.. role:: raw-latex(raw)
   :format: latex
..

3R Planar Serial Robot
======================

Mathematics involved
--------------------

.. image:: ../examples/Jacobian/images/RRR.png
   :alt: Alternative Text
   :width: 300
   :align: center

An RRR planar serial manipulator is considered as shown in the figure
above. The corresponding adjacency matrix is given by

.. math:: \bf{M} = \left[\begin{matrix}L_1 & R & O & O \\A & L_2 & R & O\\O & A & L_3 & R\\O & O & A & L_4\end{matrix}\right]

Connecting paths:
~~~~~~~~~~~~~~~~~

.. math:: \text{Path 1:}\;\;\;\;L_1-L_2-L_3-L_4

Since this has only one connecting path, if the manipulator represented
by the matrix is valid then it must be a serial manipulator. Hence,
there would be only one independent set of formulation of linear and
angular velocities, and formulation of :math:`[\bf{C}_{V}]` and
:math:`[\bf{C}_{\Omega}]` are not required.

The following are the linear and angular velocity contributions to the
end-effector from each joint of the path, which are calculated by using
the formulation shown in table :raw-latex:`\ref{velocities}` and by
using the convention that all the revolute joints of a planar
manipulator would have their axes on the xy-plane, thereby reducing the
unit vector along each axis to
:math:`\bf{\hat{n}}_{(i,j)}=\bf{\hat{k}}`, as mentioned in equation 20
of the main document.

.. math::

   \begin{matrix}
     \bf{V_{12}}=\dot{\theta}_{(1,2)} \bf{\hat{n}_{(1,2)}} \times \left( \bf{a} - \bf{r}_{(1,2)} \right) = \dot{\theta}_{(1,2)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(1,2)} \right) \\
     \bf{V_{23}}=\dot{\theta}_{(2,3)} \bf{\hat{n}_{(2,3)}} \times \left( \bf{a} - \bf{r}_{(2,3)} \right) = \dot{\theta}_{(2,3)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(2,3)} \right) \\
     \bf{V_{34}}=\dot{\theta}_{(3,4)} \bf{\hat{n}_{(3,4)}} \times \left( \bf{a} - \bf{r}_{(3,4)} \right) = \dot{\theta}_{(3,4)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(3,4)} \right)
   \end{matrix}

.. math::

   \begin{matrix}
     \bf{\Omega_{12}}=\dot{\theta}_{(1,2)} \bf{\hat{n}_{(1,2)}} = \dot{\theta}_{(1,2)} \bf{\hat{k}} \\
     \bf{\Omega_{23}}=\dot{\theta}_{(2,3)} \bf{\hat{n}_{(2,3)}} = \dot{\theta}_{(2,3)} \bf{\hat{k}} \\
     \bf{\Omega_{34}}=\dot{\theta}_{(3,4)} \bf{\hat{n}_{(3,4)}} = \dot{\theta}_{(3,4)} \bf{\hat{k}}
   \end{matrix}

Therefore, the linear and angular velocities are given by the equations
below.

.. math:: \bf{v}^{(1)}=\dot{\theta}_{(1,2)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(1,2)} \right) + \dot{\theta}_{(2,3)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(2,3)} \right) + \dot{\theta}_{(3,4)} \bf{\hat{k}} \times \left( \bf{a} - \bf{r}_{(3,4)} \right)

.. math:: \bf{\omega}^{(1)}=\dot{\theta}_{(1,2)} \bf{\hat{k}} + \dot{\theta}_{(2,3)} \bf{\hat{k}} + \dot{\theta}_{(3,4)} \bf{\hat{k}}

Since this is a planar manipulator, the case of superfluous DOF does not
come into picture.

If the actuating joint velocities vector is considered to be
:math:`\bf{\Omega_a} = \{\dot{\theta}_{(1,2)} \; \dot{\theta}_{(2,3)} \; \dot{\theta}_{(3,4)}\}^T`,
the velocity of the end-effector is given by

.. math:: \begin{Bmatrix}\bf{v} \\ \bf{\omega}\end{Bmatrix} = \begin{Bmatrix}\bf{v}^{(1)} \\ \bf{\omega}^{(1)}\end{Bmatrix} = \left[\begin{matrix}- a_{y} + r_{(1,2)y} & - a_{y} + r_{(2,3)y} & - a_{y} + r_{(3,4)y} \\a_{x} - r_{(1,2)x} & a_{x} - r_{(2,3)x} & a_{x} - r_{(3,4)x}\\1 & 1 & 1\end{matrix}\right]\begin{Bmatrix}\dot{\theta}_{(1,2)}\\\dot{\theta}_{(2,3)}\\\dot{\theta}_{(3,4)}\end{Bmatrix}

.. math::


   \Rightarrow \begin{Bmatrix}\bf{v} \\ \bf{\omega}\end{Bmatrix} = \bf{J_a} \bf{\Omega_a}

Therefore, the Jacobian of the manipulator is

.. math::


   \bf{\widetilde{J}} = \bf{J_a} = \left[\begin{matrix}- a_{y} + r_{(1,2)y} & - a_{y} + r_{(2,3)y} & - a_{y} + r_{(3,4)y}\\a_{x} - r_{(1,2)x} & a_{x} - r_{(2,3)x} & a_{x} - r_{(3,4)x}\\1 & 1 & 1\end{matrix}\right]

Since it is a serial manipulator, the matrices :math:`\bf{J_p}`,
:math:`\bf{A_a}` and :math:`\bf{A_p}` do not come into picture.