研究3-RRP球面并联机构的动力学和模拟--李瑞琴

Proceedings of 2014 Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators

July 7–8, 2014, Tianjin, China Research on Dynamics and Simulation of 3-RRP Spherical Parallel Mechanism

Ruiqin Li

School of Mechanical and Power Engineering, North University of China, Taiyuan 030051, China

e-mail: liruiqin@https://www.sodocs.net/doc/959962493.html,

Yanjun Guo

School of Mechanical and Power Engineering, North University of China, Taiyuan 030051, China

Abstract: This paper studies the dynamics of 3-RRP spherical parallel mechanism. First, the inverse kinematics is analyzed and the kinematic equations are deduced based on function principle. The dynamic equations are deduced by using Newton-Euler method. The major factor influencing driving moments is the structural parameters of the mechanism, angular velocity, force effect at the end of the actuator, etc. The simulation models of kinematics and dynamics of the 3-RRP mechanism are established with the help of simulation technique of combining ADAMS/UG and MATLAB. According to predetermined path of end-effector, the kinematic model is analyzed by using MATLAB programming. The variation rules of the displacement, angular velocity and angular acceleration of the reference point on the end-effector are obtained. The physical model established in the UG software verifies the correctness of the modeling. At last, in view of the dynamic performance of the 3-RRP mechanism, the variation rules of the driving moments and reactions of kinematic pairs are obtained. The numerical results provide important references for mechanism design of 3-DOF spherical robot.

Keywords: 3-RRP, Spherical Parallel Mechanism, Dynamics, Simulation

1 Introduction

Spherical parallel mechanism takes advantages of high stiff, big load, fast velocity and compact structure, and can rotate deftly has dexterous workspace. 3-DOF spherical parallel mechanism has wide application in waist joint, wrist joint, shoulder joint, electronic agile eye, and tracking system of spatial orientation of a satellite antenna, etc.

Researchers have studied 3-DOF spherical parallel mechanism from different views. Kong Xianwen, et al [1] studied configuration synthesis of 3-DOF spherical parallel mechanism based on screw theory. Karouia Mourad et al [2] studied configuration synthesis of asymmetrical and Non over-constrained 3-DOF spherical parallel mechanism. Luo yufeng et al [3] proposed a systematic and effective method to synthesis of 3-DOF spherical parallel mechanisms based on the Single Opened Chain (SOC) theory. 145 kind of spherical parallel mechanisms have been synthesized, which provide much more choices and accordance for structure optimization of 3-DOF spherical parallel manipulator. At present, many researches mainly focus on the research on 3-DOF 3-RRR spherical parallel mechanism.

The research on dynamics of 3-DOF spherical parallel mechanism is fewer.

LIU Shanzeng et al [4] studied kinematic and dynamic performances of a 3-DOF parallel manipulator. Zhang Junfu et al [5] established an analytical dynamic model for spherical parallel mechanism based on numeric-symbolic approach and Kane dynamic equation. The influence of adjustable parameters on dynamic of 2 DOF spherical parallel mechanisms was studied．

Li Chenggang, et al [6] established dynamic model of a spherical parallel mechanism by means of screw theory and the equation of motors moments were obtained. Through the transformation of dynamic model, the configuration space model of equation and the corresponding coefficients were presented.

Wang Yuelin, et al [7] developed a dynamic model of 3-RRR spherical parallel mechanism by using Lagrange method and designed a robust-adaptive iterative learning controller (ILC) for this mechanism.

Liu Haitao et al [8] studied a inverse dynamic model of a limb of 2-DOF spherical parallel mechanism in 5-DOF manipulator named “TriVariant-B”. and presented a method which can forecast rotor inertia, rated speed and peak torque according to the requirements of velocity and acceleration of the end-effector. Liu Haitao et al [8] verified the validity of thos method by means of a virtual prototype.

Feng Zhiyou et al [9] established inverse dynamic model of 2UPS-2RPS 4-DOF parallel mechanism based on Newton-Euler formulation. Through solving the equations, the required driving and constraint forces can be obtained when the motion of moving platform and working load were given. A computational example was also provided.

This paper takes 3-RRP spherical parallel mechanism (shortly, 3-RRP mechanism) as a research object and studies its dynamic performances.

