# Positioning Accuracy and Numerical Analysis of the Main Casting Mechanism of the Hybrid Casting Robot

Authors: Long Li; Binyang Chen; Chengjun Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The positioning accuracy is a key index to measure the performance of the robot. This paper studies the positioning accuracy of the main pouring mechanism of the hybrid truss pouring robot and analyzes that the main error sources affecting the positioning accuracy are machining error, assembly error, and thermal deformation error. Error transfer matrix is constructed to describe the influence of machining errors and assembly errors on the position and pose of the terminal, and the error parameters have physical significance. The probability distribution of sensitive errors is discussed. A joint regression prediction model based on sensitive error sets is established to determine the thermal deformation error on the basis of fully considering the contribution rate of component error. The results show that the position error has a wide range of influences on the end pose, but the angle error is more sensitive, and the probability distribution of the sensitive error is concentrated. The reliable data can be obtained without reorganizing the measurement in the calibration process. The joint regression model considering the contribution rate of component error can effectively eliminate the collinearity problem in the prediction of thermal deformation from a single heat source. Compared with the single regression model, it has better prediction accuracy and effect.


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## Body

## 1. Introduction

In this paper, a hybrid truss pouring robot is designed, and its main pouring mechanism (4-UPU parallel mechanism) is the key component to realize the pouring operation. However, under the influence of processing, installation, load, and working environment (thermal, optical, electrical, etc.), the pouring motion control model is different from its actual structure [1, 2]. These differences will reduce the positioning accuracy of the end-effector of the parallel mechanism, which makes it difficult to reflect the high working accuracy of the parallel machine tool [3, 4]. Therefore, the error analysis of the parallel mechanism is the basis to improve the positioning accuracy of the parallel robot.Based on the screw theory, Meng et al. [5] firstly modeled the Pa Quad parallel mechanism with joint disturbance and geometric error and transformed the geometric error into dimensional tolerance or geometric tolerance. The error compensation based on kinematic calibration was realized at the mechanism level. It is proved that error modeling is effective to improve operation accuracy, but this method is complicated and the actual physical meaning of the model is not clear. Tu et al. [6] derived the Stewart recovery accuracy equation by total differential method. The structural optimization problem is transformed into numerical optimization problem. Genetic algorithm (GA) is used to optimize the recovery accuracy, but the efficiency is too low. Shan and Cheng [7] made motion modeling compensation for static errors such as machining error and assembly error of 2 (3PUS + S) parallel mechanisms. The dynamic error was compensated by predictive pose compensation, and the friction force was introduced into the model to make the model more realistic. However, the improvement of motion accuracy was not obvious, and other factors were not considered. Mei et al. [8] introduced elastic deformation into pose deviation model to improve the operation of large load parallel robot.Wang et al. [9] studied the thermal deformation of a truss secondary mirror support structure and proposed a high-order sensitivity matrix method to compensate the thermal deformation error of the truss, which compensated the secondary mirror error caused by thermal deformation to a certain extent, but the compensation accuracy cannot meet the accuracy requirements of the telescope system. Li et al. [10] applied finite element analysis method to analyze the steady-state deformation law of thermodynamic coupling in two directions of large span beam of large truss extubation robot and optimized the structure. The steady-state characteristics of the optimized robot can meet the engineering needs. Wang et al. [11] proposed a control model based on positive position feedback (PPF) compensator for the vibration and quasi-static deformation of flexible spatial structure under thermal load, which can effectively suppress the multimodal vibration caused by thermal deformation and compensate the quasi-static deformation. Cui et al. [12] studied the thermal-mechanical coupling problem of spacecraft truss structure and constructed a comprehensive description model of displacement field and temperature field based on the absolute node coordinate formula. The model included the input of solar radiation and surface radiation, as well as the influence of rigid body motion and deformation on radiation absorption, revealing the nonlinear behavior of spacecraft truss structure rigid-flexible-thermal coupling system and its influence on pointing accuracy. There are few studies on extreme heat sources in the literature.On the basis of summarizing the above research, this paper analyzes the error sources of the main pouring mechanism in detail and makes it clear that the machining error is the main error source in the geometric error, and the thermal deformation error has the characteristics of centralized heat source and single heat source. The combined error transfer model including machining and installation errors is constructed, and the error is projected into the model parameters to make it have practical physical significance. At the same time, the error distribution characteristics are analyzed. Aiming at the characteristics of single heat source and high collinearity of main pouring mechanism, a joint regression model based on sensitive error set is established, and the thermal deformation error is determined based on the contribution rate of component error.


## 2. Structure Design of Main Pouring Mechanism

Industrial robots in the pouring field (mostly in series) have some problems such as small load, cumulative joint error effect, frequent supplement of pouring liquid, and limited working space. In this paper, a hybrid pouring robot is designed. As shown in Figure1, it can meet the needs of various pouring operations well and has good continuous pouring ability. The working space is not limited to a certain station [13]. The basic step of the operation is that the high-temperature molten metal liquid flows out through the melting furnace and then is transferred to the designated station. The pouring speed is set according to the demand, and the pouring liquid is poured into the pouring riser. The operation is completed and returned to the initial state, waiting for the next instruction. The maximum design load of the ladle is 500 kg, and as long as the degree of freedom of the mechanism meets the degree of freedom required for pouring, the useless degree of freedom shall be minimized. The reference working space of the ladle is as follows: the upper and lower design stroke is 0.5 m, the front and rear design stroke is 0.2 m, the left and right design stroke is 0.3 m, and the end positioning accuracy is less than 1 cm.Figure 1
Hybrid truss-type movable heavy-load pouring robot. 1, 4-WD mobile platform; 2, slewing device; 3, hoisting device; 4, forward device; 5, backward device; 6, counterweight device; 7, parallel working arm; and 8, end-effector.Among them, the main pouring mechanism is the key component to control the pouring accuracy and flow. The parallel mechanism has the advantages of compact structure, large carrying capacity, high stiffness, and high positioning accuracy, which meets the design requirements of large load and high precision for the main pouring mechanism (operating mechanism) of the hybrid pouring robot. Therefore, 4-UPU (U represents the universal joint,P represents the mobile pair) parallel structure is used in the design of the main pouring mechanism (operating mechanism). The mechanism diagram is shown in Figure 2.Figure 2
4-UPU parallel mechanism diagram.The upper platform is a fixed platform, installed on the beam of heavy-duty casting robot, and the lower platform is a moving platform, connected with the ladle. The four branched chains are UPU kinematic chains. The U-joint consists of two R-joints whose axes are perpendicular to each other (R represents the rotational joint), and the motion chain UPU is equivalent to the combination of RRPRR joints. The kinematic axes of each branch chain are expressed asSi1, Si2, Si3, Si4, and Si5 in turn (i = 1, 2, 3, 4; for simplicity, only the axis of branch 1 is indicated). Si1 is consistent with y-axis direction, and Si5 is consistent with y-axis direction, The angle between Si3 and Si4 is Θi, Si1⊥Si2, Si4⊥Si5, S12‖S22, S32‖S42.


## 3. Error Source Analysis of Main Pouring Mechanism

The main pouring mechanism is the key component to realize the pouring operation. To further improve the operation performance of the hybrid truss pouring robot, its positioning accuracy is an important index, which directly affects the correct completion of the pouring task. The main pouring mechanism is formed by multiple branches connecting the moving platform and the static platform through hinges. In the manufacturing, equipment, debugging, and other links, it is affected by factors such as machining, installation error, and external interference, which will lead to the difference between the theoretical position and the actual position of the pouring ladle during the pouring operation. This deviation will reduce the advantages of high precision of the parallel mechanism and even cause operation accidents in serious cases. A large number of error source studies show that 70% of the error of the mechanism comes from geometric error, thermal deformation error, and load error [14, 15]:(a)
Geometric error (static error) is the inherent error of structure, including machining error, assembly error, and gravity error. For the designed main pouring mechanism, the theoretical size and the actual size of the hinges and bars of the parallel mechanism are affected by the processing technology and methods. Especially in the complex branched structure, the processing error parameters are the most, and the processing error will be accumulated, transmitted, and gradually amplified. Therefore, machining error is the main research object. When the main pouring mechanism is designed, there is enough stiffness and no slender rod. The error caused by gravity deformation can be basically ignored. Under the condition of controllable assembly accuracy, the assembly error is also small.(b)
The load deformation (dynamic error) changes with the motion state of the mechanism, mainly including elastic deformation, vibration error, driving, and following error. For the main pouring mechanism, the pouring process is low-speed motion, and the elastic deformation, driving follow-up error, and natural frequency have little effect. However, the clearance of the mechanism is usually pretightening during assembly, and the motion clearance can be basically ignored, and the clearance of the mechanism can also be converted to the geometric error of the mechanism.(c)
thermal deformation error, one kind is from the mechanism of high-speed movement, mainly concentrated in the driver and high-speed movement of the joint friction caused by thermal creep mechanism, usually taking active heat dissipation. Another kind is pouring robot; the object of operation is high temperature object, at the same time affected by high temperature conduction and high temperature radiation; usually the temperature is ladder-like distribution from the heat source. For the casting organization, due to the continuous operation requirements, as well as ladle pouring liquid volume, radiation area is large, and long-time high-temperature radiation closes, although it also can package insulating layer like serial robots, but for the parallel connection of the ladle, thermal radiation according to the thermal insulation layer, a long time by the oven and foundry working environment is generally more bad. It is easy to lose the thermal insulation function due to qualitative change and even fall off. Meanwhile, the movement characteristics of the parallel mechanism are not suitable for the installation of the thermal insulation layer.In summary, the positioning accuracy of main casting mechanism includes geometric error and thermal deformation error. Among them, the geometric error is mainly machining error; thermal deformation error is a single heat source error, mainly thermal radiation.


