Kinematic Analysis of a Double-Layer Parallel Superimposed Mechanism

Kinematic Analysis of a Double-Layer Parallel Superimposed Mechanism

Zhiyong Meng

Tianjin University of Technology and Education

Tianjin China

Abstract: In order to make the parallel mechanism reach the level of the series mechanism in the working space, a new type of coupling mechanism is proposed, which is coupled in the form of the basic configuration of the parallel mechanism, and then the position positive solution of the double-layer parallel mechanism can be derived. Through the differential mapping relationship of the finite instantaneous spiral theory, the velocity of the mechanism is solved. Finally, through motion simulation, it is verified that the performance of the mechanism is consistent with the theoretical analysis.

Keywords: double-layer parallel mechanism; finite instantaneous spiral theory; differential transformations; positive kinematics

1. INTRODUCTION

   Compared with the series mechanism, the parallel mechanism has many advantages such as high stiffness, large bearing capacity, good dynamic performance and high precision, but the disadvantages of the parallel mechanism in the working space are also obvious, so that the application in practice is still very limited, and the parallel machine tool, the parallel attitude adjustment platform and the multi-degree-of-freedom motion simulator are mainly concentrated at present. However, there is still a shortage of parallel mechanisms with large working spaces, especially those with rotational freedom. Therefore, the parallel mechanism with a large working space has high research value and broad application prospects. Li Qinchuan

   [3] replaced the mechanism with the equivalent motion pair according to the relationship between the moving platform and the fixed platform, and analyzed the research status and research direction of the two-shift and one-shift parallel mechanism by relevant scholars at home and abroad. Yulin Zhou [4] of Yanshan University proposed a workspace mapping method based on bias output, which changes the institutional workspace by changing the bias angle. Miseabadesu [5] expanded the working space by optimizing the scale parameters of the 3-UPU parallel mechanism and the 3-UPS&S parallel mechanism, and the effect was remarkable. Carol Miller [6] increased the working space of the paralleling mechanism by optimizing the position of the drive unit on the frame. United States scholar Rossheim Mark [7] proposed a variety of large workspace parallel mechanism configurations and applied them to robots. Zhang Yaojun et al. [8] optimized the scale of a class of three- translational degrees of freedom parallel mechanism to increase the working space. Tao Zongjie et al. [9] proposed an effective design method based on the

   influence of member interference on the working space, and verified it in the design of the 3-RRR parallel mechanism. Chen Jianghong [10] used the idea of virtual axis and used the diamond-shaped mechanism as a branch chain to reduce the limitation of rod length on the working space, which not only improved the performance of the working space, but also increased the longitudinal expansion ratio. Bai Zhifu [11] designed a composite spherical pair with three rotating pairs from the perspective of improving the spherical pair of the parallel mechanism, so as to increase the swing angle and then expand the attitude space of the parallel mechanism. Xu Dongguang [12] used the idea of function optimization to obtain the influence of changing the scale parameters on the working space of the 6-UPS parallel mechanism, and gave a solution.

   In addition, a new mechanism formed by connecting another mechanism in series on the basis of a parallel mechanism is a hybrid mechanism [13], and this tandem method can also improve the working space. However, it is more difficult to study the hybrid mechanism, and for location analysis, the relationship between input and output is often difficult to determine, so there are multiple solutions, which makes the practicability of the mechanism lacking. The traditional spiral

   [14] method is often complex for velocity solving.

   In order to make the parallel mechanism reach the level of the series mechanism in the working space, a new type of coupling connection mechanism is proposed [15]. This configuration uses a mechanical transmission mode to transmit force and motion between parallel mechanisms, and after the coupling of two parallel mechanisms, the motion characteristics similar to those of the series mechanism are realized, and the important performance of the working space is reached or even better than that of the series mechanism [16]. For the velocity solution, the finite instantaneous spiral theory is used to make the process

   rule. Due to the symmetry of the mechanism, points A , B ,

2. DOUBLE-LAYER PAR-ALLEL MECHANISM CONFI G-URATION

   and C are all on the Z axis, and points

   G1 and Q1

   are

   symmetric about the origin. Based on this, the coordinates of

   points

   A1 ,

   C1 ,

   G1 ,

   Q1 can be set as 0 0

   a ,

   0 0

   a

   , b

   0 c

   and b

   0 c .

