In this paper, we present a task space-based local motion planner that\nincorporates collision avoidance and constraints on end-effector motion during\nthe execution of a task. Our key technical contribution is the development of a\nnovel kinematic state evolution model of the robot where the collision\navoidance is encoded as a complementarity constraint. We show that the\nkinematic state evolution with collision avoidance can be represented as a\nLinear Complementarity Problem (LCP). Using the LCP model along with Screw\nLinear Interpolation (ScLERP) in SE(3), we show that it may be possible to\ncompute a path between two given task space poses by directly moving from the\nstart to the goal pose, even if there are potential collisions with obstacles.\nThe scalability of the planner is demonstrated with experiments using a\nphysical robot. We present simulation and experimental results with both\ncollision avoidance and task constraints to show the efficacy of our approach.\n