Compurers & Slrucrures Vol. 29, No. 6, pp. 983991, 1988 0045s7949/88 53.00 + 0.00 Printed in Great Britain. 0 1988 Pergamon Press plc MODELLING OF CLEARANCES AND JOINT FLEXIBILITY EFFECTS IN MULTIBODY SYSTEMS DYNAMICS F. M. L. AMIROUCHE and T. JIA Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60608, U.S.A. zyxwvutsrqponmlkjih (Received 8 October 1987) Abstract-An effective method for modelling the dynamic response of multibody systems with flexible joints is presented. The method combines the use of finite element method with Kane’s equations to present an algorithm strictly in terms of the generalized coordinates of the system. The procedures developed outline the automatic incorporation of the joint flexibility in the equations of motion, hence a more accurate mathematical model is developed. One other advantage to the method presented lies in the explicit forms of the coefficients needed in the analysis where they are readily expressed in a form suited for computer implementation. A discussion on possible applications is also presented. INTRODUCTION Current research in multibody dynamics and especially robotics evolves around the formulation of the exact governing equations of motion that would include joints and links’ flexibility effects during motion, specified motions to certain links and end-effecters, closed-loop formations, collision avoidance, sensors and vision feedback information, among others. The design of proper controls for accurate tracking must include all the factors listed above. In order to reduce the complexity of the equations obtained, a full detailed analysis must be done. The objective of the present paper is to present the equations of motion when the joints’ deformations are present in a computer automated form as presented in [l, 21, where the coefficients are evalu- ated automatically through block arrays suitable for the dimensions given. Even though a number of efforts has been made in the past few years on the modelling of joint flexibility, it is only recently that Dubowsky et al. [3] and Soni et al. [4] have stressed the importance of the joints’ contribution in the overall motion of a robotic system. In this paper the generalized stiffness and damping forces are formulated and added to the equations of motion to account for the deformation and clearances at the joints. This paper should add a new dimension to pre- viously published work on compliance and flexibility effects by Huston [S]. The procedures developed clearly show how the equations of motion are derived including the joints’ deformation, hence more accu- rate motion is attained. The equations of motion are expressed in terms of the generalized coordinates where the elastic coordinates of the joints are related to the generalized coordinates of the system using the proper transformation matrices. ANALYTICALDEVELOPMENT Let a typical robot be denoted by n links inter- connected to one another by arbitrary joints. Let those joints be modelled as elastic bodies. Each joint has two points where the adjacent links connect. In our analysis we will investigate the contribution of the elastic deformation of the joints to the overall motion of the robot. We will try to show precisely how the flexibility effects of the joints affect the position of the end-effecters. The analysis will be based on arbitrary shaped joints that can undergo six degrees of freedom, hence it will be applicable to most designs that are readily available. The mathe- matical model of the joints involves both the system stiffness and damping, and the motor inertia. Stiffness and damping could be added on through an ad hoc technique. The objective will be to formulate the governing equations of motion, maintaining the links rigid and expressing the equations in terms of the generalized coordinates of the system. The advan- tages to the procedures developed herein stems from the isolation of scalar arrays that will be associated with the generalized coordinates and speeds for the stiffness and damping respectively. The method em- ployed herein utilizes Kane’s equations as developed by Huston et al. [l]. The generalized stiffness and damping force will be derived, when the joint is allowed twelve degrees of freedom, six at each end, three for rotation and three for translation. The equations of motion of an n-link arm robot including the effects of flexibility at the joints modelled by stiffness and damping forces could be obtained using Kane’s equations as J; +f: +.I-; +fP = 0, (1) where f; represent the generalized active forces, f: represent the generalized inertia forces, f; and fP 983