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2 Overconstrained planar parallel robots with coupled motions

In the general case, in a planar parallel robotic manipulator (PPM) with coupled motions each operational velocity depends in the general case on three actuated joint velocities: 1 1 1 2 3v v ( q ,q ,q )= & & & , 2 2 1 2 3v v ( q ,q ,q )= & & & and

1 2 3( q ,q ,q )δ δω ω= & & & . In some specific solutions, one or two operational velocities depend on just one or two actuator velocities. We note that, in this particular case, the Jacobian matrix in Eq. (6.1) is not triangular and the parallel robot always has coupled motions. They have just a few partially decoupled motions.

The overconstrained solutions of PPMs with coupled motions and q independent loops meet the condition p

i1f 3 6q< +∑ . They may have

identical limbs or limbs with different structures and could be actuated by linear or rotating motors.

The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact.

2.1 Basic solutions

The basic solutions presented in this section are either fully-parallel and non fully-parallel solutions. There are no idle mobilities in the basic solutions.

2.1.1 Fully-parallel solutions

In the fully-parallel solutions of PPMs with coupled motions F ← G1-G2-G3 presented in this section, the moving platform n≡nGi (i=1, 2, 3) is connected to the reference platform 1≡1Gi≡0 by three planar limbs with three

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, 27 DOI 10.1007/978-90-481-9831-3_2, © Springer Science + Business Media B.V. 2010

28 2 Overconstrained planar parallel robots with coupled motions

degrees of connectivity. One actuator is combined in a revolute or pris-matic pair of each limb.

The various types of planar simple and complex limbs with three degrees of connectivity used in the basic solutions illustrated in this section are presented in Figs. 2.1-2.3. The simple limbs combine only revolute and prismatic joints. One (Fig. 2.2a, c-g) or two (Figs. 2.2b, h and 2.3a) planar parallelogram loops are combined in the complex limbs. One telescopic parallelogram loop Pat is combined in the complex limb in Fig. 2.3b.

Various solutions of PPMs with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topology presented in Figs. 2.1-2.3. Only solutions with three identical limb types are illustrated in Figs. 2.4-2.9. The revolute joints in the three limbs have parallel axes and the directions of the prismatic joints are parallel to a plane perpendicular to the rotation axes of the revolute joints. The actuator is mounted on the fixed base in the solutions illustrated in Figs. 2.4-2.9 excepting the solutions in Fig.2.5a and Fig. 2.9a in which the linear actuator is mounted on a moving link.

All motions are coupled in the solutions presented in Figs. 2.4-2.9 and 1 1 1 2 3v v ( q ,q ,q )= & & & , 2 2 1 2 3v v ( q ,q ,q )= & & & and 1 2 3( q ,q ,q )δ δω ω= & & & - see Table

2.1. The prismatic joints between links 4 and 5 (Fig. 2.9a) is actuated to obtain solutions with coupled motions. If a revolute joint mounted on the fixed base and combined in the parallelogram loops is actuated, solutions with one uncoupled translational motion can be obtained and 1 1 1v v ( q )= & .

The particular case with one uncoupled translation 1 1 1v v ( q )= & can also be obtained by using the solution in Figs. 2.10. Other solutions with this partially decoupled translation can be obtained (Figs. 2.11-2.15) by combining G1-limb in Fig. 2.1f or 2.2e with other two identical or different limbs in Figs. 2.1a-e and 2.2a-d.

The particular case with partially decoupled translational motions 1 1 1 2v v ( q ,q )= & & and 2 2 1 2v v ( q ,q )= & & in Figs. 2.16-2.18 are obtained from the

solutions in Figs. 2.4a and 2.5a by superposing the axes of the revolute joints connecting limbs G1 and G2 to the moving platform. The axes of the three actuated revolute joints are also superposed in the solutions presented in Figs. 2.16b and 2.17b. In this way, the workspace can be significantly increased.

The limb topology and connecting conditions of the solutions in Figs. 2.4-2.17 are systematized in Table 2.2, as are their structural parameters in Tables 2.4-2.6.

The particular case with decoupled rotational motion 3( q )δ δω ω= & can be obtained by using three limbs with identical or different topology

2.1 Basic solutions 29

presented in Figs. 2.1g and 2.2f-h. Solutions with three identical limb types but different actuation selection are illustrated in Figs. 2.18-2.20. One actuator is mounted on the fixed base and the other two are mounted on a moving link in these solutions. The limb topology and connecting conditions of the solutions in Figs. 2.18-2.20 are systematized in Table 2.3, as are their structural parameters in Table 2.6. The first revolute joint of the three limbs have the same rotation axis. The rotation axis of the moving platform has a fixed position in these solutions.

Solutions with decoupled and unlimited rotational motion of the moving platform with different limb topologies are illustrated in Figs. 2.21-2.36. These solutions use two identical limbs (Figs. 2.1a-c and 2.2a-c) for positioning the moving platform and a different limb (Figs. 2.1g, 2.2f-g and 2.3) for rotating it. Their limb topology and connecting conditions are systematized in Table 2.3.

The rotation axis of the moving platform has a fixed position in the solutions in Figs. 2.21-2.30 and a variable position in the solutions in Figs. 2.31-2.36. The solutions using G3-limb can provide an unlimited angle of rotation of the moving platform with variable position of the rotation axis.

The structural parameters of the solutions in Figs. 2.21-2.36 are systematized in Tables 2.6-2.10.


Fig. 2.1. Simple limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( 1 2, , δv v ω )


Fig. 2.2. Complex limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( 1 2, , δv v ω ), and combining one or two planar parallelogram loops


Fig. 2.3. Complex limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( 1 2, , δv v ω ), and combining planar parallelogram loops (a) or a telescopic parallelogram loop (b)

Table 2.1. Motion coupling in overconstrained PPM with no idle mobilities presented in Figs. 2.4-2.36

No. Motion coupling Examples

Note

1 1 1 1 2 3v v ( q ,q ,q )= & & &

2 2 1 2 3v v ( q ,q ,q )= & & &

1 2 3( q ,q ,q )δ δω ω= & & &

Figs. 2.4-2.9

The two translational and one rotational motions are coupled

2 1 1 1v v ( q )= &

2 2 1 2 3v v ( q ,q ,q )= & & &

1 2 3( q ,q ,q )δ δω ω= & & &

Figs. 2.10-2.15

One decoupled translational motion

3 1 1 1 2v v ( q ,q )= & &

2 2 1 2v v ( q ,q )= & &

1 2 3( q ,q ,q )δ δω ω= & & &

Figs. 2.16-2.17

Partially decoupled translational motions

4 1 1 1 2 3v v ( q ,q ,q )= & & &

2 2 1 2 3v v ( q ,q ,q )= & & &

1( q )δ δω ω= &

Figs. 2.18-2.20

Decoupled rotational motion

5 1 1 1 2v v ( q ,q )= & &

2 2 1 2v v ( q ,q )= & &

3( q )δ δω ω= &

Figs. 2.21-2.36

Uncoupled rotational motion and partially decoupled translational motions


Table 2.2. Limb topology and connecting conditions of the overconstrained PPM with no idle mobilities presented in Figs. 2.4-2.17

No. PPM type

Limb topology

Connecting conditions

1 3-RRR (Fig. 2.4a)

R||R||R (Fig. 2.1a)

The directions of the revolute joints of the three limbs are parallel.

2 3-RRP (Fig. 2.4b)

R||R⊥ P (Fig. 2.1d)

The directions of the revolute joints of the three limbs are parallel. The directions of the prismatic joints are parallel to a plane perpendicular to the axis of the revolute joints.

3 3-RPR (Fig. 2.5a)

R⊥ P⊥ ||R (Fig. 2.1b)

Idem No. 2

4 3-PRR (Fig. 2.5b)

P⊥ R||R (Fig. 2.1c)

Idem No. 2

5 3-PRP (Fig. 2.6)

P⊥ R⊥ P (Fig. 2.1e)

Idem No. 2

6 3-PaPaR (Fig. 2.7)

Pa||Pa||R (Fig. 2.2b)

Idem No. 1

7 3-PaRR (Fig. 2.8a)

Pa||R||R (Fig. 2.2a)

Idem No. 1

8 3-PaRP (Fig. 2.8b)

Pa||R⊥ P (Fig. 2.2d)

Idem No. 2

9 3-PaPR (Fig. 2.9a)

Pa⊥ P⊥ ||R (Fig. 2.2e)

Idem No. 2

10 3-PPaR (Fig. 2.9b)

P⊥ Pa ||R (Fig. 2.2c)

Idem No. 2

11 3-PPR (Fig. 2.10)

P⊥ P ⊥⊥ R (Fig. 2.1f)

Idem No. 2

12 1PPR-2RRR (Fig. 2.11)

P⊥ P ⊥⊥ R (Fig. 2.1f) R||R||R (Fig. 2.1a)

Idem No. 2

13 1PPR-2RRP (Fig. 2.12a)

P⊥ P ⊥⊥ R (Fig. 2.1f) R||R⊥ P (Fig. 2.1d)

Idem No. 2

14 1PPR-2RPR (Fig. 2.12b)

P⊥ P ⊥⊥ R (Fig. 2.1f) R⊥ P⊥ ||R (Fig. 2.1b)

Idem No. 2


Table 2.2. (cont.)

15 1PPR-2PRR (Fig. 2.13a)

P⊥ P ⊥⊥ R (Fig. 2.1f) P⊥ R||R (Fig. 2.1c)

Idem No. 2

16 1PPR-2PRP (Fig. 2.13b)

P⊥ P ⊥⊥ R (Fig. 2.1f) P⊥ R⊥ P (Fig. 2.1e)

Idem No. 2

17 1PaPR-2PaRR (Fig. 2.14a)

Pa⊥ P⊥ ||R (Fig. 2.2e) Pa||R||R (Fig. 2.2a)

Idem No. 2

18 1PaPR-2PPaR (Fig. 2.14b)

Pa⊥ P⊥ ||R (Fig. 2.2e) P⊥ Pa||R (Fig. 2.2c)

Idem No. 2

19 1PaPR-2PaPaR (Fig. 2.15a)

Pa⊥ P⊥ ||R (Fig. 2.2e) Pa||Pa||R (Fig. 2.2b)

Idem No. 2

20 1PaPR-2PaRP (Fig. 2.15b)

Pa⊥ P⊥ ||R (Fig. 2.2e) Pa||R ⊥ P (Fig. 2.2d)

Idem No. 2

21 3RRR (Fig. 2.16a)

R||R||R (Fig. 2.1a)

Idem No. 1 The axes of the revolute joints connecting limbs G1 and G2 to the moving platform are superposed.

22 3RRR (Fig. 2.16b)

R||R||R (Fig. 2.1a)

Idem No. 21 The axes of the three actuated revolute joints are superposed.

23 3RPR (Fig. 2.17a)

R⊥ P⊥ ||R (Fig. 2.1b)

Idem 2 The axes of the revolute joints connecting limbs G1 and G2 to the moving platform are superposed.

24 2RPR-1RPR (Fig. 2.17b)

R⊥ P⊥ ||R R⊥ P⊥ ||R (Fig. 2.1b)

Idem 23 The axes of the first revolute joints of the three limbs are superposed.


Table 2.3. Limb topology and connecting conditions of the overconstrained PPM with no idle mobilities presented in Figs. 2.18-2.36

No. PPM type

Limb topology

Connecting conditions

1 2RPP-1RPP (Fig. 2.18)

R⊥ P ⊥⊥ P (Fig. 2.1g)

The directions of the revolute joints of the three limbs are parallel. The directions of the prismatic joints are parallel to a plane perpendicular to the axis of the revolute joints.

2 2RPPa-1RPPa (Fig. 2.19)

R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

3 2RPaP-1RPaP (Fig. 2.20a)

R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

4 2RPaPa-1RPaPa (Fig. 2.20b)

R||Pa||Pa (Fig. 2.2h)

Idem No. 1

5 2PRR-1RPP (Fig. 2.21a)

P⊥ R||R (Fig. 2.1c) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

6 2RRR-1RPP (Fig. 2.21b)

R||R||R (Fig. 2.1a) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

7 2RPR-1RPP (Fig. 2.21c)

R⊥ P⊥ ||R (Fig. 2.1b) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

8 2PPaR-1RPP (Fig. 2.22a)

P⊥ Pa||R (Fig. 2.2c) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

9 2PaRR-1RPP (Fig. 2.22b)

Pa||R||R (Fig. 2.2a) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

10 2PaPaR-1RPP (Fig. 2.22c)

Pa||Pa||R (Fig. 2.2b) R⊥ P ⊥⊥ P (Fig. 2.1g)

Idem No. 1

11 2PRR-1RPPa (Fig. 2.23a)

P⊥ R||R (Fig. 2.1c) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1


Table 2.3. (cont.)

