[MUSIC] In the motion planning problems we've considered so far, we've basically reduced the problem to planning on a graph, where the robot can take on various discrete positions, which we can enumerate and connect by edges. Now in the real world, most of the robots we are going to build can move continuously through space. Configuration space is a handy mathematical and conceptual tool, which was developed to help us think about these kinds of problems in a unified framework. Basically, the configuration space of a robot is the set of all configurations and/or positions that the robot can attain. This slide shows a simple example of a robot that can translate freely in the plane. Here we can quantify the positions that the robot can take on with a tuple composed of two numbers, tx and ty, which denote the coordinates of a particular reference point on the robot, with respect to a fixed coordinate frame of reference. Here are a couple of configurations that this translating robot can take on, along with the associated coordinates. In this case, we would say that our robot has 2 degrees of freedom, and we can associate the configuration space of the robot with the points on the 2D plane, namely these tx, ty coordinates. Now we'll make the story little bit more interesting by introducing fixed obstacles into our model. What these obstacles do is make certain configurations in the configuration space unattainable. This figure shows the tx, ty configurations that the robot cannot attain because of the obstacle. This set of configurations that the robot cannot inhabit Is referred to as a configuration space obstacle. Conversely, the region of configuration space that the robot can attain is referred to as the free space of the robot. On the right-hand side of this figure, we plot the configuration space obstacle, corresponding to the geometric obstacle shown in the left side of the figure. Again, the configuration space obstacle denotes the set of configurations that the robot cannot attain because of collision with the obstacle. Note that the dimensions and shape of the configuration space obstacle are obtained by considering both the obstacle and the shape of the robot. More formally, in this case, the configuration space obstacle is defined by what's known as the Minkowski sum of the obstacle and the robot shape. If we have multiple obstacles in space, we can visualize the union of all of the configuration space obstacles, and we get a picture like this. Again, the configuration of the robot corresponds to a point in the configuration space. And the dark areas correspond to configurations that the robot cannot attain. In this setting, the task of planning a path for our robot corresponds to planning a trajectory through configuration space from the starting configuration to the ending configuration. Here we are showing the motion of the robot through the space that avoids the obstacles, and the corresponding motion of the robot's coordinates in configuration space. Note that by thinking about this problem in configuration space, we are now just planning the path for a point through configuration space, avoiding the configuration space obstacles. All of the geometry of the robot and the obstacles are captured by the configuration space obstacles. This is really the beauty of formulating things in configuration space.