Accuracy Analysis of Attitude Computation-based on Optimal Coning Algorithm

To accurately evaluate the applicability of optimal coning algorithms, the direct influence of their periodic components on attitude accuracy is investigated. The true value of the change of the rotation vector is derived from the classical coning motion for analytic comparison. The analytic results show that the influence of periodic components is mostly dominant in two types of optimal coning algorithms. Considering that the errors of periodic components cannot be simply neglected, these algorithms are categorized with simplified forms. A variety of simulations are done under the classical coning motion. The numerical results are in good agreement with the analytic deductions. Considering their attitude accuracy, optimal coning algorithms of the 4-subinterval and 5-subinterval algorithms optimized with angular increments are not recommended for use for real application.

Attitude computation is a critical issue in the strapdown inertial navigation system (SINS). In the principal software function of the SINS, attitude computation is necessary for the transformation of acceleration which is twice integrated into velocity and position. Furthermore, noncommutativity of finite rotations is an inevitable phenomenon in the process of attitude computation. Therefore, how to design a special and efficient algorithm has been attracted over the past decades.

Many works appeared in the literatures describing the attitude computation based on the rotation vector proposed by Bortz1. In 1983, Miller2 firstly developed an approach which updated the attitude quaternion with the change of the rotation vector during the updating period. The change of the rotation vector consisting of the outputs of incremental gyros was optimized under the classical coning motion2 (called the optimal coning algorithm). Afterward, Ignagni3-5 provided a convenient method to simplify Miller’s derivation and made an assessment of the error inherent in the simplified form of the Bortz equation1. Several improved algorithms were also presented, additionally using the current and previous accumulated gyro outputs6 or a higher-order term when the angular rates were known analytically7. To obtain the incremental signals involved in Miller’s algorithm from the rate outputs of some modern-day gyros, hardware-based integrators are imperative. Addressing this issue, Huang and Deng8 presented an improved form including angular increments and angular rates. But in these cases, the use of hardware-based integrators has contributed to increase the cost and complexity of system. To overcome this problem, Zeng9, et al. stated a coning algorithm optimized with angular rates, which included a polynomial fitting procedure and a solution for coning compensation coefficients. Generalized method for this algorithm was detailedly introduced by Ben10, et al.

Although the mentioned kinds of algorithm worked well to minimize the residual error of one nonperiodic component, the accuracy of attitude computation in the case of multi-subinterval was rarely discussed. In this study, we analyze the influence of periodic components in the optimal coning algorithms, mainly aiming to evaluate the accuracy of attitude computation based on each algorithm.

The classical coning motion is a typical environment for the optimization and effectiveness test of the coning algorithm, which can be characterized by the unit vector

$\frac{a\left(t\right)}{a}=\left[\begin{array}{c}0\\ \mathrm{cos}\left(\gamma t\right)\\ \mathrm{sin}\left(\gamma t\right)\end{array}\right]$                (1)

where a is the coning half-angle equaling the module of the vector a, i.e., a = (a.a)1/2, γ = 2πf, and f is the coning frequency1. Based on the classical coning motion, the attitude quaternion q(t) for coordinate transformation can be formulated as

$q\left(t\right)=\left[\begin{array}{c}\mathrm{cos}\left(\frac{a}{2}\right)\\ \mathrm{sin}\left(\frac{a}{2}\right)a\left(t\right)\end{array}\right]=\left[\begin{array}{c}\mathrm{cos}\left(\frac{a}{2}\right)\\ 0\\ \mathrm{sin}\left(\frac{a}{2}\right)\mathrm{cos}\left(\gamma t\right)\\ \mathrm{sin}\left(\frac{a}{2}\right)\mathrm{sin}\left(\gamma t\right)\end{array}\right]$                (2)

and the angular rate ω(t) representing the relative angular rate between two coordinate frames can be derived as 2

$\omega \left(t\right)=2{q}^{\ast }\left(t\right)\circ \stackrel{˙}{q}\left(t\right)=\left[\begin{array}{c}-2\gamma {\mathrm{sin}}^{2}\left(\frac{a}{2}\right)\\ -\gamma \mathrm{sin}a\mathrm{sin}\left(\gamma t\right)\\ \gamma \mathrm{sin}a\mathrm{cos}\left(\gamma t\right)\end{array}\right]$                (3)

