Here's a more complicated example, once again, we are considering a robot that can translate in the plane. But now, it can also rotate, this means our robot now has three degrees of freedom. Since a rotational degree of freedom has been added to its initial two translational degrees. We can denote the configuration of our robot with a tuple, tx, ty, and theta, where tx and ty still denote the position of a reference point in the plane, and theta denotes the applied rotational angle in degrees. Once again, when we introduce obstacles into the workspace, we can think about the set of configurations that are limited. In this case, the configuration space has three dimensions, and the configuration space obstacles can be thought of as three dimensional regions in this space. This movie shows a depiction of the surface of the configuration space obstacle corresponding to the obstacles shown in the previous figure. The vertical access corresponds to the rotation theta, while the other two horizontal axes correspond to the translational parameters tx and ty Note again that in this figure, the surface that we are visualizing corresponds to the surface of the configuration space obstacle. As before, the basic problem in motion planning is to come up with a trajectory between a start point and an end point that avoids all the configuration space obstacles. This movie shows a robot moving through the space, avoiding all of the obstacles. In this second movie, we are visualizing the trajectory of the robot through configuration space as a red line. Notice how this red line snakes in and around the configuration space obstacle avoiding penetration as it moves from the start configuration to the end of configuration. It is important to understand that this idea of a configuration space where we associate coordinates with the configuration of the robot and then reason about configurations that are allowed and disallowed, and think about the motion of the robot in terms of trajectories of a point through configuration space is actually very general. Here, for example, is a plane a robot with six revolute links In principle, we can think of its motion in terms of its trajectory of a point, moving through a six dimensional configuration space. If we wanted to, we could introduce obstacles in the space and reason about the corresponding configuration obstacles. I invite you to ponder what this configuration space would actually look like.