Velocity/Position Integration Formula (I): Application to In-flight Coarse Alignment 2 roughly known initial attitude, otherwise they cannot guarantee a rapid and accurate alignment result [1][2][3][4][5]. If the SINS is stationary or quasi-stationary, analytic methods are often used to derive a coarse attitude from gyroscope/accelerometer measurements [1,3,6]. The heading angle is more difficult to determine than the two level angles and for a consumer grade SINS, is usually aided by a magnetic compass. The in-motion or in-flight alignment is necessary for many military applications and commercial aviations [7]. In such cases, the SINS is in motion, e.g., onboard a ship or aircraft, the direction of velocity/trajectory from an aided source, such as GPS, can provide a rough pitch and heading angles during a straight course. This information is generally not good enough to perform a reliable fine alignment due to the water current and air speed [3], let alone the SINS misaligning angles relative to the carrier.It may be argued that the coarse alignment difficulty confronting the navigation field is largely owed to our "local eye" on the attitude representation in three-dimension space. Nowadays, we are used to the attitude approximation by three one-dimension error angles. For example, many works have been devoted to the nonlinear angle error models to account for large heading uncertainty [4,8,9]. By so doing, most inherent characteristics of the three-dimension attitude have been lost. Our group proposed a recursive alignment approach based on attitude optimization in [10], which, for the first time in the public literature, transforms the attitude alignment problem into a continuous attitude determination problem [11] using infinite vector observations. It was rigorously proven therein that the behavior of the estimated constant initial angles can be used to detect significant sensor biases. The optimization approach is related to the so-called inertial frame method [12,13], but more theoretically solid and more robust to disturbances and noise, because it makes full use of the special algebraic property of the attitude matrix (a three-dimension orthogonal matrix with unit determinant). If the nonzero velocity rate information was externally provided, the

Nomenclature C s = coefficient of compressed coning correction algorithm c k r = combination, k elements selected from a set of r elements without regard to the order of selection f 3 ; f 4 ; : : : = coefficients in the derivatives of δφ unc t m = computer interval index, as subscript indicates parameter value at computer cycle m N = total number of samples used in the coning calculation n = number of samples in current iteration time interval o = an equivalent infinitesimal to ( ) r = order of derivatives s = index of coefficients for compressed coning algorithm T k = sampling time interval t = time α, α = integral of ω over t m−1 ; t time interval and its magnitude β = normalized coning frequency Δα N1−i , Δα N1−j = gyro data samples spaced backward in time from time t δϕ c t = coning integral over the time interval from t m−1 to t δφ cmp t = compressed coning correction algorithm δφ cmpf t = compressed frequency-series coning algorithm δφ unc t = uncompressed coning correction algorithm δφ uncE t = uncompressed Explicit coning algorithm δφ uncE3 t; δφ uncE4 t; : : := uncompressed Explicit coning algorithms taking three, four, and five samples of gyro data per update δφ uncf t = uncompressed frequency-seriesbased coning algorithm δφ uncf3 t; δφ uncf4 t; : : := uncompressed frequency-series coning correction algorithms taking three, four, and five samples of gyro data per update ς ij = coefficients of uncompressed frequency-series coning algorithms _ ΦΩ, _ Φ Eval Ω = coning rate amplitude Ω = coning frequency ω = angular rate vector _ ω, ω, ω first-, second-, third-, fourth-, fifth-, and sixth-order derivatives of ω × = skew symmetric cross-product matrix form of vector ( ) that indicates × ×

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Inertial navigation applications are usually referenced to a rotating frame. Consideration of the navigation reference frame rotation in the inertial navigation algorithm design is an important but so far less seriously treated issue, especially for the future ultra-precision navigation system of several meters per hour. This paper proposes a rigorous approach to tackle the issue of navigation frame rotation in velocity/position computation by use of the newly-devised velocity/position integration formulae in the Part I companion paper. The two integration formulae set a well-founded cornerstone for the velocity/position algorithms design that makes the comprehension of the inertial navigation computation principle more accessible to practitioners, and different approximations to the integrals involved will give birth to various velocity/position update algorithms. Two-sample velocity and position algorithms are derived to exemplify the design process. In the context of level-flight airplane examples, the derived algorithm is analytically and numerically compared to the typical algorithms existing in the literature. The results throw light on the problems in existing algorithms and the potential benefits of the derived algorithm.

With the development of high-accuracy inertial navigation system inertial sensors, such as ring laser gyroscopes and atomic spin gyroscopes, it is increasingly important to improve the strapdown inertial navigation algorithms to match such high-accuracy inertial sensors. For example, the existing inertial navigation algorithms have not taken into account the triple-cross-product term of noncommutativity error. However, theoretical analysis demonstrates that the ignored triple-cross-product term is nonignorable under coning motion with constant angular rate precession environments. In this paper, a new high-accuracy rotation vector algorithm is proposed for strapdown inertial navigation. The Taylor series about time is used for error analysis and optimization of the new algorithm. General maneuvers and coning motion with constant angular rate precession environments are considered in establishing the coefficient equations in the proposed algorithm. Error drift equations are given after the error compensation. Normalized quaternion under coning motion with constant angular rate precession environments and Savage's severe integrated angular-rate profiles are used to numerically verify the new algorithm. The results Manuscriptindicate that the new high-order attitude updating algorithm can improve inertial navigation accuracy.NOMENCLATURE δφ c (t) = coning integral over the time interval from t m−1 to t δφ hA (t) = the integral of the triple-cross product noncommutativity rate vector over the time interval from t m−1 to t related to the coning motion, which is called the triple-cross-product term part A δφ hB (t) = the integral of the triple-cross-product noncommutativity rate vector over the time interval from t m−1 to t related to the high dynamic motion, which is called the triple-cross-product term part B δφ hA (t) = algorithms for the triple-cross-product term part A δφ hB (t) = algorithms for the triple-cross-product term part B δφ hA = nonperiodic vector of the triple-cross-product term part A in one updating cycle δφ hB = nonperiodic vector of the triple-cross-product term part B in one updating cycle δφ hA = nonperiodic vector of the triple-cross-product term part A algorithms δφ hB = nonperiodic vector of the triple-cross-product term part B algorithms α N+1−i (t) = gyro data samples spaced backward in time from time t α N+1−j (t) α N+1−k (t) ς ij = coefficients of uncompressed frequency series coning algorithms and triple-cross-product term part A's algorithms η (N+i−1) = coefficients of triple-cross-product term part B's algorithms (N+j −1) (N+k−1) a, b = coning motion amplitude c = coefficient of constant angular-rate precession = coning frequency T 0 = sampling period T = attitude updating cycle N = total number of samples used in one attitude updating cycle 1178 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 2 APRIL 2015 n = number of samples in the current iteration time interval m = computer interval index, where the subscript indicates parameter value at computer cycle m o () = a...