2 Configuration of 3-RRP spherical parallel mechanism

2.1 The structure of 3-RRP mechanism

As shown in Fig 1, 3-RRP spherical parallel mechanism is composed of base B, moving platform P and three RRP limbs[10].

Fig. 1 The configuration of 3-RRP spherical parallel mechanism

3-RRP spherical parallel mechanism is a kind of spatial mechanism whose reference point of the end-effector moves along a spherical surface. The main motion characteristic is that the distance between the reference point E of the end-effector and the centre of sphere O is unchangeable when 3-RRP mechanism moves.

The base is fixed static platform B 1B 2B 3.The moving platform is composed of three sections of arc links in a spherical surface. The intersecting point E of three arc links is the reference point of the moving platform. The angles between three arc links can be not equal in order to adapt to the various working status of the platform. In this paper, for the sake of simplicity, suppose three arc links are symmetric and the angles between them are equal. The moving platform (also called end-effector) can produce flexible and continuous spherical motion.

Three limbs have the same structure and the symmetrical arrangement. Each limb is composed of revolute pair R 1, revolute pair R 2 and a spherical prismatic pair P. The spherical prismatic pair P, composed of the slider and three sections of arc links, slides relative to each other on the spherical surface. There is an angle θ between the axial line of revolute R 1 and static platform. Because of the particularity of the spherical parallel mechanism, when studying spherical parallel mechanism, some structure parameters such as link lengths cannot be considered and only need to

consider position and attitude angles of the mechanism. In order to facilitate the research, suppose 60θ=° when modeling 3-RRP mechanism. The axial line of revolute R 1 always passes through the center of sphere O when 3-RRP mechanism moves. The sliding pair P belongs to a kind of spherical sliding pair.

Defining unit vectors v i are along OB i , three arcs in the moving platform intersect the reference point E in the end-effector. The angles 123,,ααα between arcs can be selected automatically according to practical requirements in order to desired performances. Defining vector OE as s , vector OP i as r i , drive angles as i γAs shown in Fig 2.

Fig. 2 3-RRP mechanism diagram

2.2 The degrees of freedom of 3-RRP mechanism

3-RRP mechanism is typical spherical mechanism with common constraints. Hunt presented an improved Grübler-Kutzbach formula to calculate the DOF of the mechanism, as follows.

1(1)k

i

i F d n k f ==??+∑ (1)

Where,

6d λ=? is the order of the mechanism, λ is the number of common constraints of the mechanism. For 3-RRP spherical mechanism, 3λ=, so 63d λ=?=; n is the number of links, 8n =;

k is the number of kinematic pairs. 3-RRP mechanism has 3 revolute pairs and 6 prismatic pairs, 9k =.

i f

is the DOF of i th kinematic pair. The DOF of revolute pair and prismatic pair is 1.

Substitute these parameters in eq. (1), the result is

1(1)3(891)93

k

i

i F d n k f ==??+=??+=∑ (2)

Therefore, the DOF of 3-RRP spherical parallel mechanism is 3.

3 Inverse kinematic analysis of 3-RRP spherical parallel mechanism

As shown in Fig. 3, we take one of three limbs as a research object.

In order to describe 3-RRP mechanism, we need definite unit vectors, fixed coordinate system and moving coordinate system.

Defining the unit vector t i and w i are as follows.

i

i i

×=

×s r t s r (3) 1

1

i i i i i ++×=

×v v w v v (4)

Where, the unit vector t i and w i are perpendicular to planar OEP i and OB i B i+1. Establishing fixed coordinate system (,,)O O O O x y z

in the base. Let O x axis is along vector v i direction, O z

axis is along vector w i direction. O y axis is determined

by using the right-hand criterion. As shown in Fig 3.

Establishing moving coordinate system

(,,)E E E E x y z in the moving platform. Let y E is parallel

to the vector t 1, z E axis is along vector s direction. x E axis is determined by using the right-hand criterion. We can describe 3-RRP mechanism by means of fixed coordinate system, moving coordinate system and unit vector.

Defining the rotation matrix by using z E -y E -z E

Euler angles [11]:

12312312123

1231223232c c c c c s c s s c c s c s s s s c s s c E

O R θθθθθθθθθθθθθθθθθθθθθ?????=????????

