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

Ashok Kumar, Mahathi Anand, Angela Schoellig, Abdalla Swikir

arXiv:2610.19815v2cs.RO

TL;DR

Learned dynamical systems can generate flexible robot motions, but do not by themselves guarantee safe, goal-reaching, physically feasible execution. This paper adds symbolic supervisory control to monitor and modify nominal tracking inputs. The authors report competitive simulation performance and real-robot demonstrations with formal guarantees.

  • Problem

    Learned DS methods do not guarantee collision avoidance, goal reaching under perturbations, or compliance with velocity and actuator constraints.

    Source: A. DS-based Learning from Demonstrations · p. 2

  • Method

    A symbolic supervisor checks nominal tracking inputs and uses certified control interventions to preserve safety and reachability while respecting robot constraints.

    Source: §I · p. 1

  • Results

    Simulations and robot experiments are reported as competitive with state of the art, with high task success and formal guarantees, including real-world dynamic 3D environments.

    Source: B. Contributions · p. 2

  • Takeaways & Limitations

    The control-layer approach is agnostic to the DS learning method and uses finite abstractions rather than certificate functions to provide formal guarantees under stated assumptions.

    Source: A. Related Work · p. 1, B. Contributions · p. 2

Abstract

from arXiv · show

Dynamical system (DS)-based learning from demonstration generates flexible robot motions, but existing guarantees rely on certificate generation to obtain formal guarantees of safety and goal convergence. We propose a symbolic supervisory control architecture that automatically generates a fail-safe backup plan, certifying safety, reachability, and actuator feasibility directly from the robot's dynamics, independent of the learned DS. A nominal controller tracks the learned DS, while the supervisory controller monitors safety and target convergence, providing non-conservative intervention when the nominal controller violates necessary constraints. Experimental results include simulations and real robot deployment with various 2D and 3D datasets, including handwriting datasets, validating the effectiveness of our approach in providing formal guarantees with learned motion even in complex, dynamic environments.

I. INTRODUCTION

DS-based learning offers adaptable, perturbation-responsive robot motions, but learned fields alone do not ensure safety, goal reaching, or actuator feasibility. The paper adds symbolic supervisory control that intervenes at the control layer and reports simulation and robot validation.

I. INTRODUCTION

  • DS-based learning encodes demonstrations as a time-invariant state-feedback vector field that can be evaluated from any state and react to runtime perturbations.The learned field does not itself guarantee goal reaching, collision avoidance, or compliance with velocity and actuator limits.Source: §I · p. 1

A. Related Work

  • Certificate search trades expressiveness of learned dynamics against optimization tractability; this approach instead designs tracking inputs below the learned dynamics and intervenes when obstacles block the path.It also targets finite-time reachability of a goal neighborhood and uses finite abstractions rather than certificate functions.Source: A. Related Work · p. 1

B. Contributions

  • The architecture monitors learned-path safety and modifies control to reach a goal neighborhood while avoiding dynamic obstacles, without certificate functions under stated assumptions.It uses finite state abstractions to synthesize a fail-safe backup controller offline for static obstacles or online for dynamic ones.Source: B. Contributions · p. 2
  • Simulations and robot experiments are reported as showing competitive performance, high task success, formal guarantees, and flexibility in real-world dynamic 3D environments.The pipeline synthesizes nominal tracking control, supervises collision and reachability, and modifies inputs before robot execution.Source: B. Contributions · p. 2, Figure 1 · p. 2

II. PRELIMINARIES

The preliminaries frame DS learning as imitation of demonstrated trajectories through a learned vector field, while noting that imitation alone does not guarantee safe, feasible execution. The proposed alternative modifies nominal inputs through supervisory control.

A. DS-based Learning from Demonstrations

  • Learning a DS by minimizing MSE is presented as a standard approach, but the infinite-dimensional optimization is generally computationally intractable, motivating scalable function parameterizations.Examples include LPV dynamics, polynomial dynamics, and neural-network parameterizations.Source: A. DS-based Learning from Demonstrations · p. 2
  • A learned DS can fail to avoid obstacles or reach the goal under perturbations, and may generate trajectories that violate workspace, velocity, or actuator limits.Safety is only implicit in finite demonstrations, and physical feasibility is not guaranteed.Source: A. DS-based Learning from Demonstrations · pp. 2–3
  • The paper instead uses supervisory control to modify nominal inputs when tracking violates safety or robot constraints, relying on certified symbolic synthesis over an acceleration-controlled model.The approach is agnostic to the DS learning method.Source: A. DS-based Learning from Demonstrations · p. 3

B. Sample Perturbed Systems

The paper models sampled, perturbed robot behavior as a set-valued transition system, with uncertainty represented in the system dynamics. This provides the system description used for subsequent control reasoning.

