# A Position-Level Global Optimization Inverse Kinematic Solution Algorithm for Dual Redundant Robots Based on Motion Characteristics

Authors: Jingjie He
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Camera robot is an important tool for realizing and reproducing complex camera motion path in modern special film effects. This paper proposed an inverse kinematics optimization algorithm for PRRPR-S redundant degrees of freedom (DoF) camera robot. This paper analyzed the motion characteristics, in Genetic Mix (GM) method, from the idea of movement boundary composed of part robot axis. Then proposed Simplify Mix (SM) method which can stably converge to the global optimal solution in a shorter time.


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## Body

## 1. Introduction

### 1.1. Project Requirements Analysis

The PRRPR-S robot, referred to as camera robot, is an important tool of reproducing camera movement for multilayer composite film effects and realizing complex camera motion path. It is an important interactive node in film virtual manufacturing [1]. At present, the camera shooting trajectory is obtained by teaching mode, and the camera motion data in the preview system cannot be used directly. This paper studied the inverse kinematic algorithm of camera robot with global optimization ability, which makes camera move directly and transparently in a shorter time.


### 1.2. Theory Requirements Analysis

The algebraic equation form of redundancy serial robot inverse kinematics on velocity level is linear style [2], and the position-level is expressed as a nonlinear system and usually solved by numerical iteration. Liegeois [3] proposed gradient projection method based on generalized inverse matrix, which got the local optimal solution on velocity level but there was a cumulative error. Parker et al. [4] used genetic algorithm (GA) to solve the kinematic equations of redundant DoF manipulators. Sung et al. [5] took the terminal actuator positional deviation as the optimization target on the basis of the forward kinematics equation, which obtained the global optimal solution. The algorithm runs for a long time, and the multioptimization objective function is hard to construct.Masayuki et al. [6] and Singh and Claassens [7] analyzed the single redundant DoF serial robot, which determined the self-motion space, and converted the redundancy into arm angle. The method is easy to construct the objective function.The camera robot needs to get the position and attitude of the end-effector exactly in practice. And it has 8 joints, corresponding to high dimension of solution space, in other words, low effective solution proportion space. By using GA method based on forward kinematics equations, it is difficult to find valid solutions and attach other optimization goals. In addition, the randomness of the initial population makes the existence probability of the effective individuals very low, and the algorithm convergence effect is poor. Taking the axes ofr1 and r4 as redundancy, Zhang et al. [8] proposed Genetic Mix (GM) method which include motion characteristics, GA based on redundant DoF and pattern search. By presetting the initial population, GM overcomes the defects of randomness and sequence in GA and converges to the global optimal solution stably on position-level. But in literature [8], there is no discussion about key analysis of how to get redundant DoF theoretical effect solution area. This paper, first, discussed the key analysis in detail, which regarded as the most extended/contract state, and, second, proposed Simplify Mix (SM) method which reduces the time consumption further.


## 1.1. Project Requirements Analysis

The PRRPR-S robot, referred to as camera robot, is an important tool of reproducing camera movement for multilayer composite film effects and realizing complex camera motion path. It is an important interactive node in film virtual manufacturing [1]. At present, the camera shooting trajectory is obtained by teaching mode, and the camera motion data in the preview system cannot be used directly. This paper studied the inverse kinematic algorithm of camera robot with global optimization ability, which makes camera move directly and transparently in a shorter time.


## 1.2. Theory Requirements Analysis

The algebraic equation form of redundancy serial robot inverse kinematics on velocity level is linear style [2], and the position-level is expressed as a nonlinear system and usually solved by numerical iteration. Liegeois [3] proposed gradient projection method based on generalized inverse matrix, which got the local optimal solution on velocity level but there was a cumulative error. Parker et al. [4] used genetic algorithm (GA) to solve the kinematic equations of redundant DoF manipulators. Sung et al. [5] took the terminal actuator positional deviation as the optimization target on the basis of the forward kinematics equation, which obtained the global optimal solution. The algorithm runs for a long time, and the multioptimization objective function is hard to construct.Masayuki et al. [6] and Singh and Claassens [7] analyzed the single redundant DoF serial robot, which determined the self-motion space, and converted the redundancy into arm angle. The method is easy to construct the objective function.The camera robot needs to get the position and attitude of the end-effector exactly in practice. And it has 8 joints, corresponding to high dimension of solution space, in other words, low effective solution proportion space. By using GA method based on forward kinematics equations, it is difficult to find valid solutions and attach other optimization goals. In addition, the randomness of the initial population makes the existence probability of the effective individuals very low, and the algorithm convergence effect is poor. Taking the axes ofr1 and r4 as redundancy, Zhang et al. [8] proposed Genetic Mix (GM) method which include motion characteristics, GA based on redundant DoF and pattern search. By presetting the initial population, GM overcomes the defects of randomness and sequence in GA and converges to the global optimal solution stably on position-level. But in literature [8], there is no discussion about key analysis of how to get redundant DoF theoretical effect solution area. This paper, first, discussed the key analysis in detail, which regarded as the most extended/contract state, and, second, proposed Simplify Mix (SM) method which reduces the time consumption further.


## 2. Motion Characteristic Analysis

### 2.1. Camera Robot Workspace

The camera robot is shown in Figure1. There are 8 axes, including bottom linear axis r1, ring style bottom rotate axis θ2, ring style top rotate axis θ3, top linear axis r4, elbow axis θ5, rotate of wrist θ6, tilt of wrist θ7, and roll of wrist θ8. Modeled with D-H method, the robot “initial standard zero state” and coordinate system on the link are shown in Figure 2.Figure 1
Camera robot.Figure 2
Schematic diagram of camera robot mechanism.The kinematic link parameters are shown in Table1.Table 1
Robot link parameters.
Transform θ r l α 0 π 2 0 0 π 2 1 0 r 1 + r 10 0 - π 2 2 θ 2 - π 2 r 2 0 - π 2 3 θ 3 - π 2 0 l 3 - π 2 4 0 r 4 + r 40 0 π 2 5 θ 5 + π 2 0 0 - π 2 6 θ 6 r 6 0 π 2 7 θ 7 - π 2 0 0 - π 2 8 θ 8 0 0 0r 10 and r40 are fixed value by mechanism. S represents the shoulder point, E represents the elbow point, W represents the wrist point, and U represents the upper arm point (U is not an axis).The DH matrices of camera robot are as follows:(1)A0=0010100001000001,A1=100000100-10r1+r100001,A2=cos⁡θ2-π20-sin⁡θ2-π20sin⁡θ2-π20cos⁡θ2-π200-10r20001,A3=cos⁡θ3-π20-sin⁡θ3-π2l3·cos⁡θ3-π2sin⁡θ3-π20cos⁡θ3-π2l3·sin⁡θ3-π20-1000001,A4=100000-10010r4+r400001,A5=cos⁡θ5+π20-sin⁡θ5+π20sin⁡θ5+π20cos⁡θ5+π200-1000001,A6=cos⁡θ60sin⁡θ60sin⁡θ60-cos⁡θ60010r60001,A7=cos⁡θ7-π20-sin⁡θ7-π20sin⁡θ7-π20cos⁡θ7-π200-1000001,A8=cos⁡θ8sin⁡θ800-sin⁡θ8cos⁡θ80000100001.The values of Table1 are as follows: r10=1000, r2=300, l3=333, r40=2500, and r6=963. Then giving axes value, any posture of robot link can be calculated by transfer equations. For example, the posture of end-effector is(2)Te=A0·A1·A2·A3·A4·A5·A6·A7·A8.Temporarily assume that the end-effector of the camera robot can get any posture through the wrist. As the robot moving in the bottom linear rail, the reachable space ofW in the plane perpendicular to the rail represents the robot workspace. Based on the transfer matrix, as (2), the workspace coordinates of the key points and boundary equation can be calculated, as shown in Figure 3. The red curve is the outer boundary, and the green curve is the inner boundary.Figure 3
Workspace boundary of the camera robot.


### 2.2. Redundancy Setting

Considering that the industrial 6-DoF robot has mature algorithm of inverse kinematic solution, the bottom linear axis (r1) and the top linear axis (r4) are chosen as redundancy. After arbitrarily determining a set of r1r4, countable inverse solutions can be obtained. Choose the solution with the smallest value of the optimal function as the optimal solution, which reduces the 8-dimensional solution space to 2-dimensional one.For clarity, declare the axis physical motion range as physical constraint and geometric constraint as theoretical constraint. The intersection of two constraints is the range which contains effective solution, called comprehensive constraint.When using GA method with physical constraints ofr1r4 for the gene range, due to the fact that the wrist attitude adjustment mechanism may block the view of camera, the effective solution share is very low and the algorithm is inefficient. In order to solve this problem, it is necessary to further analyze dual redundancy effective range. The basic idea is that, after a given posture, the robot should have two ways as “most extended state” and “the most contract state” to achieve the specified posture.


### 2.3. Analysis ofr1 Comprehensive Constraint

Note the following:(1)
WS is the distance between shoulder point (S) and wrist point (W).(2)
pT is a 3-dimensional vector and represents the coordinate of target point position.(3)
tS is the trace of shoulder point.As for the rail can be extended to any length, the physical constraint ofr1 is not in consideration.At different heights, there are specific theoretical maximumWS, TWSmax, and the theoretical minimum WS, TWSmin, as shown in Figure 4.Figure 4
Theoretical max/minWS.


#### 2.3.1.WSmin Determination

AskpTperpendicular to tS, as dpTtSM, and the foot is ppTtSM. If dpTtSM≤TWSmin, refer to the lateral deviation of pT that is not big enough, as T2 shown in Figure 5. There is WSmin=TWSmin.Figure 5
Reaching differentpT at the specified height.IfTWSmin<dpTtSM≤TWSmax, refer to the lateral deviation of target point that is big enough, as T1 shown in Figure 5. There is WSmin=pT-S1.Thus,WSmin can be determined based on the pT.When the camera robot reachespT in the most contract state, there are three cases according to the height of the target point and the degree of lateral deviation, as shown in Figure 6. According to dpTtSM and WSmin, the limit position of S can be obtained, and then determine the limit value of r1.Figure 6
ReachingpT with WSmin.


#### 2.3.2.WSmax Determination

When the camera robot reachespT in the most extended state, W must be on the arc, in which TWSmax is the radius and the projection of S on horizon plane passed pT is the center, as shown in Figure 7.Figure 7
ReachingpT with WSmin.According todpTtSM and TWSmax, obtain the limit position of S, so as to determine the limit value of r1. In summary, according to pT as input, r1 range is(3)r1∈r1farn,r1closen∪r1closep,r1farp.


