Algorithmic Optimization of Inverse Kinematics Tables for High Degree-of-freedom Limbs

This work addresses the problem of resolving kinematic redundancy in legged robots, with the dual goals of maintaining a large reachable workspace and of achieving fast end effector motions in task space. In particular, for robots with four or more legs, gait planning allows for considerable flexibility in the orientation of a stance limb with respect to both body orientation and the ground. By appropriately commanding pitch, roll and yaw of the end effector as it moves relative to the body coordinate frame, one can increase the volume of space the feet can reach and thus allow the robot to negotiate larger terrain obstacles. At the same time, motions of the foot in task space should be done rapidly, given the joint velocities of the limbs. In this paper, we focus on RoboSimian, a robot with four identical limbs designed for dual use in manipulation and locomotion tasks, which was designed at Jet Propulsion Labs (JPL) for the DARPA Robotics Challenge (DRC). We present both heuristic guidelines and a novel, gradient-based algorithm for developing rules to set the inverse kinematics (IK) solution for the seven joint angles of a limb, allowing us to prescribe joint solutions rapidly through the use of an IK look-up table. NOMENCLATURE φee Roll (about x axis) of the end effector. ∗Address all correspondence to this author. ψee Yaw (about z axis) of the end effector. θee Pitch (about y axis) of the end effector. qi The ith joint angle, starting proximal to the body and going outward in a serial, kinematic chain. C Cost function for end effector motion, to be minimize. wφ ,i Scaling parameter (weighting) on roll, for ith joint. wψ,i Scaling parameter (weighting) on pitch, for ith joint. wθ ,i Scaling parameter (weighting) on yaw, for ith joint. xee Cartesian end effector x coordinate; body coordinate frame. yee Cartesian end effector y coordinate; body coordinate frame. zee Cartesian end effector z coordinate; body coordinate frame. Ree Polar coordinate for end effector radius, relative to limb coordinate frame. Θee Polar coordinate for end effector angle, relative to limb coordinate frame. Zee Polar coordinate for end effector height, relative to limb coordinate frame. Cartesian end effector (e.e.) coordinates and Euler angles are defined relative to the coordinate system of the chassis of the robot in space, referred to as the body coordinate system. Euler angles in this work use x-y-z ordering, i.e., with roll (φ ), pitch (θ ), and yaw (ψ) rotations performed sequentially about local axes. Polar coordinate definitions of the end effector are relative to limb coordinate frame, based at the root (proximal end) of the serial kinematic chain of a limb. The right, front limb is used in 1 Copyright c © 2014 by ASME all analyses in this work; with solutions for other limbs simply being related in an obvious manner, through symmetry. FIGURE 1. RoboSimian during the DARPA Robotics Challenge. Dexterity of the limbs allows the robot the reach a large workspace on rough terrain (top), and adding roll and pitch to the end effector enables even longer footsteps (bottom).