visible way; the fundamental equation of Udwadia-Kalaba can be obtained as

The constraints represented by (2) should be satisfied by the control force that gained from (4). However, in the simulation process, the integral error of in formula (5) increases with the time, and the motion trajectory of the mechanical system obtained eventually deviates from the given trajectory from (2). Therefore, the method of Baumgarte constraint violation stability can be used to correct (3). The corrected constraint equation can be written as This formula is the differential equation of the closed loop system of the constrained equation, in which and can be considered as control parameters. From the formula (6), it can be found that the fundamental principle of the correction equation is to correct the acceleration by the feedback of position and velocity. The structure of the control system of closed loop can be considered as Figure 1.