RoboticsChapter 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– 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.

Task P lan

Action P lan

Path P lan

TrajectoryP lan

C ontro ller

Sensor

R obot

Tasks

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

• Notation:– A: single rigid object –(the robot)– W: Euclidean space where A moves;– B1,…Bm: fixed rigid obstacles distributed in W

Configuration Space

Configuration Space is the space of all possible robot configurations.

32 RorRW

• 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.

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

110 ,, dqqqq d

q

Configuration Space of a 2D Planer Robot

For a point robot moving in 2-D plane, C-space is

Configuration Space of A is the space (C )of all possible configurations of A.

qslug

qrobot

CCfree

Cobs

2R

Point robot (free-flying, no constraints)

Configuration Space of a Robot Moving in 3D

For a point robot moving in 3-D, the C-space is

x

y

qstart

qgoal

C

Cfree

Cobs

3R

What is the difference between Euclidean space and C-space?

Configuration Space of a 2R Articulated Robot

2R manipulator

topology

Configuration space

Configuration Space

Two points in the robot’s workspace

270

360

180

90

090 18013545

Torus(wraps horizontally and vertically)

qrobot

qgoal

Configuration Space

An obstacle in the robot’s workspace

270

360

180

90

090 18013545

qslug

qrobot

a “C-space” representation

If the robot configuration is within the blue area, it will hit the obstacle

What is dimension of the C-space of puma robot (6R)?

Visualization of high dimension C-space is difficult

Motion PlanningFind a collision free path from an initial configuration to goal configuration while taking into account the constrains (geometric, physical, temporal)

A separate problem for each robot?

C-space concept provide a generalized framework to study the motion planning problem

Expand obstacle(s)

Reduce robot

not quite right ...

Robot as a Point Object

C-obstacle

Point robot

Growing Obstacles

This expansion of one planar shape by another is called the Minkowski sum

P

RP R

Used in robotics to ensure that there are free paths available.

P R = { p + r | p P and r R }

Rectangular robot which can translate only

(Dilation operation)

Minkowski Sums

What would the C-obstacle be if the rectangular robot (red) can translate and rotate in the plane.

(The blue rectangle is an obstacle.)

x

yRectangular robot which can translate and rotate

Additional Dimensions

C-obstacle in 3-DWhat would the configuration space of a 3DOF rectangular robot (red) in this world look like?

(The obstacle is blue.)

x

y

0º

180º

can we stay in 2d ?

3-D

One Slice of C-obstacle Taking one slice of the C-obstacle in which the robot is rotated 45 degrees...

x

y45 degrees

P

RP R

2-D Projection

why not keep it this simple?

x

y

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)

qgoalqrobot

What to do?

Problem Statement

Compute a collision-free path for a rigid or articulated moving object among static obstacles

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 GraphsVisibility 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).

From Cfree, a graph is defined

Converts the problem into graph search.

Dijkstra’s algorithmOrder(N^2)

N = the number of vertices in C-space

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

Visibility Graph in Action

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

start

goal

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.

start

goal

The Visibility Graph • Repeat until you’re done.

start

goal

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:

In addition, the paths they find are “semi-free,” i.e. in contact with obstacles.

shortest path

shortest path within the visibility graph

No clearance

“official” Voronoi diagram

(line segments make up the Voronoi diagram isolates a set of points)

Roadmap: Voronoi diagrams

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

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

Decomposition of the free space into trapezoidal & triangular cells

Connectivity graph representing the adjacency relation between the cells

(Sweepline algorithm)

Trapezoidal Decomposition:

Exact Cell Decomposition

Search the graph for a path (sequence of consecutive cells)

Trapezoidal Decomposition:

Exact Cell Decomposition

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

Trapezoidal Decomposition:

Obtaining the minimum number of convex cells is NP-complete.

Optimality

there may be more details in the world than the task needs to worry about...

15 cells 9 cells

Trapezoidal decomposition is exact and complete, but not optimal

Trapezoidal Decomposition:

Quadtree Decomposition:

Approximate Cell Decomposition

recursively subdivides each mixed obstacle/free (sub)region into four quarters...

Quadtree:

Further Decomposition

recursively subdivides each mixed obstacle/free (sub)region into four quarters...

Quadtree:

Quadtree Decomposition:

Even Further Decomposition

Again, use a graph-search algorithm to find a path from the start to goal

Quadtreeis this a complete path-planning algorithm?

i.e., does it find a path when one exists ?

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

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

• Compute an attractive force toward the goal

Potential Field Method

C-obstacles

Attractive potential

Potential Field Method

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

• Compute a repulsive force away from obstacles

• Compute a repulsive force away from obstacles

Potential Field Method

• Repulsive Potential

• Sum of Potential

Potential Field Method

C-obstacle

Attractive potential Repulsive potential

Sum of potentials

• After get total potential, generate force field (negative gradient)

• Let the sum of the forces control the robot

To a large extent, this is computable from sensor readings

Equipotential contours

Negative gradient

Total potential

Potential Field Method

Potential Field Method

• 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

Pros:

• 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

Cons: