CS 603 - Path Planning Rod Grupen 4/23/20 Robotics 1 4/23/20 - PowerPoint PPT Presentation

Why Path Planning? CS 603 - Path Planning Rod Grupen 4/23/20 Robotics 1 4/23/20 Robotics 2 Why Motion Planning? Origins of Motion Planning GE Virtual Prototyping Character Animation • T. Lozano-Pérez and M.A. Wesley: Structural Molecular Biology � An Algorithm for Planning Collision-Free Paths Au Autonomous Control Among Polyhedral Obstacles, � 1979. • introduced the notion of configuration space (c-space) to robotics • many approaches have been devised since then in configuration space 4/23/20 Robotics 3 4/23/20 Robotics 4 1

Representation Completeness of Planning Algorithms …given a moving object, A , initially in an unoccupied region of freespace, s , a set of stationary objects, B i , at known locations, and a goal position, g , … a co complete planner finds a path if one exists resolution complete – complete to the model resolution find a sequence of collision-free motions probabilistically complete that take A from s to g 4/23/20 Robotics 5 4/23/20 Robotics 6 Mapping to Configuration Space - Obstacles in 3D (x,y, q ) Translational Case (fixed orientation) Obstacle Robot C-Space Obstacle Reference Point changing q ? 9 4/23/20 Robotics 10 Jean-Claude Latombe 2

Exact Cell Decomposition Exact Cell Decomposition Jean-Claude Latombe Jean-Claude Latombe 4/23/20 Robotics 12 4/23/20 Robotics 13 Exact Cell Decomposition Representation – Simplicial Decomposition Schwartz and Sharir Lozano-Perez Canny 4/23/20 Robotics 14 4/23/20 Robotics 15 Jean-Claude Latombe 3

Approximate Methods: 2 n -Tree Approximate Cell Decomposition again…build a graph and search it to find a path 4/23/20 Robotics 16 4/23/20 Robotics 17 Jean-Claude Latombe Roadmap Representations Representation – Roadmaps Voronoi diagrams Visibility diagrams: a “ retraction ” unsmooth …the continuous freespace sensitive to error is represented as a network of curves… 4/23/20 Robotics 18 4/23/20 Robotics 19 Jean-Claude Latombe 4

Summary Attractive Potential Fields Oliver Brock • Exact Cell Decomposition + • Approximate Cell Decomposition u graph search u next: potential field methods state of the art techniques • Roadmap Methods • visibility graphs • Voronoi diagrams - • next: probabilistic road maps (PRM) 4/23/20 Robotics 29 30 Repulsive Potentials Electrostatic (or Gravitational) Field Oliver Brock + - - depends on direction 31 32 5

Attractive Potential A Repulsive Potential rep ( q ) = 1 Th e F pic tur x e φ att ( q ) = 1 2 k ( q − q ref ) T ( q − q ref ) ca n't be dis pla ye d. F att ( q ) = −∇ φ att ( q ) rep ( q ) = 1 x − 1 F = − k ( q − q ref ) δ 0 x 33 34 Repulsive Potential Sum Attractive and Repulsive Fields 2 ! $ ( q − q obst ) − 1 1 φ rep ( q ) = k if (( q − q obs ) < δ 0 # & # & δ 0 " % = 0 otherwise F rep ( q ) = −∇ φ ( q ) ) - # & ( q − q obs ) − 1 1 = − k if (( q − q obs ) < δ 0 + % ( + % ( * δ 0 . $ ' + + 0 otherwise , / 4/23/20 Robotics 35 36 6

Artificial Potential Function Potential Fields ● Goal: avoid local minima φ att ( q ) φ rep ( q ) φ total ( q ) ● Problem: requires global information + = ● Solution: Navigation Function = + F rep F rep Robot F att F total ( q ) = −∇ φ total F att Obstacle Goal 37 4/23/20 Robotics 38 Navigation Functions Navigation Functions Morse - navigation functions have no degenerate critical points where the robot can get stuck short of attaining the goal. Critical points are places where the gradient of Analyticity – navigation functions are analytic because φ vanishes, i.e. minima, saddle points, or maxima and they are infinitely differentiable and their Taylor series their images under φ are called critical values. converge to φ (q 0 ) as q approaches q 0 Polar – gradients (streamlines) of navigation functions terminate at a unique minima Admissibility - practical potential fields must always generate bounded torques 4/23/20 Robotics 39 4/23/20 Robotics 40 7

