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.

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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) ?


1 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.


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