# Error Analysis and Error Allocation for Turntable Systems Used in GyroWheel Calibration Tests

Authors: Yuyu Zhao; Hui Zhao; Xin Huo; Yu Yao
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

Calibration tests are of great importance to ensure rate-sensing accuracy of GyroWheel, an innovative attitude determination and control device. In the process of calibration tests, turntable errors are inevitable, which hinder the calibration accuracy and rate-sensing capability. Hence, error analysis for GyroWheel calibration tests is conducted, and the relationship between the calibration accuracy and the orientation error is established based on analytical derivation and numerical simulations. Subsequently, an error model of the turntable system is derived using rigid body kinematics, by which the relationship between the orientation error and turntable errors is described. According to sensitivity analysis and manufacturing capability, an error allocation method is proposed to determine the accuracy requirement of the test turntable, and the effectiveness of the proposed method is verified by repeated simulation tests. Based on the presented analysis and proposed method in this paper, the effects of various turntable errors on the calibration accuracy can be obtained quantitatively, and a theoretical basis for the determination of the turntable accuracy is provided, which are of great significance to guide the calibration tests and improve the calibration accuracy of GyroWheel.


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## Body

## 1. Introduction

The development of small spacecrafts has received a lot of attention in recent years [1, 2]. As GyroWheel is an innovative attitude determination and control device, it offers the potential to meet the performance, mass, and cost requirements of small spacecrafts. It provides control torques about three axes while also measuring the spacecraft angular rates about the two axes perpendicular to the spin direction, which improves the integration and efficiency of attitude control system in small spacecrafts [3, 4].The conception of GyroWheel is inspired by a dynamically tuned gyroscope (DTG). However, it has a larger rotor and tilt angles, as well as a time-varying spinning rate due to its multifunction capability. When the GyroWheel is used to measure angular rates, erroneous torques, which are often caused by design limitation and constructional deficiencies, act on the rotor of the GyroWheel. These imperfections give rise to precession of the rotor, resulting in measurement errors of angular rates [5–7].To maintain high-accuracy measurement, calibration tests are of great importance in the application of the GyroWheel. The goal of calibration tests is to fully characterize the outputs of the inertial instruments so that a nonideal behavior can be modelled and compensated [5, 7–10]. Generally, calibration tests are carried out using a turntable to provide the intended orientation of the GyroWheel. Turntable errors, including position error, wobble error, orthogonality error, and intersection error, will inevitably affect the GyroWheel system in calibration tests [11]. Although the effects of turntable errors can be reduced by using a high-accuracy turntable, the costs of calibration tests will increase significantly. To reconcile the requirement of calibration accuracy and test costs, the effects of turntable errors on calibration accuracy should be analyzed, and error allocation methods for turntable system should be investigated in consideration of the sensitivities of these errors and the manufacturing capability in practical engineering before calibration tests are carried out.Generally, error models of multi-DOF motion systems, such as machine tools and turntables, can be derived using rigid body kinematics [11–14]. Additionally, a few studies about error effect analysis for calibration tests have been conducted. An error analysis of precision centrifuge, an equipment used in calibration tests, was conducted, and its effect on the calibration accuracy of gyro accelerometers was discussed in [15, 16]. The effect of position error on calibration accuracy of inertial instruments has been analyzed in [17, 18]. As mentioned in [19], the models of a turntable’s orthogonality error and horizontality error were established, and quantitative analyses of their effects on calibration accuracy were presented. However, the existing studies have mainly analyzed a single error component’s effect on calibration accuracy, in which other error factors are not considered; obviously, this is not consistent with the actual situation. To the authors’ knowledge, little attention has been focused on the error allocation problem for the multi-DOF motion systems. By assuming that each of the errors contributes equally to the overall error, the error allocation problem for a 6-DOF parallel manipulator was discussed in [20, 21]. However, the assumption of equivalent effects of the errors is inappropriate for the turntable system. Although orthogonal experimental design has been used to determine the significant level of each error factor for a 3-DOF robotic mechanism [22], it is difficult to obtain a quantitative error allocation result. There were no relevant studies on the error allocation method for the turntable system used in calibration tests.Motivated by these facts, the error analysis and error allocation problems of the turntable used in GyroWheel’s calibration tests are investigated in this paper. According to the multiposition calibration theory and rate-sensing model of the GyroWheel, error analysis for GyroWheel calibration tests is conducted quantitatively. With numerical simulations, the relationships between the rate-sensing errors of the GyroWheel and the orientation errors are obtained. The orientation error due to manufacture imperfections and design limitations of the turntable system is modelled based on rigid body kinematics, with consideration of the interaction of these turntable errors. Then an error allocation method for the turntable system based on sensitivity analysis is proposed, and the effectiveness of the proposed method is verified by repeated simulation tests.The remainder of this paper is as follows. In Section2, the GyroWheel calibration theory is described, and the effect of orientation error on the calibration accuracy is analyzed. In Section 3, the error model of a two-axis turntable system is established based on rigid body kinematics, which develops a relationship between the turntable error components and the orientation error of the GyroWheel. In Section 4, sensitivities of the orientation error to the individual error components and the manufacturing capability are discussed. On this basis, an error allocation method is given to determine the accuracy requirement of the test turntable. And several concluding remarks are given in Section 5.


