acrod 1.1.6

Automatic Computation for Robot Design (ACRoD)

Automatic Computation for Robot Design (ACRoD)

Description

This repository is dedicated to develop functions for automatic computations for designing robotic manipulators.

Currently available functions

• Jacobian formulation for planar and spatial manipulators around a given end-effector point. (This is useful in performing optimisation of Jacobian-based performance parameters of any non-redundant robot directly from its robot-topology matrix)
  • Statement of need: Jacobian formulation is highly used in dimensional synthesis of robotic manipulators, which deals with optimal design of robot's dimensional parameters (link-lengths and joint-orientations). For a given topological structure, formulating Jacobian around a given end-effector point for designing optimal dimensional parameters would require only the topological information, as every other step can be automated. Formulating Jacobian for serial manipulators is easy but for parallel manipulators and serial-parallel hybrid manipulators it is more complicated and often tedious. ACRoD automates this task of formulating Jacobian for a given end-effector point by running all the required steps in the background. The targetted audience includes researchers and engineers working on performing dimensional synthesis of manipulators (especially for multiple manipultors in bulk for comparison) to compute optimal dimensions for operation around a single end-effector point, and those needing to verify the DOF of a topological structure (especially for those special cases of kinematic mechanisms where Chebychev–Grübler–Kutzbach criterion fails to accurately determine the DOF) by analysing the Jacobian.

Installation

The package can be installed from PyPI by using the following command via terminal.

pip install acrod

Usage

Jacobian for planar manipulators

The topological information of a robot is to be specified by using its robot-topology matrix, as defined here. For a planar 2R serial manipulator, the robot topology matrix is given by

$$\left[\begin{matrix} 9 & 1 & 0 \\ 1 & 9 & 1 \\ 0 & 1 & 9 \end{matrix}\right]$$

The corresponding Jacobian function can be formulated as follows.

Firstly, the required functions are imported as shown below.

from acrod.functions import jacobian
from numpy import matrix

The robot-topology matrix for 3R planar serial manipulator is defined and jacobian information is processed via the imported jacobian class as follows.

M = numpy.array(
        [[9, 1, 0],
         [1, 9, 1],
         [0, 1, 9]]
    )
jac = jacobian(M, robot_type = 'planar')

Jacobian function is generated as shown below.

jacobian_function = jac.get_jacobian_function()

In the process of generating the above jacobian function, other attributes of the jacobian object also are updated. Symbolic Jacobian matrices can be extracted from the attributes. Since this is a serial robot, the matrix $J_a$ itself would be the Jacobian matrix of the manipulator. The matrix $J_a$ is extracted from Ja attribute of the jacobian object as follows.

symbolic_jacobian = jac.Ja
symbolic_jacobian

In an ipynb file of JupyterLab, the above code would produce the following output.

$$\left[\begin{matrix}- a_{y} + r_{(1,2)y} & - a_{y} + r_{(2,3)y} \\ a_{x} - r_{(1,2)x} & a_{x} - r_{(2,3)x} \\ 1 & 1\end{matrix}\right]$$

The above Jacobian is based on the notations defined and described here.

Active joint velocities, in the corresponding order, can be viewed by running the following lines.

active_joint_velocities = jac.active_joint_velocities_symbolic
active_joint_velocities

In an ipynb file of JupyterLab, the above code would produce the following output.

$$\left[\begin{matrix}\dot{\theta}_{(1,2)} \\ \dot{\theta}_{(2,3)}\end{matrix}\right]$$

Robot dimensional parameters can be viewed by running the below line.

robot_dimensional_parameters = jac.parameters_symbolic
robot_dimensional_parameters

In an ipynb file of JupyterLab, the above code would produce the following output.

$$\left[\begin{matrix}r_{(1,2)x} \\ r_{(1,2)y} \\ r_{(2,3)x} \\ r_{(2,3)y}\end{matrix}\right]$$

Robot end-effector parameters can be viewed by running the below line.

robot_endeffector_parameters = jac.endeffector_variables_symbolic
robot_endeffector_parameters

In an ipynb file of JupyterLab, the above code would produce the following output.

$$\left[\begin{matrix}a_{x} \\ a_{y}\end{matrix}\right]$$

Sample computation of Jacobian for the configuration corresponding to the parameters shown below:

• End-effector point: $\textbf{a}=\hat{i}+2\hat{j}$
• Locations of joints: $\textbf{r}_{(1,2)}=3\hat{i}+4\hat{j}$ and $\textbf{r}_{(2,3)}=5\hat{i}+6\hat{j}$

For the given set of dimensional parameters of the robot, the numerical Jacobian can be computed as follows. Firstly, we need to gather the configuration parameters in Python list format, in a particular order. The robot dimensional parameters from jac.parameters_symbolic are found (as shown earlier) to be in the order of $r_{(1,2)x}$, $r_{(1,2)y}$, $r_{(2,3)x}$ and $r_{(2,3)y}$. Hence the configuration parameters are to be supplied in the same order, as a list. Thus, the computation can be performed as shown below.

end_effector_point = [1,2]
configuration_parameters = [3,4,5,6]
jacobian_at_the_given_configuration = jacobian_function(end_effector_point, configuration_parameters)
jacobian_at_the_given_configuration

The output produced by running the above code, is shown below.

array([[ 2,  4],
       [-2, -4],
       [ 1,  1]])

Examples

Some examples (along with their mathematical derivations) can be found here.

Community Guidelines

• For contribution to the software:
  • In order to contribute to the software, please consider using the pull request feature of GitHub.
• For reporting issues with the software:
  • For reporting issues or problems, please use issues.
• For support:
  • For any further support (including installation, usage, etc.), feel free to contact via suneeshjacob-at-gmail-dot-com.