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· and w=J-Tτ, where J is the Jacobian matrix, t is the platform twist, q· 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 [−τimax,τimax] [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 [−τimax,τimax]. 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 JT, 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 5R manipulator (Fig. 1). Close to such singularity a small effort w applied on the end-effector of the manipulator will create a large reaction R1 in the passive joint B. But in this pose, the actuator torques τ1 will stay under acceptable values, as it depends only on the small component F1 of the reaction R1. 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.

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

For any PPM, the reaction forces Ri in the platform passive joints (denoted as Bi in Fig. 2, i = 1,…, 3) can be related to the external wrench wT= [fT, 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
Fig. 2

Determination of the pressure angle for the planar 3-RPR robot

Fig. 2

Determination of the pressure angle for the planar 3-RPR robot

Close modal
(1)
where d¯QBiT=[-yQBi,xQBi] with xQBi and yQBi are the coordinates of vector QBi along x and y axes (the position of point Bi 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 Ri = Riri, where ri is a dimensionless unit vector and ||Ri|| = Ri, and applying the Newton–Euler equations at point B1, it comes that
(2)

swiT=[riT,d¯B1BiTri] 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.

The reactions R of the passive joints can be found from Eq. (2)
(3)
where matrix A–1 can be expressed in the form [2]
(4)

where sti is a screw corresponding to the direction of the platform twist when leg i is disconnected. Moreover, it should be noted that

  1. vi is a dimensionless vector parallel to the platform translational velocity vector vBi expressed at point Bi, i.e., vBi = λvi where λ is a scalar of which dimension is in m/s (Fig. 2);

  2. ωi is a scalar that is related to the platform rotational velocity Ωi by Ωi = λωi [6] (Fig. 2). Therefore, the dimension of ωi is in m−1. On Fig. 2, point I1 corresponds to the position of the instantaneous center of rotation of the platform when leg 1 is disconnected.

Without loss of generality, let us consider the norm R1 of the reaction force in the joint attaching the leg 1 to the platform (located at B1). Developing Eq. (3), it comes that
(5)
For a given norm 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 R1max of R1 appears when
(6)

where b1 is the distance between the application point of the external wrench, denoted as P, and point B1. β1 is the angle between vectors v1 and -ω1d¯B1P (Fig. 3).

Generalizing the approach to the other legs (i = 1, 2, 3)
(7)

Equation (7) characterizes the effort transmission between the external wrench applied on the platform and the reaction of the platform joint located at Bi. For a given mechanism configuration and a given value of Rimax, 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 Bi, technological requirements must imply that the admissible value of Ri should not be superior to a given value Radm, i.e., RimaxRadm.

It is shown in Sec. 3 that Eq. (7) can be related not only to the value of the pressure angle (the pressure angle is equal to 90 deg minus the transmission angle) but also to the value of the distance of the platform instantaneous center of rotation Ii when leg i is disconnected.

Relationships Between the Maximal Passive Joint Reaction and the Pressure Angle

By combining Eqs. (2) and (4), it comes that sw1Tst1=r1Tv1=1. Moreover, it was shown in Ref. [6] that the pressure angle of the leg 1, depicted as α1 (see Fig. 2), may be expressed as the acute angle between the directions of vectors r1 and v1 [24]. Therefore, if B1 does not coincide with I1
(8)
By definition, r1 is a unit vector. As a result, v1=(cosα1)-1. Moreover, from the definition of a planar displacement for a rigid body, v1=d1|ω1|, where d1 is the distance from the platform instantaneous center of rotation I1 to point B1 (Fig. 2). Introducing these expressions into Eq. (6), it comes that
(9a)
(9b)
Generalizing the approach to the other legs (i = 1, 2, 3)
(10a)
(10b)

where γi=1+(bi/di)2-2cosβibi/di,di is the distance between point Bi and the instantaneous center of rotation of the platform when leg i is disconnected, αi is the pressure angle of the leg i, bi is the distance between P and Bi, and βi is the angle between vectors vi and -ωid¯BiP (and as a result between vectors BiP and BiIi—Fig. 3), d¯BiP,vi, and ωi being defined at Eqs. (1) and (4). It should be mentioned that the distance between point P and Ii is equal to γidi.

Equation (10) shows that, for a given set of external force and moment applied on the platform and for di > 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. (10a), it can be shown that the mechanical system under study can be instantaneously replaced by a virtual cantilever attached to the ground at Ii (Fig. 3) lying on a virtual point contact at Bi, of which direction is parallel to the vector Ri. The external force f is applied on the point P of the cantilever.

As αi 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 Ri in passive joints should not be superior to a given value Radm, i.e., RimaxRadm.