12 2RRR-1RPPa (Fig. 2.23b)

R||R||R (Fig. 2.1a) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

13 2RPR-1RPPa (Fig. 2.23c)

R⊥ P⊥ ||R (Fig. 2.1b) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

14 2PRR-1RPaP (Fig. 2.24a)

P⊥ R||R (Fig. 2.1c) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

15 2RRR-1RPaP (Fig. 2.24b)

R||R||R (Fig. 2.1a) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

16 2RPR-1RPaP (Fig. 2.24c)

R⊥ P⊥ ||R (Fig. 2.1b) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

17 2PRR-1RPaPa (Fig. 2.25a)

P⊥ R||R (Fig. 2.1c) R||Pa||Pa (Fig. 2.2h)

Idem No. 1

18 2RRR-1RPaPa (Fig. 2.25b)

R||R||R (Fig. 2.1a) R||Pa||Pa (Fig. 2.2h)

Idem No. 1

19 2RPR-1RPaPa (Fig. 2.25c)

R⊥ P⊥ ||R (Fig. 2.1b) R||Pa||Pa (Fig. 2.2h)

Idem No. 1

20 2PPaR-1RPPa (Fig. 2.26a)

P⊥ Pa||R (Fig. 2.2c) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

21 2PaRR-1RPPa (Fig. 2.26b)

Pa||R||R (Fig. 2.2a) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

22 2PPaR-1RPaP (Fig. 2.27a)

P⊥ Pa||R (Fig. 2.2c) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1


Table 2.3. (cont)

23 2PaRR-1RPaP (Fig. 2.27b)

Pa||R||R (Fig. 2.2a) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

24 2PaPaR-1RPPa (Fig. 2.28a)

Pa||Pa||R (Fig. 2.2b) R⊥ P⊥ ||Pa (Fig. 2.2f)

Idem No. 1

25 2PaPaR-1RPaP (Fig. 2.28b)

Pa||Pa||R (Fig. 2.2b) R||Pa⊥ P (Fig. 2.2g)

Idem No. 1

26 2PPaR-1RPaPa (Fig. 2.29a)

P⊥ Pa||R (Fig. 2.2c) R||Pa||Pa (Fig. 2.2h)

Idem No. 1

27 2PaRR-1RPaPa (Fig. 2.29b)

Pa||R||R (Fig. 2.2a) R||Pa||Pa (Fig. 2.2h)

The directions of the revolute joints of the three limbs are parallel.

28 2PaPaR-1RPaPa (Fig. 2.30)

Pa||Pa||R (Fig. 2.2b) R||Pa||Pa (Fig. 2.2h)

Idem No. 27


P⊥ R||R (Fig. 2.1c) R||Pa||Pa (Fig. 2.3a)

Idem No. 1

30 2PRR-1RPaPat

(Fig. 2.31b) P⊥ R||R (Fig. 2.1c) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

31 2RRR-1RPaPa (Fig. 2.32a)

R||R||R (Fig. 2.1a) R||Pa||Pa (Fig. 2.3a)

Idem No. 27

32 2RRR-1RPaPat (Fig. 2.32b)

R||R||R (Fig. 2.1a) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

33 2RPR-1RPaPa (Fig. 2.33a)

R⊥ P⊥ ||R (Fig. 2.1b) R||Pa||Pa (Fig. 2.3a)

Idem No. 1


Table 2.3. (cont.)

34 2RPR-1RPaPat (Fig. 2.33b)

R⊥ P⊥ ||R (Fig. 2.1b) R||Pa||Pat (Fig. 2.3b)

Idem No. 1


P⊥ Pa||R (Fig. 2.2c) R||Pa||Pa (Fig. 2.3a)

Idem No. 1

36 2PPaR-1RPaPat (Fig. 2.34b)

P⊥ Pa||R (Fig. 2.2c) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

37 2PaRR-1RPaPa (Fig. 2.35a)

Pa||R||R (Fig. 2.2a) R||Pa||Pa (Fig. 2.3a)

Idem No. 27

38 2PaRR-1RPaPat (Fig. 2.35b)

Pa||R||R (Fig. 2.2a) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

39 2PaPaR-1RPaPa (Fig. 2.36a)

Pa||Pa||R (Fig. 2.2b) R||Pa||Pa (Fig. 2.3a)

Idem No. 27

40 2PaPaR-1RPaPat (Fig. 2.36b)

Pa||Pa||R (Fig. 2.2b) R||Pa||Pat (Fig. 2.3b)

Idem No. 1


Table 2.4. Structural parameters of planar parallel mechanisms in Figs. 2.4-2.10

No. Structural parameter

Solution 3-RRR , 3-RRP (Fig. 2.4a,b) 3-RPR, 3-PRR (Fig. 2.5a,b) 3-PRP (Fig. 2.6) 3-PPR (Fig. 2.10)

3-PaPaR (Fig. 2.7)

3-PaRR (Fig. 2.8a) 3-PaRP (Fig. 2.8b) 3-PaPR (Fig. 2.9a) 3-PPaR (Fig. 2.9b)

1 m 8 20 14 2 p1 3 9 6 3 p2 3 9 6 4 p3 3 9 6 5 p 9 27 18 6 q 2 8 5 7 k1 3 0 0 8 k2 0 3 3 9 k 3 3 3 10 (RG1) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 (RG2) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 12 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 13 SG1 3 3 3 14 SG2 3 3 3 15 SG3 3 3 3 16 rG1 0 6 3 17 rG2 0 6 3 18 rG3 0 6 3 19 MG1 3 3 3 20 MG2 3 3 3 21 MG3 3 3 3 22 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 23 SF 3 3 3 24 rl 0 18 9 25 rF 6 24 15 26 MF 3 3 3 27 NF 6 24 15 28 TF 0 0 0 29 1p

jj 1f

=∑ 3 9 6

30 2pjj 1

f=∑ 3 9 6

31 3pjj 1

f=∑ 3 9 6

32 pjj 1

f=∑ 9 27 18

m number of links including the fixed base, pGi number of joints in the Gi-limb, p total number of joints in the parallel mechanisma, q number of independent closed


loops in the parallel mechanismb, k1 number of simple limbs, k2 number of complex limbs, k total number of limbsc, (RGi) basis of the vector space of relative velo-cities between the moving and reference platforms in Gi-limb disconnected from the parallel mechanism, SGi connectivity between the moving and reference plat-forms in Gi-limb disconnected from the parallel mechanismd, rGi number of joint parameters that lost their independence in the closed loops combined in Gi-limb, MGi mobility of Gi-limbe, (RF) basis of the vector space of relative velocities between the moving and reference platforms in the parallel mechanismf, SF connectivity between the mobile and reference platforms in the parallel mechanismg, rl total number of joint parameters that lose their independence in the closed loops combined in the k limbsh, rF total number of joint parameters that lose their independence in the closed loops combined in the parallel mechanismi, MF mobility of the parallel mechanismj, NF number of overconstraints in the parallel mechanismk, TF degree of structural redundancy of the parallel mechanisml, fj mobility of jth joint. ap= k

Gii 1p

=∑ , bq=p-m+1, ck=k1+k2, dSGi=dim(RGi) , i=1,2,...,k, eMGi= Gip

j Gij 1f r

=−∑ , i=1,2,...,k,

f(RF)=(RG1) ∩ ... ∩ (RGk), gSF=dim(RF) , hrl=

kGii 1

r=∑ ,

i kF Gi F li 1

r S S r=

= − +∑ , j p

F i Fi 1M f r

== −∑ ,

kF FN 6q r= − ,

lF F FT M S= − .


Table 2.5. Structural parametersa of planar parallel mechanisms in Figs. 2.11-2.15


Solution 1PPR-2RRR (Fig. 2.11) 1PPR-2RRP (Fig. 2.12a) 1PPR-2RPR (Fig. 2.12b) 1PPR-2PRR (Fig. 2.13a) 1PPR-2PRP (Fig. 2.13b)

1PaPR-2PaRR 1PaPR-2PPaR (Fig. 2.14a, b) 1PaPR-2PaRP (Fig. 2.15b)

1PaPR-2PaPaR (Fig. 2.15a)


jj 1f

=∑ 3 6 6

30 2pjj 1

f=∑ 3 6 9

31 3pjj 1

f=∑ 3 6 9

32 pjj 1

f=∑ 9 18 24

aSee footnote of Table 2.4 for the nomenclature of structural parameters




Solution 3RRR (Fig. 2.16) 3RPR, 2RPR-1RPR (Fig. 2.17)2RPP-1RPP (Fig. 2.18) 2PRR-1RPP, 2RRR-1RPP, 2RPR-1RPP (Fig. 2.21)

2RPPa-1RPPa (Fig. 2.19) 2RPaP-1RPaP (Fig. 2.20a)

2RPaPa-1RPaPa (Fig. 2.20b)


jj 1f

=∑ 3 6 9

30 2pjj 1

f=∑ 3 6 9

31 3pjj 1

f=∑ 3 6 9

32 pjj 1

f=∑ 9 18 27





Solution 2PPaR-1RPP (Fig. 2.22a) 2PaRR-1RPP (Fig. 2.22b)

2PaPaR-1RPP (Fig. 2.22c)

2PRR-1RPPa, 2RRR-1RPPa 2RPR-1RPPa (Fig. 2.23a,b,c) 2PRR-1RPaP, 2RRR-1RPaP 2RPR-1RPaP (Fig. 2.24a,b,c)


jj 1f

=∑ 6 9 3

30 2pjj 1

f=∑ 6 9 3

31 3pjj 1

f=∑ 3 3 6

32 pjj 1

f=∑ 15 21 12





Solution 2PRR-1RPaPa 2RRR-1RPaPa 2RPR-1RPaPa (Fig. 2.25a,b,c)

2PPaR-1RPPa (Fig. 2.26a) 2PaRR-1RPPa (Fig. 2.26b) 2PPaR-1RPaP (Fig. 2.27a) 2PaRR-1RPaP (Fig. 2.27b)

2PaPaR-1RPPa (Fig. 2.28a) 2PaPaR-1RPaP (Fig. 2.28b)


jj 1f

=∑ 3 6 9

30 2pjj 1

f=∑ 3 6 9

31 3pjj 1

f=∑ 9 6 6

32 pjj 1

f=∑ 15 18 24





Solution 2PPaR-1RPaPa (Fig. 2.29a) 2PaRR-1RPaPa (Fig. 2.29b)

2PaPaR-1RPaPa(Fig. 2.30)

2PRR-1RPaPa (Fig. 2.31a) 2PRR-1RPaPat (Fig. 2.31b) 2RRR-1RPaPa (Fig. 2.32a) 2RRR-1RPaPat (Fig. 2.32b) 2RPR-1RPaPa (Fig. 2.33a) 2RPR-1RPaPat (Fig. 2.33b)

1 m 16 20 12 2 p1 6 9 3 3 p2 6 9 3 4 p3 9 9 9 5 p 21 27 15 6 q 6 8 4 7 k1 0 0 2 8 k2 3 3 1 9 k 3 3 3 10 (RGi)

i=1,2,3 ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω )

11 SG1 3 3 3 12 SG2 3 3 3 13 SG3 3 3 3 14 rG1 3 6 0 15 rG2 3 6 0 16 rG3 6 6 6 17 MG1 3 3 3 18 MG2 3 3 3 19 MG3 3 3 3 20 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 21 SF 3 3 3 22 rl 12 18 6 23 rF 18 24 12 24 MF 3 3 3 25 NF 18 24 12 26 TF 0 0 0 27 1p

jj 1f

=∑ 6 9 3

28 2pjj 1

f=∑ 6 9 3

29 3pjj 1

f=∑ 9 9 9

30 pjj 1

f=∑ 21 27 15





Solution 2PPaR-1RPaPa (Fig. 2.34a) 2PPaR-1RPaPat (Fig. 2.34b) 2PaRR-1RPaPa (Fig. 2.35a) 2PaRR-1RPaPat (Fig. 2.35b)

2PaPaR-1RPaPa (Fig. 2.36a) 2PaPaR-1RPaPat (Fig. 2.36b)

