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approach guarantees convergence, however, it does not take visibility constraints
into account and image features might leave the camera's field of view.
In [63] a motion generation approach called visual motion planning has been
proposed to plan optimal image paths for mobile robots under motion and visibil-
ity constraints. The constraints on the motion of the robot along with the field of
view limits are described in form of a number of equalities and inequalities. An op-
timization problem is then solved numerically using Lagrange Multipliers to obtain
optimal image paths minimizing a given weighted sum cost function (here kinetic
energy). The proposed approach has been applied only to mobile robots moving in
2D and 3D environments.
To pose the problem of path-planning for visual servoing as an optimization prob-
lem some researchers have introduced various parameterizations of camera trajecto-
ries. A polynomial parametrization of the scaled camera paths has been proposed in
[14] where the translational path is linearly interpolated and Cayley's rotation repre-
sentation is employed to rationally parameterize the rotation paths. This allows the
distance of the image trajectories from the boundary of image for a single path to
be easily calculated as the root of some polynomials. Hence, an optimization prob-
lem is then formulated to maximize the distance to the boundary of the image with
respect to all parameterized paths. By following the planned image path, the cam-
era follows a straight line in the workspace in the absence of calibration errors. In
presence of calibration errors, the camera does not follow a straight line but moves
along a different curve whose distance from the planned line grows as the calibration
errors increase.
In [12] an optimal path-planning approach is proposed which allows one to
consider constraints on the camera's field of view, workspace and joint limits,
in the form of inequalities, together with the objective of minimizing trajectory
costs including spanned image area, trajectory length, and curvature. A polynomial
parametrization is devised to represent all the camera paths connecting the initial
and desired locations (up to a scale factor) through an object reconstruction from
image measurements and, if available, the target model. Occlusion constraints and
collision avoidance for the whole robot's body cannot be represented (in the form of
inequality constraints) in their formulation. Moreover, the devised optimization is
nonconvex which may lead to multiple feasible regions and multiple locally optimal
solutions within each region and, hence, it makes it very difficult to find the global
optimal solution across all feasible regions.
In a similar work [10], a general parameterizations of trajectories from the ini-
tial to the desired location is proposed via homogeneous forms and a parameter-
dependent version of the Rodrigues formula. The constraints are modeled using
positivity conditions on suitable homogeneous forms. The solution trajectory is ob-
tained by solving a linear matrix inequality (LMI) test which is a convex optimiza-
tion. The proposed approach allows one to maximize some desired performances
such as distance of features from the boundary of the image, camera's distance from
obstacles, and similarity between the planned trajectory and a straight line.
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