The following of the paper is focused on finding the ranges on αi and di, di > 0, i.e., the parameters of the equivalent cantilever system, for which this inequality RimaxRadm is respected.

The case di = 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 |ωi|(fbi+|C|)Radm.

Introducing Eq. (10a) into the inequality RimaxRadm leads to
(11)
Please note that, as by definition, f, cos αi, di, and Radm have positive values, a necessary condition for the existence of Eq. (11) is
(12)
If not, the left term of Eq. (11) will always be superior to Radm. Let us now square the left and right sides of Eq. (11) and multiply them by di2 (from Eq. (10a), di > 0)
(13)
The obtained expression can be rewritten as
(14)

where, pi(di)=uidi2+vidi+wi, ui=Radm2cos2αi-f2, vi=-2(|C|Radmcosαi-cosβibif2), wi=C2-f2bi2.

The inequality Eq. (14) has different solutions, depending on the vanishing and signs of terms ui, vi, and wi. There are three main cases ui > 0, ui < 0, and ui = 0. Let us consider these cases.

ui > 0

ui > 0 implies that f<Radmcosαi. In this case, the polynomial pi 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
(15)
In order to obtain a root of Eq. (14) with physical values, it is necessary that the condition vi2-4uiwi0 is respected, i.e.,
(16)
Developing and simplifying, it can be shown that this polynomial in cos αi has roots with real values if and only if
(17)

For the analysis of this inequality, two following cases must be considered: wi ≤ 0 and wi > 0.

wi > 0

The condition wi > 0 implies that |C| > f bi. Here, Eq. (16) has no real roots, i.e., Eq. (16) is true for any value of αi. Thus, the condition for Eq. (11) to be true is that
(18)

where (di)1=(-vi+vi2-4uiwi)/2ui is the root of Eq. (14) solution of Eq. (11).

wi ≤ 0

The condition wi ≤ 0 implies that |C| ≤ f bi. Equation (16) is true if its roots are bounded by
(19a)
or
(19b)
After mathematical simplifications, it can be proven that if Eq. (19b) is true, then
(20)
which is in contradiction with ui > 0. Therefore, the only condition for Eq. (16) to be true is that
(21)
However, it can be demonstrated, by studying the sign of the function g=|C|cosβi+|sinβi|f2bi2-C2-fbi, that
(22)

Thus Eq. (21) implies that ui > 0, which is true. Then, Eq. (16) is always true in this section. As a result, the only condition for Eq. (11) to be true is Eq. (18).

ui < 0

ui < 0 implies that f>Radmcosαi. Introducing this into Eq. (11) leads to 0 ≤ γi ≤ 1, i.e., the cantilever allows decreasing the applied force f. Here also, two cases of coming from the analysis of Eq. (17) should be studied, i.e., wi ≤ 0 and wi > 0.

wi > 0

If wi > 0, i.e., |C| > f bi, from Eq. (12) it can be shown that
(23)
As in Sec. 3.2, ui < 0, which is equivalent to Radm cos αi < f, from Eq. (23) it comes
(24)
If Eq. (24) is true, the expression of γi at Eq. (10a) is bounded by
(25)
Introducing Eq. (25) into Eq. (11), and as |C| > f bi, it comes that
(26)
Thus
(27)

which is impossible in Sec. 3.2.

wi ≤ 0

If wi ≤ 0, i.e., |C|f bi, it could be shown after several mathematical simplifications and looking at the results of Sec. 3.1.2 that Eq. (16) is true if and only if
(28)
Once this condition is achieved, the condition for Eq. (11) to be true is that
(29)

where (di)2=(-vi-vi2-4uiwi)/2ui is the root of Eq. (14) solution of Eq. (11).

ui = 0

We have to analyze one last case: ui = 0, i.e., cosαi=f/Radm. Here also, this condition leads to 0 ≤ γi ≤ 1, i.e., the cantilever allows decreasing the applied force f. In such a case, the solution of Eq. (14) is solution of 0vidi+wi with vi=2f(cosβibif-|C|) and wi=C2-f2bi2.

Three cases will appear: vi > 0, vi < 0, and vi = 0.

vi > 0

In this case, RimaxRadm can be satisfied if and only if
(30)

vi < 0

In this case, RimaxRadm can be satisfied if and only if
(31)
As in Sec. 3.2.1, the condition wi > 0 is not compatible with the fact that ui = 0. For wi ≤ 0
(32)

Therefore, if vi < 0, the only condition is that Eq. (12) should be respected.

vi = 0

Condition Eq. (14) can be satisfied if and only if wi ≥ 0. But, as previously, wi > 0 is not compatible with the fact that ui = 0. Therefore, there exist only one possible case, wi = 0, i.e., fbi=C.

