# A Generalized Coning Correction Structure for Attitude Algorithms

Authors: Chuanye Tang; Xiyuan Chen
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A new coning correction structure is presented for attitude update coning correction. Different from the previous rate-based and increment-based coning correction structures, the new structure contains cross-product of angular rates, cross-product of angular increments, and cross-product of angular rate and increment (an angular increment may be approximated from angular rate samples). Two types of optimization methods including time Taylor-series method and frequency Taylor-series method were utilized to design the structure coefficients including the uncompressed and the compressed. Two types of algorithm error models including one applicable to coning environments and the other two applicable to maneuver environments were defined and used for analyzing or evaluating the algorithm performance. The derivation procedure of a rotation vector magnitude extraction method is included. Analysis and simulation results indicate that the new structure-based algorithm with the compressed coefficients designed by using frequency Taylor-series method gives a superior algorithm performance in coning environments and maneuver environments.


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## Body

## 1. Introduction

The so-called two-stage algorithm structure for modern strapdown inertial navigation attitude updating was introduced by Jordan [1] and Bortz [2] in 1969. One stage is based on an exact updating formula to update the attitude in an attitude updating cycle, and the other stage to be imported into the former is based on an approximate computation formula to calculate the rotation vector including an integrated gyro sensed angular rate and a coning correction calculated from gyro data. Currently there exist two classes of essentially equivalent structures for rotation vector computation: two-speed structure and single-speed structure. Because of the low-speed arithmetic capability of early computers, the coning correction was generally based on the two-speed structure [3–5] where there are some cycles executed for coning correction in a rotation vector computing cycle (or an attitude updating cycle), and this processing way allows lower attitude updating rate and computation load when algorithm accuracy holds. As the rise of arithmetic capability of modern computers, the single-speed structure [5] where there are only cycle executed for coning correction in a rotation vector computing cycle may be applicable to the coning correction substituted for the two-speed structure, and this way can meet the demand of high attitude updating rate but makes bigger computation load.In the latest several decades, attitude algorithm study activity has concentrated on designing an efficient coning correction structure and achieving the optimized structure coefficients for computing coning correction under the aforementioned rotation vector computation structures. The earliest optimization method (time Taylor-series method) for the coning correction coefficient design uses truncated Taylor series of time to approximate gyro sensed angular rate under the assumption that the body is undergoing a motion of angular rate of finite order power series law. In 1983, Miller [6] introduced the method (coning frequency Taylor-series method) of using truncated Taylor series of coning frequency to approximate coning correction error for the coning correction coefficient design under a constrained coning motion condition. In 1990 and 1996, on the basis of Miller’s idea, Ignagni [3, 4] discussed several classes of coning correction algorithm structures under the two-speed structure and realized the coning correction coefficient optimization design under an unconstrained coning motion condition. In 2010, Savage [5] further expanded Miller’s idea and raised an idea of optimizing coning correction algorithms under the single-speed structure based on least squares principle in a given composite coning environment and achieving a balanced algorithm performance over a coning frequency range. In 2013, Song et al. [7] renew the time Taylor-series method combined with Miller’s and Savage’s methods to achieve a superior maneuver accuracy, as well as retaining the optimal coning motion performance. Additionally, in 2009, Ben et al. [8] developed an angular rate-based coning correction structure similar to the angular increment-based coning correction structure used in [3–6] and introduced Miller’s idea into coning algorithm design for coning correction using cross product of angular rates rather than cross product of angular increments used in algorithms referred in [3–6].For modern strapdown inertial navigation systems, single type of angular increment gyros or angular rate gyros is generally used for constructing gyro sensor assembly [9]. Along with the advancement of gyro technology, high precise rate gyros and increment gyros may be jointly used as a portion for constructing multi type sensor assembly. This paper brings a new class of coning correction structure under the single-speed structure. The new structure contains cross product of angular rates, cross product of angular increments, and cross product of angular rate and increment for coning correction and needs both inputs of angular rate and angular increment which may be also approximated from gyro sensed angular rate (it means single type of angular rate gyros can also meet the demand of the new coning correction structure). When angular rate input is overlooked, the new structure can be simplified as some class of angular increment-based structure. In addition, the new structure includes uncompressed coning correction structure and compressed one based on coning motion characteristics. So the new structure has high flexibility and applicability which allow pure angular rate input, pure angular increment input, or composite angular rate and increment input. Two types of aforementioned coning correction optimization methods: time Taylor-series method and coning frequency Taylor-series method, were utilized to design the coning correction coefficients of the new structure. The algorithm performance is evaluated for the new structure under coning environments and maneuver environments.


## 2. Attitude Algorithm Structure

The classical attitude updating numerical computing formula in modern strapdown inertial navigation systems is given by [1, 2, 5]
(1)Cb(l)n=Cb(l-1)n·Cb(l)b(l-1),Cb(l)b(l-1)=I+f1(ϕl)(ϕl×)+f2(ϕl)(ϕl×)2,f1(ϕl)=sin|ϕl||ϕl|=∑i=1(-1)i-1|ϕl|2(i-1)(2i-1)!,f2(ϕl)=1-cos⁡|ϕl||ϕl|2=∑i=1(-1)i-1|ϕl|2(i-1)(2i)!,ϕl×=[0-ϕzϕyϕz0-ϕx-ϕyϕx0],
where n is a navigation coordinate frame, b is a body coordinate frame, Cb(l)n and Cb(l-1)n are, respectively, an attitude direction cosine matrix at the end of attitude updating cycle l and cycle l-1, Cb(l)b(l-1) and ϕl which used to update Cb(l)n from Cb(l-1)n are, respectively, an updating attitude direction cosine matrix and an updating rotation vector from the ending time of cycle l-1 to the ending time of cycle l, |ϕl| is the magnitude of vector ϕl, and ϕl× is the cross-product antisymmetry matrix composed of ϕl components.The updating rotation vectorϕl is generally computed by using a simple structure to approximate the integral of the rotation vector differential equation. A commonly used single-speed structure [1, 2, 5] is given by
(2)ϕl=αl+δϕl,αl=α(tl,tl-1),δϕl≈12∫tl-1tlα(t,tl-1)×ωdt,α(t,tl-1)=∫tl-1tωdt,
where t is a time, αl is the integral of the gyro sensed angular rate ω from time tl-1 to time tl, and δϕl denotes the coning correction.


