Safe Execution of Learned Orientation Skills with Conic Control Barrier Functions

TL;DR

This work presents an innovative approach geared toward ensuring the safe execution of learned orientation skills within constrained regions surrounding a reference trajectory, which involves learning a stable DS on SO(3), extracting time-varying conic constraints from the variability observed in expert demonstrations, and bounding the evolution of the DS with Conic Control Barrier Function (CCBF) to fulfill the constraints.

Abstract

In the field of Learning from Demonstration (LfD), Dynamical Systems (DSs) have gained significant attention due to their ability to generate real-time motions and reach predefined targets. However, the conventional convergence-centric behavior exhibited by DSs may fall short in safety-critical tasks, specifically, those requiring precise replication of demonstrated trajectories or strict adherence to constrained regions even in the presence of perturbations or human intervention. Moreover, existing DS research often assumes demonstrations solely in Euclidean space, overlooking the crucial aspect of orientation in various applications. To alleviate these shortcomings, we present an innovative approach geared toward ensuring the safe execution of learned orientation skills within constrained regions surrounding a reference trajectory. This involves learning a stable DS on SO(3), extracting time-varying conic constraints from the variability observed in expert demonstrations, and bounding the evolution of the DS with Conic Control Barrier Function (CCBF) to fulfill the constraints. We validated our approach through extensive evaluation in simulation and showcased its effectiveness for a cutting skill in the context of assisted teleoperation.

Figures (7)

• Figure 1: Setup of the experiment for assisted teleoperation. A human operator uses a haptic device to adjust the cutting skill executed by a robot. The $z$-axis of the ee frame (blue segment) coincides with the blade. The shaded blue surface indicates the incision shape.
• Figure 2: An illustration of conic constraints.
• Figure 3: Overview of the method. Dark-colored lines indicate values. Light-colored lines indicate data or functions.
• Figure 4: An illustration of time-varying conic constraints. The axes of a rotational matrix $\bm{R}_\mathrm{exc}$ always stay within the cones defined by $\bm{R}_\mathrm{ref}(t)$, which itself evolves according to the reference ds.
• Figure 5: Temporal evolution of cone angles and orientation trajectories for the L/N/W shapes (from top to bottom). Orientation is depicted in transformed frames, with square and circle markers denoting initial orientations and goals. On the left, deep/light solid lines represent the angles between constrained/unconstrained axes and reference axes, denoted as $\theta_i^c$ and $\theta_i^{uc}$ respectively. Red, green, and blue correspond to the $x$-axis, $y$-axis, and $z$-axis. Dashed black lines indicate the learned cone angles. On the right, the trajectories in solid deep/light blue correspond to constrained/unconstrained executions, and the dashed black trajectories represent the reference trajectories.