Quasistatic Modeling of Concentric Tube Robots with External Loads

Abstract

Concentric tube robots are a subset of continuum robots constructed by combining pre-curved elastic tubes. As the tubes are rotated and translated with respect to each other, their curvatures interact elastically, enabling control of the robot's tip configuration as well as the curvature along its length. This technology is projected to be useful in many types of minimally invasive medical procedures. Because these robots are flexible by design, they deflect considerably when applying forces to the external environment. Thus, in contrast to rigid-link robots, their kinematic and static force models are coupled. This paper derives a multi-tube quasistatic model that relates tube rotations and translations together with externally applied loads to robot shape and tip configuration. The model can be applied in robot design, procedure planning as well as control. For validation, the multi-tube model is compared experimentally to a computationally-efficient single-tube approximate model.

Fig. 1

Concentric tube robot comprised of four telescoping sections that can be rotated and translated with respect to each other.

Fig. 2

Tube coordinate frames are denoted F_i (s). The relative z-axis twist angle between tube frame F₀(s) and frame F_i (s) is θ_i (s).

Fig. 3

External loading on robot consists of distributed forces, f₀(s), and distributed moments, τ₀(s), as well as concentrated forces, n₀(L), and concentrated moments, m₀(L).

Fig. 4

Individual tubes comprising variable curvature tube pair.

Fig. 5

Tube pair mounted in drive system. Double exposure shows tubes in unloaded and gravity-loaded configurations. Tip-mounted disk for measuring relative tube twist is also shown. Orientation of coordinate frame F₀(0) is labeled.

Fig. 6

Disk and pointer for measuring relative angle at tip.

Fig. 7

Relative tube twist angle, α = θ₂ − θ₁, at the tip, α(L), versus the motor, α_m, for the case of no external loading using the torsionally-rigid model of [6] and the torsionally-compliant model of [8].

Fig. 8

Relative tube twist angle, α = θ₂ − θ ₁, at the tip, α(L) , versus the motor, α_m, for a 200 gram gravity load in the -y direction of Fig. 5 (in the plane of robot curvature).

Fig. 9

Tip position in the x-y plane for robot loaded with 200 g in the −y direction. Data points correspond to unloaded robot curvature varying from the configuration of Fig. 5 to zero and then to the configuration in which robot is curved downward. Diamonds are measured positions; circles are the multi-tube model predictions; squares are the single-tube model predictions. Associated positions are connected by lines.