## 3. Coning Correction Structure

For the requirement of strapdown system with multitypes of gyros, the new generalized computation algorithm formula for updating rotation vectorϕl with the form of the integrated angular rate αl and the coning correction δϕ^l is proposed as
(3)ϕl=αl+δϕ^l,αl=∑k=N-L+1NΔαk,δϕ^l=Tk2∑i=1M-1∑j=i+1Mηijωi×ωj+∑i=1N-1∑j=i+1NμijΔαi×Δαj+Tk∑i=1M∑j=1Nξijωi×Δαj,
where each Δα is an angular increment sample over a fixed time interval, the Δαs are adjacent and spaced sequentially forward in time, ΔαN-L+1 begins at time tl-1, ΔαN ends at time tl, the ωs are some angular rate samples which are adjacent, spaced sequentially forward in time, and time interval-equivalent, ωM lies at time tl, ηijs, μijs, and ξijs are coefficients depending on the coning correction structure, L is the number of angular increment samples selected to compute αl in cycle l, N, selected to be equal to or greater than L, is the number of angular increment samples selected to compute δϕ^l in cycle l, M, selected to be equal to or greater than L, is the number of angular rate samples selected to compute δϕ^l in cycle l, and Tk is the angular rate sample time interval set to compute δϕ^l in cycle l. Here, it is alternative that M can be configured to equal to N+1, and the angular increment sample interval can be configured to Tk. Right now, a sketch map, as Figure 1 where Tl equivalent to the product LTk, is a time interval from time tl-1 to time tl or the attitude updating cycle and the other signs are defined as those in (3), which has described clearly the distribution relationship of the angular increment sample series Δαs and the angular rate sample series ωs against time. This paper will mainly study the class of coning correction structure with the parameters set above, which is called the angular rate-and-increment-synchronized structure.Figure 1
Distributions ofΔα series and ω series against time.For the attitude updating algorithm with pure angular rate inputs, the integrated angular rateαl in (2) can be approximated by using an integrator to integrate the gyro sensed angular rate samples from time tl-1 to time tl, and the coning correction δϕ^l can be computed with the same structure as the angular rate-and-increment-synchronized structure used in (3). This angular rate-based computation algorithm formula for updating rotation vector has the form of the integrated angular rate αl and the coning correction δϕ^l:
(4)ϕl=αl+δϕ^l,αl=∑k=N-L+1NΔαk,Δαk=Tk∑r=0Rsrωrk,Tk=RT,δϕ^l=Tk2∑i=1M-1∑j=i+1Mηijωi×ωj+∑i=1N-1∑j=i+1NμijΔαi×Δαj+Tk∑i=1M∑j=1Nξijωi×Δαj,
where ωks are gyro sensed angular rate samples which are adjacent, spaced sequentially forward in time, and time interval-equivalent and ωs are gyro sensed angular rate subsamples which are selected from ωks, also adjacent, spaced sequentially forward in time, and time interval-equivalent; both ωRk and ωk+1 lie at time tl-(N-k)Tk (i.e., ωk+1=ωRk; both ωRN and ωM lie at time tl) and both ω0k and ωk lie at time tl-(N-k+1)Tk (i.e., ωk=ω0k; both ω0M-L and ωM-L lie at time tl-1; both ω01 and ω1 lie at time tl-NTk); Δαs are angular increment subsamples which are approximately generated from ωks, also adjacent, spaced sequentially forward in time, and time interval-equivalent (Δα1 and ΔαN, resp., end at time tl-(N-1)Tk and tl), srs are angular rate-to-increment approximation coefficients (how to determine srs will not be discussed), ηijs, μijs, ξijs, N, M, and L are defined as those in (3) above, Tk is the angular rate subsample time interval (also the angular increment subsample time interval) set to compute δϕ^l in cycle l, T is the angular rate sample time interval, and R is the whole number ratio of Tk to T. Figure 2 has described clearly the distribution relationship of the angular rate sample series ωks, the angular rate subsample series ωs, and the angular increment subsample series Δαs against time.Figure 2
Distributions ofωk series, ω series, and Δα series against time.Under the pure coning environment defined by (A.1), on the basis of those coning motion properties characterized by (A.6), (A.7), and (A.8), a compressed algorithm structure form equivalent to the uncompressed structure form for coning correction δϕ^l in (3) can be given by
(5)δϕ^l=Tk2∑m=1M-1AmωM-m×ωM+∑n=1N-1BnΔαN-n×ΔαN+Tk∑r=1NCrωM-r×ΔαN,Am=∑i=m+1Mηi-m,i,Bn=∑j=n+1Nμj-n,j,Cr=∑k=rN(ξk-r+1,k-ξk+1,k-r+1),M=N+1,ηi-m,i=1M-mAm,μj-n,j=1N-nBn,ξk-r+1,k=12(N-r+1)Cr,ξk+1,k-r+1=-12(N-r+1)Cr,
where Am is the coning correction coefficient equivalent to the sum of ηi-m,is from (3), Bn is the coning correction coefficient equivalent to the sum of μj-n,js from (3), and Cr is the coning correction coefficient equivalent to the sum of ξk-r+1,k-ξk+1,k-r+1s from (3).Setting eachηij and each ξij in (3) equivalent to zero and marking each μij as ςij will get the traditional computation algorithm formula for updating rotation vector with angular increment inputs which has the form of the integrated angular rate αl and the coning correction δϕ^l [4, 5]:
(6)ϕl=αl+δϕ^l,αl=∑k=N-L+1NΔαk,δϕ^l=∑i=1N-1∑j=i+1NςijΔαi×Δαj,
where ςijs are coefficients depending on the coning correction structure, other signs are defined as those in (3). This form is well known and here called uncompressed N subsample algorithm form with angular increments.Based on the coning motion property, a compressed algorithm structure form equivalent to the uncompressed structure form for coning correctionδϕ^l in (6) can be given by
(7)δϕ^l=∑s=1N-1JsΔαN-s×ΔαN,Js=∑i=s+1Nςi-s,i,ςi-s,i=1N-sJs,
where Js is the coning correction coefficient equivalent to the sum of ςi-s,is from (6).Setting eachμij and each ξij in (3) equivalent to zero and marking each ηij as λij will get the traditional coning correction structure with angular rate input [8]:
(8)δϕ^l=Tk2∑i=1M-1∑j=i+1Mλijωi×ωj,
where λijs are coefficients depending on the algorithm form of (8), other signs are defined as those in (3).Based on the coning motion property, the compressed coning correction algorithm form equivalent to the form as (8) can be given by
(9)δϕ^l=T2∑s=1M-1KsωM-s×ωM,Ks=∑i=s+1Mλi-s,i,λi-s,i=1N-sKs,
where Ks is the coning correction coefficients equivalent to the sum of λi-s,is in (8).


