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Task Hierarchical Control via Null-Space Projection and Path Integral Approach

Apurva Patil, Riku Funada, Takashi Tanaka, Luis Sentis

TL;DR

This work tackles hierarchical task control in redundant robots by integrating null-space projection with path integral control. By assigning high-priority tasks to the primary space and enabling a single (or few) tasks to be optimized with a stochastic, Monte Carlo-based path integral controller, the framework achieves improved global performance while leveraging simple PD-like controllers for lower-priority tasks. The approach derives a linearized PDE via a logarithmic transformation and the Feynman-Kac representation to compute optimal controls in real time, with demonstrated benefits in both single-agent and multi-agent simulations. The results show reduced susceptibility to local minima and enhanced coordination in dynamic environments, suggesting practical impact for complex, high-DOF or multi-robot systems.

Abstract

This paper addresses the problem of hierarchical task control, where a robotic system must perform multiple subtasks with varying levels of priority. A commonly used approach for hierarchical control is the null-space projection technique, which ensures that higher-priority tasks are executed without interference from lower-priority ones. While effective, the state-of-the-art implementations of this method rely on low-level controllers, such as PID controllers, which can be prone to suboptimal solutions in complex tasks. This paper presents a novel framework for hierarchical task control, integrating the null-space projection technique with the path integral control method. Our approach leverages Monte Carlo simulations for real-time computation of optimal control inputs, allowing for the seamless integration of simpler PID-like controllers with a more sophisticated optimal control technique. Through simulation studies, we demonstrate the effectiveness of this combined approach, showing how it overcomes the limitations of traditional

Task Hierarchical Control via Null-Space Projection and Path Integral Approach

TL;DR

This work tackles hierarchical task control in redundant robots by integrating null-space projection with path integral control. By assigning high-priority tasks to the primary space and enabling a single (or few) tasks to be optimized with a stochastic, Monte Carlo-based path integral controller, the framework achieves improved global performance while leveraging simple PD-like controllers for lower-priority tasks. The approach derives a linearized PDE via a logarithmic transformation and the Feynman-Kac representation to compute optimal controls in real time, with demonstrated benefits in both single-agent and multi-agent simulations. The results show reduced susceptibility to local minima and enhanced coordination in dynamic environments, suggesting practical impact for complex, high-DOF or multi-robot systems.

Abstract

This paper addresses the problem of hierarchical task control, where a robotic system must perform multiple subtasks with varying levels of priority. A commonly used approach for hierarchical control is the null-space projection technique, which ensures that higher-priority tasks are executed without interference from lower-priority ones. While effective, the state-of-the-art implementations of this method rely on low-level controllers, such as PID controllers, which can be prone to suboptimal solutions in complex tasks. This paper presents a novel framework for hierarchical task control, integrating the null-space projection technique with the path integral control method. Our approach leverages Monte Carlo simulations for real-time computation of optimal control inputs, allowing for the seamless integration of simpler PID-like controllers with a more sophisticated optimal control technique. Through simulation studies, we demonstrate the effectiveness of this combined approach, showing how it overcomes the limitations of traditional

Paper Structure

This paper contains 13 sections, 2 theorems, 51 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Null-Space Projection
  4. Integration of Null Space Projection and Path Integral Control
  5. Simulation Results
  6. Single Agent Example
  7. Obstacle-avoidance
  8. Move-to-goal
  9. Two Agents Example
  10. Obstacle-avoidance
  11. Steering the robots' centroid toward a goal position
  12. Maintaining a specific distance between two unicycles
  13. Conclusion

Key Result

Lemma 1

The solution to the linear PDE linear PDE exists. Moreover, the solution is unique in the sense that $\xi$ solving linear PDE is given by where the expectation $\mathbb{E}_P$ is taken with respect to the uncontrolled dynamics of the system SDE (i.e., equation SDE with $\widetilde{u}=0$) starting at $x,t$. $S(x,t)$ is the cost to go of the state-dependent cost of a trajectory given by

Figures (6)

  • Figure 1: Comparison of the conventional task hierarchical control approach and our proposed control approach, which integrates null-space projection with path integral control.
  • Figure 2: Results of single-agent example without the path integral controller
  • Figure 3: Results of single-agent example using the path integral controller
  • Figure 4: Results of two-agents example without the path integral controller
  • Figure 5: Results of two-agents example using the path integral controller
  • ...and 1 more figures

Theorems & Definitions (7)

  • Example 1
  • Lemma 1: Feynman-Kac lemma
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2