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Globally Optimal Inverse Kinematics as a Quadratic Program

Tomáš Votroubek, Tomáš Kroupa

TL;DR

This work shows how to compute globally optimal solutions to inverse kinematics by formulating the problem as an indefinite quadratically constrained quadratic program and demonstrates the performance on randomly generated designs and on real-world robots with up to ten revolute joints.

Abstract

We show how to compute globally optimal solutions to inverse kinematics (IK) by formulating the problem as an indefinite quadratically constrained quadratic program. Our approach makes it feasible to solve IK instances of generic redundant manipulators. We demonstrate the performance on randomly generated designs and on real-world robots with up to ten revolute joints. The same technique can be used for manipulator design by introducing kinematic parameters as variables.

Globally Optimal Inverse Kinematics as a Quadratic Program

TL;DR

This work shows how to compute globally optimal solutions to inverse kinematics by formulating the problem as an indefinite quadratically constrained quadratic program and demonstrates the performance on randomly generated designs and on real-world robots with up to ten revolute joints.

Abstract

We show how to compute globally optimal solutions to inverse kinematics (IK) by formulating the problem as an indefinite quadratically constrained quadratic program. Our approach makes it feasible to solve IK instances of generic redundant manipulators. We demonstrate the performance on randomly generated designs and on real-world robots with up to ten revolute joints. The same technique can be used for manipulator design by introducing kinematic parameters as variables.
Paper Structure (14 sections, 6 equations, 7 figures, 3 tables)

This paper contains 14 sections, 6 equations, 7 figures, 3 tables.

Table of Contents

  1. INTRODUCTION
  2. QUADRATIC PROGRAM FORMULATION
  3. Denavit–Hartenberg parameters
  4. Polynomial Formulation
  5. Lifting Algorithms
  6. Global Optimality Certificates
  7. Applicability of QCQP Lifting
  8. RESULTS
  9. Factors Influencing the Solution Time
  10. Comparison Across Solvers and Methods (7 DOF)
  11. iCub Right Arm and Torso (7--10 DOF)
  12. Quality of Local Solutions
  13. Infeasible Poses
  14. CONCLUSIONS

Figures (7)

  • Figure 1: Diagram of Denavit–Hartenberg parameters
  • Figure 2: A semi-log plot of solve-time kernel-density-estimates from 1000 samples of randomly generated designs for each parameter set.
  • Figure 3: A semi-log plot comparing the normalized cumulative distributions from 100 solve times for the KUKA LBR iiwa 7-DOF robot using our QCQP approach (Gurobi, SCIP) and the SOS-based technique.
  • Figure 4: A semi-log plot comparing the normalized cumulative distributions of solve times obtained from 100 samples of randomly generated designs with 6-radian range of motion using our approach and the SOS-based technique (rounding to two significant digits).
  • Figure 5: Normalized cumulative distributions $F(s)$ of solve times from 1000 samples of the iCub's 7-DOF right arm. The full SOS-based method with symbolic reduction did not finish in time.
  • ...and 2 more figures