If only angular motion is considered, attitude quaternion can be simply updated as follows

$q\left(t+\Delta T\right)=q\left(t\right)\circ q\left(\Delta T\right)$         (4)

where $\circ$ denotes quaternion multiplication,q(t+ΔT) and q(t) are the attitude quaternions at time t+ΔT and time t, respectively, ΔT is the updating period of attitude quaternion, and q(ΔT) is the quaternion representing the change of the attitude quaternion during time interval [t, t+ΔT], which can be obtained by2

$q\left(\Delta T\right)=\left[\begin{array}{c}\mathrm{cos}\left(\frac{\Delta \sigma }{2}\right)\\ \frac{\Delta \sigma }{\Delta \sigma }\mathrm{sin}\left(\frac{\Delta \sigma }{2}\right)\end{array}\right]$         (5)

where Δσ is the change of the rotation vector during time interval [t, t+ΔT] with module Δσ = (Δσ.Δσ)1/2. When Δσ is near zero, power series expansions should be applied to the trigonometric function coefficients of Eqn (5) to avoid singularity. In this study, a fourth-order truncation is utilized

$\begin{array}{c}\mathrm{cos}\left(\frac{\Delta \sigma }{2}\right)=1-\frac{{\left(\Delta \sigma \right)}^{2}}{8}+\frac{{\left(\Delta \sigma \right)}^{4}}{384}\\ \frac{1}{\Delta \sigma }\mathrm{sin}\left(\frac{\Delta \sigma }{2}\right)=\frac{1}{2}-\frac{{\left(\Delta \sigma \right)}^{2}}{48}\end{array}$         (6)

The theoretical value of Δσ can be derived as Eqn (7) by integrating the Bortz equation1 from time t to time t+ΔT and making some simplifications3

$\begin{array}{cc}\Delta \overline{\sigma }=\alpha +\frac{1}{2}{\int }_{t}^{t+\Delta T}\alpha \left(\tau \right)×\omega d\tau ,& \alpha \left(\tau \right)={\int }_{t}^{\tau }\omega \left(\upsilon \right)d\upsilon \end{array}$        (7)

where ω is similar to the angular rate described in Eqn (3), α(τ) is the instantaneous integration of ω from time t, and α = α(t+ΔT). Considering the classical coning environment,Eqn (7) can be rewritten as

$\begin{array}{c}\Delta \overline{\sigma }=\left[\begin{array}{c}\Delta {\overline{\sigma }}_{x}\\ \Delta {\overline{\sigma }}_{y}\\ \Delta {\overline{\sigma }}_{z}\end{array}\right]\\ =\left[\begin{array}{c}-2\gamma \Delta T{\mathrm{sin}}^{2}\left(\frac{a}{2}\right)+\frac{1}{2}{\mathrm{sin}}^{2}a\left(\gamma \Delta T-\mathrm{sin}\left(\gamma \Delta T\right)\right)\\ -2\mathrm{sin}a\mathrm{sin}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\left(\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)+{\mathrm{sin}}^{2}\left(\frac{a}{2}\right)\left(-\gamma \Delta T\mathrm{cos}\left(\frac{\gamma \Delta T}{2}\right)+2\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\right)\right)\\ 2\mathrm{sin}a\mathrm{cos}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\left(\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)+{\mathrm{sin}}^{2}\left(\frac{a}{2}\right)\left(-\gamma \Delta T\mathrm{cos}\left(\frac{\gamma \Delta T}{2}\right)+2\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\right)\right)\end{array}\right]\end{array}$        (8)

On the other hand, the quaternion q(ΔT) can be calculated from Eqns (4) and (2)

