# Topology Optimization Analysis of Separation Mechanism for the Rice Trans-Planter

Authors: Yan-li Chen; Yi Hu; Su-Yun Li; Tao Shang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A mechanical topology regeneration design method based on the basic chain was presented which was reducing invalid topology design workload and improving optimization design efficiency. And it is used for configuration topology research of planetary separation mechanism of rice transplanting machine to verify its feasibility and effectiveness, which mainly consists of crank-rocker-separation mechanism, mechanical frame, planet carrier, total input shaft, and seedlings folder, and moving trajectory equation of seedling folder point A for the crank-rocker-separation mechanism is obtained according to the quadrangle vector closure principle. The topological synthesis diagram for motion chain for the CRSM was analysis by means of generalized treatment method based on theoretical optimization conditions and topological regeneration path design. The 6 kinds of topology diagram satisfying the design task for motion chair with high motion pair were obtained and the implemented mechanism diagram of kinematic chair for separation mechanism was designed based on the basic chain topology design method. The results show that topology design for the CRSM is relatively simplified for basic chain structure. And theoretical analysis of motion trajectory for the CRSM is relatively the same as the experimental results in the curve trend and relative error of angular displacement is reasonable and effective through test-rig comparative analysis with motion trajectory of crank-rocker-separation mechanism.


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## Body

## 1. Introduction

With the rapid developing of China’s urbanization and the urgent need to improve the Chinese people’s livelihood, the agricultural engineering machinery industry plays an important role in the development process of the China in that a large number of agricultural engineering machinery equipment were applied to improve food productions and reduce labor costs. Plants and plant parts are eaten as food and around 2,000 plant species are cultivated for food [1]. Many of these plant species have several distinct cultivars, such as rice, wheat, and grain. The rice has a larger acreage and yield than other food crops in China; however, the average annual planting area is about 2800-3200 million hectares and accounts for about 40% of the national food crops [2]. Therefore, the intelligent agricultural machinery and equipment have attracted the attention for the ministry of agriculture in China.Ministry of agriculture established the national development planning for the next decade in 2005, where the rice production mechanization level will be increased to 45% by 2015 [3]. However, national mechanized planting average level only was 38.5% at 2015 and less than 20% of rice-mechanized cultivation in the southwest and southern double-season area in China [4]. To cope with these problems, some research institutions and manufactures within the world have paid to the development of the design theory and optimization method. For example, the reliability-based robust design optimization (RBRDO) was proposed to minimize the variation and ensure the levels of failure probability of the system [5]. The kriging-based genetic algorithm is applied to aerodynamic design problems to reduce the number of design variables by eliminating those that have small effect on the objective function [6, 7]. An algorithm based on teaching-learning-based optimization (TLBO) is presented to solve this highly nonlinear optimization problem in the crank-rocker mechanism system [8, 9]. The shape optimization for path synthesis of crank-rocker mechanisms using a wavelet based on the neural network was proposed to provide an approximate solution of the synthesis problem [10, 11]. Optimization design for crank-rocker mechanism based on genetic algorithm was established and solved the design requirements of rocker expectation function with MATLAB genetic algorithm toolbox [12, 13]. A structure design based on topology optimization was applicable to the thermal conductive and electromagnetic field [14–16]. A systematic method for designing fully decoupled compliant mechanisms with multiple degrees of freedom by using topology optimization was presented that an optimization formulation is posed by considering both output coupling and input coupling issues to achieve fully decoupled motion [17]. a new reliability-based topology optimization framework considering spatially varying geometric uncertainties was developed and performance measure approach is adopted to tackle the reliability constraints in the reliability-based topology optimization problem [18]. A two-phase optimization algorithm is proposed to ensure the stability of the optimization process, that is, a Level Set Method with a Bounded Diffusion for Structural Topology Optimization [19]. A new treatment of the solid boundary in the LBM is described particularly for the airfoil optimization design problem and, for a given objective function, the adjoint equation and its boundary conditions are derived analytically [20]. However, the publications devoted to topology optimization approach for the modern agricultural machinery design are relatively scarce. It is noted that the agriculture machinery can be rapidly redeveloped to get accurate motion trajectory by optimizing the structure and size parameters, if the topology optimization design can be employed to the field of intelligent rice agricultural machinery and equipment. Therefore, the survival rate of seedlings will be greatly improved, and grain yield was increasing.


