程序代写代做代考 Robot Motion Planning

Robot Motion Planning

Configuration Spaces

q=(q1,…,qn)
Configuration Space
q1
q2
q3
qn

Definition
A robot configuration is a specification of the positions of all robot points relative to a fixed coordinate system

Usually a configuration is expressed as a “vector” of position/orientation parameters

reference point
Rigid Robot Example
3-parameter representation: q = (x,y,q)
In a 3-D workspace q would be of the form (x,y,z,a,b,g)

x
y
q
robot
reference direction
workspace

Articulated Robot Example
q = (q1,q2,…,q10)

q1

q2

Protein example

Configuration Space of a Robot
Space of all its possible configurations
But the topology of this space is usually not that of a Cartesian space

C = S1 x S1

Configuration Space of a Robot
Space of all its possible configurations
But the topology of this space is usually not that of a Cartesian space

C = S1 x S1

Configuration Space of a Robot
Space of all its possible configurations
But the topology of this space is usually not that of a Cartesian space

C = S1 x S1

What is its Topology?
(S1)7xR3

q1

q2

Structure of Configuration Space
It is a manifold
For each point q, there is a 1-to-1 map between a neighborhood of q and a Cartesian space Rn, where n is the dimension of C
This map is a local coordinate system called a chart.
C can always be covered by a finite number of charts. Such a set is called an atlas

Example

reference point
Case of a Planar Rigid Robot
3-parameter representation: q = (x,y,q) with q Î [0,2p). Two charts are needed
Other representation: q = (x,y,cosq,sinq)
c-space is a 3-D cylinder R2 x S1
embedded in a 4-D space

x
y
q
robot
reference direction
workspace

Rigid Robot in 3-D Workspace
q = (x,y,z,a,b,g)

Other representation: q = (x,y,z,r11,r12,…,r33) where r11, r12, …, r33 are the elements of rotation matrix R:
r11 r12 r13
r21 r22 r23
r31 r32 r33
with:
ri12+ri22+ri32 = 1
ri1rj1 + ri2r2j + ri3rj3 = 0
det(R) = +1

The c-space is a 6-D space (manifold) embedded
in a 12-D Cartesian space. It is denoted by R3xSO(3)

Parameterization of SO(3)
Euler angles: (f,q,y)

Unit quaternion:
(cos q/2, n1 sin q/2, n2 sin q/2, n3 sin q/2)
1  2  3  4

x
y
z

f
x
y
z

x
y
z

q

x

y

z
y

Metric in Configuration Space
A metric or distance function d in C is a map
d: (q1,q2) Î C2  d(q1,q2) > 0
such that:
d(q1,q2) = 0 if and only if q1 = q2
d(q1,q2) = d (q2,q1)
d(q1,q2) < d(q1,q3) + d(q3,q2) Metric in Configuration Space Example: Robot A and point x of A x(q): location of x in the workspace when A is at configuration q A distance d in C is defined by: d(q,q’) = maxxÎA ||x(q)-x(q’)|| where ||a - b|| denotes the Euclidean distance between points a and b in the workspace Specific Examples in R2 x S1 q = (x,y,q), q’ = (x’,y’,q’) with q, q’ Î [0,2p) a = min{|q-q’| , 2p-|q-q’|} d(q,q’) = sqrt[(x-x’)2 + (y-y’)2 + a2] d(q,q’) = sqrt[(x-x’)2 + (y-y’)2 + (ar)2] where r is the maximal distance between the reference point and a robot point q q’ a Notion of a Path A path in C is a piece of continuous curve connecting two configurations q and q’: t : s Î [0,1]  t (s) Î C s’ ® s Þ d(t(s),t(s’)) ® 0 q 1 q 3 q 0 q n q 4 q 2 t(s) Other Possible Constraints on Path Finite length, smoothness, curvature, etc… A trajectory is a path parameterized by time: t : t Î [0,T]  t (t) Î C q 1 q 3 q 0 q n q 4 q 2 t(s) Obstacles in C-Space A configuration q is collision-free, or free, if the robot placed at q has null intersection with the obstacles in the workspace The free space F is the set of free configurations A C-obstacle is the set of configurations where the robot collides with a given workspace obstacle A configuration is semi-free if the robot at this configuration touches obstacles without overlap Disc Robot in 2-D Workspace Rigid Robot Translating in 2-D CB = B A = {b-a | aÎA, bÎB} a1 b1 b1-a1 Rigid Robot Translating in 2-D CB = B A = {b-a | aÎA, bÎB} a1 b1 b1-a1 Linear-Time Computation of C-Obstacle in 2-D (convex polygons) O(n+m) Rigid Robot Translating and Rotating in 2-D C-Obstacle for Articulated Robot Free and Semi-Free Paths A free path lies entirely in the free space F A semi-free path lies entirely in the semi-free space Remark on Free-Space Topology The robot and the obstacles are modeled as closed subsets, meaning that they contain their boundaries One can show that the C-obstacles are closed subsets of the configuration space C as well Consequently, the free space F is an open subset of C. Hence, each free configuration is the center of a ball of non-zero radius entirely contained in F The semi-free space is a closed subset of C. Its boundary is a superset of the boundary of F Notion of Homotopic Paths Two paths with the same endpoints are homotopic if one can be continuously deformed into the other R x S1 example: t1 and t2 are homotopic t1 and t3 are not homotopic In this example, infinity of homotopy classes q q’ t1 t2 t3 Connectedness of C-Space C is connected if every two configurations can be connected by a path C is simply-connected if any two paths connecting the same endpoints are homotopic Examples: R2 or R3 Otherwise C is multiply-connected Examples: S1 and SO(3) are multiply- connected: - In S1, infinity of homotopy classes - In SO(3), only two homotopy classes