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B) Polar coordinates. 24 2 Robust Visual Control Based on the Epipolar Geometry The Cartesian coordinates for the camera location C2 can be expressed as a function of the polar coordinates d23 and ψ2 (Fig. 4) 2 = x2 + y2 . 5) 2 = (x − x )2 + with ψ12 = φ2 − arctan (e21 /αx ) = φ1 − arctan (e12 /αx ) and d12 1 2 (y1 − y2 )2 . 1. In such a case, the focal length parameter in the x-direction is αx = 1. 3 Epipolar Control Law from Three Views Henceforth, let C1 be the initial camera location, C2 the current camera location and C3 the target camera location.
2 υdb = usm + u pp. 12) utilizes the decoupling matrix that presents a singularity problem when the camera axis of the robot is aligned with the baseline, which generates unbounded translational velocity. In order to pass through the singularity we commute to a direct sliding mode controller when |φ2 − ψ2 | is below a threshold Th . This kind of controller has been studied for output tracking through singularities [67]. 13) where kυ and kω are suitable gains and b (φ2 , ψ2 ) is a function that describes the change in sign of the translational velocity when the state trajectory crosses the singularity.
Becerra and C. 1007/978-3-319-05783-5_2, c Springer International Publishing Switzerland 2014 21 22 2 Robust Visual Control Based on the Epipolar Geometry becomes singular for some state of the robot. Indeed, unbounded velocities eventually appear because the singularity is always reached when the robot moves directly toward the target. The approach in [99] takes into account the non-holonomic nature of a wheeled robot by driving one dimension of the epipoles to zero in a smooth way. However, in order to avoid the singularity, the motion strategy steers the robot away from the target while the lateral error is corrected, and after that, the robot moves backward to the target position.