Dealing with Singularities in Robotic Systems exploiting MPC

Abstract

Kinematic singularities represent a major challenge in robotics. They occur when a robot’s joints reach a configuration in which the Jacobian matrix loses rank. As a result, certain motion directions become unattainable, or require disproportionately high joint velocities and torques. These effects can impair control performance, reduce accuracy, and, in extreme cases, make stable operation unreliable. For dealing with kinematic singularities, various approaches have been proposed. These can be broadly classified into two main categories. The first category addresses singularities by applying specialized control schemes in their vicinity. A drawback of this approach is that every possible singularity must be known in advance, which is hardly feasible in practice. The second category explicitly avoids singularities, for instance by regularizing the Jacobian matrix or constraining joint angles. However, this restriction limits the reachable workspace and may introduce larger position and orientation errors. The aim of this work is to develop a control strategy capable of deliberately approaching, traversing, and exiting singularities. Additionally, the real-time capability of the employed methods is investigated. Model Predictive Control (MPC) was chosen as the solution approach. Dynamic, parametric-dynamic, kinematic, and parametric-kinematic MPC formulations were developed and compared to a conventional inverse kinematics controller based on Jacobian matrix regularization. The methods were evaluated on five distinct singularity-inducing trajectories featuring elbow, wrist, and shoulder singularities. Evaluation was performed both in simulation and on a Franka Research 3 robot. The results demonstrate that only the dynamic MPC can handle the investigated singularities. In contrast, the conventional controller exhibits considerable trajectory errors and tends to generate null-space motions in singularities due to the heuristic Jacobian regularization. Such effects do not occur with MPC. The MPC implementation operates in real time and is capable of entering a singularity, holding it, and exiting it again, generally without incurring significant trajectory errors. This study shows that a real-time-capable MPC with an average update rate of 2 kHz can be implemented to deliberately handle singularities. For different singularities, partially adapted parameter sets were required, indicating potential for future research into a universal parameter set capable of covering all singularity types equally.