## 4. Geometric Error Compensation Models

### 4.1. Error Compensation Model of Main Pouring Mechanism

In this paper, the influence coefficient method of D-H matrix is used to explore the geometric error. All the error sources of each branch chain are integrated into the model of the whole mechanism, and then the functional relationship between the error and the end attitude position is constructed. The 4-UPU parallel mechanism in Figure2 can be seen as a multiseries mechanism acting on the same end actuator, and each series mechanism is composed of multiple kinematic pairs. Here, branch chain 1 is taken as an example, as shown in Figure 3, and the motion transfer model is established. Each axis of motion (equivalent to the zej axis of the body coordinate system) can be expressed as unit vectors Se1,0, Se1,0, …, Se1,o; common normal of adjacent motion axes (equivalent to xej axis of body coordinate system) can be expressed as unit vector ae1,O0, ae1,01, …, ae1,6o; αei,j(j+1) represents the torsional angle of adjacent motion axes; θei,j represents the rotation of adjacent common lines, aei,j(j+1) represents the distance between adjacent motion axes; dei,j represents the distance between adjacent common lines. i = 1, 2, 3, 4. j = O, 1, 2, 3, 4, 5, 6, o.Figure 3
D-H coordinate motion transfer diagram of branch 1.Figure3 shows only partial annotations for simplicity. The specific coordinate transformation parameters are shown in Table 1.Table 1
Chain 1 coordinate transformation parameters and physical significance.
Branched chainCoordinate transformation(Δ)aei,(j−1)j(Δ)αei,(j−1)j(Δ)dei,j(Δ)θei,j“Δ” physical meaningBranched chain 1e1O ⟶ e10(Δ)ae1,O0(Δ)αe1,O0(Δ)de1,0(Δ)θe1,0Installation error of upper (fixed) platforme10 ⟶ e11(Δ)ae1, 01(Δ)αe1, 01(Δ)de1,1(Δ)θe1,1Error of universal joint 1e11 ⟶ e12(Δ)ae1,12(Δ)αe1,12(Δ)de1,2(Δ)θe1,2Error of universal joint 1e12 ⟶ e13(Δ)ae1,23(Δ)αe1,23(Δ)de1,3(Δ)θe1,3The error of length rode13 ⟶ e14(Δ)ae1,34(Δ)αe1,34(Δ)de1,4(Δ)θe1,4The error of moving long rode14 ⟶ e15(Δ)ae1,45(Δ)αe1,45(Δ)de1,5(Δ)θe1,5Error of universal joint 2e15 ⟶ e16(Δ)ae1,56(Δ)αe1,56(Δ)de1,6(Δ)θe1,6Error of universal joint 2e16 ⟶ e1o(Δ)ae1,6o(Δ)αe1,6o(Δ)de1,o(Δ)θe1,oInstallation error of lower (moving) platformThe coordinate system changes between each kinematic pair of branch 1 can be written as(1)Tij=aei,jj+1Sei,j×aei,jj+1Sei,j.According to formula (1), the coordinates of any point Pe(xPe, yPe, zPe) in the (7.1)o-xyz coordinate system in the fixed coordinate system O-XYZ can be written as a transfer matrix:(2)Lo=Rn+Ti,oxpe,ype,zpeT.In Formula (2), Rn = dei,0Sei,0 + aei,01ae1,01 + dei,1Sei,1 + aei,12ae1,12 + dei,2Sei,2 + aei,23ae1,23 + dei,3Sei,3 + aei,34+aei,34 + dei,4Sei,4 + aei,45ae1,45 + dei,5Sei,5 + aei,56ae1,56 + dei,6Sei,6 + aei,6oae1,6o。.The recursion formulas ofaei,j(j+1) and Sei,j in Rn are as follows: let X (1, 0, 0), Y (0, 1, 0), and Z (0, 0, 1) in the O-XYZ coordinate system be generalized to the general form:(3)Sei,j=T00−sinαei,j−1jcosαei,j−1j,aei,jj+1=T0cosθei,1cosαei,j−1j·sinθei,jsinαei,j−1j·cosθei,j.When all the structural and kinematic parameters of the branched chain are known, namely,aei,(j−1)j, αei,(j−1)j, dei,j, θei,j, and Sei,j, aei,j(j+1), the spatial position of Pe is uniquely determined. The above derivation process is the theoretical derivation of the end pose, which does not contain any errors. According to the analysis in Section 2, in practice, the machining error and installation error of the main gating mechanism are the main causes of the end pose deviation. Assume that the theoretical and actual error values of Pe can be written as(4)△Pe_p=ΔxΔyΔz,△Pe_a=ΔαΔβΔγ.In formula (4), Δx, Δy, and Δz and Δα, Δβ, and Δγ represent the position and attitude errors of Pe in O-XYZ coordinate system, respectively; the original error groups of parameter groups ae, αe, de, and θe composed of four branched chains can be expressed as Δae, Δαe, Δde, and θe, and because the error value is small, it conforms to the linear superposition theory. A partial differential equation for Pe with respect to aei,(j−1)j, αei,(j−1)j, dei,j and θei,j, combined with Lagrangian first condition, can obtain the expression equation of the total error of the main pouring mechanism as follows:(5)△Pe=GaeΔae+GαeΔαe+GdeΔde+GθeΔθe.We have(6)Gae=GaepGaea,Gαe=GαepGαea,Gde=GdepGdea,Gθe=GθepGθea.G is called the error influence coefficient, and the derivation process of G is complicated, which is not repeated here, and only the final results are listed:(7)Gaep=aei,O0aei,01aei,12aei,23aei,34aei,45aei,56aei,6o,Gαep=aei,O0×Peaei,01×Pe−R0aei,12×Pe−R1aei,23×Pe−R2aei,34×Pe−R3aei,45×Pe−R4aei,56×Pe−R5aei,6o×Pe−R6,Gdep=Sei,0Sei,1Sei,2Sei,3Sei,4Sei,5Sei,6Sei,o,Gθep=Sei,0×Pe−R0Sei,1×Pe−R1Sei,2×Pe−R2Sei,3×Pe−R3Sei,4×Pe−S4Sei,5×Pe−R5Sei,6×Pe−R6Sei,o×Pe−Ro,Gaea=0,Gαea=aei,O0aei,01aei,12aei,23aei,34aei,45aei,56aei,6o,Gdea=0,Gθea=Sei,0Sei,1Sei,2Sei,3Sei,4Sei,5Sei,6Sei,o.From formula (5), the geometric error of the branched chain is expressed. All geometric errors can be mapped to four groups of parameters Δae, Δαe, Δde, and Δθe and can be well corresponding to the four spatial parameter groups ae, αe, de, and θe. The physical meaning of the geometric error mapping is listed in Table 1, which is simple and intuitive and convenient for calculation and error classification.The error modeling method of branches 2∼4 is consistent with that of branch 1.


### 4.2. Simplification of Geometric Error Compensation Model

(a)
In the error model,ei,0∼ei,6 are fixed in the auxiliary coordinates of the kinematic pair. Among them, Sei,0 and the Z-axis of the coordinate system O-XYZ on the platform are default coincidence and have the same origin, but X and aei,O0 do not coincide, so Δaei,O0 = Δαei,O0 = Δde1,0 = 0.(b)
The UPU structure contains five kinematic parameters of RRPRR kinematic pairs, including branched chains 1∼3, △dei,4 are active parts, and branched chains 4, Δθe4,6 are drive parts.Therefore, the kinematic error parameters Δθei,k (k = 2, 3, 5, 6) for the R pair of branched chain 1∼3 are all from the kinematic pair, and the error is actually included in the kinematic parameter θei,k (k = 2, 3, 5, 6). There is no significance in error analysis, and these four parameters can be removed. Similarly, the Δθe4,(2,3,5), △de4,4 of the driven pair of branched chain 4 can also be removed from the kinematic parameter θe4,(2,3,5), de4,4.Table2 lists the simplified parameters in branched chains 1∼4. Each branched chain has 8-9 parameters that can be ignored, which can not only reduce the processing difficulty and reduce the subsequent assembly workload but also save the calculation time.Table 2
Simplified parameters for each branch.
BranchBranchBranchBranchChain 1Chain 2Chain 3Chain 4Δaei,O00000Δαei,O00000Δde1,00000Δθei,2∼6000—Δθe4,(2,3,5)———0△de4,4———0


### 4.3. Numerical Analysis of Geometric Error Compensation Model

In this section, the numerical calculation method is used to analyze the error accuracy. Firstly, appropriate pose samples are selected in the workspace to cover the whole workspace. Then, the influence of each error source on the end pose is analyzed, and the error item is focused on.The design workspace of this paper is as follows:X: −150∼−50 mm, Y: −50∼+50 mm, Z: −825∼−425 mm, and r: −0.4884∼1.08155 rad. Constructional dimension is as follows: M = 374 mm, N = 680 mm, m = 330 mm, and n = 365 mm [16]. In order to take into account the sample capacity and calculation time, the sample size is set to 10000 poses.In general, in standard manufacturing and machining process, the error is controlled in the designed tolerance zone and the accuracy requirements of the parts of the same equipment are basically the same. For objective and fair comparison, in this paper, Δae = Δde = 0.2 mm and Δαe = Δθe = 0.004 rad are set, and the error ΔPe of each error acting independently under all 10000 positions is investigated, and the absolute value is taken.The design accuracy of the main pouring mechanism is in the order of 100 mm, and the geometric error parameters of the whole mechanism are 100. From the perspective of the cumulative error of all geometric parameters, the position error is conservative with 10−3 mm as the order of magnitude boundary, and the attitude error is conservative with 110−5 rad as the order of magnitude boundary. Taking branch 1 as an example, the results are shown in Table 3 (note: in order to express concisely, omit “△”):.Table 3
Influence of geometric parameter error on terminal accuracy.
Δx, Δy, and ΔzΔα, Δβ, and Δγ10−3∼100ae1, 01, ae1, 12, ae1, 2310−5∼100αe1, 01, αe1, 12, αe1, 23, αe1, 34, αe1, 45, αe1, 56, αe1, 6o, θe1, 0, θe1, 1, θe1, 4, θe1, oae1, 34, ae1, 45, ae1, 56ae1, 6o, αe1, 01, αe1, 12αe1, 23, αe1, 34, αe1, 45αe1, 56, de1, 1, de1, 2, de1, 3, de1, 4, de1, 5, de1, 6, de1, o, θe1, 0, θe1, 1, θe1, 410−6∼10−4αe1, 6o, θe1, o10–6ae1, 01, ae1, 12, ae1, 23, ae1, 34, ae1, 45, ae1, 56, ae1, 6o, de1, 1, de1, 2, de1, 3, de1, 4, de1, 5, de1, 6, de1, o∗The error parameter of magnitude 100 mm is as follows: X: Δαe1, 12, Δαe1, 23, Δαe1, 34, Δαe1, 56, Δθe1, 1, Δθe1, 4 and ΔY: Δαe1, 01.According to Table3, most of the geometric parameters have a big influence on the position error and a small influence on the attitude error. The position error at the end of 100 orders of magnitude is all caused by the angle error of the kinematic pair. In the case of conservative classification considering the design accuracy requirements, the influence of the geometric parameter errors of αe1,6o, θe1,o on the end pose can be ignored. Combined with Table 2, a total of eight nonsensitive error parameters are omitted in branch chain 1, which helps to reduce the independent calibration in subsequent studies. The analysis methods of branched chains 2∼4 are completely the same as those of branched chain 1, and they are not repeated.Figure4 shows the box distribution diagram of geometric parameters that affect the positioning accuracy of ladle more than 1 mm. It can be seen that they are all attitude errors and are concentrated in the X and Z directions. Most of the error distributions are relatively uniform; △αe1,56 and △αe1,4 are near the upper boundary.Figure 4
Box diagram of error distribution of some geometric parameters. (a) Error inX direction (mm). (b) Error in Y direction (mm).
(a)(b)


