I am currently reading "A Mathematical Introduction to Robotic Manipulation" by R. Murray, Z. Li, and S. Sastry to learn about kinematics of rigid bodies. Given a homogeneous transformation $g_{ab}(t)\in\mathrm{SE}(3)$ that expresses the pose of body frame $b$ relative to the inertial spatial frame $a$, the spatial velocity $\hat{V}_{ab}^s \in se(3)$ is defined as $\hat{V}_{ab}^s = \dot{g}_{ab}g_{ab}^{-1}$.
This definition makes sense, and is useful because its homogeneous coordinates $(\hat{V}_{ab}^s)^\vee = \begin{bmatrix}v_{ab}^s \\ \omega_{ab}^s\end{bmatrix}$ has geometric meaning: $v_{ab}^s$ is the velocity of a point in the body frame passing through the origin of the spatial frame, and $\omega_{ab}^s$ is the axis of rotation between the two frames as expressed in the spatial frame.
Now, I'm trying to find an equation that describes the acceleration (both the linear and angular accelerations) of the body frame in terms of the homogeneous transformation $g_{ab}(t)$. If $q_b$ is a point fixed in the body frame, then expressed in the spatial frame, it becomes $q_a(t) = g_{ab}(t)q_b$. Following a similar derivation as $\hat{V}_{ab}^s$, if I differentiate $q_a(t)$ twice, I get $\ddot{q}_a(t)=\ddot{g}_{ab}(t)q_b=\ddot{g}_{ab}(t)g_{ab}(t)^{-1}g_{ab}(t)q_b = \ddot{g}_{ab}(t)g_{ab}(t)^{-1} q_a(t)$.
My questions are:
- Does the quantity $\ddot{g}_{ab}(t)g_{ab}(t)^{-1}$ have any significance similar to how $\dot{g}_{ab}(t)g_{ab}(t)^{-1}$ is the spatial velocity?
- Is there such a thing as a spatial acceleration $\hat{A}_{ab}^s \in se(3)$ whose homogeneous coordinates represent linear and angular accelerations of the body frame?
- How can I describe the linear and rotational acceleration of the body frame in terms of $g_{ab}(t)$?