Path Planning 2
Another option to find short but pleasingly smooth paths through a navigation space is to use vector
fields. In a vector field, objects in the environment create fields around them, like gravitational or magnetic
fields. For example, an obstacle would have a repulsive field, while the goal would have an attractive field.
Figure 1: a repulsive field
Figure 2: a uniform field
These vector fields are additive, meaning that we can add them together to create more complicated behaviors. For example, to exert a force to the center of the path, we might add together two vector repulsive
fields from the walls of the hallway to look like Figure 3.
Add to that a uniform field from left to right, and the robot will move down the hallway while avoiding
the walls. Another example demonstrates a robot moving towards an attractive goal while avoiding the
repulsive obstacles in Figure 4.
For more complicated problems, you can break the space down into cells, where each cell has its own
uniform path through, then do graph search in the cells. One such result is in Figure 5.
Path Planning for Arm Motion
We’ve been learning these path planning algorithms in two dimensions for navigation, but these techniques
are also often applied to robotic arms trying to reach a position in a complicated, obstructed space. For
example, any of you who have worked on cars or plumbing are familiar with the idea that just knowing
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Figure 3: Wall avoidance
what position and orientation your hand has to be in is insufficient - you also have to get your hand there,
around the obstacles.
To think about this, suppose we’ve done our inverse kinematics, and we know what joint angles we
want our arm to have. However, there are obstacles that we have to work around - in what way should
we bend the joints so that the arm gets into position without running into the obstacle? Well, suppose the
arm has four joints. This means that we can represent the current position (and desired position) of the arm
as a vector in four-dimensional space. Some of the positions in this four-dimensional configuration space
are fine, while others are blocked by the obstacles. Well, navigating in this four-dimensional space around
obstacles is very, very similar to navigating in a two-dimensional space around obstacles, and all of these
techniques work just as well.
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Figure 4: Moving to a goal while avoiding obstacles
Figure 5: From Lindemann and LaValle, 2005
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