## 4. Algorithm I: Coning Correction Coefficient Design by Coning Frequency Taylor-Series Method

The following derives a general formula for calculating the coning correction coefficients defined in (3), (4), and (5) by using the so-called coning frequency Taylor-series method.Under the coning environment defined by (A.1) in Appendix A, integrate the cross product α(t,tl-1)×ω(t) given by (A.3) in Appendix A over the time interval [tl-1,tl] with tl=tl-1+Tl=tl-1+LTk; then substituting the integral into (2) gives the value of coning correction δϕl defined by (2) as
(10)δϕl=12∫tl-1tlα(t,tl-1)×ω(t)dt=[00δϕz]T,δϕz≡12abΩ(LTk-1Ωsin(ΩLTk)).
Additionally, substituting for ωi×ωj, Δαi×Δαj, and ωi×Δαj in (5) from (A.6), (A.7), and (A.8) in Appendix A gives the value of coning correction δϕ^l defined by (5) as
(11)δϕ^l=Tk2∑m=1M-1AmωM-m×ωM+∑n=1N-1BnΔαN-n×ΔαN+Tk∑r=1NCrωM-r×ΔαN=[00δϕ^z]T,δϕ^z=abΩ2Tk2∑m=1M-1Amsin(ΩmTk)+4absin2ΩTk2∑n=1N-1Bnsin(ΩnTk)+2abΩsinΩTk2Tk∑r=1NCrsin[Ω(r-12)Tk].With recognizing that thez-axis component of the coning correction δϕ^l or δϕl is only nonzero and dependent on the time intervals, define the z-axis component of the difference between δϕ^l and δϕl as the coning correction error e(β):
(12)e(β)=δϕ^z-δϕz.
Substituting for δϕ^z and δϕz in (12) from (10) and (11) and carrying out simple derivation then give
(13)e(β)=12ab[2β2∑m=1M-1Amsin(mβ)+8sin2β2∑n=1N-1Bnsin(nβ)+4βsinβ2×∑r=1NCrsin(rβ-12β)-Lβ+sin(Lβ)]=Φ˙(Ω)F(β),F(β)≡2βTk∑m=1M-1Amsin(mβ)+8sin2β2Tkβ∑n=1N-1Bnsin(nβ)+4sinβ2Tk∑r=1NCrsin(rβ-12β)-LTk(1-sin(Lβ)Lβ),Φ˙(Ω)≡12abΩ,β≡ΩTk,
where Φ˙(Ω) is the total rotation rate around z-axis, F(β) is the error normalized by the time interval LTk, and β is a coning frequency parameter relevant to Tk.According to the trigonometry formulasin2(β/2)=(1-cos⁡β)/2, have the relationships:
(14)8sin2β2∑n=1N-1Bnsin(nβ)=4∑n=1N-1Bnsin(nβ)-4∑n=1N-1Bnsin(nβ)cos⁡β=2∑n=1N-1Bn{2sin(nβ)-sin[(n+1)β]-sin[(n-1)β]},4βsinβ2∑r=1NCrsin(rβ-12β)=-2β∑r=1NCr{cos⁡(rβ)-cos⁡(r-1)β}.According to the relationships given by (14), (13) can be rewritten as
(15)e(β)=12ab[-Lβ+sin(Lβ)+2β2∑m=1M-1Amsin(mβ)+2∑n=1N-1Bn{2sin(nβ)-sin[(n+1)β]-sin[(n-1)β]}-2β∑r=1NCr{cos⁡(rβ)-cos⁡(r-1)β}].According to the Taylor-series expansion ofsinγ and cos⁡γ in variable γ around zero point, consider the following:
(16)sinγ=γ-γ33!+⋯+(-1)k-1γ2k-1(2k-1)!+⋯cos⁡γ=1-γ22!+⋯+(-1)k-1γ2(k-1)2(k-1)!+⋯.
The coning correction error e(β) in (15) can be reformed as a Taylor-series expansion form in parameter β around zero point:
(17)e(β)=12ab×∑k=1∞{(-1)k(2k+1)!L2k+1+2(-1)k-1(2k-1)!∑m=1M-1Amm2k-1+2(-1)k(2k+1)!×∑n=1N-1Bn[2n2k+1-(n+1)2k+1-(n-1)2k+1]-2(-1)k(2k)!∑r=1NCr[r2k-(r-1)2k]}β2k+1.The coning correction optimization design objective should be to minimize the absolute value of the coning correction errore(β), which can be achieved by setting the first M+2N-2 term coefficients of Taylor series of (17) to zeros which gives the following system of linear equations:
(18)2(-1)k-1(2k-1)!∑m=1M-1Amm2k-1+2(-1)k(2k+1)!×∑n=1N-1Bn[2n2k+1-(n+1)2k+1-(n-1)2k+1]-2(-1)k(2k)!∑r=1NCr[r2k-(r-1)2k]=(-1)k-1(2k+1)!L2k+1,k=1,2,…,M+2N-2.
Which can be simplified as
(19)∑m=1M-1Amm2k-1-1(2k+1)(2k)×∑n=1N-1Bn[2n2k+1-(n+1)2k+1-(n-1)2k+1]+12k∑r=1NCr[r2k-(r-1)2k]=12(2k+1)(2k)L2k+1,k=1,2,…,M+2N-2.
Define the following terms:
(20)K=[K1T,K2T,K3T]T,K1=(k1m,1)(M-1)×1,k1m,1≡Am,K2=(k2n,1)(N-1)×1,k2n,1≡Bn,K3=(k3r,1)N×1,k3r,1≡Cr,U=(uij)(M+2N-2)×(M+2N-2)=[U1,U2,U3],U1=(u1ij)(M+2N-2)×(M-1),u1ij≡j2i-1,U2=(u2ij)(M+2N-2)×(N-1),u2ij≡-2j2i+1-(j+1)2i+1-(j-1)2i+1(2i+1)(2i),U3=(u3ij)(M+2N-2)×N,u3ij≡j2i-(j-1)2i2i,V=(vi,1)(M+2N-2)×1,vi,1≡L2i+12(2i+1)(2i).
Then the system of linear equations (19) can be rewritten as the following matrix form:
(21)UK=V,
where U is a M+2N-2 by M+2N-2 square matrix whose ith row and jth column component is uij, V is a M+2N-2 by one column matrix formed from components vi,1s, and K is M+2N-2 by one column matrix formed from components Ams, Bns, and Crs.Solving matrix (21) results in
(22)K=U-1V,
which gives the optimized coning correction coefficients Ams, Bns, and Crs applicable to the compressed coning correction structure form defined by (5).On the basis of the solutionK of (22), the optimized coning correction coefficients ηi-m,is, μj-n,js, ξk-r+1,ks, and ξk+1,k-r+1s applicable to the uncompressed coning correction structure form defined by (3) or (4) can be gained from some relationships defined by (5) as
(23)ηi-m,i=1M-mAm,i=m+1,…,M,μj-n,j=1N-nBn,j=n+1,…,N,ξk-r+1,k=12(N-r+1)Cr,ξk+1,k-r+1=-12(N-r+1)Cr,k=r,…,N,M=N+1.


