SafeDMPs: Integrating Formal Safety with DMPs for Adaptive HRI

Robots deployed in safety-critical environments must execute motions that are simultaneously robust to disturbances and provably safe from collisions. Dynamic Movement Primitives (DMPs) offer inherent stability, temporal flexibility, and efficient trajectory generalization from single demonstrations, but they lack formal safety certificates. Conversely, Hamilton-Jacobi (HJ) Reachability analysis provides a principled framework for computing worst-case safety margins and forward-invariant safe sets, but classical grid-based methods suffer from the curse of dimensionality and are impractical for real-time control. This paper introduces HJ-SafeDMP, a framework that integrates DMPs with learned HJ Reachability-based safety value functions to achieve provably safe, robust, and computationally efficient robot motion. We learn a Control Barrier Value Function (CBVF) from offline demonstration data using a model-free, finite-difference HJ recursion and deploy it as a real-time safety filter via a closed-form control law that modulates the DMP output. Unlike optimization-based CBF-QP approaches, our method achieves safety filtering without online quadratic program solves, preserving the computational efficiency of DMPs. We further incorporate an expectile-based offline learning objective that avoids querying out-of-distribution actions, and a conformal prediction calibration step that provides finite-sample probabilistic safety coverage. Experimental evaluation on a 7-DOF robot manipulator demonstrates that HJ-SafeDMP achieves formal safety guarantees with orders-of-magnitude faster execution than optimization-based baselines, while maintaining the robustness and adaptability of DMPs for human-robot interaction.

This paper proposes a task-oriented model predictive control (ToMPC) framework for safe and efficient robotic manipulation in open workspaces. The framework unifies collision-free motion and robot-environment interaction to address diverse scenarios. Additionally, it introduces task-oriented obstacle avoidance that leverages kinematic redundancy to enhance manipulation efficiency in obstructed environments. This complex optimization problem is solved by the alternating direction method of multipliers (ADMM), which decomposes the problem into two subproblems tackled by differential dynamic programming (DDP) and quadratic programming (QP), respectively. The effectiveness of this approach is validated in simulation and hardware experiments on a Franka Panda robotic manipulator. Results demonstrate that the framework can plan motion and/or force trajectories in real time, maximize the manipulation range while avoiding obstacles, and strictly adhere to safety-related hard constraints.

A key limitation on the use of diffusion models in robotic planning is their inability to inherently enforce safety or dynamical constraints, which often results in physically infeasible or unsafe outputs. Hybrid approaches that employ model predictive control (MPC) to address this problem can be unstable, as poor trajectory initializations from the diffusion model prevent the MPC from converging to a safe and feasible solution. To overcome these challenges, we propose D-SafeMPC, which enhances the interaction between diffusion and control. Our method guides the reverse diffusion process with control barrier functions (CBFs) and control Lyapunov functions (CLFs) and employs an iterative-projection scheme where an MPC refines the trajectory at each denoising step. This steers sampling toward safe, goal-directed regions and provides reliable MPC warm starts. In simulations on a Franka manipulator across four scenarios (one static-obstacle and three dynamic-obstacle settings) and in a sim-to-real experiment on a physical Franka robot, D-SafeMPC improves safety, task success rates, and planning efficiency over state-of-the-art baselines. To facilitate reproducibility, our source code and experimental configurations are available in a repository at https://github.com/erdiphd/D-SafeMPC