Cooperative Manipulation via Internal Force Regulation: A Rigidity Theory Perspective

TL;DR

The paper addresses energy-efficient cooperative manipulation with rigid grasps by integrating distance and bearing rigidity in $\,\mathsf{SE}(3)\,$ and connecting object–agent forces to a rigidity matrix $\mathcal{R}_{\mathcal{G}}$ via a range–nullspace relation. It derives a closed-form expression for internal forces and establishes necessary and sufficient conditions for internal-force-free force distribution using an inertia-aware right inverse $G^{\ast}=M G^{\top}(G M G^{\top})^{-1}$, then presents an inverse-dynamics controller that guarantees object-tracking while ensuring (provably) zero internal forces under the proposed $G^{\ast}$. The results unify rigidity theory with cooperative manipulation, enabling energy-optimal load sharing and providing a framework to minimize internal stresses in multi-robot manipulation; simulations in realistic environments corroborate the theory. This work opens avenues for applying rigidity-based insights to pose estimation, robust control, and non-rigid grasping scenarios by linking rigidity constraints to feasible force distributions and control laws.

Abstract

This paper considers the integration of rigid cooperative manipulation with rigidity theory. Motivated by rigid models of cooperative manipulation systems, i.e., where the grasping contacts are rigid, we introduce first the notion of bearing and distance rigidity for graph frameworks in SE(3). Next, we associate the nodes of these frameworks to the robotic agents of rigid cooperative manipulation schemes and we express the object-agent interaction forces by using the graph rigidity matrix, which encodes the infinitesimal rigid body motions of the system. Moreover, we show that the associated cooperative manipulation grasp matrix is related to the rigidity matrix via a range-nullspace relation, based on which we provide novel results on the relation between the arising interaction and internal forces and consequently on the energy-optimal force distribution on a cooperative manipulation system. Finally, simulation results on a realistic environment enhance the validity of the theoretical findings.

Figures (5)

• Figure 1: Two robotic agents rigidly grasping an object.
• Figure 2: Four UR$5$ robotic arms rigidly grasping an object. The red counterpart represents a desired object pose at $t=0$.
• Figure 3: The error metrics $e_p(t)$, $e_{ O}(t)$, $e_v(t)$, respectively, top to bottom, for the two choices $G^\ast_1$ and $G^\ast_2$ and $t\in[0,15]$ seconds.
• Figure 4: The norms of the resulting control inputs, $\|\tau_i(t)\|$ for $G^\ast_1$ (with blue) and $G^\ast_2$ (with red), $\forall i\in\{1,\dots,4\}$, and $t\in[0,15]$ seconds.
• Figure 5: Left: The signal $\|h_\text{int}(t)\|$ (as computed via \ref{['eq:internal forces 1_1']}) for the two cases of $G^\ast$ and $t\in[0,15]$ seconds. Right: The signal $\|e_{\text{int}}(t)\|$, when using $G^\ast_1$ and for $t\in[0,15]$ seconds.

Theorems & Definitions (14)

• Lemma 1: Gauss' principle Udwadia92NewPerspectiveUdwadia93Constrained
• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Corollary 1
• Proposition 3
• Lemma 2
• Theorem 1
• Theorem 2