Design and Control of Concentric-Tube Robots

Abstract

A novel approach toward construction of robots is based on a concentric combination of precurved elastic tubes. By rotation and extension of the tubes with respect to each other, their curvatures interact elastically to position and orient the robot's tip, as well as to control the robot's shape along its length. In this approach, the flexible tubes comprise both the links and the joints of the robot. Since the actuators attach to the tubes at their proximal ends, the robot itself forms a slender curve that is well suited for minimally invasive medical procedures. This paper demonstrates the potential of this technology. Design principles are presented and a general kinematic model incorporating tube bending and torsion is derived. Experimental demonstration of real-time position control using this model is also described.

Fig. 1

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

Fig. 2

Dominating-stiffness tube pair. (a) When retracted, tubes conform to shape of stiff outer tube. (b) Portion of extended inner tube relaxes to its initial curvature.

Fig. 3

Balanced-stiffness tube pair. (a) Rotating tube pair with curvatures aligned. (b) Rotating-tube pair with curvatures opposed.

Fig. 4

Effect of torsional twisting when two curved tubes are combined. Tube coordinate frames are denoted by F_i(s). The relative z-axis twist angle between frames α(s) varies from a maximum α(0) at the base to a minimum α(L) at the tip. The central angles β_i are proportional to the precurvature and to the tube length L.

Fig. 5

Stress versus strain curve for NiTi showing characteristic elastic loading and unloading plateaus.

Fig. 6

Example five-tube robot design composed of three telescoping sections of variable, fixed, and variable curvature, respectively. Tube pairs comprising variable-curvature sections are rotated individually but extended simultaneously. Each section dominates the shape of those sections retracted inside it.

Fig. 7

Three-tube example illustrating that torsionally-rigid model predicts tubes of piecewise-constant curvature combine to form a robot of piecewise-constant curvature.

Fig. 8

Relative twist angle of tubes at base versus tip for three values of $L\sqrt{c}$. Only the curve with $L\sqrt{c}<\pi ∕2$ exhibits stable rotation.

Fig. 9

Three-tube concentric-tube robot.

Fig. 10

Tubes comprising the robot. Tubes 1 and 2 form a variable-curvature balanced pair that dominates tube 3. Ruler shows units in millimeters.

Fig. 11

Dimensions of tube pairs.

Fig. 12

Tube pair showing graduated disk, twist pointer, and tangent pointer.

Fig. 13

Tip versus motor twist angle for tube pair A.

Fig. 14

Tip versus motor twist angle for tube pair B.

Fig. 15

Tip versus motor twist angle for tube pair C.

Fig. 16

Tube pair B workspace. Depicted configurations are computed using the torsionally rigid model.

Fig. 17

Workspace generated by rotating and translating tube 3 with respect to the outer pair.

Fig. 18

Tube 3 tip twist angle versus motor twist angle and extended length for α_2m = 0. Circles are experimental data points. Surface denotes prediction of torsionally compliant model.

Fig. 19

Teleoperator block diagram.

Fig. 20

Teleoperated real-time position control task. Touching sequence of nine silver beads embedded in dice involves controlling both position and tangent direction of robot tip.