Gradient Projection Methods for Constrained Image-based Visual Servo - Visual Servoing via Advanced Numerical Methods - page 243

The task function approach [14] provides a formalism by which the motion
needed to accomplish the secondary task can projected onto the null space of the
primary task, yielding a total motion that satisfies the primary task while making
progress toward the secondary task. Here, we use the method of gradient projection
[13], and encode constraints using an objective function whose gradient is projected
onto the null space of the primary task. While better methods exist for individual,
specific problems, the purpose of this chapter is to give a unified treatment of gradi-
ent projection methods as they can be used to solve a variety of constrained visual
servo problems.
The methods described below have been developed in the context of redundant
robots [15, 16, 12] and visual servo control [8, 4, 10, 9]. Our development follows
that given in these references. The chapter is organized as follows. In Section 14.2,
we review null space methods, and in particular the gradient projection method,
for controlling the motion of redundant robots. Following this, we describe how
gradient projection methods can be used to avoid joint limits (Section 14.3), prevent
occlusion (Section 14.4), and keep an object of interest in the field of view (Section
14.5).

14.2 Exploiting Redundancy by Projecting onto Null Spaces

A system is said to have redundancy if it has more DOF than are necessary to per-
form a given task. Since 6 DOF are enough for full position and orientation control
of an end effector, a manipulator with 7 DOF can be said to be redundant. Systems
with 6 or fewer DOF are regarded to have redundancy if the particular task to be
performed requires fewer than 6 DOF.
In this section, we describe the mathematical formalism by which redundancy
can be exploited to make progress toward secondary goals while performing a pri-
mary task. In particular, we describe below task decomposition and gradient projec-
tion methods [12, 15, 5].

14.2.1 Task-decomposition Approach

Most complicated tasks given to a manipulator can be formulated in such a way that
the task is broken down into several subtasks with a priority order. Each subtask is
performed using the DOF that remain after all the subtasks with higher priority have
been implemented. Control problems for redundant manipulators can be approached
via the subtask method by regarding a task to be performed by a redundant manip-
ulator as the task with highest priority.
Consider a manipulator with n DOF. The joint variable of the i th joint is q i ,

( i =

2 q n ].

Assume that the first subtask can be described properly by an m 1 -dimensional vec-
tor, y 1 , which is a function of q :

, 3 ,..., n ). The manipulator configuration is denoted by the vector q =[ q 1 , q 2 ,...,

y 1 = f 1 ( q ) . (14.1)