## Abstract

In the design of a mechanism, the quality of effort transmission is a key issue. Traditionally, the effort transmissivity of a mechanism is defined as the quantitative measure of the power flowing effectiveness from the input link(s) to the output link(s). Many researchers have focused their work on the study of the effort transmission in mechanisms and diverse indices have been defined. However, the developed indices have exclusively dealt with the studies of the ratio between the input and output powers and they do not seem to have been devoted to the studies of the admissible reactions in passive joints. However, the observations show that it is possible for a mechanism to reach positions in which the transmission indices will have admissible values but the reaction(s) in passive joint(s) can reach excessively high values leading to the breakdown of the mechanism. In the present paper, a method is developed to ensure the admissible values of reactions in passive joints of planar parallel manipulators. It is shown that the increase of reactions in passive joints of a planar parallel manipulator depends not only on the transmission angle but also on the position of the instantaneous center of rotation of the platform. It allows the determination of the maximal reachable workspace of planar parallel manipulators taking into account the admissible reactions in its passive joints. The suggested method is illustrated via a 5R planar parallel mechanism and a planar 3-RPR parallel manipulator.

## Introduction

Parallel manipulators have many advantages in terms of acceleration capacities and payload-to-weight ratio [1], but one of their main drawbacks concerns the presence of singularities [2–5]. It is known that in the neighbourhood of the singular positions, the reactions in joints of a manipulator considerably grow up.

In order to have a better understanding of this phenomenon, many researchers have focused their works on the analysis of the effort transmission in parallel manipulators. One of the evident criterions for evaluation of effort transmission is the transmission angle (or pressure angle which is equal to 90 deg minus the transmission angle) [6–8]. The pressure angle is well known for characterizing the transmission quality in lower kinematic pairs, such as cams [9], but this idea was also used for effort transmission analysis in the parallel manipulators [6,8].

To evaluate the effort transmission quality, several indexes have been introduced. One of the first attempts was proposed in Ref. [10]. This paper presents a criterion named the transmission index (TI) that is based on transmission wrench screw. The TI varies between 0 and 1. If it is equal to 0, the considered link is in a dead position, i.e., it cannot move anymore. If it is equal to 1, this link has the best static properties.

In the same vein as Ref. [10], the study [11] generalizes the TI for spatial linkages and defines the global TI (GTI). The authors also prove that the GTI is equal, for prismatic and revolute joints, to the cosine of the pressure angle.

The conditioning index was also proposed [12]
for characterizing the quality of transmission between the actuators and the
end-effector. This index is based on the Jacobian matrix or its “norm,” which relate
the actuator velocities (efforts, respectively) to the platform twist (wrench,
respectively) by the following relations: $t=Jq\xb7$ and $w=J-T\tau ,$ where **J** is the Jacobian matrix, **t** is the platform twist, $q\xb7$ is the input velocities, **τ** is the actuator efforts, and **w** is the wrench applied on the platform.

All these indices have been used in many works for design and analysis of parallel mechanisms [13–20]. However, it is also known that because of the nonhomogeneity of the terms of the Jacobian matrix, the conditioning index is not well appropriated for mechanisms having both translational and rotational degrees of freedom (DOF) [21]. Moreover, all the previously mentioned indices do not take into account the real characteristics of the actuators, i.e., the fact that their input efforts are bounded between [−$\tau imax,$$\tau imax$] [21].

In order to solve this problem, in a study [22] a numerical analysis method has been developed. It has been proposed to
characterize the force workspace of robots taking into account a given fixed wrench
applied on the platform and actuator efforts comprised in the boundary interval
[−$\tau imax,$$\tau imax$].
However, this workspace depends on the given direction and norm of the external
force/moment and will change with the variation of the applied wrench. Moreover, for
many robot applications, the external wrench direction is not known, contrary to its
norm. Therefore, in Ref. [23], a way to
compute the maximal workspace taking into account the actuator effort limitations
for a given norm of the external force and moment was proposed. This approach is
based on the computation of the transmission factors of matrix **J**^{−T}, which are obtained
geometrically through the mapping of a cube by the Jacobian matrix.

