# Jacobian Derivative (Analytical?)

If I understand this correctly, given the Jacobian Derivative $$\dot{J}(q)$$ derived using the method presented from the paper by (Rhee, J. Y., & Lee, B., 2017)

where the algorithm summarized computes:

$$\overline{\underline{\bf{\text{Algorithm 1}}\textrm{ Jacobian and Jacobian Differentiation}}}\\ \begin{array}{lll} 1: & \omega_0^0 \leftarrow\left[0,0,0\right]^T & \\ 2: & z_0^0 \leftarrow\left[0,0,1\right]^T & \\ 3: & \pmb{J}_1 \leftarrow \left[\begin{matrix} z_0^0\times d_n^0 \\ z_0^0\end{matrix}\right] & \\ 4: & \bf{\text{for }} \textrm{i = 2 to n } \bf{\text{do}} & \triangleright \textrm{ Jacobian Computation loop} \\ 5: & \quad \pmb{J}_i = \left[\begin{matrix} z_{i-1}^0 \times d_{i-1,n}^0 \\ z_{i-1}^0\end{matrix}\right] & \\ 6: & \quad \omega_{i-1}^0 \leftarrow w_{i-2}^0 + z_{i-2}^0\dot{q}_{i-1} \\ 7: & \bf{\text{end for}} & \\ 8: & \beta \leftarrow J_{v,i}\dot{q}_i & \\ 9: & \bf{\text{for }} \textrm{i = n to 2 } \bf{\text{do}} & \triangleright \textrm{ Jacobian Differentiation loop}\\ 10: & \quad \dot{z}_{i-1}^0 \leftarrow \omega_{i-1}^0 \times z_{i-1}^0 & \\ 11: & \quad \alpha \leftarrow \left[0,0,0\right]^T & \\ 12: & \quad \bf{\text{for }} \textrm{j = 1 to i-1 } \bf{\text{do}} \\ 13: & \qquad \alpha \leftarrow \alpha + z_{j-1}^0 \times \left(d_n^0 - d_{i-1}^0\right)\dot{q}_j & \\ 14: & \quad \bf{\text{end for}} & \\ 15: & \quad \dot{J}_i = \left[\begin{matrix} \left(\dot{z}_{i-1}^0\right) \times (d_{n}^0- d_{i-1,n}^0) + z_{i-1}^0 \times\left(\alpha + \beta\right) \\ \dot{z}_{i-1}^0 \end{matrix}\right] & \\ 16: & \quad \beta \leftarrow \beta + J_{v,i-1}\dot{q}_{i-1} & \\ 17: & \bf{\text{end for}} & \\ 18: & \dot{J}_1 = \left[\begin{matrix} z_0^0 \times \beta \\ \left[0,0,0\right]^T \end{matrix}\right] & \\ \end{array} \\$$

My question

I want to compute the roll, pitch, yaw accelerations for the second-order differential kinematics of a robot, stating:

$$\ddot{x} = J(q)\ddot{q} + \dot{J}\dot{q}$$

How do I proceed?

Do I approach it similarly to how the analytical jacobian is computed?

$$\Gamma = [r, p, y]^\intercal$$

$$A = \begin{bmatrix} 1 &0 &sin(p) \\ 0 &cos(r) &-cos(p) sin(r)\\ 0 &sin(r) &cos(p) cos(r) \end{bmatrix}$$

$$J_a(q) = \begin{bmatrix} I &0\\ 0 &A^{-1}(\Gamma) \end{bmatrix} J(q)$$

where instead of $$J(q)$$ I use $$\dot{J}(q)$$, and the same with $$[r, p, y]^\intercal$$, I use $$[\dot{r}, \dot{p}, \dot{y}]^\intercal$$ ?

I apologize if this is too elementary, but I am genuinely curious.

• I don't understand what you're trying to accomplish, or how the first part of the question relates to the second part of the equation. You want to find some accelerations.... and what's the problem? You need $J(q)$ and $\dot{J}(q)$, but you've given in your question (lower half) the formula for $J(q)$ and you've given in your question (upper half) the algorithm for finding $\dot{J}(q)$. I can't tell if I'm missing something, or you're asking about something else. Could you please clarify the question? Apr 30 '20 at 19:11
• @Chuck , thanks for replying. I want to convert the geometric representations of $\dot{J}$ to roll pitch yaw (rpy) angular representations: $\dot{J}_a$. How do I do that? The three parts of this question was to provide context to the problem. The first part of this question (the algorithm) shows how I derive $\dot{J}(q)$. The second part of the question shows how one derives the task-space acceleration $\ddot{x}$. And finally the third part shows how one derives the analytical jacobian $J_a$, so that you can work with rpy angles (for $J$ and not $\dot{J}$). How do I do this with $\dot{J}$ ? Apr 30 '20 at 19:57
• You're asking about RPY angles, which is a particular representation of how the end-effector (I'm assuming) is oriented, but then you're trying to do something with the end-effector orientation and the Jacobian, which relates the joint speeds to the end-effector speeds based on the physical arrangement (kinematics) of the robot. The RPY representation though is just that - a representation. Why not just use the algorithm as-is and transform to/from the RPY format? Apr 30 '20 at 20:41
• @Chuck Yeah, it would be ideal if that were the case. The issue (which is not really an issue) comes from using the Tf package in ROS, where the angles retrieved are rpy. I could of course avoid the package entirely and just compute the kinematics from by retrieving joint pos/velocities, but I was just wondering if it was possible to do it like this. Btw, off topic, thanks for transcribing the algo - it revealed some mistakes I made (and fixed) from transcribing the one from the paper :)! Apr 30 '20 at 20:50
• Yeah, that's a problem I've had with images in the past - sometimes they're too blurry and don't scale well. Regarding your comment in ROS,where the angles retrieved are rpy, again, those are angles of the end effector, not the joint angles, so again I'm still not seeing what the issue is there :( Apr 30 '20 at 21:13