AN OPTIMAL MOTION PLANNING ALGORITHM FOR MANIPULATORS - American Control Conference, Proceedings of the 1995

The optimal motion generation problem is solved subject to actuator constraints while the motion is constrained to an arbitrary path. The considered objective function is a weighted time-energy function. We present some results using a mathematical programming technique. 1. Introduction Motion along a predefined path is common in robotics. The path is given from the application and a first step is to obtain a nominal motion specification. In this paper, the emphasis is put on the optimal planning of manipulators trajectories, in joint space. For rigid robots, the minimal time optimization along a predefined path can be solved using phase-plane techniques [ 6 ]. This method, however cannot be extended to the case under interest with a time-electric energy performance index. In this paper, the determination of the desired robot motion as a function of time involves a nonlinear optimal problem. We will focus on manipulators actuated by DC motors which operate over a wide speed range and have excellent control characteristics. 2. Modelling 2.1 Manipulator Model For a manipulator with n joints, the dynamic model can be expressed using the.Lagrangian cquation as: where the (n,l) vectors q, q and q are the joint position, velocity and acceleration, the (n,l) vector r is the joint input torque, G is the (n,l) gravitational force vector, B is the (n,n,n) Coriolis and Centrifugal forcc matrix, F is the viscous friction and A is the (n,n) inertial matrix. 2.2 Actuator Model For a non-redundant multi-degrees-of-freedom robot, there are usually as many actuators as the number of do -f. In a permanent magnet DC motor, the actuator dynamics, giving the voltage U as a function of the current I are: where L, R and K are square regular diagonal matrices representing the inductance, resistance and torque constant. The path describes the robot motion in space and is represented as a parameterized curve, q = q(s), where s is a scalar path parameter. The optimal path planning problem which consists in finding q(t) is then replaced by the optimal search of s(t) on the interval [O,T]. r = ~ (q h + b T w 4 + w + F m (1) (2) 3. Motion Generation Problem 3.1 Introduction Most motion generation laws are developed based on kinematical constraints, obtained for the most unfavorable configurations. Thus, to define maximal torques, accelerations and velocities, we have to ncglect some terms of the dynamic models (1) and …