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.

Fig. 1
Planar 5R manipulator close to a type 2 singular pose
Fig. 1
Planar 5R manipulator close to a type 2 singular pose
Close modal

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
f=-i=13Ri   and   C=-i=13d¯QBiTRi
(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
w=-[r1r2r30d¯B1B2Tr2d¯B1B3Tr3][R1R2R3]=-[sw1sw2sw3]R=-ATR
(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)
R=-A-Tw
(3)
where matrix A–1 can be expressed in the form [2]
A-1=[st1st2st3],   with   stiT=[viT,ωi]
(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
R1=-(v1Tf+ω1(C+d¯B1PTf))=-((v1T+ω1d¯B1PT)f+ω1C)
(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
R1max=maxf,C(R1)=fv12+(ω1b1)2-2v1|ω1b1|cosβ1+|ω1C|
(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).

Fig. 3
Instantaneous system equivalent to the planar
                                3-RPR robot platform
Fig. 3
Instantaneous system equivalent to the planar
                                3-RPR robot platform
Close modal
Generalizing the approach to the other legs (i = 1, 2, 3)
Rimax=maxf,C(Ri)=fvi2+(ωibi)2-2vi|ωibi|cosβi+|ωiC|
(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
|r1Tv1|=1=r1v1cosα1,   i.e.,   cosα1=|r1Tv1|r1v1
(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
R1max=1+(b1/d1)2-2cosβ1b1/d1f+|C|/d1cosα1,   for   d1>0
(9a)
R1max=|ω1|(fb1+|C|),   for   d1=0
(9b)
Generalizing the approach to the other legs (i = 1, 2, 3)
Rimax=γif+|C|/dicosαi,   for   di>0
(10a)
Rimax=|ωi|(fbi+|C|),   for   di=0
(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
f1+(bi/di)2-2cosβibi/diRadmcosαi-|C|/di
(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
0Radmcosαi-|C|/di|C|Radmcosαidi
(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)
f2(di2+bi2-2cosβibidi)  di2(Radmcosαi)2+C2-2di|C|Radmcosαi
(13)
The obtained expression can be rewritten as
0pi(di)
(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
-f2(di2+bi2-2cosβibidi)  di2(Radmcosαi)2+C2-2di|C|Radmcosαi
(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.,
(Radm2f2bi2)cos2αi-2(|C|Radmcosβibif2)cosαi   -sin2βibi2f4+f2C20
(16)
Developing and simplifying, it can be shown that this polynomial in cos αi has roots with real values if and only if
Radm2sin2βibi2f4wi0
(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
dimax((di)1,|C|/(Radmcosαi))
(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
cosαi|C|cosβi+|sinβi|f2bi2-C2Radmbi
(19a)
or
cosαi|C|cosβi-|sinβi|f2bi2-C2Radmbi
(19b)
After mathematical simplifications, it can be proven that if Eq. (19b) is true, then
cosαi|C|RadmbifRadm
(20)
which is in contradiction with ui > 0. Therefore, the only condition for Eq. (16) to be true is that
cosαimax(|C|cosβi+|sinβi|f2bi2-C2Radmbi,fRadm)
(21)
However, it can be demonstrated, by studying the sign of the function g=|C|cosβi+|sinβi|f2bi2-C2-fbi, that
|C|cosβi+|sinβi|f2bi2-C2RadmbifRadm,   for   ui>0   and   wi0
(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
fbiRadmcosαi<|C|Radmcosαidi
(23)
As in Sec. 3.2, ui < 0, which is equivalent to Radm cos αi < f, from Eq. (23) it comes
bi<fbiRadmcosαi,   thus   bi<di
(24)
If Eq. (24) is true, the expression of γi at Eq. (10a) is bounded by
γi|bi-di|di=di-bidi>0
(25)
Introducing Eq. (25) into Eq. (11), and as |C| > f bi, it comes that
f(di-bi)+fbidi=f<f(di-bi)+|C|diRadmcosαi
(26)
Thus
f<Radmcosαi,   orequivalently   ui>0
(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
fRadm>cosαi|C|cosβi+|sinβi|f2bi2-C2Radmbi
(28)
Once this condition is achieved, the condition for Eq. (11) to be true is that
max(|C|Radmcosαi,bi2cosβi)di(di)2
(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
-wivi=f2bi2-C22f(cosβibif-|C|)di
(30)

vi < 0

In this case, RimaxRadm can be satisfied if and only if
0di-wivi=f2bi2-C22f(cosβibif-|C|)
(31)
As in Sec. 3.2.1, the condition wi > 0 is not compatible with the fact that ui = 0. For wi ≤ 0
f2bi2-C22f(cosβibif-|C|)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).

Table 1

Conditions for limiting the maximal values of the revolute joint linked to the platform, for di> 0

|C|>fbi|C|<fbi|C|=fbi
fRadm<cosαidimax((di)1,|C|Radmcosαi)
fRadm>cosαiN/Acosαi|C|cosβi+|sinβi|f2bi2-C2Radmbi,max(|C|Radmcosαi,bi2cosβi)di(di)2
fRadm=cosαiN/Avi > 0vi < 0vi = 0vi > 0vi < 0vi = 0
dimax(|C|Radmcosαi,-wivi)N/AN/Adi|C|RadmcosαiN/Adibi,for   βi=0
|C|>fbi|C|<fbi|C|=fbi
fRadm<cosαidimax((di)1,|C|Radmcosαi)
fRadm>cosαiN/Acosαi|C|cosβi+|sinβi|f2bi2-C2Radmbi,max(|C|Radmcosαi,bi2cosβi)di(di)2
fRadm=cosαiN/Avi > 0vi < 0vi = 0vi > 0vi < 0vi = 0
dimax(|C|Radmcosαi,-wivi)N/AN/Adi|C|RadmcosαiN/Adibi,for   βi=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

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

Fig. 4
Kinematic chain of the planar five-bar robot
Fig. 4
Kinematic chain of the planar five-bar robot
Close modal
  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]
AT=[r1r2]=[cosψ1cosψ2sinψ1sinψ2]
(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
v1=[-sinψ2cosψ2]
(34)
As a result
α1=cos-1(r1Tv1r1v1)=cos-1(|sin(ψ2-ψ1)|)=cos-1(|sinɛ|)=α2=cos-1(r2Tv2r2v2)
(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)
ɛ1=sin-1(cosαmax),   ɛ2=ɛ1+π,   ɛ3=-ɛ1,   ɛ4=-ɛ1+π
(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.