## 2. GyroWheel Calibration Theory and Error Analysis for Calibration Tests

### 2.1. Overview of GyroWheel Calibration Theory

The GyroWheel is an innovative attitude determination and control instrument. A cutaway isometric view of the GyroWheel is shown in Figure1. When the GyroWheel is used to measure angular rates, erroneous torques give rise to precession of the rotor resulting in drift errors. In the null tilt condition, the rate-sensing equations of the GyroWheel are given in the following equation, with consideration of the drift errors [5]:
(1)ωx=kyiy−DxF−Dxxax−Dxyay−Dxzaz−Dxxxax2−Dxzzaz2−Dxxyaxay,ωy=kxix−DyF−Dyxax−Dyyay−Dyzaz−Dyyyay2−Dyzzaz2−Dyyxaxay,where Dx,Dy represent the drift errors of x,y axes, DxF,DyF are g insensitive terms, Dxi,Dyi,i=x,y,z, are g sensitive error coefficients, and Dxij,ij=xx,zz,xy, Dyij,ij=yy,zz,yx, are g2 sensitive error coefficients. ax,ay,az represent acceleration components of the gravity vector, ωx,ωy are external angular velocities, kx,ky are torque scale factors of the GyroWheel, and ix,iy are the currents in the torque coils.Figure 1
Cutaway view of a GyroWheel.In an effort to improve the rate-sensing accuracy of the GyroWheel, multiposition tests are performed to calibrate the GyroWheel. Multiposition tests make use of a two-axis turntable to provide the intended orientation for the GyroWheel. A schematic representation of a turntable is shown in Figure2. The earth’s rotation rate and gravitational acceleration are regarded as the nominal inputs of the GyroWheel. At each position, an equation expressed as (1) can be obtained using the inputs and outputs of the GyroWheel. Therefore, the error coefficients can be calculated when enough tests of different positions are conducted.Figure 2
Schematic representation of a two-axis turntable.Take thex-axis model of the GyroWheel as an example. According to the rate-sensing model of the GyroWheel and multiposition calibration theory [10], we have
(2)kyiy1kyiy2⋮kyiyn−ωx1ωx2⋮ωxn=1ax1ay1az1ax12az12ax1ay11ax2ay2az2ax22az22ax2ay2⋮⋮⋮⋮⋮⋮⋮1axnaynaznaxn2azn2axnaynDxFDxxDxyDxzDxxxDxzzDxxy,where n is the number of test positions, iyj,j=1,2,…,n, represent the currents in the torque coils, ωxi,i=1,2,…,n, represent the earth rate components of each test position, and axi,ayi,azi,i=1,2,…,n, represent the gravitational acceleration components of each test position.The initial orientation of the GyroWheel axes is north-west-up. Define thatωe is the earth’s rotation rate, λ is the latitude, ϕ1i,ϕ2i are the rotation angles of the turntable, and the subscript i means the ith test position. The gravitational acceleration components and the earth rate components can be expressed as follows:
(3)axi=−sinϕ1icosϕ2i,ayi=sinϕ1isinϕ2i,azi=cosϕ1i,ωxi=ωecosϕ2icosϕ1icosλ−sinϕ1isinλ,ωyi=ωesinϕ2i−cosϕ1icosλ+sinϕ1isinλ,ωzi=ωesinϕ1icosλ+cosϕ1isinλ.Denote(4)b1=kyiy1kyiy2⋮kyiyn,b2=ωx1ωx2⋮ωxn,A=1ax1ay1az1ax12az12ax1ay11ax2ay2az2ax22az22ax2ay2⋮⋮⋮⋮⋮⋮⋮1axnaynaznaxn2azn2axnayn,β=DxFDxxDxyDxzDxxxDxzzDxxy.Then (2) can be rewritten as
(5)b1−b2=Aβ.The calibration coefficient vectorβ can be solved utilizing the method of least squares:
(6)β⌢1=ATA−1ATb1−b2.


### 2.2. Error Analysis for GyroWheel Calibration Tests

Due to manufacture imperfections and design limitations of the turntable system, the orientation of the GyroWheel axes will be deviated from the intended one. As a result, there are deviations between the actual earth rate components and the ideal earth rate components, as well as the gravitational acceleration components. UseΔψ,Δθ,Δγ to denote rotational errors about y-, z-, and x-axes, respectively (Figure 3).Figure 3
Orientation error of GyroWheel axes.Oxg0yg0zg0: the intended orientation of the GyroWheel axes. Oxyggzg: the actual orientation of the GyroWheel axes.With the assumption of small angular errors, the transformation matrix describing the orientation error of the GyroWheel with respect to its ideal orientation is given below:(7)ΔR=1−ΔθΔψΔθ1−Δγ−ΔψΔγ1.The actual earth rate components and gravitational acceleration components can be calculated as follows:(8)ωxi′ωyi′ωzi′=ωxiωyiωziΔR,axi′ayi′azi′=axiayiaziΔR.In the presence of the orientation error, the regression model of multiposition calibration is given by(9)b1−b2′=A′β,where
(10)b2′=ωx1′ωx2′⋮ωxn′,A′=1ax1′ay1′az1′a′x12a′z12ax1′ay1′1ax2′ay2′az2′a′x22a′z22ax2′ay2′⋮⋮⋮⋮⋮⋮⋮1axn′ayn′azn′a′xn2a′zn2axn′ayn′.Then the true value of the calibration coefficient vector can be calculated as follows:(11)β⌢2=A′TA′−1A′Tb1−b2′.The actual calibration results are given by (6), and the deviation of the calibration results caused by the orientation error can be expressed as
(12)Δβ=β⌢1−β⌢2.The deviationΔβ results in GyroWheel’s rate-sensing error Δωx; the rate-sensing error vector at the test positions can be calculated:
(13)Δω⌢x=b1−Aβ⌢2−b2′.Define(14)Δωx=Δω⌢x∞.Δ
ω
x represents the rate-sensing error of x-axis. The rate-sensing error of y-axis, denoted by Δωy, can be calculated in the same way. Obviously, the rate-sensing errors Δωx and Δωy are affected by the orientation error described byΔψ,Δθ,Δγ. If the values of Δψ,Δθ,Δγ are given, the corresponding values of the rate-sensing errors Δωx and Δωy can be calculated by several matrix operations. To facilitate an error analysis, a numerical method is utilized. A series of rate-sensing errors are obtained with numerical simulations when a series of Δψ,Δθ,Δγ are given. The relationships between the rate-sensing errors Δωx,Δωy and the orientation error described by Δψ,Δθ,Δγ are expressed in a visualized way, as shown in Figure 4. The two surfaces describing the relationships in Figures 4(a) and 4(b) are denoted by Γ1,Γ2. From the simulation results, the following conclusions can be made:
(1)
As seen in Figures4(a)–4(f), the rate-sensing error of x-axisΔωx is caused by Δψ,Δθ, and it is not affected by Δγ. The rate-sensing error of y-axisΔωy is caused by Δθ,Δγ, and it is not affected by Δψ.(2)
As seen in Figure4(a), the surface Γ1 is symmetric about the planes Δψ=0 and Δθ=0. As seen in Figure 4(b), the surface Γ2 is symmetric about the planes Δθ=0 and Δγ=0. Therefore, Δωx is an even function of Δψ,Δθ, and Δωy is an even function of Δθ,Δγ.(3)
As seen in Figures4(a)–4(f), Δωx rises with the increasing of Δψ,Δθ, and Δωy rises with the increasing of Δθ,Δγ.Figure 4
Relationships between rate-sensing errors and orientation error.
(a)
Relationship betweenΔωx and Δψ,Δθ
(b)
Relationship betweenΔωy and Δθ,Δγ
(c)
Relationship betweenΔωx and Δψ,Δγ
(d)
Relationship betweenΔωy and Δθ,Δψ
(e)
Relationship betweenΔωx and Δθ,Δγ
(f)
Relationship betweenΔωy and Δγ,ΔψUseωm to denote the required calibration accuracy of the GyroWheel; that is, Δωx≤ωm,Δωy≤ωm. By drawing two planes defined by the implicit equations Γ3:Δωx=ωm and Γ4:Δωy=ωm in Figures 4(a) and 4(b), respectively, two lines of intersection are obtained, and the requirement of the orientation error can be determined easily and is denoted by
(15)Δψ≤Iψ,Δθ≤Iθ,Δγ≤Iγ.