### 2.4. Analysis ofr4 Comprehensive Constraint

#### 2.4.1. The Factors ofr4 Determination Analysis

After arbitrarily determiningr1 in the comprehensive constraint, there are four factors that limit the value of r4:(1)
SE, WE, WS constitute a generalized triangle, as r4∈[r4minsf1,r4maxsf1].(2)
UE moves within its physical constraint, as r4∈[r4min,r4max].(3)
θ5 moves within its physical constraint, as r4∈[r4minsf3,r4maxsf3].(4)
θ3 moves within its comprehensive constraint, as r4∈[r4minsf4,r4maxsf4].The inclusion relation of factor sets is(4)r4minsf4,r4maxsf4⊆r4minsf3,r4maxf3⊂r4minsf1,r4maxsf1.That factor 4 included in factor 3 will be approved in the following context. Thus while pT is in the workspace and S is determined, there are (5)r4∈r4min,r4max∩r4minsf4,r4maxsf4.Select an arbitraryr4 in this range which can guarantee that at least one effective inverse solution is obtained. So it is important to analyze impact of factor 4 to r4.


#### 2.4.2. Analysis of Factor 4

Given the target posture,S is fixed after selecting r1. For WE is fixed, E must be on the circle tE, whose center is W and radius is WE. Similarly, US length is fixed, so U point must be on the circle tU. ∠EUS is right angle as mechanism set. UE length is r4+r40, as shown in Figure 8. Regardless of the height of W, WS length is the main factor affecting the length of the UE.Figure 8
The physical limit state ofθ3 when WS is horizontal.TakeWS in horizontal as the benchmark situation. When θ3 rotates and E is on the connection of WS, UE gets smallest length on the WS connection. When WE is perpendicular to UE, UE takes the extreme value. Since θ5 physical constraint is [-π,0], the limits of r4 by θ3 theoretical constraint at WS in horizontal state and by θ5 physical constraint are the same.According to the geometric principal,θ3 theoretical constraint is determined:(6)θ3minTH=π2-arctan⁡WS2-r6-l32l3-arccos⁡WS2-r62+r6·l3WS2-r62+2·r6·l3·WS,θ3maxTH=π-π2-∠E3SW+∠E3SU3=π2+∠E3SW-∠E3SU3=π2+arccos⁡WS2-r62-r6·l3WS2-r62-2·r6·l3·WS-arctan⁡WS2-r6+l32l3.The general case whenWS at an arbitrary angle to the horizontal plane is shown in Figure 9.Figure 9
W S in general case.P S is perpendicular to WS. While specifying pT, the angle of WS and the horizontal plane H is ∠WSH. There are (7)θ3maxT=θ3maxTH+∠WSH,θ3minT=θ3minTH+∠WSH,θ3TN=θ3THN+∠WSH.In general, when specifying pT and S, three theoretical characteristic positions of θ3 are obtained. Since θ3 has a physical constraint, as θ3min≤θ3≤θ3max. Therefore, we need to take θ3 comprehensive constraint intervals and position of θ3THN into account.


#### 2.4.3.TRFour Function

Before interpartition analysis, discuss the function:(8)r4T=TRFourW,S,θ3.This function may get r4 that is outside the physical constraint. First, obtain θ3TH corresponding to the WS horizontal state: (9)θ3TH=θ3-∠WSH.Then, (10)∠WSU=π2-θ3TH.In △WSU, by cosine theorem, there is (11)WU=WS2+US2-2·WS·US·cos∠WSU.At the same time, there is (12)∠WUS=arccos⁡WU2+US2-WS22·WU·US,∠WUE=∠WUS-π2.In △WUE, by cosine theorem, there is (13)WE2=WU2+UE2-2·WU·UE·cos∠WUE.Considering position of line with UE, there is (14)UE=WU·cos⁡∠WUE-WE2-WU2·sin⁡∠WUE2=r4T+r40.The result is (15)r4T=UE-r40.


#### 2.4.4. Analysis ofθ3 Effective Comprehensive Constraint

After determiningW and S, discuss θ3 comprehensive constraint. Based on this, discuss r4 comprehensive constraint.The blue sector (filled with upper right oblique line) isθ3 theoretical constraint. The red sector (filled with upper left oblique line) is θ3 physical constraint. The green sector (filled with cross line) is the overlapping area, which refers to comprehensive constraint. The purple line is θ3TN position. Among them, the limit position and scale of θ3 physical constraint do not change, and there always is θ3minT<θ3TN<θ3maxT.According to the following principles, getr4maxsf4 and r4minsf4 corresponding to the value of θ3:(A)
Effectiveθ3 exists in the intersection (green sector).(B)
Withθ3TN as the boundary, the theoretical constraints on both sides are monotonically increasing space.As the standard examples, Figures10(b) and 10(c) will be described. Figures 10(a), and 10(d)–10(j) can be treated as the same way as examples.Figure 10
θ 3 comprehensive constraint and θ3THN relationship.
(a)
Relationship-1 (b)
Relationship-2 (c)
Relationship-3 (d)
Relationship-4 (e)
Relationship-5 (f)
Relationship-6 (g)
Relationship-7 (h)
Relationship-8 (i)
Relationship-9 (j)
Relationship-10Whenθ3max>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3minT<θ3TN≤θ3min. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(b). There are(16)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3minTP, obtain r4minsf4.Whenθ3=θ3maxTP, obtain r4maxsf4.Whenθ3>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3min<θ3TN<θ3maxT. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(c). There are(17)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3TN, obtain r4minsf4.There is(18)r4maxsf4=max⁡TRFourW,S,θ3minTP,TRFourW,S,θ3maxTP.In summary, on the basis of the boundary of the camera robot workspace,r1 comprehensive constraint is determined according to pT with WS characteristic as benchmark. After selecting r1, the influencing factors of r4 are analyzed synthetically, and r4 comprehensive constraint is obtained. And this is also the conception of sequence of the two redundancy in GA.