   According to the directions of the screws in the figure 1, the motion screw system of this branch can be written as:

   $11 0 1 0 ; 0 0 0

   $ 0 1 0 ; a 0 0

   12

   $13 0 1 0 ; a 0 0

   $ 1 0 0 ; 0 c 0

   14

   $

   15

   0 1 0 ; c

   0 b

   $16 0 0 1 ; 0 b 0

   $17 1 0 0 ; 0 c 0

   $ 0 1 0 ; c 0

   b

   18

   Figure 1 Mechanism Configuration

   $ 0 0 1 ; 0 b 0

   Although the double-layer parallel mechanism appears as a superposition of two 3-RSR parallel mechanisms, it can be

   19

   1

   considered as a 3-RSR parallel mechanism with three coplanar rotational constraints added at the shared linkage, due to the common linkage between the front and rear stages.

   The difference between this mechanism and traditional double- layer parallel mechanisms lies in the fact that traditional parallel stacking requires two levels of actuation, which inevitably increases the self-weight of the mechanism and reduces its dynamic performance. By modifying the kinematic pairs at the end of the mechanism to create shared kinematic pairs, the upper and lower stages of the mechanism are connected through these kinematic pairs. The kinematic pairs accurately transmit the motion output of the upper parallel mechanism to the lower mechanism as the motion input of the lower parallel mechanism. This design requires only one set of actuators placed at the base platform, reducing the overall weight of the mechanism and enhancing its stability.

   As shown in Figure 1, the origin of the branch coordinate system is selected at the center of the rotational pair of the middle platform. The Z axis is perpendicular to the middle platform and points towards the moving platform, the X axis points positively towards the center of the middle platform, and the Y axis direction is determined according to the right-hand

   Since the rank of the aforementioned motion screw system is 6, the constraint screws can only be:

   1

   $r 0 0 0 ; 0 0 0

   2

   Therefore, the branch does not impose any constraints on the platform. Additionally, since the mechanism's three branches are of the same type, the moving platform should have 6 degrees of freedom. For a 3-RSR parallel mechanism, in non- singular configurations, the movement of the platform is determined when the inputs are speified, possessing three degrees of freedom: two rotational and one translational. The linkage in the middle layer functions as a single-degree-of- freedom transmission element, which does not increase the degrees of freedom. Consequently, the overall mechanism should also have 3 degrees of freedom.

3. DIRECT KINEMATIC ANALYSIS

   the moving platform, the

   x2 axis is aligned along vector

   2 2 , and the z2 axis is perpendicular to the moving

   platform. Define as the angle between the projection of the

   moving coordinate system's

   z2 axis on the base plane and the

   positive direction of the fixed coordinate system's x axis. Let

   be the angle between the z2 axis of the moving coordinate

   system and the

   z2 axis of the fixed coordinate system. The

   distance between the centers of the moving and static platforms

   is li , and the distance from the platform center to the center of

   the rotational pair is R . All link lengths are L . Points

   G1 ,

   G2 , and

   2 , and

   G3

   3 .

   can be expressed in terms of the input angles

   1,

   R Lc1

   G 0

   1

   Ls1

   3

   1 (R Lc )

   2 2

   Figure 2 Schematic Diagram of the Mechanism

   G2

   3 (R Lc

   )

   2

   2

   As shown in Figure 2, the mechanism is composed of three parts: the static platform, the moving platform, and the branches. The branches are distributed in a 360° circular

   Ls2

   4

   2

   3

   1 (R Lc )

   arrangement. Each set of branches includes both active and passive links, where the active links are connected to the moving platform and the passive links are connected to the static platform.