## 5. Algorithm II: Coning Correction Coefficient Design by Time Taylor-Series Method

Following derives a general formula for calculating the coning correction coefficients defined in (3), (4), and (5) by using Taylor-series expansion in time t for angular rate ω around time point t0.A Taylor series truncated with the number of termsM+N for angular rate ω(t) in time t around time point t0 is given by
(24)ω(t)=ω(t=t0)+ddtω(t)|t=t011!(t-t0)+⋯+dM+N-1dtM+N-1ω(t)|t=t0×1(M+N-1)!(t-t0)M+N-1.
By setting τ=t-t0, then (24) can be rewritten as
(25)ω(τ)=ω(τ=0)+ddτω(τ)|τ=011!τ+⋯+dM+N-1dτM+N-1ω(τ)|τ=0×1(M+N-1)!τM+N-1.
Or
(26)ω(τ)=∑i=1M+Ngiτi-1,gi=di-1dτi-1ω(τ)|τ=01(i-1)!,g1=ω(τ=0),
where τ is the time increment of time t relative to time t0 and gi are the coefficients of series in (26). Based on (25) and (26), the angular rate samples ωis in time t0+τi with τi=(L-M+i)Tk can be given by
(27)ωi=ω(τi)=∑k=1M+Nτik-1gk,i=1,2,…,M.
And the angular increment samples Δαis over the time interval [t0+τi,t0+τi+Tk] with τi=(L-N+i-1)Tk can be given by
(28)Δαi=∫τiτi+Tkω(τ)dτ=∑k=1M+N1k((τi+Tk)k-τik)gk,i=1,2,…,N.
Define the following terms:
(29)G=(gk,1)(M+N)×1,gk,1≡gk,S=(sj,1)(M+N)×1=(S1T,S2T)T,S1=(ωi,1)M×1,ωi,1≡ωi,S2=(Δαi,1)N×1,Δαi,1≡Δαi,Γ=(λjk)(M+N)×(M+N)=(Γ1Γ2),Γ1=(λ¯jk)M×(M+N),λ¯jk≡τjk-1,Γ2=(λ¯¯jk)N×(M+N),λ¯¯jk≡(τj+Tk)k-τjkk.
Then equation terms (27) and (28) can be transformed to a matrix form as follows:
(30)S=ΓG,
where Γ is a M+N by M+N square matrix whose jth row kth column component is λjk, S is a M+N by one column matrix whose jth row component is sj,1, and G is a M+N by one column matrix whose kth row component is gk,1. Solving matrix equation (30) gives the following result:
(31)G=Γ-1S,Γ-1=(fkj)(M+N)×(M+N),
where Γ-1 is a M+N by M+N square matrix whose ith row jth column component is fkj. Matrix equation (31) can be rewritten in the equation term form as follows:
(32)gk=∑j=1M+Nfkjsj,1,k=1,2,…,M+N.Further, withτl=tl-t0 and τl-1=tl-1-t0, the coning correction δϕl in (2) is rewritten as
(33)δϕl=12∫τl-1τl(∫τl-1τω(τ)dτ)×ω(τ)dτ.
Substituting for ω(τ) in (33) from (26) gives
(34)∫τl-1τω(τ)dτ=∫τl-1τ(∑p=1M+Ngpτp-1)dτ=∑p=1M+N[1p(τp-τl-1p)gp],(35)(∫τl-1τω(τ)dτ)×ω(τ)={∑p=1M+N[1p(τp-τl-1p)gp]}×(∑q=1M+Ngqτq-1)=∑p=1M+N∑q=1M+N[1p(τp-τl-1p)τq-1]gp×gq.Additionally, using (32) can get
(36)gp×gq=(∑i=1M+Nfpisi,1)×(∑j=1M+Nfqjsj,1)=∑i=1M+N∑j=1M+Nfpifqjsi,1×sj,1.
Then the coning correction δϕl can be derived from (33), (35), and (36) as
(37)δϕl=12∫τl-1τl(∫τl-1τω(τ)dτ)×ω(τ)dτ=12∫τl-1τl{∑p=1M+N∑q=1M+N[1p(τp-τl-1p)τq-1]gp×gq}dτ=12∑p=1M+N∑q=1M+N1p[1p+q(τlp+q-τl-1p+q)-1qτl-1p(τlq-τl-1q)]gp×gq=∑i=1M+N∑j=1M+N{∑p=1M+N∑q=1M+Nfpifqj2p×[1p+q(τlp+q-τl-1p+q)-1qτl-1p(τlq-τl-1q)]∑p=1M+N∑q=1M+Nfpifqj2p}si,1×sj,1.
With recognizing that ωi×ωi=0, Δαi×Δαi=0, ωi×ωj=-ωj×ωi, Δαi×Δαj=-Δαj×Δαi and ωi×Δαj=-Δαj×ωi, comparing (37) to (3) or (4) can get
(38)δϕ^l=Tk2∑i=1M-1∑j=i+1Mηijωi×ωj+∑i=1N-1∑j=i+1NμijΔαi×Δαj+Tk∑i=1M∑j=1Nξijωi×Δαj,Tk2ηij=∑p=1M+N∑q=1M+Nfpifqj-fpjfqi2pγpq,μij=∑p=1M+N∑q=1M+Nfp,i+Mfq,j+M-fp,j+Mfq,i+M2pγpq,Tkξij=∑p=1M+N∑q=1M+Nfp,ifq,j+M-fp,j+Mfq,i2pγpq,γpq≡[(τlp+q-τl-1p+q)p+q-τl-1p(τlq-τl-1q)q].
Setting that tl-1=t0 and tl=tl-1+LTk will get τl-1=0 and τl=LTk; then correction coefficients ηij, μij, and ξij in (38) can be simplified as
(39)δϕ^l=Tk2∑i=1M-1∑j=i+1Mηijωi×ωj+∑i=1N-1∑j=i+1NμijΔαi×Δαj+Tk∑i=1M∑j=1Nξijωi×Δαj,ηij=∑p=1M+N∑q=1M+Nfpifqj-fpjfqi2p(p+q)(LTk)p+qTk-2,μij=∑p=1M+N∑q=1M+Nfp,i+Mfq,j+M-fp,j+Mfq,i+M2p(p+q)(LTk)p+q,ξij=∑p=1M+N∑q=1M+Nfp,ifq,j+M-fp,j+Mfq,i2p(p+q)(LTk)p+qTk-1.When used in a coning environment, the coning correctionδϕ^l can be with the compressed form of (5). So (39) can be rewritten as
(40)δϕ^l=Tk2∑m=1M-1AmωM-m×ωM+∑n=1N-1BnΔαN-n×ΔαN+Tk∑r=1NCrωM-r×ΔαN,Am=∑i=m+1Mηi-m,i=∑i=m+1M∑p=1M+N∑q=1M+Nfp,i-mfq,i-fp,ifq,i-m2p(p+q)×(LTk)p+qTk-2,Bn=∑j=n+1Nμj-n,j=∑j=n+1N∑p=1M+N∑q=1M+Nfp,j-n+Mfq,j+M-fp,j+Mfq,j-n+M2p(p+q)×(LTk)p+q,Cr=∑k=rN(ξk-r+1,k-ξk+1,k-r+1)=∑k=rN∑p=1M+N∑q=1M+N((fp,k-r+1fq,k+M-fp,k+Mfq,k-r+1)-(fp,k+1fq,k-r+1+M-fp,k_r+1+Mfq,k+1))×(2p(p+q))-1(LTk)p+qTk-1.


