Let $\xi_0(t), \xi_1(t) \in \mathbb R^6$ with $\xi_i = [\xi_{i0}, ..., \xi_{i5}]^T$.
Let
$$\exp(\xi_i) = \exp\begin{bmatrix} 0 & -\xi_{i2} & \xi_{i1} & \xi_{i3}\\ \xi_{i2} & 0 & -\xi_{i0} & \xi_{i4}\\ -\xi_{i1} & \xi_{i0} & 0 & \xi_{i5}\\ 0 & 0 & 0 & 1 \end{bmatrix} \in SE(3)$$
where $SE(3)$ is the Special Euclidean Group of dimension 3.
Given:
$$\xi_0(t),~~~~ \frac{d}{dt} \xi_0(t),~~~~ \frac{d^2}{dt^2} \xi_0(t)$$
and
$$\xi_1(t),~~~~ \frac{d}{dt} \xi_1(t),~~~~ \frac{d^2}{dt^2} \xi_1(t)$$
Let $T(t) := \exp(\xi_0(t)) \exp(\xi_1(t))$
Let a rigid object be travelling along the trajectory $T(t)$.
What is the object's body-frame acceleration $a$ and angular velocity $\omega$?