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Ideas from optimal control theory have been employed to devise image trajec-
tories for visual servoing under visibility constraints. Planning shortest path for a
differential drive robot (DDR) maintaining the visibility of a landmark using a cam-
era with limited field of view has been considered in [3]. It is shown that the set
of shortest (optimal) paths for this system consist of curve segments that are either
straight-line segments or that saturate the camera's field of view. The latter cor-
respond to exponential spirals known as T-curves. In [42] these shortest paths are
followed using a switched homography-based visual servo controller. The controls
that move the robot along these paths are devised based on the convergence of the
elements of the homography matrix relating the current image to the final desired
image. In a recent work [28], a complete motion planner for a DDR is proposed in
which optimal curve segments obtained in [3] are used as motion primitives to de-
vise locally optimal paths in an environment cluttered with obstacles. The necessary
and sufficient conditions for the feasibility of a path for the DDR in the presence of
obstacles and with visibility constraints (
i.e.
, sensing range and field of view limits)
are also provided. In their proposed planner, occlusions due to workspace obstacles
are not considered and the obstacles are assumed to be transparent.
In [56] the set of optimal curves obtained in [3] are extended and also described
in the image space, so as to enable their execution using an IBVS controller di-
rectly in the image space. Feedback control along these optimal paths in the image
is achieved through a set of Lyapunov controllers, each of which is in charge of
a specific kind of maneuver. Nonetheless, the complete characterization of all the
shortest paths and their analytic descriptions remain unsolved for a DDR.
Although the above optimization-based path-planning techniques provide a better
insight into the complexity of the problem and feasible optimal paths, they are more
or less limited to simple scenarios and systems. Introducing general robot/physical
constraints greatly adds to the complexity of the optimization problem and, hence,
accounting for such constraints in the above frameworks would greatly increase the
time complexity of such techniques.
11.3.3
Potential Field-based Path-planning
In the field of robot path-planning, potential field method has been proposed as
a promising local and fast obstacle avoidance strategy to plan safe and real-time
motions for a robot in a constrained environment [33]. The main idea is to construct
an artificial potential field defined as the sum of attractive potentials, pulling the
robot towards the desired location, and repulsive potentials, pushing the robot away
from various constraints such as the obstacles or robot's joint limits. A driving force
computed along the negated gradient of the potential field moves the robot towards
the goal location.
Mezouar and Chaumette [49] introduced robust image-based control based on
the Potential Field method for a robotic arm with eye-in-hand configuration. In their
proposed approach, two types of constraints are considered: field of view and robot's
joint limits. To obtain valid robot trajectories, the motion of the robot is first planned
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