How do you calculate this integral term in this PI Controller Formula?

Question

This formula is the PI Control given in Eqn. 11.6, Pg. 419 of Chapter 11 in book Modern Robotics by Kevin M Lynch and Frank C Park.

Here, Vb is the twist ==> Vb = (angular velocity, linear velocity) ===> (6, 1) Matrix
X is the SE(3) representation consisting of (Rotation, Position) ===> (4, 4) Matrix
(Xe is the error term)

This book heavily uses Screw Theory, Product of Exponentials and Lie Algebra.

Can anyone tell me how to integrate the integral term ? What is the formula to integrate the SE(3) Matrix Xe(t) ?

Answer 1

The term $X_e$ is not a matrix in SE(3) but a twist, as defined in the paragraph following the equation where it states that "... the configuration error $X_e(t)$ is not simply $X_d(t)-X(t)$, since it does not make sense to subtract elements of SE(3)."

It's actually the twist that, if followed for a unit time step, would shift $X$ to $X_d$. As stated in the book at the end of the paragraph: "The se(3) representation of this twist, expressed in the end-effector frame, is $[X_e]=log(X^{-1}X_d)$."

Now to answer your question: To calculate the integral term, you simply need to integrate the twist. Since it is a coordinate vector with 6 elements, it is directly integrable over time.