Path-Planning for Visual Servoing: A Review and Issues - Visual Servoing via Advanced Numerical Methods - page 187

on a Jacobian estimator was used to track the planned image trajectories without
using any knowledge of the system or camera calibration.
Park and Chung proposed an image space path-planning approach for an eye-to-
hand system using uncalibrated stereo cameras in a vision-based grasping scenario
[54]. They generate a number of intermediate views of the robot's gripper along a
straight line between the initial image and the final desired image in the projective
space with the help of epipolar geometry and without using any 3D information re-
garding either the gripper or the target object. These intermediate views constitute
the desired image trajectories. The robot is then controlled along the image trajec-
tories using the IBVS technique presented in [22]. When followed by the robot,
the planned trajectories allow the robot's gripper to track a straight line in the 3D
workspace and through out its motion a selected set of features on the gripper are
kept in the camera's field of view.
Projective homography . To avoid explicit computation of feasible camera paths
which relies on the knowledge of the camera calibration and target model, a number
of approaches have been developed using the projective geometry [23] relationship
established between the initial and desired images. Working in projective space al-
lows one to partially parameterize the Euclidean displacement of the camera without
explicit reconstruction of the Euclidean components.
Projective homography matrix has been employed in the context of path-planning
for visual servoing. Projective homography captures the relationship between the
images taken from different views of the same scene. Given the projective homoge-
neous coordinates p =( u 1) T of a 3D point P in the current

and desired images, respectively, the projective homography matrix G , also called
collineation matrix, is defined (up to an scale

1) T and p ∗ =( u ∗ , v ∗ ,

, v ,

α g )as

g p = Gp ∗ .

α (11.3)

The projective homography matrix can be estimated form the knowledge of several
features such as points, lines, and contours matched between two images [13], [26],
and [46].
In [50] a calibration-free path-planning approach is proposed which consists of
interpolating for the collineation matrix G between the initial and desired images
to obtain closed-form analytical collineation paths. The image feature trajectories
are then derived and followed using an IBVS technique. The proposed approach
guarantees convergence to the desired location, however, the convergence does not
hold in presence of visibility constraints such as field of view limits. This approach
has been extended in [58] to take visibility constraints into account by guiding the
image of an arbitrary selected reference point on the target along a straight line in
the image which guarantees that the reference point remains in the camera's field
of view. However, the camera will not follow a straight line anymore and the other
features may still leave camera's field of view. A depth modulation approach has
been proposed to keep the visibility of other features by controlling the camera
backwards along an optical ray whenever a feature reaches the borders of camera's
field of view.