### 2.3. Example

A GyroWheel with the required calibration accuracy of0.01°/h is taken as an example. The nominal values of the calibration coefficients are listed in Table 1. According to the above analysis, the line of intersection between Γ1 and Γ3, and the line of intersection between Γ2 and Γ4 are given in Figure 5.Table 1
Nominal values of calibration coefficients.
D(x)F (deg/h)
−93.3
D(y)F (deg/h)
105.3
D(x)x (deg/h/g)
−43.2
D(y)x (deg/h/g)
2.2
D(x)y (deg/h/g)
3.5
D(y)y (deg/h/g)
−33.2
D(x)z (deg/h/g)
−1.2
D(y)z (deg/h/g)
−0.1
D(x)xx (deg/h/g2)
2.1
D(y)yy (deg/h/g2)
−10.6
D(x)zz (deg/h/g2)
−0.089
D(y)zz (deg/h/g2)
−10.38
D(x)xy (deg/h/g2)
17.4
D(y)xy (deg/h/g2)
47.8Figure 5
Lines of intersection.
(a)
x-axis
(b)
y-axisAs seen in Figure5, the rectangular areas in black represent the ranges of Δψ,Δθ,Δγ, in which the requirement of the calibration accuracy can be met. To maximize the areas of the two rectangles in Figure 5, the requirement of the orientation error is determined as follows:
(16)Δψ≤0.027°,Δθ≤0.027°,Δγ≤0.027°.


## 2.1. Overview of GyroWheel Calibration Theory

The GyroWheel is an innovative attitude determination and control instrument. A cutaway isometric view of the GyroWheel is shown in Figure1. When the GyroWheel is used to measure angular rates, erroneous torques give rise to precession of the rotor resulting in drift errors. In the null tilt condition, the rate-sensing equations of the GyroWheel are given in the following equation, with consideration of the drift errors [5]:
(1)ωx=kyiy−DxF−Dxxax−Dxyay−Dxzaz−Dxxxax2−Dxzzaz2−Dxxyaxay,ωy=kxix−DyF−Dyxax−Dyyay−Dyzaz−Dyyyay2−Dyzzaz2−Dyyxaxay,where Dx,Dy represent the drift errors of x,y axes, DxF,DyF are g insensitive terms, Dxi,Dyi,i=x,y,z, are g sensitive error coefficients, and Dxij,ij=xx,zz,xy, Dyij,ij=yy,zz,yx, are g2 sensitive error coefficients. ax,ay,az represent acceleration components of the gravity vector, ωx,ωy are external angular velocities, kx,ky are torque scale factors of the GyroWheel, and ix,iy are the currents in the torque coils.Figure 1
Cutaway view of a GyroWheel.In an effort to improve the rate-sensing accuracy of the GyroWheel, multiposition tests are performed to calibrate the GyroWheel. Multiposition tests make use of a two-axis turntable to provide the intended orientation for the GyroWheel. A schematic representation of a turntable is shown in Figure2. The earth’s rotation rate and gravitational acceleration are regarded as the nominal inputs of the GyroWheel. At each position, an equation expressed as (1) can be obtained using the inputs and outputs of the GyroWheel. Therefore, the error coefficients can be calculated when enough tests of different positions are conducted.Figure 2
Schematic representation of a two-axis turntable.Take thex-axis model of the GyroWheel as an example. According to the rate-sensing model of the GyroWheel and multiposition calibration theory [10], we have
(2)kyiy1kyiy2⋮kyiyn−ωx1ωx2⋮ωxn=1ax1ay1az1ax12az12ax1ay11ax2ay2az2ax22az22ax2ay2⋮⋮⋮⋮⋮⋮⋮1axnaynaznaxn2azn2axnaynDxFDxxDxyDxzDxxxDxzzDxxy,where n is the number of test positions, iyj,j=1,2,…,n, represent the currents in the torque coils, ωxi,i=1,2,…,n, represent the earth rate components of each test position, and axi,ayi,azi,i=1,2,…,n, represent the gravitational acceleration components of each test position.The initial orientation of the GyroWheel axes is north-west-up. Define thatωe is the earth’s rotation rate, λ is the latitude, ϕ1i,ϕ2i are the rotation angles of the turntable, and the subscript i means the ith test position. The gravitational acceleration components and the earth rate components can be expressed as follows:
(3)axi=−sinϕ1icosϕ2i,ayi=sinϕ1isinϕ2i,azi=cosϕ1i,ωxi=ωecosϕ2icosϕ1icosλ−sinϕ1isinλ,ωyi=ωesinϕ2i−cosϕ1icosλ+sinϕ1isinλ,ωzi=ωesinϕ1icosλ+cosϕ1isinλ.Denote(4)b1=kyiy1kyiy2⋮kyiyn,b2=ωx1ωx2⋮ωxn,A=1ax1ay1az1ax12az12ax1ay11ax2ay2az2ax22az22ax2ay2⋮⋮⋮⋮⋮⋮⋮1axnaynaznaxn2azn2axnayn,β=DxFDxxDxyDxzDxxxDxzzDxxy.Then (2) can be rewritten as
(5)b1−b2=Aβ.The calibration coefficient vectorβ can be solved utilizing the method of least squares:
(6)β⌢1=ATA−1ATb1−b2.


