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To determine the domain of attraction, we determined all the orientations of the robot for which our empirical computation of the time derivative of the Lyapunov function was positive. We then start at the level surface V = c for c very large and slowly decrease c until arriving at a level surface where V˙ < 0 for every point in its interior except at the goal. The volume defined by the interior of the level surface defines the domain of attraction for this 3 degrees of freedom system. Figure 5(a) shows this level surface and nearby points where V˙ > 0.

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Using the modified notation, the signal at any translated position is related to the signal at the goal I(w, x(t)) = I(w − x(t), 0) = I0 (w − x(t)). 3) = K(w)I0 (w − x(t))dw, I and the kernel-projected value at the goal is ξ 0 = ξ at x0 . The dimension n refers to the number of kernels being used such that the kernels represent a function K : R2 → Rn . 3) can be written as ξ= I K(w ¯ + x(t))I0 (w)d ¯ w. 4) show that the the kernel-projected value is the same for the fixed kernel with an image taken after the camera has been moved and a kernel shifted in the opposite direction with the goal image.

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Active Dreaming: The Key to Power by Robert Balthazaar

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