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CS 326A: Motion Planning. Non-Holonomic Motion Planning. Coordination for Multiple Robots (Notes for HW#2). n robots R1, …, Rn, with configuration spaces C1, …, Cn, sharing the same workspace Problem: Plan coordinated motion so that each robot achieves its own goal configuration. - PowerPoint PPT Presentation
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CS 326A: Motion Planning
Non-Holonomic Motion Planning
Coordination for Multiple Robots (Notes for HW#2)
n robots R1, …, Rn, with configuration spaces C1, …, Cn, sharing the same workspace
Problem: Plan coordinated motion so that each robot achieves its own goal configuration.
Centralized planning: Plan the coordinated motion in C1xC2x…xCn (but very high dimensional space)
Decoupled planning: Plan the motion of each robot ignoring the other robots; then coordinate their motions so that no two robots collide
Prioritized planning: Plan the motion of one robot ignoring the other robots; then plan the trajectory of a second robot in its configurationxtime space treating the first robot as a moving obstacle; then plan the trajectory of a third robot …
Coordination Space 2 robots R1 and R2 2 paths: i : si [0,1] Ci (i=1,2) 2-D coordination space
Generalize to n robots n-D coordination space
0 1
1
s1
s2
Variants of Decoupled Planning
#1: Coordinate the n paths in n-D coordination space
#2:Coordinate paths of R1 and R2 in a 2-D coordination diagram ( path of “R1-R2”) , then coordinate paths of R1-R2 and R3 in 2-D coordination diagram, etc…
s1
Under-Actuated Robots
Fewer controls than dimensions in configuration space What is a degree of freedom: number of dimensions of
C-space (global) or number of controls (local)?
How can m controls make it possible to span a C-space with n > m dimensions?
By exploiting mechanics properties:- Rolling-with-no-sliding contact (friction), e.g.: car, bicycle, roller skate- Conservation of angular momentum: satellite robot, under-actuated robot, cat- Others: submarine, plane, object pushing
Why is it useful?- Fewer actuators (less weight)- Design simplicity- Convenience (think about driving a car with 3 controls!)
Example: Car-Like Robot
yy
xxConfiguration space is 3-dimensional: q = (x, y, )
But control space is 2-dimensional: (v, ) with |v| = sqrt[(dx/dt)2+(dy/dt)2]
L
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
dx sin – dy cos = 0
Example: Car-Like Robot
q = (x,y,)q’= dq/dt = (dx/dt,dy/dt,d/dt)dx sin – dy cos = 0 is a particular form of f(q,q’)=0
A robot is nonholonomic if its motion is constrained by a non-integrable equation of the form f(q,q’) = 0
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
dx sin – dy cos = 0
yy
xx
L
Example: Car-Like Robot
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
dx sin – dy cos = 0
yy
xx
L
Lower-bounded turning radius
How Can This Work?Tangent Space/Velocity
Space
x
y
(x,y,)
(dx,dy,d)
(dx,dy)
yy
xx
L
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
x
y
(x,y,)
(dx,dy,d)
(dx,dy)dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
yy
xx
L
How Can This Work?Tangent Space/Velocity
Space
Lie Bracket
Maneuver made of 4 motions
X (t)
Y
-X
-Y
Lie Bracket
Maneuver made of 4 motionsFor example:
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
X: Going straight
Y: Turning, angle
0,sin,cos X
tan,sin,cos
LY
T
T
Maneuver made of 4 motionsFor example:
X (t)
Y
-X
-Y
[X,Y] (t2 )
Lie bracket
Lie Bracket
X: Going straight
Y: Turning, angle
0,sin,cos X
tan,sin,cos
LY
T
T
[X,Y] = dY.X – dX.Y
X1/x X1/y X1/
dX = X2/x X2/y X1/
X2/x X2/y X2/
X (t)
Y
-X
-Y
[X,Y] (t2 )
Lie bracket
Lie Bracket
[X,Y] Lin(X,Y) the motion constraint is nonholonomic
Tractor-Trailer Example
4-D configuration space 2-D control/velocity space two independent
velocity vectors X and Y U = [X,Y] Lin(X,Y) V = [X,U] Lin(X,Y,U)
Nonholonomic Path Planning Approaches
Two-phase planning (path deformation):• Compute collision-free path ignoring nonholonomic
constraints• Transform this path into a nonholonomic one• Efficient, but possible only if robot is “controllable”• Need for a “good” set of maneuvers
Direct planning (control-based sampling):• Use “control-based” sampling to generate a tree of
milestones until one is close enough to the goal (deterministic or randomized)
• Robot need not be controllable• Applicable to high-dimensional c-spaces
Path DeformationHolonomic path
Nonholonomic path
Type 1 Maneuver
Allows sidewise motion
dq
dq
(x1, y1, 0+)
(x3, y3, 0)
(x2, y2, 0+)
(x0, y0, 0)
d
(x,y)
q
CYL(x,y,,)
= 2tand = 2(1/cos1) > 0
(x,y,)
When 0, so does d and the cylinder becomes arbitrarily small
Type 2 Maneuver
Allows pure rotation
Combination
Coverage of a Path by Cylinders
x
y
+q q’
Path Examples
Drawbacks of Two-phase Planning
Final path can be far from optimal
Not applicable to robots that are not locally controllable (e.g., car that can only move forward)
Reeds and Shepp Paths
Reeds and Shepp Paths
CC|C0 CC|C C|CS0C|C
Given any two configurations,the shortest RS paths betweenthem is also the shortest path
Example of Generated Path
Holonomic
Nonholonomic
Path Optimization
Nonholonomic Path Planning Approaches
Two-phase planning (path deformation):• Compute collision-free path ignoring nonholonomic
constraints• Transform this path into a nonholonomic one• Efficient, but possible only if robot is “controllable”• Need for a “good” set of maneuvers
Direct planning (control-based sampling):• Use “control-based” sampling to generate a tree of
milestones until one is close enough to the goal (deterministic or randomized)
• Robot need not be controllable• Applicable to high-dimensional c-spaces
Control-Based Sampling
Previous sampling technique: Pick each milestone in some region
Control-based sampling:1. Pick control vector (at random or not)2. Integrate equation of motion over short
duration (picked at random or not)3. If the motion is collision-free, then the
endpoint is the new milestone
Tree-structured roadmaps Need for endgame regions
Example
1. Select a milestone m2. Pick v, , and t3. Integrate motion from m new milestone m’
dx/dt = v cosdy/dt = v sin
ddt = (v/L) tan
| <
Example Indexing array: A 3-D grid is placed over the configuration space. Each milestone falls into one cell of the grid. A maximum number of milestones is allowed in each cell (e.g., 2 or 3).
Asymptotic completeness: If a path exists, the planner is guaranteed to find one if the resolution of the grid is fine enough.
Computed Paths
Computed Paths
max=45o, min=22.5o
Car That Can Only Turn Left
max=45o
Tractor-trailer
Application
SummaryTwo planning approaches:
Path deformation: Fast but paths can be far from optimal. Restricted to “controllable” robots.
Control-based sampling: Can generate better paths, but slower. Can be scaled to higher dimensional space using probabilistic sampling techniques (next lecture)
