Beyond Constant Curvature: A New Mechanics Model for Unidirectional Notched-Tube Continuum Wrists

Abstract

This paper presents a new mechanics model for unidirectional notched-tube continuum wrists, a class of mechanisms frequently used to implement distal steering in needle-sized surgical robotic instruments. Existing kinematic models available for these devices are based on the simplifying assumption that, during actuation, all the notches undergo the same amount of deflection, so that the shape of a wrist can be approximated by an arc of constant curvature. This approach is analytically attractive, but, as we show in this paper, it can sometimes fail to provide good tracking accuracy. In this article, we provide a new model that relaxes the assumption above, and we report experimental evidence showing its superior accuracy. We model wrist deflection using Castigliano's second theorem, with the addition of a capstan friction term that accounts for frictional losses on the actuation tendon. Because notched-tube wrists are typically made of Nickel-Titanium (Nitinol), which has nonlinear stress-strain characteristics, we use a technique to obtain a local linearized approximation of the material modulus, suitable for use in the deflection model. The result of our modeling is a system of nonlinear equations that can be solved numerically to predict the wrist configuration based on the applied actuation force. Experimental results on physical specimens show that this improved model provides a more accurate estimate of wrist kinematics than prior models assuming constant curvature bending.

Keywords: Notched-tube Joints; Steerable Needles; Surgical Robotics.

Fig. 1.

Notched-tube wrists, typically manufactured from superelastic Nickel-Titanium (Nitinol), are articulated by pulling a tendon attached to the tip, causing the wrist to bend. When tendon friction and material nonlinearity are accounted for, wrist kinematics can deviate from constant curvature assumptions, resulting in significant tracking error for large deflections.

Fig. 2.

Actuation of two different notched-tube wrists. In wrist (a), which features five identical notches, the most proximal notch bends first (due to tendon friction, which makes the moment highest at the base of the device), while the most distal notch is the last one to reach the hard stop. Wrist (b), which features variable notch depths with the tip notch being deepest, exhibits the opposite behavior, i.e., the most distal notch is the one that closes first since it is the least stiff of the notches (tip-first bending). In this paper, we propose a mechanics model capable of explaining these different behaviors.

Fig. 3.

A notched-tube wrist is an open kinematic chain consisting of a sequence of interleaving notches and uncut sections. Throughout this paper, we use c_j to indicate the length of the j^th uncut section, while h_j and g_j respectively denote the length and depth of the j^th notch. Wrist actuation is performed by applying a pulling force F_p on the tendon. Kinematic models provide a mapping between the tendon displacement Δl and the tube pose T_wrist ∈ SE(3). A consequence of assuming a frictionless tendon in prior models is that each notch undergoes the same amount of deflection θ_j.

Fig. 4.

Detail of a single notch. The bending angle of the j^th notch is denoted with θ_j, the curvature is k_j, and the arc length is s_j. The distance between the centerline of the tube and the neutral bending plane is ${\overline{y}}_j$.

Fig. 5.

Diagram of the forces and moments being experienced at each notch due the pulling force, and the capstan loss model being applied at each corner of the notches.

Fig. 6.

Illustrative stress-strain curve of Nitinol as per Eq. (13).

Fig. 7.

Iterative secant modulus approximation of superelastic material behavior: (top) modulus approximation for a tube in the superelastic range, (bottom) convergence plot.

Fig. 8.

Experimental setup. Notched-tube wrists were mounted in a collet, and a linear slider was used to pull the actuation tendon. Tension on the tendon was measured by means of a force sensor, while the wrist deflection was recorded with a digital single-lens reflex camera.

Fig. 9.

Experimental Results. (Top Row) Model output for the three different wrist designs considered in the experiments. Wrists A and B were designed to exhibit proximal-first bending, while wrist C displayed tip-first bending. Note how the model correctly captures the notch closing sequence in all cases. (Bottom Row) Comparison of model output versus experimental data.

Fig. 10.

Comparison between the wrist shapes observed experimentally and the wrist shapes predicted by the model. The lines in the figure above represent the wrist center line. This data shows that the proposed model describes wrist deflection well across the entire range of motion.

Fig. 11.

Comparison of the accuracy of the proposed model versus three other models using simplifying assumptions. This figure shows three sample wrist shapes observed during experimentation, along with the wrist shapes predicted by the four different models. The model we propose in this paper predicts the wrist shape almost exactly, while models assuming simplifying assumptions generally provide a higher tracking error.

Fig. 12.

Tip tracking accuracy (Root-Mean-Square Error) of three different models calculated on Wrists A, B, and C.