## 6. Algorithm Performance Evaluation

### 6.1. Coning Performance Evaluation

In a coning environment, the algorithm error for evaluating the performance of the algorithm with the form of (5) was defined as the average rotation rate e˙E(β) equivalent to e(β) in (13) divided by Tl (Tl=LTk) as follows:
(41)e˙E(β)=Φ˙E(Ω)F˙(β),F˙(β)≡2Lβ∑m=1M-1Amsin(mβ)+8Lβsin2β2∑n=1N-1Bnsin(nβ)+4Lsinβ2∑r=1NCrsin(rβ-12β)-(1-sin(Lβ)Lβ),
where Φ˙E(Ω) is the total rotation rate used to evaluate coning algorithm performance and F˙(β) is the normalized error equivalent to F(β) in (13) divided by Tl. Note that Φ˙E(Ω) is different from Φ˙(Ω) which can be used to design coning correction coefficients [5].Setting eachAm and each Cr in (41) equivalent to zero and marking each Bn as Js will get an algorithm error applicable to evaluating the algorithm with the form of (7) as follows:
(42)e˙E(β)=Φ˙E(Ω)×[8Lβsin2β2∑s=1N-1Jssin(sβ)-(1-sin⁡(Lβ)Lβ)],
where Jss are coefficients designed based on the algorithm structure of (7).Setting eachBn and each Cr in (41) equivalent to zero and marking each Am as Ks will get an algorithm error applicable to evaluating the algorithm with the form of (9) as follows:
(43)e˙E(β)=Φ˙E(Ω)×[2Lβ∑s=1M-1Kssin(sβ)-(1-sin⁡(Lβ)Lβ)],
where Kss are coefficients designed based on the algorithm structure of (9).Each uncompressed algorithm has the same coning accuracy as its compressed version, because each uncompressed algorithm is equivalent to its compressed version under coning conditions.


### 6.2. Maneuver Performance Evaluation

Under the maneuver environment defined by (26), setting that τl-1=0 and τl=LTk to (37) will get
(44)δϕl=12∑p=1M+N∑q=1M+N1p(p+q)Lp+qTkp+qgp×gq.In addition, cross productsωi×ωj, Δαi×Δαj, and ωi×Δαj are respectively derived from (27) and (28) as
(45)ωi×ωj=∑p=1M+N∑q=1M+N(L-M+i)p-1iiiiiiiiiiiiiiii×(L-M+j)q-1gp×gqTkp+q-2,Δαi×Δαj=∑p=1M+N∑q=1M+N1pqiiiiiiiiiiiiiiiiiiii×{gqTkp+q[(L-N+i)p-(L-N+j-1)p]iiiiiiiiiiiiiiiiiiiiiiiii·[(L-N+q)q-(L-N+q-1)q]gpiiiiiiiiiiiiiiiiiiiiiiiii×gqTkp+q},ωi×Δαj=∑p=1M+N∑q=1M+N1q(L-M+i)p-1iiiiiiiiiiiiiiiiii×[(L-N+j)q-(L-N+j-1)q]gpiiiiiiiiiiiiiiiiii×gqTkp+q-1.
Substituting for ωi×ωj, Δαi×Δαj, and ωi×Δαj in (3) or (4) from (45) gets
(46)δϕ^l=∑p=1M+N∑q=1M+N{∑i=1M-1∑j=i+1Mηij(L-M+i)p-1(L-M+j)q-1+1pq∑i=1N-1∑j=i+1Nμij×[(L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]+1q∑i=1M∑j=1Nξij(L-M+i)p-1×[(L-N+j)q-(L-N+j-1)q]∑i=1M-1∑j=i+1Mηij(L-M+i)p-1(L-M+j)q-1}×gp×gqTkp+q.In a maneuver environment, the algorithm error for analyzing the performance of the algorithm with the form of (3) or (4) was defined as the difference of δϕ^l of (46) and δϕl of (44) with gp×gp=0 and gp×gq=-gp×gp as follows:
(47)em(Tk)=δϕ^l-δϕl=∑p=1M+N∑q=p+1M+Nzpqgp×gqTkp+q,zpq=∑i=1M-1∑j=i+1Mηij×[(L-M+i)p-1(L-M+j)q-1-(L-M+i)q-1(L-M+j)p-1]+1pq∑i=1N-1∑j=i+1Nμij×{[[(L-N+j)q-(L-N+j-1)q](L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]-[(L-N+i)q-(L-N+i-1)q]×[(L-N+j)p-(L-N+j-1)p]}+1pq∑i=1M∑j=1Nξij×{p(L-M+i)p-1×[(L-N+j)q-(L-N+j-1)q]-q(L-M+i)q-1×[(L-N+j)p-(L-N+j-1)p]}-q-p2pq(p+q)Lp+q.SettingM, each ηij, and each ξij in (48) equivalent to zero will get an algorithm error applicable to analyzing the algorithm with the form of (6) as follows:
(48)em(Tk)=δϕ^l-δϕl=∑p=1N∑q=p+1Nzpqgp×gqTkp+q,zpq=1pq∑i=1N-1∑j=i+1Nμij×{[[(L-N+j)q-(L-N+j-1)q](L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]-[(L-N+i)q-(L-N+i-1)q]×[(L-N+j)p-(L-N+j-1)p]}-q-p2pq(p+q)Lp+q.SettingN, each μij, and each ξij in (48) equivalent to zero will get an algorithm error applicable to analyzing the algorithm with the form of (8) as follows:
(49)em(Tk)=δϕ^l-δϕl=∑p=1M∑q=p+1Mzpqgp×gqTkp+q,zpq=∑i=1M-1∑j=i+1Mηij[(L-M+i)p-1(L-M+j)q-1-(L-M+i)q-1(L-M+j)p-1]-q-p2pq(p+q)Lp+q.In a maneuver environment, the algorithm error for evaluating the performance of all concerned algorithms was defined as the rotation vector magnitude erroreM(t) equivalent to the direction cosine attitude matrix error as
(50)eM(t)=[c32(t)]2+[c13(t)]2+[c21(t)]2,C(t)=CA(t)[CT(t)]T,
where C(t) is the direction cosine attitude matrix error at time t, cij(t) is the ith row and jth column component of matrix C(t), CA(t) is (1) computed direction cosine attitude matrix at time t using those investigated algorithms, and CT(t) is the true direction cosine attitude matrix at time t. In this paper, an approximated CT(t) obtained by using (1) at a more than 100 kHz rotation vector updating rate was used instead of a true CT(t) (see Appendix B for the derivation of (50)).