Table 1 summarized all conditions for obtaining RimaxRadm, for di > 0 (for di = 0, the solution is directly given at Eq. (9b). 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 Rimax (Table 2 2).

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

The DexTAR is a five-bar mechanism [27] (Fig. 4) of which dimensions are

  1. lAB = lDE = lAB = lDE = 0.23 m

  2. a = 0.1375 m

In the following of the paper, it is considered that the leg 1 is composed of the segments 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 AT of Eq. (2) is equal to [1]
(33)
Moreover, disconnecting leg 1 (2, respectively) from the end-effector, for a fixed position of the actuator 2 (1, respectively), the direction v1 (v2, respectively) of the translational velocity vector of the end-point of leg 2 (1, respectively) is orthogonal to the direction of the segment DP (BP, respectively) (Fig. 4). Therefore
(34)
As a result
(35)

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

  1. ε is denoted as the transmission angle of the robot [8];

  2. fixing angle ε is equivalent to fixing the shape of the triangle BPD and the robot can be shown as a 1-DOF planar four-bar mechanism (Fig. 5 
    Fig. 5

    The four equivalent four-bars mechanisms, for a fixed value of αmax

    Fig. 5

    The four equivalent four-bars mechanisms, for a fixed value of αmax

    Close modal
    ).

Taking only into account the each revolute joint can admit a maximal force Radm, 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<Radmcosα. Knowing f and Radm, this remains to fixing the maximal value αmax of α.

For a fixed angle αmax, four possible values of ε are possible (Fig. 5)
(36)

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 αmax 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 αmax, the smaller the workspace. When αmax is large, the workspace has only one aspect. For a smaller αmax, 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:

  1. for the base, Rb = 0.35 m

  2. for the platform, Rp = 0.1 m

For 3-RPR robots, it can be shown that the matrix AT of Eq. (2) expressed at point Bi is equal to [26]
(37)
with
and
(38a)
(38b)
(38c)

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 B1 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]

  1. 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+Rp2-2RbRpcosφ

  2. for orientations φ=±cos-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.

In the remainder of this example, we take into account only the maximal admissible value Radm of the reactions of the revolute joints located at Bi. 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 Rmax 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.

