Velocity modeling demands formulating a specific linear map between two vector spaces at a given configuration, i.e., velocity modeling involves the linear map between the velocity state, or twist about a screw, and the actuator rates. The velocity state of the moving platform as observed from the fixed platform, notated as a six-dimensional vector

${\mathit{V}}_{C}$, is defined as follows:

where

$\mathit{\omega}$ is the angular velocity vector of the moving platform as measured from the fixed platform, while

${\mathit{v}}_{C}$ is the linear velocity vector of point

C expressed in the

$O\_XYZ$ reference frame. Furthermore, the velocity state

${\mathit{V}}_{C}$ may be written through each limb as a linear combination of the screws representing the kinematic pairs of the

i-th leg as follows:

where the assignation of screws is as follows:

${}^{0}{\$}_{i}^{1}$ denotes the screw associated with the lower revolute joint connecting the limb to the fixed platform,

${}^{1}{\$}_{i}^{2}$ denotes the screw representing the actuated prismatic joint along the limb, the screws

${}^{2}{\$}_{i}^{3}$,

${}^{3}{\$}_{i}^{4}$ and

${}^{4}{\$}_{i}^{5}$ are screws with concurrent primal parts reproducing the effect of the spherical joint and

${}^{5}{\$}_{i}^{6}$ represents the screw associated with the upper passive prismatic joint. On the other hand, with the purpose of simplifying the analysis, let us consider that

${S}_{i}$ is a line in Plücker coordinates directed from

${B}_{i}$ to

${D}_{i}$. Note that this line is reciprocal to all the screws in the same limb except the screw representing the actuated prismatic joint. Let

${\$}_{1}=({\widehat{\mathit{s}}}_{1},{\mathit{s}}_{O1})$ and

${\$}_{2}=({\widehat{\mathit{s}}}_{2},{\mathit{s}}_{O2})$ be two elements of the Lie algebra

$se\left(3\right)$ of the Euclidean group

$SE\left(3\right)$. The Klein form, notated as

$\{*;*\}$, is a bilinear symmetric form defined as follows:

Furthermore, it is said that the screws

${\$}_{1}$ and

${\$}_{2}$ are reciprocal if

$\{{\$}_{1};{\$}_{2}\}=0$. Thus, the application of the Klein form between the line

${S}_{i}$ and both sides of Equation (

14) with the cancellation of terms leads to:

where

${\dot{q}}_{i}={}_{1}{\omega}_{2}^{i}$ is the

i-th generalized speed. Expression (

15) provides enough information to obtain the input-output equation of velocity. However, the handling of non-square Jacobian matrices is imminent. The expansion of the Jacobians of limited-dof parallel manipulators to full rank Jacobian matrices brings important advantages [

40]. Dealing with the contribution, in order to avoid the tedious handling of non-square matrices, consider that a standard basis for parallel manipulators performing the spherical motion is given by

$\{{\$}_{1},{\$}_{2},{\$}_{3}\}$ where

${\$}_{1}=[\widehat{\mathit{i}},\mathbf{0}]$,

${\$}_{2}=[\widehat{\mathit{j}},\mathbf{0}]$ and

${\$}_{3}=[\widehat{\mathit{k}},\mathbf{0}]$. Therefore, by resorting to the concept of the reciprocal screw through the Klein form, three equations may be written as follows: