A General Euler Angle Error Model of Strapdown Inertial Navigation Systems

Attitude error models play an important role in analyzing the characteristics of navigation error propagation for the design and operation of strapdown inertial navigation systems (SINS). However, the majority of existing attitude error models focus on misalignment, rather than Euler angle errors. Misalignment cannot directly describe attitude error propagation, which is an indirect measurement. To solve the problem, a general Euler angle error model of SINS is proposed. Based on Euler angle error propagation analysis, relative Euler angle errors, and convected Euler angle errors are introduced to compose the general Euler angle error model. Simulation experiments are carried out to verify the proposed model.


Introduction
Inertial navigation systems (INS) can generally be classified into gimbaled inertial navigation systems (GINS) and strapdown inertial navigation systems (SINS) [1][2][3].For SINS, gyro output is used to maintain a digital computational platform as its reference frame [4,5].Compared with GINS, SINS have many advantages, such as small size, light weight, low cost, easy maintenance, etc., gradually replacing GINS [6,7].Whether in GINS or in SINS, attitude error models play an important role in analyzing the characteristics of navigation error propagation, implementing the Kalman filter for aided navigation systems, and detecting failures in real-time [8,9].
Attitude error formulations of GINS, including those of Psi-angle, Phi-angle, were developed for local level platform navigators [10].With the development of SINS, Silva et al. derived attitude error formulations for stationary SINS [11].The static model had a better application in research for its simple structure, but it did not apply to a dynamic base.Hao et al. established the dynamic error model based on the static model, where vehicle location changes would directly affect the error characteristics of SINS [12].Assuming that the propagation characteristics of the navigation errors are linear, some linearized attitude error formulations are also modeled [13,14].Goshen and Bar presented a SINS error model which contains all of the known models in the same framework [8].Chen et al. proved that these linearized error models are equivalent to each other [15], but linearized error models are not effective for nonlinear systems where large navigation errors, such as in-flight alignment, are generated [16].Savage introduced a rotation vector error and brought in white noise for error differential equations to analyze the large attitude errors of SINS [17].Then, quaternion and dual quaternion errors were developed to describe attitude error propagation [18,19].As quaternions and Euler angles are not in one-to-one correspondence, both the converting relation and the corresponding relation have to be taken into consideration during an attitude control process [20].The SINS attitude error formulations above mainly focus on misalignment for a specific computational frame rather than Euler angle errors.However, misalignment error formulations cannot describe attitude error propagation directly [21].
Euler angles are the most intuitive measurements of attitude movements, especially large-angle attitude movements.Real-time attitude information can be obtained by using Euler angles to express the attitude movement in controlled areas [22,23], then appropriate optimization can be made.An adaptive Euler-angle-based unscented Kalman filter (UKF) is utilized to analyze attitude in real-time [24].The manipulator control process is a large-angle attitude movement process which is very sensitive to Euler angles [25,26].The controllers for each degree of freedom are Euler angles.There are some specific applications, such as 6-DOF robot manipulators with Euler wrists [27,28], and robotic-wheelchair controllers combined with multi-sensors to obtain accurate orientation in the form of Euler angles [29].Additionally, Jing proposed a quasi-Euler angle to control the attitude, where the terminal attitude of a satellite can be determined [20].The control approaches mentioned above are based on Euler angles, and the Euler angle method turns out to be superior to the others in the applications because it shows better numerical accuracy, stability, and robustness [30,31].
The misalignment angle is characterized by the calculation of the three-axis deviation angle between the computational frame and the navigation frame, which cannot directly reflect and optimize a body's attitude movement.The initial attitude of the navigation system is in Euler angle form [32]. Since the output attitudes of SINS are Euler angles, it is necessary to analyze the Euler angle error model which can reflect attitude measurement accuracy more directly.
To address the above issues, a general Euler angle error modeling method of SINS is proposed.Firstly, based on the operation principles of SINS, the Euler angle solution formulation is given.Then the differential formulation of relative Euler angle errors induced by angular velocity errors can be derived.Secondly, the differential formulation of convected Euler angle errors induced by an inaccurate attitude matrix is proposed based on the Euler angle differential equations.Then, general Euler angle error formulations of SINS are modeled, and the mechanisms and characteristics of Euler angle error propagation in SINS are analyzed.Finally, some simulation experiments are carried out to verify the proposed models.The general Euler angle error model can be used to analyze the attitude movements in real-time, and an optimal control strategy can be designed based on the general Euler angle error model.