All the previously mentioned approaches analyze the quality of the effort
transmission by taking into account the input torque limitations only. However,
there are such positions of parallel manipulators for which the limitations of input
torques can be satisfied while the limitations of reactions in passive joints are
not. To have a better understanding of this phenomenon, let us consider a simple
example via a planar 5*R* manipulator (Fig. 1). Close to such singularity a small effort **w** applied on the end-effector of the manipulator will create a large reaction **R _{1}** in the passive joint

*B*. But in this pose, the actuator torques

*τ*

_{1}will stay under acceptable values, as it depends only on the small component

**F**of the reaction

_{1}**R**. Thus, for the mentioned case, the transmission indices will have acceptable values but they will not give any information about the inadequately high values of reactions in passive joints of the mechanism.

_{1}Therefore, the development of criteria for limitation of the passive joints' reactions is also an important consideration in effort transmission field. It is a complementary condition to the transmission indices for characterizing the quality of effort transmission especially in the neighbourhood of singularity.

This paper focuses on the study of the effort transmission in planar parallel mechanisms (PPM) from the above point of view. It aims at proposing a new criterion for taking into account the passive joint reactions and at finding the relationships between this criterion and the previously developed indices, especially the transmission angle.

First, the expressions of the maximal platform joint reactions as a function of the parameters of the wrench applied on the platform is presented, i.e., the maximal force norm and the absolute value of the moment. Then, it is disclosed that the maximal values of platform joint reactions depend not only on the value of the transmission angle but also on the position of the instantaneous center of rotation of the platform. Moreover, the obtained results allow one to define the ranges for the admissible values of the transmission angles and distance to the instantaneous center of rotation that ensure a good effort transmission quality. Finally, two illustrative examples and simulation results are presented.

## A Criterion for Evaluation of the Passive Joints Reactions

**R**

*in the platform passive joints (denoted as*

_{i}*B*in Fig. 2,

_{i}*i*= 1,…, 3) can be related to the external wrench

**w**

^{T}^{ }= [

**f**

*,*

^{T}*C*]

*(*

^{T}**f**is the external force and

*C*the scalar value of the external moment applied on the platform) applying the Newton–Euler equations at any point

*Q*

**QB**

*along*

_{i}**and**

*x***axes (the position of point**

*y**B*may vary if the passive joints linked to the platform are prismatic joints). It should be mentioned that in Fig. 2 and the following figures, the double arrows indicate the direction of the platform reaction forces but not their norm. Considering that

_{i}**R**

*=*

_{i}*R*

_{i}**r**

*, where*

_{i}**r**

*is a dimensionless unit vector and ||*

_{i}**R**

*|| =*

_{i}*R*, and applying the Newton–Euler equations at point

_{i}*B*

_{1}, it comes that

$swiT=[riT,d\xafB1BiTri]$ being unit screws corresponding to the direction of the platform joint reaction wrenches.

Matrix **A** used in Eq. (2) is defined in Refs. [1,3] as the parallel Jacobian matrix than can be found through the
differentiation of the loop closure equations of the robot with respect to the
platform coordinates. As a result, this matrix is always invertible if the robot is
not in a type 2 singularity.

**A**

^{–1}can be expressed in the form [2]

where **s _{t}**

*is a screw corresponding to the direction of the platform twist when leg*

_{i}*i*is disconnected. Moreover, it should be noted that

- —
**v**is a dimensionless vector parallel to the platform translational velocity vector_{i}**v**expressed at point_{Bi}*B*, i.e.,_{i}**v**=_{Bi}*λ***v**where_{i}*λ*is a scalar of which dimension is in m/s (Fig. 2); - —
*ω*is a scalar that is related to the platform rotational velocity Ω_{i}by Ω_{i}=_{i}*λω*[6] (Fig. 2). Therefore, the dimension of_{i}*ω*is in m_{i}^{−1}. On Fig. 2, point*I*_{1}corresponds to the position of the instantaneous center of rotation of the platform when leg 1 is disconnected.

