Notation and Nomenclature
=========================

Notation for describing the parameters of spatial manipulators
--------------------------------------------------------------

The notations presented in this document are modified versions of the
notations mentioned in :ref:`[1] <references_notation>`, which can be formulated directly
from the `robot-topology matrix`_, a is a
modified form of matrix-based representation of robots mentioned in :ref:`[2] <references_notation>`. The notations are explained below in detail.

.. _robot-topology matrix: robot_topology_matrix.html

The robot-topology matrix considered in this study consists of the
diagonal elements corresponding to the links of the manipulator, in
which the first diagonal element corresponds to the base link and the
last diagonal element corresponds to the end-effector link of the
manipulator. The upper off-diagonal elements correspond to the joints
that the corresponding links are connected with. The types of joints
considered are revolute, prismatic, cylindrical, spherical, universal
and plane joints. The parameters required to describe each joint are
discussed below in detail.

The position of the joint of a manipulator connecting the two links
:math:`i` and :math:`j`, is given by equation (1).

.. math:: \mathbf{r}_{(i,j)}=r_{(i,j)x}\mathbf{\hat{i}}+r_{(i,j)y}\mathbf{\hat{j}}+r_{(i,j)z}\mathbf{\hat{k}} \tag{1}

In order to fully describe a revolute joint or a prismatic joint or a
cylindrical joint, apart from its position
(:math:`\mathbf{r}_{(i,j)}`), the orientation of the axis of its
appropriate motion (:math:`\mathbf{\hat{n}}_{(i,j)}`) needs to be
specified. For a revolute joint, the appropriate motion would be
revolute motion, and for a prismatic joint, the appropriate motion would
be translatory motion. On the other hand, for a cylindrical joint, the
appropriate motion consists of both revolute and translatory motions
along the same axis. While specifying one axis is sufficient for a
revolute, prismatic or cylindrial joint (apart from the position), it is
required to specify two mutually perpendicular axes for a universal
joint, namely :math:`\mathbf{\hat{m}}_{(i,j)}` and
:math:`\mathbf{\hat{n}}_{(i,j)}`, as shown in figure below. And in order to specify a helical
joint, apart from its position (:math:`\mathbf{r}_{(i,j)}`) and the
orientation of its axis (:math:`\mathbf{\hat{n}}_{(i,j)}`), the pitch
(:math:`p_{ij}`) of the helix is to be specified. Alternatively, the
helix angle can also be specified, from which the pitch of the helix can
be calculated. Finally, in order to specify a plane joint, apart from
its position, the orientation of the axis perpendicular to the plane
(:math:`\mathbf{\hat{m}}_{(i,j)}`) is to be specified.

.. image:: ../misc/universaljoint.png
   :alt: Alternative Text
   :width: 300
   :align: center

In case of universal joints, the vectors
:math:`\mathbf{\hat{m}}_{(i,j)}` and :math:`\mathbf{\hat{n}}_{(i,j)}`
are unit vectors, each of which describes the corresponding axis of
rotation/translation. The components in the global frame of reference
are shown in equations (2) and (3).

.. math:: \mathbf{\hat{n}}_{(i,j)}=n_{(i,j)x}\mathbf{\hat{i}}+n_{(i,j)y}\mathbf{\hat{j}}+n_{(i,j)z}\mathbf{\hat{k}} \tag{2}

.. math:: \mathbf{\hat{m}}_{(i,j)}=m_{(i,j)x}\mathbf{\hat{i}}+m_{(i,j)y}\mathbf{\hat{j}}+m_{(i,j)z}\mathbf{\hat{k}} \tag{3}

Since these are unit vectors, they have to satisfy the equations shown
in (4) and (5). This can be achieved by writing the elements of the unit
vector in terms of two independent variables, as shown in equations (9)
and (12). And since :math:`\mathbf{\hat{m}}_{(i,j)}` occurs only in
case of universal joint wherein it is always perpendicular to its
companion axis :math:`\mathbf{\hat{n}}_{(i,j)}`, the elements of those
unit vectors should also satisfy equation (6). Also in case of plane
joints, :math:`\mathbf{\hat{m}}_{(i,j)}` is the unit vector normal to
the plane, and :math:`\mathbf{\hat{n}}_{(i,j)}` is the unit vector
along which instantaneous planar translation takes place, and hence the
inner product of the two unit vectors :math:`\mathbf{\hat{m}}_{(i,j)}`
and :math:`\mathbf{\hat{n}}_{(i,j)}` should always be zero. Therefore,
equations (4), (5) and (6) are applicable in the case of plane joints as
well.

.. math:: n_{(i,j)x}^2+n_{(i,j)y}^2+n_{(i,j)z}^2=1 \tag{4}

.. math:: m_{(i,j)x}^2+m_{(i,j)y}^2+m_{(i,j)z}^2=1 \tag{5}

.. math:: m_{(i,j)x}n_{(i,j)x}+m_{(i,j)y}n_{(i,j)y}+m_{(i,j)z}n_{(i,j)z}=0 \tag{6}

.. math:: n_{(i,j)x} = \sin{\left(\beta_{(i,j)}\right)}\cos{\left(\phi_{(i,j)}\right)} \tag{7}

.. math:: n_{(i,j)y} = \sin{\left(\beta_{(i,j)}\right)}\sin{\left(\phi_{(i,j)}\right)} \tag{8}

.. math:: n_{(i,j)z} = \cos{\left(\beta_{(i,j)}\right)} \tag{9}

.. math:: m_{(i,j)x} = \sin{\left(\alpha_{(i,j)}\right)}\cos{\left(\delta_{(i,j)}\right)} \tag{10}

