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Motion Accuracy and Computational Effort in QP-based Robot Control

Sélim Chefchaouni, Mehdi Benallegue, Adrien Escande, Pierre-Brice Wieber

TL;DR

The paper analyzes the often-overlooked link between QP solver accuracy and actual robot motion in a QP-based whole-body controller for humanoids. By using a RHPS-1 dynamic simulation, it demonstrates that motion accuracy targets are achievable even with substantially degraded QP solutions, enabling dramatic reductions in computation through infrequent updates of QP matrices and, optionally, the QP solution itself. The key finding is that updating the QP's defining matrices at a reduced rate (around 9 Hz) yields no noticeable loss in tracking while reducing computation by more than 27×, highlighting a practical path to energy- and cost-efficient embedded control. These results suggest that high-precision QP solving is unnecessary for many manipulation tasks and motivate further validation on more dynamic scenarios and real hardware.

Abstract

Quadratic Programs (QPs) have become a mature technology for the control of robots of all kinds, including humanoid robots. One aspect has been largely overlooked, however, which is the accuracy with which these QPs should be solved. QP solvers aim at providing solutions accurate up to floating point precision ($\approx10^{-8}$). Considering physical quantities expressed in SI or similar units (meters, radians, etc.), such precision seems completely unrelated to both task requirements and hardware capacity. Typically, humanoid robots never achieve, nor are capable of achieving sub-millimeter precision in manipulation tasks. With this observation in mind, our objectives in this paper are two-fold: first examine how the QP solution accuracy impacts the resulting robot motion accuracy, then evaluate how a reduced solution accuracy requirement can be leveraged to reduce the corresponding computational effort. Experiments with a dynamic simulation of RHPS-1 humanoid robot indicate that computational effort can be divided by more than 27 while maintaining the desired motion accuracy.

Motion Accuracy and Computational Effort in QP-based Robot Control

TL;DR

The paper analyzes the often-overlooked link between QP solver accuracy and actual robot motion in a QP-based whole-body controller for humanoids. By using a RHPS-1 dynamic simulation, it demonstrates that motion accuracy targets are achievable even with substantially degraded QP solutions, enabling dramatic reductions in computation through infrequent updates of QP matrices and, optionally, the QP solution itself. The key finding is that updating the QP's defining matrices at a reduced rate (around 9 Hz) yields no noticeable loss in tracking while reducing computation by more than 27×, highlighting a practical path to energy- and cost-efficient embedded control. These results suggest that high-precision QP solving is unnecessary for many manipulation tasks and motivate further validation on more dynamic scenarios and real hardware.

Abstract

Quadratic Programs (QPs) have become a mature technology for the control of robots of all kinds, including humanoid robots. One aspect has been largely overlooked, however, which is the accuracy with which these QPs should be solved. QP solvers aim at providing solutions accurate up to floating point precision (). Considering physical quantities expressed in SI or similar units (meters, radians, etc.), such precision seems completely unrelated to both task requirements and hardware capacity. Typically, humanoid robots never achieve, nor are capable of achieving sub-millimeter precision in manipulation tasks. With this observation in mind, our objectives in this paper are two-fold: first examine how the QP solution accuracy impacts the resulting robot motion accuracy, then evaluate how a reduced solution accuracy requirement can be leveraged to reduce the corresponding computational effort. Experiments with a dynamic simulation of RHPS-1 humanoid robot indicate that computational effort can be divided by more than 27 while maintaining the desired motion accuracy.
Paper Structure (22 sections, 22 equations, 6 figures, 3 tables)

This paper contains 22 sections, 22 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Representation
  4. Task-Based Approach
  5. Control Constraints
  6. Equations of Motion
  7. Feet Motion
  8. Friction Cones
  9. Joint Limits
  10. Control Objectives
  11. Left Hand Motion
  12. Center of Mass
  13. Body Posture
  14. QP convexity
  15. QP Formulation
  16. ...and 7 more sections

Figures (6)

  • Figure 1: Snapshot of simulated RHPS-1 robot controlled with a QP-based whole-body motion control law. The green square represents the desired left hand trajectory.
  • Figure 2: (a) Control scheme for a position and speed controlled robot based on an acceleration obtained from a QP. (b) Same scheme with noise $\sigma$ added to the QP solution to model solution inaccuracy.
  • Figure 3: Tracking errors as functions of noise level $I_\sigma$ added to the QP solution: (a) hand position and orientation, (b) CoM position and (c) joint positions.
  • Figure 4: Tracking errors and computational effort over $1~\left[s\right]$ of robot motion for varying QP matrices update ratios $r$.
  • Figure 5: Tracking errors and computational effort over $1~\left[s\right]$ of robot motion for varying control frequencies $f$, feedback gains $k$ and matrices update ratios $r$ as specified in Table \ref{['tab:my_label2']}. An additional point with standard values ($f=200~[Hz]$, $r=1$) is added for comparison.
  • ...and 1 more figures