Image-based ControlConvergence issues

CMPUT 610

Winter 2001

Martin Jagersand

Notes:

Please hand in project proposal on Thu. See me if there are any problems. Some on-line search engines for papers:

– http://www.cora.whizbang.com/– http://citeseer.nj.nec.com/cs

Recall: Visual Servoing

Observed features: Motor variables: Local linear model:

y = y1 y2 . . . ym[ ]T

x = x1 x2. . . xn[ ]T

É y = J É x

Recall again: Visual Servoing Steps

1. Solve:

2. Update and move:

3. Read actual visual move

4. Update Jacobian:

yã à yk = J É x

xk+1 = xk + É x

Jêk+1 = Jêk +É xTÉ x

(É y à JêkÉ x)É xT

Recall again: Visual Servoing Steps

1. Solve:

2. Update and move:

3. Read actual visual move

4. Update Jacobian:

yã à yk = J É x

xk+1 = xk + É x

Jêk+1 = Jêk +É xTÉ x

(É y à JêkÉ x)É xT

How to specify a visual task?

Visual specifications

Point to Point task “error”:

yãE = yã à y0 y0

yã

y0

E =y1...y16

2

4

3

5

ã

ày1...y16

2

4

3

5

0

Why 16 elements?

Visual specifications 2

Point to Line

Line:

E pl(y; l) = yl ál l

yr álr

ô õ

l l = y3 à y1

y4 à y2

ô õ

yl =y5

y6

ô õ

y1

y2

ô õ

y3

y4

ô õ

ParallelComposition example

E (y ) =wrench

y - y4 7

y - y2 5

y • (y y )8 3 4

y • (y y )6 1 2

(plus e.p. checks)

Serial CompositionSolving whole real tasks

Task primitive/”link”

1. Acceptable initial (visual) conditions

2. Visual or Motor constraints to be maintained

3. Final desired condition Task =

A = (E init;M;E final)

A1A2...Ak

“Natural” primitive links

1. Transportation Coarse primitive for large movements <= 3DOF control of object centroid Robust to disturbances

2. Fine Manipulation– For high precision control of both position and

orientation– 6DOF control based on several object features

Example: Pick and place type of movement

3. Alignment??? To match transport final to fine manipulation initial conditions

More primitives

4. Guarded move– Move along some direction until an external

contraint (e.g. contact) is satisfied.

5. Open loop movements: When object is obscured Or ballistic fast movements Note can be done based on previously estimated

Jacobians

Solving the puzzle…

Summary

Servoing alone does not solve whole tasks– Parallel composition: Stacking of visual constraints

to be simultaneously satisfied– Serial composition: Linking together several small

movements into a chain of continuous movements

Convergence?

Convex function? Convex domain? Convergent method?