CS 326A: Motion Planning

CS 326A: Motion Planningai.stanford.edu/~latombe/cs326/2007/index.htm

Non-Holonomic Motion Planning

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 generate 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)