#### 9.1 and 9.2. Point-to-Point Trajectories (Part 2 of 2)

This video introduces third-order polynomial, fifth-order polynomial, trapezoidal, and S-curve time scalings to turn a path into a trajectory.

In the previous video, we learned that a trajectory can be represented as theta-of-s-of-t. We also found expressions for simple paths theta-of-s. In this video, we study time scalings s-of-t that turn a path into a trajectory.

One simple time scaling is a third-order polynomial time scaling, where s is a cubic function of time. The time scaling is defined by the four coefficients of time, a_zero through a_three. The time derivative of the time scaling is shown here. To solve for the coefficients, we apply the four terminal constraints, which say that s is zero at time zero and one at time capital T, and that s-dot is zero at times zero and capital T for motions that begin and end at rest. Solving for the four coefficients using these four constraints, we get these values. Now we can plot s as a function of t, as well as s-dot and s-double-dot. Notice that s is a cubic, s-dot is a parabola, and s-double-dot is a line. s-dot begins and ends at zero, but s-double-dot jumps discontinuously to six over capital-T-squared at time zero.

If we would prefer a smoother motion, where the acceleration at the beginning and end of the motion are zero, we can use a fifth-order polynomial time scaling. A fifth-order polynomial gives us two more coefficients to choose, and we use them to satisfy two more terminal constraints, that the acceleration is zero at times zero and capital T. Now s-double-dot is a cubic, allowing s-double-dot to be zero at the beginning and end of the motion.

Another popular time scaling in motion control is the trapezoidal time scaling, named for its s-dot plot, shown here. First the robot follows a constant acceleration s-double-dot, then it coasts at a constant s-dot, then it follows a constant deceleration to rest.

Like the third-order polynomial time scaling, this time scaling has discontinuous jumps in the acceleration. If this is undesirable, we can use an S-curve time scaling, shown here as an s-dot plot. An S-curve has seven segments. In the first segment, the robot follows a constant jerk. Jerk is the time derivative of acceleration. Then it follows a constant acceleration, followed by a constant negative jerk, followed by a coasting period at constant velocity. Then the robot slows down, symmetrically to the first three segments. The acceleration is zero at the beginning and end of the motion.

Of course, the actual speed and acceleration of the robot at any time depends on the distance of the path and the total duration of the motion capital T, not just the form of the time scaling.

In the next video we will see how to control the shape of a robot's path by having it pass through a set of timed via points.