## 2. System Structure and Theoretical Analysis for the Separation Mechanism of Rice Transplanting Machine

### 2.1. System Components and Working Principle

Figure1 shows a structure diagram of the separation mechanism for rice transplanting machine; it can be seen that the crank-rocker separation mechanism is as research objective because this mechanism has a relatively simple and common features and can have a great improvement in terms of topology optimization.Figure 1
Configuration diagram of the planetary separation mechanism of rice transplanting machine. 1-mechanical frame; 2-input-shaft; 3-fixed plate of transmission wheel; 4-center wheel; 5-planet carrier; 6-I # transmission wheel; 7-II # transmission wheel; 8-connecting rod; 9-sun gear; 10-middle idler gear; 11-planet wheel; 12- total input shaft.The working principle is that the power flow was transmitted to input shaft 2 via the total input shaft 12 and bevel gear. The power flow was divided into two parts by fixed plate of transmission wheel 3; one part was transmitted to crank-rocker mechanism though center wheel 4 and transmission shafts 6 and 7, which is constituted by connecting rod 8 and sun wheel 9. Planetary wheel 11 is cyclical oscillation affected by sun gear 9 and middle idler gear 10. It was driven by other parts where the planet carrier 5 was uniform rotation and planetary wheel 11 is circular motion around the input shaft2 with planet carrier 5 rotating. Obviously, the planting target trajectory of the insertion arms attached to planetary wheel 11 is the complex synthetic motion between circular motion and cyclical oscillation.


### 2.2. Mathematical Model

It can be seen from Figure1 that the double insertion arms of planetary separation mechanism are symmetrically distributed and have a same law of motion when the horizontal movement of the separation mechanism is not executed. An insertion arm was used to study in order to simplify the analysis procession, and the phase angle of other insertion arm can be delayed 180 degrees [21, 22]. Obviously, if the planet carrier 5 is regarded as a lever a and the insertion arm as a lever b, the simplified kinematics analysis diagram of crank-rocker-separation mechanism (CRSM) is shown in Figure 2 and the plane motion diagram of two-bar mechanism is shown in Figure 3.Figure 2
Simplified kinematics analysis diagram of crank-rocker separation mechanism.Figure 3
The plane motion diagram of two-bar mechanism.The trajectory equation of point A is obtained:(1)xy=acos⁡θ+bcos⁡φasin⁡θ+bsin⁡φ=afθ+bfφθ∈θ1,θ2φ∈φ1,φ2(2)θt=θ0+ωtt∈0,Tφt=φ0+ωtt∈0,TObviously,θ,φ∈0,2π, a>b; the trajectory range of point A is as follows:(3)PA∈x2+y2=a-b2,x2+y2=a+b2,where ω is angular velocity of rod A, θ is rotation angle between the rod a and x-axis, counterclockwise is positive direction, θ0 is initial angle, φ is rotation angle between the rod b and x-axis, counterclockwise is positive direction, φ0 is initial angle,T is the rotation period of rod a, the XO1Y is the absolute coordinate system, the point O1 is rotation center of planet carrier 5 for separation mechanism, the XO2Y is the relative coordinate system on the planet carrier 5, the point O2 is rotation center of planet wheel 11, and line O2X is always parallel to line O2X.Assuming the lever joystick of crank-rocker separation mechanism is as the x-axis, the kinematics analysis diagram of crank-rocker-separation mechanism is shown in Figure4.Figure 4
Kinematics analysis diagram of crank-rocker separation mechanism.According to the quadrangle vector closure principle, the following equation is obtained.(4)L1⃑+L2⃑=L3⃑+L4⃑By substituting (1) into (4),(5)L1cos⁡αi+α0+L2cos⁡βi=L4+L3cos⁡γi+γ0L1sin⁡αi+α0+L2sin⁡βi=L3sin⁡γi+γ0The rotation angle equation of link L3 is obtained:(6)γ=γi+γ0=2arctan⁡K2±K12+K22-K32K1-K3K1=L4-L1cos⁡αi+α0K2=-L1sin⁡αi+α0K3=K12++K22+L32-L222L3According to the transmission relationship, the angle of the joystick changes the same as the angle of the lever b; that is,αi=ωt/2,Δγ=Δϕ⇒dγ/dt=dϕ/dt.(7)φt=φ0+dγdttt∈0,TTherefore, the trajectory equation of holding point A for the crank-rocker separation mechanism is obtained:(8)xy=acos⁡θ0+ωt+bcos⁡φ0+dγdttasin⁡θ0+ωt+bsin⁡φ0+dγdttwhere Li (i=1-4) is the length of four links for crank-rocker separation mechanism, respectively; α, β, γ is rotation angle between the linkL1,L2,L3 and x-axis, respectively; counterclockwise is positive direction; α0, β0, γ0 is initial angle, respectively.


## 2.1. System Components and Working Principle

Figure1 shows a structure diagram of the separation mechanism for rice transplanting machine; it can be seen that the crank-rocker separation mechanism is as research objective because this mechanism has a relatively simple and common features and can have a great improvement in terms of topology optimization.Figure 1
Configuration diagram of the planetary separation mechanism of rice transplanting machine. 1-mechanical frame; 2-input-shaft; 3-fixed plate of transmission wheel; 4-center wheel; 5-planet carrier; 6-I # transmission wheel; 7-II # transmission wheel; 8-connecting rod; 9-sun gear; 10-middle idler gear; 11-planet wheel; 12- total input shaft.The working principle is that the power flow was transmitted to input shaft 2 via the total input shaft 12 and bevel gear. The power flow was divided into two parts by fixed plate of transmission wheel 3; one part was transmitted to crank-rocker mechanism though center wheel 4 and transmission shafts 6 and 7, which is constituted by connecting rod 8 and sun wheel 9. Planetary wheel 11 is cyclical oscillation affected by sun gear 9 and middle idler gear 10. It was driven by other parts where the planet carrier 5 was uniform rotation and planetary wheel 11 is circular motion around the input shaft2 with planet carrier 5 rotating. Obviously, the planting target trajectory of the insertion arms attached to planetary wheel 11 is the complex synthetic motion between circular motion and cyclical oscillation.