### 4.4. Distribution Probability of Geometric Error

The range of △Pe reflects the influence degree of each geometric error on the end pose, and the analysis of the distribution probability of △Pe can further show the characteristics of each error. In this paper, only seven parameters which have the greatest impact on △Pe in branch 1 of Section 4.3 are taken as examples. The distribution probability of end (moving platform) position error caused by them is analyzed. The result is shown in Figure 5.Figure 5
Distribution probability statistics of the influence of geometric parameter error on the end position.
(a)(b)(c)(d)In Figure5, the probability distribution of the influence of △αe1,23, △αe1,34, △θe1,1, and △αe1,01 on the terminal position error is uniform, and the direction of each error is consistent in the working space, indicating that the influence range on the terminal position error is large and has a certain direction. The probability distribution of △αe1,01 and △θe1,1 is basically the same.The main direction of the influence of △αe1,23, △αe1,34, and △θe1,1 on the pose of the end is X, and the main direction of the influence of △αe1,01 on the pose of the end is Y. In processing and installation, the accuracy of X or Y direction should be focused on to reduce the position error of the end of the main pouring mechanism.The probability distribution of the influence of △αe1,12 on the end position error is uniform, but the error is relatively concentrated and has a certain direction, and the influence range on the end position error is much smaller. The main direction of the influence of △αe1,12 on the end pose is X, and the accuracy of the geometric parameter X direction is mainly controlled when machining and installation, and the error measurement results are reliable.The end position errors caused by △αe1,56 and △θe1,1 are mainly concentrated in −1.3175 mm and 1.3175 mm; in particular, the distribution probability of △αe1,56 is very concentrated. In error measurement, this error usually does not need to be measured and calculated many times, which is easy to obtain.The position error of branched chain 1 in the order of 100 mm and the attitude error in the order of 10−3 rad are all derived from the axis deflection angle and torsion angle error of the kinematic pair, and the angle error is controlled when the key point is in the processing and assembly.


## 4.1. Error Compensation Model of Main Pouring Mechanism

In this paper, the influence coefficient method of D-H matrix is used to explore the geometric error. All the error sources of each branch chain are integrated into the model of the whole mechanism, and then the functional relationship between the error and the end attitude position is constructed. The 4-UPU parallel mechanism in Figure2 can be seen as a multiseries mechanism acting on the same end actuator, and each series mechanism is composed of multiple kinematic pairs. Here, branch chain 1 is taken as an example, as shown in Figure 3, and the motion transfer model is established. Each axis of motion (equivalent to the zej axis of the body coordinate system) can be expressed as unit vectors Se1,0, Se1,0, …, Se1,o; common normal of adjacent motion axes (equivalent to xej axis of body coordinate system) can be expressed as unit vector ae1,O0, ae1,01, …, ae1,6o; αei,j(j+1) represents the torsional angle of adjacent motion axes; θei,j represents the rotation of adjacent common lines, aei,j(j+1) represents the distance between adjacent motion axes; dei,j represents the distance between adjacent common lines. i = 1, 2, 3, 4. j = O, 1, 2, 3, 4, 5, 6, o.Figure 3
D-H coordinate motion transfer diagram of branch 1.Figure3 shows only partial annotations for simplicity. The specific coordinate transformation parameters are shown in Table 1.Table 1
Chain 1 coordinate transformation parameters and physical significance.
Branched chainCoordinate transformation(Δ)aei,(j−1)j(Δ)αei,(j−1)j(Δ)dei,j(Δ)θei,j“Δ” physical meaningBranched chain 1e1O ⟶ e10(Δ)ae1,O0(Δ)αe1,O0(Δ)de1,0(Δ)θe1,0Installation error of upper (fixed) platforme10 ⟶ e11(Δ)ae1, 01(Δ)αe1, 01(Δ)de1,1(Δ)θe1,1Error of universal joint 1e11 ⟶ e12(Δ)ae1,12(Δ)αe1,12(Δ)de1,2(Δ)θe1,2Error of universal joint 1e12 ⟶ e13(Δ)ae1,23(Δ)αe1,23(Δ)de1,3(Δ)θe1,3The error of length rode13 ⟶ e14(Δ)ae1,34(Δ)αe1,34(Δ)de1,4(Δ)θe1,4The error of moving long rode14 ⟶ e15(Δ)ae1,45(Δ)αe1,45(Δ)de1,5(Δ)θe1,5Error of universal joint 2e15 ⟶ e16(Δ)ae1,56(Δ)αe1,56(Δ)de1,6(Δ)θe1,6Error of universal joint 2e16 ⟶ e1o(Δ)ae1,6o(Δ)αe1,6o(Δ)de1,o(Δ)θe1,oInstallation error of lower (moving) platformThe coordinate system changes between each kinematic pair of branch 1 can be written as(1)Tij=aei,jj+1Sei,j×aei,jj+1Sei,j.According to formula (1), the coordinates of any point Pe(xPe, yPe, zPe) in the (7.1)o-xyz coordinate system in the fixed coordinate system O-XYZ can be written as a transfer matrix:(2)Lo=Rn+Ti,oxpe,ype,zpeT.In Formula (2), Rn = dei,0Sei,0 + aei,01ae1,01 + dei,1Sei,1 + aei,12ae1,12 + dei,2Sei,2 + aei,23ae1,23 + dei,3Sei,3 + aei,34+aei,34 + dei,4Sei,4 + aei,45ae1,45 + dei,5Sei,5 + aei,56ae1,56 + dei,6Sei,6 + aei,6oae1,6o。.The recursion formulas ofaei,j(j+1) and Sei,j in Rn are as follows: let X (1, 0, 0), Y (0, 1, 0), and Z (0, 0, 1) in the O-XYZ coordinate system be generalized to the general form:(3)Sei,j=T00−sinαei,j−1jcosαei,j−1j,aei,jj+1=T0cosθei,1cosαei,j−1j·sinθei,jsinαei,j−1j·cosθei,j.When all the structural and kinematic parameters of the branched chain are known, namely,aei,(j−1)j, αei,(j−1)j, dei,j, θei,j, and Sei,j, aei,j(j+1), the spatial position of Pe is uniquely determined. The above derivation process is the theoretical derivation of the end pose, which does not contain any errors. According to the analysis in Section 2, in practice, the machining error and installation error of the main gating mechanism are the main causes of the end pose deviation. Assume that the theoretical and actual error values of Pe can be written as(4)△Pe_p=ΔxΔyΔz,△Pe_a=ΔαΔβΔγ.In formula (4), Δx, Δy, and Δz and Δα, Δβ, and Δγ represent the position and attitude errors of Pe in O-XYZ coordinate system, respectively; the original error groups of parameter groups ae, αe, de, and θe composed of four branched chains can be expressed as Δae, Δαe, Δde, and θe, and because the error value is small, it conforms to the linear superposition theory. A partial differential equation for Pe with respect to aei,(j−1)j, αei,(j−1)j, dei,j and θei,j, combined with Lagrangian first condition, can obtain the expression equation of the total error of the main pouring mechanism as follows:(5)△Pe=GaeΔae+GαeΔαe+GdeΔde+GθeΔθe.We have(6)Gae=GaepGaea,Gαe=GαepGαea,Gde=GdepGdea,Gθe=GθepGθea.G is called the error influence coefficient, and the derivation process of G is complicated, which is not repeated here, and only the final results are listed:(7)Gaep=aei,O0aei,01aei,12aei,23aei,34aei,45aei,56aei,6o,Gαep=aei,O0×Peaei,01×Pe−R0aei,12×Pe−R1aei,23×Pe−R2aei,34×Pe−R3aei,45×Pe−R4aei,56×Pe−R5aei,6o×Pe−R6,Gdep=Sei,0Sei,1Sei,2Sei,3Sei,4Sei,5Sei,6Sei,o,Gθep=Sei,0×Pe−R0Sei,1×Pe−R1Sei,2×Pe−R2Sei,3×Pe−R3Sei,4×Pe−S4Sei,5×Pe−R5Sei,6×Pe−R6Sei,o×Pe−Ro,Gaea=0,Gαea=aei,O0aei,01aei,12aei,23aei,34aei,45aei,56aei,6o,Gdea=0,Gθea=Sei,0Sei,1Sei,2Sei,3Sei,4Sei,5Sei,6Sei,o.From formula (5), the geometric error of the branched chain is expressed. All geometric errors can be mapped to four groups of parameters Δae, Δαe, Δde, and Δθe and can be well corresponding to the four spatial parameter groups ae, αe, de, and θe. The physical meaning of the geometric error mapping is listed in Table 1, which is simple and intuitive and convenient for calculation and error classification.The error modeling method of branches 2∼4 is consistent with that of branch 1.


