Path-Planning for Visual Servoing: A Review and Issues - Visual Servoing via Advanced Numerical Methods - page 188

If the camera calibration is known, one could derive further information regarding
the camera transformation. In particular, one can compute the Euclidean homogra-
phy matrix H (up to a scale α h )as

α h H = K − 1 GK (11.4)

where K is a nonsingular matrix and contains the camera intrinsic parameters. The
Euclidean homography (from a set of planar features) can be decomposed to obtain
the corresponding (Euclidean) camera transformation parameters as

R

n ∗

t
d ∗ ,

⇒ ,

H (11.5)

where R and t denote the translation and rotation from the desired to the current
camera frame, and d ∗ is the distance of the plane containing the features from the
desired camera frame and n ∗ is the normal to the plane expressed with respect to the
desired frame.
Decomposition of Euclidean homography has been employed by some researchers
to plan for image paths corresponding to feasible (yet unknown) camera paths with-
out explicit reconstruction of the camera paths in the Cartesian space.
A shortest path approach has been proposed in [36] which avoids the use of 3D re-
construction by using homography-based partial pose estimation. The proposed ap-
proach moves the in-hand camera directly along the direction (obtained through the
homography decomposition) towards the desired pose in the 3D workspace while
maintaining the visibility of (only) a virtual point located at the origin of the target
object. The virtual point is used to control two degrees of rotation of the camera
(around x

− −

axes) and the third rotation axis (around camera optical axis)
is controlled using the rotation matrix retrieved from homography. This technique
yields a straight line trajectory for the virtual point and, hence, keeps the virtual
point always in the camera's field of view. However, the camera can get too close
to the target so that some features may get lost. Switching between visual servoing
strategies or using repulsive potentials can be employed to avoid such situations,
however, without ensuring straight line trajectories.
In [1] a similar approach has been proposed based on homography decomposition
in which helicoidal shape paths (instead of straight path) are chosen as the reference
path to represent camera translation from the initial position to the desired position.
One should note that since the homography is known only up to an unknown scale,
the actual camera path is not completely known and one can only determine its
shape. However, regardless of the value of unknown scale factor, the entire image
path will remain the same and since the control is defined directly in the image,
the positioning task can be successfully accomplished given a feasible image path.
In [4] a particular decomposition of homography is used to interpolate a path for a
planar object with known model from the initial image to the desired final image.
Given the known object model, the interpolated desired path is then transformed to
a camera path by using 3D reconstruction. The camera path can then be checked for
workspace boundary singularities.

and y