~ Pergamon Mech. Math. Theory Vol. 30, No. 5, pp. 749-764, 1995 Copyright © 1995ElsevierScienceLtd 0094-114X(94)E0005-5 Printed in Great Britain.All rights reserved 0094-114X/95 $9.50+ 0.00 EFFICIENT MODELING OF RIGID LINK ROBOT DYNAMIC PROBLEMS WITH FRICTION¢ CHARLES DHANARAJ and ANAND M. SHARAN Faculty of Engineering and Applied Sciences, Memorial University of Newfoundland, St John's, Newfoundland, Canada AIB 3X5 (Received 26 January 1990; in rev&ed form 14 December 1993; received for publication 7 February 1994) Abstract--ln industrial manipulators for high torque applications, friction is quite significant and has to be incorporated in the dynamic model to ensure control robustness. Such a dynamic model has to be efficient to facilitate real-time computations. The paper discusses a modified Newton-Euler algorithm for symbolic implementation which proves to be computationally more efficient than the earlier published results. The paper also describes a method for modeling friction at the joints as well as in the transmissions in the robotic manipulators. NOMENCLATURE {a}~--acceleration of the origin of the ith link co- ordinate frame {a},,--acceleration of the center of gravity of the ith link co-ordinate frame d---distance between the support bearings D,, D,, D.--inertia constants {f}i--reaction force at the ith joint {f~--reaction force at the ith joint referred in jth co-ordinate frame f--frictional force F--frictional force {g }--gravity vector [l]7-centroidal inertia tensor of the ith link /--subscript to denote the link number k~, k,, k.--kinematic parameters m--reduction ratio of the harmonic drive n--the total number of links in the manipulator {n },--reaction moment at the ith join {N}~inertial moment of the ith link {p}~--position vector of the i + lth origin r--radius of the journal {r}--position vector of a point q~, qi,/]i--position, velocity and acceleration of the link movement (angular for revolute joints and linear for prismatic joints) [Rli--Transformation matrix from ith to i- lth frame {s }--position vector of the center of gravity of the ith link x, y, z--subscripts to denote the x, y and z components of a vector {z}--unit vector along the local z direction (axis of motion) {~o}i--absolute angular velocity of the ith link {ct},--absolute angular acceleration of the ith link [2l,--acceleration difference matrix r,--basic dynamic torque of the ith link (zf)J--frictional torque at the joint (zr)~frictional torque in the transmission r~,--input torque of the harmonic drive rout--output torque of the harmonic drive rn--break-away torque of the harmonic drive p---co-efficient of kinetic friction jl--absolute value { }--vector or column matrix [ ]--matrix 1. INTRODUCTION As robotic manipulator systems become increasingly common in industrial and space applications, an accurate manipulator dynamics that govern their operations become essential to ensure control robustness. The formulation of the dynamic equations have to be such that they can be solved very efficiently in terms of multiplications, additions, divisions, etc. Various formulations have been proposed in the literature, which are: (a) the conventional Lagrangian formulation [1]; (b) the recursive Lagrangian formulation [2]; (c) the recursive Newton-Euler (NE) formulation [3]; (d) Kane's formulation [4]; and (e) tensor formulation [5]. Though the NE formulation is computation- ally less intensive compared to the Lagrange's formulation, the structure of the equations are not conducive for control implementation. Symbolic computations have been proposed to increase the computational efficiency, where the multiplications with zeros and ones in real-time are avoided tDue to circumstances beyond the Publisher's control this paper appears in print without author corrections. 749