Fig. 6
Variation of the robot joint reaction (in Newton) within the workspace
                            for f = 100 N
Fig. 6
Variation of the robot joint reaction (in Newton) within the workspace
                            for f = 100 N
Close modal

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.

Fig. 7
Workspace shape as a function of the maximal pressure angle αmax
Fig. 7
Workspace shape as a function of the maximal pressure angle αmax
Close modal

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:

Fig. 8
Kinematics of the PAMINSA
Fig. 8
Kinematics of the PAMINSA
Close modal
  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]
AT=[sw1sw2sw3]=[r1r2r3C1C2C3]
(37)
with
riT=[-sinqicosqi]
and
C1=0,   C2=3Rp[-sinφcosφ]Tr2,   C3=3Rp[-sin(π/3+φ)cos(π/3+φ)]Tr3,   if   Bi=B1
(38a)
C1=3Rp[-sin(φ+π)cos(φ+π)]Tr1,   C2=0,   C3=3Rp[-sin(2π/3+φ)cos(2π/3+φ)]Tr3,   if   Bi=B2
(38b)
C1=3Rp[-sin(φ+4π/3)cos(φ+4π/3)]Tr1,   C2=3Rp[-sin(-π/3+φ)cos(-π/3+φ)]Tr2,   C3=0,   if   Bi=B3
(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]

Fig. 9
Variations of the joint reaction (in Newton) at point B1 within the workspace for several
                            platform orientations ϕ, for f = 100 N
                            and m = 5 Nm
Fig. 9
Variations of the joint reaction (in Newton) at point B1 within the workspace for several
                            platform orientations ϕ, for f = 100 N
                            and m = 5 Nm
Close modal
  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.

Table 2

Norm of the reaction effort inside of the leg joints of planar parallel robots

Type of legsJoint reactionsType of legsJoint reactions
RPRWAiT=[fAiT,τ]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|τ|=ρiRi,|CCi|=liRiRPRWAiT=[fAiT,0]T,WCiT=[fCiT,0]TwithfAi=fCi=|τ|=Ri
RRRWAiT=[fAiT,τ]T,RRRWAiT=[fAiT,0]T,WCiT=[fCiT,τ]TwithfAi=fCi=Ri,|τ|=li2Ri|sinɛi|
WCiT=[fCiT,0]TwithfAi=fCi=Ri,|τ|=li1Ri|cosɛi|
PRRWAiT=[fAiT,CAi]T,WCiT=[fCiT,0]TwithfAi=fCi=Ri,|CAi|=li1Ri|sinɛi||τ|=Ri|cosɛi|PRRWAiT=[fAiT,0]T,WCiT=[fCiT,τ]TwithfAi=fCi=Ri,|τ|=li2Ri|sinɛi|
PPRWAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Ri|ρi-li1cosɛi||CCi|=Rili2,|τ|=Ri|sinɛi|PPRWAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Ri|li1-ρicosɛi||CCi|=Rili2|cosɛi|,|τ|=Ri|sinɛi|
PRPRRP
WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Rili1|sinɛi|,|CCi|=Riρi,WAiT=[fAiT,τ]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CCi|=Riρi,|τ|=Rili1|sinɛi|
Type of legsJoint reactionsType of legsJoint reactions
RPRWAiT=[fAiT,τ]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|τ|=ρiRi,|CCi|=liRiRPRWAiT=[fAiT,0]T,WCiT=[fCiT,0]TwithfAi=fCi=|τ|=Ri
RRRWAiT=[fAiT,τ]T,RRRWAiT=[fAiT,0]T,WCiT=[fCiT,τ]TwithfAi=fCi=Ri,|τ|=li2Ri|sinɛi|
WCiT=[fCiT,0]TwithfAi=fCi=Ri,|τ|=li1Ri|cosɛi|
PRRWAiT=[fAiT,CAi]T,WCiT=[fCiT,0]TwithfAi=fCi=Ri,|CAi|=li1Ri|sinɛi||τ|=Ri|cosɛi|PRRWAiT=[fAiT,0]T,WCiT=[fCiT,τ]TwithfAi=fCi=Ri,|τ|=li2Ri|sinɛi|
PPRWAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Ri|ρi-li1cosɛi||CCi|=Rili2,|τ|=Ri|sinɛi|PPRWAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Ri|li1-ρicosɛi||CCi|=Rili2|cosɛi|,|τ|=Ri|sinɛi|
PRPRRP
WAiT=[fAiT,CAi]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CAi|=Rili1|sinɛi|,|CCi|=Riρi,WAiT=[fAiT,τ]T,WCiT=[fCiT,CCi]TwithfAi=fCi=Ri,|CCi|=Riρi,|τ|=Rili1|sinɛi|

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.

Fig. 10
Constant orientation workspace as a function of Rmax and the platform orientation ϕ
Fig. 10
Constant orientation workspace as a function of Rmax and the platform orientation ϕ
Close modal

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.

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