4R-4P Serial-Parallel Hybrid Robot

Figure

Alternative Text

A figure of RRRRPPPP planar serial-parallel hybrid manipulator is shown above. The corresponding adjacency matrix is given by

\[\begin{split}\bf{M} = \left[\begin{matrix}L_1 & P & P & O & O & O & O\\A & L_2 & O & P & O & O & O\\A & O & L_3 & P & O & O & O\\O & O & O & L_4 & R & R & O\\O & O & O & A & L_5 & O & R\\O & O & O & O & O & L_6 & R\\O & O & O & O & O & O & L_7\end{matrix}\right]\end{split}\]

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 RRRRPPPP planar serial-parallel hybrid manipulator shown above, the robot topology matrix is given by

\[\begin{split}\left[\begin{matrix}9 & 2 & 2 & 0 & 0 & 0 & 0\\1 & 9 & 0 & 2 & 0 & 0 & 0\\1 & 0 & 9 & 2 & 0 & 0 & 0\\0 & 0 & 0 & 9 & 1 & 1 & 0\\0 & 0 & 0 & 1 & 9 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 9 & 1\\0 & 0 & 0 & 0 & 0 & 0 & 9\end{matrix}\right]\end{split}\]

The corresponding Jacobian function can be formulated as follows.

Firstly, the required functions are imported as shown below.

from acrod.jacobian import Jacobian
from numpy import array

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

M = array(
        [[9, 2, 2, 0, 0, 0, 0],
         [1, 9, 0, 2, 0, 0, 0],
         [1, 0, 9, 2, 0, 0, 0],
         [0, 0, 0, 9, 1, 1, 0],
         [0, 0, 0, 1, 9, 0, 1],
         [0, 0, 0, 0, 0, 9, 1],
         [0, 0, 0, 0, 0, 0, 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 non-serial robot, there would be four matrices required to compute the Jacobian, which are \(J_a\), \(J_p\), \(A_a\) and \(A_p\). These can be extracted by the attributes Ja, Jp, Aa and Ap, respectively, as shown below.

symbolic_Ja = jac.Ja
symbolic_Ja

Output in Jupyter notebook:

\[\begin{split}\left[\begin{matrix}\cos{\left(\phi_{(1,2)} \right)} & 0 & - a_{y} + r_{(4,5)y}\\\sin{\left(\phi_{(1,2)} \right)} & 0 & a_{x} - r_{(4,5)x}\\0 & 0 & 1\end{matrix}\right]\end{split}\]
symbolic_Jp = jac.Jp
symbolic_Jp

Output in Jupyter notebook:

\[\begin{split}\left[\begin{matrix}\cos{\left(\phi_{(2,4)} \right)} & 0 & 0 & - a_{y} + r_{(5,7)y} & 0\\\sin{\left(\phi_{(2,4)} \right)} & 0 & 0 & a_{x} - r_{(5,7)x} & 0\\0 & 0 & 0 & 1 & 0\end{matrix}\right]\end{split}\]
symbolic_Aa = jac.Aa
symbolic_Aa

Output in Jupyter notebook:

\[\begin{split}\left[\begin{matrix}0 & 0 & a_{y} - r_{(4,5)y}\\0 & 0 & - a_{x} + r_{(4,5)x}\\- \cos{\left(\phi_{(1,2)} \right)} & \cos{\left(\phi_{(1,3)} \right)} & 0\\- \sin{\left(\phi_{(1,2)} \right)} & \sin{\left(\phi_{(1,3)} \right)} & 0\\0 & 0 & -1\end{matrix}\right]\end{split}\]
symbolic_Ap = jac.Ap
symbolic_Ap

Output in Jupyter notebook:

\[\begin{split}\left[\begin{matrix}0 & 0 & - a_{y} + r_{(4,6)y} & a_{y} - r_{(5,7)y} & - a_{y} + r_{(6,7)y}\\0 & 0 & a_{x} - r_{(4,6)x} & - a_{x} + r_{(5,7)x} & a_{x} - r_{(6,7)x}\\- \cos{\left(\phi_{(2,4)} \right)} & \cos{\left(\phi_{(3,4)} \right)} & 0 & 0 & 0\\- \sin{\left(\phi_{(2,4)} \right)} & \sin{\left(\phi_{(3,4)} \right)} & 0 & 0 & 0\\0 & 0 & 1 & -1 & 1\end{matrix}\right]\end{split}\]

The above matrices are 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.

\[\begin{split}\left[\begin{matrix}\dot{d}_{(1,2)} \\ \dot{d}_{(1,3)} \\ \dot{\theta}_{(4,5)}\end{matrix}\right]\end{split}\]

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.

\[\begin{split}\left[\begin{matrix}\phi_{(1,2)}\\\phi_{(1,3)}\\\phi_{(2,4)}\\\phi_{(3,4)}\\r_{(4,5)x}\\r_{(4,5)y}\\r_{(4,6)x}\\r_{(4,6)y}\\r_{(5,7)x}\\r_{(5,7)y}\\r_{(6,7)x}\\r_{(6,7)y}\end{matrix}\right]\end{split}\]

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.

