Encoding Desired Deformation Profiles in Endoscope-Like Soft Robots

Abstract

Prior models of continuously flexible robots typically assume uniform stiffness, and in this paper we relax this assumption. Geometrically varying stiffness profiles provide additional design freedom to influence the motions and workspaces of continuum robots. These results are timely, because with recent rapid advancements in multimaterial additive manufacturing techniques, it is now straightforward to create more complex stiffness profiles in robots. The key insight of this paper is to project forces and moments applied to the robot onto its center of stiffness (i.e. the Young's modulus-weighted center of each cross section). We show how the center of stiffness can be thought of as analogous to a "precurved backbone" in a robot with uniform stiffness. This analogy enables a large body of prior work in Cosserat Rod modeling of such robots to be applied directly to those with stiffness variations. We experimentally validate this approach using multimaterial, soft, tendon-actuated robots. Lastly, to illustrate how these results can be used in practice, we investigate how stiffness variation can improve performance in a neurosurgical task.

Fig. 1.

An example of a continuum robot with an inhomogeneous stiffness profile and one straight tendon. Shifting the stiffness center along arclength enables the actuated curvature to dramatically change directions along arclength when actuated with a single straight-routed tendon.

Fig. 2.

(a) In a generic multimaterial cross section, the stiffness center frame (red) may be different from the geometric center frame (black). The stiffness center is offset by $p_{nc}$, with principal axes of the cross section rotated by $R_{nc}$. (b) The path of the stiffness center may vary nonlinearly along the arclength of the device. This path can be thought of as a virtual homogeneous and symmetric beam with precurvature.

Fig. 3.

A cross section of the two-material design parameterization used to vary the placement of the stiffness center as a function of arclength, where the dark blue denotes a stiffer material than the light blue.

Fig. 4.

Results of Finite Element Simulation of a 10 mm segment of the constant offset stiffness center cross section. A small annular sector of the cross section is made of elastomer with a higher stiffness (shown in blue). On the left, a 0.5 N normal load is applied at the geometric center which causes the prototype to bend to the left. On the right, the same load is applied to the stiffness center, which we calculated based on Equation 19. With the load at the stiffness center, the prototype experiences pure compression and no bending.

Fig. 5.

Tensile testing of 3D printed samples with various mix ratios of elastic and rigid resin; * indicates materials that were used in the tendon manipulator prototypes.

Fig. 6.

Experimental setup to measure the robot shape. Inset: Point cloud data is converted to an arclength parameterized curve by sequentially fitting cylinders and interpolating with a spline.

Fig. 7.

Comparison of the model predicted backbone shape with laser-scanned backbone shape for the constant stiffness center prototype. The CAD rendering on the left shows the geometry of the soft tendon robot with a constant bi-material pattern in the cross-section. On the right, we show the model-predicted shape and the measured shape of the TDCR at these configurations. The offset stiffness center in this design effectively biases the workspace toward the other direction.

Fig. 8.

Comparison of the model predicted backbone shape with laser-scanned backbone shape for the helical stiffness center prototype. The CAD rendering on the left shows the geometry of the soft tendon robot with a helically varying bi-material pattern in the cross-section, where the stiff segment becomes smaller towards the tip. On the right, we show the model-predicted shape and the measured shape of the TDCR at these configurations. With a straight tendon, pre curvature of the stiffness center causes the direction and magnitude of bending to change along arclength.

Fig. 9.

Improving the navigability of a tendon-actuated flexible endoscope. (a) Conceptual image of a tendon-actuated endoscope with a helically varying stiffness profile over arclength. Two views of the workspace with the clinical endoscope (b, c) and the arclength-varying stiffness design (d, e). Reachable tip positions shown in black and associated backbone shape in green, unreachable points shown in red.