Robotics Chapter 5 – Path and Trajectory Planning Dr. Amit Goradia

Topics Introduction – 2 hrs Coordinate transformations – 6 hrs Forward Kinematics - 6 hrs Inverse Kinematics - 6 hrs Velocity Kinematics - 2 hrs Trajectory Planning - 6 hrs Robot Dynamics (Introduction) - 2 hrs Force Control (Introduction) - 1 hrs Task Planning - 6 hrs

Robot Motion Planning Path planning Trajectory planning, Geometric path Issues: obstacle avoidance, shortest path Trajectory planning, “interpolate” or “approximate” the desired path by a class of polynomial functions Generate a sequence of time-based “control set points” for the control of manipulator from the initial configuration to its destination.

The World is Comprised of… Obstacles Already occupied spaces of the world In other words, robots can’t go there Free Space Unoccupied space within the world Robots “might” be able to go here To determine where a robot can go, we need to discuss what a Configuration Space is

Configuration Space Notation: A: single rigid object –(the robot) W: Euclidean space where A moves; B1,…Bm: fixed rigid obstacles distributed in W • FW – world frame (fixed frame) • FA – robot frame (moving frame rigidly associated with the robot) Configuration q of A is a specification of the physical state (position and orientation) of A w.r.t. a fixed environmental frame FW. Configuration Space is the space of all possible robot configurations.

Definitions Configuration: Specification of all the variables that define the system completely Example: Configuration of a dof robot is Configuration space (C-space): Set of all configurations Free configuration: A configuration that does not collide with obstacles Free space ( F ) : Set of all free configurations It is a subset of C

Configuration Space of a 2D Planer Robot Configuration Space of A is the space (C )of all possible configurations of A. Point robot (free-flying, no constraints) qslug qrobot C Cfree Cobs For a point robot moving in 2-D plane, C-space is

Configuration Space of a Robot Moving in 3D Z x y qstart qgoal C Cfree Cobs For a point robot moving in 3-D, the C-space is What is the difference between Euclidean space and C-space?

Configuration Space of a 2R Articulated Robot 2R manipulator Configuration space topology

Two points in the robot’s workspace Configuration Space 360 qrobot 270 b 180 b a 90 qgoal a 45 90 135 180 Two points in the robot’s workspace Torus (wraps horizontally and vertically)

Configuration Space qrobot qslug If the robot configuration is within the blue area, it will hit the obstacle 360 qrobot 270 b 180 b a 90 qslug a 45 90 135 180 An obstacle in the robot’s workspace a “C-space” representation What is dimension of the C-space of puma robot (6R)? Visualization of high dimension C-space is difficult

Motion Planning Find a collision free path from an initial configuration to goal configuration while taking into account the constrains (geometric, physical, temporal) C-space concept provide a generalized framework to study the motion planning problem A separate problem for each robot?

Robot as a Point Object Expand obstacle(s) Reduce robot not quite right ...

Growing Obstacles C-obstacle Point robot

Minkowski Sums This expansion of one planar shape by another is called the Minkowski sum  Rectangular robot which can translate only P  R R P P  R = { p + r | p  P and r  R } (Dilation operation) Used in robotics to ensure that there are free paths available.

Additional Dimensions What would the C-obstacle be if the rectangular robot (red) can translate and rotate in the plane. (The blue rectangle is an obstacle.) y Rectangular robot which can translate and rotate x

C-obstacle in 3-D What would the configuration space of a 3DOF rectangular robot (red) in this world look like? (The obstacle is blue.) 3-D y 180º 0º x can we stay in 2d ?

One Slice of C-obstacle Taking one slice of the C-obstacle in which the robot is rotated 45 degrees... P  R R y 45 degrees P x

2-D Projection y x why not keep it this simple?

Projection problems qinit qgoal too conservative!

Motion Planning Methods The motion planning problem consists of the following: Input Output geometric descriptions of a robot and its environment (obstacles) initial and goal configurations a path from start to finish (or the recognition that none exists) qrobot qgoal Problem Statement Compute a collision-free path for a rigid or articulated moving object among static obstacles What to do?

Motion Planning Methods (1) Roadmap approaches (2) Cell decomposition (3) Potential Fields (4) Bug algorithms Goal reduce the N-dimensional configuration space to a set of one-D paths to search. Goal account for all of the free space Goal Create local control strategies that will be more flexible than those above Limited knowledge path planning

Roadmap: Visibility Graphs Visibility graphs: In a polygonal (or polyhedral) configuration space, construct all of the line segments that connect vertices to one another (and that do not intersect the obstacles themselves). Formed by connecting all “visible” vertices, the start point and the end point, to each other. For two points to be “visible” no obstacle can exist between them Paths exist on the perimeter of obstacles From Cfree, a graph is defined Converts the problem into graph search. Dijkstra’s algorithm Order(N^2) N = the number of vertices in C-space

Visibility Graph in Action First, draw lines of sight from the start and goal to all “visible” vertices and corners of the world. goal start

Visibility Graph in Action Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight. goal start

The Visibility Graph Repeat until you’re done. goal start Since the map was in C-space, each line potentially represents part of a path from the start to the goal.