$q\left(\Delta T\right)={q}^{\ast }\left(t\right)\circ q\left(t+\Delta T\right)\text{=}\left[\begin{array}{c}1-2{\mathrm{sin}}^{2}\left(\frac{a}{2}\right){\mathrm{sin}}^{2}\left(\frac{\gamma \Delta T}{2}\right)\\ -{\mathrm{sin}}^{2}\left(\frac{a}{2}\right)\mathrm{sin}\left(\gamma \Delta T\right)\\ -\mathrm{sin}a\mathrm{sin}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\\ \mathrm{sin}a\mathrm{cos}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\end{array}\right]$        (9)

where * denotes conjugate operator2 For brevity, our discussions are confined to the single-speed-N-subinterval coning algorithm. Thus dividing time interval [t, t+ΔT] into N subintervals of equal width Δt = ΔT/N, angular increments αi (i = 1, 2, …, N) and angular rates ωi (i = 0, 1, …, N) can be acquired from Eqn (3) to optimize the following coning algorithms. Here Δt is just the sampling time.

3.1  Coning Algorithms Optimized with Angular Increments

Under a simplified coning motion, Ignagni3 validated that the value of the cross product of two angular increments depended only on their spacing. A similar property can be derived under the classical coning motion, just referring to the nonperiodic component of αi×αj. Using this property, the change of the rotation vector during time interval [t, t+ΔT] is approximated

Eqn (10) $\Delta \stackrel{˜}{\sigma }=\sum _{i=1}^{N}{\alpha }_{i}+\sum _{i=1}^{N-1}{K}_{i}\left({\alpha }_{i}×{\alpha }_{N}\right)$        (10)

where Ki (i = 1, 2, …, N−1) are constant coefficients. If γΔT < 1, the coefficients Ki can be optimized by using power series expansions and minimizing the error of the nonperiodic component between Eqns (8) and (10).
The optimal coning algorithms in the form of Eqn (10) are listed in Table 1, where N = 1, 2, 3, 4, 5. In this study, the absolute value of the residual error along nonperiodic component is defined as the error drift of algorithm (EDOA).

3.2 Coning Algorithms Optimized with Angular Rates

Considering the given rate signals, the definite integral of a fitted polynomial is used to determine the accumulated angular increment αs during time interval [t, t+ΔT]. Furthermore, noticing that ωi×ωj under the classical coning motion has a property similar to αi×αj, the change of the rotation vector is approximately assumed as follows

$\Delta \stackrel{˜}{\sigma }={\alpha }_{s}+{\left(\Delta T\right)}^{2}\sum _{i=0}^{N-1}{M}_{i}\left({\omega }_{i}×{\omega }_{N}\right)$        (11)

where Mi (i = 0, 1,..., N−1) are constant coefficients. Finally, taking the same as what has been done in the derivation of Ki, the coefficients Mi can be optimally solved.9,10
The coning algorithms in the optimal expression of Eqn (11) and the corresponding EDOAs are shown in Table 2, where N = 1, 2, 3, 4.

Comparing the EDOAs in Table 2 with those in Table 1 indicates that the coning algorithms optimized with angular rates may be superior in the accuracy of attitude computation on condition γΔT < 1. This conjecture is based on the conventional assumption6 that the errors of periodic components in the optimal coning algorithms are negligible. However, almost no attention has been paid to examining this negligibility up till now.

4.1  Change of Rotation Vector

Based on Eqns (5) and (9), the true value of the change of the rotation vector can be deduced as

$\Delta \sigma =\left[\begin{array}{c}\Delta {\sigma }_{x}\\ \Delta {\sigma }_{y}\\ \Delta {\sigma }_{z}\end{array}\right]=\left(1+\xi \right)\Delta \Phi$        (12)
Where
$\Delta \Phi =\left[\begin{array}{c}\Delta {\Phi }_{x}\\ \Delta {\Phi }_{y}\\ \Delta {\Phi }_{z}\end{array}\right]=\left[\begin{array}{c}-2{\mathrm{sin}}^{2}\left(\frac{a}{2}\right)\mathrm{sin}\left(\gamma \Delta T\right)\\ -2\mathrm{sin}a\mathrm{sin}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\\ 2\mathrm{sin}a\mathrm{cos}\left(\gamma \left(t+\frac{\Delta T}{2}\right)\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\end{array}\right]$        (13)