## 4.2. Simplification of Geometric Error Compensation Model

(a)
In the error model,ei,0∼ei,6 are fixed in the auxiliary coordinates of the kinematic pair. Among them, Sei,0 and the Z-axis of the coordinate system O-XYZ on the platform are default coincidence and have the same origin, but X and aei,O0 do not coincide, so Δaei,O0 = Δαei,O0 = Δde1,0 = 0.(b)
The UPU structure contains five kinematic parameters of RRPRR kinematic pairs, including branched chains 1∼3, △dei,4 are active parts, and branched chains 4, Δθe4,6 are drive parts.Therefore, the kinematic error parameters Δθei,k (k = 2, 3, 5, 6) for the R pair of branched chain 1∼3 are all from the kinematic pair, and the error is actually included in the kinematic parameter θei,k (k = 2, 3, 5, 6). There is no significance in error analysis, and these four parameters can be removed. Similarly, the Δθe4,(2,3,5), △de4,4 of the driven pair of branched chain 4 can also be removed from the kinematic parameter θe4,(2,3,5), de4,4.Table2 lists the simplified parameters in branched chains 1∼4. Each branched chain has 8-9 parameters that can be ignored, which can not only reduce the processing difficulty and reduce the subsequent assembly workload but also save the calculation time.Table 2
Simplified parameters for each branch.
BranchBranchBranchBranchChain 1Chain 2Chain 3Chain 4Δaei,O00000Δαei,O00000Δde1,00000Δθei,2∼6000—Δθe4,(2,3,5)———0△de4,4———0


## 4.3. Numerical Analysis of Geometric Error Compensation Model

In this section, the numerical calculation method is used to analyze the error accuracy. Firstly, appropriate pose samples are selected in the workspace to cover the whole workspace. Then, the influence of each error source on the end pose is analyzed, and the error item is focused on.The design workspace of this paper is as follows:X: −150∼−50 mm, Y: −50∼+50 mm, Z: −825∼−425 mm, and r: −0.4884∼1.08155 rad. Constructional dimension is as follows: M = 374 mm, N = 680 mm, m = 330 mm, and n = 365 mm [16]. In order to take into account the sample capacity and calculation time, the sample size is set to 10000 poses.In general, in standard manufacturing and machining process, the error is controlled in the designed tolerance zone and the accuracy requirements of the parts of the same equipment are basically the same. For objective and fair comparison, in this paper, Δae = Δde = 0.2 mm and Δαe = Δθe = 0.004 rad are set, and the error ΔPe of each error acting independently under all 10000 positions is investigated, and the absolute value is taken.The design accuracy of the main pouring mechanism is in the order of 100 mm, and the geometric error parameters of the whole mechanism are 100. From the perspective of the cumulative error of all geometric parameters, the position error is conservative with 10−3 mm as the order of magnitude boundary, and the attitude error is conservative with 110−5 rad as the order of magnitude boundary. Taking branch 1 as an example, the results are shown in Table 3 (note: in order to express concisely, omit “△”):.Table 3
Influence of geometric parameter error on terminal accuracy.
Δx, Δy, and ΔzΔα, Δβ, and Δγ10−3∼100ae1, 01, ae1, 12, ae1, 2310−5∼100αe1, 01, αe1, 12, αe1, 23, αe1, 34, αe1, 45, αe1, 56, αe1, 6o, θe1, 0, θe1, 1, θe1, 4, θe1, oae1, 34, ae1, 45, ae1, 56ae1, 6o, αe1, 01, αe1, 12αe1, 23, αe1, 34, αe1, 45αe1, 56, de1, 1, de1, 2, de1, 3, de1, 4, de1, 5, de1, 6, de1, o, θe1, 0, θe1, 1, θe1, 410−6∼10−4αe1, 6o, θe1, o10–6ae1, 01, ae1, 12, ae1, 23, ae1, 34, ae1, 45, ae1, 56, ae1, 6o, de1, 1, de1, 2, de1, 3, de1, 4, de1, 5, de1, 6, de1, o∗The error parameter of magnitude 100 mm is as follows: X: Δαe1, 12, Δαe1, 23, Δαe1, 34, Δαe1, 56, Δθe1, 1, Δθe1, 4 and ΔY: Δαe1, 01.According to Table3, most of the geometric parameters have a big influence on the position error and a small influence on the attitude error. The position error at the end of 100 orders of magnitude is all caused by the angle error of the kinematic pair. In the case of conservative classification considering the design accuracy requirements, the influence of the geometric parameter errors of αe1,6o, θe1,o on the end pose can be ignored. Combined with Table 2, a total of eight nonsensitive error parameters are omitted in branch chain 1, which helps to reduce the independent calibration in subsequent studies. The analysis methods of branched chains 2∼4 are completely the same as those of branched chain 1, and they are not repeated.Figure4 shows the box distribution diagram of geometric parameters that affect the positioning accuracy of ladle more than 1 mm. It can be seen that they are all attitude errors and are concentrated in the X and Z directions. Most of the error distributions are relatively uniform; △αe1,56 and △αe1,4 are near the upper boundary.Figure 4
Box diagram of error distribution of some geometric parameters. (a) Error inX direction (mm). (b) Error in Y direction (mm).
(a)(b)


## 4.4. Distribution Probability of Geometric Error

The range of △Pe reflects the influence degree of each geometric error on the end pose, and the analysis of the distribution probability of △Pe can further show the characteristics of each error. In this paper, only seven parameters which have the greatest impact on △Pe in branch 1 of Section 4.3 are taken as examples. The distribution probability of end (moving platform) position error caused by them is analyzed. The result is shown in Figure 5.Figure 5
Distribution probability statistics of the influence of geometric parameter error on the end position.
(a)(b)(c)(d)In Figure5, the probability distribution of the influence of △αe1,23, △αe1,34, △θe1,1, and △αe1,01 on the terminal position error is uniform, and the direction of each error is consistent in the working space, indicating that the influence range on the terminal position error is large and has a certain direction. The probability distribution of △αe1,01 and △θe1,1 is basically the same.The main direction of the influence of △αe1,23, △αe1,34, and △θe1,1 on the pose of the end is X, and the main direction of the influence of △αe1,01 on the pose of the end is Y. In processing and installation, the accuracy of X or Y direction should be focused on to reduce the position error of the end of the main pouring mechanism.The probability distribution of the influence of △αe1,12 on the end position error is uniform, but the error is relatively concentrated and has a certain direction, and the influence range on the end position error is much smaller. The main direction of the influence of △αe1,12 on the end pose is X, and the accuracy of the geometric parameter X direction is mainly controlled when machining and installation, and the error measurement results are reliable.The end position errors caused by △αe1,56 and △θe1,1 are mainly concentrated in −1.3175 mm and 1.3175 mm; in particular, the distribution probability of △αe1,56 is very concentrated. In error measurement, this error usually does not need to be measured and calculated many times, which is easy to obtain.The position error of branched chain 1 in the order of 100 mm and the attitude error in the order of 10−3 rad are all derived from the axis deflection angle and torsion angle error of the kinematic pair, and the angle error is controlled when the key point is in the processing and assembly.


## 5. Analysis of Thermal Deformation Error Compensation Model

Thermal deformation error accounts for about 40%∼70% of the total error and is usually nonlinear and time-varying, and the distribution is extremely complex. Prediction and compensation are the most effective methods to solve the problem of equipment thermal error. For the main pouring mechanism, the heat source is a single molten metal liquid, and the temperature is above 1700°C. Compared with other heat sources, such as motor and gap friction, the influence of heat error can be ignored. However, in multipose measurement, there is a strong correlation between postures, which is easy to produce collinear and coupling problems and reduce the robustness of the thermal error model.In this section, in view of the above problems, firstly, the fuzzy clustering method is used to classify the temperature variables, and then the grey correlation degree method is used to select several groups of data with the highest correlation degree in various categories as samples. Finally, the principal component analysis is used to eliminate the influence of collinearity among independent variables, and the final error model is given by equation transformation.


### 5.1. The Establishment Steps of Thermal Deformation Error Compensation Model

#### 5.1.1. Fuzzy Clustering

The fuzzy similarity matrixRF = [rF,i,j] (i, j denotes the sequence of pose observation sets) is constructed by using correlation coefficient. Let XF = {xF1, xF2, …, xFi} be the set of observed values at i temperature measuring points. xFi = (xF,i1,, xF,i2, …, xF,in) represents the n temperature observations of the ith pose. The values of rF,i,j are obtained by the following formula:(8)rF,i,j=∑k=1nxF,ik−xF,i¯xF,jk−xF,j¯∑k=1nxF,ik−xF,i¯2∑k=1nxF,jk−xF,j¯2.We have(9)xF,i¯=1n∑k=1nxF,ik,xF,j¯=1n∑k=1nxF,jk.After a finite number of operations, when the set residual condition is satisfied,RF2b=RF2b+1 (b = 1, 2, …), and the fuzzy equivalent matrix t(RF) =  RF2b. The temperature measurement pose is grouped by extracting the threshold λF for the point, and the temperature observation values can be divided into two groups.


#### 5.1.2. Grey Relativity

Grey theory is to find out the numerical relationship between the factors of the system. If the trend of the two factors is similar, the grey correlation between them is very high; otherwise, it is low. The formula of grey correlation degree is as follows:(10)ryG,k,xG,ik=miniminkyG,k−xG,ik+ηGmaximaxkyG,k−xG,ikyG,k−xG,ik+ηGmaximaxkyG,k−xG,ik.In equation (10), yG,k is the temperature step, xG,i is the ith observation set of temperature measuring points, yG,k and xG,ik are the kth observation values of temperature step and temperature measuring point i, respectively, and ηG is the resolution coefficient, ηG ∈ [0,1]. Normally, ηG = 0.5 s [17].The greater the grey correlation degree is, the more similar the change trend of temperature measurement error and temperature measurement point are. However, the selection of temperature-sensitive points directly according to the grey correlation degree of error-sensitive pose cannot well reduce the collinearity between errors.


#### 5.1.3. Variance Inflation Factor

Variance expansion factor (VIF) is one of the most commonly used indicators to measure the collinearity between variables. The calculation method of VIF is as follows:(11)VIFi=11−RV,i2=11−RV,j2=VIFj.IfxV,i and xV,j are error sets of position measurement points i and j (i, j = 1, 2, 10, …), the VIF values of xv,i and xv,j are obtained by formula (11).RV,i2 (RV,j2) is the regression model coefficient, xV,i(xV,j) is the dependent variable, xV,j(xV,i) is the independent variable, and the value range is 0∼1. It is generally assumed that there is a serious collinearity between input variables when VIF ≥ 10 [18].