## 2.1. Camera Robot Workspace

The camera robot is shown in Figure1. There are 8 axes, including bottom linear axis r1, ring style bottom rotate axis θ2, ring style top rotate axis θ3, top linear axis r4, elbow axis θ5, rotate of wrist θ6, tilt of wrist θ7, and roll of wrist θ8. Modeled with D-H method, the robot “initial standard zero state” and coordinate system on the link are shown in Figure 2.Figure 1
Camera robot.Figure 2
Schematic diagram of camera robot mechanism.The kinematic link parameters are shown in Table1.Table 1
Robot link parameters.
Transform θ r l α 0 π 2 0 0 π 2 1 0 r 1 + r 10 0 - π 2 2 θ 2 - π 2 r 2 0 - π 2 3 θ 3 - π 2 0 l 3 - π 2 4 0 r 4 + r 40 0 π 2 5 θ 5 + π 2 0 0 - π 2 6 θ 6 r 6 0 π 2 7 θ 7 - π 2 0 0 - π 2 8 θ 8 0 0 0r 10 and r40 are fixed value by mechanism. S represents the shoulder point, E represents the elbow point, W represents the wrist point, and U represents the upper arm point (U is not an axis).The DH matrices of camera robot are as follows:(1)A0=0010100001000001,A1=100000100-10r1+r100001,A2=cos⁡θ2-π20-sin⁡θ2-π20sin⁡θ2-π20cos⁡θ2-π200-10r20001,A3=cos⁡θ3-π20-sin⁡θ3-π2l3·cos⁡θ3-π2sin⁡θ3-π20cos⁡θ3-π2l3·sin⁡θ3-π20-1000001,A4=100000-10010r4+r400001,A5=cos⁡θ5+π20-sin⁡θ5+π20sin⁡θ5+π20cos⁡θ5+π200-1000001,A6=cos⁡θ60sin⁡θ60sin⁡θ60-cos⁡θ60010r60001,A7=cos⁡θ7-π20-sin⁡θ7-π20sin⁡θ7-π20cos⁡θ7-π200-1000001,A8=cos⁡θ8sin⁡θ800-sin⁡θ8cos⁡θ80000100001.The values of Table1 are as follows: r10=1000, r2=300, l3=333, r40=2500, and r6=963. Then giving axes value, any posture of robot link can be calculated by transfer equations. For example, the posture of end-effector is(2)Te=A0·A1·A2·A3·A4·A5·A6·A7·A8.Temporarily assume that the end-effector of the camera robot can get any posture through the wrist. As the robot moving in the bottom linear rail, the reachable space ofW in the plane perpendicular to the rail represents the robot workspace. Based on the transfer matrix, as (2), the workspace coordinates of the key points and boundary equation can be calculated, as shown in Figure 3. The red curve is the outer boundary, and the green curve is the inner boundary.Figure 3
Workspace boundary of the camera robot.


## 2.2. Redundancy Setting

Considering that the industrial 6-DoF robot has mature algorithm of inverse kinematic solution, the bottom linear axis (r1) and the top linear axis (r4) are chosen as redundancy. After arbitrarily determining a set of r1r4, countable inverse solutions can be obtained. Choose the solution with the smallest value of the optimal function as the optimal solution, which reduces the 8-dimensional solution space to 2-dimensional one.For clarity, declare the axis physical motion range as physical constraint and geometric constraint as theoretical constraint. The intersection of two constraints is the range which contains effective solution, called comprehensive constraint.When using GA method with physical constraints ofr1r4 for the gene range, due to the fact that the wrist attitude adjustment mechanism may block the view of camera, the effective solution share is very low and the algorithm is inefficient. In order to solve this problem, it is necessary to further analyze dual redundancy effective range. The basic idea is that, after a given posture, the robot should have two ways as “most extended state” and “the most contract state” to achieve the specified posture.


## 2.3. Analysis ofr1 Comprehensive Constraint

Note the following:(1)
WS is the distance between shoulder point (S) and wrist point (W).(2)
pT is a 3-dimensional vector and represents the coordinate of target point position.(3)
tS is the trace of shoulder point.As for the rail can be extended to any length, the physical constraint ofr1 is not in consideration.At different heights, there are specific theoretical maximumWS, TWSmax, and the theoretical minimum WS, TWSmin, as shown in Figure 4.Figure 4
Theoretical max/minWS.


