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.
$$ V_b (t) = [ Ad_{X^{-1} X_{desired}}] V_{desired}(t) + K_d X_{error}(t) + K_i \int_0^t X_{error}(t) dt $$
Vb is the twist ==> Vb = (angular velocity, linear velocity) ===> (6, 1) Matrix
X is the SE(3) representation consisting of (Rotation, Position) ===> (4 x 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 Xerror(t)