Hybrid Quadratic Programming -- Pullback Bundle Dynamical Systems Control

TL;DR

The paper addresses reactive motion control on non-Euclidean spaces by marrying geometry-aware Pullback Bundle Dynamical Systems (PBDS) with a Quadratic Program (QP) inverse dynamics controller. PBDS embeds sub-manifold constraints directly into the task-space dynamics, while the QP solves for $\ddot{\mathbf{q}}$, $\boldsymbol{\tau}$, and a slack $\boldsymbol{\xi}$ to ensure torque-feasible, dynamically consistent motion, with the objective $\tfrac{1}{2}\mathbf{z}^T \mathcal{W} \mathbf{z} + w^T \mathbf{z}$. A key contribution is the analytical pullback of second-order DSs across multiple manifolds, yielding $\\ddot{\mathbf{q}} = (\\sum_i J_i^T W_i J_i)^{-1} (\\sum_i J_i^T W_i b_i)$, and handling constraints via a QP structured around $\mathbf{z}=[\\ddot{\mathbf{q}},\\boldsymbol{\tau},\\boldsymbol{\\xi}]^T$. Preliminary simulations on a KUKA arm with $S^2$-based constraints and obstacles demonstrate feasibility and stability, supporting the approach's potential for real-time, geometry-aware robotic control, albeit with observed limitations from joint limits and the need for deeper convergence analysis.

Abstract

Dynamical System (DS)-based closed-loop control is a simple and effective way to generate reactive motion policies that well generalize to the robotic workspace, while retaining stability guarantees. Lately the formalism has been expanded in order to handle arbitrary geometry curved spaces, namely manifolds, beyond the standard flat Euclidean space. Despite the many different ways proposed to handle DS on manifolds, it is still unclear how to apply such structures on real robotic systems. In this preliminary study, we propose a way to combine modern optimal control techniques with a geometry-based formulation of DS. The advantage of such approach is two fold. First, it yields a torque-based control for compliant and adaptive motions; second, it generates dynamical systems consistent with the controlled system's dynamics. The salient point of the approach is that the complexity of designing a proper constrained-based optimal control problem, to ensure that dynamics move on a manifold while avoiding obstacles or self-collisions, is "outsourced" to the geometric DS. Constraints are implicitly embedded into the structure of the space in which the DS evolves. The optimal control, on the other hand, provides a torque-based control interface, and ensures dynamical consistency of the generated output. The whole can be achieved with minimal computational overhead since most of the computational complexity is delegated to the closed-form geometric DS.

Figures (4)

• Figure 1:
• Figure 2: Tree diagram of the manifolds structure.
• Figure 3: Simulated Environment: (a) the end-effector KUKA IWAA 14 follows a second order DS on the sphere while avoiding obstacles located on the sub-manifold; (b) manifolds tree to generate desired accelerations.
• Figure 4: Pullback second order DS on $S^2$ (a) $3$ sampled trajectories; the color gradient represents the potential function used to generate the elastic force towards the desired attractor. (b) $x$, $y$, $z$ position and velocity signals. (c) End-effector trajectory tracking of the geometric DS.