Gradient Projection Methods for Constrained Image-based Visual Servo - Visual Servoing via Advanced Numerical Methods - page 248

where we can note the use of the Jacobian pseudo-inverse. Classical methods
propose generally to use simply the transpose of the Jacobian, leading to q =

( ∂ ∂ q ) T

−

∇ Φ V Φ . Since the pseudo-inverse provides the least-square solution, the re-
sulting artificial force is the most efficient one at equivalent norm.
Considering now several minimization problems V i = V i

Φ i ,where

Φ

i are different

parameterizations. The global cost function can be written:

∑ i γ i V i

V =

Φ i ,

(14.29)

where the scale factors γ i are used to adjust the relative influence of the different

forces. The force realizing a trade-off between these constraints is thus:

∑ i γ i ∂Φ i

†

∑ i γ i g i

∇ Φ i V i g =

Φ i = Φ i .

(14.30)

∂ q

As to the final control law, the gradient g is used as the second task. It has thus
to be projected onto the null space of the first task. Substituting q 1 = J 1 y 1 d and
P 1 =( I

J 1 J 1 ) into (14.12), the complete control law is finally −

q d = q 1 − κ P 1 g . (14.31)

is too small, the gradient
force may be too small to respect the constraints. Besides, if

The choice of the parameter κ is very important. If κ

κ is too high, some

overshoot can occur in the computed velocity.

14.3 Gradient Projection Method for Avoiding Joint Limits

In this section we describe how the gradient projection method can be used to avoid
reaching joint limits while performing visual servo tasks. The mathematical devel-
opment below is based on the work reported in [4]. We present a set of simulation
results that illustrate the performance of the method.

14.3.1 Basic Algorithm Design

The cost function proposed in [4] for joint-limit avoidance is defined directly in the
configuration (joint) space. It reaches its maximal value at the joint limits, and is
constant far from the limits, so that the gradient is zero. The robot lower and upper
joint limits for each axis i are denoted q min
i

and q max
i

, and the robot configuration q

is said to be admissible if, for all i ,wehave q min
i

q max
i

≤ q i ≤ .

Activation thresholds, q min
li

and q max

li are defined in terms of the joint limits so
that when the joint value q i crosses an activation threshold the cost function begins
to grow rapidly. These activation levels can be defined in terms of the joint limits,
the interval q i = q max
i

q mi i , and the fraction of the interval over which the cost
function should remain nearly constant

−