2

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

References

1.
Merlet
,
J. P.
,
2006
,
Parallel Robots
, 2nd ed.,
Springer
,
New York
.
2.
Bonev
,
I. A.
,
Zlatanov
,
D.
, and
Gosselin
,
C. M.
,
2003
, “
Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory
,”
ASME J. Mech. Des.
,
125
(
3
), pp.
573
581
.10.1115/1.1582878
3.
Daniali
,
M. H. R.
,
Zsombor-Murray
,
P. J.
, and
Angeles
,
J.
,
1995
, “
Singularity Analysis of Planar Parallel Manipulators
,”
Mech. Mach. Theory
,
30
(
5
), pp.
665
678
.10.1016/0094-114X(94)00071-R
4.
Gosselin
,
C. M.
, and
Angeles
,
J.
,
1990
, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
281
290
.10.1109/70.56660
5.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
,
2002
, “
Constraint Singularities of Parallel Mechanisms
,”
IEEE International Conference on Robotics and Automation
, Washington, DC, May 11–15.
6.
Arakelian
,
V.
,
Briot
,
S.
, and
Glazunov
,
V.
,
2008
, “
Increase of Singularity-Free Zones in the Workspace of Parallel Manipulators Using Mechanisms of Variable Structure
,”
Mech. Mach. Theory
,
43
(
9
), pp.
1129
1140
.10.1016/j.mechmachtheory.2007.09.005
7.
Alba-Gomez
,
O.
,
Wenger
,
P.
, and
Pamanes
,
A.
,
2005
, “
A Consistent Kinetostatic Indices for Planar 3-DOF Parallel Manipulators, Application to the Optimal Kinematic Inversion
,”
Proceedings of the ASME 2005 IDETC/CIE Conference
, Long Beach, CA, Sept. 24–28.
8.
Balli
,
S.
, and
Chand
,
S.
,
2002
, “
Transmission Angle in Mechanisms
,”
Mech. Mach. Theory
,
37
, pp.
175
195
.10.1016/S0094-114X(01)00067-2
9.
Angeles
,
J.
, and
López-Cajún
,
C.
,
1991
,
Optimization of Cam Mechanisms
,
Kluwer Academic Publishers B. V.
,
Dordrecht
.
10.
Sutherland
,
G.
, and
Roth
,
B.
,
1973
, “
A Transmission Index for Spatial Mechanisms
,”
ASME J. Eng. Ind.
,
95
, pp.
589
597
.10.1115/1.3438195
11.
Chen
,
C.
, and
Angeles
,
J.
,
2007
, “
Generalized Transmission Index and Transmission Quality for Spatial Linkages
,”
Mech. Mach. Theory
,
42
(
9
), pp.
1225
1237
.10.1016/j.mechmachtheory.2006.08.001
12.
Gosselin
,
C. M.
, and
Angeles
,
J.
,
1991
, “
A Global Performance Index for the Kinematic Optimization of Robotic Manipulators
,”
J. Mech. Des.
,
113
(
3
), pp.
220
226
.10.1115/1.2912772
13.
Rakotomanga
,
N.
,
Chablat
,
D.
, and
Caro
,
S.
,
2008
, “
Kinetostatic Performance of a Planar Parallel Mechanism With Variable Actuation
,”
11th International Symposium on Advances in Robot Kinematics
, Kluwer Academic Publishers, Batz-sur-mer, France, June.
14.
Ranganath
,
R.
,
Nair
,
P. S.
,
Mruthyunjaya
,
T. S.
, and
Ghosal
,
A.
,
2004
, “
A Force–Torque Sensor Based on a Stewart Platform in a Near-Singular Configuration
,”
Mech. Mach. Theory
,
39
(
9
), pp.
971
998
.10.1016/j.mechmachtheory.2004.04.005
15.
Stocco
,
L.
,
Salcudean
,
S.
, and
Sassani
,
F.
,
1998
, “
Fast Constrained Global Minimax Optimization of Robot Parameters
,”
Robotica
,
16
, pp.
595
605
.10.1017/S0263574798000435
16.
Hayward
,
V.
,
Choksi
,
J.
,
Lanvin
,
G.
, and
Ramstein
,
C.
,
1994
, “
Design and Multi-Objective Optimization of a Linkage for a Haptic Interface
,”
Advances in Robot Kinematics
, Springer, pp.
352
359
.
17.
Frisoli
,
A.
,
Prisco
,
M.
,
Salsedo
,
F.
, and
Bergamasco
,
M.
,
1999
, “
A Two Degrees-of-Freedom Planar Haptic Interface With High Kinematic Isotropy
,”
Proceedings of the 8th IEEE International Workshop on Robot and Human Interaction (RO-MAN'99)
, pp.
297
302
.
18.
Liu
,
X. -J.
,
Wu
,
C.
, and
Wang
,
J.
,
2012
, “
A New Approach for Singularity Analysis and Closeness Measurement to Singularities of Parallel Manipulators
,”
ASME J. Mech. Rob.
,
4
, p.
041001
.10.1115/1.4007004
19.
Takeda
,
Y.
, and
Funabashi
,
H.
,
1995
, “
Motion Transmissibility of In-Parallel Actuated Manipulators
,”
Trans. JSME Int. J., Ser. C
,
38
(
4
), pp.
749
755
.
20.
Takeda
,
Y.
, and
Funabashi
,
H.
,
1996
, “
Kinematic and Static Characteristics of In-Parallel Actuated Manipulators at Singular Points and in Their Neighborhood
,”
Trans. JSME Int. J., Ser. C
,
39
(
1
), pp.
85
93
.
21.
Merlet
,
J.-P.
,
2006
, “
Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots
,”
ASME J. Mech Des.
,
128
(
1
), pp.
199
206
.10.1115/1.2121740
22.
Hubert
,
J.
, and
Merlet
,
J.-P.
,
2009
, “
Static of Parallel Manipulators and Closeness to Singularity
,”
J. Mech. Rob.
,
1
(
1
).
23.
Briot
,
S.
,
Pashkevich
,
A.
, and
Chablat
,
D.
,
2010
,
Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications
,”
Proceedings of the 2010 IEEE International Conference on Robotics and Automation (ICRA 2010
), Anchorage, Alaska, May 3–8.
24.
Terminology for the Mechanism and Machine Science
,”
2003
,
Mech. Mach. Theory
,
38
.
25.
Campos
,
L.
,
Bourbonnais
,
F.
,
Bonev
,
I. A.
, and
Bigras
,
P.
,
2010
, “
Development of a Five-Bar Parallel Robot With Large Workspace
,”
Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE
, Montréal, Québec, Canada, Aug. 15–18.
26.
Briot
,
S.
,
Arakelian
,
V.
, and
Guégan
,
S.
,
2008
, “
Design and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator With High-Load Carrying Capacity
,”
J. Mech. Des.
,
130
(
12
), p.
122303
.10.1115/1.2991137
27.
Chablat
,
D.
,
Wenger
,
P.
, and
Angeles
,
J.
,
1998
, “
The Isoconditioning Loci of a Class of Closed-Chain Manipulators
,”
Proceedings of the IEEE International Conference on Robotics and Automation
, May, pp.
1970
1976
.