Learning Kinematics from direct Self-Observation using Nearest-Neighbour Methods
-
Upload
cijat -
Category
Technology
-
view
1.424 -
download
1
description
Transcript of Learning Kinematics from direct Self-Observation using Nearest-Neighbour Methods
L K S-O N-NM
Lionel Ott, Hannes Schulz
University of Freiburg, ACS
November 2008
O
1 I
2 M
3 M-F A
4 R
O
1 I
2 M
3 M-F A
4 R
Introduction Motivation Model-Free Approach Results
C S K F
World-Space Configuration-Space
Introduction Motivation Model-Free Approach Results
C S K F
World-Space Configuration-Space
Introduction Motivation Model-Free Approach Results
C S K F
f−1(x, y) 7→ (q1, q2)
f (q1, q2) 7→ (x, y)
Inverse Kinematics
Forward Kinematics
World-Space Configuration-Space
Introduction Motivation Model-Free Approach Results
S E
Represent f/f−1 as closed mathematicalformula derived from robot model
Calibration:Determine parameters of f , f−1
BenefitsAnalytical solutionHigh accuracy
DrawbacksRequires a CAD model of the robotRequires a lot of a priori knowledgeabout the individual jointsEnvironment needs to be representedseparately
Introduction Motivation Model-Free Approach Results
RW
model-basedminimal data
model-freedata driven
Denavit-Hartenberg
Nearest-Neighbour Methods
Body Scheme Learning
Artificial Neural Networks
O
1 I
2 M
3 M-F A
4 R
Introduction Motivation Model-Free Approach Results
WM-F?
Robot with unknownkinematic structure
Old, bent robot
Robot with unknownactuator models
Unknown whichconfigurations lead to (self-)collisions
Introduction Motivation Model-Free Approach Results
D A
Zora
Introduction Motivation Model-Free Approach Results
D A
Zora
Introduction Motivation Model-Free Approach Results
D A
AR-Toolkit Markers
Zora
Introduction Motivation Model-Free Approach Results
D A
Zora
Introduction Motivation Model-Free Approach Results
D A
ZoraZoraSend Joint Pos
Observe 3D Pos
Introduction Motivation Model-Free Approach Results
N
q Joint-space coordinates (q1, q2, . . . , qn)
x World-space coordinates (x1, x2, x3)
S = (s1, s2, . . . sm)= (< q1, x1 >, . . . , < qm, xm >)visually observed samples
O
1 I
2 M
3 M-F A
Nearest Neighbour Method
Locally Weighted Regression Method
Planning
4 R
O
1 I
2 M
3 M-F A
Nearest Neighbour Method
Locally Weighted Regression Method
Planning
4 R
Introduction Motivation Model-Free Approach Results
N-NM
p(q, x)
Join
tPos
itio
n
World Position
World Query
x
F
s = arg mins∈S
∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2
2
Achoose closest sample to thetarget in world-space
go to the selected position qs
Pindependent of number of joints
the more samples the better
Introduction Motivation Model-Free Approach Results
N-NM
p(q, x)
Join
tPos
itio
n
World Position
World Neighbours
x
F
s = arg mins∈S
∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2
2
Achoose closest sample to thetarget in world-space
go to the selected position qs
Pindependent of number of joints
the more samples the better
Introduction Motivation Model-Free Approach Results
N-NM
p(q, x)
Join
tPos
itio
n
World Position
Nearest Neighbour
x
F
s = arg mins∈S
∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2
2
Achoose closest sample to thetarget in world-space
go to the selected position qs
Pindependent of number of joints
the more samples the better
Introduction Motivation Model-Free Approach Results
N-NM
p(q, x)
Join
tPos
itio
n
World Position
q = arg maxq
p(q|x)
x
q
F
s = arg mins∈S
∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2
2
Achoose closest sample to thetarget in world-space
go to the selected position qs
Pindependent of number of joints
the more samples the better
Introduction Motivation Model-Free Approach Results
N-NM
p(q, x)
Join
tPos
itio
n
World Position
q = arg maxq
p(q|x)
x
q
F
s = arg mins∈S
∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2
2
Achoose closest sample to thetarget in world-space
go to the selected position qs
Pindependent of number of joints
the more samples the better
O
1 I
2 M
3 M-F A
Nearest Neighbour Method
Locally Weighted Regression Method
Planning
4 R
Introduction Motivation Model-Free Approach Results
LW RM
p(q, x)
Join
tPos
itio
n
World Position
Joint Neighbours
x
Nearest WorldNeighbour
Introduction Motivation Model-Free Approach Results
LW RM
p(q, x)
Join
tPos
itio
n
World Positionx
Introduction Motivation Model-Free Approach Results
LW RM
p(q, x)
Join
tPos
itio
n
World Positionx
q = arg maxq
p(q|x)
q
AAssume that small jointmovements move end-effector ona straight line
Introduction Motivation Model-Free Approach Results
LW RM
p(q, x)
Join
tPos
itio
n
World Positionx
Distance-Weighted
AAssume that small jointmovements move end-effector ona straight line
Introduction Motivation Model-Free Approach Results
LW RM
p(q, x)
Join
tPos
itio
n
World Positionx
q = arg maxq
p(q|x)
q
A L
w0 +w1x1 +w2x2 +w3x3 = f−1(x)
L-S S
w = arg min ||Xw − Q ||22⇒ w = (XT X)−1XT Q
⇒ w = (XT X)+XT Q
Use SVD for pseudo-inverse(XT X)+
Introduction Motivation Model-Free Approach Results
M J A
q = arg maxq1...qn
p(q1, . . . , qn |x)
= arg maxq1...qn
p(q1|x)p(q2|x, q1) · · · p(qn |x, q1, . . . , qn−1)
≈
arg maxq1
p(q1|x)
arg maxq2
p(q2|x, q1)
. . .
