This paper suggests three methods which are useful to the path planning of cyclic tasks for a redundant manipulator. The first method is based on the necessary condition for optimality to optimize any performance indices while performing a given continuous path task. To overcome the shortcomings of the previous researches where the equation of optimal solution depends on the performance index, a unified necessary condition is derived and a unified approach for the equation of solution is proposed. This method is also applicable to obtain a globally optimal joint trajectory. Since only necessary condition for optimality is used in this method, it yields extremal solutions as well as an optimal solution. Therefore, several features of extremal solutions are expressed so that an optimal solution can be obtained by using the features through the topological liftings of the path. On the other hand, the second method is based on Fourier series approximation to optimize joint velocity norm while performing a given continuous path task. Since a cyclic joint trajectory is practical for a cyclic task, an optimal joint trajectory can be represented as Fourier series and searched by using the numerical method. This method shows that it provides a globally optimal solution without topological liftings of a path. The last method is also based on Fourier series approximation to generate a smooth joint trajectory while a manipulator carries out a point-to-point task. Smoothness of a joint trajectory often requires first-order or second-order continuity. However, the continuity of the joint trajectory is meaningful when the frequency components due to structural resonant frequency or infinite frequency components of the joint trajectory is considered. This method provides a smooth and first-order (or more) continuous joint trajectory.
