Solution:

When the mobile polygon is allowed to rotate, the configuration-space obstacles are no longer polygonal.

The surfaces of configuration space obstacles are curved in the orientation dimension. To handle rotations, enlarged polygons which enclose the mobile part over a range of orientations is employed.

The configuration space obtained is no longer a convex polygon but an enlarged a polygon that is curved and encloses the mobile part over a range of orientations.

The advantage of the rotational approach of generating the configuration space obstacle is, it converts the problem of finding a path for a part that rotates and translates into a simple problem of finding a path for an enlarged part that only translates.

Since the rotated polygonal part will be contained in a circular sector of radius d and angle ∆ɸ ≥ (ɸ₁ -ɸ₀). If a path is found, then the mobile part can assume any orientation in the interval {ɸ₀, ɸ₁} at each point along the path.