Learning from Demonstrations Meets Symbolic Control - A Supervisory Control Architecture for Formal Certification of Learned Motion

Dynamical Systems (DS) are reactive motion policies representing vector fields trained with theoretical guarantees of stability and convergence. To ensure safety during deployment in unknown environments they must be locally reshaped, either through modulation or geometric control barrier function strategies. However, depending on the geometry of the obstacles and the complexity of the DS, these local strategies can lead the system to unavoidable collisions or spurious attractors. In this work, we certify safety with a value function drawn from the notion of backward reachability tube, which measures the worst-case safety along a rollout trajectory of the nominal DS. Usually, such a value function is intractable for a controlled system due to curse of dimensionality. We show that in the DS-based learning-from-demonstration setting, the absence of a control input collapses the reachability problem to a deterministic rollout, and the presence of certain stability conditions truncates the infinite horizon to a finite one, resulting in a well-defined value function. We further show that the value function we devised is the maximal forward-invariant subset of the obstaclefree region for the nominal DS flow. The application of this certificate function is validated across five DS constructions - analytical, Neural ODE, diffeomorphic latent space, LPV-DS, SE(3)and validate it on a Franka manipulator. Modulation and geometric CBFs also suffer from saddle point in cases of headon approach towards an unsafe zone. We show that CBF-on-V avoids this pitfall entirely.

We present STT-LfD, a unified Learning from Demonstration (LfD) framework that integrates motion learning with control for unknown Euler-Lagrange systems. Unlike traditional decoupled approaches that track a fixed reference, the proposed method treats demonstrations as a data-driven safety specification. Using heteroscedastic Gaussian Processes, STT-LfD learns Spatiotemporal Tubes (STTs) as an intent envelope that capture time-varying precision requirements of a task. A closed-form feedback controller then enforces these learned constraints while respecting actuator limits, without requiring explicit system identification. The approach preserves the temporal structure of demonstrations, remains computationally efficient, and avoids explicit system identification. Hardware experiments on a mobile robot and a 7-DOF manipulator show that it outperforms baselines in robustness to disturbances and computational speed.

Safe model-based reinforcement learning (RL) often bridges control-theoretic analysis and RL for robots to safely explore (partially) unknown system dynamics while deriving control actions for task efficiency. The control performance and safety assurance typically rely on prior knowledge of partially modeled nominal system dynamics and the data-driven models that compensate for residual model uncertainties. However, existing methods often overlook the structure of residual model uncertainties (e.g., components affine in control), which could lead to overly conservative robot behaviors or invalid safety guarantees under the safe learning-based controllers. This paper proposes a safe reinforcement learning framework that learns control-affine dynamics with a certifiable data-driven safe policy using control barrier functions (CBF). Specifically, we first use Control-Affine Random Fourier Features (ARFF) to model robot dynamics in a control-affine form, which offers computational efficiency that scales with dataset size and reduces potential model bias for model-based reinforcement learning. Then, a model-free, efficient uncertainty quantification method using adaptive conformal prediction (ACP) is applied to quantify the uncertainty in the safety constraint arising from the learned control-affine dynamics. This allows for data-driven safety assurance amenable to principled and efficient controller synthesis with CBF. Simulation results on the cartpole and the 3D quadrotor platforms demonstrate the effectiveness of the proposed framework.