arg maxqn
p(qn |x, q1, . . . , qn−1)
O
1 I
2 M
3 M-F A
Nearest Neighbour Method
Locally Weighted Regression Method
Planning
4 R
Introduction Motivation Model-Free Approach Results
E: P
So far,
We can move to arbitrary positions
No control over path to position
. Plan path between positions
. Need waypoints in between
C R PCAD model of the robot
CAD model of theenvironment
Sample virtualrepresentation
M-F R PGathered samples implicitlyrepresent environment androbot model
No further sampling needed
Introduction Motivation Model-Free Approach Results
E: P
So far,
We can move to arbitrary positions
No control over path to position
. Plan path between positions
. Need waypoints in between
C R PCAD model of the robot
CAD model of theenvironment
Sample virtualrepresentation
M-F R PGathered samples implicitlyrepresent environment androbot model
No further sampling needed
Introduction Motivation Model-Free Approach Results
M-F P P
Roadmap: Connect pairs of observed sampleswith world-distance < dmax
s0: NN of start point in config-space
sn: NN of end point in world space
Plan path via A∗, minimizing
n−1∑i=0
(α
∣∣∣∣∣∣xi − xi+1∣∣∣∣∣∣
2 + β∣∣∣∣∣∣qi − qi+1
∣∣∣∣∣∣22
)
s0
sn
squared distance results in smallest possible steps
use world-space distance as heuristic
α = 1/dmax , β: normalize by maximum distance possible
Introduction Motivation Model-Free Approach Results
M-F P P
Roadmap: Connect pairs of observed sampleswith world-distance < dmax
s0: NN of start point in config-space
sn: NN of end point in world space
Plan path via A∗, minimizing
n−1∑i=0
(α
∣∣∣∣∣∣xi − xi+1∣∣∣∣∣∣
2 + β∣∣∣∣∣∣qi − qi+1
∣∣∣∣∣∣22
)
s0
sn
squared distance results in smallest possible steps
use world-space distance as heuristic
α = 1/dmax , β: normalize by maximum distance possible
O
1 I
2 M
3 M-F A
4 R
Introduction Motivation Model-Free Approach Results
CW V E
N N
0
50
100
150
200
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Acc
ura
cy (
mm
)
Number of samples
Nearest Neighbour - Inverse Kinematics: Accuracy
6 DoF4 DoF2 DoF
NN: Independent of joint count when workspace volume constant
Introduction Motivation Model-Free Approach Results
CW V E
LW R
0
50
100
150
200
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Acc
ura
cy (
mm
)
Number of samples
Linear Weighted Regression - Inverse Kinematics: Accuracy
6 DoF4 DoF2 DoF
LWR: Increased accuracy with more joints
Introduction Motivation Model-Free Approach Results
J T E
I J T
0
50
100
150
200
250
300
Acc
ura
cy (
mm
)
Method
Accuracy vs. Robot Complexity (500 Samples)
LWR (Model A)
LWR (Model B)
NN (Model A)
NN (Model B)
M A 5 rotational joints
M B 2 rotational and 3 prismaticjoints
Introduction Motivation Model-Free Approach Results
S Z
E D
0
20
40
60
80
100
120
140
160
180
0 500 1000 1500 2000 2500 3000 3500 4000
Acc
ura
cy (
mm
)
Sample (Sorted by Accuracy)
Comparison by Sample: Locally Weighted Regression vs. Nearest Neighbour
Locally Weighted Regression 3-DoFLocally Weighted Regression 4-DoFLocally Weighted Regression 6-DoF
Nearest Neighbour 3-DoFNearest Neighbour 4-DoFNearest Neighbour 6-DoF
Due to dimensionality and covered space, both methodsworsen with DoF
LWR always better than NN
Introduction Motivation Model-Free Approach Results
R-W R
A R R
0
50
100
150
200
250
Acc
ura
cy (
mm
)
Method
Real World Data: Locally Weighted Regression vs. Nearest Neighbour
Camera noise levelLocally Weighted Regression
Nearest Neighbour
57 mm accuracy with 250 samples,
Significantly better than Nearest Neighbour(paired-samples t-test, p < 0.0001, df = 1999)
Introduction Motivation Model-Free Approach Results
M-F P
Introduction Motivation Model-Free Approach Results
M-F P
Self-CollisionEnvironment
Roadmap avoids obstacles
Resulting path short in bothspaces
Introduction Motivation Model-Free Approach Results
C
Model-free learning of kinematic function fromself-observation
Shown to work on (almost) arbitrary robots
Robust in presence of (real-world) noise
Motion planning in world and configuration space
C NN LWR
0
50
100
150
200
250
300
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Acc
ura
cy (
mm
)
Sample (Sorted by Accuracy)
Comparison by Sample: Locally Weighted Regression vs. Nearest Neighbour 3-DoF
Locally Weighted RegressionNearest Neighbour
J N N E A
45
50
55
60
65
70
75
80
10 15 20 25 30 35
Acc
ura
cy (
mm
)
Number of Joint Neighbours
LWR: Changing Number of Joint Neighbours (6-DoF)
500 samples 750 samples
1000 samples1250 samples1500 samples