Task-Space Singularity Avoidance for Control Affine Systems Using Control Barrier Functions
Authors: Kimia Forghani, Suraj Raval, Lamar Mair, Axel Krieger, Yancy Diaz-Mercado
Organizations: Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 · Division of Magnetic Manipulation and Particle Research, Weinberg Medical Physics, Inc., North Bethesda, MD 20852 · Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218
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.
Figures & tables
Fig. 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.
Fig. 2: Comparison of joint angles and velocities: (Left) without the CBF, θ2→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.
Fig. 3: The magnetic actuation system (left), and the corresponding simulation workspace in MATLAB (right).
Fig. 4: 3D scatter plot of configurations where the smallest singular value falls below a threshold for the magnetic system in Fig. 3 . Color indicates singular value magnitude. Coordinates represent 2D position and orientation θ of the magnetic agent.
Fig. 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.
Fig. 6: Safe magnetic needle trajectory (time-colored dots) for a suture pattern in configuration space that avoids singular configurations (green).
This paper presents a modular control barrier function (CBF) framework for safe free-flying robotic spacecraft operations during tumbling target capture. Motivated by latest ESA guidelines for safe close proximity operations, safety zones and requirements are translated into dedicated CBFs. The 13-DoF system is decomposed into translational, attitude, and robotic subsystems, each equipped with a safety filter that minimally modifies nominal control inputs in a lightweight quadratic program. The filters enforce a conical approach corridor, collision avoidance zone, attitude line-of-sight pointing, angular velocity limits, robotic joint limits, link-base collision avoidance, and actuator constraints. Dynamic coupling between subsystems is handled by treating upstream safe control commands as known interconnection inputs in the downstream safety filters, preserving modularity while supporting system-level safety. The framework is validated in an on-orbit servicing scenario, including final approach, angular rate synchronization, and tumbling target grasping, using the high-fidelity astrodynamics simulator Basilisk. Monte Carlo simulation results demonstrate runtime efficiency and operational safety for various tumbling rates.
Alexander Meinert, Peter Stadler, Niklas Baldauf +1
Space Applications Group, e:fs TechHub GmbH, Gaimersheim, Germany · Faculty of Electrical Engineering and Information Technology, Ingolstadt University of Applied Sciences, Germany
Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially for systems with low computational resources, higher-order dynamics, and moving close to obstacles of complex shape. A common solution to this problem is to use a purely distance-based CBF. In this paper, we study Hyperplane CBFs (H-CBFs), where a hyperplane separates the agent from the obstacle. First, we note that the common distance-based CBF is a special case of an H-CBF where the hyperplane is a supporting hyperplane of the obstacle that is orthogonal to a line between the agent and the closest point of the obstacle. We then show that a less conservative CBF can be found by optimising over the orientation of the supporting hyperplane, in order to find the Least Restrictive Hyperplane CBF. This enables us to maintain the safety guarantees while allowing controls that are closer to the desired ones, especially when moving fast and passing close to obstacles. We illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles, and show that the proposed approach reduces the average gap between desired and safe controls by an order of magnitude.
Mattias Trende, Petter Ögren
The authors are with the Robotics, Perception and Learning Lab., School of Electrical Engineering and Computer Science, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden
Safe navigation often relies on well-defined conditions based on the shape of robots and obstacles, and can be challenging when they have irregular geometries. While Control Barrier Functions (CBFs) offer an efficient mechanism to enforce safe set forward invariance, common shape surrogates (e.g., spheres or super-ellipsoids) either are overly conservative in unstructured scenes or require many local primitives, which inflates constraint counts and degrades real-time performance. In this paper, we introduce a novel geometry-aware Control Barrier Function (CBF) based on Bernstein-Polynomial Signed Distance Fields (BP-SDFs). It provides a unified way to represent the obstacles and robots, so as to represent the barrier function with a unified minimum distance. Benefiting from the differentiability of the Bernstein polynomials, one can easily enforce the control constraints in a closed loop. We validate the method's efficiency and performance to guarantee safety in single-robot navigation and heterogeneous multi-robot collision avoidance via simulations under different environments.
Siwon Jo, Yanze Zhang, Yupeng Yang +1
GRASP Laboratory, University of Pennsylvania, Philadelphia, PA, USA · Department of Computer Science, University of Illinois Chicago, Chicago, IL, USA · Department of Computer Science, University of North Carolina at Charlotte, Charlotte, NC, USA