   Coordinate systems are established as follows: a fixed coordinate system O xyz on the static platform and a

   G3

   3

   )

   2

   (R Lc

   2

   Ls3

   5

   moving coordinate system

   O1 x1 y1z1

   on the moving

   platform. The origins O O1 and O2

   are the centers of the

   static and moving platforms, respectively. The x axis is

   aligned along vector

   OA1 , the z axis is perpendicular to the

   static platform and points vertically upwards. The

   x1 axis is

   aligned along vector

   11 , the

   z1 axis is perpendicular to

   The vectors

   G1G2 and

   G1G3 are further determined as

   arctan k (m 0, k 0)

   follows:

   1 (3R 2Lc

   Lc )

   m

   arctan k

   m

   m 0

   2 1 2 k

   arctan m 0, k 0)

   G1G2

   3 (R Lc )

   2

   2

   m

   10

   L(s2 s1)

   6

   2

   m2 k 2

   2 arctan

   p

   1 (3R 2Lc Lc )

   11

   2 1 3 Where

   GG 3 (R Lc )

   1 3

   2

   L(s

   3

   3 s1)

   7

   m=3/2(RLs_3+RLs_2-2Rs_1-L^2 c_2 s_3-L^2 c_3 s_2

   +Lc_2 s_1+Lc_3 s_1)

   k=1/2(3RLs_2-2L^2 c_1 s_2-L^2 c_3 s_2+Lc_3 s_1-3RLs_3

   +2L^2 c_1 s_3+L^2 c_2 s_3-Lc_2 s_1))

   The cross product of the vectors

   G1G2

   and

   G1G3 yields a

   p 3 (6R2 4RLc

   • 2RLc

   • 4RLc

   3L2c c )

   vector that is normal to the plane

   4

   G1G2G3 , with the direction

   3 1 2 2 3

   pointing from the static platform to the moving platform. Using

   The rotation matrix

   O R representing the orientation of the

   O

   1

   the coordinates of point

   G1 and the normal vector, the point-

   moving coordinate system relative to the fixed coordinate

   normal form equation of the plane G1G2G3 can be obtained.

   system can be expressed as:

   s1v1 c1

   s1c1v1

   c1s1

   This allows for the calculation of the distance from point O

   O R s c v

   c2 v c

   s s

   O1

   1 1 1 1 1 1 1 1

   to the plane

   G1G2G3 .

   m 2 k 2 p 2

   1 1 1

   d (m1 (R Lc1) pLs1)

   c1s1

   s1s1

   12

   c1

   s sin , c cos , v 1 cos is the

   2

   Therefore, the solution parameters are:

   8

   angle between the

   OO2 axis of the moving coordinate

   m2 k 2 p2

   1

   l OO 2d 2 (m(R Lc1 ) pLs1)

   system and the z axis of the fixed coordinate system.

   The coordinates of Q1 , Q2 and Q3 relative to the fixed

   9

   coordinate system

   O1 are:

   R Lc1

   18

   O1Q 0

   1

   Therefore, the solution parameters are:

   Ls1

   13

   m 2 k 2 p 2

   2 2 2

   1 (R Lc )

   l2 O1O2

   2d2

   2 (m2 (R Lc1) p2 Ls1)

   2 2

   19

   m

   O1Q 3 (R Lc )

   arctan k2

   2

   2 2

   Ls2

   2

   2

   20

   14

   1

   2 2 arctan

   m 2 k 2

   2 2

   p

   2

   2 (R Lc3)

   21

   O1Q 3 (R Lc )

   Where

   m2 , p2 , k2

   are all functions of

   i and can be

   3 2 3

   Ls3

   15

   The vectors are further determined as:

   1 3

   1 (3R 2Lc Lc )

   2

   indirectly converted into functions of i . The vector OO1 can be expressed as:

   OO1 (l1 sin1 cos1,l1 sin1 sin1,l1 cos1)

   22

   The vector OO in the O coordinate system can be

   1 2 1

   Q Q 3 (R Lc ) expressed as:

   1 2

   2 3

   L(s s )

   O OO (l sin

   cos ,l sin sin ,l cos )