The Hessian Harmonic Functions multivariable control function, f(q 0 ,q 1 ,...,q n ) if the trace of the Hessian (the Laplacian) is 0 then function f is a harmonic function laminar fluid flow, steady state temperature distribution, if the Hessian is positive semi-definite over electromagnetic fields, current flow in conductive media the domain Q, then the function f is convex over Q 4/23/20 Robotics 41 4/23/20 Robotics 42 Properties of Harmonic Functions Properties of Harmonic Functions Mean-Value - up to truncation error, the value of the Min-Max Property - harmonic potential at a point in a lattice is the ...in any compact neighborhood of freespace, the average of the values of its 2n Manhattan neighbors. minimum and maximum of the function must occur on the boundary. analog & numerical methods ¼ ¼ ¼ ¼ 4/23/20 Robotics 43 4/23/20 Robotics 44 8

Numerical Relaxation Harmonic Relaxation: Numerical Methods Gauss-Seidel Jacobi iteration Successive Over Relaxation 4/23/20 Robotics 45 4/23/20 Robotics 46 Properties of Harmonic Functions Minima in Harmonic Functions for some i, if ∂ 2 φ / ∂ x i2 > 0 (concave upward), then Hitting Probabilities - if we denote p(x) at state x as there must exist another dimension, j, where the probability that starting from x, a random walk ∂ 2 φ / ∂ x j2 < 0 to satisfy Laplace’s constraint. process will reach an obstacle before it reaches a goal—p(x) is known as the hitting probability greedy descent on the harmonic function minimizes the hitting probability. therefore, if you’re not at a goal, there is always a way downhill... ...there are no local minima... 4/23/20 Robotics 47 4/23/20 Robotics 48 9

Configuration Space Harmonic Functions for Path Planning 4/23/20 Robotics 49 4/23/20 Robotics 50 Harmonic Functions for Path Planning Reactive Admittance Control 4/23/20 Robotics 51 4/23/20 Robotics 52 10

Probabilistic Roadmaps (PRM) ● Construction – Generate random configurations – Eliminate if they are in collision – Use local planner to connect configurations ● Expansion ok, back to graphical methods… – Identify connected components – Resample gaps – Try to connect components ● Query – Connect initial and final configuration to roadmap – Perform graph search 4/23/20 Robotics 54 4/23/20 Robotics 55 Probabilistic Roadmaps (PRM) Sampling Phase Oliver Brock ● Construction R = (V,E) – – repeat n times: – generate random configuration – add to V if collision free – attempt to connect to neighbors using local planner , unless in same connected component of R 4/23/20 Robotics 56 4/23/20 Robotics 58 11

Local Planner Path Extraction ● Connect start and goal configuration to roadmap using local planner q 2 ● Perform graph search on roadmap ● Computational cost of searching negligible d compared to construction of roadmap q 1 tests up to a specified resolution d ! 4/23/20 Robotics 59 4/23/20 Robotics 61 Another Local Planner Summary: PRM • Algorithmically very simple • Surprisingly efficient even in high-dimensional C- spaces • Capable of addressing a wide variety of motion planning problems • One of the hottest areas of research • Allows probabilistic performance guarantees perform random walk of predetermined length; choose new direction randomly after hitting obstacle; attempt to connect to roadmap after random walk 4/23/20 Robotics 62 4/23/20 Robotics 65 12

Variations of the PRM Lazy PRM • Lazy PRMs observation: pre-computation of roadmap takes a long time and does not respond well in dynamic • Rapidly-exploring Random Trees environments 4/23/20 Robotics 68 4/23/20 Robotics 69 Lazy PRM Rapidly-Exploring Random Trees (RRT) Oliver Brock Oliver Brock 4/23/20 Robotics 70 4/23/20 Robotics 74 13

Rapidly-Exploring Random Trees (RRT) 4/23/20 Robotics 75 Steven LaValle 14