Embed Size (px)
DESCRIPTION
Commonly, the kinematic function of robotic manipulators is derived from the robot model analytically and can be represented in closed form. However, there are cases in which a model is not apriori available. We propose an approach that enables an autonomous robot to estimate the kinematic and inverse kinematic function on-the-fly directly from self-observations. As observations, we sample pairs of randomly chosen joint configurations and the resulting world positions. For approximating the kinematic and inverse kinematic function, we use instance-based learning techniques, such as Nearest Neighbour (NN) and Locally Weighted Regression (LWR). The sampled pairs not only contain information about the kinematics, but also implicitly encode the connectivity and reachability both in the configuration and world space. The robot can take advantage of this information to build roadmaps with a combined cost function. We present an analysis of our approach as well as the results obtained from experiments on a real robot and from simulation. We show in our talk, that with our approach, it becomes possible to accurately control robots with unknown kinematic models of various complexity and joint types from little data obtained through self-observation.
Citation preview
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