*R*

_{1}of the reaction force in the joint attaching the leg 1 to the platform (located at

*B*

_{1}). Developing Eq. (3), it comes that

*f*of the external force

**f**and a given value

*C*of the external moment, and for any direction of vector

**f**, the maximal value

*R*

_{1max}of

*R*

_{1}appears when

where *b*_{1} is the distance between the application point of
the external wrench, denoted as *P*, and point *B*_{1}. *β*_{1} is the angle
between vectors **v _{1}** and $-\omega 1d\xafB1P$ (Fig. 3).

*i*= 1, 2, 3)

Equation (7) characterizes the effort
transmission between the external wrench applied on the platform and the reaction of
the platform joint located at *B _{i}*. For a given mechanism
configuration and a given value of

*R*

_{i}_{max}, it is thus possible to find the admissible ranges for

*f*and

*C*, i.e., for the parameters of the external wrench applied on the platform. Moreover, in order to avoid the breakdown of the platform joint located at

*B*, technological requirements must imply that the admissible value of

_{i}*R*should not be superior to a given value

_{i}*R*

_{adm}, i.e., $Rimax\u2264Radm$.

## Relationships Between the Maximal Passive Joint Reaction and the Pressure Angle

*α*

_{1}(see Fig. 2), may be expressed as the acute angle between the directions of vectors

**r**

_{1}and

**v**

_{1}[24]. Therefore, if

*B*

_{1}does not coincide with

*I*

_{1}

**r**is a unit vector. As a result, $\Vert v1\Vert =(cos\alpha 1)-1$. Moreover, from the definition of a planar displacement for a rigid body, $\Vert v1\Vert =d1|\omega 1|,$ where

_{1}*d*

_{1}is the distance from the platform instantaneous center of rotation

*I*

_{1}to point

*B*

_{1}(Fig. 2). Introducing these expressions into Eq. (6), it comes that

*i*= 1, 2, 3)

where $\gamma i=1+(bi/di)2-2cos\beta ibi/di,$*d _{i}* is the distance between point

*B*and the instantaneous center of rotation of the platform when leg

_{i}*i*is disconnected,

*α*is the pressure angle of the leg

_{i}*i*,

*b*is the distance between

_{i}*P*and

*B*, and

_{i}*β*is the angle between vectors

_{i}**v**

*and $-\omega id\xafBiP$ (and as a result between vectors*

_{i}**B**and

*P*_{i}**B**—Fig. 3), $d\xafBiP,$

*I*_{i}_{i}**v**

*, and*

_{i}*ω*being defined at Eqs. (1) and (4). It should be mentioned that the distance between point

_{i}*P*and

*I*is equal to

_{i}*γ*.

_{i}d_{i}Equation (10) shows that, for a
given set of external force and moment applied on the platform and for *d _{i}* > 0, the reactions in passive joints
depend not only on the pressure angle but also on the position of the instantaneous
center of rotation of the platform when one of the legs is disconnected. To the best
of our knowledge, this property has not been rigorously formulated and demonstrated
before. It allows not only giving a qualitative evaluation of the effort
transmission in PPM but also disclosing the geometrical interpretation of the
problem: from the Eq. (10

*a*), it can be shown that the mechanical system under study can be instantaneously replaced by a virtual cantilever attached to the ground at

*I*(Fig. 3) lying on a virtual point contact at

_{i}*B*, of which direction is parallel to the vector

_{i}**R**

*. The external force*

_{i}**f**is applied on the point

*P*of the cantilever.

As *α _{i}* is a criterion used for the kinetostatic design of
robots [8,13,17,20], it is of interest to find its boundaries with respect to
the technological requirements that imply that the admissible reaction

*R*in passive joints should not be superior to a given value

_{i}*R*

_{adm}, i.e., $Rimax\u2264Radm$.

The following of the paper is focused on finding the ranges on *α _{i}* and

*d*,

_{i}*d*> 0, i.e., the parameters of the equivalent cantilever system, for which this inequality $Rimax\u2264Radm$ is respected.

_{i}The case *d _{i}* = 0 is discarded, as the pressure angle
cannot be defined. However, it can be shown that the joint reaction stays under
acceptable value if and only if $|\omega i|(fbi+|C|)\u2264Radm$.

*a*) into the inequality $Rimax\u2264Radm$ leads to

*f*, cos

*α*,

_{i}*d*, and

_{i}*R*

_{adm}have positive values, a necessary condition for the existence of Eq. (11) is

where, $pi(di)=uidi2+vidi+wi$, $ui=Radm2cos2\alpha i-f2$, $vi=-2(|C|Radmcos\alpha i-cos\beta ibif2)$, $wi=C2-f2bi2$.

The inequality Eq. (14) has different
solutions, depending on the vanishing and signs of terms *u _{i}*,

*v*, and

_{i}*w*. There are three main cases

_{i}*u*> 0,

_{i}*u*< 0, and

_{i}*u*= 0. Let us consider these cases.

_{i}*u*_{i} > 0

*u*> 0 implies that $f<Radmcos\alpha i$. In this case, the polynomial

_{i}*p*has got two roots but only one corresponds to the real mechanism, i.e., to a solution of Eq. (11). The other root is solution of

_{i}*α*has roots with real values if and only if

_{i}For the analysis of this inequality, two following cases must be considered: *w _{i}* ≤ 0 and

*w*> 0.

_{i}#### w_{i} > *0*

#### w_{i} ≤ *0*

*w*≤ 0 implies that $|C|$ ≤

_{i}*f b*. Equation (16) is true if its roots are bounded by

_{i}*b*) is true, then

*u*> 0. Therefore, the only condition for Eq. (16) to be true is that

_{i}*u*_{i} < 0

#### w_{i} > *0*

*w*> 0, i.e., $|C|$ >

_{i}*f b*, from Eq. (12) it can be shown that

_{i}which is impossible in Sec. 3.2.

#### w_{i} ≤ *0*

*u*_{i} = 0

We have to analyze one last case: *u _{i}* = 0, i.e., $cos\alpha i=f/Radm$.
Here also, this condition leads to 0 ≤

*γ*≤ 1, i.e., the cantilever allows decreasing the applied force

_{i}**f**. In such a case, the solution of Eq. (14) is solution of $0\u2264vidi+wi$ with $vi=2f(cos\beta ibif-|C|)$ and $wi=C2-f2bi2$.

Three cases will appear: *v _{i}* > 0,

*v*< 0, and

_{i}*v*= 0.

_{i}#### v_{i} > *0*

#### v_{i} < *0*

*w*> 0 is not compatible with the fact that

_{i}*u*= 0. For

_{i}*w*≤ 0

_{i}Therefore, if *v _{i}* < 0, the only condition is
that Eq. (12) should be
respected.

#### v_{i} = *0*

Condition Eq. (14) can be
satisfied if and only if *w _{i}* ≥ 0. But, as
previously,

*w*> 0 is not compatible with the fact that

_{i}*u*= 0. Therefore, there exist only one possible case,

_{i}*w*= 0, i.e., $fbi=C$.

_{i}Table 1 summarized all conditions for
obtaining $Rimax\u2264Radm,$ for *d _{i}* > 0 (for

*d*= 0, the solution is directly given at Eq. (9

_{i}*b*). It should be mentioned that for planar parallel robots, the reactions in the other joints that are not linked to the platform can be found using linear relationships with respect to

*R*

_{i}_{max}(Table 2

^{2}).