## 2.2. Error Analysis for GyroWheel Calibration Tests

Due to manufacture imperfections and design limitations of the turntable system, the orientation of the GyroWheel axes will be deviated from the intended one. As a result, there are deviations between the actual earth rate components and the ideal earth rate components, as well as the gravitational acceleration components. UseΔψ,Δθ,Δγ to denote rotational errors about y-, z-, and x-axes, respectively (Figure 3).Figure 3
Orientation error of GyroWheel axes.Oxg0yg0zg0: the intended orientation of the GyroWheel axes. Oxyggzg: the actual orientation of the GyroWheel axes.With the assumption of small angular errors, the transformation matrix describing the orientation error of the GyroWheel with respect to its ideal orientation is given below:(7)ΔR=1−ΔθΔψΔθ1−Δγ−ΔψΔγ1.The actual earth rate components and gravitational acceleration components can be calculated as follows:(8)ωxi′ωyi′ωzi′=ωxiωyiωziΔR,axi′ayi′azi′=axiayiaziΔR.In the presence of the orientation error, the regression model of multiposition calibration is given by(9)b1−b2′=A′β,where
(10)b2′=ωx1′ωx2′⋮ωxn′,A′=1ax1′ay1′az1′a′x12a′z12ax1′ay1′1ax2′ay2′az2′a′x22a′z22ax2′ay2′⋮⋮⋮⋮⋮⋮⋮1axn′ayn′azn′a′xn2a′zn2axn′ayn′.Then the true value of the calibration coefficient vector can be calculated as follows:(11)β⌢2=A′TA′−1A′Tb1−b2′.The actual calibration results are given by (6), and the deviation of the calibration results caused by the orientation error can be expressed as
(12)Δβ=β⌢1−β⌢2.The deviationΔβ results in GyroWheel’s rate-sensing error Δωx; the rate-sensing error vector at the test positions can be calculated:
(13)Δω⌢x=b1−Aβ⌢2−b2′.Define(14)Δωx=Δω⌢x∞.Δ
ω
x represents the rate-sensing error of x-axis. The rate-sensing error of y-axis, denoted by Δωy, can be calculated in the same way. Obviously, the rate-sensing errors Δωx and Δωy are affected by the orientation error described byΔψ,Δθ,Δγ. If the values of Δψ,Δθ,Δγ are given, the corresponding values of the rate-sensing errors Δωx and Δωy can be calculated by several matrix operations. To facilitate an error analysis, a numerical method is utilized. A series of rate-sensing errors are obtained with numerical simulations when a series of Δψ,Δθ,Δγ are given. The relationships between the rate-sensing errors Δωx,Δωy and the orientation error described by Δψ,Δθ,Δγ are expressed in a visualized way, as shown in Figure 4. The two surfaces describing the relationships in Figures 4(a) and 4(b) are denoted by Γ1,Γ2. From the simulation results, the following conclusions can be made:
(1)
As seen in Figures4(a)–4(f), the rate-sensing error of x-axisΔωx is caused by Δψ,Δθ, and it is not affected by Δγ. The rate-sensing error of y-axisΔωy is caused by Δθ,Δγ, and it is not affected by Δψ.(2)
As seen in Figure4(a), the surface Γ1 is symmetric about the planes Δψ=0 and Δθ=0. As seen in Figure 4(b), the surface Γ2 is symmetric about the planes Δθ=0 and Δγ=0. Therefore, Δωx is an even function of Δψ,Δθ, and Δωy is an even function of Δθ,Δγ.(3)
As seen in Figures4(a)–4(f), Δωx rises with the increasing of Δψ,Δθ, and Δωy rises with the increasing of Δθ,Δγ.Figure 4
Relationships between rate-sensing errors and orientation error.
(a)
Relationship betweenΔωx and Δψ,Δθ
(b)
Relationship betweenΔωy and Δθ,Δγ
(c)
Relationship betweenΔωx and Δψ,Δγ
(d)
Relationship betweenΔωy and Δθ,Δψ
(e)
Relationship betweenΔωx and Δθ,Δγ
(f)
Relationship betweenΔωy and Δγ,ΔψUseωm to denote the required calibration accuracy of the GyroWheel; that is, Δωx≤ωm,Δωy≤ωm. By drawing two planes defined by the implicit equations Γ3:Δωx=ωm and Γ4:Δωy=ωm in Figures 4(a) and 4(b), respectively, two lines of intersection are obtained, and the requirement of the orientation error can be determined easily and is denoted by
(15)Δψ≤Iψ,Δθ≤Iθ,Δγ≤Iγ.


## 2.3. Example

A GyroWheel with the required calibration accuracy of0.01°/h is taken as an example. The nominal values of the calibration coefficients are listed in Table 1. According to the above analysis, the line of intersection between Γ1 and Γ3, and the line of intersection between Γ2 and Γ4 are given in Figure 5.Table 1
Nominal values of calibration coefficients.
D(x)F (deg/h)
−93.3
D(y)F (deg/h)
105.3
D(x)x (deg/h/g)
−43.2
D(y)x (deg/h/g)
2.2
D(x)y (deg/h/g)
3.5
D(y)y (deg/h/g)
−33.2
D(x)z (deg/h/g)
−1.2
D(y)z (deg/h/g)
−0.1
D(x)xx (deg/h/g2)
2.1
D(y)yy (deg/h/g2)
−10.6
D(x)zz (deg/h/g2)
−0.089
D(y)zz (deg/h/g2)
−10.38
D(x)xy (deg/h/g2)
17.4
D(y)xy (deg/h/g2)
47.8Figure 5
Lines of intersection.
(a)
x-axis
(b)
y-axisAs seen in Figure5, the rectangular areas in black represent the ranges of Δψ,Δθ,Δγ, in which the requirement of the calibration accuracy can be met. To maximize the areas of the two rectangles in Figure 5, the requirement of the orientation error is determined as follows:
(16)Δψ≤0.027°,Δθ≤0.027°,Δγ≤0.027°.


## 3. Error Analysis of Turntable System

### 3.1. Overview of Turntable Errors

There are four main sources of errors in a turntable system; they are position error, wobble error, orthogonality error, and intersection error.Position error is defined as the difference between the actual and the intended rotation angles of the turntable’s rotating shaft, which is influenced by servo precision and measurement accuracy of the turntable system.Wobble error can be divided into axial wobble error, radial wobble error, and angular wobble error, as illustrated in Figure6, where Δsa is the axial wobble error, Δsr is the radial wobble error, and Δα is the angular wobble error. In practical engineering, Δsa and Δsr are small enough to be ignored; thus, wobble error refers specifically to angular wobble error Δα in this paper [11, 23, 24].Figure 6
Schematic of a wobble error.Orthogonality error and intersection error are induced by mechanical imperfections of the turntable structure and misalignment of turntable elements.For the two-axis turntable given in Figure2, a summary of the error components involved is shown in Table 2.Table 2
Error components of a two-axis turntable system.
Location of error components
Symbols
Definitions
Outer axis
Δ
ϕ
y
1
Position error of outer axis
Δ
α
x
1
Wobble error component of outer axis aboutx-axis
Δ
α
z
1
Wobble error component of outer axis aboutz-axis
Inner axis
Δ
ϕ
z
2
Position error of inner axis
Δ
α
x
2
Wobble error component of inner axis aboutx-axis
Δ
α
y
2
Wobble error component of inner axis abouty-axis
Between outer and inner axes
Δ
ε
x
Orthogonality error component aboutx-axis
Δ
ε
y
Orthogonality error component abouty-axis
Δ
η
Intersection error