## 2.2. Mathematical Model

It can be seen from Figure1 that the double insertion arms of planetary separation mechanism are symmetrically distributed and have a same law of motion when the horizontal movement of the separation mechanism is not executed. An insertion arm was used to study in order to simplify the analysis procession, and the phase angle of other insertion arm can be delayed 180 degrees [21, 22]. Obviously, if the planet carrier 5 is regarded as a lever a and the insertion arm as a lever b, the simplified kinematics analysis diagram of crank-rocker-separation mechanism (CRSM) is shown in Figure 2 and the plane motion diagram of two-bar mechanism is shown in Figure 3.Figure 2
Simplified kinematics analysis diagram of crank-rocker separation mechanism.Figure 3
The plane motion diagram of two-bar mechanism.The trajectory equation of point A is obtained:(1)xy=acos⁡θ+bcos⁡φasin⁡θ+bsin⁡φ=afθ+bfφθ∈θ1,θ2φ∈φ1,φ2(2)θt=θ0+ωtt∈0,Tφt=φ0+ωtt∈0,TObviously,θ,φ∈0,2π, a>b; the trajectory range of point A is as follows:(3)PA∈x2+y2=a-b2,x2+y2=a+b2,where ω is angular velocity of rod A, θ is rotation angle between the rod a and x-axis, counterclockwise is positive direction, θ0 is initial angle, φ is rotation angle between the rod b and x-axis, counterclockwise is positive direction, φ0 is initial angle,T is the rotation period of rod a, the XO1Y is the absolute coordinate system, the point O1 is rotation center of planet carrier 5 for separation mechanism, the XO2Y is the relative coordinate system on the planet carrier 5, the point O2 is rotation center of planet wheel 11, and line O2X is always parallel to line O2X.Assuming the lever joystick of crank-rocker separation mechanism is as the x-axis, the kinematics analysis diagram of crank-rocker-separation mechanism is shown in Figure4.Figure 4
Kinematics analysis diagram of crank-rocker separation mechanism.According to the quadrangle vector closure principle, the following equation is obtained.(4)L1⃑+L2⃑=L3⃑+L4⃑By substituting (1) into (4),(5)L1cos⁡αi+α0+L2cos⁡βi=L4+L3cos⁡γi+γ0L1sin⁡αi+α0+L2sin⁡βi=L3sin⁡γi+γ0The rotation angle equation of link L3 is obtained:(6)γ=γi+γ0=2arctan⁡K2±K12+K22-K32K1-K3K1=L4-L1cos⁡αi+α0K2=-L1sin⁡αi+α0K3=K12++K22+L32-L222L3According to the transmission relationship, the angle of the joystick changes the same as the angle of the lever b; that is,αi=ωt/2,Δγ=Δϕ⇒dγ/dt=dϕ/dt.(7)φt=φ0+dγdttt∈0,TTherefore, the trajectory equation of holding point A for the crank-rocker separation mechanism is obtained:(8)xy=acos⁡θ0+ωt+bcos⁡φ0+dγdttasin⁡θ0+ωt+bsin⁡φ0+dγdttwhere Li (i=1-4) is the length of four links for crank-rocker separation mechanism, respectively; α, β, γ is rotation angle between the linkL1,L2,L3 and x-axis, respectively; counterclockwise is positive direction; α0, β0, γ0 is initial angle, respectively.