#### 5.1.4. Principal Component Regression (PCR)

Principal component regression is to use the principal component of the original variable instead of the original variable for regression analysis. Most information of the original index can be retained and the influence of collinearity between variables can be eliminated [19, 20].①
Standardized dataXP∗ = (xP,1∗, xP,2∗, …, xP,k∗) is obtained from raw data XP = (xP,1, xP,2, …, xP,k), where xP,i∗ = (xP,i − xP¯)/s (i = 1, 2, …, k); xP¯=∑i=1kxP,i/k;(12)s=∑i=1kxp,i−xP¯2k.②
Correlation matrix for standardized dataRP.(13)RP=ρP,11ρP,12⋯ρP,1kρP,21ρP,22⋯ρP,2k⋮⋮⋯⋮ρP,k1ρP,k1⋯ρP,kk.We haveρP,ij = cov(XP,i∗, XP,j∗)/DXP,i∗DXP,j∗ and cov(XP,i∗,XP,j∗) = E{[XP,i∗ − E(XP,i∗)]·[XP,j∗ − E(XP,j∗)]}, where i, j represent the sequence of pose error sets.③
Obtaining principal components from correlation matrixRP.We useλP,1 ≥ λP,2 ≥ … ≥ λP,k and uP,1 ≥ uP,2 ≥ … ≥ uP,k as the eigenvalues and eigenvectors of RP, so the principal component composition of XP∗ can be written as(14)ZP,1=uP,11xP,1∗+uP,21xP,2∗+⋯+uP,k1xP,k∗⋮ZP,k=uP,1kxP,1∗+uP,2kxP,2∗+⋯+uP,kkxP,k∗..In equation (14), Var (ZP,k) = λP,k and Cov(ZP,k) = 0 (i ≠ j). The cumulative variance contribution of the gth principal component (Vccg) is Vccg=∑igλp,j/∑ikλp,i,g=1,2,...,k, where Vccg  ≥ 85%; it is believed that it can meet the requirements of principal component selection.④
Using standardized explanatory variables and principal componentsZP,1, ZP,2, …, ZP,k for regression modeling.(15)y^P∗=a^P,1ZP,1+a^P,2ZP,2+⋯+a^P,gZP,g.⑤
Combining formulas (14) and (15), we get the standardized variables and the model of xP,1∗, xP,2∗, …, xP,k∗.(16)y^P∗=β^P,1xP,1∗+β^P,2xP,2∗+⋯+β^P,kxP,k∗.The relationship betweenβi coefficient and parameter bi (regression coefficient belonging to the original model, the regression coefficients of xP,1, xP,2, …, xP,k) is(17)bP,i=SP,ySP,iβP,ii=1,2,…,k,bP,0=y¯P−∑i=1kbP,ix¯P,i.In formula (17), SP,y is the standard deviation of the original data yP, and SP,i is the standard deviation of the original data xP,i.


### 5.2. Numerical Analysis of Thermal Deformation Error

Working at high temperatures is very dangerous. In this section, the algorithm is verified by numerical analysis software, and the structure size of the main pouring mechanism is consistent with Section3. In high strength continuous operation, the ladle is usually equipped with insulation system, and the pouring temperature is usually relatively stable. Therefore, the error of the main pouring mechanism in the stable temperature field is analyzed [21].


#### 5.2.1. Heat Conduction Analysis

The thermal conductivity of dry air is 12.5 W/m·°C, the thermal conductivity of steel is 60 W/m·°C, and the thermal conductivity of composite insulation layer is 0.055 W/m·°C. The default heat transfer between components is through contact without considering the heat diffusion. Selecting the liquid as the only heat source, the temperature is 1700°C. The external environment is set to 20°C, the convective heat transfer coefficient (film coefficient) is 12.5 W/m·°C, and all the surfaces except the liquid are selected. The temperature distribution of the main casting mechanism is obtained by solving the calculation only considering the heat conduction effect between the casting parts, as shown in Figure6.Figure 6
Temperature distribution of heat conduction.Figure6 shows the profile of the temperature distribution in the ladle. Under the condition of only considering the heat conduction in the figure, the pouring temperature in the ladle presents a dense ladder-like distribution, and the temperature decreases gradually from the inside to the outside. It can be seen that the temperature is not conducted to the branch chain of the ladle. The edge temperature of the insulation layer is basically close to room temperature, which is a good isolation of the influence of the high-temperature molten pouring on the equipment itself. Therefore, when the deformation of temperature ladle is affected, the influence of heat transfer of ladle on equipment is not considered.


#### 5.2.2. Thermal Radiation + Heat Conduction Analysis

In the above analysis, the thermal radiation is added, and the shape coefficient is 0.8. The surface with the relative angle less than 90° between the upper surface of the molten casting and the parts of the main pouring mechanism is selected as the radiation surface, and the other selected surfaces are still used as the heat exchange surface. The radiation temperature distribution of the main pouring mechanism is obtained by solving the calculation, as shown in Figure7.Figure 7
Temperature distribution of thermal radiation surface.It can be seen from Figure7 that, in the steady state, the local temperature near the pouring is nearly 500°C, and most local temperature is 300°C, which is very huge for the whole mechanism. Further analysis shows that the isotropic thermal expansion coefficient of steel and composite insulation material is 1.2 × 10−5/°C and 2 × 10−6/°C, and the total temperature deformation error is obtained.In Figure8, in the gravity environment, the maximum cumulative error caused by thermal radiation is about 9 mm, and the average error is about 3 mm, and the error has a certain direction because of the arrangement of the mechanism. Therefore, it is necessary to compensate the error caused by thermal radiation and heat conduction.Figure 8
Total temperature deformation.The discussion of thermal error compensation is based on the homogeneity of thermal expansion coefficient, and the ladle is a regular round table, so the thermal expansion is considered to be proportional. In order to simplify the analysis process, take the central part of the ladle, that is, the coordinate centero point of the moving platform analyzed by the parallel mechanism.The molten metal liquid is the only heat source, so the molten metal liquid at different temperatures is selected as the sample. In order to take into account the efficiency and effectiveness of numerical analysis, the dip angle of the ladle is set to 10°C. Ten sample points in the space are randomly selected to form the test points, as shown in Tables4 and 5. The error data of each pose are obtained by numerical analysis, and the results are shown in Figure 9.Table 4
Temperature test points.
Temperature (°C)16001650170017501800185019001950200020502100Table 5
Random points.
Serial numberPositionX (mm)Y (mm)Z (mm)P1−107−1−648P2−95−12−622P3−1093−565P4−10813−614P5−97−10−604P6−85−13−611P7−982−579P8−1100−657P9−86−7−575P1−10412−585Figure 9
Results of thermal error analysis.
(a)(b)(c)Figure9 is the trend of the steady-state error of each temperature. It can be seen that the steady-state error increases gradually with the increase of temperature, which has strong linear regression characteristics. The analysis data also have multiple sets, which are suitable for principal component analysis. In this chapter, the regression model of thermal error in X direction is established with X direction as an example.(1) Fuzzy Clustering Algorithm. Firstly, the temperature error point set in the direction of thermal error X is selected, and the temperature measurement points are grouped by fuzzy clustering analysis. The results are as follows.(2) Grey Correlation Degree and Variance Expansion Factor. In Figure 10, the transverse axis is the grey relation between the thermal error set and the temperature set, that is, the influence weight of the temperature set on the error set. According to the selection principle of error set for the purpose of reducing collinearity, P8 and P5 are suitable to be selected as sensitive error sets. Further analysis shows that although the variance expansion factor of P5 is slightly lower than that of P8, the temperature set has the greatest influence on P8. Therefore, P8 is selected as the error sensitive set in Group 2.Figure 10
Group 2 grey correlation degree and variance expansion factor of errors.(3) Principal Component Regression. Select sensitive variables P1 and P8 to obtain eigenvalues, eigenvectors, and cumulative contribution rate of correlation matrix RP; see Table 6:Table 6
Eigenvalues, eigenvectors, and cumulative contribution rate.
EigenvalueEigenvectorThe contribution rate (%)λP,1 = 1.9398uP,1 = [0.7071–0.7071]96.99λP,2 = 0.0602uP,2 = [0.7071 0.7071]100Expressions of principal componentsZP,1 and ZP,2 are obtained from Table 7.(18)ZP,1=0.7071·P1∗−0.7071·P8∗,ZP,2=0.7071·P1∗+0.7071·P8∗.Table 7
Grouping results.
Group 1Group 2P1P2∼P9At the same time, Table7 shows that the cumulative contribution rate of variance of principal component ZP,1 is 96.99% ≥ 85%, so the number of principal components is 1. Regression modeling of standardized explanatory variable TEMP∗ and principal component ZP,1 is as follows:(19)TEMP∗=−0.6707.Zp,1.In equation (19), there is only one principal component, so there is no collinearity problem. When there are multiple principal components in the equation, these principal components are not correlated after calculation. Bring formula (18) into formula (19):(20)TEMP∗=−0.4743.P1∗+0.4743.p8H∗.According to formula (16), equation (20) is restored to a principal component regression equation composed of original data TEMP, P1, and P8.(21)TEMP=−538.7242·P1+500.0238·P8+1894.8.Thus, the principal component regression equation about temperature set and two sensitive sets is established. However, relying solely on equation (21), the required temperature and error relationship cannot be solved.Further analysis shows that P1 set is in Group 1, only one effective set, P2∼P10 set is in Group 2, P8 is the representative of error sensitivity in Group 2, and there is a certain linear relationship between P1, P8, and TEMP. Therefore, we establish a linear regression equation between TEMP and P1 and between TEMP and P8 and bring it into the principal component regression equation. P8PCR is obtained by TEMP and P1, and vice versa. Then, the factor contribution rate of equation (21) is analyzed to determine the weight coefficients of P1 and P8, and the final compensation error Pend is obtained.In summary, we construct the temperature and error compensation equation as(22)P1=TEMP+237.61135.4,P1PCR=TEMP−1894.8+538.7242·P8500.0238,P8=TEMP+143.61054,P8PCR=TEMP−1894.8+538.7242·P1500.0238,Pend=0.53∗P1PCR+0.47·P8PCR.(4) Accuracy Analysis of Principal Component Regression. Linear regression is a common method to predict linear problems [22]. P1, P8 (linear regression), and Pend (PCR) are used as the predicted values, respectively, and compared with the original data P2∼P7, P9, and P10. The results are shown in Figures 11 and 12.Figure 11
Comparison of predicted mean values of P1, P8, and Pend models.Figure 12
Comparison of prediction standard deviations of P1, P8, and Pend models.Figures11 and 12 are the comparison of the mean and standard deviation of the predicted model and the original error data set, respectively. Since the principal component regression algorithm can eliminate the influence of collinearity among variables, the robustness of the Pend model prediction is greatly improved compared with the prediction model of P8, and it is also significantly improved compared with P1, indicating that the regression model established in this paper is effective. In the practical application of compensation, in order to improve the sensitivity of temperature error, a large number of data of position and attitude points in the workspace should be collected and processed to avoid local optimization of the model and improve the advantages of principal component regression algorithm, so that the model has better predictability and robustness to the global. The derivation process of Y and Z direction thermal error modeling is consistent with the X direction thermal error modeling process, which is no longer redundant.