### 2.3.1.WSmin Determination

AskpTperpendicular to tS, as dpTtSM, and the foot is ppTtSM. If dpTtSM≤TWSmin, refer to the lateral deviation of pT that is not big enough, as T2 shown in Figure 5. There is WSmin=TWSmin.Figure 5
Reaching differentpT at the specified height.IfTWSmin<dpTtSM≤TWSmax, refer to the lateral deviation of target point that is big enough, as T1 shown in Figure 5. There is WSmin=pT-S1.Thus,WSmin can be determined based on the pT.When the camera robot reachespT in the most contract state, there are three cases according to the height of the target point and the degree of lateral deviation, as shown in Figure 6. According to dpTtSM and WSmin, the limit position of S can be obtained, and then determine the limit value of r1.Figure 6
ReachingpT with WSmin.


### 2.3.2.WSmax Determination

When the camera robot reachespT in the most extended state, W must be on the arc, in which TWSmax is the radius and the projection of S on horizon plane passed pT is the center, as shown in Figure 7.Figure 7
ReachingpT with WSmin.According todpTtSM and TWSmax, obtain the limit position of S, so as to determine the limit value of r1. In summary, according to pT as input, r1 range is(3)r1∈r1farn,r1closen∪r1closep,r1farp.


## 2.3.1.WSmin Determination

AskpTperpendicular to tS, as dpTtSM, and the foot is ppTtSM. If dpTtSM≤TWSmin, refer to the lateral deviation of pT that is not big enough, as T2 shown in Figure 5. There is WSmin=TWSmin.Figure 5
Reaching differentpT at the specified height.IfTWSmin<dpTtSM≤TWSmax, refer to the lateral deviation of target point that is big enough, as T1 shown in Figure 5. There is WSmin=pT-S1.Thus,WSmin can be determined based on the pT.When the camera robot reachespT in the most contract state, there are three cases according to the height of the target point and the degree of lateral deviation, as shown in Figure 6. According to dpTtSM and WSmin, the limit position of S can be obtained, and then determine the limit value of r1.Figure 6
ReachingpT with WSmin.


## 2.3.2.WSmax Determination

When the camera robot reachespT in the most extended state, W must be on the arc, in which TWSmax is the radius and the projection of S on horizon plane passed pT is the center, as shown in Figure 7.Figure 7
ReachingpT with WSmin.According todpTtSM and TWSmax, obtain the limit position of S, so as to determine the limit value of r1. In summary, according to pT as input, r1 range is(3)r1∈r1farn,r1closen∪r1closep,r1farp.


## 2.4. Analysis ofr4 Comprehensive Constraint

### 2.4.1. The Factors ofr4 Determination Analysis

After arbitrarily determiningr1 in the comprehensive constraint, there are four factors that limit the value of r4:(1)
SE, WE, WS constitute a generalized triangle, as r4∈[r4minsf1,r4maxsf1].(2)
UE moves within its physical constraint, as r4∈[r4min,r4max].(3)
θ5 moves within its physical constraint, as r4∈[r4minsf3,r4maxsf3].(4)
θ3 moves within its comprehensive constraint, as r4∈[r4minsf4,r4maxsf4].The inclusion relation of factor sets is(4)r4minsf4,r4maxsf4⊆r4minsf3,r4maxf3⊂r4minsf1,r4maxsf1.That factor 4 included in factor 3 will be approved in the following context. Thus while pT is in the workspace and S is determined, there are (5)r4∈r4min,r4max∩r4minsf4,r4maxsf4.Select an arbitraryr4 in this range which can guarantee that at least one effective inverse solution is obtained. So it is important to analyze impact of factor 4 to r4.


### 2.4.2. Analysis of Factor 4

Given the target posture,S is fixed after selecting r1. For WE is fixed, E must be on the circle tE, whose center is W and radius is WE. Similarly, US length is fixed, so U point must be on the circle tU. ∠EUS is right angle as mechanism set. UE length is r4+r40, as shown in Figure 8. Regardless of the height of W, WS length is the main factor affecting the length of the UE.Figure 8
The physical limit state ofθ3 when WS is horizontal.TakeWS in horizontal as the benchmark situation. When θ3 rotates and E is on the connection of WS, UE gets smallest length on the WS connection. When WE is perpendicular to UE, UE takes the extreme value. Since θ5 physical constraint is [-π,0], the limits of r4 by θ3 theoretical constraint at WS in horizontal state and by θ5 physical constraint are the same.According to the geometric principal,θ3 theoretical constraint is determined:(6)θ3minTH=π2-arctan⁡WS2-r6-l32l3-arccos⁡WS2-r62+r6·l3WS2-r62+2·r6·l3·WS,θ3maxTH=π-π2-∠E3SW+∠E3SU3=π2+∠E3SW-∠E3SU3=π2+arccos⁡WS2-r62-r6·l3WS2-r62-2·r6·l3·WS-arctan⁡WS2-r6+l32l3.The general case whenWS at an arbitrary angle to the horizontal plane is shown in Figure 9.Figure 9
W S in general case.P S is perpendicular to WS. While specifying pT, the angle of WS and the horizontal plane H is ∠WSH. There are (7)θ3maxT=θ3maxTH+∠WSH,θ3minT=θ3minTH+∠WSH,θ3TN=θ3THN+∠WSH.In general, when specifying pT and S, three theoretical characteristic positions of θ3 are obtained. Since θ3 has a physical constraint, as θ3min≤θ3≤θ3max. Therefore, we need to take θ3 comprehensive constraint intervals and position of θ3THN into account.


