single robot motion planning
DESCRIPTION
Single Robot Motion Planning. Liang-Jun Zhang COMP790-058 Sep 22, 2008. Motion planning is the ability for an agent to compute its own motions in order to achieve certain goals. All autonomous robots and digital actors should eventually have this ability. Piano Mover’s Problem. - PowerPoint PPT PresentationTRANSCRIPT
2
Motion planning is the ability for an agent to compute its own motions in order to achieve
certain goals. All autonomous robots and digital actors should eventually have this ability
6
Goal of Motion Planning
• Compute motion strategies, e.g.:– geometric paths – time-parameterized trajectories– sequence of sensor-based motion commands
• To achieve high-level goals, e.g.:– go from A to B without colliding with obstacles– assemble product P– build map of environment E– find object O
8
Basic Problem Statement:
Compute a collision-free path for a rigid or articulated object among static obstacles
Inputs:•Geometry of moving object and obstacles•Kinematics of moving object (degrees of freedom)•Initial and goal configurations (placements)
Output:Continuous collision-free path connecting the initial and goal configurations
9
Types of Path Constraints
• Local constraints – Collision-free paths
• Differential constraints– A car cannot move sideways– Have bound curvature
• Global constraints– Shortest or optimal paths
Path Planning
Motion Planning
11
Outline (Mon & Wed)
• Path planning for a point robot
• Configuration space
• Approximate cell decomposition
• Sampling-based motion planning
13
Visibility Graphs
Introduced in the Shakey project at SRI in the late 60s
Can produce shortest paths in a point robot in 2D
g
s
14
Simple (Naïve) Algorithm1. Install all obstacles vertices in VG, plus the start
and goal positions2. For every pair of nodes u, v in VG3. If segment(u,v) is an obstacle edge then4. insert (u,v) into VG5. else6. for every obstacle edge e7. if segment(u,v) intersects e8. then goto 29. insert (u,v) into VG10.Search VG using BFS (any other graph search
scheme)
15
Complexity
• A point robot in 2D using visibility graphs Simple algorithm: O(n3) time Rotational sweep: O(n2 log n) Optimal algorithm: O(n2) Space: O(n2)
g
s
16
Motion Planning Framework
• Motion planning: a search problem in continuous space
Discretization
Continuous representation
Graph search
17
g
s
Issues With Visibility Graphs
Difficult to extend from point robots to rigid
or articulated robots
A L-shaped robot
18
Outline (Mon & Wed)
• Path planning for a point robot
• Configuration space
• Approximate cell decomposition
• Sampling-based motion planning
20
Example: rigid robot in 2-D workspace
• 3-parameter specification: q = (x, y, ) with [0, 2).– 3-D configuration space
robot
workspace
reference point
x
y
reference direction
21
Configuration Space (C-Space)• The configuration of a moving
object is a specification of the position of every point on the object. – Usually a configuration is expressed
as a vector of position & orientation parameters: q = (q1, q2,…,qn).
• Configuration space – C-space– The set of all possible configurations.– A configuration is a point in C-space.
q=(q1, q2,…,qn)
qq11
qq22
qq33
qqnn
22
Dimension of C-space
• The dimension of a configuration space is the minimum number of parameters needed to specify the configuration of the object completely.
• It is also called the number of degrees of freedom (dofs) of a moving object.
24
Example: rigid robot in 3-D workspace
• q = (position, orientation) = (x, y, z, ???)
• Number of dofs = 6
• Euler angles
xx
y
zz
xxyy
zz
x
y
z
xx
yy
zz
1 2 3 4
25
C = S1 x S1
θ
Φ
Topology of C-Space
• The topology of C is usually not that of a Cartesian space Rn.
0 2
2
θ
Φ
θ
Φ
28
An articulated object is a set of rigid bodies connected at the joints.
An Articulated Robot Puma 560
Number of dofs = 6
29
Obstacles in C-space
Workspace Configuration Space
xyRobot
Start
Goal
Free
Obstacle C-obstacle
A 2D Translating Robot
30
Obstacles in C-space• A configuration q is collision-free, or free, if a
moving object placed at q does not intersect any obstacles in the workspace.
• The free space F is the set of free configurations.
• A configuration space obstacle (C-obstacle) is the set of configurations where the moving object collides with workspace obstacles.
36
Problem: Computing C-obstacles
• Input:– Polygonal moving object translating in 2-D
workspace – Polygonal obstacles
• Output: configuration space obstacles represented as polygons
42
Minkowski Sum
• The Minkowski sum of two sets A and B, denoted by AB, is defined as A B = { a+b | a A, bB }
• Similarly, the Minkowski difference is defined as
A – B = { a–b | aA, bB }
p
q
44
Configuration Space Obstacle• If P is an obstacle in the workspace and M is a
translating object. Then the C-space obstacle corresponding to P is P – M.
P -MObstacle
PRobot
M
C-obstacle
Classic result by Lozano-Perez and Wesley 1979
45
Minkowski Sum of Convex Polygons
• The Minkowski sum of two convex polygons A and B of m and n vertices respectively is a convex polygon A + B of m + n vertices.– The vertices of A + B are the “sums” of
vertices of A and B.
46
Complexity of Minkowksi Sum
• 2D convex polygons: O(n+m) • 2D non-convex polygons: O(n2m2)
– Decompose into convex polygons (e.g., triangles or trapezoids), compute the Minkowski sums, and take the union
• 3-D convex polyhedra: O(nm)• 3-D non-convex polyhedra: O(n3m3)
47
Complexity of Computing C-obstacles
• 3D rigid robots with both translational and rotational DOF– 6D C-space– Arrangement of non-linear surfaces– High combinatorial complexity
• Conclusion– Explicit computation of the boundary of C-obstacle is
difficult and impractical for robots more than 3 DOFs