Path-Planning for Visual Servoing: A Review and Issues - Visual Servoing via Advanced Numerical Methods - page 189

Projective invariance . Malis [45] proposed an image-based path-planning ap-
proach in an invariant space defined through a projective transformation. The basic
idea of using projective invariance is to create a task function which is invariant to
camera intrinsic parameters and only depends on the position of the camera with
respect to the observed object and on its 3D structure. This allows one to generate a
path for a feature vector in the invariant space (independent of camera's intrinsic pa-
rameters) which, when followed, results in a straight line path for the camera in the
workspace. The visibility of the features is (partially) achieved using a motorized
zooming mechanism available on the vision system.
The main advantage of direct path-planning in image space is the independence
of such approaches from camera calibration and/or object model. On the other hand,
since the planning is done directly in the image space, robot/physical constraints
cannot be handled through such approaches and these techniques are shown to be
ineffective in complex visual servoing scenarios.

11.3.2 Optimization-based Path-planning

Planning optimal paths has absorbed a great amount of interest in robotics com-
munity. In a visual servoing task, there might be many different paths, which when
followed, will result in successful accomplishment of the same task. This motivates
optimization techniques aimed at finding the optimal path with respect to various
costs such as distance from the image boundary, length of the path traversed by the
robot, energy expenditure, etc .
In an early work [60], a path-planning framework is proposed based on the con-
cept of perceptual control manifold (PCM) defined on the product of the robot's
joint space and the space of all image features related to a target object. PCM can
be considered as a mapping which relates a robot configuration to the vector of im-
age features visible at that configuration. Given the model of the camera, the object,
and the robot kinematic model, the PCM needs to be computed only once (in an
eye-to-hand configuration) and is then applicable to any manipulation task. Con-
straints such as the camera's field of view and the robot joint limits and/or physical
obstacles are mapped into the PCM to yield a subset of PCM as the feasible solu-
tion space. This mapping could be quite time consuming considering the number of
constraints and the robot's degrees of freedom. Various optimization criteria such
as minimum velocity, minimum interception time, and minimum robot movement
have been considered to plan optimal paths in the feasible subset of the PCM. The
proposed approach has been considered for the task of intercepting a moving target
(with a known trajectory) using the visual feedbacks obtained from a fixed camera
which simultaneously views both the robot's end-effector and the moving target.
In [50] closed-form collineation paths corresponding to minimum energy and
minimum acceleration camera paths are planned in the image space. The proposed
strategy is then generalized to the case where a number of relay (intermediate)
images are available in addition of the initial and desired images. The proposed