(5)

Fig. 3 The parameters of the i th limb

The unit vector s is determined by angles 1θ and

2θ.Therefore, inverse kinematic analysis can be

defined as follows: according to the given direction

and the other kinematic parameters, obtain drive angles

i γ(i =1, 2, 3).

The inverse kinematic analysis can be expressed by

equivalent angular axes. This is a special form

Rodriguez equation. The equivalent angular axes can

be expressed in the form of

330(,)cos (1cos )sin 00z y T z x y x e e e e e e ηηηη×??

???=+?+???

?????

Q e I ee (6)

Where, the unit vector e is spin axis, η is spin angle of unit vector e ,,,x y z e e e is coordinate components of unit vector e .

Because y E direction is parallel to t 1 direction, z E direction is along vector s direction, the following equations can be obtained.

121220c s 0s s 1c O E O

E E E z θθθθθ????????===????????????

s R R (7)

12313112313230c c s s c 1s c s c c 0s s O E O

E E E y θθθθθθθθθθθθ??????????===+????????????t R R (8) According to equivalent axial angle expression and

the moving platform structure, the vectors t 2 and t 3 can

be obtained. From equation (6), the following equations can be obtained.

2313333131(,)cos (1cos )sin ()T αααα×==+?+×t Q s t I ss t s t (9) 32123312121(,)cos (1cos )sin ()

T αααα×=?=+??×t Q s t I t ss t s t (10)

Because i t is perpendicular to vector s , eq. (9) and (10) can be simplified as follows.

23131cos sin ()αα=+×t t s t (11) 32121cos sin ()αα=?×t t s t (12)

In 3-RRP mechanism, the motions of three spherical prismatic pairs can be regarded as moving on the spherical surface around one axis which is through spherical center.

In the inverse kinematic analysis, the positions of drive angles are unknown. The positions are defined by unit vector r i . Vector r i can be determined by the angle i γ between vector v i and unit vector w i along forward

direction. Therefore, unit vector r i can be expressed as: 33(,)cos (1cos )sin ()i i i i

T i i i i i i i i i γγγγ×==+?+×r Q w v I v w w v w v (13) Because vector v i is perpendicular to vector w i , eq. (13) can be simplified as follow.

cos sin ()i i i i i i γγ=+×r v w v ，1,2,3i = (14) Thus, unit vector i r can be defined as a function about drive angles i γ. In order to obtain drive angles i γ, from i t

perpendicular to i r , three independent relationships can be obtained:

0T i i =r t ， 1,2,3i = (15)

Substituting eq. (8), (11), (12), (14) into (15), the following equation can be obtained.

cos sin ()0T T i i i i i i i γγ+×=v t w v t ，

1,2,3i = (16)

Thus, the solutions of inverse kinematics are as follows.

tan()()T i i

i T i i i

arc γ=?×v t w v t ，1,2,3i = (17)

it is noteworthy that T i i v t and ()T i i i ×w v t are all numerical value. Therefore, inverse kinematic questions can be solved. At the same time, the angles i β also can be calculated. i β is the angles between vector s and vector r k .

1cos ()T i i β?=r s (18)

4 The motion equations of 3-RRP spherical parallel mechanism

4.1 Kinematic equations of 3-RRP mechanism

According to function principle ，all the work of external force is equal to the sum of kinetic energy of 3-RRP mechanism, that is

W T Δ=Δ (19)

When selecting a rotating link as equivalent link, let the driving moment and moment of resistance acting on the equivalent link as d M and f M , respectively. When the rotating angle rotate from 1?to 2?, the corresponding Moment of inertia varies from J 1 to J 2. Angular velocity varies from ω1 to ω2. Therefore, according to the function principle of the mechanism motion, the following equation holds,

21

2

2

222211111122

d f W Md M d M d J J ?

?

?????ωω??Δ=∫=?=?∫∫ (20)

In order to facilitate the expression, equation (20) can be expressed by differential motion equation

21

()()2

d f Md M M d d J ??ω=?= (21)

From the following eq.

,d d d d d dt d dt

?ωωωω

??== eq. (21) can be expressed as

22d f dJ d M M M J d dt ωω?

=?=

+ (22)

Eq. (22) is the moment expression of the

mechanism moment.

