The extra degrees of freedom of a redundant manipulator can be used to improve dexterity, to avoid singularity and to optimize certain performance criterion such as energy consumption. However, the motion of a redundant manipulator is usually not conservative while it is tracking a repetitive task. That is, it requires re-calculation of its motion at each increment in a repetitive task. Therefore, conservative motion has been a research topic of redundant manipulators. In this paper, two trajectory-planning methods based on the pseudo-inverse method and Euler Taylor-series are presented. These two methods enable us to obtain a convergent configuration(s) after repetitive motion cycles and the motions become conservative after the manipulator reaches the convergent configuration(s). The required number of iterations could be exceedingly large for certain trajectories. Two methods which are able to efficiently identify the convergent configuration(s) are also presented. This study is based on a three-linked planar manipulator tracking four trajectories: circles, ellipses, squares and line segments.