1 m 16 20 2 p1 6 9 3 p2 6 9 4 p3 9 9 5 p 21 27 6 q 6 8 7 k1 0 0 8 k2 3 3 9 k 3 3 10 (RG1) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 (RG2) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 12 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 13 SG1 3 3 14 SG2 3 3 15 SG3 3 3 16 rG1 3 6 17 rG2 3 6 18 rG3 6 6 19 MG1 3 3 20 MG2 3 3 21 MG3 3 3 22 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 23 SF 3 3 24 rl 12 18 25 rF 18 24 26 MF 3 3 27 NF 18 24 28 TF 0 0 29 1p

jj 1f

=∑ 6 9

30 2pjj 1

f=∑ 6 9

31 3pjj 1

f=∑ 9 9

32 pjj 1

f=∑ 21 27



Fig. 2.4. Overconstrained PPMs with coupled motions of types 3-RRR (a) and 3-RRP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology R||R||R (a) and R||R⊥ P (b)


Fig. 2.5. Overconstrained PPMs with coupled motions of types 3-RPR (a) and 3-PRR (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology R⊥ P⊥ ||R (a) and P⊥ R||R (b)


Fig. 2.6. Overconstrained PPM with coupled motions of type 3-PRP defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology P⊥ R⊥ P

Fig. 2.7. Overconstrained PPM with coupled motions of type 3-PaPaR defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=24, limb topology Pa||Pa||R


Fig. 2.8. Overconstrained PPMs with coupled motions of types 3-PaRR (a) and 3-PaRP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, limb topology Pa||R||R (a) and Pa||R⊥ P (b)


Fig. 2.9. Overconstrained PPMs with coupled motions of types 3-PaPR (a) and 3-PPaR (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, limb topology Pa⊥ P⊥ ||R (a) and P⊥ Pa||R (b)


Fig. 2.10. Overconstrained PPM with one decoupled translation of type 3-PPR defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology P⊥ P ⊥⊥ R

Fig. 2.11. Overconstrained PPM with one decoupled translation of type 1PPR-2RRR defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology P⊥ P ⊥⊥ R and R||R||R


Fig. 2.12. Overconstrained PPMs with one decoupled translation of types 1PPR-2RRP (a) and 1PPR-2RPR (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology P⊥ P ⊥⊥ R and R||R⊥ P (a), P⊥ P ⊥⊥ R and R⊥ P⊥ ||R (b)


Fig. 2.13. Overconstrained PPMs with one decoupled translation of types 1PPR-2PRR (a) and 1PPR-2PRP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology P⊥ P ⊥⊥ R and P⊥ R||R (a), P⊥ P ⊥⊥ R and P⊥ R⊥ P (b)


Fig. 2.14. Overconstrained PPMs with one decoupled translation of types 1PaPR-2PaRR (a) and 1PaPR-2PPaR (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=15, limb topology Pa⊥ P⊥ ||R and Pa||R||R (a), Pa⊥ P⊥ ||R and P⊥ Pa||R (b)


Fig. 2.15. Overconstrained PPMs with one decoupled translation of types 1PaPR-2PaPaR (a) and 1PaPR-2PaRP (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=21 (a), NF=15 (b), limb topology Pa⊥ P⊥ ||R and Pa||Pa||R (a), Pa⊥ P⊥ ||R and Pa||R ⊥ P (b)


Fig. 2.16. 3RRR-type overconstrained PPMs with partially decoupled translations defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=6, limb topology R||R||R


Fig. 2.17. Overconstrained PPMs with partially decoupled translations of types 3RPR (a) and 2RPR-1RPR (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=6, limb topology R⊥ P⊥ ||R (a), R⊥ P⊥ ||R and R⊥ P⊥ ||R (b)


Fig. 2.18. Overconstrained PPM with decoupled rotation of type 2RPP-1RPP defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology R⊥ P ⊥⊥ P

Fig. 2.19. Overconstrained PPM with decoupled rotation of type 2RPPa-1RPPa, NF=21 defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, limb topology R⊥ P⊥ ||Pa


Fig. 2.20. Overconstrained PPMs with decoupled rotation of types 2RPaP-1RPaP (a) and 2RPaPa-1RPaPa (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, (a), NF=24 (b), limb topology R||Pa⊥ P (a) and R||Pa||Pa (b)


Fig. 2.21. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPP (a), 2RRR-1RPP (b) and 2RPR-1RPP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology R⊥ P ⊥⊥ P and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.22. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPP (a), 2PaRR-1RPP (b) and 2PaPaR-1RPP (c) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=12 (a) and (b), NF=18 (c), limb topology R⊥ P ⊥⊥ P and P⊥ Pa||R (a), Pa||R||R (b), Pa||Pa||R (c)


Fig. 2.23. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPPa (a), 2RRR-1RPPa (b) and 2RPR-1RPPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9, limb topology R⊥ P⊥ ||Pa and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.24. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaP (a), 2RRR-1RPaP (b) and 2RPR-1RPaP (c) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=9, limb topology R||Pa⊥ P and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.25. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaPa (a), 2RRR-1RPaPa (b) and 2RPR-1RPaPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology R||Pa||Pa and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.26. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPPa (a) and 2PaRR-1RPPa (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, limb topology R⊥ P⊥ ||Pa and P⊥ Pa||R (a), Pa||R||R (b)


Fig. 2.27. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPaP (a) and 2PaRR-1RPaP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=15, limb topology R||Pa⊥ P and P⊥ Pa||R (a), Pa||R||R (b)


Fig. 2.28. Overconstrained PPMs with uncoupled rotation of types 2PaPaR-1RPPa (a) and 2PaPaR-1RPaP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=21, limb topology Pa||Pa||R and R⊥ P⊥ ||Pa (a), R||Pa⊥ P (b)


Fig. 2.29. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPaPa (a) and 2PaRR-1RPaPa (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=18, limb topology R||Pa||Pa and P⊥ Pa||R (a), Pa||R||R (b)


Fig. 2.30. 2PaPaR-1RPaPa-type overconstrained PPM with uncoupled rotation defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=24, limb topology Pa||Pa||R and R||Pa||Pa


Fig. 2.31. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaPa (a) and 2PRR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology P⊥ R||R and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.32. Overconstrained PPMs with uncoupled rotation of types 2RRR-1RPaPa (a) and 2RRR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology R||R||R and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.33. Overconstrained PPMs with uncoupled rotation of types 2RPR-1RPaPa (a) and 2RPR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology R⊥ P⊥ ||R and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.34. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPaPa (a) and 2PPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=18, limb topology P⊥ Pa||R and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.35. Overconstrained PPMs with uncoupled rotation of types 2PaRR-1RPaPa (a) and 2PaRR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=18, limb topology Pa||R||R and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.36. Overconstrained PPMs with uncoupled rotation of types 2PaPaR-1RPaPa (a) and 2PaPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=24, limb topology Pa||Pa||R and R||Pa||Pa (a), R||Pa||Pat (b)


2.1.2 Non fully-parallel solutions

The non fully-parallel solutions presented in Figs. 2.37-2.43 are obtained from the fully-parallel solutions in Figs. 2.21, 2.23-2.25, 2.31-2.33 by superposing the last revolute joints of limbs G1 and G2. In this way, these two limbs compose just one complex limb denoted by G1-G2 in which two actuators are combined. For example, the solutions in Fig. 2.37 combine one simple and one complex limb. Simple limb G3 is of type RPP and complex limb G1-G2 is of types (PRRRP)R (Fig. 2.37a), (RRRRR)R (Fig. 2.37b) and (RPRPR)R (Fig. 2.37c). One closed loop defined in the brackets is combined in each complex limb G1-G2.

The solutions in Figs. 2.38-2.43 combine two complex limbs G1-G2 and G3. The rotation axis of the moving platform has a fixed position in the solutions in Figs. 2.38-2.40 and a variable position in Figs. 2.41-2.43.

The structural parameters of the solutions presented in Figs. 2.37-2.43 are systematized in Tables 2.10 and 2.11.

The basis of the vector space of the relative velocities between the moving and the fixed platforms in the complex limb G1-G2 isolated from the parallel mechanism is denoted by (RG1-G2) in Tables 2.10 and 2.11. The connectivity between the moving and the fixed platforms of this complex limb isolated from the parallel mechanism is denoted by SG1-G2 (see Table 2.10-2.11).

The non fully-parallel solutions presented in Figs. 2.37-2.43 have the same number of overconstraints as their fully-parallel counterparts in Figs. 2.21, 2.23-2.25, 2.31-2.33.




Solution (PRRRP)R-RPP (Fig. 2.37a) (RRRRR)R-RPP (Fig. 2.37b) (RPRPR)R-RPP (Fig. 2.37c)

(PRRRP)R-RPPa (Fig. 2.38a) (RRRRR)R-RPPa (Fig. 2.38b) (RPRPR)R-RPPa (Fig. 2.38c) (PRRRP)R-RpaP (Fig. 2.39a) (RRRRR)R-RPaP (Fig. 2.39b) (RPRPR)R-RPaP (Fig. 2.39c)

1 m 8 10 2 p1 6 6 3 p2 3 6 4 p 9 12 5 q 2 3 6 k1 1 0 7 k2 1 2 8 k 2 2 9 (RG1-G2) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 10 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 SG1-G2 3 3 12 SG3 3 3 13 rG1-G2 3 3 14 rG3 0 3 15 MG1-G2 3 3 16 MG3 3 3 17 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 18 SF 3 3 19 rl 3 6 20 rF 6 9 21 MF 3 3 22 NF 6 9 23 TF 0 0 24 1p

jj 1f

=∑ 6 6

25 2pjj 1

f=∑ 3 6

26 pjj 1

f=∑ 9 12





Solution (PRRRP)R-RpaPa (Fig. 2.40a) (RRRRR)R-RPaPa (Fig. 2.40b) (RPRPR)R-RPaPa (Fig. 2.40c) (PRRRP)R-RPaPa (Fig. 2.41a) (PRRRP)R-RPaPat (Fig. 2.41b) (RRRRR)R-RPaPa (Fig. 2.42a) (RRRRR)R-RPaPat (Fig. 2.42b) (RPRPR)R-RPaPa (Fig. 2.43a) (RPRPR)R-RPaPat (Fig. 2.43b)

1 m 12 2 p1 6 3 p2 9 4 p 15 5 q 4 6 k1 0 7 k2 2 8 k 2 9 (RG1-G2) ( 1 2, , δv v ω ) 10 (RG3) ( 1 2, , δv v ω ) 11 SG1-G2 3 12 SG3 3 13 rG1-G2 3 14 rG3 6 15 MG1-G2 3 16 MG3 3 17 (RF) ( 1 2, , δv v ω ) 18 SF 3 19 rl 9 20 rF 12 21 MF 3 22 NF 12 23 TF 0 24 1p

jj 1f

=∑ 6

25 2pjj 1

f=∑ 9

26 pjj 1

f=∑ 15



Fig. 2.37. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPP (a), (RRRRR)R-RPP (b) and (RPRPR)R-RPP (c), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6


Fig. 2.38. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPPa (a), (RRRRR)R-RPPa (b) and (RPRPR)R-RPPa (c), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9


Fig. 2.39. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPaP (a), (RRRRR)R-RPaP (b) and (RPRPR)R-RPaP (c), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9


Fig. 2.40. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPaPa (a), (RRRRR)R-RPaPa (b) and (RPRPR)R-RPaPa (c), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12


Fig. 2.41. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPaPa (a) and (PRRRP)R-RPaPat (b), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12


Fig. 2.42. Overconstrained PPMs with uncoupled rotation of types (RRRRR)R-RPaPa (a) and (RRRRR)R-RPaPat (b), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12


Fig. 2.43. Overconstrained PPMs with uncoupled rotation of types (RPRPR)R-RPaPa (a) and (RPRPR)R-RPaPat (b), defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12

2.2 Derived solutions 87

2.2 Derived solutions

Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 2.4-2.43 by using joints with idle mobilities. A large set of solutions can be obtained by introducing idle mobilities in the limbs. The joints combining idle mobilities are denoted by an asterisk. The idle mobilities which can be combined in a parallelogram loop are system-atized in Fig. 1.2 and Table 1.1. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint S*. These idle mobilities can be introduced outside or inside the loops com-bined in the limbs. The rotational mobility of the revolute joint denoted by R* is an idle mobility. In the cylindrical joint denoted by C* in Figs. 2.45, 2.46b, 2.54b, 2.55a, 2.61, 2.62a the translation is the actuated motion and the rotation is the idle mobility. The rotation is also the idle mobility in the cylindrical joint denoted by C* in Figs 2.65a, 2.70, 2.71 and 2.72a.