   3 1

   1

   1 2 2 2 2 2 2 2 2 2

   23

   16

   1 (3R 2Lc Lc )

   The vector

   OO2 can be expressed as:

   2 1 3

   OO OO OR O O

   l ,l ,l

   2 1

   O1 1 2

   x y z

   Q Q

   3 (R Lc )

   24

   1 3

   2 3

   L(s s )

   The distance between the centers of the moving platform and

   3 1

   17

   the static platform is:

   l 2 l2 l2

   x y z

   L

   25

   The distance from point

   O1 to the plane Q1Q2Q3 is:

   The direct kinematic analysis of the mechanism is validated through simulation, with the link length L 0.07848m

   (m (R Lc ) p Ls )

   and platform radius r 0.04497m measured. The motion

   m 2 k 2 p 2

   2 2 2

   d2 2 1 2 1

   trajectory is described by the following equation:

   26

   0.8t2 8t 95

   p(t)

   0

   0 t 5

   5 t 15

4. SPEED ANALYSIS

According to the differential mapping in the finite instantaneous spiral theory, the velocity model can be solved by differential equations

0 0 t 5

St ,i S f ,PM St ,1

St ,2

St ,3 29

(t) 0.32t3 7.4t 2 48t 100 5 t 10

0 10 t 15 5 5

S S

,i 1, 2,3 30

27

k 1

t ,i,k i,k t ,i,k k 1

0 0 t 5

In the structure, the velocity model of the moving platform is:

(t) 0 5 t 10

s s

3.6t 2 72t 360 10 t 15

St ,i

11,i r

1,i

s

12,iT r

1,i

s

11,i

11,i 12,i

1,i

28

The motion trajectory of the center point of the moving

In the structure, the velocity model of the moving platform is:

4

platform is shown in Figure 3, and its positional solution is shown in Figure 4.

St ,i

k 1

qi,k St ,i,k ,i 1,3 32

where St ,i,k

is the unit instantaneous motion i axis of the first

joint in the first rod, and Si,k as its unit direction vector, is its

intensity.

Sw,i f

a,i,3S

wa,i,3

f

2

kc1

S

c,i,kc wc,i,kc

,i 1,3 33

S ri,3 si,3 ,S ri,2 si,2 , S

si,2 si,1 34

wa,i,3 s wc,i,1 s wc,i,2 0

i,3 i,2

S ri,3 si,3 ,S ri,2 si,2 , S

si,2 si,1

wa,i,3 s wc,i,1 s wc,i,2 0

i,3 i,2

34

Figure 3 Motion Trajectory

JwSt Jqq 35

J Jwa ,J

Jqa

,q qa ,q i

t t t T

w J q

o 0 a

1,3 1,3 2,4 3,3

wc

31

36

Jwa

S

wa,1,3

Swa,2,4

T

S

wa,3,3

wc S

wc,1,1

Swc,2,1

T

S

wc,3,1

, J

37

Figure 4 Position Solution

S T S

Jq

wa,1,3

t ,1,3

T

S

wa,2,4

St ,2,4

S

T

wa,3,3

St ,3,3

0

38

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   5 CONCLUSION

   1A configuration idea of connecting the parallel mechanism in series in the form of a mechanism unit is proposed, and the

   multi-level coupling connection mechanism with the 3-RSR parallel mechanism as the structural unit is proposed, and the motion space is improved after two superimpositions. This kind of mechanism has both a large working space for a series mechanism and a dynamic performance of a parallel mechanism.

   1. The position analysis of the single-layer 3-RSR

      mechanism is carried out by using the allocation principle, and the velocity is solved by using the finite instantaneous spiral theory, and the simulation verification is carried out, which lays a foundation for the research of multi-level coupling and linking mechanism.In terms of research content, the neural network model established in this study calculates the mean square error (MSE) to verify the model's reliability. The optimal parameters are found using the Bayesian optimization algorithm. Finally, the reliability of the model is verified through path planning analysis.

   2. The configuration proposed in this paper is obviously

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