### 3.2. Error Modeling for Turntable System

The error model of the turntable system can be developed based on rigid-body kinematics. The two-axis turntable system, as shown in Figure2, consists of three bodies: the base, the outer frame, and the inner frame. In addition, the GyroWheel being attached to the inner frame should be considered. In an effort to derive the error model of the turntable, five coordinate systems are introduced. They are the base coordinate system F0: O0X0Y0Z0, the outer axis coordinate system F1: O1X1Y1Z1, the reference coordinate system of the inner axis F2: O2X2Y2Z2, the inner axis coordinate system F3: O3X3Y3Z3 and the GyroWheel coordinate system F4: OgXgYgZg. F0, F1, F3, and F4 are four body-fixed coordinate systems, which are attached to the base, the outer frame, the inner frame, and the GyroWheel, respectively. F2 is an interim coordinate system; the relation between F2 and F1 is determined by the orthogonality error and intersection error of the turntable system. An illustration of these coordinate systems is shown in Figure 7.Figure 7
An illustration of the coordinate systems.Actually, the base coordinate systemF0 is fixed with respect to the earth. The outer axis of the turntable is aligned along the y-axis of the outer axis coordinate system F1. The inner axis of the turntable is aligned along the z-axis of the inner axis coordinate system F3. Since the GyroWheel is attached to the inner frame of the turntable, the GyroWheel coordinate system F4 is coincident with F3. The relations between these coordinate systems and the error components of the turntable system are expressed in Table 3. ϕ1,ϕ2 are the rotation angles about the outer axis and inner axis, respectively.Table 3
Rotational and translational motion relations between adjacent coordinate systems.
i
F
i → Fi+1
0
A rotationϕ1 about y-axis, followed by a rotation Δϕy1 about y-axisA rotation Δαz1 about z-axis and then a rotation Δαx1 about x-axis
1
A translationΔη along x-axis, followed by a rotation Δεy about y-axis, and then a rotation Δεx about x-axis
2
A rotationϕ2 about z-axis, followed by a rotation Δαy2 about y-axis; a rotation Δϕz2 about z-axis, and then a rotation Δαx2 about x-axisThe relative position and orientation of one coordinate system with respect to another coordinate system can be modeled using a homogeneous transformation matrix (HTM), denoted asTij,i,j=0,1,2,3,4. The presuperscript represents the coordinate system we are transferring from, and the postsubscript represents the coordinate system we want the results to be represented in.Hence, the transform relation betweenF1 and F0 can be calculated with the assumption of small angular errors:
(17)T01=cosϕ10sinϕ100100−sinϕ10cosϕ100001⋅1−Δαz1Δϕy10Δαz11−Δαx10−Δϕy1Δαx1100001.The transform relation betweenF2 and F1 can be calculated as follows:
(18)T12=10Δεy001−Δεx0−ΔεyΔεx100001⋅100Δη010000100001.And the transform relation betweenF3 and F2 is given below:
(19)T23=cosϕ2−sinϕ200sinϕ2cosϕ200010100001⋅1−Δϕz2Δαy20Δϕz21−Δαx20−Δαy2Δαx2100001.As stated above, the transform relation betweenF4 and F3 is written in the following form:
(20)T34=I4×4.We introduce homogeneous transformation matrixesTd,Tt to denote the ideal and actual transform relations between the GyroWheel and the base coordinate system. Td is obtained through a series of perfect rotations ϕ1,ϕ2 and is given by the following:
(21)Td=cosϕ10sinϕ100100−sinϕ10cosϕ100001⋅cosϕ2−sinϕ200sinϕ2cosϕ200010100001.In presence of the turntable errors,Tt can be calculated by matrix multiplication:
(22)Tt=∏i=03Tii+1.The top left corner3×3 matrixes of Td,Tt denoted by Rd,Rt represent the ideal and actual direction cosine matrixes. Since Rd,Rt are orthogonal matrixes, the orientation error of the GyroWheel caused by the turntable errors can be expressed as follows:
(23)ΔR=RdTRt.According to (23) and (7), we have the following:
(24)Δψ=ΔR13,Δθ=ΔR21,Δγ=ΔR32,where ΔRij represents the element in the ith row and jth column of matrix ΔR. Substituting (15)–(21) into (22) yields the following expression:
(25)Δψ=fψϕ,χ,Δθ=fθϕ,χ,Δγ=fγϕ,χ,where fψ,fθ,fγ are nonlinear functions of the rotation angles and the turntable error components, ϕ=ϕ1ϕ2T represents the rotation angle vector, and χ=Δϕy1Δϕz2Δαx1Δαz1Δαx2Δαy2ΔεxΔεyT represents the turntable error vector. According to (25), the intersection error has no effect on the orientation error of the GyroWheel.To simplify the analysis, (25) is rewritten into the following form with small angle approximation of the errors:
(26)Δψ=fψϕ2,χψ=Δϕy1cosϕ2−Δαx1sinϕ2+Δαy2−Δεxsinϕ2+Δεycosϕ2,Δθ=fθχθ=Δϕz2+Δαz1,Δγ=fγϕ2,χγ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2,where
(27)χγ=Δϕy1Δαx1Δαx2ΔεxΔεyT,χθ=Δϕz2Δαz1T,χψ=Δϕy1Δαx1Δαy2ΔεxΔεyT.


## 3.1. Overview of Turntable Errors

There are four main sources of errors in a turntable system; they are position error, wobble error, orthogonality error, and intersection error.Position error is defined as the difference between the actual and the intended rotation angles of the turntable’s rotating shaft, which is influenced by servo precision and measurement accuracy of the turntable system.Wobble error can be divided into axial wobble error, radial wobble error, and angular wobble error, as illustrated in Figure6, where Δsa is the axial wobble error, Δsr is the radial wobble error, and Δα is the angular wobble error. In practical engineering, Δsa and Δsr are small enough to be ignored; thus, wobble error refers specifically to angular wobble error Δα in this paper [11, 23, 24].Figure 6
Schematic of a wobble error.Orthogonality error and intersection error are induced by mechanical imperfections of the turntable structure and misalignment of turntable elements.For the two-axis turntable given in Figure2, a summary of the error components involved is shown in Table 2.Table 2
Error components of a two-axis turntable system.
Location of error components
Symbols
Definitions
Outer axis
Δ
ϕ
y
1
Position error of outer axis
Δ
α
x
1
Wobble error component of outer axis aboutx-axis
Δ
α
z
1
Wobble error component of outer axis aboutz-axis
Inner axis
Δ
ϕ
z
2
Position error of inner axis
Δ
α
x
2
Wobble error component of inner axis aboutx-axis
Δ
α
y
2
Wobble error component of inner axis abouty-axis
Between outer and inner axes
Δ
ε
x
Orthogonality error component aboutx-axis
Δ
ε
y
Orthogonality error component abouty-axis
Δ
η
Intersection error