## 6.1. Coning Performance Evaluation

In a coning environment, the algorithm error for evaluating the performance of the algorithm with the form of (5) was defined as the average rotation rate e˙E(β) equivalent to e(β) in (13) divided by Tl (Tl=LTk) as follows:
(41)e˙E(β)=Φ˙E(Ω)F˙(β),F˙(β)≡2Lβ∑m=1M-1Amsin(mβ)+8Lβsin2β2∑n=1N-1Bnsin(nβ)+4Lsinβ2∑r=1NCrsin(rβ-12β)-(1-sin(Lβ)Lβ),
where Φ˙E(Ω) is the total rotation rate used to evaluate coning algorithm performance and F˙(β) is the normalized error equivalent to F(β) in (13) divided by Tl. Note that Φ˙E(Ω) is different from Φ˙(Ω) which can be used to design coning correction coefficients [5].Setting eachAm and each Cr in (41) equivalent to zero and marking each Bn as Js will get an algorithm error applicable to evaluating the algorithm with the form of (7) as follows:
(42)e˙E(β)=Φ˙E(Ω)×[8Lβsin2β2∑s=1N-1Jssin(sβ)-(1-sin⁡(Lβ)Lβ)],
where Jss are coefficients designed based on the algorithm structure of (7).Setting eachBn and each Cr in (41) equivalent to zero and marking each Am as Ks will get an algorithm error applicable to evaluating the algorithm with the form of (9) as follows:
(43)e˙E(β)=Φ˙E(Ω)×[2Lβ∑s=1M-1Kssin(sβ)-(1-sin⁡(Lβ)Lβ)],
where Kss are coefficients designed based on the algorithm structure of (9).Each uncompressed algorithm has the same coning accuracy as its compressed version, because each uncompressed algorithm is equivalent to its compressed version under coning conditions.


## 6.2. Maneuver Performance Evaluation

Under the maneuver environment defined by (26), setting that τl-1=0 and τl=LTk to (37) will get
(44)δϕl=12∑p=1M+N∑q=1M+N1p(p+q)Lp+qTkp+qgp×gq.In addition, cross productsωi×ωj, Δαi×Δαj, and ωi×Δαj are respectively derived from (27) and (28) as
(45)ωi×ωj=∑p=1M+N∑q=1M+N(L-M+i)p-1iiiiiiiiiiiiiiii×(L-M+j)q-1gp×gqTkp+q-2,Δαi×Δαj=∑p=1M+N∑q=1M+N1pqiiiiiiiiiiiiiiiiiiii×{gqTkp+q[(L-N+i)p-(L-N+j-1)p]iiiiiiiiiiiiiiiiiiiiiiiii·[(L-N+q)q-(L-N+q-1)q]gpiiiiiiiiiiiiiiiiiiiiiiiii×gqTkp+q},ωi×Δαj=∑p=1M+N∑q=1M+N1q(L-M+i)p-1iiiiiiiiiiiiiiiiii×[(L-N+j)q-(L-N+j-1)q]gpiiiiiiiiiiiiiiiiii×gqTkp+q-1.
Substituting for ωi×ωj, Δαi×Δαj, and ωi×Δαj in (3) or (4) from (45) gets
(46)δϕ^l=∑p=1M+N∑q=1M+N{∑i=1M-1∑j=i+1Mηij(L-M+i)p-1(L-M+j)q-1+1pq∑i=1N-1∑j=i+1Nμij×[(L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]+1q∑i=1M∑j=1Nξij(L-M+i)p-1×[(L-N+j)q-(L-N+j-1)q]∑i=1M-1∑j=i+1Mηij(L-M+i)p-1(L-M+j)q-1}×gp×gqTkp+q.In a maneuver environment, the algorithm error for analyzing the performance of the algorithm with the form of (3) or (4) was defined as the difference of δϕ^l of (46) and δϕl of (44) with gp×gp=0 and gp×gq=-gp×gp as follows:
(47)em(Tk)=δϕ^l-δϕl=∑p=1M+N∑q=p+1M+Nzpqgp×gqTkp+q,zpq=∑i=1M-1∑j=i+1Mηij×[(L-M+i)p-1(L-M+j)q-1-(L-M+i)q-1(L-M+j)p-1]+1pq∑i=1N-1∑j=i+1Nμij×{[[(L-N+j)q-(L-N+j-1)q](L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]-[(L-N+i)q-(L-N+i-1)q]×[(L-N+j)p-(L-N+j-1)p]}+1pq∑i=1M∑j=1Nξij×{p(L-M+i)p-1×[(L-N+j)q-(L-N+j-1)q]-q(L-M+i)q-1×[(L-N+j)p-(L-N+j-1)p]}-q-p2pq(p+q)Lp+q.SettingM, each ηij, and each ξij in (48) equivalent to zero will get an algorithm error applicable to analyzing the algorithm with the form of (6) as follows:
(48)em(Tk)=δϕ^l-δϕl=∑p=1N∑q=p+1Nzpqgp×gqTkp+q,zpq=1pq∑i=1N-1∑j=i+1Nμij×{[[(L-N+j)q-(L-N+j-1)q](L-N+i)p-(L-N+i-1)p]×[(L-N+j)q-(L-N+j-1)q]-[(L-N+i)q-(L-N+i-1)q]×[(L-N+j)p-(L-N+j-1)p]}-q-p2pq(p+q)Lp+q.SettingN, each μij, and each ξij in (48) equivalent to zero will get an algorithm error applicable to analyzing the algorithm with the form of (8) as follows:
(49)em(Tk)=δϕ^l-δϕl=∑p=1M∑q=p+1Mzpqgp×gqTkp+q,zpq=∑i=1M-1∑j=i+1Mηij[(L-M+i)p-1(L-M+j)q-1-(L-M+i)q-1(L-M+j)p-1]-q-p2pq(p+q)Lp+q.In a maneuver environment, the algorithm error for evaluating the performance of all concerned algorithms was defined as the rotation vector magnitude erroreM(t) equivalent to the direction cosine attitude matrix error as
(50)eM(t)=[c32(t)]2+[c13(t)]2+[c21(t)]2,C(t)=CA(t)[CT(t)]T,
where C(t) is the direction cosine attitude matrix error at time t, cij(t) is the ith row and jth column component of matrix C(t), CA(t) is (1) computed direction cosine attitude matrix at time t using those investigated algorithms, and CT(t) is the true direction cosine attitude matrix at time t. In this paper, an approximated CT(t) obtained by using (1) at a more than 100 kHz rotation vector updating rate was used instead of a true CT(t) (see Appendix B for the derivation of (50)).


