Formulation and optimization of cubic polynomial joint trajectories for industrial robots

Abstract

Because of physical constraints, the optimum control of industrial robots is a difficult problem. An alternative approach is to divide the problem into two parts: optimum path planning for off-line processing followed by on-line path tracking. The path tracking can be achieved by adopting the existing approach. The path planning is done at the joint level. Cubic spline functions are used for constructing joint trajectories for industrial robots. The motion of the robot is specified by a sequence of Cartesian knots, i.e., positions and orientations of the hand. For an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -joint robot, these Cartesian knots are transformed into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> sets of joint displacements, with one set for each joint. Piecewise cubic polynomials are used to fit the sequence of joint displacements for each of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> joints. Because of the use of the cubic spline function idea, there are only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - 2</tex> equations to be solved for each joint, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is the number of selected knots. The problem is proved to be uniquely solvable. An algorithm is developed to schedule the time intervals between each pair of adjacent knots such that the total traveling time is minimized subject to the physical constraints on joint velocities, accelerations, and jerks. Fortran programs have been written to implement: 1) the procedure for constructing the cubic polynomial joint trajectories; and 2) the algorithm for minimizing the traveling time. Results are illustrated by means of a numerical example.