We recall that the notation Pacs is associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same link. In the cylindrical joints of the Pacs-type parallelogram loops (Figs. 2.84-2.95 and 2.104-2.109) the translational motion is an idle mobility. We note that in the Pass-type parallelogram loop, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of the binary link around the axis passing by the centre of the two spherical joints. This internal mobility gives one degree of structural redundancy (see Table 2.13). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. This mobility is restricted by the con-straints of the parallel mechanism and remains just a potential mobility. For example in Fig. 2.68c, this rotational motion is internal mobility of links 3A and 3B, and idle mobility for the ternary links 7A and 7B.

Examples of solutions with identical limbs and 3-18 overconstraints derived from the basic solutions in Figs. 2.4-2.10 are illustrated in Figs. 2.44-2.52.

The solutions with one decoupled translation 1 1 1v v ( q )= & and 3-15 over-constraints illustrated in Figs. 2.53-2.59 are derived from the basic solu-tions in Figs. 2.11-2.15. The solutions with partially decoupled translational motions and three overconstraints illustrated in Figs. 2.60-2.61 are derived from the solutions in Figs. 2.16 and 2.17. The solutions with decoupled


rotational motion 3( q )δ δω ω= & and 2-22 overconstraints illustrated in Figs. 2.62-2.95 are derived from the solutions in Figs. 2.18-2.36.

The non fully-parallel solutions with 2-8 overconstraints illustrated in Figs. 2.96-2.109 are derived from the solutions in Figs. 2.37-2.43.

The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 2.44-109 are given in Tables 2.14 and 2.15. The limb topology and connecting conditions of these solutions are systematized in Tables 2.16-2.21, as are their structural parameters in Tables 2.22-2.40.

Table 2.13. Links with internal mobilities and the degree of structural redundancy TF of overconstrained SPMs with uncoupled planar motion of the moving platform

No. Parallel mechanism Figure TF

Link with internal rotational mobility in limb G1 G2 G3

1 Figs. 2.47b, 2.50b, 2.51a

3 3A 3B 3C

2 Fig. 2.50a 6 3A, 6A 3B, 6B 3C, 6C 3 Fig.2.51b 3 4A 4B 4C 4 Figs. 2.68c, 2.79b,

2.82 2 3A 3B -

5 Figs. 2.72, 2.98 1 - - 5C 6 Figs. 2.73, 2.99 1 - - 4C 7 Figs. 2.74, 2.100 2 - - 3C, 6C


Table 2.14. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 2.44-2.95

No. Parallel mechanism

Basis (RG1)

(RG2)

(RG3)

1 Figs. 2.44-2.46, 2.60

( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,β δv v ω ω )

2 Figs. 2.47a, 2.48, 2.49, 2.56, 2.57

( 1 2, , , ,α β δv v ω ω ω ) ( 1 2 3, , , ,α δv v v ω ω ) ( 1 2 3, , , ,β δv v v ω ω )

3 Figs. 2.47b, 2.50, 2.51, 2.72-2.74

( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω )

4 Figs. 2.52, 2.53, 2.54a, 2.55b

( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,α δv v ω ω )

5 Figs. 2.54b, 2.61b

( 1 2, , ,β δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω )

6 Fig. 2.55a ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,β δv v ω ω ) 7 Fig. 2.58 ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) 8 Figs. 2.59,

2.61a, 2.63, 2.64, 2.80b, 2.81b

( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω )

9 Figs. 2.62, 2.80a, 2.81a, 2.82

( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω )

10 Figs. 2.65, 2.69-2.71, 2.75-2.78, 2.79a, 2.84-2.86

( 1 2 3, , , δv v v ω ) ( 1 2 3, , , δv v v ω ) ( 1 2, , δv v ω )

11 Figs. 2.66, 2.87-2.89

( 1 2, , , ,α β δv v ω ω ω ) ( 1 2, , , ,α β δv v ω ω ω ) ( 1 2, , δv v ω )

12 Figs. 2.67 ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , , ,α β δv v ω ω ω ) 13 Figs. 2.68a,c ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω ) ( 1 2, , δv v ω ) 14 Figs. 2.68b ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , δv v ω ) 15 Figs. 2.79b,

2.83a ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,α δv v ω ω ) ( 1 2, , , ,α β δv v ω ω ω )

16 Figs. 2.83b ( 1 2, , ,α δv v ω ω ) ( 1 2, , ,β δv v ω ω ) ( 1 2, , , ,α β δv v ω ω ω ) 17 Figs. 2.90,

2.93 ( 1 2 3, , , ,β δv v v ω ω ) ( 1 2 3, , , ,α δv v v ω ω ) ( 1 2, , δv v ω )


Table 2.14. (cont.)

18 Figs. 2.91, 2.94

( 1 2 3, , , ,α δv v v ω ω ) ( 1 2 3, , , ,β δv v v ω ω ) ( 1 2, , δv v ω )

19 Figs. 2.92, 2.95

( 1 2 3, , , , ,α β δv v v ω ω ω ) ( 1 2 3, , , , ,α β δv v v ω ω ω ) ( 1 2, , δv v ω )

Table 2.15. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 2.96-2.109

No. Parallel mechanism

Basis (RG1-G2)

(RG3)

1 Figs. 2.96 ( 1 2 3, , , δv v v ω ) ( 1 2, , δv v ω ) 2 Figs. 2.97, 2.101-2.103 ( 1 2, , ,β δv v ω ω ) ( 1 2, , δv v ω ) 3 Figs. 2.98, 2.99, 2.100,

2.104-2.106 ( 1 2, , δv v ω ) ( 1 2, , δv v ω )

4 Figs. 2.107-109 ( 1 2 3, , , ,β δv v v ω ω ) ( 1 2, , δv v ω )

Table 2.16. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.44-2.46

No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology

1 3-RRR (Fig. 2.4a)

6 3-RR*RR (Fig. 2.44a)

3 R⊥ R*⊥ ||R||R

2 3-RRP (Fig. 2.4b)

6 3-RRC* (Fig. 2.44b)

3 R||R⊥ C*

3 3-RPR (Fig. 2.5a)

6 3-RC*R (Fig. 2.45a)

3 R⊥ C*⊥ ||R

4 3-PRR (Fig. 2.5b)

6 3-C*RR (Fig. 2.45b)

3 C*⊥ R||R

5 3-PRP (Fig. 2.6)

6 3-PRC* (Fig. 2.46a)

3 P⊥ R⊥ C*

6 3-C*RP (Fig. 2.46b)

3 C*⊥ R⊥ P



No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology

1 3-PaPaR (Fig. 2.7)

24 3-PaPaS* (Fig. 2.48a)

18 Pa||PaS*

2 3-PassPassR (Fig. 2.50a)

6 Pass||Pass||R

3 3-PaRR (Fig. 2.8a)

15 3-PaRS* (Fig. 2.47a)

9 Pa||RS*

4 3-PassRR (Fig. 2.47b)

6 Pass||R||R

5 3-PaRP (Fig. 2.8b)

15 3-PaS*P (Fig. 2.48b)

9 PaS*P

6 3-PassRP (Fig. 2.50b)

6 Pass||R⊥ P

7 3-PaPR (Fig. 2.9a)

15 3-PaPS* (Fig. 2.49a)

9 Pa⊥ PS*

8 3-PassPR (Fig. 2.51a)

6 Pass ⊥ P⊥ ||R

9 3-PPaR (Fig. 2.9b)

15 3-PPaS* (Fig. 2.49b)

9 P⊥ PaS*

10 3-PPassR (Fig. 2.51b)

6 P⊥ Pass||R

11 3-PPR (Fig. 2.10)

6 3-PC*R (Fig. 2.52)

3 P⊥ C* ⊥⊥ R

12 1PPR-2RRR (Fig. 2.11)

6 1PC*R-2RR*RR (Fig. 2.53)

3 P⊥ C* ⊥⊥ R R⊥ R*⊥ ||R||R

13 1PPR-2RRP (Fig. 2.12a)

6 1PC*R-2RRC* (Fig. 2.54a)

3 P⊥ C* ⊥⊥ R R||R⊥ C*

14 1PPR-2RPR (Fig. 2.12b)

6 1PC*R-2RC*R (Fig. 2.54b)

3 P⊥ C* ⊥⊥ R R⊥ C*⊥ ||R



No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology

1 1PPR-2PRR (Fig. 2.13a)

6 1C*PR-2C*RR (Fig. 2.55a)

3 C*⊥ P ⊥⊥ R C*⊥ R||R

2 1PPR-2PRP (Fig. 2.13b)

6 1PC*R-2PRC* (Fig. 2.55b)

3 P⊥ C* ⊥⊥ R P⊥ R⊥ C*

3 1PaPR-2PaRR (Fig. 2.14a)

15 1PaPS*-2PaRS* (Fig. 2.56a)

9 Pa⊥ PS* Pa||RS*

4 1PassPR-2PassRR (Fig. 2.58a)

3 Pass ⊥ P⊥ ||R Pass||R||R

5 1PaPR-2PPaR (Fig. 2.14b)

15 1PaPS*-2PPaS* (Fig. 2.56b)

9 1Pa⊥ PS* P⊥ PaS*

6 1PassPR-2PPassR (Fig. 2.58b)

3 Pass ⊥ P⊥ ||R P⊥ Pass||R

7 1PaPR-2PaPaR (Fig. 2.15a)

21 1PaPS*-2PaPaS* (Fig. 2.57a)

15 Pa⊥ PS* Pa||PaS*

8 1PassPR-2PaPassR (Fig. 2.59a)

9 Pass ⊥ P⊥ ||R Pass||R⊥ P

9 1PaPR-2PaRP (Fig. 2.15b)

15 1PaPS*-2PaS*P (Fig. 2.57b)

9 Pa⊥ PS* PaS*P

10 1PassPR-2PassRP (Fig. 2.59b)

3 Pass ⊥ P⊥ ||R Pass||R⊥ P

11 3RRR (Fig. 2.16a,b)

6 3RR*RR (Fig. 2.60a,b)

3 R⊥ R*⊥ ||R||R

12 3RPR (Fig. 2.17a)

6 3RC*R (Fig. 2.61a)

3 R⊥ C*⊥ ||R

13 2RPR-1RPR (Fig. 2.17b)

6 2RC*R-1RC*R (Fig. 2.61b)

3 R⊥ C*⊥ ||R R⊥ C*⊥ ||R

14 2RPP-1RPP (Fig. 2.18)

6 2RPC*-1RPC* (Fig. 2.62)

3 R⊥ P ⊥⊥ C*

15 2RPPa-1RPPa (Fig. 2.19)

15 2RC*Pa-1RC*Pa (Fig. 2.63)

12 R⊥ C*⊥ ||Pa

16 2RPaP-1RPaP (Fig. 2.20a)

15 2RPaC*-1RPaC* (Fig. 2.64a)

12 R||Pa⊥ C* R||Pa⊥ C*

17 2RPaPa-1RPaPa (Fig. 2.20b)

24 2RPaPass-1RPaPass

(Fig. 2.64b) 12 R||Pa||Pass

R||Pa||Pass



No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology

1 2PRR-1RPP (Fig. 2.21a)

6 2PRC*-1RPP (Fig. 2.65a)

4 P⊥ R||C* R⊥ P ⊥⊥ P

2 6 2PS*R-1RPP (Fig. 2.66a)

2 PS*R R⊥ P ⊥⊥ P

3 2RRR-1RPP (Fig. 2.21b)

6 2RRC*-1RPP (Fig. 2.65b)

4 R||R||C* R⊥ P ⊥⊥ P

4 6 2RS*R-1RPP (Fig. 2.66b)

2 RS*R R⊥ P ⊥⊥ P

5 2RPR-1RPP (Fig. 2.21c)

6 2RPC*-1RPP (Fig. 2.65c)

4 R⊥ P⊥ ||C* R⊥ P ⊥⊥ P

6 6 2S*PR-1RPP (Fig. 2.66c)

2 SP⊥ R R⊥ P ⊥⊥ P

7 2PPaR-1RPP (Fig. 2.22a)

