Dynamic Movement Primitives With Control Barrier Functions for Constrained Trajectory Planning

Abstract

Dynamic Movement Primitives (DMPs) form a robust framework for trajectory generation based on imitation learning, aiming to replicate the shape of reference trajectories from demonstrations closely. DMPs have been extensively employed for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">trajectory planning</i> in robotic systems. However, they cannot safely guarantee complex <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear constraints</i>, which is essential <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">at the control level</i>. On the other hand, Control Barrier Functions (CBFs) are used to modulate the input of control-affine dynamic systems subject to state-dependent constraints, guaranteeing that the system remains within predefined safe sets while converging towards target states. This paper proposes Constrained Movement Primitives (CMPs), a novel framework that integrates DMPs with CBFs to generate safe-by-construction trajectories subject to nonlinear constraints. We represent DMPs in control-affine form and combine them with the closed-form input provided by CBFs, overcoming the limitations of existing iterative optimisation methods for constrained DMPs. We demonstrate that CBFs preserve the goal convergence guarantees of DMPs. Moreover, we validate our approach in simulation and on a real mobile robot subject to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear kinodynamic constraints</i>, concerning maximum Cartesian velocity, obstacle avoidance, and maximum centrifugal acceleration to avoid slippery over curved trajectories.