When selecting a sliding link as equivalent link, suppose d F and f F as equivalent forces of the driving

force and resistance force, respectively; m 1 and m 2 as

corresponding equivalent masses of the sliding link in the position s 1 and s 2, respectively; v 1, v 2 as velocities in the positions s 1, s 2, respectively; then the energy

equation and the force equation of the motion of 3-RRP mechanism are as follows:

21

22222211

111122

d f s

W Fds

s s s F ds F ds m v m v s s Δ==∫?=?∫∫ (23)

22d f v dm dv

F F F m ds dt

=?=+ (24)

Because the kinematic equations of a spatial mechanism under the action of all known force are the same as those of a planar mechanism, we can solve the true motion of the equivalent link only according to the characteristics of spatial mechanism to solve equivalent force/moment, equivalent mass, equivalent moment of inertia.

4.2 Kinematic equations of 3-RRP mechanism

Suppose the radius of 3-RRP mechanism

300r =mm ，the trajectory of the reference point E of the end-effector is spiral line. Its equation is

3cos(2)x t = 3sin(2)y t = 5z t =

By using MATLAB programming calculation and analyzing the mathematical model of 3-RRP mechanism, the variation rules of displacement, angular velocity, angular acceleration curves of the reference point E of the end-effector around x , y , z axis, as shown in Fig. 4.

(a) Displacement of the reference point in the moving platform

along the coordinate x , y , z , respectively

time(s)

d i s p l a c

e m e n t (m m )

y x z

(b) Angular velocity of the reference point in the moving

platform along the coordinate x , y , z , respectively

(c) Angular acceleration of the reference point in the moving

platform along the coordinate x , y , z , respectively

Fig. 4 Displacement, angular velocity, angular acceleration curves of the reference point in the moving platform

4.3 Dynamic Simulation

Because 3-RRP mechanism belong to spatial mechanism, direct modeling in ADAMS has a certain difficulty. Here, modeling follows the steps below.

Firstly, establish a virtual 3-dimensional model in UG . Secondly, this 3-diemnsional model is derived as Parasolid (*.xmt_txt, *x_t,*xmt_bin,*x_b) file format.

Then, open ADAMS file, exert constraints and drives on 3-RRP mechanism by using geometric modeling tools in the main toolbar in ADAMS/View. The constraint types are mainly revolute pairs and prismatic pair. The established 3-dimensional model is as shown in Fig. 5.

The drive of 3-RRP mechanism in ADAMS is set up as the same as that in MATLAB. This kind of set can be used to verify the forward kinematics is verified. From Fig. 5, the motion trajectory of 3-RRP mechanism

is spiral line.

Fig. 5 The trajectory of the reference point of the 3-RRP

mechanism under the given parameters

In order to measure the variation rules of the

displacement, angular velocity and angular acceleration of the end-effector, suppose the simulation time is 5s, the step length is 200, click the simulation button in the main tool kit to start simulation. After finishing simulation, using Build →Measure →Point to point →New function to bulid path, measure the motion of the reference point E (marker point )of the

end-effector, the displacement, velocity and

acceleration curves can be obtained, as shown in Fig. 6.

Fig. 6 The displacement, velocity and acceleration curves of the

reference point of the end-effector

In the post processing interface, select source (Objects)—Filter(Body)—Object— Characteristic (CM Angular Velocity)—Component(z), the system automatically draws angular velocity diagram of the selected objects. In the characteristic point, select CM Angular Veolcity, CM Angular Acceleration around x , y , z -direction, respectively, the angular velocity curves and angular acceleration curves can be drawn. The results are as shown in Fig. 7. Compared with the calculation in MATLAB, the results are consistent. This verifies the established mathematical analytical model is correct.

time(s)

x

y z a n g u l a r v e l o c i t y (°/s )

time(s)

x y z a n g u l a r a c c e l e r a t i o n (°/s 2)

time (s)

a c c e l e r a t i o n (m m /s 2)

l e n g t h (m m )

v e l o c i t y (m m /s )

(a) Displacement of the reference point of the moving platform

around the coordinate x , y , z , respectively.

(b) Angular velocity of the reference point of the moving

platform around the coordinate x , y , z , respectively.