12 2PPaR-1RC*C* (Fig. 2.67a)

10 P⊥ Pa||R R⊥ C* ⊥⊥ C*

8 2PPassR-1RPP (Fig. 2.68a)

4 P⊥ Pass||R R⊥ P ⊥⊥ P

9 2PaRR-1RPP (Fig. 2.22b)

12 2PaRR-1RC*C* (Fig. 2.67b)

10 Pa||R||R R⊥ C* ⊥⊥ C*

10 2PassRR-1RPP (Fig. 2.68b)

4 Pass||R||R R⊥ P ⊥⊥ P

11 2PaPaR-1RPP (Fig. 2.22c)

18 2PaPaR-1RC*C* (Fig. 2.67c)

16 Pa||Pa||R

12 2PassPassR-1RPP (Fig. 2.68c)

4 Pass||Pass||R R⊥ P ⊥⊥ P

13 2PRR-1RPPa (Fig. 2.23a)

9 2PRC*-1RPPa (Fig. 2.69a)

7 P⊥ R||C* R⊥ P⊥ ||Pa

14 2PRR-1RPPass

(Fig. 2.72a) 6 P⊥ R||R

R⊥ P⊥ ||Pass 15 2RRR-1RPPa

(Fig. 2.23b) 9 2RRC*-1RPPa

(Fig. 2.69b) 7 R||R||C*

R⊥ P⊥ ||Pa 16 2RRR-1RPPass

(Fig. 2.72b) 6 R||R||R

R⊥ P⊥ ||Pass 17 2RPR-1RPPa

(Fig. 2.23c) 9 2RPC*-1RPPa

Fig. 2.69c) 7 R⊥ P⊥ ||C*

R⊥ P⊥ ||Pa 18 2RPR-1RPPass

(Fig. 2.72c) 6 R⊥ P⊥ ||R

R⊥ P⊥ ||Pass 19 2PRR-1RPaP

(Fig. 2.24a) 9 2PRC*-1RPaP

(Fig. 2.70a) 7 P⊥ R||C*

R||Pa⊥ P


Table 2.19. (cont.)

20 2PRR-1RPassP (Fig. 2.73a)

6 P⊥ R||R R||Pass ⊥ P

21 2RRR-1RPaP (Fig. 2.24b)

9 2RRC*-1RPaP (Fig. 2.70b)

7 R||R||C* R||Pa⊥ P

22 2RRR-1RPassP (Fig. 2.73b)

6 R||R||R R||Pass ⊥ P

23 2RPR-1RPaP (Fig. 2.24c)

9 2RPC*-1RPaP (Fig. 2.70c)

7 R⊥ P⊥ ||C* R||Pa⊥ P

24 2RPR-1RPassP (Fig. 2.73c)

6 R⊥ P⊥ ||R R||Pass ⊥ P


12 2PRC*-1RPaPa (Fig. 2.71a)

10 P⊥ R||C* R||Pa||Pa

26 2PRR-1RPassPass

(Fig. 2.74a) 6 P⊥ R||R

R||Pass||Pass 27 2RRR-1RPaPa

(Fig. 2.25b) 12 2RRC*-1RPaPa

(Fig. 2.71b) 10 R||R||C*

R||Pa||Pa 28 2RRR-1RPassPass

(Fig. 2.74b) 6 R||R||R

R||Pass||Pass 29 2RPR-1RPaPa

(Fig. 2.25c) 12 2RPC*-1RPaPa

(Fig. 2.71c) 10 R⊥ P⊥ ||C*

R||Pa||Pa 30 2RPR-1RPassPass

(Fig. 2.74c) 6 R⊥ P⊥ ||R

R||Pass||Pass 31 2PPaR-1RPPa

(Fig. 2.26a) 15 2PPaC*-1RPPa

(Fig. 2.75a) 13 P⊥ Pa||C*

R⊥ P⊥ ||Pa 32 2PPassR-1RPPass

(Fig. 2.80a) 3 P⊥ Pass||R

R⊥ P⊥ ||Pass 33 2PaRR-1RPPa

(Fig. 2.26b) 15 2PaRC*-1RPPa

(Fig. 2.75b) 13 Pa||R||C*

R⊥ P⊥ ||Pa 34 2PassRR-1RPPass

(Fig. 2.80b) 3 Pass||R||R

R⊥ P⊥ ||Pass 35 2PPaR-1RPaP

(Fig. 2.27a) 15 2PPaC*-1RPaP

(Fig. 2.76a) 13 P⊥ Pa||C*

R||Pa⊥ P 36 2PPassR-1RPassP


R⊥ Pass ⊥ ||P 37 2PaRR-1RPaP

(Fig. 2.27b) 15 2PaRC*-1RPaP

(Fig. 2.76b) 13 Pa||R||C*

R||Pa⊥ P 38 2PassRR-1RPassP


R⊥ Pass ⊥ ||P 39 2PaPaR-1RPPa

(Fig. 2.28a) 21 2PaPaC*-1RPPa

(Fig. 2.77a) 19 Pa||Pa||C*

R⊥ P⊥ ||Pa 40 2PassPassR-1RPPass

(Fig. 2.82a) 3 Pass||Pass||R

R⊥ P⊥ ||Pass 41 2PaPaR-1RPaP

(Fig. 2.28b) 21 2PaPaC*-1RPaP

(Fig. 2.77b) 19 Pa||Pa||C*

R||Pa⊥ P


Table 2.19. (cont.)

42 2PassPassR-1RPassP (Fig. 2.82b)

3 Pass||Pass||R R||Pass ⊥ P


18 2PPaC*-1RPaPa (Fig. 2.78a)

16 P⊥ Pa||C* R||Pa||Pa

44 2PPassR-1RPassPass


R||Pass||Pass 45 2PaRR-1RPaPa

(Fig. 2.29b) 18 2PaRC*-1RPaPa

(Fig. 2.78b) 16 Pa||R||C*

R||Pa||Pa 46 2PassRR-1RPassPass


R||Pass||Pass 47 2PaPaR-1RPaPa

(Fig. 2.30) 24 2PaPaC*-1RPaPa

(Fig. 2.79a) 22 Pa||Pa||C*

R||Pa||Pa 48 2PassPassR-1RPassPass

(Fig. 2.79b) 2 Pass||Pass||R

R||Pass||Pass 49 2PRR-1RPaPa

(Fig. 2.31a) 12 2PRC*-1RPacsPacs

(Fig. 2.84a) 4 P⊥ R||C*

R||Pacs||Pacs 50 2PS*R-1RPacsPacs

(Fig. 2.87a) 2 PS*R

R||Pacs||Pacs 51 2PRR-1RPaPat

(Fig. 2.31b) 12 2PRC*-1RPacsPatcs

(Fig. 2.84b) 4 P⊥ R||C*

R||Pacs||Pat cs 52 2PS*R-1RPacsPatcs

(Fig. 2.87b) 2 PS*R

R||Pacs||Patcs 53 2RRR-1RPaPa

(Fig. 2.32a) 12 2RRC*-1RPacsPacs

(Fig. 2.85a) 4 R||R||C*

R||Pacs||Pacs 54 2RS*R-1RPacsPacs

(Fig. 2.88a) 2 RS*R

R||Pacs||Pacs 55 2RRR-1RPaPat

(Fig. 2.32b) 12 2RRC*-1RPacsPatcs

(Fig. 2.85b) 4 R||R||C*

R||Pacs||Patcs 56 2RS*R-1RPacsPatcs

(Fig. 2.88b) 2 R||R||C*

R||Pacs||Patcs 57 2RPR-1RPaPa

(Fig. 2.33a) 12 2RPC*-1RPacsPacs

(Fig. 2.86a) 4 R⊥ P⊥ ||C*

R||Pacs||Pacs 58 2S*PR-1RPacsPacs

(Fig. 2.89a) 2 S*P⊥ R

R||Pacs||Pacs 59 2RPR-1RPaPat

(Fig. 2.33b) 12 2RPC*-1RPacsPatcs

(Fig. 2.86b) 4 R⊥ P⊥ ||C*

R||Pacs||Patcs 60 2S*PR-1RPacsPatcs

(Fig. 2.89b) 2 S*P⊥ R

R||Pacs||Patcs



No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology


18 2PPassC*-1RPaPa (Fig. 2.90a)

8 P⊥ Pass||C* R||Pa||Pa

62 2PPassC*-1RPacsPacs

(Fig. 2.93a) 2 P||Pass||C*

R||Pacs||Pacs 63 2PPaR-1RPaPat

(Fig. 2.34b) 18 2PPassC*-1RPaPat

(Fig. 2.90b) 8 P⊥ Pass||C*

R||Pa||Pat 64 2PPassC*-1RPacsPatcs

(Fig. 2.93b) 2 P||Pass||C*

R||Pacs||Patcs 65

2PaRR-1RPaPa (Fig. 2.35a)

18 2PassRC*-1RPaPa (Fig. 2.91a)

8 Pass||R||C* R||Pa||Pa

66 2PassRC*-1RPacsPacs

(Fig. 2.94a) 2 Pass||R||C*

R||Pacs||Pacs 67 2PaRR-1RPaPat

(Fig. 2.35b) 18 2PassRC*-1RPaPat

(Fig. 2.91b) 8 Pass||R||C*

R||Pa||Pat 68 2PassRC*-1RPacsPatcs

(Fig. 2.94b) 2 Pass||R||C*

R||Pacs||Patcs 69 2PaPaR-1RPaPa

(Fig. 2.36a) 24 2PassPassC*-1RPaPa

(Fig. 2.92a) 6 Pass||Pass||C*

R||Pa||Pa 70 2PassPassC*-1RPacsPacs

(Fig. 2.95a) 3 Pass||Pass||C*

R||Pacs||Pacs 71 2PaPaR-1RPaPat

(Fig. 2.36b) 24 2PassPassC*-1RPaPat

(Fig. 2.92b) 6 Pass||Pass||C*

R||Pa||Pat 72 2PassPassC*-1RPacsPatcs

(Fig. 2.95b) 3 Pass||Pass||C*

R||Pacs||Patcs



No. Basic PPM Type

NF

Derived PPM Type

NF

Limb topology

1 (PRRRP)R-RPP (Fig. 2.37a)

6 (PRRRP)C*-RPP (Fig. 2.96a)

5 (PRRRP)C* R⊥ P ⊥⊥ P

2 (PS*RS*P)R-RPP (Fig. 2.97a)

2

3 (RRRRR)R-RPP (Fig. 2.37b)

6 (RRRRR)C*-RPP (Fig. 2.96b)

5 (RRRRR)C* R⊥ P ⊥⊥ P

4 (RS*RS*R)R-RPP (Fig. 2.97b)

2

5 (RPRPR)R-RPP (Fig. 2.37c)

6 (RPRPR)C*-RPP (Fig. 2.96c)

5 (RPRPR)C* R⊥ P ⊥⊥ P

6 (S*PRPS*)R-RPP (Fig. 2.97c)

2 (S*PRPS*)R R⊥ P ⊥⊥ P

7 (PRRRP)R-RPPa (Fig. 2.38a)

9 (PRRRP)R-RPPass

(Fig. 2.98a) 6 (PRRRP)R

R⊥ P⊥ ||Pass 8 (PS*RS*P)R-RPPa

(Fig. 2.101a) 5 (PS*RS*P)

R⊥ P⊥ ||Pa 9 (RRRRR)R-RPPa

(Fig. 2.38b) 9 (RRRRR)R-RPPass

(Fig. 2.98b) 6 (RRRRR)R

R⊥ P⊥ ||Pass 10 (RS*RS*R)R-RPPa

(Fig. 2.101b) 5 (RS*RS*R)R

R⊥ P⊥ ||Pa 11 (RPRPR)R-RPPa

(Fig. 2.38c) 9 (RPRPR)R-RPPass

(Fig. 2.98c) 6 (RPRPR)R

R⊥ P⊥ ||Pass 12 (S*PRPS*)R-RPPa

(Fig. 2.101c) 5 (S*PRPS*)R

R⊥ P⊥ ||Pa 13 (PRRRP)R-RPaP

(Fig. 2.39a) 9 (PRRRP)R-RPassP

(Fig. 2.99a) 6 PRRRP)R

R||Pass ⊥ P 14 (PS*RS*P)R-RPaP

(Fig. 2.102a) 5 (PS*RS*P)R

R||Pa⊥ P 15 (RRRRR)R-RPaP

(Fig. 2.39b) 9 (RRRRR)R-RPassP


R||Pass ⊥ P 16 (RS*RS*R)R-RPaP

(Fig. 2.102b) 5 (RS*RS*R)R

R||Pa⊥ P 17 (RPRPR)R-RPaP

(Fig. 2.39c) 9 (RPRPR)R-RPassP


R||Pass ⊥ P 18 (S*PRPS*)R-RPaP

(Fig. 2.102c) 5 (S*PRPS*)R

R||Pa⊥ P 19 (PRRRP)R-RPaPa

(Fig. 2.40a) 12 (PRRRP)R-RPassPass


R||Pass||Pass 20 (PS*RS*P)R-RPaPa

(Fig. 2.103a) 8 (PS*RS*P)R

R||Pa||Pa


Table 2.21. (cont.)

