Contact Instability of the Direct Drive Robot When Constrained by a Rigid Environment

Robot manipulations require mechanical interaction with the environment (i.e., the object being manipulated) which can constrain the robot endpoint in some or all directions. A sufficient condition for the stability of robot manipulators in constrained maneuvers is derived in the work presented here. Attention is focused on the class of direct drive robots whose rigid links dominate the robot's dynamic behavior; compared to the robot, the environment is assumed to be infinitely rigid. The stability of the manipulatorenvironment system is investigated, and a bound for stable manipulation is determined. This bound is verified experimentally on the Minnesota direct drive robot. 1. UNCONSTRAINED ANALYSIS The dynamic behavior of direct drive robots with n degrees of freedom is expressed by equation 1: M(e)!j+C(e,9)='t"-JTf (1) where ~. ~ , e are vectors containing the joints' accelerations, velocities, and positions; M(e) is the inertia matrix; C(e.~) is the vector representing the coriolis, centrifugal; and gravity forces; 't" is the vector of joint torques; JT is the Jacobian transpose matrix; and f is the vector of external forces applied at the robot endpoint (Hollerbach 1980). Trajectory control of the manipulator is performed by a digital implementation of a feedforward torque controller, which torque is given by: 't" = Kp (ed -e) + K" (~d -~) + "" (ed) §d + ~ (ed. ~d) (2) where 't" is the vector of joint torques; (ed -e) is the error between the command position, ed, and the actual position, e, and (~d-~) is the error between the respective velocities; Kp is an n"n matrix containing the position gains; K" is an n"n matrix containing the velocity gains; r1 (ed) and e (ed.~ d), which can be found experimentally or analytically, are educated guesses for M(e) and C(e.~). The nonlinear feedforward terms, ",,(ed) and e(ed.~d)' cancel the nonlinear effects of M(e) and C(e.~) in the robot's dynamics and result in a nearly uncoupled linear system (Spong and Vidyasagar, 1985). In feedforward torque control, the robot trajectory is specified in joint coordinates, and the joint positions, velocities, and accelerations for a given trajectory are computed and stored before the trajectory is executed. The Kln(,) operator in the diagram represents the forward kinematics, while Kln-1(,) represents the inverse kinematics. When the trajectory is specified in Cartesian space as a function of time, e(t), inverse kinematics and numerical differentiation transform it into the joint space, ed(t).