## 3. Topology Structures Analysis and Optimization Design for Planetary Distribution Mechanism of Transplanting Machine

### 3.1. Topological Regeneration Design Path and Processing Steps

It is all known that the mechanism topology regeneration design method is one of the famous innovative design methods and the key is that synthesis analysis of the topological structure of the kinematic chain that can reduce the workload and get an available solution.In order to get more feasible institutions, the completely topological structure forms of kinematic chain corresponding to the initial mechanism were obtained by means theoretical analysis.(i) For single-hinged kinematic chain on the plane, the structure formula of kinematic chain is obtained.(9)F=3n-1-2PL=P-n+1∑i=2imaxni=nimax≤L+1∑i=2imaxini=2Pimax≤L+1(ii) For kinematic chain with high-pair on the plane, the structure formula of kinematic chain is obtained.(10)F=3n-1-2Plow-pair-Phigh-pairL=Plow-pair+Phigh-pair-n+1∑i=2imaxni=nimax≤L+1∑i=2imaxini=3n-1-F+Phigh-pairimax≤L+1whereF is degrees of freedom of the mechanism;N is number of components;P is the number of motion pairs in the mechanism;Plow-pair is the number of low motion pairs in the mechanism;Phigh-pair is the number of high motion pairs in the mechanism; ni is number of components which hasi motion pair elements;L is the number of closed kinematic chains; imax is the maximum number of the motion pair elements in the components.Therefore, processing steps of topological regeneration design for topological analysis and regeneration creations were as follows [23].Step 1.
Calculate the variable parametersP andL according to the parameters n, F, andPhigh-pair.Step 2.
Determiningimax, the components number with each motion is calculated that distribution plan from n2, n3, …, nimax is obtained, respectively.Step 3.
Neglecting the high motion pairs, a topology thumbnail on the basis of distribution plan fromn2,n3,…, nimax was drawn.Step 4.
The n2 second-degree nodes are arranged to the topological thumbnails, ensuring any loops length is at least 4 and any loops length with high motion pairs is at least 3.Step 5.
Deleting stiffened and isomorphic types of mechanisms.


### 3.2. Generalized Treatment and Analysis

Since the axes of respective components for the separation mechanism are parallel to each other and the insertion arm portions are symmetrical, gear train diagram of separation mechanism according to the configuration diagram of separation mechanism is shown in Figure5. It can be seen that gear train for separation mechanism is a planar kinematic chain with one degree of freedom that is composed of eight components and four high motion pairs.Figure 5
Diagram of gear train for separation mechanism.
(a)
Simplified component diagram (b)
Motion chain diagram (c)
Topology diagramObviously, the solid dot is expressed as high motion pairs, the hollow dot is expressed as low motion pairs, and line indicates the connecting member in the motion chain diagram. The full line is expressed as high motion pairs, the broken line is expressed as low motion pairs, and dot indicates the connecting member in the topology diagram.Admittedly, gear 2 and gear 6 are idler wheels in terms of functionality. If gear 2 and gear 6 will be ignored, this system is a planar kinematic chain with one degree of freedom that is composed of six components and two high motion pairs. Figure6 shows simplified diagram of separation mechanism with six components and two high motion pairs.Figure 6
Simplified diagram of separation mechanism with six components and two high motion pairs.
(a)
Motion chain diagram (b)
Topology diagramIf the simplified diagram is further processed, an abbreviated topology diagram is obtained that the points 0, 4, and 7 are removed in Figure6(b), respectively. The Figure 7 shows the abbreviated topology diagram of separation mechanism with 6 components and 2 high motion pairs.Figure 7
Abbreviated topology diagram of separation mechanism with 6 components and 2 high motion pairs.


### 3.3. Topological Structure Optimization Conditions and Topological Synthesis

#### 3.3.1. Optimization Conditions for Motion Chain

According to the theoretical analysis, the topological structure optimization conditions are as follows.(i)
The number of mechanism components is n=6; the degree of freedom F=1.(ii)
Execution component must have more than two sets of components.(iii)
The input shaft rotates at a constant speed and the actuating member should normally oscillate periodically.(iv)
There may be at least one pair of gears and speed ratio is 1:2.(v)
The topological degree of the mechanical frame is two at least and the topological degree of the original motion pieces is three at least.


#### 3.3.2. Topological Synthesis for Motion Chain

(  1) Allocation Scheme. From (10)(11)F=3n-1-2Plow-pair-Phigh-pair=36-1-2×6-2=1L=Plow-pair+Phigh-pair-n+1=6+2-6+1=3∑i=2imaxni=n⟹n2+n3+n4=6∑i=2imaxini=3n-1-F+Phigh-pair⟹2n2+3n3+4n4=16imax≤L+1=4The allocation scheme is as follows:Option I:n2=2,n3=4,n4=0Option II:n2=3,n3=2,n4=1Option III:n2=4,n3=0,n4=2Since there is at least one high pair, the other high pair is replaced by one member and two rotating pairs; the allocation scheme is as follows according to (10).Option I:n2=3,n3=4,n4=0Option II:n2=4,n3=2,n4=1Option III:n2=5,n3=0,n4=2(  2) Abbreviated Topology Diagram Ignored High-Pair. The permutation order is noted as 26042, 26123, 26204,17043, 17124, and 17205 according to the number of high motion pairs, components, four members, three members, and two members [24, 25]. The abbreviated topology diagram ignored high-pair and two-degree point is shown in Figure 8.Figure 8
Abbreviated topology diagram ignored high-pair.(  3) Topology Diagram within Two-Degree Point Layout. All topology diagrams within two-degree point layout are shown in Figure 9.Figure 9
All topology diagrams within the two-degree point layout.The topology diagram of final allocation method is shown in Figure10, which was recorded as 26042a, 26042b, 26042c, 26123a, 26123b, and 26123c according to the optimization conditions, respectively.Figure 10
The topology diagram of final allocation method ignored high-pair.Obviously, according to the design requirements, the topology design object contains high motion pair, which is to be configured. There must be a basic chain in the design mechanism, and the base chain in the separation mechanism for the rice trans-planter is shown in Figure11.Figure 11
The topology diagram of basic chain.According to the design task, the topological object is to contain the high motion pair where topological diagram of motion chain with basic chain was shown in Figure12.Figure 12
The topology diagram of motion chain with high motion pair.Obviously, there is redundant power transmission line after determining the basic chain for 26042c and 26123c. Removing the redundant chain, the original kinematic chain contains 5 components and 1 high pair with 1DOF and the deformed kinematic chain is denoted by 15023a and 15023b, respectively.