## 7. Algorithm Example and Performance Evaluation

To show the properties of several algorithms, evaluated results computed by using the optimized coning correction coefficients designed by using coning frequency Taylor-series method and time Taylor-series method were produced, compared, and analyzed under coning environments and maneuver conditions. To analyze and compare conveniently, signsM and N in (3), (4), and (5) were, respectively, rewritten as Mir and Nir (the number of subsamples in the new structure is Mir+Nir), sign N in (6) and (7) was rewritten as Ni, and sign M in (8) and (9) was rewritten as Mr. Two terms of parameters were set: (1) Mir=3, Nir=2, L=2 (two sequential Δαs samples in the αl calculation), and Tk=10 ms (i.e., attitude updating time interval Tl equals to 10 ms) and (2) Ni=5, Mr=5, L=4 (four sequential Δαs samples in the αl calculation), and Tk=5 ms (i.e., attitude updating time interval Tl equals to 20 ms).(1)
FSIRc indicates the coning algorithm based on the compressed form of (5) taking the coefficients designed by using frequency Taylor-series method (see Algorithm I).(2)
TSIRc indicates the coning algorithm based on the compressed form of (5) taking the coefficients designed by using time Taylor-series method (see Algorithm II).(3)
FSIc indicates the coning algorithm based on the compressed form of (7) taking the coefficients designed by using frequency Taylor-series method (see [5]).(4)
TSIc indicates the coning algorithm based on the compressed form of (7) taking the coefficients designed by using time Taylor-series method (derived from TSIuc and (7)).(5)
FSRc indicates the coning algorithm based on the compressed form of (9) taking the coefficients designed by using frequency Taylor-series method (see [7]).(6)
FSIRuc indicates the coning algorithm based on the uncompressed form of (3) or (4) taking the coefficients designed by using frequency Taylor-series method (see Algorithm I).(7)
TSIRuc indicates the coning algorithm based on the uncompressed form of (3) or (4) taking the coefficients designed by using time Taylor-series method (see Algorithm II).(8)
FSIuc indicates the coning algorithm based on the uncompressed form of (6) taking the coefficients designed by using frequency Taylor-series method (derived from FSIc and (7)).(9)
TSIuc indicates the coning algorithm based on the uncompressed form of (6) taking the coefficients designed by using time Taylor-series method (see [5]).(10)
FSRuc indicates the coning algorithm based on the uncompressed form of (8) taking the coefficients designed by using frequency Taylor-series method (see [7]).Tables1, 2, 3, 4, 5, and 6 give the coefficients of the ten concerned algorithms above. The coefficients in all tables were given in the form of whole number ratio approximating the decimals of limited significant digits (the number of significant digits is sixteen in this paper) in order to both ignore the numerical truncation error effect on the algorithm performance and simplify writing.Table 1
Algorithm coefficients for the compressed form of (5).
Algorithms
Coefficients
A
1
A
2
B
1
C
1
C
2
FSIRc
−64/385
2/385
160/231
169/385
−23/385
TSIRc
−16/315
−1/315
8/21
58/105
2/45Table 2
Algorithm coefficients for the compressed form of (7).
Algorithms
Coefficients
J
1
J
2
J
3
J
4
FSIc
656/315
262/315
8/15
−1/315
TSIc
1471/644
465/748
552/877
−101/4903Table 3
Algorithm coefficients for the compressed form of (9).
Algorithms
Coefficients
K
1
K
2
K
3
K
4
FSRc
2942/1601
2049/2398
291/524
193/6392Table 4
Algorithm coefficients for the uncompressed form of (3) or  (4).
Algorithms
Coefficients
η
12
η
13
η
23
μ
12
ξ
11
ξ
12
ξ
21
ξ
22
ξ
31
ξ
32
FSIRuc
−32/385
2/385
−32/385
160/231
169/1540
−23/770
−169/1540
169/1540
23/770
−169/1540
TSIRuc
−8/315
−1/315
−8/315
8/21
23/315
1/45
−64/315
64/315
−1/45
−23/315Table 5
Algorithm coefficients for the compressed form of (6).
Algorithms
Coefficients
ς
12
ς
13
ς
14
ς
15
ς
23
ς
24
ς
25
ς
34
ς
35
ς
45
FSIuc
164/315
262/945
4/15
−1/315
164/315
262/945
4/15
164/315
262/945
164/315
TSIuc
101/4903
−52/4725
52/4725
−101/4903
472/675
632/1613
974/1575
482/675
421/1748
1669/1963Table 6
Algorithm coefficients for the compressed form of (8).
Algorithms
Coefficients
ς
12
ς
13
ς
14
ς
15
ς
23
ς
24
ς
25
ς
34
ς
35
ς
45
FSRuc
1471/3202
683/2398
291/1048
193/6392
1471/3202
683/2398
291/1048
1471/3202
683/2398
1471/3202Under coning environments withΦ˙E(Ω) set to unity over all frequencies, computing algorithm error e˙E(β) in (41) with FSIRc and TSIRc coefficients in Table 1, algorithm error e˙E(β) in (42) with FSIc and TSIc coefficients in Table 2, and algorithm error e˙E(β) in (43) with FSRc coefficients in Table 3 results in a plot of normalized algorithm error e˙E(β) versus coning frequency β/(2πT) as in Figure 3, where each of five denotations indicates both versions of its compressed and uncompressed algorithms.Figure 3
