학위논문 (박사)-- 서울대학교 대학원 : 기계항공공학부, 2015. 8. 박종우. ; Minimally invasive surgery can involve navigating inside small cavities or reaching around sensitive tissues. Robotic instruments based on concentric tube technology are well suited to these tasks since they are slender and can be designed to take on shapes of high and varying curvature along their length. One limitation of these robots, however, is that elastic instabilities can arise when manipulating the robots by rotating or translating the bases of the tubes. As the tubes rotate and translate with respect to each other, elastic potential energy associated with tube bending and twisting can accumulate ; if a configuration is not locally elastically stable, then a dangerous snapping motion may occur as energy is suddenly released. To enhance the elastic stability of the concentric tube robots, this paper presents two researches: i) optimal design of tube pair, ii) local stability test to avoid unstable configurations. While prior work has considered tubes of piecewise-constant pre-curvature, the first research in this paper proposes varying tube pre-curvature as a function of arc length as a means to enhance stability. Stability conditions for a planar tube pair are derived and used to define an optimal design problem. This framework enables solving for pre-curvature functions that achieve a desired tip orientation range while maximizing stability and respecting bending strain limits. Analytical and numerical examples of the approach are provided. The second research provide a local stability condition and test to determine if a configuration is a stable equilibrium or not. This condition applies to arbitrary robot designs with any external loads. The local stability test based on this condition is validated by comparison with known stability results, and its utility is demonstrated by application to stable path planning. Though those two researches address the elastic instability issue of concentric tube robots, they both are based on the theoretical kinematics of the robots. Robot control requires the rapid computation of this kinematics, which involves solving complex mechanics-based models. Furthermore, shape computation based on kinematic input variables can be inaccurate due to parameter errors and model simplification. An alternate approach is to compute the shape in real time from a set of sensors positioned along the length the robot that provide measurements of local curvature, e.g., optical fiber Bragg gratings. In this point of view, the third research in this paper proposes a general framework for selecting the number and placement of such sensors with respect to arc length so as to compute the forward kinematic solution accurately and quickly. The approach is based on defining numerically efficient shape reconstruction models parameterized by sensor number and location. Optimization techniques are used to solve for the sensor locations that minimize shape and tip error between a reconstruction model and a mechanics-based model. As a specific example, several reconstruction models are proposed and compared for concentric tube robots. These results indicate that the choice of reconstruction model as well as sensor placement can have a substantial effect on shape accuracy. ; Contents Abstract 3 List of Tables 11 List of Figures 13 1 Introduction 1 1.1 Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Achieving Elastic Stability Through Precurvature Optimization 4 1.1.2 Optimizing Curvature Sensor Placement for Fast, Accurate Shape Sensing of Continuum Robots . . . . . . . . . . . . . 6 1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Achieving Elastic Stability Through Precurvature Optimization 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Kinematics of General Concentric-Tube Robot . . . . . . . . . . . . 14 2.3 Kinematics of Planar Tube Pair . . . . . . . . . . . . . . . . . . . . 16 2.4 Stability Condition for Planar Tube Pairs . . . . . . . . . . . . . . . 17 2.4.1 Ignoring Straight Transmission Length Inside Collar . . . . . 17 2.4.2 Considering Straight Transmission Length Inside Collar . . . 19 2.5 Evaluating Stability for Specific Pre-curvature Functions . . . . . . 22 2.5.1 Constant Pre-curvature . . . . . . . . . . . . . . . . . . . . . 23 2.5.2 Partially Constant Pre-curvature . . . . . . . . . . . . . . . . 23 2.5.3 Pre-curvature Function, û_y = b/(s+a). . . . . . . . . . . . . . . 25 2.6 Formulation as an Optimal Design Problem . . . . . . . . . . . . . 27 2.6.1 Numerical Solution of the Optimal Design Problem . . . . . 29 2.6.2 Analytic Solution of the Optimal Design Problem . . . . . . 31 2.6.3 Feasibility of Optimal Design Problem . . . . . . . . . . . . 36 2.7 Hardware Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Elastic Stability of Concentric Tube Robots Subject to External Loads 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Concentric Tube Robot Modeling . . . . . . . . . . . . . . . . . . . 54 3.3 Elastostatic Kinematic Model . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 Concentric Tube Robot with No External Load . . . . . . . 58 3.3.2 Concentric Tube Robot with External Load . . . . . . . . . 59 3.3.3 Generalized Force Representation of External loads . . . . . 63 3.4 Evaluating Local Elastic Stability . . . . . . . . . . . . . . . . . . . 65 3.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.2 Jacobi Non-Conjugacy Test for Conservative External Loads 66 3.4.3 Physical Interpretation of Jacobi Non-Conjugacy Condition 68 3.4.4 Non-Conservative External Loads . . . . . . . . . . . . . . . 69 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5.1 Example 1: Stability of an Unloaded Constant-precurvature Tube Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5.2 Example 2: Stability of a Constant Precurvature Tube Pair Subject to Elastic Forces . . . . . . . . . . . . . . . . . . . . 73 3.5.3 Example 3: Stability of a Constant-precurvature Tube Pair Subject to Constant World-frame Loads . . . . . . . . . . . . 77 3.5.4 Example 4: Application to Stable Path Planning . . . . . . . 80 4 Optimizing Curvature Sensor Placement for Shape Sensing 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Shape Estimation for Continuum Robots . . . . . . . . . . . . . . . 94 4.2.1 Kinematics of Continuum Robots . . . . . . . . . . . . . . . 94 4.2.2 Shape Reconstruction Models . . . . . . . . . . . . . . . . . . 95 4.2.3 Optimal Sensor Location . . . . . . . . . . . . . . . . . . . . 98 4.3 Case Study: Concentric Tube Robots . . . . . . . . . . . . . . . . . 100 4.3.1 Kinematics of Concentric Tube Robots . . . . . . . . . . . . 101 4.3.2 Section-Based Principal Component Analysis Model . . . . . 102 4.3.3 Section-based Polynomial Regression Model . . . . . . . . . 104 4.4 Numerical Experiments for a Concentric Tube Robot . . . . . . . . 105 4.4.1 Selection of Basis Functions . . . . . . . . . . . . . . . . . . 107 4.4.2 Number and Location of Sensors . . . . . . . . . . . . . . . . 109 4.4.3 Comparison between Reconstruction Models . . . . . . . . . 109 5 Conclusion 115 Bibliography 118 국문초록 125 ; Doctor
Concentric Tube Robots: Stability Analysis, Optimal Design, and Shape Sensing ; 컨센트릭 튜브 로봇: 안정성 분석, 최적 디자인, 자세 측정
2015-01-01
Theses
Electronic Resource
English
DDC: | 629 |
Real-Time Shape Estimation for Concentric Tube Continuum Robots with a Single Force/Torque Sensor
BASE | 2021
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