Modelling and Simulation in Engineering

Volume 2012 (2012), Article ID 501251, 9 pages

doi:10.1155/2012/501251

Research Article

High-Order Dimension Synthesis of Planar/Spatial Mechanisms with One-DoF by CAD Variational Geometry

Yi Lu,¹ Yang Lu,² Nijia Ye,² and Lin Yuan¹

¹Parallel Robot and Mechatronic System Laboratory, Hebei Province, Key Laboratory, Advanced Forging and Stamping Technology and Science, Ministry of National Education, and College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China

²Yingkou Equipment Development and Manufacturing Co., Ltd., Liaoning, Yingkou 115004, China

Received 3 April 2012; Accepted 17 September 2012

Academic Editor: Javier Otamendi

Copyright © 2012 Yi Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proposes a (computer aided design) CAD variational geometry approach for the high-order dimension synthesis of one-DoF mechanisms based on the given velocity/acceleration of a moving platform along a prescribed trajectory. The objective of this approach is to determine the reasonable dimensions of the mechanisms when given the velocity or/and the acceleration of the moving platform along a prescribed trajectory. First, some concepts and mathematical foundations are explained for constructing the velocity/acceleration simulation mechanism of a general mechanism. Second, the inverse velocity/acceleration simulation mechanisms of the planar/spatial four-bar mechanisms with one-DoF are constructed by the CAD variational geometry. Third, when given the position and the velocity/acceleration of the coupler along a prescribed trajectory, all the reasonable dimensions of the planar/spatial four-bar mechanisms are solved from their simulation mechanisms.

1. Introduction

In industrial practice, problems are frequently encountered that require the design of mechanisms that will generate a prescribed planar/spatial trajectory [1, 2]. The problems may become further complicated if the velocity and acceleration of the motion are critical, as might be the case where possible damage to the mechanism may result from large velocity and accelerations. In some other cases, the combination of both position and velocity/acceleration are desired [1, 3]. In this aspect, Alba et al. [4] proposed a general analytic method for the synthesis of 2D and 3D mechanisms by considering velocity, acceleration, and jerk. Chang [5] synthesized adjustable four-bar mechanisms generating circular arcs with specified tangential velocities. Avilés et al. [6] proposed second-order methods for the optimum synthesis of multibody systems. Arsenault and Boudreau [7] synthesized planar parallel mechanisms while considering workspace, dexterity, stiffness, and singularity avoidance. Perry et al. [8] conducted 2 position, 2 velocity synthesis of a spherical mechanism with translating center. Using a global optimization algorithm, Prebil et al. [9] studied dimension synthesis of a four-bar mechanism in a hydraulic support. Lin et al. [10] proposed motion-curve synthesis and geometric design of a planar mechanism with intermittent variable speed. Lin and Huang [11] conducted dimension synthesis for higher-order kinematic parameters and self adjustability design of planar linkage mechanisms. In terms of CAD (computer aided design) mechanism synthesis, mainly based on analytical approaches, some suitable programs are studied and compiled [12–18]. By using CAD variational geometry, Lu solved the forward position, forward velocity/accelerations of some parallel mechanisms [19], and developed a planar four-bar simulation mechanism for the approximate dimensional synthesis [20]. However, these analytical methods are not easy to learn. The reasonable solutions are difficulty to determine from multisolutions, the processes of programming compile are complicated, and synthesis results are not straightforward.

In order to overcome these conflicts, this paper focuses on a CAD variational geometry approach for the high-order dimension synthesis of mechanisms based on the given velocity or acceleration along the prescribed trajectory. All the reasonable dimensions of the planar or spatial four-bar mechanisms with one-DoF are determined by the CAD variational geometry when given the position and the velocity/acceleration of the platform along the prescribed trajectory.

2. Concepts and Mathematical Foundations

2.1. Some Concepts of the CAD Variational Geometry

A simulation mechanism, which includes the vectors of velocity, acceleration, angular velocity, or angular acceleration, is designated as a velocity/acceleration simulation mechanism. The purposes of designing the velocity/acceleration simulation mechanism are to solve all dimensions of links when given the velocity and acceleration of the output link. Some concepts on CAD variationalgeometry are explained as follows.