$\xi =\frac{\frac{\Delta \sigma }{2}}{\mathrm{sin}\left(\frac{\Delta \sigma }{2}\right)}-1=\frac{{\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{a}{2}\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\right)}{\mathrm{sin}\left(\frac{a}{2}\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\sqrt{1-{\left(\mathrm{sin}\left(\frac{a}{2}\right)\mathrm{sin}\left(\frac{\gamma \Delta T}{2}\right)\right)}^{2}}}-1$        (14)

Yan13, et al. took the vector ΔΦ denoted by Eqn (13) as the true value of the change of the rotation vector to analyze the residual error along nonperiodic component of Miller’s algorithm. This approximation seems necessary for further derivation in this case. However, in the accuracy analysis of attitude computation, the difference between the vectors Δσ and ΔΦ in Eqn (12) (i.e., the coefficient ξ) cannot be ignored.
As described in Eqn (14), the coefficient ξ is a function of the coning frequency f = γ/(2π) and the coning half-angle a when the updating period ΔT is invariant. This functional relation is illustrated in Fig. 1, where f varies in a wide range from 0.1 Hz to 10 Hz, a changes from 0.1° to 15°, and ΔT is 0.01 s. The surface in Fig. 1 shows that the coefficient ξ increases rapidly, and reaches its maximum value of 1.05098×10−3 as the coning frequency and coning half-angle grow. To a great extent, it demonstrates that the coefficient ξ is non-ignorable.

4.2 Categorization of Algorithms

Equation (8) is used to express the theoretical limit of the change of the rotation vector under the classical coning motion. As shown in Eqns (8) and (12), there is a difference between the theoretical value and the true value. To clarify this difference, we take advantage of the vector ΔΦ denoted by Eqn (13) and define the following parameters

${\overline{\xi }}_{x}=\frac{\Delta {\overline{\sigma }}_{x}}{\Delta {\Phi }_{x}}-1$, ${\overline{\xi }}_{y}=\frac{\Delta {\overline{\sigma }}_{y}}{\Delta {\Phi }_{y}}-1$, ${\overline{\xi }}_{z}=\frac{\Delta {\overline{\sigma }}_{z}}{\Delta {\Phi }_{z}}-1$        (15)

Substituting Eqn (15) into Eqn (8) yields

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\stackrel{˜}{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\overline{\xi }}_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\overline{\xi }}_{z}\right)\Delta {\Phi }_{z}\end{array}\right]$         (16)

Equation (16) shows that$\nabla \stackrel{-}{\sigma }$ can be written in a form similar to Eqn (12). So the deviation of$\nabla \stackrel{-}{\sigma }$ from the true value Δσ can be substantiated by comparing the parameters ${\overline{\xi }}_{x}$ , ${\overline{\xi }}_{y}$ and ${\overline{\xi }}_{z}$ with the coefficient ξ. For example, in an ideal coning environment where f = 2 Hz, a = 1°, and ΔT = 0.01 s, the approximate results can be obtained that ${\overline{\xi }}_{x}=2.00795×{10}^{-7}$ , ${\overline{\xi }}_{y}={\overline{\xi }}_{z}=2.00478×{10}^{-7}$ , and $\xi =2.00162×{10}^{-7}$ . It can be seen that the order of magnitude of the deviations along all three axes is 10−10.

This procedure can also be used to analyze the algorithms listed in Tables 1 and 2. For the convenience of further discussion, these algorithms are uniformly denoted by

$\Delta \stackrel{˜}{\sigma }=\alpha +\Delta {\sigma }_{s}$        (17)

Where
$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\Delta {\stackrel{˜}{\sigma }}_{x}\\ \Delta {\stackrel{˜}{\sigma }}_{y}\\ \Delta {\stackrel{˜}{\sigma }}_{z}\end{array}\right]$, $\alpha$, α=[ α x $\Delta {\sigma }_{s}=\left[\begin{array}{c}\Delta {\sigma }_{sx}\\ \Delta {\sigma }_{sy}\\ \Delta {\sigma }_{sz}\end{array}\right]$        (18)

and Δσs and α are the sum of all cross products and the rest, respectively. Furthermore, five parameters are defined as follows

${\eta }_{y}=\frac{{\alpha }_{y}}{\Delta {\Phi }_{y}}-1$        (19)