21 (RRRRR)R-RPaPa (Fig. 2.40b)

12 (RRRRR)R-RPassPass

(Fig. 2.100b) 6 RRRRR)R

R||Pass||Pass 22 (RS*RS*R)R-RPaPa

(Fig. 2.103b) 8 (RS*RS*R)R

R||Pa||Pa 23 (RPRPR)R-RPaPa

(Fig. 2.40c) 12 (RPRPR)R-RPassPass


R||Pass||Pass 24 (S*PRPS*)R-RPaPa

(Fig. 2.103c) 8 (S*PRPS*)R

R||Pa||Pa 25 (PRRRP)R-RPaPa

(Fig. 2.41a) 12 (PRRRP)R-RPacsPacs


R||Pacs||Pacs 26 (PS*RS*P)R-RPacsPacs

(Fig. 2.107a) 2 (PS*RS*P)R

R||Pacs||Pacs 27 (PRRRP)R-RPaPat

(Fig. 2.41b) 12 (PRRRP)R-RPacsPatcs

(Fig. 2.104b) 6 (PRRRP)R

R||Pacs||Patcs 28 (PS*RS*P)R-RPacsPatcs

(Fig. 2.107b) 2 (PS*RS*P)R

R||Pacs||Patcs 29 (RRRRR)R-RPaPa

(Fig. 2.42a) 12 (RRRRR)R-RPacsPacs

(Fig. 2.105a) 6 (RRRRR)R

R||Pacs||Pacs 30 (RS*RS*R)R-RPacsPacs

(Fig. 2.108a) 2 (RS*RS*R)R

R||Pacs||Pacs 31 (RRRRR)R-RPaPat

(Fig. 2.42b) 12 (RRRRR)R-RPacsPatcs


R||Pacs||Patcs 32 (RS*RS*R)R-RPacsPatcs

(Fig. 2.108b) 2 (RS*RS*R)R

R||Pacs||Patcs 33 (RPRPR)R-RPaPa

(Fig. 2.43a) 12 (RPRPR)R-RPacsPacs

(Fig. 2.106a) 6 (RPRPR)R

R||Pacs||Pacs 34 (S*PRPS*)R-RPacsPacs

(Fig. 2.109a) 2 (S*PRPS*)R

R||Pacs||Pacs 35 (RPRPR)R-RPaPat

(Fig. 2.43b) 12 (RPRPR)R-RPacsPatcs

(Fig. 2.106b) 6 (RPRPR)R

R||Pacs||Patcs 36 (S*PRPS*)R-RPacsPatcs

(Fig. 2.109b) 2 (S*PRPS*)R

R||Pacs||Patcs




Solution 3-RR*RR (Fig. 2.44a)

3-RRC* (Fig. 2.44b) 3-RC*R (Fig. 2.45a) 3-C*RR (Fig. 2.45b) 3-PRC* (Fig. 2.46a) 3-C*RP (Fig. 2.46b)

1 m 11 8 2 p1 4 3 3 p2 4 3 4 p3 4 3 5 p 12 9 6 q 2 2 7 k1 3 3 8 k2 0 0 9 k 3 3 10 (RGi)

(i=1,2,3) See Table 2.14 See Table 2.14

11 SG1 4 4 12 SG2 4 4 13 SG3 4 4 14 rG1 0 0 15 rG2 0 0 16 rG3 0 0 17 MG1 4 4 18 MG2 4 4 19 MG3 4 4 20 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 21 SF 3 3 22 rl 0 0 23 rF 9 9 24 MF 3 3 25 NF 3 3 26 TF 0 0 27 1p

jj 1f

=∑ 4 4

28 2pjj 1

f=∑ 4 4

29 3pjj 1

f=∑ 4 4

30 pjj 1

f=∑ 12 12





Solution 3-PaRS* (Fig. 2.47a) 3-PaS*P (Fig. 2.48b) 3-PaPS* (Fig. 2.49a) 3-PPaS* (Fig. 2.49b)

3-PassRR (Fig. 2.47b)

3-PaPaS* (Fig. 2.48a)

1 m 14 14 20 2 p1 6 6 9 3 p2 6 6 9 4 p3 6 6 9 5 p 18 18 27 6 q 5 5 8 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi)

(i=1,2,3) See Table 2.14 See Table 2.14 See Table 2.14


jj 1f

=∑ 8 10 11

28 2pjj 1

f=∑ 8 10 11

29 3pjj 1

f=∑ 8 10 11

30 pjj 1

f=∑ 24 30 33





Solution 3-PassPassR (Fig. 2.50a)

3-PassRP (Fig. 2.50b) 3-PassPR (Fig. 2.51a) 3-PPassR (Fig. 2.51b)

3-PC*R (Fig. 2.52)

1 m 20 14 8 2 p1 9 6 3 3 p2 9 6 3 4 p3 9 6 3 5 p 27 18 9 6 q 8 5 2 7 k1 0 0 3 8 k2 3 3 0 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 17 10 4

28 2pjj 1

f=∑ 17 10 4

29 3pjj 1

f=∑ 17 10 4

30 pjj 1

f=∑ 51 30 12





Solution 1PC*R-2RR*RR (Fig. 2.53)

1PC*R-2RRC* (Fig. 2.54a) 1PC*R-2RC*R (Fig. 2.54b) 1C*PR-2C*RR (Fig. 2.55a) 1PC*R-2PRC* (Fig. 2.55b)

1PaPS*-2PaRS* (Fig. 2.56a) 1PaPS*-2PPaS* (Fig. 2.56b)

1 m 10 8 14 2 p1 3 3 6 3 p2 4 3 6 4 p3 4 3 6 5 p 11 9 18 6 q 2 2 5 7 k1 3 3 0 8 k2 0 0 3 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 4 4 8

28 2pjj 1

f=∑ 4 4 8

29 3pjj 1

f=∑ 4 4 8

30 pjj 1

f=∑ 12 12 24



Table 2.26. Structural parametersa of planar parallel mechanisms in Figs. 2.57 and 2.58


Solution 1PaPS*-2PaPaS* (Fig. 2.57a)

1PaPS*-2PaS*P (Fig. 2.57b)

1PassPR-2PassRR (Fig. 2.58a) 1PassPR-2PPassR (Fig. 2.58b)

1 m 18 14 14 2 p1 6 6 6 3 p2 9 6 6 4 p3 9 6 6 5 p 24 18 18 6 q 7 5 5 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 8 8 10

28 2pjj 1

f=∑ 11 8 10

29 3pjj 1

f=∑ 11 8 10

30 pjj 1

f=∑ 30 24 30





Solution 1PassPR-2PaPassR (Fig. 2.59a)

1PassPR-2PassRP (Fig. 2.59b)

3RR*RR (Fig. 2.60)

1 m 18 14 11 2 p1 6 6 4 3 p2 9 6 4 4 p3 9 6 4 5 p 24 18 12 6 q 7 5 2 7 k1 0 0 3 8 k2 3 3 0 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 10 10 4

28 2pjj 1

f=∑ 13 10 4

29 3pjj 1

f=∑ 13 10 4

30 pjj 1

f=∑ 36 30 12





Solution 3RC*R (Fig. 2.61a) 2RC*R-1RC*R (Fig. 2.61b) 2RPC*-1RPC* (Fig. 2.62)

2RC*Pa-1RC*Pa (Fig. 2.63) 2RPaC*-1RPaC* (Fig. 2.64a)

2RPaPass-1RPaPass (Fig. 2.64b)

1 m 8 14 20 2 p1 3 6 9 3 p2 3 6 9 4 p3 3 6 9 5 p 9 18 27 6 q 2 5 8 7 k1 3 0 3 8 k2 0 3 0 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 4 7 13

28 2pjj 1

f=∑ 4 7 13

29 3pjj 1

f=∑ 4 7 13

30 pjj 1

f=∑ 12 21 39





Solution 2PRC*-1RPP (Fig. 2.65a) 2RRC*-1RPP (Fig. 2.65b) 2RPC*-1RPP(Fig. 2.65c)

2PS*R-1RPP (Fig. 2.66a) 2RS*R-1RPP (Fig. 2.66b) 2S*PR-1RPP (Fig. 2.66c)

2PPaR-1RC*C* (Fig. 2.67a) 2PaRR-1RC*C* (Fig. 2.67b)

2PaPaR-1RC*C* (Fig. 2.67c)

1 m 8 8 12 16 2 p1 3 3 6 9 3 p2 3 3 6 9 4 p3 3 3 3 3 5 p 9 9 15 21 6 q 2 2 4 6 7 k1 3 3 2 2 8 k2 0 0 1 1 9 k 3 3 3 3 10 (RGi)

(i=1,2,3) See Table 2.14See Table 2.14See Table 2.14 See Table 2.14

11 SG1 4 5 3 3 12 SG2 4 5 3 3 13 SG3 3 3 5 5 14 rG1 0 0 3 6 15 rG2 0 0 3 6 16 rG3 0 0 0 0 17 MG1 4 5 3 3 18 MG2 4 5 3 3 19 MG3 3 3 5 5 20 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 21 SF 3 3 3 3 22 rl 0 0 6 12 23 rF 8 10 14 20 24 MF 3 3 3 3 25 NF 4 2 10 16 26 TF 0 0 0 0 27 1p

jj 1f

=∑ 4 5 6 9

28 2pjj 1

f=∑ 4 5 6 9

29 3pjj 1

f=∑ 3 3 5 5

30 pjj 1

f=∑ 11 13 17 23





Solution 2PPassR-1RPP (Fig. 2.68a) 2PassRR-1RPP (Fig. 2.68b)

2PassPassR-1RPP (Fig. 2.68c)

2PRC*-1RPPa (Fig. 2.69a) 2RRC*-1RPPa (Fig. 2.69b) 2RPC*-1RPPa (Fig. 2.69c) 2PRC*-1RPaP (Fig. 2.70a) 2RRC*-1RPaP (Fig. 2.70b) 2RPC*-1RPaP (Fig. 2.70c)

1 m 12 16 10 2 p1 6 9 3 3 p2 6 9 3 4 p3 3 3 6 5 p 15 21 12 6 q 4 6 3 7 k1 1 1 2 8 k2 2 2 1 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 10 17 4

28 2pjj 1

f=∑ 10 17 4

29 3pjj 1

f=∑ 3 3 6

30 pjj 1

f=∑ 23 37 14





Solution 2PRC*-1RPaPa (Fig. 2.71a) 2RRC*-1RPaPa (Fig. 2.71b) 2RPC*-1RPaPa (Fig. 2.71c)

2PRR-1RPPass (Fig. 2.72a) 2RRR-1RPPass (Fig. 2.72b) 2RPR-1RPPass (Fig. 2.72c) 2PRR-1RPassP (Fig. 2.73a) 2RRR-1RPassP (Fig. 2.73b) 2RPR-1RPassP (Fig. 2.73c)

2PRR-1RPassPass (Fig. 2.74a) 2RRR-1RPassPass (Fig. 2.74b) 2RPR-1RPassPass (Fig. 2.74c)

1 m 12 10 12 2 p1 3 3 3 3 p2 3 3 3 4 p3 9 6 9 5 p 15 12 15 6 q 4 3 4 7 k1 2 2 2 8 k2 1 1 1 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 4 3 3

28 2pjj 1

f=∑ 4 3 3

29 3pjj 1

f=∑ 9 10 17

30 pjj 1

f=∑ 17 16 23





Solution 2PPaC*-1RPPa (Fig. 2.75a) 2PaRC*-1RPPa (Fig. 2.75b) 2PPaC*-1RpaP (Fig. 2.76a) 2PaRC*-1RPaP (Fig. 2.76b)