## 3.2. Error Modeling for Turntable System

The error model of the turntable system can be developed based on rigid-body kinematics. The two-axis turntable system, as shown in Figure2, consists of three bodies: the base, the outer frame, and the inner frame. In addition, the GyroWheel being attached to the inner frame should be considered. In an effort to derive the error model of the turntable, five coordinate systems are introduced. They are the base coordinate system F0: O0X0Y0Z0, the outer axis coordinate system F1: O1X1Y1Z1, the reference coordinate system of the inner axis F2: O2X2Y2Z2, the inner axis coordinate system F3: O3X3Y3Z3 and the GyroWheel coordinate system F4: OgXgYgZg. F0, F1, F3, and F4 are four body-fixed coordinate systems, which are attached to the base, the outer frame, the inner frame, and the GyroWheel, respectively. F2 is an interim coordinate system; the relation between F2 and F1 is determined by the orthogonality error and intersection error of the turntable system. An illustration of these coordinate systems is shown in Figure 7.Figure 7
An illustration of the coordinate systems.Actually, the base coordinate systemF0 is fixed with respect to the earth. The outer axis of the turntable is aligned along the y-axis of the outer axis coordinate system F1. The inner axis of the turntable is aligned along the z-axis of the inner axis coordinate system F3. Since the GyroWheel is attached to the inner frame of the turntable, the GyroWheel coordinate system F4 is coincident with F3. The relations between these coordinate systems and the error components of the turntable system are expressed in Table 3. ϕ1,ϕ2 are the rotation angles about the outer axis and inner axis, respectively.Table 3
Rotational and translational motion relations between adjacent coordinate systems.
i
F
i → Fi+1
0
A rotationϕ1 about y-axis, followed by a rotation Δϕy1 about y-axisA rotation Δαz1 about z-axis and then a rotation Δαx1 about x-axis
1
A translationΔη along x-axis, followed by a rotation Δεy about y-axis, and then a rotation Δεx about x-axis
2
A rotationϕ2 about z-axis, followed by a rotation Δαy2 about y-axis; a rotation Δϕz2 about z-axis, and then a rotation Δαx2 about x-axisThe relative position and orientation of one coordinate system with respect to another coordinate system can be modeled using a homogeneous transformation matrix (HTM), denoted asTij,i,j=0,1,2,3,4. The presuperscript represents the coordinate system we are transferring from, and the postsubscript represents the coordinate system we want the results to be represented in.Hence, the transform relation betweenF1 and F0 can be calculated with the assumption of small angular errors:
(17)T01=cosϕ10sinϕ100100−sinϕ10cosϕ100001⋅1−Δαz1Δϕy10Δαz11−Δαx10−Δϕy1Δαx1100001.The transform relation betweenF2 and F1 can be calculated as follows:
(18)T12=10Δεy001−Δεx0−ΔεyΔεx100001⋅100Δη010000100001.And the transform relation betweenF3 and F2 is given below:
(19)T23=cosϕ2−sinϕ200sinϕ2cosϕ200010100001⋅1−Δϕz2Δαy20Δϕz21−Δαx20−Δαy2Δαx2100001.As stated above, the transform relation betweenF4 and F3 is written in the following form:
(20)T34=I4×4.We introduce homogeneous transformation matrixesTd,Tt to denote the ideal and actual transform relations between the GyroWheel and the base coordinate system. Td is obtained through a series of perfect rotations ϕ1,ϕ2 and is given by the following:
(21)Td=cosϕ10sinϕ100100−sinϕ10cosϕ100001⋅cosϕ2−sinϕ200sinϕ2cosϕ200010100001.In presence of the turntable errors,Tt can be calculated by matrix multiplication:
(22)Tt=∏i=03Tii+1.The top left corner3×3 matrixes of Td,Tt denoted by Rd,Rt represent the ideal and actual direction cosine matrixes. Since Rd,Rt are orthogonal matrixes, the orientation error of the GyroWheel caused by the turntable errors can be expressed as follows:
(23)ΔR=RdTRt.According to (23) and (7), we have the following:
(24)Δψ=ΔR13,Δθ=ΔR21,Δγ=ΔR32,where ΔRij represents the element in the ith row and jth column of matrix ΔR. Substituting (15)–(21) into (22) yields the following expression:
(25)Δψ=fψϕ,χ,Δθ=fθϕ,χ,Δγ=fγϕ,χ,where fψ,fθ,fγ are nonlinear functions of the rotation angles and the turntable error components, ϕ=ϕ1ϕ2T represents the rotation angle vector, and χ=Δϕy1Δϕz2Δαx1Δαz1Δαx2Δαy2ΔεxΔεyT represents the turntable error vector. According to (25), the intersection error has no effect on the orientation error of the GyroWheel.To simplify the analysis, (25) is rewritten into the following form with small angle approximation of the errors:
(26)Δψ=fψϕ2,χψ=Δϕy1cosϕ2−Δαx1sinϕ2+Δαy2−Δεxsinϕ2+Δεycosϕ2,Δθ=fθχθ=Δϕz2+Δαz1,Δγ=fγϕ2,χγ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2,where
(27)χγ=Δϕy1Δαx1Δαx2ΔεxΔεyT,χθ=Δϕz2Δαz1T,χψ=Δϕy1Δαx1Δαy2ΔεxΔεyT.


## 4. Error Allocation of the Turntable System

### 4.1. Sensitivity Analysis and Constraints Analysis

To ensure the orientation accuracy and perform error allocation of the turntable system, sensitivities of the orientation error to the individual errors and the manufacturing capability should be considered.According to (23), sensitivities of the orientation error can be analyzed:
(28)Cψi=∂fψ∂χi,Cθi=∂fθ∂χi,Cγi=∂fγ∂χi,i=1,2,…,8.χi represents the ith element of the turntable error vector χ, and Cψi,Cθi,Cγi represent the sensitivities of the orientation error to the individual error components. Given that Cψi,Cθi,Cγi vary with the rotation angles, we here consider the average values, which are given in Figure 8. The values represent how sensitive the orientation error is to the individual error components. For example, if the orientation error is more sensitive to error component χi than to error component χj, χi is required to be smaller than χj to achieve a reasonable error allocation.Figure 8
Sensitivities of orientation error to individual error components.
(a)
Δ
ψ
(b)
Δ
θ
(c)
Δ
γAdditionally, with the limitation of technological level and manufacturing costs in practical engineering, the manufacturing capability determines the maximum accuracy that the turntable can reach. Therefore, it gives the lower bounds of the error components of the turntable system, namely, the error components should satisfy the constraints:(29)χi≥LBi,i=1,2,…,8.L
B
i are the lower bounds of the error components.