### 2.4.3.TRFour Function

Before interpartition analysis, discuss the function:(8)r4T=TRFourW,S,θ3.This function may get r4 that is outside the physical constraint. First, obtain θ3TH corresponding to the WS horizontal state: (9)θ3TH=θ3-∠WSH.Then, (10)∠WSU=π2-θ3TH.In △WSU, by cosine theorem, there is (11)WU=WS2+US2-2·WS·US·cos∠WSU.At the same time, there is (12)∠WUS=arccos⁡WU2+US2-WS22·WU·US,∠WUE=∠WUS-π2.In △WUE, by cosine theorem, there is (13)WE2=WU2+UE2-2·WU·UE·cos∠WUE.Considering position of line with UE, there is (14)UE=WU·cos⁡∠WUE-WE2-WU2·sin⁡∠WUE2=r4T+r40.The result is (15)r4T=UE-r40.


### 2.4.4. Analysis ofθ3 Effective Comprehensive Constraint

After determiningW and S, discuss θ3 comprehensive constraint. Based on this, discuss r4 comprehensive constraint.The blue sector (filled with upper right oblique line) isθ3 theoretical constraint. The red sector (filled with upper left oblique line) is θ3 physical constraint. The green sector (filled with cross line) is the overlapping area, which refers to comprehensive constraint. The purple line is θ3TN position. Among them, the limit position and scale of θ3 physical constraint do not change, and there always is θ3minT<θ3TN<θ3maxT.According to the following principles, getr4maxsf4 and r4minsf4 corresponding to the value of θ3:(A)
Effectiveθ3 exists in the intersection (green sector).(B)
Withθ3TN as the boundary, the theoretical constraints on both sides are monotonically increasing space.As the standard examples, Figures10(b) and 10(c) will be described. Figures 10(a), and 10(d)–10(j) can be treated as the same way as examples.Figure 10
θ 3 comprehensive constraint and θ3THN relationship.
(a)
Relationship-1 (b)
Relationship-2 (c)
Relationship-3 (d)
Relationship-4 (e)
Relationship-5 (f)
Relationship-6 (g)
Relationship-7 (h)
Relationship-8 (i)
Relationship-9 (j)
Relationship-10Whenθ3max>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3minT<θ3TN≤θ3min. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(b). There are(16)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3minTP, obtain r4minsf4.Whenθ3=θ3maxTP, obtain r4maxsf4.Whenθ3>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3min<θ3TN<θ3maxT. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(c). There are(17)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3TN, obtain r4minsf4.There is(18)r4maxsf4=max⁡TRFourW,S,θ3minTP,TRFourW,S,θ3maxTP.In summary, on the basis of the boundary of the camera robot workspace,r1 comprehensive constraint is determined according to pT with WS characteristic as benchmark. After selecting r1, the influencing factors of r4 are analyzed synthetically, and r4 comprehensive constraint is obtained. And this is also the conception of sequence of the two redundancy in GA.


## 2.4.1. The Factors ofr4 Determination Analysis

After arbitrarily determiningr1 in the comprehensive constraint, there are four factors that limit the value of r4:(1)
SE, WE, WS constitute a generalized triangle, as r4∈[r4minsf1,r4maxsf1].(2)
UE moves within its physical constraint, as r4∈[r4min,r4max].(3)
θ5 moves within its physical constraint, as r4∈[r4minsf3,r4maxsf3].(4)
θ3 moves within its comprehensive constraint, as r4∈[r4minsf4,r4maxsf4].The inclusion relation of factor sets is(4)r4minsf4,r4maxsf4⊆r4minsf3,r4maxf3⊂r4minsf1,r4maxsf1.That factor 4 included in factor 3 will be approved in the following context. Thus while pT is in the workspace and S is determined, there are (5)r4∈r4min,r4max∩r4minsf4,r4maxsf4.Select an arbitraryr4 in this range which can guarantee that at least one effective inverse solution is obtained. So it is important to analyze impact of factor 4 to r4.


## 2.4.2. Analysis of Factor 4

Given the target posture,S is fixed after selecting r1. For WE is fixed, E must be on the circle tE, whose center is W and radius is WE. Similarly, US length is fixed, so U point must be on the circle tU. ∠EUS is right angle as mechanism set. UE length is r4+r40, as shown in Figure 8. Regardless of the height of W, WS length is the main factor affecting the length of the UE.Figure 8
The physical limit state ofθ3 when WS is horizontal.TakeWS in horizontal as the benchmark situation. When θ3 rotates and E is on the connection of WS, UE gets smallest length on the WS connection. When WE is perpendicular to UE, UE takes the extreme value. Since θ5 physical constraint is [-π,0], the limits of r4 by θ3 theoretical constraint at WS in horizontal state and by θ5 physical constraint are the same.According to the geometric principal,θ3 theoretical constraint is determined:(6)θ3minTH=π2-arctan⁡WS2-r6-l32l3-arccos⁡WS2-r62+r6·l3WS2-r62+2·r6·l3·WS,θ3maxTH=π-π2-∠E3SW+∠E3SU3=π2+∠E3SW-∠E3SU3=π2+arccos⁡WS2-r62-r6·l3WS2-r62-2·r6·l3·WS-arctan⁡WS2-r6+l32l3.The general case whenWS at an arbitrary angle to the horizontal plane is shown in Figure 9.Figure 9
W S in general case.P S is perpendicular to WS. While specifying pT, the angle of WS and the horizontal plane H is ∠WSH. There are (7)θ3maxT=θ3maxTH+∠WSH,θ3minT=θ3minTH+∠WSH,θ3TN=θ3THN+∠WSH.In general, when specifying pT and S, three theoretical characteristic positions of θ3 are obtained. Since θ3 has a physical constraint, as θ3min≤θ3≤θ3max. Therefore, we need to take θ3 comprehensive constraint intervals and position of θ3THN into account.


