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 mechanical manipulators. The motion of the manipulator is specified by a sequence of Cartesian knots, i.e. positions and orientations of the hand. For a 6-joint manipulator, these Cartesian knots are transformed into six sets of joint values, with each set corresponding to a joint. Piece-wise cubic polynomials are used to fit the sequence of joint values for each of the six joints. The problem is proved to be uniquely solvable. Furthermore, an algorithm is developed to schedule the time periods 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 given as an illustration.