#### 3.3.3. Implementation Analysis for Motion Chain

The implemented mechanism diagram of kinematic chair for separation mechanism was shown in Figure13 according to the working principle and optimization conditions, respectively.Figure 13
The implemented mechanism diagram of kinematic chair for separation mechanism.
(a)
26042a (b)
26042b (c)
26123a (d)
26123b (e)
15023a (f)
15023b


## 3.1. Topological Regeneration Design Path and Processing Steps

It is all known that the mechanism topology regeneration design method is one of the famous innovative design methods and the key is that synthesis analysis of the topological structure of the kinematic chain that can reduce the workload and get an available solution.In order to get more feasible institutions, the completely topological structure forms of kinematic chain corresponding to the initial mechanism were obtained by means theoretical analysis.(i) For single-hinged kinematic chain on the plane, the structure formula of kinematic chain is obtained.(9)F=3n-1-2PL=P-n+1∑i=2imaxni=nimax≤L+1∑i=2imaxini=2Pimax≤L+1(ii) For kinematic chain with high-pair on the plane, the structure formula of kinematic chain is obtained.(10)F=3n-1-2Plow-pair-Phigh-pairL=Plow-pair+Phigh-pair-n+1∑i=2imaxni=nimax≤L+1∑i=2imaxini=3n-1-F+Phigh-pairimax≤L+1whereF is degrees of freedom of the mechanism;N is number of components;P is the number of motion pairs in the mechanism;Plow-pair is the number of low motion pairs in the mechanism;Phigh-pair is the number of high motion pairs in the mechanism; ni is number of components which hasi motion pair elements;L is the number of closed kinematic chains; imax is the maximum number of the motion pair elements in the components.Therefore, processing steps of topological regeneration design for topological analysis and regeneration creations were as follows [23].Step 1.
Calculate the variable parametersP andL according to the parameters n, F, andPhigh-pair.Step 2.
Determiningimax, the components number with each motion is calculated that distribution plan from n2, n3, …, nimax is obtained, respectively.Step 3.
Neglecting the high motion pairs, a topology thumbnail on the basis of distribution plan fromn2,n3,…, nimax was drawn.Step 4.
The n2 second-degree nodes are arranged to the topological thumbnails, ensuring any loops length is at least 4 and any loops length with high motion pairs is at least 3.Step 5.
Deleting stiffened and isomorphic types of mechanisms.


## 3.2. Generalized Treatment and Analysis

Since the axes of respective components for the separation mechanism are parallel to each other and the insertion arm portions are symmetrical, gear train diagram of separation mechanism according to the configuration diagram of separation mechanism is shown in Figure5. It can be seen that gear train for separation mechanism is a planar kinematic chain with one degree of freedom that is composed of eight components and four high motion pairs.Figure 5
Diagram of gear train for separation mechanism.
(a)
Simplified component diagram (b)
Motion chain diagram (c)
Topology diagramObviously, the solid dot is expressed as high motion pairs, the hollow dot is expressed as low motion pairs, and line indicates the connecting member in the motion chain diagram. The full line is expressed as high motion pairs, the broken line is expressed as low motion pairs, and dot indicates the connecting member in the topology diagram.Admittedly, gear 2 and gear 6 are idler wheels in terms of functionality. If gear 2 and gear 6 will be ignored, this system is a planar kinematic chain with one degree of freedom that is composed of six components and two high motion pairs. Figure6 shows simplified diagram of separation mechanism with six components and two high motion pairs.Figure 6
Simplified diagram of separation mechanism with six components and two high motion pairs.
(a)
Motion chain diagram (b)
Topology diagramIf the simplified diagram is further processed, an abbreviated topology diagram is obtained that the points 0, 4, and 7 are removed in Figure6(b), respectively. The Figure 7 shows the abbreviated topology diagram of separation mechanism with 6 components and 2 high motion pairs.Figure 7
Abbreviated topology diagram of separation mechanism with 6 components and 2 high motion pairs.


## 3.3. Topological Structure Optimization Conditions and Topological Synthesis

### 3.3.1. Optimization Conditions for Motion Chain

According to the theoretical analysis, the topological structure optimization conditions are as follows.(i)
The number of mechanism components is n=6; the degree of freedom F=1.(ii)
Execution component must have more than two sets of components.(iii)
The input shaft rotates at a constant speed and the actuating member should normally oscillate periodically.(iv)
There may be at least one pair of gears and speed ratio is 1:2.(v)
The topological degree of the mechanical frame is two at least and the topological degree of the original motion pieces is three at least.