${\eta }_{y}=\frac{{\alpha }_{y}}{\Delta {\Phi }_{y}}-1$, ${\eta }_{z}=\frac{{\alpha }_{z}}{\Delta {\Phi }_{z}}-1$        (20)

${\mu }_{y}=\frac{\Delta {\sigma }_{sy}}{{\alpha }_{y}}$, ${\mu }_{z}=\frac{\Delta {\sigma }_{sz}}{{\alpha }_{z}}$       (21)

Substituting Eqns (19-21) into Eqn (17), the relationship between$∇ σ −$ and ΔΦ can be described as

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\stackrel{˜}{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\eta }_{y}+{\mu }_{y}+{\eta }_{y}{\mu }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\eta }_{z}+{\mu }_{z}+{\eta }_{z}{\mu }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]$       (22)

So comparing the parameters ${\stackrel{˜}{\xi }}_{x}$ ,${\eta }_{y}+{\mu }_{y}+{\eta }_{y}{\mu }_{y}$ and ${\eta }_{z}+{\mu }_{z}+{\eta }_{z}{\mu }_{z}$ of each algorithm with the coefficient ξ, we can find its deviation from the true value Δσ.

As shown in Table 3 ${\stackrel{˜}{\xi }}_{x}$ , gradually approaches the parameter ${\overline{\xi }}_{x}$ calculated above as the number of subintervals increases, for both two types of algorithms. This result proves that the limit of the parameter ${\stackrel{˜}{\xi }}_{x}$ is not ξ but ${\stackrel{˜}{\xi }}_{x}$ . At the same time, low-magnitude oscillations occur in the parameters μy and μz, and the absolute values of the parameters ηy and ηz of the second type decline rapidly. Note that the deviations of the coning algorithms along y and z axes are determined by these four parameters.

Taking the limit denoted by Eqn (8) into account, the algorithms involved in Table 3 can be categorized into four groups as compared with the true value Δσ:
1) ${\stackrel{˜}{\xi }}_{x}-\xi \gg \left({\eta }_{y}+{\mu }_{y}+{\eta }_{y}{\mu }_{y}\right)-\xi$ and ${\stackrel{˜}{\xi }}_{x}-\xi \gg \left({\eta }_{z}+{\mu }_{z}+{\eta }_{z}{\mu }_{z}\right)-\xi$ .The 1-subinterval and 2-subinterval algorithms optimized with angular increments are ascribed to this group. Accordingly, Eqn (22) can be simplified as

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\stackrel{˜}{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\overline{\xi }}_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\overline{\xi }}_{z}\right)\Delta {\Phi }_{z}\end{array}\right]=\left[\begin{array}{c}\Delta {\stackrel{˜}{\sigma }}_{x}\\ \Delta {\overline{\sigma }}_{y}\\ \Delta {\overline{\sigma }}_{z}\end{array}\right]$       (23)

2)${\eta }_{y}\gg {\mu }_{y}$ and ${\eta }_{z}\gg {\mu }_{z}$ . This refers to the 1-subinterval algorithm optimized with angular rates. The corresponding approximation of Eqn (22) is

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\overline{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\eta }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\eta }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]=\left[\begin{array}{c}\Delta {\overline{\sigma }}_{x}\\ \left(1+{\eta }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\eta }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]$       (24)

3) The parameters ηy and μy are not negligible as compared with ηz and μz, respectively. The 2-subinterval and 3-subinterval algorithms optimized with angular rates are included in this group, where the simplified form of Eqn (22) can be expressed as

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\overline{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\eta }_{y}+{\mu }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\eta }_{z}+{\mu }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]=\left[\begin{array}{c}\Delta {\overline{\sigma }}_{x}\\ \left(1+{\eta }_{y}+{\mu }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\eta }_{z}+{\mu }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]$       (25)

4) ${\eta }_{y}\ll {\mu }_{y}$ and ${\eta }_{z}\ll {\mu }_{z}$ . This group is composed of the 3-subinterval, 4-subinterval, and 5-subinterval algorithms optimized with angular increments, and the 4-subinterval algorithm optimized with angular rates. In this case, the approximated expression for Eqn (22) is