$|C|>fbi$ | $|C|<fbi$ | $|C|=fbi$ | ||||||
---|---|---|---|---|---|---|---|---|

$fRadm<cos\alpha i$ | $di\u2265max((di)1,|C|Radmcos\alpha i)$ | |||||||

$fRadm>cos\alpha i$ | N/A | $cos\alpha i\u2265|C|cos\beta i+|sin\beta i|f2bi2-C2Radmbi,max(|C|Radmcos\alpha i,bi2cos\beta i)\u2264di\u2264(di)2$ | ||||||

$fRadm=cos\alpha i$ | N/A | v > 0_{i} | v < 0_{i} | v = 0_{i} | v > 0_{i} | v < 0_{i} | v = 0_{i} | |

$di\u2265max(|C|Radmcos\alpha i,-wivi)$ | N/A | N/A | $di\u2265|C|Radmcos\alpha i$ | N/A | $di\u2265bi,for\u2003\u2003\u2003\beta i=0$ |

$|C|>fbi$ | $|C|<fbi$ | $|C|=fbi$ | ||||||
---|---|---|---|---|---|---|---|---|

$fRadm<cos\alpha i$ | $di\u2265max((di)1,|C|Radmcos\alpha i)$ | |||||||

$fRadm>cos\alpha i$ | N/A | $cos\alpha i\u2265|C|cos\beta i+|sin\beta i|f2bi2-C2Radmbi,max(|C|Radmcos\alpha i,bi2cos\beta i)\u2264di\u2264(di)2$ | ||||||

$fRadm=cos\alpha i$ | N/A | v > 0_{i} | v < 0_{i} | v = 0_{i} | v > 0_{i} | v < 0_{i} | v = 0_{i} | |

$di\u2265max(|C|Radmcos\alpha i,-wivi)$ | N/A | N/A | $di\u2265|C|Radmcos\alpha i$ | N/A | $di\u2265bi,for\u2003\u2003\u2003\beta i=0$ |

Let us now consider two illustrative examples.

## Illustrative Examples

Let us now consider, for given external wrenches, the evolution of the maximal joint
reactions within the workspace of two given planar robots: the DexTAR robot, which
is a planar five-bar mechanism developed at the ETS [25] and a 3-*RPR* robot, which is
the planar model of the PAMINSA manipulator developed at the INSA of Rennes [26].

### The DexTAR Robot

- —
*l*=_{AB}*l*=_{DE}*l*=_{AB}*l*= 0.23 m_{DE} - —
*a*= 0.1375 m

*AB*and

*BP*and that leg 2 is composed of segments

*ED*and

*DP*. The active joints are located at points

*A*and

*E*. For five-bar mechanisms, it can be shown that the matrix

**A**

*of Eq. (2) is equal to [1]*

^{T}**v**(

_{1}**v**, respectively) of the translational velocity vector of the end-point of leg 2 (1, respectively) is orthogonal to the direction of the segment

_{2}*DP*(

*BP*, respectively) (Fig. 4). Therefore

where *ε* is the angle between segments *BP* and *DP* (Fig. 4). Please
note that

Taking only into account the each revolute joint can admit a maximal force *R*_{adm}, and as for such mechanisms no moments are
applied on the controlled point, the only condition for nonbreakdown of the
mechanism under a force *f* applied at *P* is
given by $f<Radmcos\alpha $.
Knowing *f* and *R*_{adm}, this remains to
fixing the maximal value *α*_{max} of *α*.

*α*

_{max}, four possible values of

*ε*are possible (Fig. 5)

Therefore, four four-bar mechanisms can be studied, depending on the assembly
mode of the five-bar robot (Fig. 5).
Therefore, the constant pressure angle loci, and as a result, the constant joint
reaction loci, can be found algebraically by studying the displacement of points *P* of the four-bar mechanisms. These borders are portions of
sextic curves [1]. The variations of the
joint reaction within the workspace of the DexTAR robot, on which is applied a
force equal to 100 N, are presented in Fig. 6 for the four working modes of the robot. In this picture, the
dotted black lines correspond to the type 1 singularities and the full black
lines to the type 2 singularities [4],
i.e., the maximal workspace boundaries. It can be shown that, the closer the
robot from type 2 singular configurations, the higher the joint reactions.