The Proposed General Euler Angle Error Model
The general Euler angle error model contains the relative Euler angle error model and the convected Euler angle error model.The relative Euler angle error model indicates the relationship between Euler angle errors and angular velocity errors of the body frame with respect to the navigation frame denoted in the body frame.The convected Euler angle error model indicates the relationship between Euler angle errors and Euler angles.

Attitude Solution by Euler Angle Differential Equations
Replacing the inertial platform of GINS, the digital computational platform of SINS, the computational frame inside the navigation computer, is theoretically equivalent to the local level navigation frame when there are no navigation errors.Gyro output is used to maintain the digital computational platform, and the specific force measurements from the accelerometer triads are resolved, then the velocity and position are acquired by double integration as shown in Figure 1.However, no matter which reference frame is chosen, all quantities must be transformed to the The body attitude matrix can be calculated by Euler angles ϕ , θ , and γ in turn along the head, pitch, and roll axes.These Euler angles can be resolved in real-time using the angular velocities of the body frame with respect to the navigation frame denoted in the body frame, and the angular velocity can be calculated by: n en ω is the angular velocity vector of the navigation frame with respect to the Earth frame denoted in the navigation frame.They can be respectively calculated by: As shown in Figure 2, in order to accurately derive the Euler angle error differential equations, the transformation relation between the angular velocity of the body frame with respect to the navigation frame denoted in the body frame and the Euler angle velocities in head, pitch, and roll with respect to the navigation frame, can be expressed as: The body attitude matrix can be calculated by Euler angles ϕ, θ, and γ in turn along the head, pitch, and roll axes.These Euler angles can be resolved in real-time using the angular velocities of the body frame with respect to the navigation frame denoted in the body frame, and the angular velocity can be calculated by: where ω b ib is the angular velocity vector of the body frame with respect to the inertial frame denoted in the body frame, and can be directly measured by gyros, and ω n in is the angular velocity vector of the navigation frame with respect to the inertial frame denoted in the navigation frame ω n in = ω n ie + ω n en .ω n ie is the angular velocity vector of the Earth frame with respect to the inertial frame denoted in the navigation frame.ω n en is the angular velocity vector of the navigation frame with respect to the Earth frame denoted in the navigation frame.They can be respectively calculated by: As shown in Figure 2, in order to accurately derive the Euler angle error differential equations, the transformation relation between the angular velocity of the body frame with respect to the navigation frame denoted in the body frame and the Euler angle velocities in head, pitch, and roll with respect to the navigation frame, can be expressed as: where ω b nbi is the angular velocity of the body frame with respect to the navigation frame denoted in the body frame along i axis, R 2 y (γ), R 1 x (θ), and R n z (ϕ) respectively represent attitude transformation matrixes induced by Euler angles γ, θ, and ϕ along the roll, pitch, and head.Equation (3) can be further rewritten by Euler angles:  From Equation (4), the Euler angle velocities induced by the angular velocities of the body frame with respect to the navigation frame denoted in the body frame can be written as: According to Equation ( 5), the Euler angles can be calculated by introducing the integral operation: ( ) ( ) From Equation ( 4), the Euler angle velocities induced by the angular velocities of the body frame with respect to the navigation frame denoted in the body frame can be written as: According to Equation ( 5), the Euler angles can be calculated by introducing the integral operation: where ϕ 0 , θ 0 , and γ 0 respectively represent the initial head, pitch, and roll of the SINS.According to Equations ( 6)-( 8), the body attitude matrix from the navigation frame to the body frame can be expressed by Euler angles ϕ, θ, and γ in the head, pitch, and roll: where T ij is the element of the transformation matrix C b n with i = 1, 2, 3; j = 1, 2, 3.