$\Delta \stackrel{˜}{\sigma }=\left[\begin{array}{c}\left(1+{\overline{\xi }}_{x}\right)\Delta {\Phi }_{x}\\ \left(1+{\mu }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\mu }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]=\left[\begin{array}{c}\Delta {\overline{\sigma }}_{x}\\ \left(1+{\mu }_{y}\right)\Delta {\Phi }_{y}\\ \left(1+{\mu }_{z}\right)\Delta {\Phi }_{z}\end{array}\right]$       (26)

As shown in Eqns (24-26), the difference between the optimal coning algorithm and the theoretical value $\Delta \overline{\sigma }$ is not revealed by its component on x axis any longer. In other words, the influence of periodic components in these algorithms is dominant. It should be noted that all algorithms optimized with angular rates are included in the last three groups.

To validate the influence of periodic components, a variety of pitch-error simulations are carried out based on the two types of algorithms and their simplified forms denoted by Eqns (23-26).
In the simulation, the pitch error is defined as $\delta \theta =|\theta -\stackrel{˜}{\theta }|$ , where θ and $\stackrel{˜}{\theta }$ are the pitches calculated from

In the simulation, the pitch error is defined as , where θ and are the pitches calculated from Eqns (2) and (4) at the same time, respectively. Table 4 shows the ideal coning environments for simulation. The updating period of attitude quaternion is 0.01 s, and the duration of simulation is 36 s.

The numerical results in the coning environment 3# are illustrated in Figs. 2 and 3. For brevity’s sake, we omit the results in the other coning environments because of their high degree of similarity to those in the environment3.

Figure 2. Pitch errors (semilog plot in vertical axis) based on the coning algorithms optimized with angular increments (CAOWAI) and their simplified forms (SF) for: (a) N = 1, b) N = 2, (c) N = 3, (d) N = 4, and (e) N = 5.

In Fig. 2, the pitch errors based on the simplified forms (SF) are close to those based on the coning algorithms optimized with angular increments (CAOWAI), except for the case N = 3. This discrepancy is related to the corresponding parameter ${\stackrel{˜}{\xi }}_{x}$

in Table 3, which reveals that the optimization of nonperiodic component in the 3-subinterval algorithm is inadequate. In contrast, the curves in Fig. 3 show good agreement between the pitch errors based on the coning algorithms optimized with angular rates (CAOWAR) and their simplified forms (SF).
Fig. 2 reveals that when N exceeds 3

the CAOWAIs do not have substantial improvement in pitch accuracy any longer. Thus the 4-subinterval and 5-subinterval algorithms are not recommended on the consideration of their complex structures (see Table 1). On the other hand, comparing with Fig. 2, the CAOWAR with identical N is not obviously superior in pitch accuracy. This is not consistent with the conjecture stated at the end of section 3. The explanation for this phenomenon is that the periodic components in CAOWARs are dominant factors, just as what has been discussed in section 4. So it is necessary to further optimize these periodic components.

conjecture stated at the end of section 3. The explanation for this phenomenon is that the periodic components in CAOWARs are dominant factors, just as what has been discussed earlier. So it is necessary to further optimise these periodic components.

This study analyzes the accuracy of attitude computation based on two types of optimal coning algorithms under the classical coning motion. After deriving the true value of the change of the rotation vector, we explore the influence of periodic components in these algorithms using analytical comparison and categorization. Analytical results indicate that the influence of periodic components is dominant in these algorithms except for the 1-subinterval and 2-subinterval ones optimized with angular increments. Moreover, numerical tests are constructed, and the results agree well with the analyses. Allowing for the comparison of accuracy, the 4-subinterval and 5-subinterval algorithms optimized with angular increments are not recommended for use. To improve the accuracy of attitude computation, future work will concentrate on the optimization of periodic components in existing coning algorithms.

This work was supported in part by the National Basic Research Program of China (973 Program) (2009CB724002), the National Natural Science Foundation of China (50975049), the Specialized Research Fund for the Doctoral Program of Higher Education (20110092110039) and Ocean special funds for scientific research on public causes (201205035-09) and the Aeronautical Science Fund of China (20090869008). The authors would like to thank Tianhong Pan for his insightful comments and constructive suggestions. Thanks also to Feng Li for his help in partial work of the numerical simulations.

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