In Ref. [25], it is shown that the DexTAR
is able to pass through type 1 singularities [4]. For one given assembly mode, a position of the end-effector is
able to be attained by two working modes. Taking this result into account, the
borders of the force workspace for a given assembly mode are computed. As
previously, these borders can also be found algebraically, by studying the
displacement of points *P* of the four-bar mechanisms. For one
given assembly mode, a point of the sextic curves will belong to the border of
the force workspace if the pressure angle of the mechanism at this end-effector
position is always superior or equal to *α*_{max} for any
of the working modes. If not, this point can be reached by at least one of the
working modes, i.e., it is not a workspace boundary. Some examples of the force
workspace, for several values of *α*, are presented in Fig. 7. Obviously, the smaller *α*_{max}, the smaller the workspace. When *α*_{max} is large, the workspace has only one
aspect. For a smaller *α*_{max}, several aspects will
appear.

### The 3-RPR Robot

The PAMINSA (Fig. 8) is a parallel
manipulator with 4 DOF (Schoenflies motions) of which translation along the
vertical axis is decoupled from the displacement in the horizontal plane. When
the vertical translational motion is locked up, the PAMINSA is fully equivalent
to a 3-*RPR* manipulator (Fig. 8(b)) with equilateral
platform and base triangles, of which circumcircles have the following
radii:

- —
for the base,

*R*= 0.35 m_{b} - —
for the platform,

*R*= 0.1 m_{p}

For this mechanism, the way to compute the pressure angle and the distance between the observed joint and the platform instantaneous center of rotation is explained in Ref. [6].

The variations of the joint reaction at point *B*_{1} within the workspace for several platform orientations *ϕ* are
presented in Fig. 9 for *f* = 100 N and *C* = 5 Nm. The dotted lines
correspond to the type 2 singularities that appear if [26]

- —
for a given orientation

*ϕ*of the platform, the point*P*is located on a circle*C*(*ϕ*) centered in*O*, of radius equal to $Rb2+Rp2-2RbRpcos\phi $ - —
for orientations $\phi =\xb1cos-1(Rp/Rb),$ the robot is in singular configuration for any position of

*P*.

It can be observed that, the closer from type 2 singularities, the higher the joint reaction. Moreover, the lowest values of the joint reactions appear in the center of the workspace.

Let us analyze the force workspace of the robot. On the contrary of the DexTAR
for which the obtained expressions are symbolic, and the force workspace
boundaries can be obtained algebraically, for this robot, a numerical method has
to be used. The method consists in discretising the workspace using polar
coordinates (*r*, *θ*). For one given angle *θ*, the algorithm tests for all rising values of *r* that the manipulator can support the applied wrench (see
Tables 1 and 2). In the case where the manipulator cannot support the
applied wrench, the boundary of the force workspace is defined by the previous
computational point.

Type of legs | Joint reactions | Type of legs | Joint reactions |
---|---|---|---|

RPR | $WAiT=[fAiT,\tau ]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=\rho iRi,|CCi|=liRi$ | RPR | $WAiT=[fAiT,0]T,WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =|\tau |=Ri$ |

RRR | $WAiT=[fAiT,\tau ]T,$ | RRR | $WAiT=[fAiT,0]T,WCiT=[fCiT,\tau ]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li2Ri|sin\u025bi|$ |

$WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li1Ri|cos\u025bi|$ | |||

PRR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=li1Ri|sin\u025bi||\tau |=Ri|cos\u025bi|$ | PRR | $WAiT=[fAiT,0]T,WCiT=[fCiT,\tau ]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li2Ri|sin\u025bi|$ |

