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Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \theta_i \\\hline \\1 & 0 & \pi/2 & 0 & \theta_1 \\2 & a_2 & 0 & 0 & \theta_2 \\3 & 0 & \pi/2 & 0 & \theta_3 \\4 & 0 & -\pi/2 & d_4 & \theta_4 \\5 & 0 & \pi/2 & 0 & \theta_5 \\6 & 0 & 0 & d_6 & \theta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\theta_i} & -s_{\theta_i} c_{\alpha_i} & s_{\theta_i} s_{\alpha_i} & a_i c_{\theta_i} \\s_{\theta_i} & c_{\theta_i} c_{\alpha_i} & -c_{\theta_i} s_{\alpha_i} & a_i s_{\theta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$

Here are general instructions how to compute Jacobian:

It is easiest to compute the Jacobian for linear velocity and angular velocity separately.

For linear velocity:

The time derivative of $\boldsymbol{p}_e(\boldsymbol{q})$ is:

$$ \dot{\boldsymbol{p}}_e = \sum\limits_{i=1}^n \frac{\partial \boldsymbol{p}_e}{\partial q_i} \dot{q}_i = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{P_i} \dot{q}_i $$

This shows how $\dot{\boldsymbol{p}}_e$ can be obtained by summing the contributions from each joint. Note for revolute joints, $q_i = \vartheta_i$.

$$ \dot{q}_i \boldsymbol{\jmath}_{P_i} = \boldsymbol{\omega}_{i-1,i} \times r_{i-1,e} = \dot{\vartheta}_i \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$ Simplifying: $$ \boldsymbol{\jmath}_{P_i} = \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$

Now for angular velocity: $$ \omega_e = \omega_n = \sum\limits_{i=1}^n \omega_{i-1,i} = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{O_i} \dot{q}_i $$ in detail: $$ \dot{q}_i \boldsymbol{\jmath}_{O_i} = \vartheta_i \boldsymbol{z}_{i-1} $$ and then: $$ \boldsymbol{\jmath}_{O_i} = \boldsymbol{z}_{i-1} $$

So now the Jacobian can be partitioned into (3x1) column vectors $\boldsymbol{\jmath}_{P_i}$ and $\boldsymbol{\jmath}_{O_i}$ as such:

$$ \boldsymbol{J} = \begin{bmatrix} \boldsymbol{\jmath}_{P_1} & & \boldsymbol{\jmath}_{P_n}\\ & \ldots & \\ \boldsymbol{\jmath}_{O_1} & & \boldsymbol{\jmath}_{O_n} \end{bmatrix} $$

Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.

Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \vartheta_i \\\hline \\1 & 0 & \pi/2 & 0 & \vartheta_1 \\2 & a_2 & 0 & 0 & \vartheta_2 \\3 & 0 & \pi/2 & 0 & \vartheta_3 \\4 & 0 & -\pi/2 & d_4 & \vartheta_4 \\5 & 0 & \pi/2 & 0 & \vartheta_5 \\6 & 0 & 0 & d_6 & \vartheta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\vartheta_i} & -s_{\vartheta_i} c_{\alpha_i} & s_{\vartheta_i} s_{\alpha_i} & a_i c_{\vartheta_i} \\s_{\vartheta_i} & c_{\vartheta_i} c_{\alpha_i} & -c_{\vartheta_i} s_{\alpha_i} & a_i s_{\vartheta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$

Here are general instructions how to compute Jacobian:

It is easiest to compute the Jacobian for linear velocity and angular velocity separately.

For linear velocity:

The time derivative of $\boldsymbol{p}_e(\boldsymbol{q})$ is:

$$ \dot{\boldsymbol{p}}_e = \sum\limits_{i=1}^n \frac{\partial \boldsymbol{p}_e}{\partial q_i} \dot{q}_i = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{P_i} \dot{q}_i $$

This shows how $\dot{\boldsymbol{p}}_e$ can be obtained by summing the contributions from each joint. Note for revolute joints, $q_i = \vartheta_i$.

$$ \dot{q}_i \boldsymbol{\jmath}_{P_i} = \boldsymbol{\omega}_{i-1,i} \times r_{i-1,e} = \dot{\vartheta}_i \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$ Simplifying: $$ \boldsymbol{\jmath}_{P_i} = \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$

Now for angular velocity: $$ \omega_e = \omega_n = \sum\limits_{i=1}^n \omega_{i-1,i} = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{O_i} \dot{q}_i $$ in detail: $$ \dot{q}_i \boldsymbol{\jmath}_{O_i} = \vartheta_i \boldsymbol{z}_{i-1} $$ and then: $$ \boldsymbol{\jmath}_{O_i} = \boldsymbol{z}_{i-1} $$

So now the Jacobian can be partitioned into (3x1) column vectors $\boldsymbol{\jmath}_{P_i}$ and $\boldsymbol{\jmath}_{O_i}$ as such:

$$ \boldsymbol{J} = \begin{bmatrix} \boldsymbol{\jmath}_{P_1} & & \boldsymbol{\jmath}_{P_n}\\ & \ldots & \\ \boldsymbol{\jmath}_{O_1} & & \boldsymbol{\jmath}_{O_n} \end{bmatrix} $$

Courtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.

Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \theta_i \\\hline \\1 & 0 & \pi/2 & 0 & \theta_1 \\2 & a_2 & 0 & 0 & \theta_2 \\3 & 0 & \pi/2 & 0 & \theta_3 \\4 & 0 & -\pi/2 & d_4 & \theta_4 \\5 & 0 & \pi/2 & 0 & \theta_5 \\6 & 0 & 0 & d_6 & \theta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\theta_i} & -s_{\theta_i} c_{\alpha_i} & s_{\theta_i} s_{\alpha_i} & a_i c_{\theta_i} \\s_{\theta_i} & c_{\theta_i} c_{\alpha_i} & -c_{\theta_i} s_{\alpha_i} & a_i s_{\theta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$

Here are general instructions how to compute Jacobian:

And this should be the Jacobian for your specific arm:

Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.

Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \theta_i \\\hline \\1 & 0 & \pi/2 & 0 & \theta_1 \\2 & a_2 & 0 & 0 & \theta_2 \\3 & 0 & \pi/2 & 0 & \theta_3 \\4 & 0 & -\pi/2 & d_4 & \theta_4 \\5 & 0 & \pi/2 & 0 & \theta_5 \\6 & 0 & 0 & d_6 & \theta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\theta_i} & -s_{\theta_i} c_{\alpha_i} & s_{\theta_i} s_{\alpha_i} & a_i c_{\theta_i} \\s_{\theta_i} & c_{\theta_i} c_{\alpha_i} & -c_{\theta_i} s_{\alpha_i} & a_i s_{\theta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$

Here are general instructions how to compute Jacobian:

It is easiest to compute the Jacobian for linear velocity and angular velocity separately.

For linear velocity:

The time derivative of $\boldsymbol{p}_e(\boldsymbol{q})$ is:

$$ \dot{\boldsymbol{p}}_e = \sum\limits_{i=1}^n \frac{\partial \boldsymbol{p}_e}{\partial q_i} \dot{q}_i = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{P_i} \dot{q}_i $$

This shows how $\dot{\boldsymbol{p}}_e$ can be obtained by summing the contributions from each joint. Note for revolute joints, $q_i = \vartheta_i$.

$$ \dot{q}_i \boldsymbol{\jmath}_{P_i} = \boldsymbol{\omega}_{i-1,i} \times r_{i-1,e} = \dot{\vartheta}_i \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$ Simplifying: $$ \boldsymbol{\jmath}_{P_i} = \boldsymbol{z}_{i-1} \times ( \boldsymbol{p}_e - \boldsymbol{p}_{i-1} ) $$

Now for angular velocity: $$ \omega_e = \omega_n = \sum\limits_{i=1}^n \omega_{i-1,i} = \sum\limits_{i=1}^n \boldsymbol{\jmath}_{O_i} \dot{q}_i $$ in detail: $$ \dot{q}_i \boldsymbol{\jmath}_{O_i} = \vartheta_i \boldsymbol{z}_{i-1} $$ and then: $$ \boldsymbol{\jmath}_{O_i} = \boldsymbol{z}_{i-1} $$

So now the Jacobian can be partitioned into (3x1) column vectors $\boldsymbol{\jmath}_{P_i}$ and $\boldsymbol{\jmath}_{O_i}$ as such:

$$ \boldsymbol{J} = \begin{bmatrix} \boldsymbol{\jmath}_{P_1} & & \boldsymbol{\jmath}_{P_n}\\ & \ldots & \\ \boldsymbol{\jmath}_{O_1} & & \boldsymbol{\jmath}_{O_n} \end{bmatrix} $$

Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.

Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

Here are general instructions how to compute Jacobian:

And this should be the Jacobian for your specific arm:

Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.

Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.

Here are the DH parameters and transformation matrix:

DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \theta_i \\\hline \\1 & 0 & \pi/2 & 0 & \theta_1 \\2 & a_2 & 0 & 0 & \theta_2 \\3 & 0 & \pi/2 & 0 & \theta_3 \\4 & 0 & -\pi/2 & d_4 & \theta_4 \\5 & 0 & \pi/2 & 0 & \theta_5 \\6 & 0 & 0 & d_6 & \theta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\theta_i} & -s_{\theta_i} c_{\alpha_i} & s_{\theta_i} s_{\alpha_i} & a_i c_{\theta_i} \\s_{\theta_i} & c_{\theta_i} c_{\alpha_i} & -c_{\theta_i} s_{\alpha_i} & a_i s_{\theta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$

Here are general instructions how to compute Jacobian:

And this should be the Jacobian for your specific arm:

Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.

Hope that helps.