Modeling of Relative Euler Angle Errors
The coordinate systems and transformation relation for SINS are shown in Figure 3.The misalignment are computational platform angle errors with respect to the navigation frame in the navigation system [33].Generally, it is a small angle from the navigation frame o n x n y n z n to the SINS computational frame o c x c y c z c , and can be expressed by a variable Φ = φ E φ N φ V T with φ i being misalignment along the east, north, and vertical directions.The coordinate transformation matrix from the navigation frame o n x n y n z n to the computational frame o c x c y c z c is referred to as an attitude matrix, and can be written as: where φ E , φ N , φ V are small angles, and cos(φ According to Equations ( 9) and ( 10), the body attitude matrix from the computational frame to the body frame can be calculated as: where ϕ c , θ c , and γ c are the head, pitch, and roll with respect to the computational frame o c x c y c z c , and C ij is an element of the transformation matrix C b c with i = 1, 2, 3; j = 1, 2, 3. ( ) where The The computational angular velocity of the computational frame with respect to the inertial frame denoted in the computational frame has errors, and can be expressed as: where  Since the misalignment Φ from the navigation frame o n x n y n z n to the computational frame o c x c y c z c , according to Equation ( 11), the transformation relation among attitude matrixes can be expressed as: where δC n b represents the error of attitude transformation matrix from the body frame o b x b y b z b to the navigation frame o n x n y n z n .The differential equation of attitude transformation matrix for SINS can be given by: .
The SINS output includes not only sensing the angular velocity ω b ib , but also harmful measurement error δω b ib and random drift of gyros ε b , and can be expressed as: The computational angular velocity of the computational frame with respect to the inertial frame denoted in the computational frame has errors, and can be expressed as: where δω n in is the angular velocity error of the navigation frame with respect to the inertial frame denoted in the navigation frame δω n in = δω n ie + δω n en , and δω n ie is the angular velocity error of the Earth frame with respect to the inertial frame denoted in the navigation frame.δω n en is the angular velocity error of the navigation frame with respect to the earth frame denoted in the navigation frame.
According to Equation ( 15), the differential equation of attitude transformation matrix from the body frame o b x b y b z b to the computational frame o c x c y c z c can be given by: .Substituting Equations ( 16), ( 17) and (19) into Equation (18), the differential equation of the attitude transformation matrix can be rewritten as: .
Substituting Equation (15) into Equation (20), supposing that the attitude error is slight and ignoring the term as small as the second order and the high-order terms of error products, it is easy to obtain the differential equation of attitude transformation matrix errors: Substituting Equation ( 14) into Equation ( 21), the differential equation of attitude transformation matrix errors can be rewritten as: The differential equation of misalignment of SINS can be further rewritten as: . .
The differential equation of misalignment errors can be expanded as: According to Equations ( 16) and ( 17 The angular velocity error vector of the navigation frame with respect to the inertial frame denoted in the body frame can be rewritten as: (27) where ω n in is the angular velocity vector of the navigation frame with respect to the inertial frame denoted in the navigation frame, and can be found in Equation (2).δω n in is an angular velocity error vector of the navigation frame with respect to the inertial frame denoted in the navigation frame, as shown in Equation (17).According to Equation ( 27), the angular velocity error of the navigation frame with respect to the inertial frame denoted in the body frame is rewritten as: where δω b inx , δω b iny , δω b inz are angular velocity errors of the navigation frame with respect to the inertial frame denoted in the body frame.
According to Equations ( 5) and ( 28), the differential equation of the relative Euler angle error caused by angular velocity errors of the body frame with respect to the navigation frame denoted in the body frame, can be written as: According to Equation ( 29), the relative Euler angle error can be written as: where δθ ⊗ , δγ ⊗ , δϕ ⊗ are Euler angle errors caused by δω b nbx , δω b nby , δω b nbz , and these Euler angle errors are called the relative Euler angle errors.