PPR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Ri|\rho i-li1cos\u025bi||CCi|=Rili2,|\tau |=Ri|sin\u025bi|$ | PPR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Ri|li1-\rho icos\u025bi||CCi|=Rili2|cos\u025bi|,|\tau |=Ri|sin\u025bi|$ |

PRP | RRP | ||

$WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Rili1|sin\u025bi|,|CCi|=Ri\rho i,$ | $WAiT=[fAiT,\tau ]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CCi|=Ri\rho i,|\tau |=Rili1|sin\u025bi|$ |

Type of legs | Joint reactions | Type of legs | Joint reactions |
---|---|---|---|

RPR | $WAiT=[fAiT,\tau ]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=\rho iRi,|CCi|=liRi$ | RPR | $WAiT=[fAiT,0]T,WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =|\tau |=Ri$ |

RRR | $WAiT=[fAiT,\tau ]T,$ | RRR | $WAiT=[fAiT,0]T,WCiT=[fCiT,\tau ]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li2Ri|sin\u025bi|$ |

$WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li1Ri|cos\u025bi|$ | |||

PRR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,0]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=li1Ri|sin\u025bi||\tau |=Ri|cos\u025bi|$ | PRR | $WAiT=[fAiT,0]T,WCiT=[fCiT,\tau ]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|\tau |=li2Ri|sin\u025bi|$ |

PPR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Ri|\rho i-li1cos\u025bi||CCi|=Rili2,|\tau |=Ri|sin\u025bi|$ | PPR | $WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Ri|li1-\rho icos\u025bi||CCi|=Rili2|cos\u025bi|,|\tau |=Ri|sin\u025bi|$ |

PRP | RRP | ||

$WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CAi|=Rili1|sin\u025bi|,|CCi|=Ri\rho i,$ | $WAiT=[fAiT,\tau ]T,WCiT=[fCiT,CCi]Twith\Vert fAi\Vert =\Vert fCi\Vert =Ri,|CCi|=Ri\rho i,|\tau |=Rili1|sin\u025bi|$ |

In the remainder of this example, we take into account only the maximal
admissible value *R*_{adm} of the reactions of the
revolute joints located at *B _{i}*. It is applied on the
platform a force of norm

*f*= 100 N and a moment of norm

*C*= 5 Nm. The shape of the force workspace, for one given assembly mode and for several values of

*R*

_{max}and platform orientation

*ϕ*is shown in Fig. 10.

It can be observed that, the greater the value of *ϕ*, i.e., the
closer from the orientation for which the robot is in singular configuration for
any position of the platform center, the smaller the workspace.

## Conclusions

This paper extends the previous works on the quality of the effort transmission in
isostatic planar closed-loop mechanisms. The traditional transmission indices only
show the ratio between input and output powers but they do not show an unacceptable
high increase of the reactions in the passive joints. In this study, it is disclosed
that the increase of reactions in passive joints depends not only on the
transmission angle but also on the position of the instantaneous center of rotation
of the platform. This is the first time that such a kinetostatic property is
rigorously formulated and clearly demonstrated. The obtained results allow expanding
the knowledge about the effort transmission quality. They are complementary
conditions to the traditional transmission indices and allow avoiding a breakdown
close to the singularity. In this aim, the boundaries for the admissible values of
the transmission angle and of the distance to the instantaneous center of rotation
are computed. The DexTAR and the 3-*RPR* robot
have been studied as illustrative examples. The effort transmission in these
manipulators has been studied as well as their reachable workspaces taking into
account the limitation of the efforts in passive joints.

Finally, it should be noted that in this paper, the disclosed properties have been only devoted to the study of planar parallel mechanisms. It is quite possible that such a concise and accurate criterion can be also obtained for spatial case. In our future work we will try to extend this approach to spatial parallel mechanisms, but it is a real challenge.

In this table, the dark joints correspond to the actuated joints.