CSCE 452/752: Homework 6

Robert tries to implement a potential field to control his robot, but he makes a mistake with the obstacle part of the potential function. Instead of the standard equation, he uses this equation: \[ U^{(i)}_{\rm obst}(x) = \begin{cases} \frac{1}{2} \eta \left( \frac{1}{Q_i^*} - \frac{1}{d_i(x)} \right)^2 & {\rm if\quad} d_i(x) > Q_i^* \\ 0 & {\rm if\quad} d_i(x) \leq Q_i^* \\ \end{cases} \] Will Robert's variation on the potential field method work as well as the standard method discussed in class?

YES NO

If you answered YES, explain why.

If you answered NO, explain why not and describe how the robot would behave.

This potential function will NOT work as effectively as the standard one.

When the robot is far from the obstacle (the $d_i(x) > Q_i^*$ case), the potential increases as $d_i(x)$ increases, creating a (mild) attractive component. (That is: The robot is pulled toward to the obstacle.)

When the robot is near the obstacle (the $d_i(x) \le Q_i^*$ case), the potential is 0, so the obstacle plays no role in the robot's movements. Whether there is a collision or not depends on the influence from other terms of the potential function.

Note that the inversion in the formula, $\frac{1}{Q_i^*} - \frac{1}{d_i(x)} $ instead of $ \frac{1}{d_i(x)} - \frac{1}{Q_i^*} $, is irrelevant because the difference is eliminated by squaring.

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