Visibility Graph Drawbacks Visibility graphs do not preserve their optimality in higher dimensions: shortest path shortest path within the visibility graph In addition, the paths they find are “semi-free,” i.e. in contact with obstacles. No clearance

Roadmap: Voronoi diagrams “official” Voronoi diagram (line segments make up the Voronoi diagram isolates a set of points) Generalized Voronoi Graph (GVG): locus of points equidistant from the closest two or more obstacle boundaries, including the workspace boundary. Property: maximizing the clearance between the points and obstacles.

Roadmap: Voronoi diagrams GVG is formed by paths equidistant from the two closest objects maximizing the clearance between the obstacles. This generates a very safe roadmap which avoids obstacles as much as possible

Voronoi Diagram: Metrics Many ways to measure distance; two are: L1 metric (x,y) : |x| + |y| = const L2 metric (x,y) : x2 +y2 = const

Note the lack of curved edges Voronoi Diagram (L1) Note the lack of curved edges

Voronoi Diagram (L2) Note the curved edges

Motion Planning Methods Roadmap approaches Visibility Graph Voronoi Diagram Cell decomposition Exact Cell Decomposition (Trapezoidal) Approximate Cell Decomposition (Quadtree) Potential Fields Hybrid local/global

Exact Cell Decomposition Trapezoidal Decomposition: Decomposition of the free space into trapezoidal & triangular cells Connectivity graph representing the adjacency relation between the cells (Sweepline algorithm)

Exact Cell Decomposition Trapezoidal Decomposition: Search the graph for a path (sequence of consecutive cells)

Exact Cell Decomposition Trapezoidal Decomposition: Transform the sequence of cells into a free path (e.g., connecting the mid-points of the intersection of two consecutive cells)

Optimality Trapezoidal Decomposition: 15 cells 9 cells Obtaining the minimum number of convex cells is NP-complete. Trapezoidal decomposition is exact and complete, but not optimal there may be more details in the world than the task needs to worry about...

Approximate Cell Decomposition Quadtree Decomposition: recursively subdivides each mixed obstacle/free (sub)region into four quarters... Quadtree:

Further Decomposition Quadtree Decomposition: recursively subdivides each mixed obstacle/free (sub)region into four quarters... Quadtree:

Even Further Decomposition Quadtree Decomposition: • The rectangle cell is recursively decomposed into smaller rectangles • At a certain level of resolution, only the cells whose interiors lie entirely in the free space are used • A search in this graph yields a collision free path Again, use a graph-search algorithm to find a path from the start to goal is this a complete path-planning algorithm? i.e., does it find a path when one exists ? Quadtree

Probablistic Roadmap Methods What is a PRM (Probablistic Roadmap Method) A probabilistic road map is a discrete representation of a continuous configuration space generated by randomly sampling the free configurations of the C-space and connecting those points into a graph Complete path planning in high dimensional C- spaces is very complex PRM methods boost performance by trading completeness for probabilistic completeness Two phase approach: Learning phase, Query phase

Probabilistic Roadmap Methods Probabilistic techniques to incrementally build a roadmap in free space of robot Efficiency-driven Robots with many dofs (high-dim C-spaces) Static environments s g s ~ g

Learning Phase Learning phase: Construction: randomly sample free space and create a list of nodes in free space. Connect all the nearest neighbors using a fast local planner. Store the graph whose nodes are configurations and edges are paths computed by local planner Expansion step: Find “Difficult” nodes and expand the graph around them using random walk techniques

Query Phase Find a path from the start and goal positions to two nodes of the roadmap Search the graph to find a sequence of edges connecting those nodes in the roadmap Concatenating successive segments gives a feasible path for robot.

Motion Planning Methods Roadmap approaches Cell decomposition Exact Cell Decomposition (Trapezoidal) Approximate Cell Decomposition (Quadtree) Potential Fields Hybrid local/global

Potential Field Method Potential Field (Working Principle) – The goal location generates an attractive potential – pulling the robot towards the goal – The obstacles generate a repulsive potential – pushing the robot far away from the obstacles – The negative gradient of the total potential is treated as an artificial force applied to the robot -- Let the sum of the forces control the robot C-obstacles

Potential Field Method Compute an attractive force toward the goal C-obstacles Attractive potential

Potential Field Method Compute a repulsive force away from obstacles Repulsive Potential Create a potential barrier around the C-obstacle region that cannot be traversed by the robot’s configuration It is usually desirable that the repulsive potential does not affect the motion of the robot when it is sufficiently far away from C-obstacles

Potential Field Method Compute a repulsive force away from obstacles Repulsive Potential

Potential Field Method Sum of Potential Attractive potential Repulsive potential C-obstacle Sum of potentials

Potential Field Method After get total potential, generate force field (negative gradient) Let the sum of the forces control the robot Negative gradient Equipotential contours Total potential To a large extent, this is computable from sensor readings

Potential Field Method Pros: Spatial paths are not preplanned and can be generated in real time Planning and control are merged into one function Smooth paths are generated Planning can be coupled directly to a control algorithm Cons: Trapped in local minima in the potential field Because of this limitation, commonly used for local path planning Use random walk, backtracking, etc to escape the local minima