.. math:: m_{(i,j)y} = \sin{\left(\alpha_{(i,j)}\right)}\sin{\left(\delta_{(i,j)}\right)} \tag{11}

.. math:: m_{(i,j)z} = \cos{\left(\alpha_{(i,j)}\right)} \tag{12}

This reduces the six parameters, i.e., :math:`m_{(i,j)x}`,
:math:`m_{(i,j)y}`, :math:`m_{(i,j)z}`, :math:`n_{(i,j)x}`,
:math:`n_{(i,j)y}` and :math:`n_{(i,j)z}` into four parameters, i.e.,
:math:`\alpha_{(i,j)}`, :math:`\delta_{(i,j)}`, :math:`\beta_{(i,j)}`
and :math:`\phi_{(i,j)}`. But there are only three independent
parameters. By putting these in the equality constraint shown in
equation (6), we get

.. math::

   \begin{array}{cc}\left(\sin{\left(\alpha_{(i,j)}\right)}\cos{\left(\delta_{(i,j)}\right)}\right)\left(\sin{\left(\beta_{(i,j)}\right)}\cos{\left(\phi_{(i,j)}\right)}\right)
   \\
   +\left(\sin{\left(\alpha_{(i,j)}\right)}\sin{\left(\delta_{(i,j)}\right)}\right)\left(\sin{\left(\beta_{(i,j)}\right)}\sin{\left(\phi_{(i,j)}\right)}\right)
   \\
   +\left(\cos{\left(\alpha_{(i,j)}\right)}\right)\left(\cos{\left(\beta_{(i,j)}\right)}\right)=0\end{array}

.. math:: \Rightarrow \tan{\left(\alpha_{(i,j)}\right)}\tan{\left(\beta_{(i,j)}\right)}\cos{\left(\delta_{(i,j)}-\phi_{(i,j)}\right)}+1=0

.. math:: \Rightarrow \tan{\left(\alpha_{(i,j)}\right)}=-\frac{1}{\tan{\left(\beta_{(i,j)}\right)}\cos{\left(\delta_{(i,j)}-\phi_{(i,j)}\right)}}

.. math:: 0\leq\beta_{(i,j)},\phi_{(i,j)},\delta_{(i,j)}\leq 2\pi \tag{13}

Regarding helical joints, in the context of this present study,
single-threaded screws are considered with the convention that
right-handed threading has positive pitch. Right-handed threading in the
context of this study is defined such that when two links are connected
by such a joint then the relative rotation of a link with respect to the
other about an axis produces translation in the same direction of that
axis. If :math:`p_{ij}` is the pitch of a helical joint connecting the
links :math:`i` and :math:`j`, then the angular displacement of the
screw is related to the linear displacement of the screw by equation
(14).

.. math:: d_{(i,j)} = \frac{p_{(i,j)}}{2\pi}\theta_{(i,j)} \tag{14}

Notation for planar manipulators
--------------------------------

Planar manipulators are a special case of spatial manipulators and hence
the notation of planar manipulators is in some sense a subset of that of
spatial manipulators. In planar manipulators, it is assumed that all the
motion exists in xy-plane and hence the z-coordinate is 0 for all the
position vectors of locations of joints. Thus, for planar manipulators,
the equation (1) reduces to the equation (15).

.. math:: \mathbf{r}_{(i,j)}=r_{(i,j)x}\mathbf{\hat{i}}+r_{(i,j)y}\mathbf{\hat{j}} \tag{15}

In this study, only two types of joints, namely revolute and prismatic
are considered. The axis of each revolute joint is always perpendicular
to the plane, and hence, for revolute joints of planar manipulators, the
equation (2) reduces to the equation (17). And the axis of each
prismatic joint should lie within the plane, and hence the z-coordinate
of the unit vector along the axis of each prismatic joint would be zero.
Thus, for prismatic joints of planar manipulators, the equation (2)
reduces to the equation (16), and correspondingly, :math:`n_{(i,j)z}`
being zero conventionally implies :math:`\beta_{(i,j)}=\frac{\pi}{2}`
and :math:`\sin{\beta_{(i,j)}}=1`, thereby reducing equations (7) and (8) to
equations (18) and (19) respectively.

.. math:: \mathbf{\hat{n}}_{(i,j)} = n_{(i,j)x}\mathbf{\hat{i}}+n_{(i,j)y}\mathbf{\hat{j}} \tag{16}

.. math:: \mathbf{\hat{n}}_{(i,j)} = \mathbf{\hat{k}} \tag{17}

.. math:: n_{(i,j)x} = \cos{\left(\phi_{(i,j)}\right)} \tag{18}

.. math:: n_{(i,j)y} = \sin{\left(\phi_{(i,j)}\right)} \tag{19}

Since all the motion lies entirely in the xy-plane, the z-component of
linear velocity along with the x & the y components of the angular
velocity of the end-effector would be zeros, thereby reducing the size
of the Jacobian from six rows to three rows.

.. _references_notation:

References
----------

[1] Jacob, Akkarapakam Suneesh, and Bhaskar Dasgupta. “Dimensional
synthesis of spatial manipulators for velocity and force transmission
for operation around a specified task point.” arXiv preprint
arXiv:2210.04446 (2022).

[2] Jacob, Akkarapakam Suneesh, Bhaskar Dasgupta, and Rituparna Datta.
“Enumeration of spatial manipulators by using the concept of Adjacency
Matrix.” arXiv preprint arXiv:2210.03327 (2022).