Concentric Tube Robot Design and Optimization Based on Task and Anatomical Constraints

Abstract

Concentric tube robots are catheter-sized continuum robots that are well suited for minimally invasive surgery inside confined body cavities. These robots are constructed from sets of pre-curved superelastic tubes and are capable of assuming complex 3D curves. The family of 3D curves that the robot can assume depends on the number, curvatures, lengths and stiffnesses of the tubes in its tube set. The robot design problem involves solving for a tube set that will produce the family of curves necessary to perform a surgical procedure. At a minimum, these curves must enable the robot to smoothly extend into the body and to manipulate tools over the desired surgical workspace while respecting anatomical constraints. This paper introduces an optimization framework that utilizes procedureor patient-specific image-based anatomical models along with surgical workspace requirements to generate robot tube set designs. The algorithm searches for designs that minimize robot length and curvature and for which all paths required for the procedure consist of stable robot configurations. Two mechanics-based kinematic models are used. Initial designs are sought using a model assuming torsional rigidity. These designs are then refined using a torsionally-compliant model. The approach is illustrated with clinically relevant examples from neurosurgery and intracardiac surgery.

Fig. 1

Concentric tube robot comprised of three curved telescoping sections that can be rotated and translated with respect to each other. The first section, comprising two tubes, is a variable curvature section.

Fig. 2

Follow-the-leader robot extension. Robot cross sections, described by g_r(s_r), move along desired curve, described by g_c(s_c) with arc length velocity, v.

Fig. 3

Tip position workspace for robot Design 1 of Table II showing xz-plane slices. Complete workspace is generated by rotation of slice about z axis. (a) Fixed-fixed curvature sections, (b) fixed-variable curvature sections, (c) variable-fixed curvature sections. Dark shaded area in each plot is workspace of variable-variable curvature design. Red ○ are tip positions used to generate Fig. 4

Fig. 4

Solution sets of orientations for three tip positions labeled in Fig. 3. Cut planes show cross sections of solution sets. (a) Fixed-fixed design, (b) Fixed-variable design, (c) Variable-fixed design.

Fig. 5

Workspace of variable-fixed design. (a) Configuration 1 (dotted) is unstable and snaps. Configuration 2 is stable. Region of workspace containing unstable configurations is indicated in blue. (b)–(c) S-surfaces of Configuration 1, (d)–(e) S-surfaces of Configuration 2.

Fig. 6

Navigation and manipulation tasks. (a) Navigation – telescopic extension and steering of proximal sections from entry frame, E, to frame A. (b) Manipulation – distal sections move from A to set of tip task frames, B_i located at surgical sites.

Fig. 7

Robot design optimization framework.

Fig. 8

Robotic cauterization of the choroid plexus. Robot enters right lateral ventricle and also crosses over into left ventricle to perform cauterization. Red ○ define the tip task frame set $\mathcal{B}$ indicating the cauterization points in the right lateral ventricle.

Fig. 9

Cauterization targets and entry waypoints specified on the anatomical model of the hydrocephalic ventricles.

Fig. 10

Architecture-dependent optimized robot designs. (a) Variable-variable-fixed curvature, (b) Variable-fixed-fixed curvature, (c) Fixed-variable-fixed curvature, (d) Fixed-fixed-fixed curvature. Red lines in (c) and (d) indicate violation of anatomical constraints.

Fig. 11

Comparison of torsionally rigid and torsionally compliant models: (a) Front view, and (b) Side view.

Fig. 12

Stable and unstable configurations for a target point. Unstable configuration is shown dotted.

Fig. 13

Percutaneous robotic PFO closure. Inset: Target points intended to allow treatment for a range of anatomical sizes.

Fig. 14

Anatomical constraints for vascular navigation.

Fig. 15

Telescopic extension through the vasculature using the optimized navigation sections. Configurations shown are solutions to the anatomically-constrained inverse kinematics problem using the: (a) torsionally rigid model, (b) torsionally compliant model.

Fig. 16

Anatomical model together with initial and optimized designs for the navigation and manipulation portions of the robot. Inset: close up of manipulation sections and target points.