Motion Planning

Artificial Potential Fields

Artificial Potential Fields are another approach to guiding robots into obstacle-filled environments. The basic idea here is to construct a smooth function over the extent of the configuration space, which has high values when the robot is near to an obstacle and lower values when it’s further away.

Attractive Potential Field

An attractive potential function, $f_a(x)$, can be constructed by considering the distance between the current position of the robot x and the desired goal location $x_g$

$f_a(x)=k(||x-x_g|{|}^2)$ where k is a constant scaling parameter

Repulsive Potential Field

Now in addition to getting to the goal, we also want the robot to avoid the obstacles in the environment. A repulsive function in the plane, $f_r(x)$, can be constructed based on a function, $p(x)$, that returns the distance to the closest obstacle from a given point in configuration space, x.

$f_r(x)=n(\frac{1}{p(x)}-\frac{1}{d_0}{)}^2$ if $p(x)\le d_0$ $f_r(x)=0$ if $p(x)$ > $d_0$ n is a constant scaling parameter $d_0$ is a parameter that controls the influence of the repulsive potential

Getting to the goal

Gradient-Based Control Strategy While robot position is not close enough to goal

• Choose direction of robot velocity based on the gradient of the artificial potential field: $v\propto -\nabla f(x)=-(\frac{\delta f(x)}{\delta x_1},\frac{\delta f(x)}{\delta x_2})$
• Choose an appropriate robot speed, $\Vert v\Vert$

To summarize the algorithm, the robot continually evaluates the gradient of the artificial potential field and steps in that direction until it is close enough to the desired goal. One way to visualize the action of this algorithm or control strategy is to think about a marble rolling down the energy surface towards the goal location.

Issues with Local Minima

the simple gradient based control strategy may or may not converge the goal depending upon where the robot starts. If we go back to our marble analogy, these local minima correspond to local valleys other than the goal, where the robot can get stuck.