### 4.2. Error Allocation Method for Turntable Systems

For the error allocation problem of the turntable system, we expect to find the upper boundsUBi,i=1,2,⋯,8. The turntable system should reach the required orientation accuracy at any rotation angles when the error components satisfy the constraints, namely,
(30)fγϕ,χ≤Iγ,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ,∀ϕ∈Ωϕ,∀χ∈Ωχ,where Ωϕ=ϕ∣0≤ϕi≤2π,i=1,2 and Ωχ=χ∣LBi≤χi≤UBi,i=1,2,…,8.With the simplification of the orientation error model given by (26), the error allocation of the turntable system can be described by the following optimization problems:
(31)minχγJγ=∑i=15Wγi21χγi2,s.t.fγϕ2,χγ≤Iγ,∀ϕ2∈0,π2,χγi≥LBγi,i=1,2,…,5,(32)minχθJθ=∑i=12Wθi21χθi2,s.t.fθχθ≤Iθ,χθi≥LBθi,i=1,2,(33)minχψJψ=∑i=15Wψi21χψi2,s.t.fψϕ2,χψ≤Iψ,∀ϕ2∈3π2,2π,χψi≥LBψi,i=1,2,…,5,whereχγi,χθi,χψi are the ith elements of the vectors χγ,χθ,χψ, respectively. Wγ,Wθ,Wψ are weighted coefficients; the values of which are the multiplicative inverse of the corresponding sensitivities given in Figure 8. By solving the optimization problems shown in (31)–(35), the optimal solutions χγ∗,χθ∗,χψ∗ are obtained respectively. Then, the upper bounds of the turntable errors can be given by
(34)UB1=minχγ1∗,χψ1∗,UB2=χθ1∗,UB3=minχγ2∗,χψ2∗,UB4=χθ2∗,UB5=χγ3∗,UB6=χψ3∗,UB7=minχγ4∗,χψ4∗,UB8=minχγ5∗,χψ5∗.According to (31)–(36), the error allocation of the turntable system can be accomplished. The error allocation results UBi,i=1,2,…,8, are proven below.Lemma 1.
Consider a turntable system with the orientation error model shown in (26). The error allocation results UBi,i=1,2,…,8, are obtained from (31)–(36). ∀ϕ∈Ωϕ,∀χ∈Ωχ, where Ωϕ=ϕ∣0≤ϕi≤2πi=12 and Ωχ=χ∣LBi≤χi≤UBii=12⋯8, it is always true that fγϕ,χ≤Iγ,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ.Proof
Case 1. ∀ϕ2∈0,π/2, then
(35)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤χγ1∗sinϕ2+χγ2∗cosϕ2+χγ3∗+χγ4∗cosϕ2+χγ5∗sinϕ2=fγϕ2,χγ∗≤Iγ.
Case 2.
∀
ϕ
2
∈
π
/
2
,
π, denote ϕ⌢2=π−ϕ2; then ∀ϕ⌢2∈0,π/2,
(36)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.
Case 3. ∀ϕ2∈π,3π/2, denote ϕ⌢2=ϕ2−π; then ∀ϕ⌢2∈0,π/2,
(37)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.
Case 4. ∀ϕ2∈3π/2,2π, denote ϕ⌢2=2π−ϕ2; then ∀ϕ⌢2∈0,π/2,
(38)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.In summary,∀ϕ∈Ωϕ,∀χ∈Ωχ, we have fγϕ,χ≤Iγ, where Ωϕ=ϕ∣0≤ϕi≤2π,i=1,2 and Ωχ=χ∣LBi≤χi≤UBi,i=1,2,…,8.Similarly,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ can be demonstrated as well.


### 4.3. Example

Take the GyroWheel with the required calibration accuracy of0.01°/h as an example. According to the analysis in Section 2, the requirement of the orientation accuracy is given in (16). Using the error allocation method proposed in this section, as shown in (31)–(36), the allocation results are given in Table 4.Table 4
Error allocation results of the turntable system.
Error sources
Symbols
Accuracy requirements (arc minute)
Position error
Δ
ϕ
y
1
0.4235
Δ
ϕ
z
2
0.7800
Wobble error
Δ
α
x
1
0.4234
Δ
α
z
1
0.7800
Δ
α
x
2
0.2825
Δ
α
y
2
0.2824
Orthogonality error
Δ
ε
x
0.4234
Δ
ε
y
0.4233As seen in Table4, the x-axis wobble error component of the inner axis Δαx2 and the y-axis wobble error component of the inner axis Δαy2 have higher accuracy requirements than the other error components. This is due to the fact that the orientation error is more sensitive to Δαx2 and Δαy2, as shown in Figure 8. According to Figure 8, only Δθ is sensitive to the position error of the inner axis Δϕz2 and the z-axis wobble error component of the outer axis Δαz1, and Δψ and Δγ are insensitive to Δϕz2 and Δαz1. Meanwhile, Δθ is affected only by Δϕz2 and Δαz1; thus, Δϕz2 and Δαz1 have lower accuracy requirements. The error allocation results provide a basis for the selection of the suitable turntable and are of great significance to guide the calibration tests and improve the calibration accuracy of the GyroWheel.Given that the error allocation of the turntable system is accomplished based on certain simplifications, Monte Carlo simulations are performed to verify the effectiveness of the error allocation results for the original orientation error model as shown in (25). According to the allocation results given in Table 4, values of the error components satisfying the accuracy requirements LBi≤χi≤UBi,i=1,2,…,8 are generated randomly by utilizing MATLAB. 10,000 times of numerical simulations are performed, and the orientation error is calculated for each simulation test. The distribution of Δψ,Δθ,Δγ describing the orientation error for the 10,000 times of simulations is shown in Figure 9.Figure 9
Distribution of orientation error.
(a)
Δ
ψ
(b)
Δ
θ
(c)
Δ
γIt is seen in Figure9 that the angular errors Δψ,Δθ,Δγ always satisfy the corresponding accuracy requirements in the repeated random simulation tests, which verifies the validity of the proposed error allocation method in this section.


## 4.1. Sensitivity Analysis and Constraints Analysis

To ensure the orientation accuracy and perform error allocation of the turntable system, sensitivities of the orientation error to the individual errors and the manufacturing capability should be considered.According to (23), sensitivities of the orientation error can be analyzed:
(28)Cψi=∂fψ∂χi,Cθi=∂fθ∂χi,Cγi=∂fγ∂χi,i=1,2,…,8.χi represents the ith element of the turntable error vector χ, and Cψi,Cθi,Cγi represent the sensitivities of the orientation error to the individual error components. Given that Cψi,Cθi,Cγi vary with the rotation angles, we here consider the average values, which are given in Figure 8. The values represent how sensitive the orientation error is to the individual error components. For example, if the orientation error is more sensitive to error component χi than to error component χj, χi is required to be smaller than χj to achieve a reasonable error allocation.Figure 8
Sensitivities of orientation error to individual error components.
(a)
Δ
ψ
(b)
Δ
θ
(c)
Δ
γAdditionally, with the limitation of technological level and manufacturing costs in practical engineering, the manufacturing capability determines the maximum accuracy that the turntable can reach. Therefore, it gives the lower bounds of the error components of the turntable system, namely, the error components should satisfy the constraints:(29)χi≥LBi,i=1,2,…,8.L
B
i are the lower bounds of the error components.