Normalized algorithm errore˙E(β) versus coning frequency β/(2πT) (Hz).The coefficientsz12, z13, z14, z23, z15, z24, z16, z25, and z34 in (47), (48), and (49) are, respectively, calculated with corresponding algorithm coefficients from Table 1 to Table 6.In addition, the maneuver environment set for algorithm performance evaluation was the extreme 2s angular rate profile pictured in Figure4. Then the rotation vector magnitude error eM(t) in (50) was computed over Figure 4’s maneuver profile by using corresponding coefficients from Table 1 to Table 6. Maxima of rotation vector magnitude errors of ten algorithms over the 2s maneuver profile are presented in Table 8.Figure 4
Angular rate (deg/s) versus time (s).Figure3 shows the relative coning accuracy of ten concerned algorithms, rather than its absolute coning accuracy over the 1 Hz~100 Hz coning frequency range. The new FSIR algorithm has almost the best coning performance over the most presented frequencies. All algorithms (FSIR, FSI, and FSR) designed by using coning frequency Taylor-series method are more accurate than those (TSIR and TSI) designed by using time Taylor-series over the most presented frequencies. If the traditional FSI algorithm had adequate coning accuracy, the FSIR algorithm would have more adequate coning accuracy.By analyzing Table7 which is associated to (47), (48), and (49), it is found that z13 is the main attribution to maneuver error of all concerned compressed algorithms (FSIRc, TSIRc, FSIc, TSIc, and FSRc); z14, z16, z13, and z12 are, respectively, the main attribution to maneuver error of FSIRuc, that of TSIRuc, that of FSIuc, and that of TSIuc, and z12, z14, and, z23 are the main attributions to maneuver error of FSRuc. Under the general fact that one low-order power series term is bigger than one high-order, the power series terms with the main attributions supply the main maneuver error of concerned algorithms. According to the values of the main attributions in Table 7, the FSIRc algorithm has the best maneuver performance compared to other compressed algorithms; each uncompressed algorithm except FSIuc has an much evident maneuver accuracy improvement compared with its compressed algorithm version.Table 7
Attributions to maneuver errorem(Tk).
Algorithms
z
12
z
13
z
14
z
23
z
15
z
24
z
16
z
25
z
34
FSIRc
0
-
8.66
e
-
2
-
2.15
e
-
1
-
7.16
e
-
2
-
4.77
e
-
1
-
2.94
e
-
1
-
1.02
e
+
0
-
8.03
e
-
1
-
3.00
e
-
1
TSIRc
0
-
3.47
e
-
1
-
8.38
e
-
001
-
2.79
e
1
-
1.70
e
+
0
-
7.62
e
-
1
-
3.28
e
+
0
-
1.68
e
+
0
-
5.08
e
-
1
FSIc
0
-
1.65
e
+
1
-
6.38
e
+
1
-
2.13
e
+
1
-
2.46
e
+
2
-
9.05
e
+
1
-
9.08
e
+
2
-
3.66
e
+
2
-
9.20
e
+
1
TSIc
5.50
e
-
7
-
1.66
e
+
1
-
6.37
e
+
1
-
2.12
e
+
1
-
2.46
e
+
2
-
9.05
e
+
1
-
9.09
e
+
2
-
3.66
e
+
2
-
9.20
e
+
1
FSRc
-
1.23
e
-
6
-
1.06
e
+
1
-
5.12
e
+
1
-
1.71
e
+
1
-
2.05
e
+
2
-
8.53
e
+
1
-
7.80
e
+
2
-
3.51
e
+
2
-
9.75
e
+
1
FSIRuc
0
0
-
6.47
e
-
2
-
2.16
e
-
2
-
2.59
e
-
1
-
1.29
e
-
1
-
7.08
e
-
1
-
3.72
e
-
1
-
1.15
e
-
1
TSIRuc
0
0
0
0
0
0
2.54
e
-
2
0
0
FSIuc
0
-
5.33
e
+
0
-
2.14
e
+
1
-
7.13
e
+
0
-
8.46
e
+
1
-
2.95
e
+
1
-
3.19
e
+
2
-
1.20
e
+
2
-
2.79
e
+
1
TSIuc
7.44
e
-
8
6.43
e
-
7
3.08
e
-
6
6.35
e
-
7
1.26
e
-
5
3.51
e
-
6
-
1.02
e
+
0
1.41
e
-
5
3.92
e
-
6
FSRuc
-
1.23
e
-
6
-
4.91
e
-
6
5.16
e
+
0
1.72
e
+
0
4.13
e
+
1
2.06
e
+
1
2.38
e
+
2
1.3
e
+
2
4.19
e
+
1Table 8
Maximum of rotate vector magnitude errors in the set maneuver environment.
Algorithms
Maximum-error of compressed  algorithms,μrad
Maximum-error of uncompressed  algorithms,μrad
FSIR
6.46
e
-
1
8.43
e
-
2
TSIR
2.56
e
-
0
8.33
e
-
2
FSI
7.58
e
-
0
2.45
e
-
0
TSI
7.60
e
-
0
8.34
e
-
2
TSR
4.88
e
-
0
8.12
e
-
2Table8 indicates that the maximum-error of FSIRc is the smallest in those of all compressed algorithms, the maximum-error of each uncompressed algorithm except FSIuc is smaller about 1 to 2 orders than its compressed algorithm version. The simulation results in Table 8 are basically consistent with the analytic conclusions above. According to the application experience that the algorithm maneuvering error less than 1% of 21.3 μrad from sensors is compatible with the overall INS accuracy requirement of 0.01 deg/h [5], the 6.46e-1 μrad maximum-error of FSIRc algorithm is less than 1% of 200 μrad and compatible with the overall INS accuracy requirement of 0.1 deg/h (a lower system accuracy), whereas the maximum-errors of other compressed algorithms are not. All uncompressed algorithms except FSIuc have much adequate maneuver accuracy for the overall requirement of 0.1 deg/h and are just compatible with the overall requirement of 0.01 deg/h.


## 8. Conclusions

The new rate-and-increment-synchronized coning correction structure is presented for achieving higher coning correction performance in coning and maneuver environments. As the traditional increment-based structure, the correction coefficient of the new structure permits to be designed using the frequency Taylor-series method and time Taylor-series method. Compared with the traditional increment-based and rate-based compressed algorithms, the new algorithm with the compressed coefficients optimized by using frequency Taylor-series method gives a superior coning performance, as well as a higher maneuver accuracy. Uncompressed algorithms based on the new structure are much higher accurate than its compressed version under a maneuver environment.



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*Source: 614378-2014-09-25.xml*