## 5.1. The Establishment Steps of Thermal Deformation Error Compensation Model

### 5.1.1. Fuzzy Clustering

The fuzzy similarity matrixRF = [rF,i,j] (i, j denotes the sequence of pose observation sets) is constructed by using correlation coefficient. Let XF = {xF1, xF2, …, xFi} be the set of observed values at i temperature measuring points. xFi = (xF,i1,, xF,i2, …, xF,in) represents the n temperature observations of the ith pose. The values of rF,i,j are obtained by the following formula:(8)rF,i,j=∑k=1nxF,ik−xF,i¯xF,jk−xF,j¯∑k=1nxF,ik−xF,i¯2∑k=1nxF,jk−xF,j¯2.We have(9)xF,i¯=1n∑k=1nxF,ik,xF,j¯=1n∑k=1nxF,jk.After a finite number of operations, when the set residual condition is satisfied,RF2b=RF2b+1 (b = 1, 2, …), and the fuzzy equivalent matrix t(RF) =  RF2b. The temperature measurement pose is grouped by extracting the threshold λF for the point, and the temperature observation values can be divided into two groups.


### 5.1.2. Grey Relativity

Grey theory is to find out the numerical relationship between the factors of the system. If the trend of the two factors is similar, the grey correlation between them is very high; otherwise, it is low. The formula of grey correlation degree is as follows:(10)ryG,k,xG,ik=miniminkyG,k−xG,ik+ηGmaximaxkyG,k−xG,ikyG,k−xG,ik+ηGmaximaxkyG,k−xG,ik.In equation (10), yG,k is the temperature step, xG,i is the ith observation set of temperature measuring points, yG,k and xG,ik are the kth observation values of temperature step and temperature measuring point i, respectively, and ηG is the resolution coefficient, ηG ∈ [0,1]. Normally, ηG = 0.5 s [17].The greater the grey correlation degree is, the more similar the change trend of temperature measurement error and temperature measurement point are. However, the selection of temperature-sensitive points directly according to the grey correlation degree of error-sensitive pose cannot well reduce the collinearity between errors.


### 5.1.3. Variance Inflation Factor

Variance expansion factor (VIF) is one of the most commonly used indicators to measure the collinearity between variables. The calculation method of VIF is as follows:(11)VIFi=11−RV,i2=11−RV,j2=VIFj.IfxV,i and xV,j are error sets of position measurement points i and j (i, j = 1, 2, 10, …), the VIF values of xv,i and xv,j are obtained by formula (11).RV,i2 (RV,j2) is the regression model coefficient, xV,i(xV,j) is the dependent variable, xV,j(xV,i) is the independent variable, and the value range is 0∼1. It is generally assumed that there is a serious collinearity between input variables when VIF ≥ 10 [18].


### 5.1.4. Principal Component Regression (PCR)

Principal component regression is to use the principal component of the original variable instead of the original variable for regression analysis. Most information of the original index can be retained and the influence of collinearity between variables can be eliminated [19, 20].①
Standardized dataXP∗ = (xP,1∗, xP,2∗, …, xP,k∗) is obtained from raw data XP = (xP,1, xP,2, …, xP,k), where xP,i∗ = (xP,i − xP¯)/s (i = 1, 2, …, k); xP¯=∑i=1kxP,i/k;(12)s=∑i=1kxp,i−xP¯2k.②
Correlation matrix for standardized dataRP.(13)RP=ρP,11ρP,12⋯ρP,1kρP,21ρP,22⋯ρP,2k⋮⋮⋯⋮ρP,k1ρP,k1⋯ρP,kk.We haveρP,ij = cov(XP,i∗, XP,j∗)/DXP,i∗DXP,j∗ and cov(XP,i∗,XP,j∗) = E{[XP,i∗ − E(XP,i∗)]·[XP,j∗ − E(XP,j∗)]}, where i, j represent the sequence of pose error sets.③
Obtaining principal components from correlation matrixRP.We useλP,1 ≥ λP,2 ≥ … ≥ λP,k and uP,1 ≥ uP,2 ≥ … ≥ uP,k as the eigenvalues and eigenvectors of RP, so the principal component composition of XP∗ can be written as(14)ZP,1=uP,11xP,1∗+uP,21xP,2∗+⋯+uP,k1xP,k∗⋮ZP,k=uP,1kxP,1∗+uP,2kxP,2∗+⋯+uP,kkxP,k∗..In equation (14), Var (ZP,k) = λP,k and Cov(ZP,k) = 0 (i ≠ j). The cumulative variance contribution of the gth principal component (Vccg) is Vccg=∑igλp,j/∑ikλp,i,g=1,2,...,k, where Vccg  ≥ 85%; it is believed that it can meet the requirements of principal component selection.④
Using standardized explanatory variables and principal componentsZP,1, ZP,2, …, ZP,k for regression modeling.(15)y^P∗=a^P,1ZP,1+a^P,2ZP,2+⋯+a^P,gZP,g.⑤
Combining formulas (14) and (15), we get the standardized variables and the model of xP,1∗, xP,2∗, …, xP,k∗.(16)y^P∗=β^P,1xP,1∗+β^P,2xP,2∗+⋯+β^P,kxP,k∗.The relationship betweenβi coefficient and parameter bi (regression coefficient belonging to the original model, the regression coefficients of xP,1, xP,2, …, xP,k) is(17)bP,i=SP,ySP,iβP,ii=1,2,…,k,bP,0=y¯P−∑i=1kbP,ix¯P,i.In formula (17), SP,y is the standard deviation of the original data yP, and SP,i is the standard deviation of the original data xP,i.


## 5.1.1. Fuzzy Clustering

The fuzzy similarity matrixRF = [rF,i,j] (i, j denotes the sequence of pose observation sets) is constructed by using correlation coefficient. Let XF = {xF1, xF2, …, xFi} be the set of observed values at i temperature measuring points. xFi = (xF,i1,, xF,i2, …, xF,in) represents the n temperature observations of the ith pose. The values of rF,i,j are obtained by the following formula:(8)rF,i,j=∑k=1nxF,ik−xF,i¯xF,jk−xF,j¯∑k=1nxF,ik−xF,i¯2∑k=1nxF,jk−xF,j¯2.We have(9)xF,i¯=1n∑k=1nxF,ik,xF,j¯=1n∑k=1nxF,jk.After a finite number of operations, when the set residual condition is satisfied,RF2b=RF2b+1 (b = 1, 2, …), and the fuzzy equivalent matrix t(RF) =  RF2b. The temperature measurement pose is grouped by extracting the threshold λF for the point, and the temperature observation values can be divided into two groups.


## 5.1.2. Grey Relativity

Grey theory is to find out the numerical relationship between the factors of the system. If the trend of the two factors is similar, the grey correlation between them is very high; otherwise, it is low. The formula of grey correlation degree is as follows:(10)ryG,k,xG,ik=miniminkyG,k−xG,ik+ηGmaximaxkyG,k−xG,ikyG,k−xG,ik+ηGmaximaxkyG,k−xG,ik.In equation (10), yG,k is the temperature step, xG,i is the ith observation set of temperature measuring points, yG,k and xG,ik are the kth observation values of temperature step and temperature measuring point i, respectively, and ηG is the resolution coefficient, ηG ∈ [0,1]. Normally, ηG = 0.5 s [17].The greater the grey correlation degree is, the more similar the change trend of temperature measurement error and temperature measurement point are. However, the selection of temperature-sensitive points directly according to the grey correlation degree of error-sensitive pose cannot well reduce the collinearity between errors.


## 5.1.3. Variance Inflation Factor

Variance expansion factor (VIF) is one of the most commonly used indicators to measure the collinearity between variables. The calculation method of VIF is as follows:(11)VIFi=11−RV,i2=11−RV,j2=VIFj.IfxV,i and xV,j are error sets of position measurement points i and j (i, j = 1, 2, 10, …), the VIF values of xv,i and xv,j are obtained by formula (11).RV,i2 (RV,j2) is the regression model coefficient, xV,i(xV,j) is the dependent variable, xV,j(xV,i) is the independent variable, and the value range is 0∼1. It is generally assumed that there is a serious collinearity between input variables when VIF ≥ 10 [18].