### 3.3.2. Topological Synthesis for Motion Chain

(  1) Allocation Scheme. From (10)(11)F=3n-1-2Plow-pair-Phigh-pair=36-1-2×6-2=1L=Plow-pair+Phigh-pair-n+1=6+2-6+1=3∑i=2imaxni=n⟹n2+n3+n4=6∑i=2imaxini=3n-1-F+Phigh-pair⟹2n2+3n3+4n4=16imax≤L+1=4The allocation scheme is as follows:Option I:n2=2,n3=4,n4=0Option II:n2=3,n3=2,n4=1Option III:n2=4,n3=0,n4=2Since there is at least one high pair, the other high pair is replaced by one member and two rotating pairs; the allocation scheme is as follows according to (10).Option I:n2=3,n3=4,n4=0Option II:n2=4,n3=2,n4=1Option III:n2=5,n3=0,n4=2(  2) Abbreviated Topology Diagram Ignored High-Pair. The permutation order is noted as 26042, 26123, 26204,17043, 17124, and 17205 according to the number of high motion pairs, components, four members, three members, and two members [24, 25]. The abbreviated topology diagram ignored high-pair and two-degree point is shown in Figure 8.Figure 8
Abbreviated topology diagram ignored high-pair.(  3) Topology Diagram within Two-Degree Point Layout. All topology diagrams within two-degree point layout are shown in Figure 9.Figure 9
All topology diagrams within the two-degree point layout.The topology diagram of final allocation method is shown in Figure10, which was recorded as 26042a, 26042b, 26042c, 26123a, 26123b, and 26123c according to the optimization conditions, respectively.Figure 10
The topology diagram of final allocation method ignored high-pair.Obviously, according to the design requirements, the topology design object contains high motion pair, which is to be configured. There must be a basic chain in the design mechanism, and the base chain in the separation mechanism for the rice trans-planter is shown in Figure11.Figure 11
The topology diagram of basic chain.According to the design task, the topological object is to contain the high motion pair where topological diagram of motion chain with basic chain was shown in Figure12.Figure 12
The topology diagram of motion chain with high motion pair.Obviously, there is redundant power transmission line after determining the basic chain for 26042c and 26123c. Removing the redundant chain, the original kinematic chain contains 5 components and 1 high pair with 1DOF and the deformed kinematic chain is denoted by 15023a and 15023b, respectively.


### 3.3.3. Implementation Analysis for Motion Chain

The implemented mechanism diagram of kinematic chair for separation mechanism was shown in Figure13 according to the working principle and optimization conditions, respectively.Figure 13
The implemented mechanism diagram of kinematic chair for separation mechanism.
(a)
26042a (b)
26042b (c)
26123a (d)
26123b (e)
15023a (f)
15023b


## 3.3.1. Optimization Conditions for Motion Chain

According to the theoretical analysis, the topological structure optimization conditions are as follows.(i)
The number of mechanism components is n=6; the degree of freedom F=1.(ii)
Execution component must have more than two sets of components.(iii)
The input shaft rotates at a constant speed and the actuating member should normally oscillate periodically.(iv)
There may be at least one pair of gears and speed ratio is 1:2.(v)
The topological degree of the mechanical frame is two at least and the topological degree of the original motion pieces is three at least.


## 3.3.2. Topological Synthesis for Motion Chain

(  1) Allocation Scheme. From (10)(11)F=3n-1-2Plow-pair-Phigh-pair=36-1-2×6-2=1L=Plow-pair+Phigh-pair-n+1=6+2-6+1=3∑i=2imaxni=n⟹n2+n3+n4=6∑i=2imaxini=3n-1-F+Phigh-pair⟹2n2+3n3+4n4=16imax≤L+1=4The allocation scheme is as follows:Option I:n2=2,n3=4,n4=0Option II:n2=3,n3=2,n4=1Option III:n2=4,n3=0,n4=2Since there is at least one high pair, the other high pair is replaced by one member and two rotating pairs; the allocation scheme is as follows according to (10).Option I:n2=3,n3=4,n4=0Option II:n2=4,n3=2,n4=1Option III:n2=5,n3=0,n4=2(  2) Abbreviated Topology Diagram Ignored High-Pair. The permutation order is noted as 26042, 26123, 26204,17043, 17124, and 17205 according to the number of high motion pairs, components, four members, three members, and two members [24, 25]. The abbreviated topology diagram ignored high-pair and two-degree point is shown in Figure 8.Figure 8
Abbreviated topology diagram ignored high-pair.(  3) Topology Diagram within Two-Degree Point Layout. All topology diagrams within two-degree point layout are shown in Figure 9.Figure 9
All topology diagrams within the two-degree point layout.The topology diagram of final allocation method is shown in Figure10, which was recorded as 26042a, 26042b, 26042c, 26123a, 26123b, and 26123c according to the optimization conditions, respectively.Figure 10
The topology diagram of final allocation method ignored high-pair.Obviously, according to the design requirements, the topology design object contains high motion pair, which is to be configured. There must be a basic chain in the design mechanism, and the base chain in the separation mechanism for the rice trans-planter is shown in Figure11.Figure 11
The topology diagram of basic chain.According to the design task, the topological object is to contain the high motion pair where topological diagram of motion chain with basic chain was shown in Figure12.Figure 12
The topology diagram of motion chain with high motion pair.Obviously, there is redundant power transmission line after determining the basic chain for 26042c and 26123c. Removing the redundant chain, the original kinematic chain contains 5 components and 1 high pair with 1DOF and the deformed kinematic chain is denoted by 15023a and 15023b, respectively.