The dimensions in the simulation mechanisms are classified into the driving dimension, the driven dimension, and the fixed dimension. The driving dimensions are given to some input parameters, such as the initial position/orientation of the input link, the input velocity of the input link, the position/orientation of the output link, the output velocity of the output link, the acceleration of the output link, and its orientation. The driven dimensions are given to some key links. The fixed dimensions are given to the prescribed path and the platform. When varying or modifying some driving dimensions, all the driven dimensions are solved automatically. In addition, when varying or modifying some fixed dimensions, the driven dimensions are varied automatically [1]. Thus, the suitable dimensions of some key links can be solved.

The simulation mechanism constructed by the CAD variational geometry includes three different constraints: the completed constraint, the lacking constraint, and the over constraint. When a mechanism is in completed constraint, its number of DoF is the same as the prescribed value. In this case, when modifying some driving dimensions, all the driven dimensions are varied automatically and can be solved correctly. When a mechanism has a lacking constraint, its number of DoF is larger than the prescribed value. In this case, when modifying some driving dimensions, some driven dimensions are varied automatically, but the correct driven dimensions can not be solved. However, some key dimensions of the mechanism with the lacking constraint can be modified easily by transforming some driven dimensions into driving dimensions. When a simulation mechanism is in over constraint, its number of DoF is less than the prescribed value. In this case, the mechanism becomes a structure, and each of the driving dimensions can not be modified unless deleting one of the driving dimensions. Therefore, based on the 3 different constraints, the number of DoF of the mechanism can be determined.

2.2. Some Mathematical Foundations

A general one-DoF mechanism includes an actuator, a base , a moving platform , and some branches for connecting with , as shown Figure 1 (solid line). Let be a coordinate O-XYZ fixed on ground at . Let be a parallel constraint, and a perpendicular constraint. Let be a point on , and be a prescribed trajectory in .

Figure 1: A general one-DoF mechanism.

Suppose the actuator drives an input link to move from an initial position angular to an angular and brings to move along at the given position and velocity and acceleration . In this case, it is a significant and challenging issue to solve all reasonable dimensions of the mechanism.

Since the pose of is varied continually with time , a point on is moved along from point to point during time increment . An average output velocity of during coincides with a vector from to . Its formula is derived as below:

As in the velocity/acceleration simulation mechanism, is satisfied. Thus, the velocity of point on can be solved as below:

Similarly, an output velocity of point on at position during coincides with a line from point to point without coincident with . The velocity of point on at is derived as

In order to solve the acceleration of at point , constitute an auxiliary velocity vector , connect its one end to , set and . Thus, an average acceleration of at during is parallel to a line from the end point of to the end point of . Its formula is derived as below:

As in the velocity/acceleration simulation mechanism, is satisfied. Thus, the acceleration of at point can be solved as below:

Let . From (2), (3), it leads to

From (2) and (5), it leads to

Based on (2), a forward velocity simulation mechanism can be constructed by the CAD variational geometry. When all the dimensions of links and the input velocity of this mechanism are given, and when gradually reduces from an initial value to a value being close to 0, can be reduced automatically from an initial value to a value being close to 0. Thus, forward velocity can be solved and its vector is along the tangent lines of at [19].

Similarly, based on (6) and (7), an inverse velocity simulation mechanism can be constructed by the CAD variational geometry. When the initial position and the input velocity of input link and the output velocity of at point are given, and when gradually reduces from an initial value to a value being close to 0, must be reduced automatically from an initial value to a value being close to 0. Thus, all the driven dimensions of links in length or angular can be solved [19]. After that, the first order (velocity) dimension synthesis of the mechanisms can be conducted based on its output velocity. The first and second order (velocity and acceleration) dimension synthesis can be conducted based on its output velocity and the output acceleration.

Therefore, the mathematical foundations for design velocity/acceleration simulation mechanism are the finite differential theory and the differential geometry. When is 0.01s or less, the output velocity and the output acceleration solved by the CAD variational geometry are coincident with analytic solutions [19].

3. Order Dimension Synthesis of Planar Four-Bar Mechanism

Based on Section 2, the order (velocity) dimension synthesis of a planar four-bar mechanism (see Figure 2) can be conducted in a 2D sketch. In this case, all dimensions of a planar four-bar mechanism with one-DOF can be solved by CAD variational geometry when given the initial orientation and the input velocity of input link, and the output position and output velocity of output link. The synthesis processes are described as follows.