## 4.2. Error Allocation Method for Turntable Systems

For the error allocation problem of the turntable system, we expect to find the upper boundsUBi,i=1,2,⋯,8. The turntable system should reach the required orientation accuracy at any rotation angles when the error components satisfy the constraints, namely,
(30)fγϕ,χ≤Iγ,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ,∀ϕ∈Ωϕ,∀χ∈Ωχ,where Ωϕ=ϕ∣0≤ϕi≤2π,i=1,2 and Ωχ=χ∣LBi≤χi≤UBi,i=1,2,…,8.With the simplification of the orientation error model given by (26), the error allocation of the turntable system can be described by the following optimization problems:
(31)minχγJγ=∑i=15Wγi21χγi2,s.t.fγϕ2,χγ≤Iγ,∀ϕ2∈0,π2,χγi≥LBγi,i=1,2,…,5,(32)minχθJθ=∑i=12Wθi21χθi2,s.t.fθχθ≤Iθ,χθi≥LBθi,i=1,2,(33)minχψJψ=∑i=15Wψi21χψi2,s.t.fψϕ2,χψ≤Iψ,∀ϕ2∈3π2,2π,χψi≥LBψi,i=1,2,…,5,whereχγi,χθi,χψi are the ith elements of the vectors χγ,χθ,χψ, respectively. Wγ,Wθ,Wψ are weighted coefficients; the values of which are the multiplicative inverse of the corresponding sensitivities given in Figure 8. By solving the optimization problems shown in (31)–(35), the optimal solutions χγ∗,χθ∗,χψ∗ are obtained respectively. Then, the upper bounds of the turntable errors can be given by
(34)UB1=minχγ1∗,χψ1∗,UB2=χθ1∗,UB3=minχγ2∗,χψ2∗,UB4=χθ2∗,UB5=χγ3∗,UB6=χψ3∗,UB7=minχγ4∗,χψ4∗,UB8=minχγ5∗,χψ5∗.According to (31)–(36), the error allocation of the turntable system can be accomplished. The error allocation results UBi,i=1,2,…,8, are proven below.Lemma 1.
Consider a turntable system with the orientation error model shown in (26). The error allocation results UBi,i=1,2,…,8, are obtained from (31)–(36). ∀ϕ∈Ωϕ,∀χ∈Ωχ, where Ωϕ=ϕ∣0≤ϕi≤2πi=12 and Ωχ=χ∣LBi≤χi≤UBii=12⋯8, it is always true that fγϕ,χ≤Iγ,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ.Proof
Case 1. ∀ϕ2∈0,π/2, then
(35)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤χγ1∗sinϕ2+χγ2∗cosϕ2+χγ3∗+χγ4∗cosϕ2+χγ5∗sinϕ2=fγϕ2,χγ∗≤Iγ.
Case 2.
∀
ϕ
2
∈
π
/
2
,
π, denote ϕ⌢2=π−ϕ2; then ∀ϕ⌢2∈0,π/2,
(36)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.
Case 3. ∀ϕ2∈π,3π/2, denote ϕ⌢2=ϕ2−π; then ∀ϕ⌢2∈0,π/2,
(37)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.
Case 4. ∀ϕ2∈3π/2,2π, denote ϕ⌢2=2π−ϕ2; then ∀ϕ⌢2∈0,π/2,
(38)fγϕ,χ=Δϕy1sinϕ2+Δαx1cosϕ2+Δαx2+Δεxcosϕ2+Δεysinϕ2≤Δϕy1sinϕ⌢2+Δαx1cosϕ⌢2+Δαx2+Δεxcosϕ⌢2+Δεysinϕ⌢2≤χγ1∗sinϕ⌢2+χγ2∗cosϕ⌢2+χγ3∗+χγ4∗cosϕ⌢2+χγ5∗sinϕ⌢2=fγϕ⌢2,χγ∗≤Iγ.In summary,∀ϕ∈Ωϕ,∀χ∈Ωχ, we have fγϕ,χ≤Iγ, where Ωϕ=ϕ∣0≤ϕi≤2π,i=1,2 and Ωχ=χ∣LBi≤χi≤UBi,i=1,2,…,8.Similarly,fθϕ,χ≤Iθ,fψϕ,χ≤Iψ can be demonstrated as well.


## 4.3. Example

Take the GyroWheel with the required calibration accuracy of0.01°/h as an example. According to the analysis in Section 2, the requirement of the orientation accuracy is given in (16). Using the error allocation method proposed in this section, as shown in (31)–(36), the allocation results are given in Table 4.Table 4
Error allocation results of the turntable system.
Error sources
Symbols
Accuracy requirements (arc minute)
Position error
Δ
ϕ
y
1
0.4235
Δ
ϕ
z
2
0.7800
Wobble error
Δ
α
x
1
0.4234
Δ
α
z
1
0.7800
Δ
α
x
2
0.2825
Δ
α
y
2
0.2824
Orthogonality error
Δ
ε
x
0.4234
Δ
ε
y
0.4233As seen in Table4, the x-axis wobble error component of the inner axis Δαx2 and the y-axis wobble error component of the inner axis Δαy2 have higher accuracy requirements than the other error components. This is due to the fact that the orientation error is more sensitive to Δαx2 and Δαy2, as shown in Figure 8. According to Figure 8, only Δθ is sensitive to the position error of the inner axis Δϕz2 and the z-axis wobble error component of the outer axis Δαz1, and Δψ and Δγ are insensitive to Δϕz2 and Δαz1. Meanwhile, Δθ is affected only by Δϕz2 and Δαz1; thus, Δϕz2 and Δαz1 have lower accuracy requirements. The error allocation results provide a basis for the selection of the suitable turntable and are of great significance to guide the calibration tests and improve the calibration accuracy of the GyroWheel.Given that the error allocation of the turntable system is accomplished based on certain simplifications, Monte Carlo simulations are performed to verify the effectiveness of the error allocation results for the original orientation error model as shown in (25). According to the allocation results given in Table 4, values of the error components satisfying the accuracy requirements LBi≤χi≤UBi,i=1,2,…,8 are generated randomly by utilizing MATLAB. 10,000 times of numerical simulations are performed, and the orientation error is calculated for each simulation test. The distribution of Δψ,Δθ,Δγ describing the orientation error for the 10,000 times of simulations is shown in Figure 9.Figure 9
Distribution of orientation error.
(a)
Δ
ψ
(b)
Δ
θ
(c)
Δ
γIt is seen in Figure9 that the angular errors Δψ,Δθ,Δγ always satisfy the corresponding accuracy requirements in the repeated random simulation tests, which verifies the validity of the proposed error allocation method in this section.


## 5. Conclusions

Error analysis and error allocation problems in calibration tests are studied for GyroWheel in this paper. The relationships between the rate-sensing errors of GyroWheel and the orientation error are obtained, and the orientation error induced by the turntable errors is modelled based on rigid body kinematics. A practical error allocation method is developed to determine the accuracy requirement of the test turntable. The salient features and contributions of this work are as follows:(1)
The error analysis for the GyroWheel calibration tests and the turntable system provides a way to express the effects of various turntable errors on the calibration accuracy quantitatively. Of all the error components, the wobble error of the inner axis contributes most to the calibration error.(2)
The proposed error allocation method provides a theoretical basis for the selection of the suitable turntable used in the calibration tests. With the considerations of the sensitivities of the orientation error to the individual error components, the requirements of these error components are determined according to the different sensitivities. Compared with the traditional allocation method assuming the equivalent effects of the errors, the proposed method is more reasonable and is beneficial to reducing the costs of calibration tests.Although the study is focused on a two-axis turntable for GyroWheel calibration, it can be easily applied to other inertial instruments and other types of turntables, and it holds true for the cases of other calibration equipment such as centrifuges as well.To further enhance the calibration accuracy of the GyroWheel, future research will focus on the compensation of the turntable errors.



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*Source: 6524782-2017-11-28.xml*