## 3.3.3. Implementation Analysis for Motion Chain

The implemented mechanism diagram of kinematic chair for separation mechanism was shown in Figure13 according to the working principle and optimization conditions, respectively.Figure 13
The implemented mechanism diagram of kinematic chair for separation mechanism.
(a)
26042a (b)
26042b (c)
26123a (d)
26123b (e)
15023a (f)
15023b


## 4. Simulation Analysis and Example Certification for Distribution Mechanism of Transplanting Machine

### 4.1. Parameterized Settings and Cosimulation

From the above topology diagram, it can be seen that the other five kinds of structure have common characteristic besides the cam mechanism that member 5 is oscillated and rotated around the center of drive member 1 [26]. The simulation conditions of crank-rocker separation mechanism are shown in Table 1.Table 1
Main parameters of the CRSM.
Name Parameter value YG Rotation center of member 1 300, 0, 0 QB Coordinate center of the CRSM 0, 0, 0 LG1 Coordinate 1 of connecting rod 100 sin ⁡ ( J D ), 100cos⁡(JD), 0 LG2 Coordinate 2 of connecting rod 100 + G C sin ⁡ ( J D ), 100+GCcos⁡(JD), 0 PP1 The coordinates of component 5 -600, 0, 0 JD Initial angle of CRSM 0.25π GC The length of the connecting rod 300mm LM The length of the drive member 1 400mm RC Rotation center of the member 5 -100, 0, 0Figure14 shows the motion trajectory of crank-rocker separation mechanism at various lengths of GC. It can be seen that the position of the inserting seeding point is almost constant horizontal coordinates with increase of length of connecting rod and the position of the removing seeding point is lower with increase of length of connecting rod.Figure 14
The motion trajectory of crank-rocker separation mechanism.Figure15 shows angle curve of the CRSM under different conditions; that is, JD∈[0,π] and GC∈[100,1000]. It can be seen that the amplitude of β angular variable for member 5 is very large when the JD and GC get different values. Obviously, the maximum amplitude curves of βmax angular variable were done by using the cosimulation between Recurdyn and Matlab software which is to obtain optimization parameters according to the mechanical properties and size constraintsFigure 15
The angle curve of CRSM under different conditions.
(a)
The curve of angleβ at JD=0.1π and GC=500mm (b)
The curve of angleβ at JD=0.25π and GC=300mm (c)
The curve of angleβ at JD=0.4π and GC=700mm (d)
The curve of angleβmax under the different conditions for JD∈[0,π] and GC∈[100,1000]


### 4.2. Bench Test

The position diagram of the inserting seeding point and the removing seeding point of bench test for the distribution mechanism are as follows.It can be seen from Figure16 that the length LM and angle α of member 1 are 25.5mm and 5°, the length LM and angle α of member 5 are 25.5mm and 5° under the limit conditions of inserting seeding in Figure (a), respectively. Similarly, the length LM and angle α of member 1 are 28.86mm and 86° and the length LM and angle α of member 5 are 36.32mm and 19° under the limit conditions of removing seeding in Figure (b), respectively.Figure 16
The position diagram of the inserting seeding point and the removing seeding point for the distribution mechanism.
(a) (b)Obviously, the average ratio of the member 5 to the member 1 is 1.3635, the swing angle of the member 1 is 89 degrees, and the swing angle of the member 5 is 41 degrees. The length of equivalent member 1 and equivalent member 1 is 400mm and 545mm, respectively, and swing angleβ of rocker member 5 is 41 degrees and the swing angle α of crank member 1 is 178 degrees according to the mathematical model of chapter 2.2. Therefore, the array of angular variables Z is defined to analyse the various combinations forms for JD and GC; that is, Z(JD,GC)∈[39,43]; the optimal solution is chosen among the 72 approximate solutions according to the extreme angle α of crank member 1 which is close to 178 degrees; that is, Z(JD,GC)= Z(0.3π,200). The motion trajectory of crank-rocker separation mechanism at Z(0.3π,200) is done in Figure 17. It can be seen that the theoretical analysis of motion trajectory for the CRSM is the same as the experimental results in the curve trend by contrast with Figures 16 and 17. And the extreme angle relative error is 8 degrees meeting the design requirements where angle α of crank member1 is close to 178 degrees in test and angle α of crank member 1 is 186 degrees in theoryFigure 17
The Motion trajectory of CRSM at Z(0.3π,200).