(1)Construct a horizontal line and a vertical line , connect their one end to original point . Thus a coordinate frame is constructed.(2)Construct a spline for prescribed trajectory in .(3)Construct a coupler by some subprocesses as follows.

(a)Construct 2 lines and , connect the one end of with , and set . (b)Give an initial driving dimension (206) in length, an initial driving dimension (60) in length, the distance from to the one end of an initial driving dimension (70).(c)Construct a moving platform (coupler) from and by the block command.

(4)Copy and paste platform by the block copy command and paste to form a new platform .(5)Construct a circle with central point and a circle with central point .(6)Construct a prescribe path by spline command and fix it by the fixing command.(7)Connect the 2 ends of in with at and at , respectively.(8)Connect the free end of in with at point .(9)Construct a line , connect its 2 ends to at and , respectively. Thus, 2 revolute joints at points and are formed, respectively.(10)Construct a line , connect its 2 ends to at and , respectively. Thus, 2 revolute joints at points and are formed, respectively. Thus, the first simulation mechanism of four-bar mechanism is constructed from (, , , , and 4 revolute joints ).(11)Repeat steps 7–10, except that () are replaced by (). Thus, the second simulation mechanism of the planar four-bar mechanism is constructed from (, and 4 revolute joints ).(12)Construct a line for velocity , connect its one end to point , coincide with point , and give an initial driving dimension (50) in length.(13)Give the vector increment of (the distance between and a driving dimension .(14)Construct a line for input angular velocity, connect its one end to point , set , give an initial driving dimension (20).(15)Construct a line for time increment, connect its one end to point , set , give a driven dimension. Give a driven dimension by a constraint equation .(16)Give the angular between and a horizontal line a driven dimension for initial angular of .(17)Give the angular between and a horizontal line a driven dimension for the angular of link by a constraint equation .(18)Give the distance from each of points to a driven dimension for the vertical component of and to a driven dimension for the horizontal component of .(19)Give the distance from point to a driving dimension for the horizontal component of , give the distance from point to a driven dimension for the vertical component of .(20)Give each of circles a driven dimension for the diameter of .

(a)

(b)

Figure 2: A planar four-bar velocity simulation mechanism versus and .

When given (the output velocity , the input angular velocity , the initial angular of link , the platform parameters , , ), and gradually reducing the vector increment of point (the distance between and ) from 10.1 to 0.1, is reduced automatically from 0.202 to 0.002 . At the same time, twin simulation mechanisms of planar four-bar mechanism are coincident to each other. Thus, the driven dimensions of () are solved automatically, as shown in Figure 2(b).

The solved results of () corresponding to and 0.1 mm are listed in Table 1 the first and the second rows. When and varying input velocity, the solved results of () corresponding to and 80 are listed in Table 1 the third and the fourth rows.

Table 1: The solved results of the planar four-bar linkage mechanism.

Given values
Solved values

°/s
mm/s
mm
mm
s
mm
mm
mm
(°)
(°)
mm
mm
mm
mm
mm
mm

50
45.17
10.1
0.202
206
60
70
34.5
38.54
215.99
217.15
−280.04
24.96
21.09
−131.35

20
50
45.18
0.1
0.002
206
60
70
34.5
34.54
215.99
214.01
−277.87
34.4
20.63
−128.38

20
60
45
0.1
0.0017
206
60
70
34.5
34.53
215.99
217.15
274.3
45.62
11.91
−131.93

20
80
45
0.1
0.0013
206
60
70
34.5
34.53
215.99
217.15
268.01
60.38
1.67
−134.04

When given (the output velocity , the input angular velocity , the initial angular of link , the platform parameters , , ), the velocity simulation mechanism of the planar four-bar mechanism may include a lacking constraint. In this case, when transforming one of the driven dimensions () into a driving dimension, the simulation mechanism with the lacking constraint can be transformed into one with the completed constraint. Thus, more reasonable dimensions of mechanism can be solved easily.