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

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: \(\textbf{r}_{(4,5)}=2\hat{i}+8\hat{j}\), \(\textbf{r}_{(4,6)}=6\hat{i}+8\hat{j}\), \(\textbf{r}_{(5,7)}=3\hat{i}+10\hat{j}\) and \(\textbf{r}_{(6,7)}=5\hat{i}+10\hat{j}\)

  • Orientations of joints: \(\phi_{(1,2)}=2\pi/3\), \(\phi_{(1,3)}=\pi/3\), \(\phi_{(2,4)}=\pi/3\) and \(\phi_{(3,4)}=2\pi/3\)

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 \(\phi_{(1,2)}\), \(\phi_{(1,3)}\), \(\phi_{(2,4)}\), \(\phi_{(3,4)}\), \(r_{(4,5)x}\), \(r_{(4,5)y}\), \(r_{(4,6)x}\), \(r_{(4,6)y}\), \(r_{(5,7)x}\), \(r_{(5,7)y}\), \(r_{(6,7)x}\) and \(r_{(6,7)y}\). Hence the configuration parameters are to be supplied in the same order, as a list. Thus, the computation can be performed as shown below.

from numpy import pi

end_effector_point = [1,2]
configuration_parameters = [2*pi/3,pi/3,pi/3,2*pi/3,2,8,6,8,3,10,5,10]
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([[ -0.5      ,   0.5      , -10.       ],
       [  0.8660254,   0.8660254,   3.       ],
       [  0.       ,   0.       ,  -1.       ]])

Accessing each matrix individually:

Each of \(J_a\), \(J_p\), \(A_a\) and \(A_p\) functions can be accessed as shown below

numerical_Ja = jac.Ja_func(end_effector_point, configuration_parameters)
numerical_Ja

Output:

array([[-0.5      ,  0.       ,  6.       ],
       [ 0.8660254,  0.       , -1.       ],
       [ 0.       ,  0.       ,  1.       ]])

numerical_Jp = jac.Jp_func(end_effector_point, configuration_parameters)
numerical_Jp

Output:

array([[ 0.5      ,  0.       ,  0.       ,  8.       ,  0.       ],
       [ 0.8660254,  0.       ,  0.       , -2.       ,  0.       ],
       [ 0.       ,  0.       ,  0.       ,  1.       ,  0.       ]])

numerical_Aa = jac.Aa_func(end_effector_point, configuration_parameters)
numerical_Aa

Output:

array([[ 0.       ,  0.       , -6.       ],
       [ 0.       ,  0.       ,  1.       ],
       [ 0.5      ,  0.5      ,  0.       ],
       [-0.8660254,  0.8660254,  0.       ],
       [ 0.       ,  0.       , -1.       ]])

numerical_Ap = jac.Ap_func(end_effector_point, configuration_parameters)
numerical_Ap

Output:

array([[ 0.       ,  0.       ,  6.       , -8.       ,  8.       ],
       [ 0.       ,  0.       , -5.       ,  2.       , -4.       ],
       [-0.5      , -0.5      ,  0.       ,  0.       ,  0.       ],
       [-0.8660254,  0.8660254,  0.       ,  0.       ,  0.       ],
       [ 0.       ,  0.       ,  1.       , -1.       ,  1.       ]])

And the computation \(J_a-J_pA^{-1}_pA_a\) is given by

import numpy
numerical_Ja-numpy.matmul(numpy.matmul(numerical_Jp,numpy.linalg.inv(numerical_Ap)),numerical_Aa)

which gives the same output as jacobian_at_the_given_configuration.

Some other attributes

To get the computed list of all connecting paths from the base link to the end-effector link, the below script can be used:

jac.P

Output:

[[0, 1, 3, 4, 6], [0, 1, 3, 5, 6], [0, 2, 3, 4, 6], [0, 2, 3, 5, 6]]

which gives the list of all connecting paths (only link numbers are shown, indexed from 0).

Independent paths pertaining to linear and angular velocities:

Out of the above paths, the computed list of independent paths pertaining to linear velocities and angular velocities, can be accessed by the below scripts:

For independent paths pertaining to linear velocities:

jac.P_tilde

Output:

[[0, 1, 3, 4, 6], [0, 1, 3, 5, 6], [0, 2, 3, 4, 6]]

For independent paths pertaining to angular velocities:

jac.P_tilde_omega

Output:

[[0, 1, 3, 4, 6], [0, 1, 3, 5, 6]]

The above scripts give the lists of all the independent connecting paths pertaining to linear and angular velocities (only link numbers are shown, indexed from 0).