2PaPaC*-1RPPa(Fig. 2.77a) 2PaPaC*-1RPaP(Fig. 2.77b)

2PPaC*-1RPaPa (Fig. 2.78a) 2PaRC*-1RPaPa (Fig. 2.78b)

1 m 14 18 16 2 p1 6 9 6 3 p2 6 9 6 4 p3 6 6 9 5 p 18 24 21 6 q 5 7 6 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 7 10 7

28 2pjj 1

f=∑ 7 10 7

29 3pjj 1

f=∑ 6 6 9

30 pjj 1

f=∑ 20 26 23





Solution 2PaPaC*-1RPaPa (Fig. 2.79a)

2PassPassR-1RPassPass

(Fig. 2.79b)

2PPassR-1RPPass 2PassRR-1RPPass (Fig. 2.80a,b) 2PPassR-1RPassP 2PassRR-1RPassP (Fig. 2.81a,b)

1 m 20 20 14 2 p1 9 9 6 3 p2 9 9 6 4 p3 9 9 6 5 p 27 27 18 6 q 8 8 5 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 10 17 10

28 2pjj 1

f=∑ 10 17 10

29 3pjj 1

f=∑ 9 17 10

30 pjj 1

f=∑ 29 51 30





Solution 2PassPassR- 1RPPass (Fig. 2.82a) 2PassPassR- 1RPassP (Fig. 2.82b)

2PPassR- 1RPassPass (Fig. 2.83a) 2PassRR- 1RPassPass

(Fig. 2.83b)

2PRC*-1RPacsPacs (Fig. 2.84a) 2PRC*-1RPacsPatcs (Fig. 2.84b) 2RRC*-1RPacsPacs (Fig. 2.85a)

2RRC*-1RPacsPatcs (Fig. 2.85b) 2RPC*-1RPacsPacs (Fig. 2.86a) 2RPC*-1RPacsPatcs (Fig. 2.86b)

1 m 18 16 12 2 p1 9 6 3 3 p2 9 6 3 4 p3 6 9 9 5 p 24 21 15 6 q 7 6 4 7 k1 0 0 2 8 k2 3 3 1 9 k 3 3 3 10 (RGi)



jj 1f

=∑ 17 10 4

28 2pjj 1

f=∑ 17 10 4

29 3pjj 1

f=∑ 10 17 15

30 pjj 1

f=∑ 44 37 23





Solution 2PS*R-1RPacsPacs (Fig. 2.87a) 2PS*R-1RPacsPatcs (Fig. 2.87b) 2RS*R-1RPacsPacs (Fig. 2.88a) 2RS*R-1RPacsPatcs (Fig. 2.88b) 2S*PR-1RPacsPacs (Fig. 2.89a) 2S*PR-1RPacsPatcs (Fig. 2.89b)

2PPassC*-1RPaPa (Fig. 2.90a) 2PPassC*-1RPaPat (Fig. 2.90b) 2PassRC*-1RPaPa (Fig. 2.91a) 2PassRC*-1RPaPat (Fig. 2.91b)

1 m 12 16 2 p1 3 6 3 p2 3 6 4 p3 9 9 5 p 15 21 6 q 4 6 7 k1 2 0 8 k2 1 3 9 k 3 3 10 (RGi)

(i=1,2,3) See Table 2.14 See Table 2.14

11 SG1 5 5 12 SG2 5 5 13 SG3 3 3 14 rG1 0 6 15 rG2 0 6 16 rG3 12 6 17 MG1 5 5 18 MG2 5 5 19 MG3 3 3 20 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 21 SF 3 3 22 rl 12 18 23 rF 22 28 24 MF 3 3 25 NF 2 8 26 TF 0 0 27 1p

jj 1f

=∑ 5 11

28 2pjj 1

f=∑ 5 11

29 3pjj 1

f=∑ 15 9

30 pjj 1

f=∑ 25 31





Solution 2PassPassC*- 1RPaPa (Fig. 2.92a) 2PassPassC*- 1RPaPat

(Fig. 2.92b)

2PPassC*-1RPacsPacs (Fig. 2.93a) 2PPassC*-1RPacsPatcs (Fig. 2.93b) 2PassRC*-1RPacsPacs (Fig. 2.94a) 2PassRC*-1RPacsPatcs (Fig. 2.94b)

2PassPassC*-1RPacsPacs

(Fig. 2.95a) 2PassPassC*-1RPacsPatcs

(Fig. 2.95b) 1 m 20 16 20 2 p1 9 6 9 3 p2 9 6 9 4 p3 9 9 9 5 p 27 21 27 6 q 8 6 8 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi)

(i=1,2,3) See Table 2.14See Table 2.14 See Table 2.14


jj 1f

=∑ 18 11 18

28 2pjj 1

f=∑ 18 11 18

29 3pjj 1

f=∑ 9 15 12

30 pjj 1

f=∑ 45 37 48





Solution (PRRRP)C*-RPP (Fig. 2.96a) (RRRRR)C*-RPP (Fig. 2.96b) (RPRPR)C*-RPP (Fig. 2.96c)

(PS*RS*P)R-RPP (Fig. 2.97a) (RS*RS*R)R-RPP (Fig. 2.97b) (S*PRPS*)R-RPP (Fig. 2.97c)

1 m 8 8 2 p1 6 6 3 p2 3 3 4 p 9 9 5 q 2 2 6 k1 1 1 7 k2 1 1 8 k 2 2 9 (RG1-G2) ( 1 2 3, , , δv v v ω ) ( 1 2, , ,β δv v ω ω ) 10 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 SG1-G2 4 4 12 SG3 3 3 13 rG1-G2 3 6 14 rG3 0 0 15 MG1-G2 4 4 16 MG3 3 3 17 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 18 SF 3 3 19 rl 3 6 20 rF 7 10 21 MF 3 3 22 NF 5 2 23 TF 0 0 24 1p

jj 1f

=∑ 7 10

25 2pjj 1

f=∑ 3 3

26 pjj 1

f=∑ 10 13



Table 2.38. Structural parametersa of planar parallel mechanisms in Figs. 2.98- 2.100


Solution (PRRRP)R-RPPass (Fig. 2.98a) (RRRRR)R-RPPass (Fig. 2.98b) (RPRPR)R-RPPass (Fig. 2.98c) PRRRP)R-RPassP (Fig. 2.99a) (RRRRR)R-RPassP (Fig. 2.99b) (RPRPR)R-RPassP (Fig. 2.99c)

(PRRRP)R-RPassPass

(Fig. 2.100a) (RRRRR)R-RPassPass

(Fig. 2.100b) (RPRPR)R-RPassPass

(Fig. 2.100c) 1 m 10 12 2 p1 6 6 3 p2 6 9 4 p 12 15 5 q 3 4 6 k1 0 0 7 k2 2 2 8 k 2 2 9 (RG1-G2) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 10 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 SG1-G2 3 3 12 SG3 3 3 13 rG1-G2 3 3 14 rG3 6 12 15 MG1-G2 3 3 16 MG3 4 5 17 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 18 SF 3 3 19 rl 9 15 20 rF 12 18 21 MF 4 5 22 NF 6 6 23 TF 1 2 24 1p

jj 1f

=∑ 6 6

25 2pjj 1

f=∑ 10 17

26 pjj 1

f=∑ 16 23





Solution (PS*RS*P)R-RPPa (Fig. 2.101a) (RS*RS*R)R-RPPa (Fig. 2.101b) (S*PRPS*)R-RPPa (Fig. 2.101c) (PS*RS*P)R-RPaP (Fig. 2.102a) (RS*RS*R)R-RPaP (Fig. 2.102b) (S*PRPS*)R-RPaP (Fig. 2.102c)

(PS*RS*P)R-RPaPa (Fig. 2.103a) (RS*RS*R)R-RPaPa (Fig. 2.103b) (S*PRPS*)R-RPaPa (Fig. 2.103c)

1 m 10 12 2 p1 6 6 3 p2 6 9 4 p 12 15 5 q 3 4 6 k1 0 0 7 k2 2 2 8 k 2 2 9 (RG1-G2) ( 1 2, , ,β δv v ω ω ) ( 1 2, , ,β δv v ω ω ) 10 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 SG1-G2 4 4 12 SG3 3 3 13 rG1-G2 6 6 14 rG3 3 6 15 MG1-G2 4 4 16 MG3 3 3 17 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 18 SF 3 3 19 rl 9 12 20 rF 13 16 21 MF 3 3 22 NF 5 8 23 TF 0 0 24 1p

jj 1f

=∑ 10 10

25 2pjj 1

f=∑ 6 9

26 pjj 1

f=∑ 16 19





Solution (PRRRP)R-RPacsPacs

(Fig. 2.104a) (PRRRP)R-RPacsPatcs

(Fig. 2.104b) (RRRRR)R-RPacsPacs

(Fig. 2.105a) (RRRRR)R-RPacsPatcs

(Fig. 2.105b) (RPRPR)R-RPacsPacs

(Fig. 2.106a) (RPRPR)R-RPacsPatcs

(Fig. 2.106b)

(PS*RS*P)R-RPacsPacs

(Fig. 2.107a) (PS*RS*P)R-RPacsPatcs

(Fig. 2.107b) (RS*RS*R)R-RPacsPacs

(Fig. 2.108a) (RS*RS*R)R-RPacsPatcs

(Fig. 2.108b) (S*PRPS*)R-RPacsPacs

(Fig. 2.109a) (S*PRPS*)R-RPacsPatcs

(Fig. 2.109b) 1 m 12 12 2 p1 6 6 3 p2 9 9 4 p 15 15 5 q 4 4 6 k1 0 0 7 k2 2 2 8 k 2 2 9 (RG1-G2) ( 1 2, , δv v ω ) ( 1 2, , ,β δv v ω ω ) 10 (RG3) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 11 SG1-G2 3 4 12 SG3 3 3 13 rG1-G2 3 6 14 rG3 12 12 15 MG1-G2 3 4 16 MG3 3 3 17 (RF) ( 1 2, , δv v ω ) ( 1 2, , δv v ω ) 18 SF 3 3 19 rl 15 18 20 rF 18 22 21 MF 3 3 22 NF 6 2 23 TF 0 0 24 1p

jj 1f

=∑ 6 10

25 2pjj 1

f=∑ 9 15

26 pjj 1

f=∑ 15 25



Fig. 2.44. Overconstrained PPMs with coupled motions of types 3-RR*RR (a) and 3-RRC* (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ R*⊥ ||R||R (a) and R||R⊥ C* (b)


Fig. 2.45. Overconstrained PPMs with coupled motions of types 3-RC*R (a) and 3-C*RR (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ C*⊥ ||R (a) and C*⊥ R||R (b)


Fig. 2.46. Overconstrained PPMs with coupled motions of types 3-PRC* (a) and 3-C*RP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology P⊥ R⊥ C* (a) and C*⊥ R⊥ P (b)


Fig. 2.47. Overconstrained PPMs with coupled motions of types 3-PaRS* (a) and 3-PassRR (b) defined by SF=3, (RF)=( 1 2, , δv v ω ) and MF=3, NF=9, TF=0 (a), MF=6, NF=6, TF=3 (b), limb topology Pa||RS* (a) and Pass||R||R (b)


Fig. 2.48. Overconstrained PPMs with coupled motions of types 3-PaPaS* (a) and 3-PaS*P (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=18 (a), NF=9 (b), limb topology Pa||PaS* (a) and PaS*P (b)


Fig. 2.49. Overconstrained PPMs with coupled motions of types 3-PaPS* (a) and 3-PPaS* (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9, limb topology Pa⊥ PS* (a) and P⊥ PaS* (b)


Fig. 2.50. Overconstrained PPMs with coupled motions of types 3-PassPassR (a) and 3-PassRP (b) defined by SF=3, (RF)=( 1 2, , δv v ω ), NF=6 and MF=9, TF=6 (a) MF=6, TF=3 (a), limb topology Pass||Pass||R (a) and Pass||R⊥ P (b)


Fig. 2.51. Overconstrained PPMs with coupled motions of types 3-PassPR (a) and 3-PPassR (b) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=6, NF=6, TF=3, limb topology Pass ⊥ P⊥ ||R (a) and P⊥ Pass||R (b)


Fig. 2.52. Overconstrained PPM with one decoupled translation of type 3-PC*R defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology P⊥ C* ⊥⊥ R

