A robot controller typically accepts a steady stream of desired robot configurations, reads joint sensors to determine the robot's actual configuration, and updates the actuator commands to follow the desired configuration. This process can happen thousands of times a second. A robot configuration as a function of time is called a trajectory. We can write a trajectory as theta of t, where the time t goes from zero to capital T. This figure shows a trajectory for a robot with two degrees of freedom. We could also plot this trajectory directly in the configuration space, also known as the C-space. As the time t increases from zero to capital T, the configuration follows the path shown here. In some cases it is desirable to separate the C-space path from the speed at which it is followed. For example, we might plan a geometric path for a mobile robot to avoid obstacles on the floor, without worrying about how fast the path is followed. We define a path to be a curve in configuration space as a function of a path parameter, s, that goes from zero to one. As s increases from zero, the robot moves from the start configuration at theta-of-zero to the end configuration at theta-of-one. A path can be turned into a trajectory by defining a function s of t, which maps the time range zero to capital T to the path parameter range zero to one. This function is called a time scaling. The time scaling controls how fast the path is followed. Now, with a trajectory theta of s of t, the time derivative theta-dot is determined by the chain rule to be d-theta d-s times s-dot. The acceleration theta-double-dot is determined by the product rule and the chain rule to be d-theta d-s times s-double-dot plus d-squared-theta d-s-squared times s-dot-squared. Since the dynamics depend on theta-double-dot, for the dynamics to be well defined, the second derivatives of both theta-of-s and s-of-t must exist. In this chapter we consider the problem of planning paths and trajectories without considering obstacles. In Chapter 10, on motion planning, we address the case of obstacles in the environment. Let's start by planning a path for a robot arm in its C-space. The simplest type of path is a straight-line path from an initial configuration theta-start to a final configuration theta-end, as described by this equation. As s goes from 0 to 1, the configuration goes from theta-start to theta-end. A straight-line path in joint space is shown here, for a 2R robot with 180 degrees of motion about its first joint and 150 degrees of motion about its second joint. The path can also be visualized in the workspace, as shown here. Note that the endpoint of the robot does not follow a straight line. If we prefer a straight-line motion of the end-effector in Cartesian space, we can define X to be the coordinates of the end-effector and define a straight-line path as shown here. Then we have to use inverse kinematics to solve for the robot configuration at each point along the path. As you can see here, some straight-lines in Cartesian space cannot be executed, as they pass outside the workspace. One advantage of planning straight-line motions in joint space is that the joint limits are usually independent of each other, so the set of feasible joint configurations is convex, unlike the workspace. A straight line between two points in a convex space always remains inside the space. We can also plan a path between two rigid-body frames represented in SE(3). We could try directly extending our straight-line definition in joint space, but of course this does not make sense; we there is no meaning to subtracting two elements of SE(3). Let's find the screw path where the frame follows a constant twist from X_start to X_end. First, let's express the frame X_end relative to X_start. Each of X_start and X_end is implicitly defined in a space frame {s}, so by our subscript cancellation rule we find that X_start-inverse times X_end is the configuration of the {end} frame in the {start} frame. The log of this is the little-se(3) representation of the twist, expressed in the {start} frame, that takes the {start} frame to the {end} frame in unit time. So our formula for the path is X-of-s equals X_start times the matrix exponential of s times the log of X_start-inverse times X_end as s goes from zero to one. The matrix exponential is multiplied on the right since the twist is expressed in the {start} frame, not the space frame. The path parameter s determines how far we follow the twist that takes the {start} frame to the {end} frame. The final screw path can be visualized as shown here. This is a "straight-line path" in the sense that the twist is constant throughout the motion. Another type of path between X_start and X_end is one that decouples the rotation and translation. The origin of the frame follows a straight line, and the rotation is about an axis fixed in the frame. For this type of path, the position coordinates p follow a straight-line path for coordinates, as discussed earlier, while the orientation satisfies a formula similar to that for a constant twist, except now just for the orientation components. These two types of paths are the most natural "simple" paths between two frames. Now that we can plan paths between two configurations, in the next video we study common time scalings that turn a path into a trajectory.