In the next few videos, we consider the following problem:

Given a desired path theta-of-s, find the time-optimal time scaling along this path, considering the dynamics of the robot and torque limits at the robot joints. Minimum-time motions can be used to maximize the productivity of a robot. You could imagine trying to optimize other criteria, like the amount of energy consumed by the actuators, but in these next few videos we will focus on time-optimal trajectories.

Recall that the dynamics of a robot can be written as M times theta-double-dot plus a velocity-product term plus a gravity term equals the joint force and torque vector tau. Here I've written the velocity-product term using the Christoffel symbol matrix Gamma to emphasize that it is quadratic in the joint velocity vector.

Keeping in mind that we are only interested in the dynamics of the robot when it is on the path theta-of-s, we can rewrite theta-dot and theta-double-dot as a function of the derivatives of the path with respect to s and the derivatives of s with respect to time. Plugging these expressions into the dynamics, we get the expression shown here. Since the path is given in advance, the derivatives of theta with respect to s are also known in advance, and only s, s-dot, and s-double-dot are variables. We can therefore write this equation as the vector equation m-of-s times s-double-dot plus c-of-s times s-dot-squared plus g-of-s equals tau. Each of m-of-s, c-of-s, and g-of-s is a vector function of s, where c-of-s times s-dot-squared is a velocity-product term, g-of-s is the gravity term, and m-of-s plays the role of a mass. Some elements of the m-vector may be negative, however.

This equation is the dynamics of the robot when it is restricted to move along the path theta-of-s. This equation says nothing about the dynamics when the robot is off the path.

Now that we've expressed the dynamics in terms of the single path parameter s, as opposed to the joint vector theta, we have to consider the limits on the forces or torques that the robot's actuators can produce. The limits at the i'th joint can be written as tau_i is greater than tau_i-min and less than tau_i-max. For example, tau_i-min could be minus-five newton-meters and tau_i-max could be plus-five newton-meters. But in general the limits are a function of theta and theta-dot. In particular, the maximum torque that can be produced by an electric motor typically decreases as the velocity increases, until eventually it becomes zero.

Remembering that we can express theta and theta-dot in terms of s and s-dot when the robot is restricted to the path, we can rewrite the actuator limits as a function of s and s-dot. If we substitute the i'th component of the path-restricted dynamics in for tau_i, we get these constraints. The i'th actuator therefore places limits on the possible accelerations s-double-dot along the path when the robot is at the state (s, s-dot). To determine the i'th actuator's limits on s-double-dot, we subtract c_i times s-dot-squared and g_i from all three expressions, then divide by m_i. Since m_i could be positive or negative, there are two possible cases: if m_i is positive, then L_i, the lower limit on s-double-dot, and U_i, the upper limit on s-double-dot, are given by these equations. If m_i is negative, then L_i and U_i are given by these equations. These equations tell us the maximum and minimum accelerations s-double-dot along the path that joint i will allow at the state (s, s-dot).

If we calculate L_i and U_i for all the joints, then L of (s, s-dot), the minimum feasible acceleration s-double-dot at the state (s, s-dot), is just the maximum of the lower limits over all the joints. Similarly, U of (s, s-dot), the maximum feasible acceleration s-double-dot, is just the minimum of the upper limits over all the joints. We can now express the constraints on the robot's acceleration along the path compactly as s-double-dot is greater than L of (s, s-dot) and less than U of (s, s-dot). At some states, L may actually be greater than U, and in this case there is no feasible acceleration that keeps the robot on the path. In other words, if the robot found itself at such a state, it would immediately have to leave the path; the actuators are not strong enough to keep the robot on the path. This typically happens when the robot is moving at high speed.

Now that we have reduced the actuator constraints to s-double-dot is greater than L and less than U, we can mathematically express the time-optimal scaling problem as follows:

Given a path theta-of-s, the initial state where both s and s-dot are zero, and the final state where s is one and s-dot is zero, we want to find a monotonically increasing twice-differentiable time scaling that (a) satisfies the terminal conditions and (b) minimizes the total travel time capital T while satisfying the actuator constraints.

This problem lends itself to a nice graphical interpretation, as we will see in the next video.