jacobian of Abb irb140 robot

Question

Can someone please help me with the jacobian matrix equations for Abb irb140 robot. Or an easy way by which I can derive it given the DH parameters. I need it to implement some form of control that am working on. Thanks

Answer 1

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.

Answer 2

Not exaclty sure how to easily/automatically derive it from DH parmameters. Is using DH actually necessery for your project? What are the advantages of using the Denavit-Hartenberg representation?

Do you have a CAD model for the robot?

If you are using Solidworks, you could generate a URDF (Unified Robot Description Format) model based on the CAD model using sw_urdf_exporter. Then use the tool pykdl_utils to automatically fill the FK and jacobian requests with the current joint angles.

Answer 3

Similar to Pikey's answer, OpenRAVE lets you input the kinematics of your arm in an XML format (different than URDF), and will give you the Jacobian, FK, IK, motion planning, etc for your arm.

If you want to do it the traditional way, many standard robotics textbooks outline how to go from DH parameters to the geometrical Jacobian. I like this book: "Robotics: Modelling, Planning and Control" by Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, Giuseppe Oriolo. For all but the simplest of arms, you will want to use a symbolic algebra package to help with this...