cs.ROApr 10, 2026

Kinematics of continuum planar grasping

Authors: Udit Halder, Nicolas Echeverria Zambrano, Xincheng Li

Organizations: Department of Mechanical and Aerospace Engineering, University of South Florida.

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Abstract

This paper presents an analytical framework to study the geometry arising when a soft continuum arm grasps a planar object. Both the arm centerline and the object boundary are modeled as smooth curves. The grasping problem is formulated as a kinematic boundary following problem, in which the object boundary acts as the arm's 'shadow curve'. This formulation leads to a set of reduced kinematic equations expressed in terms of relative geometric shape variables, with the arm curvature serving as the control input. An optimal control problem is formulated to determine feasible arm shapes that achieve optimal grasping configurations, and its solution is obtained using Pontryagin's Maximum Principle. Based on the resulting optimal grasp kinematics, a class of continuum grasp quality metrics is proposed using the algebraic properties of the associated continuum grasp map. Feedback control aspects in the dynamic setting are also discussed. The proposed methodology is illustrated through systematic numerical simulations.

Figures & tables

Fig. 1: (a) Schematic of a continuum arm grasping a planar object with smooth boundary. The boundary of the object γ\gamma is parameterized by its arclength sos_{\text{o}} . The arm is described by the position of its centerline r(s)∈\mathdsR2\mathbf{r}(s)\in{\mathds{R}}^{2} where s∈[0,L]s\in[0,L] is the arclength of the centerline. The proposed control is given by the contact vector ρ(s)\bm{\rho}(s) and the contact angle α(s)\alpha(s) defined in ( 4 ). (b) An illustration showing the arm: i. approaching the target when 0<α<π/20<\alpha<\pi/2 ; ii. departing the target when π/2<α<π\pi/2<\alpha<\pi .
Fig. 2: Optimal grasping configurations obtained by solving the optimal control problem ( 9 ). The problem is solved for three different smooth boundary objects: (a) circle, (b) ellipse and (c) deformed circle. A rendering of the arm grasping each object is shown at the top of each column. Below this, ρ(s)/ρd\rho(s)/\rho_{d} (solid blue line) and α(s)\alpha(s) (solid orange line) are plotted against the normalized arc length s/Ls/L , with their expected convergence values indicated by blue and orange dashed lines, respectively. The arm curvature κ(s)\kappa(s) (solid blue line) and object curvature κo(s)\kappa_{\text{o}}(s) (dashed black line) are shown in the bottom plots.
Fig. 3: Grasp quality maps for the metrics Q1Q_{1} , Q2Q_{2} and Q3Q_{3} for the circular, elliptical, and deformed circular objects. The quality values are normalized by the maximum quality for each object, where darker regions denote higher quality. The purple point indicates the optimal starting location, and the corresponding arm configuration is shown as a purple dashed curve.
Fig. 4: Dynamic simulation results for a soft arm grasping a fixed circular object. The top section shows six time instances of the arm in transparent purple. The bottom section shows a series of time snapshots of ρ/ρd\rho/\rho_{d} and α\alpha in blue. The black dashed lines represent the theoretical static solutions, corresponding to the curvature ( 20 ). Black arrows illustrate the time progression of ρ/ρd\rho/\rho_{d} and α\alpha .
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