Fig. 2.53. Overconstrained PPM with one decoupled translation of type 1PC*R-2RR*RR defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology P⊥ C* ⊥⊥ R and R⊥ R*⊥ ||R||R


Fig. 2.54. Overconstrained PPMs with one decoupled translation of types 1PC*R-2RRC* (a) and 1PC*R-2RC*R (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology P⊥ C* ⊥⊥ R and R||R⊥ C* (a), P⊥ C* ⊥⊥ R and R⊥ C*⊥ ||R (b)


Fig. 2.55. Overconstrained PPMs with one decoupled translation of types 1C*PR-2C*RR (a) and 1PC*R-2PRC* (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=3, limb topology P⊥ C* ⊥⊥ R and C*⊥ R||R (a), P⊥ C* ⊥⊥ R and P⊥ R⊥ C* (b)


Fig. 2.56. Overconstrained PPMs with one decoupled translation of types 1PaPS*-2PaRS* (a) and 1PaPS*-2PPaS* (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9, limb topology Pa⊥ PS* and Pa||RS* (a), Pa⊥ PS* and P⊥ PaS* (b)


Fig. 2.57. Overconstrained PPMs with one decoupled translation of types 1PaPS*-2PaPaS* (a) and 1PaPS*-2PaS*P (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=15 (a), NF=9 (b), limb topology Pa⊥ PS* and Pa||PaS* (a), Pa⊥ PS* and PaS*P (b)


Fig. 2.58. Overconstrained PPMs with one decoupled translation of types 1PassPR-2PassRR (a) and 1PassPR-2PPassR (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology Pass ⊥ P⊥ ||R and Pass||R||R (a), Pass ⊥ P⊥ ||R and P⊥ Pass||R (b)


Fig. 2.59. Overconstrained PPMs with one decoupled translation of types 1PassPR-2PaPassR (a) and 1PassPR-2PassRP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=9 (a), NF=3 (b), limb topology Pass ⊥ P⊥ ||R and Pa||Pass||R (a), Pass ⊥ P⊥ ||R and Pass||R⊥ P (b)


Fig. 2.60. 3RR*RR-type overconstrained PPMs with partially decoupled translations defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ R*⊥ ||R||R


Fig. 2.61. Overconstrained PPMs with partially decoupled translations of types 3RC*R (a) and 2RC*R-1RC*R (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ C*⊥ ||R (a), R⊥ C*⊥ ||R and R⊥ C*⊥ ||R (b)


Fig. 2.62. Overconstrained PPM with decoupled rotation of type 2RPC*-1RPC* defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ P ⊥⊥ C*

Fig. 2.63. Overconstrained PPM with decoupled rotation of type 2RC*Pa-1RC*Pa defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology R⊥ C*⊥ ||Pa


Fig. 2.64. Overconstrained PPMs with decoupled rotation of types 2RPaC*-1RPaC* (a) and 2RPaPass-1RPaPass (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=12, limb topology R||Pa⊥ C* (a) and R||Pa||Pass (b)


Fig. 2.65. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPP (a), 2RRC*-1RPP (b) and 2RPC*-1RPP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=4, limb topology R⊥ P ⊥⊥ P and P⊥ R||C* (a), R||R||C* (b), R⊥ P⊥ ||C* (c)


Fig. 2.66. Overconstrained PPMs with uncoupled rotation of types 2PS*R-1RPP (a), 2RS*R-1RPP (b) and 2S*PR-1RPP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology R⊥ P ⊥⊥ P and PS*R (a), RS*R (b), SP⊥ R (c)


Fig. 2.67. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RC*C* (a), 2PaRR-1RC*C* (b) and 2PaPaR-1RC*C* (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=10 (a), (b) NF=16 (c), limb topology R⊥ C* ⊥⊥ C* and P⊥ Pa||R (a), Pa||R||R (b), Pa||Pa||R (c)


Fig. 2.68. Overconstrained PPMs with uncoupled rotation of types 2PPassR-1RPP (a), 2PassRR-1RPP (b) and 2PassPassR-1RPP (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), NF=4 and MF=3, TF=0 (a and b), MF=5, TF=2 (c), limb topology R⊥ P ⊥⊥ P and P⊥ Pass||R (a), Pass||R||R (b), Pass||Pass||R (c)


Fig. 2.69. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPPa (a), 2RRC*-1RPPa (b) and 2RPC*-1RPPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=7, limb topology R⊥ P⊥ ||Pa and P⊥ R||C* (a), R||R||C* (b), R⊥ P⊥ ||C* (c)


Fig. 2.70. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPaP (a), 2RRC*-1RPaP (b) and 2RPC*-1RPaP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=7, limb topology R||Pa⊥ P and P⊥ R||C* (a), R||R||C* (b), R⊥ P⊥ ||C* (c)


Fig. 2.71. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPaPa (a), 2RRC*-1RPaPa (b) and 2RPC*-1RPaPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=10, limb topology R||Pa||Pa and P⊥ R||C* (a), R||R||C* (b), R⊥ P⊥ ||C* (c)


Fig. 2.72. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPPass (a), 2RRR-1RPPass (b) and 2RPR-1RPPass (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=4, NF=6, TF=1, limb topology R⊥ P⊥ ||Pass and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.73. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPassP (a), 2RRR-1RPassP (b) and 2RPR-1RPassP (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=4 NF=6, TF=1, limb topology R||Pass ⊥ P and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.74. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPassPass (a), 2RRR-1RPassPass (b) and 2RPR-1RPassPass (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=5, NF=6, TF=2, limb topology R||Pass||Pass and P⊥ R||R (a), R||R||R (b), R⊥ P⊥ ||R (c)


Fig. 2.75. Overconstrained PPMs with uncoupled rotation of types 2PPaC*-1RPPa (a) and 2PaRC*-1RPPa (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=13, limb topology R⊥ P⊥ ||Pa and P⊥ Pa||C* (a), Pa||R||C* (b)


Fig. 2.76. Overconstrained PPMs with uncoupled rotation of types 2PPaC*-1RPaP (a) and 2PaRC*-1RPaP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=13, limb topology R||Pa⊥ P and P⊥ Pa||C* (a), Pa||R||C* (b)


Fig. 2.77. Overconstrained PPMs with uncoupled rotation of types 2PaPaC*-1RPPa (a) and 2PaPaC*-1RPaP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=19, limb topology Pa||Pa||C* and R⊥ P⊥ ||Pa (a), R||Pa⊥ P (b)


Fig. 2.78. Overconstrained PPMs with uncoupled rotation of types 2PPaC*-1RPaPa (a) and 2PaRC*-1RPaPa (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=16, limb topology R||Pa||Pa and P⊥ Pa||C* (a), Pa||R||C* (b)


Fig. 2.79. Overconstrained PPMs with uncoupled rotation of types 2PaPaC*-1RPaPa (a) and 2PassPassR-1RPassPass (b) defined by SF=3, (RF)=( 1 2, , δv v ω ) and MF=3, NF=22, TF=0 (a), MF=5, NF=2, TF=2 (b), limb topology Pa||Pa||C* and R||Pa||Pa (a), Pass||Pass||R and R||Pass||Pass (b)


Fig. 2.80. Overconstrained PPMs with uncoupled rotation of types 2PPassR-1RPPass (a) and 2PassRR-1RPPass (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ P⊥ ||Pass and P⊥ Pass||R (a), Pass||R||R (b)


Fig. 2.81. Overconstrained PPMs with uncoupled rotation of types 2PPassR-1RPassP (a) and 2PassRR-1RPassP (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology R⊥ Pass ⊥ ||P and P⊥ Pass||R (a), Pass||R||R (b)


Fig. 2.82. Overconstrained PPMs with uncoupled rotation of types 2PassPassR -1RPPass (a) and 2PassPassR-1RPassP (b) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=5, NF=3, TF=2, limb topology Pass||Pass||R and R⊥ P⊥ ||Pass (a), R||Pass ⊥ P (b)


Fig. 2.83. Overconstrained PPMs with uncoupled rotation of types 2PPassR-1RPassPass (a) and 2PassRR-1RPassPass (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology R||Pass||Pass and P⊥ Pass||R (a), Pass||R||R (b)


Fig. 2.84. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPacsPacs (a) and 2PRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=4, limb topology P⊥ R||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.85. Overconstrained PPMs with uncoupled rotation of types 2RRC*-1RPacsPacs (a) and 2RRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=4, limb topology R||R||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.86. Overconstrained PPMs with uncoupled rotation of types 2RPC*-1RPacsPacs (a) and 2RPC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=4, limb topology R⊥ P⊥ ||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.87. Overconstrained PPMs with uncoupled rotation of types 2PS*R-1RPacsPacs (a) and 2PS*R-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology PS*R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.88. Overconstrained PPMs with uncoupled rotation of types 2RS*R-1RPacsPacs (a) and 2RS*R-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology RS*R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.89. Overconstrained PPMs with uncoupled rotation of types 2S*PR-1RPacsPacs (a) and 2S*PR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology S*P⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.90. Overconstrained PPMs with uncoupled rotation of types 2PPassC*-1RPaPa (a) and 2PPassC*-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=8, limb topology P⊥ Pass||C* and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.91. Overconstrained PPMs with uncoupled rotation of types 2PassRC*-1RPaPa (a) and 2PassRC*-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=8, limb topology Pass||R||C* and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.92. Overconstrained PPMs with uncoupled rotation of types 2PassPassC*-1RPaPa (a) and 2PassPassC*-1RPaPat (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6, limb topology Pass||Pass||C* and R||Pa||Pa (a), R||Pa||Pat (b)


Fig. 2.93. Overconstrained PPMs with uncoupled rotation of types 2PPassC*-1RPacsPacs (a) and 2PPassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology P||Pass||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.94. Overconstrained PPMs with uncoupled rotation of types 2PassRC*-1RPacsPacs (a) and 2PassRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2, limb topology Pass||R||C*and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.95. Overconstrained PPMs with uncoupled rotation of types 2PassPassC*-1RPacsPacs (a) and 2PassPassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=3, limb topology Pass||Pass||C*and R||Pacs||Pacs (a), R||Pacs||Patcs (b)


Fig. 2.96. Overconstrained PPMs with uncoupled rotation of types (PRRRP)C*-RPP (a), (RRRRR)C*-RPP (b) and (RPRPR)C*-RPP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=5


Fig. 2.97. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)R-RPP (a), (RS*RS*R)R-RPP (b) and (S*PRPS*)R-RPP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2


Fig. 2.98. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPPass (a), (RRRRR)R-RPPass (b) and (RPRPR)R-RPPass (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=4, NF=6, TF=1


Fig. 2.99. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPassP (a), (RRRRR)R-RPassP (b) and (RPRPR)R-RPassP (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=4, NF=6, TF=1


Fig. 2.100. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPassPass (a), (RRRRR)R-RPassPass (b) and (RPRPR)R-RPassPass (c) defined by SF=3, (RF)=( 1 2, , δv v ω ), MF=5, NF=6, TF=2


Fig. 2.101. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)R-RPPa (a), (RS*RS*R)R-RPPa (b) and (S*PRPS*)R-RPPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=5


Fig. 2.102. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)R-RPaP (a), (RS*RS*R)R-RPaP (b) and (S*PRPS*)R-RPaP (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=5


Fig. 2.103. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)R-RPaPa (a), (RS*RS*R)R-RPaPa (b) and (S*PRPS*)R-RPaPa (c) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=8


Fig. 2.104. Overconstrained PPMs with uncoupled rotation of types (PRRRP)R-RPacsPacs (a) and (PRRRP)R-RPacsPatcs (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=6


Fig. 2.105. Overconstrained PPMs with uncoupled rotation of types (RRRRR)R-RPacsPacs (a) and (RRRRR)R-RPacsPatcs (b) defined by MF=SF=3, (RF)= ( 1 2, , δv v ω ), TF=0, NF=6


Fig. 2.106. Overconstrained PPMs with uncoupled rotation of types (RPRPR)R-RPacsPacs (a) and (RPRPR)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=6


Fig. 2.107. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)R-RPacsPacs (a) and (PS*RS*P)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2


Fig. 2.108. Overconstrained PPMs with uncoupled rotation of types (RS*RS*R)R-RPacsPacs (a) and (RS*RS*R)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2


Fig. 2.109. Overconstrained PPMs with uncoupled rotation of types (S*PRPS*)R-RPacsPacs (a) and (S*PRPS*)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( 1 2, , δv v ω ), TF=0, NF=2

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