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

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.