Statics of continuum planar grasping

TL;DR

This work develops a control-theoretic framework for the statics of continuum planar grasping, modeling the static equilibrium of a planar object under distributed contact as a linear time-varying system with arclength acting as time. The continuum grasp map G(s) relates local contact forces f(s) to the resultant wrench w_L, enabling reachability and force-closure analysis; minimum-force grasping is formulated as a constrained optimal control problem and solved via Pontryagin’s Maximum Principle, linking the optimum to the controllability Gramian. A continuum-grasp quality metric is defined as the inverse of the worst-case minimum-force resistance, extending Ferrari-Canny concepts to continuous contact; the quality is then maximized by choosing the grasp arc start and length. Numerical results on circles, ellipses, and deformed circles illustrate the tradeoffs between grasp length, friction, and geometry, and demonstrate that longer, well-placed continuum grasps improve quality while sometimes yielding nonuniform or partial-continuum contact patterns. The framework sets the stage for extending to 3D, incorporating soft-arm mechanics and contact compliance, and validating with soft robotic prototypes and tactile feedback.

Abstract

Continuum robotic grasping, inspired by biological appendages such as octopus arms and elephant trunks, provides a versatile and adaptive approach to object manipulation. Unlike conventional rigid-body grasping, continuum robots leverage distributed compliance and whole-body contact to achieve robust and dexterous grasping. This paper presents a control-theoretic framework for analyzing the statics of continuous contact with a planar object. The governing equations of static equilibrium of the object are formulated as a linear control system, where the distributed contact forces act as control inputs. To optimize the grasping performance, a constrained optimal control problem is posed to minimize contact forces required to achieve a static grasp, with solutions derived using the Pontryagin Maximum Principle. Furthermore, two optimization problems are introduced: (i) for assigning a measure to the quality of a particular grasp, which generalizes a (rigid-body) grasp quality metric in the continuum case, and (ii) for finding the best grasping configuration that maximizes the continuum grasp quality. Several numerical results are also provided to elucidate our methods.

Figures (4)

• Figure 1: A schematic of continuum grasping of a planar object. The boundary $\gamma$ of the object is parameterized by its arclength $s \in [0, L_\text{o}]$. At each $s$, there is a moving frame on the boundary $\{\text{t} (s), \text{n} (s) \}$. A soft arm grasps the object by making continuous contact in the interval $[0, L]$. The contact forces per unit length $(f^\text{t} (s), f^\text{n} (s))$ are shown in the top right inset. The constraints of the contact forces due to friction or the friction cone $FC$ is depicted by the orange cone and is elaborated in the bottom right inset.
• Figure 2: Minimum force grasping as a solution of problem \ref{['eq:optimal_control_problem_1']}. For each of the objects (unit circle, ellipse, deformed circle), each graph represents the minimum contact forces required to resist the corresponding external wrench $w_e$. The tangent forces are shown in solid blue lines, the normal forces are shown in solid red lines, and the friction constraints are shown by dashed black lines. The objects are also shown in orange, and the optimal region to be grasped is depicted by solid black lines overlaid on the boundaries. The optimal grasp effort $J^*(w_e)$ is also indicated for each case.
• Figure 3: Normalized grasp quality $\mathcal{Q}(0, L)$ as a function of the grasp length $L$, for each of the objects. The grasp quality increases as the grasp length increases.
• Figure 4: Maximizing grasp quality by solving optimization problem \ref{['eq:maximize_grasp_quality']}. For a fixed $L = 0.5L_\text{o}$, normalized grasp quality $\mathcal{Q}(s_0, L)$ is first calculated as a function of $s_0 \in [0, L_\text{o}]$. These values are then overlaid on top of each of the objects, with the colorbar shown on the right. Therefore, the more pink a point is on the object, the better is the grasp quality if the grasp were to be started at the point. Every point on the circle is equivalent in terms of grasp quality, whereas it is not true for the other two objects.

Theorems & Definitions (11)

• Remark 1
• Definition 1
• Remark 2
• Remark 3
• Proposition III.1
• proof
• Remark 4
• Proposition IV.1
• proof
• Remark 5