4. High-Order Dimension Synthesis of Planar Four-Bar Mechanism

Based on Section 2, the high-order (velocity/acceleration) dimension synthesis of a planar four-bar mechanism (see Figure 3) can be conducted. In this case, all the dimensions of the planar four-bar mechanism with one-DoF can be solved by the CAD variational geometry when the input velocity, the output velocity, and the output acceleration are given. The processes of the high-order dimension synthesis are described as follows:

(1)Repeat steps 1–6 in Section 3.(2)Repeat steps 7–10 in Section 3, except that () are replaced by (). In addition, is not connected with . Thus, the third simulation mechanism of planar four-bar mechanism is constructed from (). Point is attached on .(3)Construct a line for velocity , connect its one end to point , coincide with point .(4)Construct an auxiliary line , connect its one end to point , and set .(5)Construct a line for acceleration , connect its one end to the free end of vector at point , give the angular between line and line an initial driving dimension (140°) for orientation, give acceleration line an initial driving dimension (60) in length for magnitude.(6)Connect the free end of line to line at point . Give the distance between and an initial driving dimension (15.1) for velocity increment .(7)Give a driven dimension by a constraint equation .(8)Repeat steps 16 to 20 in Section 3.(9)Give the angular between and a horizontal line a driven dimension for the angular of link by a constraint equation .

(a)

(b)

Figure 3: A planar four-bar velocity/acceleration simulation mechanism versus (a) , ; (b) , .

Thus, when given some driving dimensions (the output acceleration and it orientation , the output velocity , the input angular velocity , the initial angular of link , and the platform parameters , , ), and gradually reducing the vector increment (the distance between and ) from 15.1 to 0.1, is reduced automatically from 0.202 to 0.0008. At the same time, three simulation mechanisms of planar four-bar mechanism are coincident to each other. Thus, the driven dimensions of () are solved automatically, as shown in Figure 3(b).

The solved results of () versus and 0.1 mm/s are listed in Table 2 the first and the second rows. When varying orientation of acceleration , the solved results of () versus are listed in Table 2 the third rows. When varying and , the solved results of () versus are listed in Table 2 the fourth rows. When varying velocity , and , the solved results of () versus are listed in Table 2 the fifth rows.

Table 2: Solved results of planar four-bar mechanism as (, , , ) are the same as that in Table 1.

Given values
Solved values

mm/s
mm/s²
(°)
mm
mm/s
(°)
s
mm
(°)
mm
mm
mm
mm
mm
mm

60
120
140
50
15.1
34.5
0.202
4.97
37.02
201.07
286.17
−280.04
17.22
−44.95
−160.88

60
120
140
50
0.1
34.5
0.0008
0.03
34.52
201.07
185.97
−280.04
17.22
−24.20
−115.11

60
120
120
50
0.1
34.5
0.0008
0.03
34.52
201.07
163.19
−280.04
16.49
18.51
−105.97

60
140
120
50
0.1
34.5
0.0008
0.03
34.52
201.07
191.89
−280.04
16.49
25.68
−118.40

70
140
120
50
0.1
34.5
0.0008
0.03
34.52
201.07
228.89
−280.04
16.49
34.91
−134.43

5. Order Synthesis of Spatial Four-Bar Mechanism with One-DoF

A spatial four-bar mechanism with one-DoF is shown in Figure 4(a). It includes a base , a crank link , a coupler (moving platform ), an oscillator , 2 revolute joints and , a universal joint , and a spherical joint . Here, is composed of 2 crossed revolute joints and . Some structural constraints (, , , , , , ) are satisfied.

(a)

(b)

(c)

Figure 4: A spatial four-bar mechanism with one-DOF and its velocity simulation mechanism versus .

The order (velocity) dimension synthesis of a spatial four-bar mechanism (see Figure 4(b)) can be conducted in a 3D sketch based on the Section 2. In this case, all the dimensions of the spatial four-bar mechanism with one-DoF can be solved by the CAD variational geometry when the initial orientation and the input velocity of input link, and the output position and output velocity of output link are given. The order dimension synthesis processes of this mechanism are described as follows:

(1)Construct 3 lines for , connect their one ends to original point , set the 3 lines in , respectively. Thus a coordinate frame - is constructed.(2)Construct a spatial spline for prescribed trajectory in by the spline command.(3)Construct 2 spatial circles with central point , and connect with .(4)Construct a line , connect its 2 ends to and , set and .(5)Construct 2 lines , connect their one ends to , the other end of to , and set .(6)Construct 2 lines , connect their 2 ends to at and , set and . Thus, 2 revolute joints at point are constructed.(7)Construct a line for axis of , connect its one end with point , and set .(8)Construct a platform . Subprocesses are:

(a)Construct 2 lines and , connect the one end of with , and set .(b)give an initial driving dimension (200) in length, a driven dimension in length, the distance from to the one end of a driven dimension.