## 4.1. Parameterized Settings and Cosimulation

From the above topology diagram, it can be seen that the other five kinds of structure have common characteristic besides the cam mechanism that member 5 is oscillated and rotated around the center of drive member 1 [26]. The simulation conditions of crank-rocker separation mechanism are shown in Table 1.Table 1
Main parameters of the CRSM.
Name Parameter value YG Rotation center of member 1 300, 0, 0 QB Coordinate center of the CRSM 0, 0, 0 LG1 Coordinate 1 of connecting rod 100 sin ⁡ ( J D ), 100cos⁡(JD), 0 LG2 Coordinate 2 of connecting rod 100 + G C sin ⁡ ( J D ), 100+GCcos⁡(JD), 0 PP1 The coordinates of component 5 -600, 0, 0 JD Initial angle of CRSM 0.25π GC The length of the connecting rod 300mm LM The length of the drive member 1 400mm RC Rotation center of the member 5 -100, 0, 0Figure14 shows the motion trajectory of crank-rocker separation mechanism at various lengths of GC. It can be seen that the position of the inserting seeding point is almost constant horizontal coordinates with increase of length of connecting rod and the position of the removing seeding point is lower with increase of length of connecting rod.Figure 14
The motion trajectory of crank-rocker separation mechanism.Figure15 shows angle curve of the CRSM under different conditions; that is, JD∈[0,π] and GC∈[100,1000]. It can be seen that the amplitude of β angular variable for member 5 is very large when the JD and GC get different values. Obviously, the maximum amplitude curves of βmax angular variable were done by using the cosimulation between Recurdyn and Matlab software which is to obtain optimization parameters according to the mechanical properties and size constraintsFigure 15
The angle curve of CRSM under different conditions.
(a)
The curve of angleβ at JD=0.1π and GC=500mm (b)
The curve of angleβ at JD=0.25π and GC=300mm (c)
The curve of angleβ at JD=0.4π and GC=700mm (d)
The curve of angleβmax under the different conditions for JD∈[0,π] and GC∈[100,1000]


## 4.2. Bench Test

The position diagram of the inserting seeding point and the removing seeding point of bench test for the distribution mechanism are as follows.It can be seen from Figure16 that the length LM and angle α of member 1 are 25.5mm and 5°, the length LM and angle α of member 5 are 25.5mm and 5° under the limit conditions of inserting seeding in Figure (a), respectively. Similarly, the length LM and angle α of member 1 are 28.86mm and 86° and the length LM and angle α of member 5 are 36.32mm and 19° under the limit conditions of removing seeding in Figure (b), respectively.Figure 16
The position diagram of the inserting seeding point and the removing seeding point for the distribution mechanism.
(a) (b)Obviously, the average ratio of the member 5 to the member 1 is 1.3635, the swing angle of the member 1 is 89 degrees, and the swing angle of the member 5 is 41 degrees. The length of equivalent member 1 and equivalent member 1 is 400mm and 545mm, respectively, and swing angleβ of rocker member 5 is 41 degrees and the swing angle α of crank member 1 is 178 degrees according to the mathematical model of chapter 2.2. Therefore, the array of angular variables Z is defined to analyse the various combinations forms for JD and GC; that is, Z(JD,GC)∈[39,43]; the optimal solution is chosen among the 72 approximate solutions according to the extreme angle α of crank member 1 which is close to 178 degrees; that is, Z(JD,GC)= Z(0.3π,200). The motion trajectory of crank-rocker separation mechanism at Z(0.3π,200) is done in Figure 17. It can be seen that the theoretical analysis of motion trajectory for the CRSM is the same as the experimental results in the curve trend by contrast with Figures 16 and 17. And the extreme angle relative error is 8 degrees meeting the design requirements where angle α of crank member1 is close to 178 degrees in test and angle α of crank member 1 is 186 degrees in theoryFigure 17
The Motion trajectory of CRSM at Z(0.3π,200).


## 5. Conclusions

The conclusions can be drawn as follows.(1) The system structure and working principle of separation mechanism for rice transplanter is illustrated and moving trajectory equation of holding point A for the crank-rocker separation mechanism is obtained according to the quadrangle vector closure principle.(2) Topological regeneration design path and processing steps for theoretical optimization conditions are proposed and the topological synthesis diagram for motion chain for the CRSM was analysed by means of generalized treatment method.(3) The 6 kinds of topology diagram satisfying the design task for motion chair with high motion pair were obtained and the implemented mechanism diagram of kinematic chair for separation mechanism was designed based on the basic chain design method.(4) A test-rig was set up based on the working principles of the CRSM system; the results show that topology design for the CRSM is relatively simplified for basic chain structure and theoretical analysis of motion trajectory for the CRSM is the same as the experimental results in the curve trend and relative error of angular displacement is reasonable and effective through comparative analysis with motion trajectory of crank-rocker separation mechanism.



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*Source: 6951647-2018-07-30.xml*