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Compliant Hierarchical Control for Arbitrary Equality and Inequality Tasks with Strict and Soft Priorities

Gianluca Garofalo

TL;DR

This work tackles the problem of enforcing arbitrary numbers of equality and inequality tasks in redundant robots while preserving the system's natural inertia. It introduces Weighted Hierarchical Quadratic Problems (WHQP) with weights and slack variables to support soft priorities, and leverages Complete Orthogonal Decomposition (COD) to extract the active task set and construct an inertially decoupled coordinate transform. The resulting control law operates in transformed coordinates, yielding a passivity-based, compliant hierarchy that unifies optimization-based and passivity-based approaches and remains robust to singularities. Validation in simulation on a 7-DOF Panda demonstrates effective handling of both hard and soft priorities, including inequality constraints, without reshaping the robot’s inertia.

Abstract

When a robotic system is redundant with respect to a given task, the remaining degrees of freedom can be used to satisfy additional objectives. With current robotic systems having more and more degrees of freedom, this can lead to an entire hierarchy of tasks that need to be solved according to given priorities. In this paper, the first compliant control strategy is presented that allows to consider an arbitrary number of equality and inequality tasks, while still preserving the natural inertia of the robot. The approach is therefore a generalization of a passivity-based controller to the case of an arbitrary number of equality and inequality tasks. The key idea of the method is to use a Weighted Hierarchical Quadratic Problem to extract the set of active tasks and use the latter to perform a coordinate transformation that inertially decouples the tasks. Thereby unifying the line of research focusing on optimization-based and passivity-based multi-task controllers. The method is validated in simulation.

Compliant Hierarchical Control for Arbitrary Equality and Inequality Tasks with Strict and Soft Priorities

TL;DR

This work tackles the problem of enforcing arbitrary numbers of equality and inequality tasks in redundant robots while preserving the system's natural inertia. It introduces Weighted Hierarchical Quadratic Problems (WHQP) with weights and slack variables to support soft priorities, and leverages Complete Orthogonal Decomposition (COD) to extract the active task set and construct an inertially decoupled coordinate transform. The resulting control law operates in transformed coordinates, yielding a passivity-based, compliant hierarchy that unifies optimization-based and passivity-based approaches and remains robust to singularities. Validation in simulation on a 7-DOF Panda demonstrates effective handling of both hard and soft priorities, including inequality constraints, without reshaping the robot’s inertia.

Abstract

When a robotic system is redundant with respect to a given task, the remaining degrees of freedom can be used to satisfy additional objectives. With current robotic systems having more and more degrees of freedom, this can lead to an entire hierarchy of tasks that need to be solved according to given priorities. In this paper, the first compliant control strategy is presented that allows to consider an arbitrary number of equality and inequality tasks, while still preserving the natural inertia of the robot. The approach is therefore a generalization of a passivity-based controller to the case of an arbitrary number of equality and inequality tasks. The key idea of the method is to use a Weighted Hierarchical Quadratic Problem to extract the set of active tasks and use the latter to perform a coordinate transformation that inertially decouples the tasks. Thereby unifying the line of research focusing on optimization-based and passivity-based multi-task controllers. The method is validated in simulation.
Paper Structure (17 sections, 1 theorem, 34 equations, 4 figures, 1 algorithm)

This paper contains 17 sections, 1 theorem, 34 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Moore-Penrose Inverse
  4. Notation
  5. Projected Stack
  6. Weighted Hierarchical Quadratic Problem
  7. Efficient Computation via COD
  8. Summary
  9. Robot Model and Control Objective
  10. Control objective
  11. Compliant Hierarchical Control Laws
  12. Summary and Discussion
  13. Validation
  14. Conclusion
  15. Optimality conditions
  16. ...and 2 more sections

Key Result

Lemma 1

The WMPI $A^{\dagger_{_{W_1,W_0}}}$ can be computed as ${ W_0^{-1} A^\top R_1^\top (R_1 A W_0^{-1} A^\top R_1^\top )^\dagger R_1^\top }$.

Figures (4)

  • Figure 1: Simulated Franka Emika Panda robot with relevant frames, desired trajectory (purple spiral) and allowed region (green square).
  • Figure 2: Non-trivial part of the homogeneous transformation matrices for the case of tracking weight much bigger than the regulation weight. The desired orientation in orange is perfectly tracked (actual values in blue), since the orientation is the first task in the stack. The position is tracked as long as the desired trajectory is within the admisable box.
  • Figure 3: Non-trivial part of the homogeneous transformation matrices for the case of tracking weight much smaller than the regulation weight. While there is no change in the orientation, since it is the first task in the stack, the position of the TCP now converges to the center of the spiral.
  • Figure 4: Non-trivial part of the homogeneous transformation matrices for the case of tracking weight equal to the regulation weight. Also in this case, there is no change for the orientation. Nevertheless, the position of the TCP is in between the tracking of the spiral and reaching its center.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof