Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator

The Inverse Kinematics (IK) problem is concerned with finding robot control parameters to bring the robot into a desired position under the kinematics and joint limit constraints. We present a globally optimal solution to the IK problem for a general serial 7DOF manipulator with revolute joints and a polynomial objective function. We show that the kinematic constraints due to rotations can be all generated by the second-degree polynomials. This is an important result since it significantly simplifies the further step where we find the optimal solution by Lasserre relaxations of nonconvex polynomial systems. We demonstrate that the second relaxation is sufficient to solve a general 7DOF IK problem. Our approach is certifiably globally optimal. We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are, in practice, able to compute the optimal IK or certify infeasibility in 99.9% tested poses. We also demonstrate that by the same approach, we are able to solve the IK problem for any generic (random) manipulator with seven revolute joints.