Modeling of Convected Euler Angle Errors
For SINS, there are Euler angle errors between head ϕ c , pitch θ c , and roll γ c with respect to the computational frame o c x c y c z c and head ϕ, pitch θ, and roll γ with respect to the navigation frame o n x n y n z n .These Euler angle errors are different from misalignment, and can be expressed as: According to Equations ( 5) and ( 31), the differential equations of Euler angle errors in head, pitch, and roll can be obtained.The differential equation of the pitch angle error is written by: By Taylor expansion, sin(δγ) ≈ δγ, cos(δγ) ≈ 1 − (δγ) 2 /2, and the differential equation of pitch angle error can be rewritten as: Ignoring the term as small as the second order, the differential equation of pitch angle errors can be written as: Similarly, the differential equation of the roll angle error can be obtained as: Based on Taylor expansion, tan(θ c − δθ) = tan(θ c ) − sec 2 (θ c )δθ, sin(δθ) ≈ δθ, and the differential equation of roll angle error can be rewritten as: Ignoring the term as small as second order, the differential equation of the roll angle error can be rewritten as: Assuming cos(δθ) ≈ 1 − (δθ) 2 /2, cos(δγ) ≈ 1 − (δγ) 2 /2, sin(δθ) ≈ δθ, sin(δγ) ≈ δγ, and the differential equation of the head angle error can be written as: Based on Taylor expansion, sec(θ c − δθ) ≈ sec(θ c ) − sec(θ c ) tan(θ c )δθ, and the differential equation of the head angle error can be rewritten as: When ignoring the term as small as second order, the differential equation of head angle error can be rewritten as: According to Equations ( 34), ( 37) and ( 40), the differential equation of the convected Euler angle errors can be obtained as: According to Equation (41), the convected Euler angle errors can be written as: From Equations ( 41) and ( 42), some conclusions can be obtained.Firstly, the convected Euler angle errors of SINS are affected by ω b nbx and ω b nbz , the angular velocity of the body frame with respect to the navigation frame is denoted in the body frame along the X and Z axes, not ω b nby along the Y axis.Secondly, the convected Euler angle errors are affected by the initial pitch error δθ 0 and roll error δγ 0 , rather than the initial head error δϕ 0 .Finally, the pitch magnitude greatly affects the propagation rules of convected Euler angle errors in the head and roll, but not pitch.
Since the state is an error vector, Equation ( 43) is an approximate expression which ignores the term as small as second order.According to Equations ( 29) and ( 41), the differential equation of the general Euler angle errors, including relative Euler angle errors and convected Euler angle errors, can be modeled as: According to Equation (43), the general Euler angle errors can be calculated by introducing the integral operation: The singularity of the Euler angle error method can be avoided by different Euler angle rotation orders.Similarly, the differential equation of general Euler angle errors and conclusions can be obtained.

Simulation Experiments
The Euler angles are used for large-angle attitude movements, which are direct outputs of SINS.Compared with misalignment, the Euler angle error model is more important in some applications.The general Euler angle error model can be classified as the relative Euler angle error model or the convected Euler angle error model.In order to prove the validity of the proposed model, simulation experiments are carried out.
The simulation experiments are designed in large-angle attitude movements.Firstly, the proposed Euler angle error model can be established according to Equations ( 41)-(44).Then, the initial attitude, the initial Euler angle errors, and the rotational angular velocity can be set to describe large-angle attitude movements.Finally, the simulation experiments are carried out to verify the accuracy of the proposed model.

Simulation and Analysis of Convected Euler Angle Errors
In order to validate the proposed convected Euler angle error models (Equations ( 41) and ( 42)) and their conclusions, four simulation experiments are carried out, where the initial attitude of SINS is supposed by θ 0 γ 0 ϕ 0 = −45 • −45 • −45 • .In the first group simulation, there is the coupling of initial Euler angle errors in three axes δθ 0 δγ 0 δϕ 0 = 0.1 • 0.1 • 0.1 • and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame along the Y axis ω b nby = 1 • /s.The second group simulation presents the coupling of initial head errors δϕ 0 = 0.1 • , the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame The coupling of initial Euler angle errors in three axes δθ 0 δγ 0 δϕ 0 = 0.1 • 0.1 • 0.1 • and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame along the X axis ω b nbx = 1 • /s are given in the third group simulation.In the fourth group simulation, there is the coupling of initial Euler angle errors in three axes δθ 0 δγ 0 δϕ 0 = 0.1 • 0.1 • 0.1 • and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame, ω b nb = 1 • /s 1 • /s 1 • /s .The results of the four group simulation experiments are given in Figures 4-7, respectively.The convected Euler angle error model can precisely describe the attitude error propagation rules of SINS.When the SINS' attitude errors are regarded as error references, the root mean square error (RMSE) of the attitude angle error calculated by the convected Euler angle error model is superior to 0.4441", as shown in Figures 4-7 and Table 1.The angular velocity in the Y axis ω b nby and the initial head error δϕ 0 does not affect the convected Euler angle errors, which are greatly affected by the pitch magnitude.The conclusions above have been validated.
When the SINS' attitude errors are regarded as error references, the root mean square error (RMSE) of the attitude angle error calculated by the convected Euler angle error model is superior to 0.4441″, as shown in Figures 4-7 and Table 1.The angular velocity in the Y axis b nby ω and the initial head error 0 δϕ does not affect the convected Euler angle errors, which are greatly affected by the pitch magnitude.The conclusions above have been validated.
The second group II simulation with the initial head error δϕ 0 .
The fourth group IV simulation with coupling of the initial Euler angle errors and rotation in three axes.

