Download pdf - 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

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Introduction Motivation Model-Free Approach Results

C S K F

World-Space Configuration-Space


C S K F



C S K F

f−1(x, y) 7→ (q1, q2)

f (q1, q2) 7→ (x, y)

Inverse Kinematics

Forward Kinematics



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


RW

model-basedminimal data

model-freedata driven

Denavit-Hartenberg

Nearest-Neighbour Methods

Body Scheme Learning

Artificial Neural Networks

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WM-F?

Robot with unknownkinematic structure

Old, bent robot

Robot with unknownactuator models

Unknown whichconfigurations lead to (self-)collisions


D A

Zora


D A

Zora


D A

AR-Toolkit Markers

Zora


D A

Zora


D A

ZoraZoraSend Joint Pos

Observe 3D Pos


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

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Nearest Neighbour Method

Locally Weighted Regression Method

Planning

4 R

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Planning

4 R


N-NM

p(q, x)

Join

tPos

itio

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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


N-NM

p(q, x)

Join

tPos

itio

n

World Position

World Neighbours

x

F

s = arg mins∈S

∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2

2






N-NM

p(q, x)

Join

tPos

itio

n

World Position

Nearest Neighbour

x

F

s = arg mins∈S

∣∣∣∣∣∣xtarget − xs∣∣∣∣∣∣2

2






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






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





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Planning

4 R


LW RM

p(q, x)

Join

tPos

itio

n

World Position

Joint Neighbours

x

Nearest WorldNeighbour


LW RM

p(q, x)

Join

tPos

itio

n

World Positionx


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


LW RM

p(q, x)

Join

tPos

itio

n

World Positionx

Distance-Weighted

AAssume that small jointmovements move end-effector ona straight line


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)+


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)

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Planning

4 R


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


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


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


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

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CW V E

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

Acc

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Number of samples

Nearest Neighbour - Inverse Kinematics: Accuracy

6 DoF4 DoF2 DoF

NN: Independent of joint count when workspace volume constant


CW V E

LW R

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Acc

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Number of samples

Linear Weighted Regression - Inverse Kinematics: Accuracy

6 DoF4 DoF2 DoF

LWR: Increased accuracy with more joints


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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


S Z

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0 500 1000 1500 2000 2500 3000 3500 4000

Acc

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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


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A R R

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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)


M-F P


M-F P

Self-CollisionEnvironment

Roadmap avoids obstacles

Resulting path short in bothspaces


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

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

Acc

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Sample (Sorted by Accuracy)

Comparison by Sample: Locally Weighted Regression vs. Nearest Neighbour 3-DoF

Locally Weighted RegressionNearest Neighbour

J N N E A

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Acc

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Number of Joint Neighbours

LWR: Changing Number of Joint Neighbours (6-DoF)

500 samples 750 samples

1000 samples1250 samples1500 samples