The Kinematic Synthesis of Custom-Segmented Continuum Robots Including Segments of Constant Curvature

Date of Award

5-5-2024

Degree Name

Ph.D. in Engineering

Department

Department of Mechanical and Aerospace Engineering

Advisor/Chair

Andrew P. Murray

Abstract

This dissertation explores the kinematic synthesis of continuum robots and the design of a specific continuum robot for use in laparoscopy. The kinematic synthesis of continuum robots is studied based on either the desired target end-effector pose or the backbone shape of the robot. Initiating with an investigation into the mathematical relationships among positions and orientations at the segment tips, this study focuses on piecewise constant curvature (PCC) continuum robots with up to three segments. For a continuum robot with more than three segments, a method is introduced for generating the backbone of a continuum robot that closely approximates a given spatial curve. Furthermore, the kinematic synthesis methodology is investigated for designing a chain of three-dimensional bodies to match a set of arbitrary spatial curves. For a one-segment PCC, a reachability criterion is proposed, simplifying the calculation of the neighboring orientation. For a two-segment PCC, a reachability criterion is proposed and the redundancy of its inverse kinematics solution is found, establishing a circle of tip locations. For a three-segment PCC, the redundancy of the inverse kinematics includes tips that lie on a sphere providing a closed-form solution to the inverse kinematics problem. These relationships are derived from the unique characteristics of the bisecting plane of a single segment. The degenerate cases for the solutions are also addressed. These outcomes stem from a specific PCC parametrization, with implications extending to the general PCC model. Note that this study is grounded solely in simulation. Shape-changing mechanisms provide the theoretical tools for the synthesis of spatial chains of rigid bodies that position themselves along curves of any complexity. This study investigates the kinematic synthesis methodology for designing a chain of three-dimensional bodies to match a set of arbitrary spatial curves. The bodies synthesized can be one of three types: a rigid segment, a helical segment with constant curvature and torsion but varying length, and a growth segment that maintains its geometry but may be scaled to become larger or smaller. To realize mechanical chains, only rigid and helical segments are used. After designing the segments, they may be aligned with the original spatial curves with their ends connected via an optimization. For two curves, these connections may be made with revolute joints to obtain high accuracy. For three or more curves, spherical joint connections allow for the best accuracy. To compare curves as is useful in morphometry, all three segment types may be employed. In this case, an accurate description of the changes between curves is important, and optimizing to connect the segments is not needed. The study proposes the process of redefining curves to enable the application of the outlined techniques, as well as the methodologies for synthesizing the three segment types. Examples include a continuum robot problem and the morphometric analyses of cochlear curves and the lambdoidal suture. This work extends the established planar techniques for synthesizing mechanisms and addressing morphometric issues that are motivated by curves in two dimensions. For the shape inverse of a continuum robot with more than three segments, the approach combines the kinematic model of the continuum robot, which operates under the constant curvature assumption, with the design profile and target profile concepts from established spatial shape-changing mechanism theory. This approach involves an analysis of the accessibility of the terminal orientation of the spatial curve, which subsequently identifies a rapid synthesis method. This method facilitates the creation of a backbone with a predetermined number of segments that closely align with the tangential direction of the reference points of the target profile while simultaneously approximating the target profile itself. Optimization techniques demonstrate the efficacy of using the generated backbone of the rapid approximation method as the initial design for the optimization. In a situation where a close approximation is not needed at interim steps, optimization is not needed to provide a displacement sequence to move the robot. Finally, a novel soft robot laparoscope design is presented that enables accurate and safe deployment of the photodynamic therapy (PDT) optical fiber for pancreatic tumor treatment. To achieve this, the proposed soft robot is able to produce 3 degrees of freedom (DoF) motions: linear translation, in-plane bending, and axial rotation. A microcamera is integrated with the soft robot to provide intraoperative image guidance during the procedure. The robot forward kinematics model was obtained via a generalized calibration method, and the differential kinematics was derived analytically. Note that the proposed calibration method can be applied to general models including the PCC model. Benchtop characterization results indicate that the soft robot is able to achieve 1.4±0.4 mm position and 1.5±1.1 degrees orientation accuracy, respectively. The robot was experimentally validated in two in vivo mice models, and the results indicate that the proposed system successfully reduced the tumor size from 15.3 mm³ to 12.1 mm³ and 11.8 mm³ to 4.4 mm³.

Keywords

Design, Mechanical Engineering, Robotics

Rights Statement

Copyright 2024, author

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