## 2.4.3.TRFour Function

Before interpartition analysis, discuss the function:(8)r4T=TRFourW,S,θ3.This function may get r4 that is outside the physical constraint. First, obtain θ3TH corresponding to the WS horizontal state: (9)θ3TH=θ3-∠WSH.Then, (10)∠WSU=π2-θ3TH.In △WSU, by cosine theorem, there is (11)WU=WS2+US2-2·WS·US·cos∠WSU.At the same time, there is (12)∠WUS=arccos⁡WU2+US2-WS22·WU·US,∠WUE=∠WUS-π2.In △WUE, by cosine theorem, there is (13)WE2=WU2+UE2-2·WU·UE·cos∠WUE.Considering position of line with UE, there is (14)UE=WU·cos⁡∠WUE-WE2-WU2·sin⁡∠WUE2=r4T+r40.The result is (15)r4T=UE-r40.


## 2.4.4. Analysis ofθ3 Effective Comprehensive Constraint

After determiningW and S, discuss θ3 comprehensive constraint. Based on this, discuss r4 comprehensive constraint.The blue sector (filled with upper right oblique line) isθ3 theoretical constraint. The red sector (filled with upper left oblique line) is θ3 physical constraint. The green sector (filled with cross line) is the overlapping area, which refers to comprehensive constraint. The purple line is θ3TN position. Among them, the limit position and scale of θ3 physical constraint do not change, and there always is θ3minT<θ3TN<θ3maxT.According to the following principles, getr4maxsf4 and r4minsf4 corresponding to the value of θ3:(A)
Effectiveθ3 exists in the intersection (green sector).(B)
Withθ3TN as the boundary, the theoretical constraints on both sides are monotonically increasing space.As the standard examples, Figures10(b) and 10(c) will be described. Figures 10(a), and 10(d)–10(j) can be treated as the same way as examples.Figure 10
θ 3 comprehensive constraint and θ3THN relationship.
(a)
Relationship-1 (b)
Relationship-2 (c)
Relationship-3 (d)
Relationship-4 (e)
Relationship-5 (f)
Relationship-6 (g)
Relationship-7 (h)
Relationship-8 (i)
Relationship-9 (j)
Relationship-10Whenθ3max>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3minT<θ3TN≤θ3min. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(b). There are(16)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3minTP, obtain r4minsf4.Whenθ3=θ3maxTP, obtain r4maxsf4.Whenθ3>θ3maxT, θ3maxT≥θ3min>θ3minT and θ3min<θ3TN<θ3maxT. There is an intersection of the lower subinterval of θ3 physical constraint and the higher subinterval of θ3 theoretical constraint, as shown in Figure 10(c). There are(17)θ3minTP=θ3min,θ3maxTP=θ3maxT.And whenθ3=θ3TN, obtain r4minsf4.There is(18)r4maxsf4=max⁡TRFourW,S,θ3minTP,TRFourW,S,θ3maxTP.In summary, on the basis of the boundary of the camera robot workspace,r1 comprehensive constraint is determined according to pT with WS characteristic as benchmark. After selecting r1, the influencing factors of r4 are analyzed synthetically, and r4 comprehensive constraint is obtained. And this is also the conception of sequence of the two redundancy in GA.


## 3. Optimized Method SM

### 3.1. Optimized Objective Function

The optimization goal is to improve the motion stability of the end-effector; this paper sets the motor shaft to move as little as possible with the higher load inertia. The optimization objective functionF is(19)F=W·X,where W is the weight vector and X represents the movement distance vector of axis.


