THE basis for all advanced manipulator control is therelationship between the cartesian coordinates of theendef fector and the manipulator joint coordinates. A directmethod for assigning link coordinate systems and obtaining theendeffector position and orientation in terms of jointcoordinates is reviewed (direct kinematics). An analyticaltechnique for solving the kinematic equations (inversekinematics) for n degrees of freedom manipulator has beendeveloped.Uecause of the physical constraints and the highlynonlinear robot dynamics, the optimum control of industrialrobots is a diff icul t problem. An al ternati ve approach is todivide the problem into two parts: optimum path planning foroff-line processing followed by on-line path tracking. The pathtracking can be achieved by adopting the inverse kinematictechnlque. The path planning is done at the joint level. Cubicand 4th degree spline functions are used for constructing jointtrajectories for industrial robots. The motion of the robot isspecified by a sequence of cartesian knots, i.e., positions andorientations of the endeffector. For an N-joint robot, thesecartesian knots are transformed into N sets of jointdisplacements, with one set for each joint, by using theinverse kinematic technique. Piecewise cubic and 4th degreepolynomials are used to fit the sequence of joint displacementsfor each of the N joints. Because of the use of the splinefunctions idea, There are only (n-2) equations to be solved for