B. Sample Perturbed Systems

  • A sample-perturbed system is represented by state and input sets together with a set-valued transition map from state-input pairs to possible successor states.The continuous-time inclusion induces sampled discrete-time behavior under held inputs.Source: B. Sample Perturbed Systems · p. 3
  • Measurement uncertainty is represented in the system model.The supplied passage identifies this representation but does not state its form.Source: B. Sample Perturbed Systems · p. 3

C. Acceleration Model based on Operational Space Dynamics

The robot dynamics are expressed in operational space and recast using a virtual end-effector acceleration input. Bounded residual and implementation errors are then represented as additive disturbance, with the input set reflecting actuation limits.

C. Acceleration Model based on Operational Space Dynamics

  • The manipulator’s joint-space rigid-body dynamics define the starting model, with inertia, Coriolis and centrifugal terms, gravity, commanded torque, and external torque identified.The end-effector position is mapped from joint configuration as x = h(q), and its velocity is given by the Jacobian relation.Source: C. Acceleration Model based on… · p. 3
  • Operational-space dynamics use Λ(q) as inertia and η(q, ˙q) to group task-space bias terms, with commanded task-space wrench and external effects represented separately.The model is then parameterized using a virtual input for desired end-effector acceleration.Source: C. Acceleration Model based on… · p. 3
  • Residual, discretization, measurement, and torque-realization errors are bounded as additive disturbance, while the compact input set represents actuation limits after torque realization.The robot state combines end-effector position and velocity.Source: C. Acceleration Model based on… · p. 3

D. Symbolic Controller Synthesis

The section specifies reach-and-avoid behavior in discrete time and uses symbolic control to transfer safety and goal-reaching guarantees from an abstract system to the sampled system.

D. Symbolic Controller Synthesis

  • The reach-and-avoid specification requires the sampled system to reach the goal region while remaining in the safe region.The specification is described in discrete time, following the sampled perturbed system’s behavior.Source: D. Symbolic Controller Synthesis · p. 3
  • Symbolic control constructs a finite abstraction that over-approximates the system through a quantizer and represents it as a graph-based relation.Source: D. Symbolic Controller Synthesis · p. 3
  • A feedback refinement relation allows abstract safe inputs to be used on the original system while preserving the reach-and-avoid guarantees.Source: D. Symbolic Controller Synthesis · p. 3

III. PROPOSED METHOD

The method aims to follow learned reference paths when possible while supervising motion to preserve safety, reachability, obstacle avoidance, and path recovery.

III. PROPOSED METHOD

  • The controller aims to track DS-derived paths and velocities where possible while reaching close to the goal, even if the learned dynamics lack a goal fixed point.Source: §III · p. 3
  • When needed, the controller safely modifies the DS reference path to navigate around obstacles and recovers the original path when obstacles clear.Source: §III · p. 3
  • The method targets formal reach-and-avoid guarantees with dynamical-system learning in the loop.Source: §III · p. 3

A. Reach-and-Avoid Characterization

The section characterizes safe and goal regions for reach-and-avoid control, including obstacle clearance, velocity limits, and a goal-position tolerance.

A. Reach-and-Avoid Characterization

B. Nominal Controller Extraction

The nominal controller derives spatial references from a learned DS trajectory, localizes progress along the path, and tracks it while balancing path-following and smoothness objectives.

B. Nominal Controller Extraction

  • Arc-length parameterization produces evenly spaced path waypoints, while the learned vector field does not guarantee that the path reaches the goal.Desired velocity profiles are stored at path samples and interpolated between them.Source: B. Nominal Controller Extraction · p. 4
  • At each control step, nearest-point projection estimates progress, with fresh localization because interventions or perturbations may invalidate earlier progress.Source: B. Nominal Controller Extraction · p. 4
  • A safety Boolean determines whether return to the reference path is permissible, and an optional speed envelope can slow motion around bends and nonsmooth curves.The envelope upper-bounds preferred velocity magnitude.Source: B. Nominal Controller Extraction · p. 4
  • The nominal controller uses a high return gain to encourage path return without obstruction, then optimizes for path following and minimal acceleration variation.The cost prioritizes closeness to the DS path, limits deviation from the nominal controller, and favors smooth acceleration.Source: B. Nominal Controller Extraction · p. 4
  • The controller uses a velocity-tracking feedback gain to form the acceleration command.Source: B. Nominal Controller Extraction · p. 4

C. Supervisory Control Methodology

The methodology synthesizes a symbolic reach-avoid policy from a finite abstraction, then checks nominal motion at runtime and overrides it when it cannot be certified. An online variant moves synthesis into runtime to address dynamic obstacles.

