On the Mathematical Modeling of Slender Biomedical Continuum Robots

Abstract

The passive, mechanical adaptation of slender, deformable robots to their environment, whether the robot be made of hard materials or soft ones, makes them desirable as tools for medical procedures. Their reduced physical compliance can provide a form of embodied intelligence that allows the natural dynamics of interaction between the robot and its environment to guide the evolution of the combined robot-environment system. To design these systems, the problems of analysis, design optimization, control, and motion planning remain of great importance because, in general, the advantages afforded by increased mechanical compliance must be balanced against penalties such as slower dynamics, increased difficulty in the design of control systems, and greater kinematic uncertainty. The models that form the basis of these problems should be reasonably accurate yet not prohibitively expensive to formulate and solve. In this article, the state-of-the-art modeling techniques for continuum robots are reviewed and cast in a common language. Classical theories of mechanics are used to outline formal guidelines for the selection of appropriate degrees of freedom in models of continuum robots, both in terms of number and of quality, for geometrically nonlinear models built from the general family of one-dimensional rod models of continuum mechanics. Consideration is also given to the variety of actuators found in existing designs, the types of interaction that occur between continuum robots and their biomedical environments, the imposition of constraints on degrees of freedom, and to the numerical solution of the family of models under study. Finally, some open problems of modeling are discussed and future challenges are identified.

Keywords: continuum robots; dynamics; mechanics; medical robotics; soft robots; statics.

FIGURE 1

(A) Concentric tube robots are comprised of hard (metallic) tubes which are precurved and nested inside one another. Rotating and translating the tubes results in motion. (B) Tendon-driven robots use one or more tendons or cables to provide internal actuation forces that bend a flexible, slender rod. (C) Pneumatic soft continuum robots use soft air muscles, which extend or contract with internal air pressure, to create bending in a composite structure. The supports could be hard or soft materials. (D) A fully soft pneumatic gripper uses asymmetry introduced by an inextensible fabric layer and an asymmetric air volume to create four slender fingers which bend to wrap around objects.

FIGURE 2

Mathematical setup of the curve-based kinematic description of slender continuum robots.

FIGURE 3

Simulation of a cantilevered rod under a combined bending and twisting concentrated moment, forming a helix.

FIGURE 4

Convergence of the PCC discretization to the exact flexural strains of the helical rod shape depicted in Figure 3. Note that the exact flexural strain components are not constant functions of arc length.

FIGURE 5

Flowchart depicting the modeling decisions to be made when selecting a model type for a biomedical continuum robot.

FIGURE 6

Error in reproducing the correct behavior under cantilevered loading conditions for three-DOF kinematic hypotheses of the PCC, PRB, and spectral types.

FIGURE 7

(A) Schematic diagram for the beam on an elastic foundation as a model for a continuum robot interacting with soft tissue. (B) Example with $\beta =3$ , showing the shape of the displacement that must be approximated.

FIGURE 8

Approximation errors for best polynomial fits in the L² norm to the solution for the linear beam on an elastic foundation problem. Higher-order polynomials permit greater elastic foundation stiffnesses.