Simulation and Analysis of General Euler Angle Errors
In order to validate the proposed general Euler angle error models (Equations ( 43) and ( 44

Simulation and Analysis of General Euler Angle Errors
In order to validate the proposed general Euler angle error models (Equations ( 43) and (44)), three group simulation experiments are carried out, where the initial attitude and initial Euler angle errors of the SINS are respectively supposed as θ 0 γ 0 ϕ 0 = −45 • −45 • −45 • , δθ 0 δγ 0 δϕ 0 = 0.1 • 0.1 • 0.1 • .In the three experiments, the SINS is driven to rotate with 1 • /s in turn along the X, Y, and Z axes. The

Discussion
There are some limitations of the proposed method.These above-mentioned experiments raised the idea that the proposed method can achieve sufficient accuracy for SINS.However, it does not perform well when there are three singularities.Therefore, the singularity problem of the Euler angle error method cannot be avoided by different Euler angle rotation orders.In practice, it is rare that the Euler angles ϕ , θ , and γ , in turn along with head, pitch, and roll axes, have singularities at 90 degrees at the same time.Meanwhile, Euler angles are the direct output of the attitude change, and Euler angle errors can more accurately and rapidly characterize the attitude error propagation.The computational burden of the proposed method is reduced by avoiding attitude transformation compared with the conventional misalignment method.Consequently, the proposed method is

Discussion
There are some limitations of the proposed method.These above-mentioned experiments raised the idea that the proposed method can achieve sufficient accuracy for SINS.However, it does not perform well when there are three singularities.Therefore, the singularity problem of the Euler angle error method cannot be avoided by different Euler angle rotation orders.In practice, it is rare that the Euler angles ϕ, θ, and γ, in turn along with head, pitch, and roll axes, have singularities at 90 degrees at the same time.Meanwhile, Euler angles are the direct output of the attitude change, and Euler angle errors can more accurately and rapidly characterize the attitude error propagation.The computational burden of the proposed method is reduced by avoiding attitude transformation compared with the conventional misalignment method.Consequently, the proposed method is applicable in most cases.Future work will address its application in a nonlinear environment.

Conclusions
The existing misalignment angle error model cannot directly describe the attitude error propagation, especially in large-angle attitude movements.The misalignment angle is a small angle, which is characterized by the calculation of the three-axis deviation angle between the computational frame and the navigation frame.Compared with the proposed general Euler angle error model, it does not reflect and optimize the body's attitude movement directly.Euler angles are the most intuitive measurements of attitude movements.The Euler angle errors of SINS, different from the misalignment of platform inertial navigation systems, comprise two parts: relative Euler angle errors and convected Euler angle errors.The proposed convected Euler angle error model and the general Euler angle error model, as well as their conclusions, have been validated by the simulation experiments.The general Euler angle error model can accurately describe attitude error propagation rules, which contribute to the design and operation of SINS.The attitude can be measured in real-time, then attitude movements can be optimized in large-angle attitude movements.
The convected Euler angle error, rad

Subscripts i
The inertial frame b The body frame n The navigation frame e The Earth frame c The computational frame

ωω
is the angular velocity vector of the body frame with respect to the inertial frame denoted in the body frame, and can be directly measured by gyros, and n in ω is the angular velocity vector of the navigation frame with respect to the inertial frame denoted in the navigation frame is the angular velocity vector of the Earth frame with respect to the inertial frame denoted in the navigation frame.

Figure 2 .
Figure 2. The Euler angles transformation relation for SINS.

Figure 2 .
Figure 2. The Euler angles transformation relation for SINS.