(9)Connect the end point of with spline at point .(10)Connect the 2 ends of to at point and point , set and . Thus, a universal joint at point and a spherical joint at are constructed. Finally, the first spatial four-bar mechanism with one-DOF is constructed from .(11)Repeat steps 6–10, except that the constraint equations (, , ) are satisfied and are replaced by , respectively. Thus, the second simulation mechanism of spatial four-bar mechanism is constructed from .(12)Construct an auxiliary line , connect its one end to , and set .(13)Give the angular between line and a driven dimension.

When given some driving dimensions (the output velocity , the input angular velocity , the initial angular of link , the platform parameters , ), and gradually reducing the vector increment of point (the distance between and ) from 12.1 to 0.1, is reduced automatically from 0.202 to 0.00016. At the same time, twin simulation mechanisms of spatial four-bar mechanism are coincident to each other. Thus, the driven dimensions of are solved automatically, as shown in Figure 4(c). The solved results of versus and 0.01 are listed in Table 3.

Table 3: The solved results of a spatial four-bar linkage mechanism with one-DoF.

Given values
Solved values

°/s
mm/s
mm
mm
mm
mm
mm
mm
(°)
s
mm
(°)
(°)
mm
mm
mm
mm
mm

20
60
−100
−16
77
12.1
200
100
66
0.202
50
70.03
91.89
167.16
110
−172.61
32.28
124.02

20
60
−100
−16
77
0.01
200
100
66
0.00016
45.94
66.003
101.74
127.69
100
−175
43.59
97.88

20
100
−100
−16
77
0.01
200
100
66
0.0001
45.93
66.002
101.73
127.67
100
−175
43.57
97.88

In fact, when given (the output velocity , the input angular velocity , the initial angular of link , the platform parameters ), the velocity simulation mechanism has a lack constraint. In this case, one of can be given a driving dimension. Thus, more reasonable dimensions of mechanism can be solved easily.

Similarly, an acceleration simulation mechanism of the spatial four-bar linkage mechanism with one-DOF can be constructed by using the processes in Section 4. Finally, when given some driving dimensions (the output velocity , output acceleration, the input angular velocity , the initial angular of link , the platform parameters , ), the driven dimensions of can be solved automatically.

6. Conclusions

The high-order dimension synthesis of one-DoF mechanisms can be conducted by the CAD variational geometry when the output velocity and acceleration are given.

The reasonable dimensions of the planar four-bar mechanism with one-DoF can be solved by the CAD variational geometry when the position, the velocity, and the acceleration of the coupler along prescribed trajectory are given.

The reasonable dimensions of the spatial four-bar mechanism with one-DoF can be solved by the CAD variational geometry when the position and the velocity of the coupler are given along prescribed trajectory.

When the mechanism is in lacking constraint, one of the driven dimensions can be modified by transforming it into a driving dimension. Thus, more reasonable dimensions may be solved easily.

This approach has not only the merits of fairly quick and straightforward, but also the advantages of accuracy and repeatability. It can be applied for the high-order dimension synthesis of other mechanisms and the spatial parallel mechanisms.

Comparing with the existing analytical methods, the proposed CAD variational geometry approach for high-order dimension synthesis of mechanism has its special merits as follows.

(1)The whole dimension synthesis processes are simple, straightforward, and easy to learn.(2)Since no any mathematic equations are required, the complicate processes of the programming and determining reasonable solutions from multisolutions of velocity or acceleration are avoided.(3)When given velocity and acceleration, the reasonable results of dimensional synthesis of mechanisms can be obtained and inspected easily by modifying one or some driving dimensions.

Acknowledgments

The authors would like to acknowledge (1) Project 51175447 supported by National Natural Science Foundation of China (NSFC); (2) key planned Project of Hebei China application foundation no. 11962127D; (3) Project 2011-002 of State Key Laboratory of Robotics.

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