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Task-Space Singularity Avoidance for Control Affine Systems Using Control Barrier Functions

Kimia Forghani, Suraj Raval, Lamar Mair, Axel Krieger, Yancy Diaz-Mercado

Abstract

Singularities in robotic and dynamical systems arise when the mapping from control inputs to task-space motion loses rank, leading to an inability to determine inputs. This limits the system's ability to generate forces and torques in desired directions and prevents accurate trajectory tracking. This paper presents a control barrier function (CBF) framework for avoiding such singularities in control-affine systems. Singular configurations are identified through the eigenvalues of a state-dependent input-output mapping matrix, and barrier functions are constructed to maintain a safety margin from rank-deficient regions. Conditions for theoretical guarantees on safety are provided as a function of actuator dynamics. Simulations on a planar 2-link manipulator and a magnetically actuated needle demonstrate smooth trajectory tracking while avoiding singular configurations and reducing control input spikes by up to 100x compared to the nominal controller.

Task-Space Singularity Avoidance for Control Affine Systems Using Control Barrier Functions

Abstract

Singularities in robotic and dynamical systems arise when the mapping from control inputs to task-space motion loses rank, leading to an inability to determine inputs. This limits the system's ability to generate forces and torques in desired directions and prevents accurate trajectory tracking. This paper presents a control barrier function (CBF) framework for avoiding such singularities in control-affine systems. Singular configurations are identified through the eigenvalues of a state-dependent input-output mapping matrix, and barrier functions are constructed to maintain a safety margin from rank-deficient regions. Conditions for theoretical guarantees on safety are provided as a function of actuator dynamics. Simulations on a planar 2-link manipulator and a magnetically actuated needle demonstrate smooth trajectory tracking while avoiding singular configurations and reducing control input spikes by up to 100x compared to the nominal controller.
Paper Structure (22 sections, 3 theorems, 38 equations, 6 figures)

This paper contains 22 sections, 3 theorems, 38 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND AND PRELIMINARIES
  3. Dynamics and Task Output
  4. Control Barrier Functions
  5. CONTROL BARRIER FUNCTION CONSTRUCTION
  6. Quadratic Problem
  7. Singularity Points
  8. Barrier Function
  9. CASE STUDIES
  10. Analytical CBF
  11. System description
  12. Control Barrier Function
  13. Feasibility Guarantee
  14. Quadratic Program
  15. Simulation
  16. ...and 7 more sections

Key Result

Lemma 1

Let $\phi \in \mathbb{R}^{d \times m}$ be a differentiable matrix-valued function. Then the matrix $M(x) = \phi(x)\phi(x)^\top$ is symmetric, positive semidefinite, and differentiable.

Figures (6)

  • Figure 1: Two similar end-effector trajectories (dashed lines) for a two-link manipulator. The red trajectory in the fourth pose passes through a singular configuration when the middle joint angle becomes zero (flat links), while the green trajectory avoids singularity by slightly modifying the joint angles.
  • Figure 2: Comparison of joint angles and velocities: (Left) without the CBF, $\theta_2 \to 0$ at $t=5s$ induces spikes in joint velocities and abrupt changes in joint angles. (Right) with the CBF, the singularity is avoided, the joint angle trajectory remains smooth, and the joint velocities remain low.
  • Figure 3: The magnetic actuation system (left), and the corresponding simulation workspace in MATLAB (right).
  • Figure 4: 3D scatter plot of configurations where the smallest singular value falls below a threshold for the magnetic system in Fig. \ref{['fig:magneto']}. Color indicates singular value magnitude. Coordinates represent 2D position and orientation $\theta$ of the magnetic agent.
  • Figure 5: Snapshots of a simulated suturing task. The needle (gray) follows the planned trajectory (blue) while tilting from the reference orientation (gray shadow) to avoid singular configurations. Orange arrows indicate motion direction. The needle: a) approaches the first penetration site, b) tightens the suture while adjusting orientation, c) advances along the incision returning to the reference angle, d–f) completes the stitch.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Remark 1
  • Lemma 1: magnus1985matrix
  • Lemma 2: lancaster1985theory
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3