3 ESTIMATION OF THE MOTION FIELD OF THE REFERENCE POINT 121

FIGURE 4.2

Motion of a rigid object.

/

!

t r

r t

a , n

~'"'"/:i

s r

!

!

/

!

i !

!

i

zw

Xw< ;

Yw

A reference point of a moving object is defined as a 3-D point such that

1. Its position relative to the object does not change over time.

2. The point has a purely translational motion and no rotation.

3. With the constraint that the translational motion of the reference point is smooth, its

distance from the centroid of the object is kept as small as possible.

The definition of the reference point consists of two important aspects. First, the choice of

the reference point should be such that its motion is smooth and therefore predictable.

Second, because the tracking and grasping will be designed with respect to the reference point,

the point should be kept close to the centroid of the object in order to achieve stable grasping.

The rotational motion of the object can be described as follows. Let rffs rar be a coordinate

frame attached to the object centered at the reference point with each axis being parallel to

the corresponding sides of the hexahedral object (Figure 4.2). When the object moves, the

orientation of the coordinate frame changes (with respect to the WCF) according to (see also

[163)

with

[fir gr ar-] __f~[ff s r a r] (4.20)

0 --(93 (92

~'~ -- (93 0 --(91

(4.21)

--092 (91 0

where co

= [(91 (92 0-)3] T

is the angular velocity.

3 ESTIMATION OF THE MOTION FIELD OF THE REFERENCE POINT

3.1 Formulation of the 3-D Reference Point Problem

The three assumptions that a reference point should satisfy can be described mathematically

as follows. Let (x~, y~, z~) (j = 1, 2,..., or be the coordinates of 3-D feature points in the

objects in the WCF. We assume these points can be observed by each camera. Since rnrsra r

r r r T

is the coordinate frame whose origin is at the reference point xr= [Xw

Yw Zw],

by

122 CHAPTER 4 /VISUALLY GUIDED TRACKING AND MANIPULATION

assumption (1), there exist

kjl, kj2, kj3ER 1

such that

Xw Xw kjl

_ZUwJJ Z w kj3

(4.22)

holds for any time t. Differentiating (4.22) and using (4.20), the 3-D motion of the feature

point is described by

i u;] F]

Xw Xw XUw j X w

9

9 uj

LzwJ LzT]

Zw

(4.23)

Using (4.1), Eq. (4.2) can be written in the CCFi as

Xci

where ~ =

RiR ~.

Clearly, ~ is skew symmetric. Hence

0

--(~)3 (~-)2

~'~ -- (-793 0 --(J)l

-- (-~)2 (-~)1 0

(4.25)

In assumption (3), the smoothness of the motion is interpreted as the 3-D velocity of the

reference point changing slowly over any two adjacent frames. Let any two adjacent frames

be denoted by k = 1, 2. Then it is required that

,~i(1) = ,~,(2) (4.26)

and the objective function

f/

2

*~" *r .r .r

J2 -(zci(1))2

[(Xci(1) -- xc,(2)) z +

(y~i(1) -- vc/(2)) 2]

(4.27)

is minimized. Furthermore, the requirement that the reference point be as close to the

centroid as possible can be satisfied by minimizing

L 2

= ~ ~ " Zc,(k)) ]

y~(k)) + (Zci(k) -

(4.28)

J1 k=x (z~/~)) 2 [(Xc~(k) -

x~,(k)) 2 + (yci(k) - ~ 2 ~ ~ 2

where weighting factors z~i(1) and z~i(2) indicate that the object is paid greater attention

when it moves closer to the i th camera.

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