C. Supervisory Control Methodology

  • The nominal controller tracks desired position and velocity paths and recovers the original path after deviations, but does not ensure safety or goal reaching unless the learned DS already has those properties.The symbolic supervisor is introduced to verify these requirements at runtime.Source: C. Supervisory Control Methodology · p. 4
  • The offline architecture runs once before execution for static obstacles, while an online version updates symbolic grids and requests finite-horizon reach-avoid control at runtime for dynamic obstacles.The online implementation is described as an adaptation, with detailed analysis referred to [18, Algorithm 1, 2].Source: C. Supervisory Control Methodology · p. 5
  • The offline synthesis discretizes state and input spaces, overapproximates sampled successors under disturbance and measurement bounds, and computes a winning set with admissible controls.The resulting policy is stored as a binary decision diagram or lookup table.Source: C. Supervisory Control Methodology · p. 5
  • At each decision, the supervisor predicts and quantizes nominal motion, applying it if its successor cells are nonempty and contained in the winning set; otherwise it selects a safe, feasible backup input.The backup is chosen to be minimally restrictive among the candidates available at the quantized robot state.Source: C. Supervisory Control Methodology · p. 5
  • The guarantees depend on a sound abstraction, bounded realization error, matching workspace and obstacle models with configured inflation, and a safety margin covering end-effector and tool geometry.For online synthesis, the server must return a feasible winning set, and environmental changes must not outpace its update frequency.Source: C. Supervisory Control Methodology · p. 6

IV. EXPERIMENTAL EVALUATION

The evaluation spans 2D LASA handwriting simulations using several learned dynamical-system models and a 3D Franka FR3 deployment. Across simulations and robot trials, the supervisory controller is evaluated for task performance, safety, trajectory fidelity, and recovery from perturbations.

Experimental setup

  • In MuJoCo, evaluation used 2D LASA handwriting datasets with LPV, polynomial, and neural-network dynamics models; offline tests used static obstacles, while online tests used dynamic obstacles.Source: §IV · p. 6

MuJoCo Simulations with 2D LASA Datasets

  • In offline simulations, the supervisor tracked learned paths until obstacles made them unsafe, then modified paths and returned to them after clearance.CLF-CBF could encounter QP infeasibility and collisions or make conservative modifications that did not converge to the desired path.Source: §IV · p. 6, Figure 2 · p. 7
  • In online simulations with dynamic obstacles and perturbations, the supervisor navigated toward the goal, while CLF-CBF failed through runtime collisions.Online synthesis replanned at 5Hz, with the resulting backup controller applied at 1kHz.Source: §IV · p. 6, Figure 3 · p. 7

Benchmark Comparisons with CLF-CBF

  • Zero position, velocity, or input-acceleration constraint violations were reported across all 300 benchmark trials.The trials combined learned DS models and tracking tasks; the benchmark included nominal motion, obstacles, perturbations, and combined cases.Source: §IV · pp. 6–7
  • The supervisory method had lower MSE and DTW and shorter navigation time than CLF-CBF, but greater acceleration variation and lower trajectory smoothness.The authors attribute the smoothness trade-off to discretization of the acceleration space; Table II aggregates successful trials across 10 LASA datasets and three DS model families.Source: §IV · p. 7, Table II · p. 8

Robot Experiments

  • On the Franka FR3, offline and online executions recovered from perturbations; online execution also avoided a virtual obstacle and reached the goal within 10mm tolerance.The representative executions ended with position errors of (8.83,9.14,9.57,9.23)mm, within the stated tolerance in all directions.Source: Figure 5 · p. 7, Figure 4 · p. 7, §IV · p. 8

V. CONCLUSION

The paper concludes that symbolic supervisory control can verify and modify controllers tracking learned dynamical systems, independently of the learning method and under mild system and environment assumptions. It also identifies joint-limit violations as a limitation of the task-space abstraction.

Conclusion

  • The proposed supervisor verifies and modifies a nominal controller that tracks learned dynamical systems, with formally correct actions under mild assumptions.The conclusion describes the results as promising and reports competitive performance against the state of the art.Source: §V · p. 8
  • The approach is agnostic to the dynamical-system learning method.The conclusion presents this as a property of the supervisory control methodology.Source: §V · p. 8
  • The low-dimensional task-space abstraction does not account for joint-space dynamics, and the method therefore occasionally causes joint-limit violations.The authors identify extending symbolic synthesis to higher-dimensional joint spaces through scalable, compositional approaches as future work.Source: §V · p. 8

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