68 CHAPTER 2 / OBSERVER-BASED VISUAL SERVOING

is obtained, where M =

[M r r r Mr]r

1

M2 M3 .

Since M and

Jrob

are

square, J becomes full

rank if both M and

CJro b

are invertible and N is full rank. To check the singularity of M, it

is straightforward to see that

det M =

1 x I Yx

1 x 2 Y2

1 x a Y3

1

9 1

1

X2 Y2

x3 Y3

x4 Y4

1

9 1

1

x3 Y3

x4 Y4

Xl Yl

1

9 1

1

x4 Y4

Xl Yl

X2 Y2

(2.22)

Thus M is invertible because any three feature points are not collinear. Since we are

interested in the robot motion only in the nonsingular region,

CJro b

is invertible. On the other

hand, if p2 + q2 4= 0, the first six rows of N are linearly independent. If p = q = 0, the first

four and the last two rows are linearly independent. Thus N is full rank. Therefore, the image

Jacobian J is proved to be full rank.

Performance improvement by using redundant features is discussed in [17] and [25]. The

smallest and the largest singular values of the image Jacobian play a central role for

performance imprvement.

4 NONLINEAR CONTROL LAW

This section introduces a nonlinear controller and a nonlinear observer. An example of a

two-link direct drive robot is also given.

4.1 Controlled Variable

Our goal is to track the object so as to keep the features of the object at the reference features.

The models of robot, object motion, and camera are given by (2.2), (2.3), and (2.6),

respectively. On the basis of these models, it is natural to adopt the features as the controlled

variables, joint angles and joint velocities as the state, and the joint torque as the input. For

a minimum set of features (n =

m/2),

this selection is appropriate. 2 However, for redundant

features, the system becomes uncontrollable because the features cannot move in

R 2n

arbitrarily. To resolve this problem, one has to solve nonlinear geometric constraints on the

features that represent the rigidness of the object. Since these constraints are difficult to solve,

we linearize the constraints at the reference point and reduce the dimension of the feature

vector to the dimension of the joint space.

Let ~# be a manifold, that is, the set of all admissible features (2.7), and consider a nominal

point

r a

that satisfies ~e =

t(ra).

Define a matrix B as follows:

Bde=f~J(q*,p*)

if n~> m, (2.23)

if

n = m

Note that J is a function of q and p; in this equation, p* and q* are a typical position of the

object and a typical configuration of the robot that satisfy

Scorn(q*)- Sobj(P*)= ra.

The

matrix B is the image Jacobian at the nominal point if the features are redundant. If the

features are minimum, then B is the identity matrix. The controlled variable is defined by

2There is no other choice.

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