## 5.1.4. Principal Component Regression (PCR)

Principal component regression is to use the principal component of the original variable instead of the original variable for regression analysis. Most information of the original index can be retained and the influence of collinearity between variables can be eliminated [19, 20].①
Standardized dataXP∗ = (xP,1∗, xP,2∗, …, xP,k∗) is obtained from raw data XP = (xP,1, xP,2, …, xP,k), where xP,i∗ = (xP,i − xP¯)/s (i = 1, 2, …, k); xP¯=∑i=1kxP,i/k;(12)s=∑i=1kxp,i−xP¯2k.②
Correlation matrix for standardized dataRP.(13)RP=ρP,11ρP,12⋯ρP,1kρP,21ρP,22⋯ρP,2k⋮⋮⋯⋮ρP,k1ρP,k1⋯ρP,kk.We haveρP,ij = cov(XP,i∗, XP,j∗)/DXP,i∗DXP,j∗ and cov(XP,i∗,XP,j∗) = E{[XP,i∗ − E(XP,i∗)]·[XP,j∗ − E(XP,j∗)]}, where i, j represent the sequence of pose error sets.③
Obtaining principal components from correlation matrixRP.We useλP,1 ≥ λP,2 ≥ … ≥ λP,k and uP,1 ≥ uP,2 ≥ … ≥ uP,k as the eigenvalues and eigenvectors of RP, so the principal component composition of XP∗ can be written as(14)ZP,1=uP,11xP,1∗+uP,21xP,2∗+⋯+uP,k1xP,k∗⋮ZP,k=uP,1kxP,1∗+uP,2kxP,2∗+⋯+uP,kkxP,k∗..In equation (14), Var (ZP,k) = λP,k and Cov(ZP,k) = 0 (i ≠ j). The cumulative variance contribution of the gth principal component (Vccg) is Vccg=∑igλp,j/∑ikλp,i,g=1,2,...,k, where Vccg  ≥ 85%; it is believed that it can meet the requirements of principal component selection.④
Using standardized explanatory variables and principal componentsZP,1, ZP,2, …, ZP,k for regression modeling.(15)y^P∗=a^P,1ZP,1+a^P,2ZP,2+⋯+a^P,gZP,g.⑤
Combining formulas (14) and (15), we get the standardized variables and the model of xP,1∗, xP,2∗, …, xP,k∗.(16)y^P∗=β^P,1xP,1∗+β^P,2xP,2∗+⋯+β^P,kxP,k∗.The relationship betweenβi coefficient and parameter bi (regression coefficient belonging to the original model, the regression coefficients of xP,1, xP,2, …, xP,k) is(17)bP,i=SP,ySP,iβP,ii=1,2,…,k,bP,0=y¯P−∑i=1kbP,ix¯P,i.In formula (17), SP,y is the standard deviation of the original data yP, and SP,i is the standard deviation of the original data xP,i.


## 5.2. Numerical Analysis of Thermal Deformation Error

Working at high temperatures is very dangerous. In this section, the algorithm is verified by numerical analysis software, and the structure size of the main pouring mechanism is consistent with Section3. In high strength continuous operation, the ladle is usually equipped with insulation system, and the pouring temperature is usually relatively stable. Therefore, the error of the main pouring mechanism in the stable temperature field is analyzed [21].


### 5.2.1. Heat Conduction Analysis

The thermal conductivity of dry air is 12.5 W/m·°C, the thermal conductivity of steel is 60 W/m·°C, and the thermal conductivity of composite insulation layer is 0.055 W/m·°C. The default heat transfer between components is through contact without considering the heat diffusion. Selecting the liquid as the only heat source, the temperature is 1700°C. The external environment is set to 20°C, the convective heat transfer coefficient (film coefficient) is 12.5 W/m·°C, and all the surfaces except the liquid are selected. The temperature distribution of the main casting mechanism is obtained by solving the calculation only considering the heat conduction effect between the casting parts, as shown in Figure6.Figure 6
Temperature distribution of heat conduction.Figure6 shows the profile of the temperature distribution in the ladle. Under the condition of only considering the heat conduction in the figure, the pouring temperature in the ladle presents a dense ladder-like distribution, and the temperature decreases gradually from the inside to the outside. It can be seen that the temperature is not conducted to the branch chain of the ladle. The edge temperature of the insulation layer is basically close to room temperature, which is a good isolation of the influence of the high-temperature molten pouring on the equipment itself. Therefore, when the deformation of temperature ladle is affected, the influence of heat transfer of ladle on equipment is not considered.


### 5.2.2. Thermal Radiation + Heat Conduction Analysis

In the above analysis, the thermal radiation is added, and the shape coefficient is 0.8. The surface with the relative angle less than 90° between the upper surface of the molten casting and the parts of the main pouring mechanism is selected as the radiation surface, and the other selected surfaces are still used as the heat exchange surface. The radiation temperature distribution of the main pouring mechanism is obtained by solving the calculation, as shown in Figure7.Figure 7
Temperature distribution of thermal radiation surface.It can be seen from Figure7 that, in the steady state, the local temperature near the pouring is nearly 500°C, and most local temperature is 300°C, which is very huge for the whole mechanism. Further analysis shows that the isotropic thermal expansion coefficient of steel and composite insulation material is 1.2 × 10−5/°C and 2 × 10−6/°C, and the total temperature deformation error is obtained.In Figure8, in the gravity environment, the maximum cumulative error caused by thermal radiation is about 9 mm, and the average error is about 3 mm, and the error has a certain direction because of the arrangement of the mechanism. Therefore, it is necessary to compensate the error caused by thermal radiation and heat conduction.Figure 8
Total temperature deformation.The discussion of thermal error compensation is based on the homogeneity of thermal expansion coefficient, and the ladle is a regular round table, so the thermal expansion is considered to be proportional. In order to simplify the analysis process, take the central part of the ladle, that is, the coordinate centero point of the moving platform analyzed by the parallel mechanism.The molten metal liquid is the only heat source, so the molten metal liquid at different temperatures is selected as the sample. In order to take into account the efficiency and effectiveness of numerical analysis, the dip angle of the ladle is set to 10°C. Ten sample points in the space are randomly selected to form the test points, as shown in Tables4 and 5. The error data of each pose are obtained by numerical analysis, and the results are shown in Figure 9.Table 4
Temperature test points.
Temperature (°C)16001650170017501800185019001950200020502100Table 5
Random points.
Serial numberPositionX (mm)Y (mm)Z (mm)P1−107−1−648P2−95−12−622P3−1093−565P4−10813−614P5−97−10−604P6−85−13−611P7−982−579P8−1100−657P9−86−7−575P1−10412−585Figure 9
Results of thermal error analysis.
(a)(b)(c)Figure9 is the trend of the steady-state error of each temperature. It can be seen that the steady-state error increases gradually with the increase of temperature, which has strong linear regression characteristics. The analysis data also have multiple sets, which are suitable for principal component analysis. In this chapter, the regression model of thermal error in X direction is established with X direction as an example.(1) Fuzzy Clustering Algorithm. Firstly, the temperature error point set in the direction of thermal error X is selected, and the temperature measurement points are grouped by fuzzy clustering analysis. The results are as follows.(2) Grey Correlation Degree and Variance Expansion Factor. In Figure 10, the transverse axis is the grey relation between the thermal error set and the temperature set, that is, the influence weight of the temperature set on the error set. According to the selection principle of error set for the purpose of reducing collinearity, P8 and P5 are suitable to be selected as sensitive error sets. Further analysis shows that although the variance expansion factor of P5 is slightly lower than that of P8, the temperature set has the greatest influence on P8. Therefore, P8 is selected as the error sensitive set in Group 2.Figure 10
Group 2 grey correlation degree and variance expansion factor of errors.(3) Principal Component Regression. Select sensitive variables P1 and P8 to obtain eigenvalues, eigenvectors, and cumulative contribution rate of correlation matrix RP; see Table 6:Table 6
Eigenvalues, eigenvectors, and cumulative contribution rate.
EigenvalueEigenvectorThe contribution rate (%)λP,1 = 1.9398uP,1 = [0.7071–0.7071]96.99λP,2 = 0.0602uP,2 = [0.7071 0.7071]100Expressions of principal componentsZP,1 and ZP,2 are obtained from Table 7.(18)ZP,1=0.7071·P1∗−0.7071·P8∗,ZP,2=0.7071·P1∗+0.7071·P8∗.Table 7
Grouping results.
Group 1Group 2P1P2∼P9At the same time, Table7 shows that the cumulative contribution rate of variance of principal component ZP,1 is 96.99% ≥ 85%, so the number of principal components is 1. Regression modeling of standardized explanatory variable TEMP∗ and principal component ZP,1 is as follows:(19)TEMP∗=−0.6707.Zp,1.In equation (19), there is only one principal component, so there is no collinearity problem. When there are multiple principal components in the equation, these principal components are not correlated after calculation. Bring formula (18) into formula (19):(20)TEMP∗=−0.4743.P1∗+0.4743.p8H∗.According to formula (16), equation (20) is restored to a principal component regression equation composed of original data TEMP, P1, and P8.(21)TEMP=−538.7242·P1+500.0238·P8+1894.8.Thus, the principal component regression equation about temperature set and two sensitive sets is established. However, relying solely on equation (21), the required temperature and error relationship cannot be solved.Further analysis shows that P1 set is in Group 1, only one effective set, P2∼P10 set is in Group 2, P8 is the representative of error sensitivity in Group 2, and there is a certain linear relationship between P1, P8, and TEMP. Therefore, we establish a linear regression equation between TEMP and P1 and between TEMP and P8 and bring it into the principal component regression equation. P8PCR is obtained by TEMP and P1, and vice versa. Then, the factor contribution rate of equation (21) is analyzed to determine the weight coefficients of P1 and P8, and the final compensation error Pend is obtained.In summary, we construct the temperature and error compensation equation as(22)P1=TEMP+237.61135.4,P1PCR=TEMP−1894.8+538.7242·P8500.0238,P8=TEMP+143.61054,P8PCR=TEMP−1894.8+538.7242·P1500.0238,Pend=0.53∗P1PCR+0.47·P8PCR.(4) Accuracy Analysis of Principal Component Regression. Linear regression is a common method to predict linear problems [22]. P1, P8 (linear regression), and Pend (PCR) are used as the predicted values, respectively, and compared with the original data P2∼P7, P9, and P10. The results are shown in Figures 11 and 12.Figure 11
Comparison of predicted mean values of P1, P8, and Pend models.Figure 12
Comparison of prediction standard deviations of P1, P8, and Pend models.Figures11 and 12 are the comparison of the mean and standard deviation of the predicted model and the original error data set, respectively. Since the principal component regression algorithm can eliminate the influence of collinearity among variables, the robustness of the Pend model prediction is greatly improved compared with the prediction model of P8, and it is also significantly improved compared with P1, indicating that the regression model established in this paper is effective. In the practical application of compensation, in order to improve the sensitivity of temperature error, a large number of data of position and attitude points in the workspace should be collected and processed to avoid local optimization of the model and improve the advantages of principal component regression algorithm, so that the model has better predictability and robustness to the global. The derivation process of Y and Z direction thermal error modeling is consistent with the X direction thermal error modeling process, which is no longer redundant.


