3.2 Obstacles and the Configuration Space

Let’s define a problem of determining a continuous mapping, $c:[0,1]\to Q$ such that no configuration in the path causes a collision between the robot and an obstacle.

A configuration space obstacle $Q{O}_i$ is defined to be the set of configurations where a robot intersects and obstacle $W{O}_i$ in the workspace.

$Q{O}_i=q\in Q|R(q)\cap W{O}_i!=\varnothing$ The free space is the set of configurations where the robot does not intersect any obstacles $Q_{free}$.

a free path is a continuous mapping $c:[0,1]\to Q_{free}$.

3.2.1 Circular Mobile Robot

Consider a robot that is a circle and has no rotational orientation. given some radius its configuration is easy to define around some (x,y) at its center point. Thus its configuration space is ${\mathbb{R}}^2$

There is a good visualization showing how c-space changes wrt. the robot, and how the workspace is different than the robot.

3.2.2 Two-joint Planar Arm

The two-joint arm has a different configuration space than the circular mobile robot, since its rigid links would self-collide in the center of the configuration. The two-joint arm’s configuration is a circle with a smaller circle cut out of the center - denoted $T^2$.

Here is a visualization of the work space and the configuration space of a two link arm.