Figure 3 .
Figure 3.The coordinate systems and transformation relation for SINS.Since the misalignment Φ from the navigation frame n n n n o x y z to the computational frame c c c c o x y z , according to Equation (11), the transformation relation among attitude matrixes can be expressed as: of attitude transformation matrix from the body frame b b b b o x y z to the navigation frame n n n n o x y z .The differential equation of attitude transformation matrix for SINS can be given by: SINS output includes not only sensing the angular velocity b ib ω , but also harmful measurement error b ib δω and random drift of gyros b ε , and can be expressed as: in δω is the angular velocity error of the navigation frame with respect to the inertial frame denoted in the navigation frame angular velocity error of the Earth frame with respect to the inertial frame denoted in the navigation frame.n en δω is the angular velocity error of the navigation frame with respect to the earth frame denoted in the navigation frame.

Figure 3 .
Figure 3.The coordinate systems and transformation relation for SINS.
), the angular velocity error vector of the body frame o b x b y b z b with respect to the navigation frame o n x n y n z n denoted in the body frame o b x b y b z b is a main factor leading to the Euler angle error, and can be expressed by: RMSE of attitude angle error(〞) RMSE of head error calculated by convected model RMSE of pitch error calculated by convected model RMSE of roll error calculated by convected model 0 0 0 = -45 -45 -45

Figure 4 .
Figure 4.The first group I simulation with rotation in the Y axis

Figure 4 .
The first group I simulation with rotation in the Y axis ω b nby .RMSE: root mean square error.Appl.Sci.2018, RMSE of attitude angle error(〞) RMSE of head error calculated by convected model RMSE of pitch error calculated by convected model RMSE of roll error calculated by convected model 0 0 0 = -45 -45 -45

Figure 5 .
Figure 5.The second group II simulation with the initial head error

Figure 5 .
Figure 5.The second group II simulation with the initial head error 0 δϕ .
Attitude angle error(〞) Head error of SINS Pitch error of SINS Roll error of SINS Head error calculated by convected model Pitch error calculated by convected model Roll error calculated by convected model angle error(〞) RMSE of head error calculated by convected model RMSE of pitch error calculated by convected model RMSE of roll error calculated by convected model 0 0 0 = -45 -45 -45

1 δθFigure 7 .
Figure 7.The fourth group IV simulation with coupling of the initial Euler angle errors and rotation in three axes.

Figure 8 .
Figure 8.The Euler angle error propagation with coupling of three initial Euler angle errors and rotation in the X axis.

Figure 9 .
Figure 9.The Euler angle error propagation with coupling of three initial Euler angle errors and rotation in the Y axis.

Figure 9 .
Figure 9.The Euler angle error propagation with coupling of three initial Euler angle errors and rotation in the Y axis.

Figure 10 .
Figure 10.The Euler angle error propagation with the coupling of three initial Euler angle errors and rotation in the Z axis.

Figure 10 .
Figure 10.The Euler angle error propagation with the coupling of three initial Euler angle errors and rotation in the Z axis.

Table 1 .
The third group III simulation with rotation in the X axis Attitude angle errors (RMSE (root mean square error)) solved by the convected Euler angle error model.The third group III simulation with rotation in the X axis ω b nbx .

Table 1 .
Attitude angle errors (RMSE (root mean square error)) solved by the convected Euler angle error model.
results of three group simulation experiments are given in Figures 8-10.The general Euler angle error model can precisely describe the attitude error propagation rules of the SINS.When the SINS' attitude error is regarded as the reference, the RMSE of attitude angle errors calculated by the general Euler angle error model is shown in Figures 8-10 and Table 2.The RMSE of the proposed Euler angle error model is superior to 0.3195", which is accurate enough.The simulation results prove the validity of the general Euler angle error model proposed.The general Euler angle error model can accurately describe attitude error propagation rules, and it can be applied in the attitude movements.The proposed Euler angle error model can direct the response attitude movement status, especially in large-angle attitude movements.The attitude movement optimization can also be analyzed.

Table 2 .
Attitude angle errors (RMSE) calculated by the general Euler angle error model.
* (") Pitch Error δθ * (") Roll Error δγ * (") Maximum Error (") The Euler angle error propagation with coupling of three initial Euler angle errors and rotation in the X axis.The Euler angle error propagation with coupling of three initial Euler angle errors and rotation in the X axis.