Introduction to Robot Motion Planning

Example A robot arm is to build an assembly from a set of parts. Tasks for the robot: Grasping: position gripper on objectGrasping: position gripper on object design a path to this position design a path to this position Trasferring: determine geometry path for armTrasferring: determine geometry path for arm avoide obstacles + clearance avoide obstacles + clearance PositioningPositioning

Information required Knowledge of spatial arrangement of wkspace. E.g., location of obstaclesKnowledge of spatial arrangement of wkspace. E.g., location of obstacles Full knowledge full motion planningFull knowledge full motion planning Partial knowledge combine planning and executionPartial knowledge combine planning and execution motion planning = collection of problems

Basic Problem A simplified version of the problem assumes Robot is the only moving object in the wkspaceRobot is the only moving object in the wkspace No dynamics, no temporal issuesNo dynamics, no temporal issues Only non-contact motionsOnly non-contact motions MP = pure “geometrical” problem

Components of BMPP A: single rigid object - the robot - moving in Euclidean space W (the wkspace).A: single rigid object - the robot - moving in Euclidean space W (the wkspace). W = R N, N=2,3 W = R N, N=2,3 B i, I=1,…,q. Rigid objects in W. The obstaclesB i, I=1,…,q. Rigid objects in W. The obstaclesAssume Geometry of A and B i is perfectly knownGeometry of A and B i is perfectly known Location of B i is knownLocation of B i is known No kinematic constraints on A: a “free flying” objectNo kinematic constraints on A: a “free flying” object

Components of BMPP (cont.) The Problem:The Problem: - Given: an initial position and orientation a goal position and orientation - Generate: continuous path t from initial postion to goal t is a continuous sequence of position and orientation

Configuration Space Idea:Idea: represent robot as point in space map obstacles into this space transform problem from planning object motion to planning point motion Notion of CS A at a given position is a compact in W. attached frame F A B i closed subset of W. F w F w is a frame fixed in W

Notion of CS (cont) Def: configuration of an object Is the position of every point of the object relative to F W Configuration q of A : is the postion T and orientation O of F A w.r.t. F W Def: configuration space of A T Is the space T of all configurations of A A(q) : subset of W occupied by A at q a(q) : is a point in A(q)

Information Required Example: T: N-dimensional vector O: NxN rotation matrix In this case, q = (T,O), a subset of R N(N+1) Note that C is locally like R 3 or R 6. Notice: no global correspondence

Notion of Path Need a notion of continuity Define a distance function d : C x C -> R + – –Example : d(q,q’) = max a in A ||a(q) - a(q’)|| Def: A path of A from q init to q goal Is a continuous map t : [0,1] -> C s.t. t(0) = q init and t (1) = q goal Prop.: t is continuous if for each s o in (0,1), lim d(s,s o ) = 0 when s -> s o

Obstacles in Configuration Space B i maps in C to a region CB i = {q in C, s.t. A(q)  B i   } Obstacles in C are called C-obstacles. C-obstacle region: Free space: C free = C - q is a free configuration if q belongs to C free Def: Free Path. Is a path between q init and q goal, t: [0,1] -> C free

Example of a C-Obstacle The robot is a triange that translates but not rotates BiBi A CB i

Obstacles in C (cont.) Def: Connected Component q 1, q 2 belong to the same connected component of C free iff they are connected by a free path Objective of Motion Planning: generate a free path between 2 configurations if one exists or report that no free path exists.

Examples of C-Obstacles Translational Case: A is a single point -> no orientation W = R N = C A is a disk or dimensioned object allow to translate freely but without rotation. C-Obstacles: Are the obstacles “grow” by the shape of A

Planning Approaches 3 approaches: road maps, cell decomposition and potential field 1- Roadmap Captures connectivity of C free in a network of 1-D curves called “the roadmaps.” Once a roadmap is constructed: use a standard path. Roadmap Construction Methods: 1) Visibility Graph, 2) Voronoi Diagram, 3) Freeway Net and 4) Silhouette.

a- Visibility Graphs Mainly 2-D C-space and polygonal C-obstacles q init, q goal C-obstacle vertices b- Retractions: Voronoi Diagram V.D.:V.D.: set of all configurations whose minimal distance to C-obstacle region is achieved with at least 2 points in the boundary of CB nodes

c. Cell decomposition Decompose the robot’s free space into simple regions called cellsDecompose the robot’s free space into simple regions called cells A path within a cell -> easy to generateA path within a cell -> easy to generate Connectivity graph: non-directed graph representing adjacency relation between cellsConnectivity graph: non-directed graph representing adjacency relation between cells Nodes = cells extracted from free spaceNodes = cells extracted from free space 2 nodes connected by a link iff cells are adjacent2 nodes connected by a link iff cells are adjacent channel: resulting sequence of linkschannel: resulting sequence of links Cell Decomp: Exact or ApproximateCell Decomp: Exact or Approximate

d- Potential Fields In principle, we can discretize the C - space by using a greed. Then search for path. Computational expensive => need heuristics. Example of heuristics: potential fields Idea: robot “attracted” by q goal and “repulsed” by CB i ’s – –Potential methods can be very efficient, but – –Can be trapped in local minima!

Interaction with Sensing Robot with no prior knowledge about environment cannot plan Instead, rely on sensory information E.g., robot with proximity sensor can attempt to create repulsive potential. This can result on falling into local minima. Absence of info: reactive scheme due to Lumelsky. Algorithm guaranteed to reach q goal whenever C -obstacles are bounded by a simple closed curve of finite length Works in 2-D