Clothoid Based Spline-RRT with Bézier Approximation

Transcription

1 Clothoid Based Spline-RRT with Bézier Approximation CHEN YONG - School of Mechanical & Aerospace Engineering - Institute for Media Innovation Supervisors: Assoc Prof. Cai Yiyu - School of Mechanical & Aerospace Engineering Prof. Daniel Thalmann - Institute for Media Innovation 1

2 Outline of the Presentation 1. Problem formulation 2. Overview of the relevant research 3. Methodology 4. Results and discussions 2

3 Problem Formulation Planar Path A piecewise curve as a superset of n segments Path Smoothing Parametric continuity Geometric Continuity 3

4 Problem Formulation Curvature continuity 4

5 Problem Formulation Clothoid A curve whose curvature changes linearly with its curve length (Euler Spiral) Advantage: Shortest path satisfying Maximum Principle (optimal control theory) Disadvantage: No closed form due to Fresnel integrals 5

6 Relevant Research Online Approximation RBC Algorithm (Montés, N., 2008) Circular Interpolation (Brezak, M., 2014) Offline Approximation Method Cons 1 Continuous function approximation (Wang, Lazhu Z., 2001) Degree can be 26th order 2 3 C 2 Hermite interpolation via s-power series (Sánchez-Reyes, 2003) G 3 Bézier approximation with numerical search (Cross, 2012; L Lu., 2013) 4 G 2+ deterministic approximation (Cross, 2015) Complicated coefficients calculation Numerical search procedure is expensive; not robust Not accurate due to linear approximation Note: 1, 2, 3, 4 can only deal with unit-lenth clothoids 6

7 Methodology Elementary Clothoid Basic Clothoid General Clothoid 7

8 Methodology (a) Bézier curves as primitives (b) The Bézier approximation to the basic clothoid G 3 Continuity Constraints 8

9 Coordinate (Position) Error [1] Polynomial approximation to clothoids via s-power series [2] Rational Approximations for the Fresnel Integrals [3] Real-time Approximation of Clothoids With Bounded Error for Path Planning Applications 9

10 Orientation and Curvature Errors Orientation Error Curvature Error (4, 5, 6, 7) [4] G3 quintic polynomial approximation for Generalised Cornu Spiral segments [5] Smooth polynomial approximation of spiral arcs [6] A note on quintic polynomial approximation of generalized Cornu spiral segments [7] Efficient robust approximation of the generalised Cornu spiral 10

11 Circular Interpolation (Brezak, M., 2014) 11

12 RBC Algorithm (Montés, N., 2008) 9-th order RBC 12

13 Accuracy 13

14 Efficiency 14

15 Scalability Position error with respect to clothoid scaling 15

16 Differential System Extension of Dubins car (8) Feasible Path: G 2 curvature continuous with curvature derivative (sharpness) bounded [8] From Reeds and Shepp's to Continuous-Curvature Paths 16

17 Local Path Case 1: A pair of clothoids, a circular arc and line segments 17

18 Local Path Case 1 18

19 Local Path Case 2: Degenerated case with small deflection 19

20 Comparison Bezier based planner (9) Max solve time: Max curvature: 10 Diagonal length of environment: Max distance each step: 0.4 Robot dimensions: (0.127, ) Length of robot: Start state 0: (-1.15, -1.1) Start state 1: ( , -0.9) Goal state: (1.16, 1.1) RRT: Created 1876 states Solution found in seconds Geometric path with 22 states [9] Optimal path planning based on spline-rrt star for fixed-wing UAVs operating in three-dimensional environments 20

21 Comparison Proposed method Max solve time: Max curvature: 10 Diagonal length of environment: Max distance each step: 0.4 Robot dimensions: (0.127, ) Length of robot: Start state 0: (-1.15, -1.1) Start state 1: ( , -0.9) Goal state: (1.16, 1.1) RRT: Created 522 states Solution found in seconds Geometric path with 20 states 21

22 Thank You! 22