### 3.2. Algorithm Optimization

According to the camera robot motion characteristics,r1 can be determined first and then the theoritical effect range of r4 can be calculated. This is the sequence of two variables. GM method used the dual redundancy as genes. By setting the initial population with average distribution in effect solution area, GM can eliminate the large iterative calculation from GA nonlinear constraints caused by sequence of dual redundancy and overcome the low effective proportion of stochastic initial population. Then through the pattern search algorithm, the composite algorithm can converge to the global optimal solution stably.In the experiment, it is found that the optimal solution of GM did not change or change little in the process of iterative computation. That is, the analysis of camera robot motion characteristics with average initial population distribution greatly weakened the global optimization effect of GA in GM. So it is desirable to solve in a shorter time by motion characteristics with pattern search. This paper proposed SM method. To verify the effect of it, set the experiment.The robot is in the initial standard zero state. 4 to 8 meters in the positive direction of the rail, set 45 target postures evenly. Specifically, five planes are set in the space, and 9 target postures are distributed in each plane, as shown in Figure11, where red line represents the coordinate system z-axis and black line represents the coordinate system x-axis.Figure 11
Target postures in workspace.Assume that bothvsGM and vsSM are 8-dimensional vectors and are solutions for each method, which can both achieve the target posture.Set the physical constraints of each axis as(20)vAR=vAR1vAR2⋯vAR8,where(21)vARi=xjmax-xjmin,i=1,2,…,8,xj=r1,θ2,θ3,r4,θ5,θ6,θ7,θEE.Define(22)nDR=∏i=18vsGMi-vsSMivARi.n D R can be understood as the proportion of the generalized distance between different solutions in the entire workspace.Take the experiment as follows.Select 20 values inr1 comprehensive effective subinterval (specified pT as boundary, the robot at the same side with the current state). According to 20 r1 values, the corresponding r4 comprehensive constraints are calculated. Select the middle value as r4 value, forming an individual with the corresponding r1.Calculate the optimal solutions by GM and SM methods, respectively. The result is shown in Figure12. The x-axis is the target posture number, and y-axis is nDR. The solid point on the corresponding coordinate of the x-axis indicates that the GM gets a better solution, and the triangle represents that the SM gets a better solution. The cross symbol means that the two methods get the same solution.Figure 12
Compared result of two methods.For 45 target postures, GM costs 6328.32 s and SM costs 3410.07 s. There is only one posture with big distance appearing on the 30th posture, where the ratio of two values of optimal objective function is 12%, and GM method gets a better solution. The robot states are shown in Figure13, the black body is the result of the GA, and the blue body is the result of the SM. The actual difference of two solutions is not big. The solutions of two methods for the other 44 target postures are close enough to the global optimal solution (verified by a certain accuracy of the traversal algorithm). nDR is nearly to 0.Figure 13
Maximum deviation of two methods.In summary, SM method is more convenient in practical application than GM.


## 3.1. Optimized Objective Function

The optimization goal is to improve the motion stability of the end-effector; this paper sets the motor shaft to move as little as possible with the higher load inertia. The optimization objective functionF is(19)F=W·X,where W is the weight vector and X represents the movement distance vector of axis.


## 3.2. Algorithm Optimization

According to the camera robot motion characteristics,r1 can be determined first and then the theoritical effect range of r4 can be calculated. This is the sequence of two variables. GM method used the dual redundancy as genes. By setting the initial population with average distribution in effect solution area, GM can eliminate the large iterative calculation from GA nonlinear constraints caused by sequence of dual redundancy and overcome the low effective proportion of stochastic initial population. Then through the pattern search algorithm, the composite algorithm can converge to the global optimal solution stably.In the experiment, it is found that the optimal solution of GM did not change or change little in the process of iterative computation. That is, the analysis of camera robot motion characteristics with average initial population distribution greatly weakened the global optimization effect of GA in GM. So it is desirable to solve in a shorter time by motion characteristics with pattern search. This paper proposed SM method. To verify the effect of it, set the experiment.The robot is in the initial standard zero state. 4 to 8 meters in the positive direction of the rail, set 45 target postures evenly. Specifically, five planes are set in the space, and 9 target postures are distributed in each plane, as shown in Figure11, where red line represents the coordinate system z-axis and black line represents the coordinate system x-axis.Figure 11
Target postures in workspace.Assume that bothvsGM and vsSM are 8-dimensional vectors and are solutions for each method, which can both achieve the target posture.Set the physical constraints of each axis as(20)vAR=vAR1vAR2⋯vAR8,where(21)vARi=xjmax-xjmin,i=1,2,…,8,xj=r1,θ2,θ3,r4,θ5,θ6,θ7,θEE.Define(22)nDR=∏i=18vsGMi-vsSMivARi.n D R can be understood as the proportion of the generalized distance between different solutions in the entire workspace.Take the experiment as follows.Select 20 values inr1 comprehensive effective subinterval (specified pT as boundary, the robot at the same side with the current state). According to 20 r1 values, the corresponding r4 comprehensive constraints are calculated. Select the middle value as r4 value, forming an individual with the corresponding r1.Calculate the optimal solutions by GM and SM methods, respectively. The result is shown in Figure12. The x-axis is the target posture number, and y-axis is nDR. The solid point on the corresponding coordinate of the x-axis indicates that the GM gets a better solution, and the triangle represents that the SM gets a better solution. The cross symbol means that the two methods get the same solution.Figure 12
Compared result of two methods.For 45 target postures, GM costs 6328.32 s and SM costs 3410.07 s. There is only one posture with big distance appearing on the 30th posture, where the ratio of two values of optimal objective function is 12%, and GM method gets a better solution. The robot states are shown in Figure13, the black body is the result of the GA, and the blue body is the result of the SM. The actual difference of two solutions is not big. The solutions of two methods for the other 44 target postures are close enough to the global optimal solution (verified by a certain accuracy of the traversal algorithm). nDR is nearly to 0.Figure 13
Maximum deviation of two methods.In summary, SM method is more convenient in practical application than GM.


## 4. Conclusion

In this paper, the motion characteristics analysis of PRRPR-S robot is discussed; GM method overcomes the defects of dual redundancy sequence and stochastic of GA. The experiment in Section3 proved that GA in GM has little effect. SM gets good result directly and more quickly (about half the time consumed in GM).Even though the kinematic solution method is for PRRPR-S robot, by using the idea of subworkspace and motion path, combining with pattern search, any other type of redundant robot can get the inverse kinematic solution. This is the significance of SM method.



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*Source: 9167837-2018-03-26.xml*