(c) Angular acceleration of the reference point of the moving

platform around the coordinate x , y , z , respectively

Fig. 7 The motion graphs of the reference point of the moving

platform

In general, the curve direction change point of angle velocity and angle acceleration occurred in the region of velocity mutation. Acceleration and force of these regions on rod impact is relatively large. They are the main factors affecting the mechanism motion balance in design. But from Fig. 7, the angular velocity and angular acceleration of the moving platform are smooth curve. This shows the 3-RRP mechanism has better motion stability and also is easy for real-time control. Therefore, in the design process, we should consider the influence of the impacts acting on the links, and choose a suitable size and material.

4.4 Dynamic Simulation

Under the same conditions, the displacement curve

of drivers in the 3-RRP mechanism is

3cos(2)3sin(2)5x t y t z t =??

=??=?

In this case, the end-effector moves along spiral trajectory. When the end-effector doesn’t bear external forces, the variation of the driving powers acting on three motors are shown as Fig. 8, and the variation of the moment along z-direction is shown as Fig. 9.

Fig. 8 Variation of driving powers acting on three motors

Fig. 9 Variation of moments acting on z-direction of motors

When the reference point of the end-effector is applied a constant force, whose magnitude is F=100N and the force is perpendicular to the static platform, the variation of the driving powers acting on three motors is shown as Fig. 10, and the variation of the moment along z-direction is shown as Fig. 11.

Fig. 10 The variation of the driving powers acting on three motors under the constant force

time (s)

l e n g t h (m m )

time(s)

a n g u l a r v e l o c i t y （°/s ）

time(s)

a n g u l a r a c c e l e r a t i o n （°/s 2）

time(s)

p o w e r （N ?m m /s ）

time(s)

m o m e n t （N ?m m ）

time(s)

p o w e r （N ?m m /s ）

Fig. 11 The variation of the moments acting on three motors under the constant force

Without considering the friction between the links, from Fig. 8-11, in the circumstance of the end-effector not being exerted external force, motor drive powers are mainly used to overcome the links gravities.

In the circumstance of the end-effector being exerted external force, motor drive powers and drive moments all increase.

If other conditions remain unchanged, assume that the quality of each actuator link is 1 kg, each intermediate passive link is 0.25 kg, and the moving platform is 0.8 kg. Thus we can obtain the variation of the driving powers acting on the three motors is shown as Fig. 12, and the variation of the moment along z-direction is shown as Fig. 13.

Fig. 12 The variation of driving powers acting on three

motors

Fig. 13 The variation of moments acting on z-direction of

motors

Also we assume that the quality of each actuator link is 3 kg, each intermediate passive link is 0.75 kg,

the moving platform is 2.4 kg. Thus we can obtain the variation of the driving powers acting on the three motors is shown as Fig. 14, and the variation of the moment along z-direction is shown as Fig. 15.

Fig. 14 The variation of driving powers acting on three motors

Fig. 15 The variation of moments acting on z-direction of motors

From the variation of Fig. 12 and 14, Fig. 13 and 15, without considering the friction between the links, with the increase of the quality of the link, the overall variation of the power consumption and the torque is consistent. Only with the increasing of the quality of the link, the power consumption and the torque is increased correspondingly.

The dynamic performance index can comprehensively assess the ratio relationship between force and moment. From the reference [12]：

p =

(25) Where, a N is the numbers of the driving joints; i τ is the force/moment of i th close-loop joint; max i τ is the maximum force/moment of the i th joint .

From equation (25), the driving performance of the manipulator becomes better as p becomes smaller.

5 Conclusions

The Conclusion can be drawn as follows:

(1) The dynamic model of 3-RRP spherical parallel

mechanism is established. The correction of the dynamic model is verified by using ADASM/UG and MATLAB software.

time(s)

p o w e r （N ?m m /s ）

time(s)

time (s)

time(s)

time(s)

m o m e n t （N ?m m ）

m o m e n t （N ?m m ）

p o w e r （N ?m m /s ）

m o m e n t （N ?m m ）

(2)Under the circumstance of the reference point of the

end-effector with exerted forces or without exerted

forces, the variation rules of driving powers and

moments of the driving links are obtained.

(3)The results in this paper can be used to instruct the

application of 3-RRP mechanism. 3-RRP mechanism can be used to a tracking system of

spherical motion trajectory, etc.

Acknowledgements

This research is sponsored by the National Natural Science Foundation of China (Grant No. 51275486) and the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No. 20111420110005).

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