## 5.2.1. Heat Conduction Analysis

The thermal conductivity of dry air is 12.5 W/m·°C, the thermal conductivity of steel is 60 W/m·°C, and the thermal conductivity of composite insulation layer is 0.055 W/m·°C. The default heat transfer between components is through contact without considering the heat diffusion. Selecting the liquid as the only heat source, the temperature is 1700°C. The external environment is set to 20°C, the convective heat transfer coefficient (film coefficient) is 12.5 W/m·°C, and all the surfaces except the liquid are selected. The temperature distribution of the main casting mechanism is obtained by solving the calculation only considering the heat conduction effect between the casting parts, as shown in Figure6.Figure 6
Temperature distribution of heat conduction.Figure6 shows the profile of the temperature distribution in the ladle. Under the condition of only considering the heat conduction in the figure, the pouring temperature in the ladle presents a dense ladder-like distribution, and the temperature decreases gradually from the inside to the outside. It can be seen that the temperature is not conducted to the branch chain of the ladle. The edge temperature of the insulation layer is basically close to room temperature, which is a good isolation of the influence of the high-temperature molten pouring on the equipment itself. Therefore, when the deformation of temperature ladle is affected, the influence of heat transfer of ladle on equipment is not considered.


## 5.2.2. Thermal Radiation + Heat Conduction Analysis

In the above analysis, the thermal radiation is added, and the shape coefficient is 0.8. The surface with the relative angle less than 90° between the upper surface of the molten casting and the parts of the main pouring mechanism is selected as the radiation surface, and the other selected surfaces are still used as the heat exchange surface. The radiation temperature distribution of the main pouring mechanism is obtained by solving the calculation, as shown in Figure7.Figure 7
Temperature distribution of thermal radiation surface.It can be seen from Figure7 that, in the steady state, the local temperature near the pouring is nearly 500°C, and most local temperature is 300°C, which is very huge for the whole mechanism. Further analysis shows that the isotropic thermal expansion coefficient of steel and composite insulation material is 1.2 × 10−5/°C and 2 × 10−6/°C, and the total temperature deformation error is obtained.In Figure8, in the gravity environment, the maximum cumulative error caused by thermal radiation is about 9 mm, and the average error is about 3 mm, and the error has a certain direction because of the arrangement of the mechanism. Therefore, it is necessary to compensate the error caused by thermal radiation and heat conduction.Figure 8
Total temperature deformation.The discussion of thermal error compensation is based on the homogeneity of thermal expansion coefficient, and the ladle is a regular round table, so the thermal expansion is considered to be proportional. In order to simplify the analysis process, take the central part of the ladle, that is, the coordinate centero point of the moving platform analyzed by the parallel mechanism.The molten metal liquid is the only heat source, so the molten metal liquid at different temperatures is selected as the sample. In order to take into account the efficiency and effectiveness of numerical analysis, the dip angle of the ladle is set to 10°C. Ten sample points in the space are randomly selected to form the test points, as shown in Tables4 and 5. The error data of each pose are obtained by numerical analysis, and the results are shown in Figure 9.Table 4
Temperature test points.
Temperature (°C)16001650170017501800185019001950200020502100Table 5
Random points.
Serial numberPositionX (mm)Y (mm)Z (mm)P1−107−1−648P2−95−12−622P3−1093−565P4−10813−614P5−97−10−604P6−85−13−611P7−982−579P8−1100−657P9−86−7−575P1−10412−585Figure 9
Results of thermal error analysis.
(a)(b)(c)Figure9 is the trend of the steady-state error of each temperature. It can be seen that the steady-state error increases gradually with the increase of temperature, which has strong linear regression characteristics. The analysis data also have multiple sets, which are suitable for principal component analysis. In this chapter, the regression model of thermal error in X direction is established with X direction as an example.(1) Fuzzy Clustering Algorithm. Firstly, the temperature error point set in the direction of thermal error X is selected, and the temperature measurement points are grouped by fuzzy clustering analysis. The results are as follows.(2) Grey Correlation Degree and Variance Expansion Factor. In Figure 10, the transverse axis is the grey relation between the thermal error set and the temperature set, that is, the influence weight of the temperature set on the error set. According to the selection principle of error set for the purpose of reducing collinearity, P8 and P5 are suitable to be selected as sensitive error sets. Further analysis shows that although the variance expansion factor of P5 is slightly lower than that of P8, the temperature set has the greatest influence on P8. Therefore, P8 is selected as the error sensitive set in Group 2.Figure 10
Group 2 grey correlation degree and variance expansion factor of errors.(3) Principal Component Regression. Select sensitive variables P1 and P8 to obtain eigenvalues, eigenvectors, and cumulative contribution rate of correlation matrix RP; see Table 6:Table 6
Eigenvalues, eigenvectors, and cumulative contribution rate.
EigenvalueEigenvectorThe contribution rate (%)λP,1 = 1.9398uP,1 = [0.7071–0.7071]96.99λP,2 = 0.0602uP,2 = [0.7071 0.7071]100Expressions of principal componentsZP,1 and ZP,2 are obtained from Table 7.(18)ZP,1=0.7071·P1∗−0.7071·P8∗,ZP,2=0.7071·P1∗+0.7071·P8∗.Table 7
Grouping results.
Group 1Group 2P1P2∼P9At the same time, Table7 shows that the cumulative contribution rate of variance of principal component ZP,1 is 96.99% ≥ 85%, so the number of principal components is 1. Regression modeling of standardized explanatory variable TEMP∗ and principal component ZP,1 is as follows:(19)TEMP∗=−0.6707.Zp,1.In equation (19), there is only one principal component, so there is no collinearity problem. When there are multiple principal components in the equation, these principal components are not correlated after calculation. Bring formula (18) into formula (19):(20)TEMP∗=−0.4743.P1∗+0.4743.p8H∗.According to formula (16), equation (20) is restored to a principal component regression equation composed of original data TEMP, P1, and P8.(21)TEMP=−538.7242·P1+500.0238·P8+1894.8.Thus, the principal component regression equation about temperature set and two sensitive sets is established. However, relying solely on equation (21), the required temperature and error relationship cannot be solved.Further analysis shows that P1 set is in Group 1, only one effective set, P2∼P10 set is in Group 2, P8 is the representative of error sensitivity in Group 2, and there is a certain linear relationship between P1, P8, and TEMP. Therefore, we establish a linear regression equation between TEMP and P1 and between TEMP and P8 and bring it into the principal component regression equation. P8PCR is obtained by TEMP and P1, and vice versa. Then, the factor contribution rate of equation (21) is analyzed to determine the weight coefficients of P1 and P8, and the final compensation error Pend is obtained.In summary, we construct the temperature and error compensation equation as(22)P1=TEMP+237.61135.4,P1PCR=TEMP−1894.8+538.7242·P8500.0238,P8=TEMP+143.61054,P8PCR=TEMP−1894.8+538.7242·P1500.0238,Pend=0.53∗P1PCR+0.47·P8PCR.(4) Accuracy Analysis of Principal Component Regression. Linear regression is a common method to predict linear problems [22]. P1, P8 (linear regression), and Pend (PCR) are used as the predicted values, respectively, and compared with the original data P2∼P7, P9, and P10. The results are shown in Figures 11 and 12.Figure 11
Comparison of predicted mean values of P1, P8, and Pend models.Figure 12
Comparison of prediction standard deviations of P1, P8, and Pend models.Figures11 and 12 are the comparison of the mean and standard deviation of the predicted model and the original error data set, respectively. Since the principal component regression algorithm can eliminate the influence of collinearity among variables, the robustness of the Pend model prediction is greatly improved compared with the prediction model of P8, and it is also significantly improved compared with P1, indicating that the regression model established in this paper is effective. In the practical application of compensation, in order to improve the sensitivity of temperature error, a large number of data of position and attitude points in the workspace should be collected and processed to avoid local optimization of the model and improve the advantages of principal component regression algorithm, so that the model has better predictability and robustness to the global. The derivation process of Y and Z direction thermal error modeling is consistent with the X direction thermal error modeling process, which is no longer redundant.


## 6. Obtaining the Actual Pose of the Moving Platform of the Main Pouring Mechanism

As shown in Figure13, the solving process of the actual position and orientation of the moving platform of the main pouring mechanism is as follows. First, the theoretical position of the input and the temperature of the pouring liquid are obtained. Then, the theoretical output of each drive is calculated through the kinematic model. The geometric error compensation model is used to calculate the geometric error of the end position and orientation and the thermal deformation error compensation model is used to calculate the thermal deformation. The actual position of the moving platform of the main pouring mechanism can be obtained by “+” of the three phases.Figure 13
Solution process of actual position of moving platform of main pouring mechanism.The actual parameters of the geometric error compensation model can be obtained by measuring method, neural network identification, and least squares parameter estimation. The actual sample points of the thermal deformation error compensation model can be obtained by using three-coordinate measuring instruments and temperature sensors.


## 7. Conclusion

(1)
The geometric error, load error, and thermal deformation error of the main pouring mechanism are analyzed in detail. The influence of load error on the low-speed mechanism is ignored. It is clear that the machining error in the geometric error is the main error source, and the thermal deformation error has the characteristics of heat source concentration and single heat source.(2)
The combined error transfer model including machining and installation errors is constructed, and the error is projected into the model parameters to make it have practical physical significance. Taking branch 1 as an example, it is found that the influence range of position error is wider than that of attitude error, but the position error of 100 mm magnitude and attitude error of 10−3 rad magnitude are derived from the axis deflection angle and torsional angle error of the kinematic pair, indicating that the geometric error is sensitive to the angle error. The error distribution probability and analysis of its error distribution characteristics provide a research basis for the later error parameter calibration.(3)
In view of the characteristics of single heat source and high collinearity of the main pouring mechanism, a joint regression model based on sensitive error set is established by using fuzzy clustering, grey correlation degree, and principal component analysis to correlate all sensitive errors with temperature. Furthermore, on the basis of considering the contribution rate of component error, a method to determine the final error by principal component contribution rate is proposed. Numerical analysis verifies that the thermal deformation error analysis method proposed in this paper has better prediction accuracy and effect than the single thermal deformation error regression model.



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*Source: 6140729-2022-04-13.xml*