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Velocity/Position Integration Formula (I): Application to In-flight Coarse Alignment

Yuanxin Wu, Xianfei Pan

arXiv:1207.1550v1cs.RO

TL;DR

Obtaining a rough initial attitude during motion or maneuvering is difficult, while fine Kalman alignment depends heavily on coarse alignment. This paper uses optimization based on velocity/position integration formulae and reports potential heading alignment to one-degree accuracy in ten seconds when the GPS lever arm is handled.

  • Problem

    Acquiring a rough initial angle while the carrier is moving or maneuvering is difficult, yet fine Kalman alignment relies heavily on coarse alignment.

  • Method

    The paper uses an optimization-based in-flight coarse alignment approach founded on velocity/position integration formulae and absolute velocity/position sensors.

  • Results

    With the GPS lever arm well handled, the SINS heading can potentially be aligned to one-degree accuracy in ten seconds.

  • Takeaways & Limitations

    The proposed algorithms could support applications requiring SINS alignment on the run.

Abstract

from arXiv · show

The in-flight alignment is a critical stage for airborne INS/GPS applications. The alignment task is usually carried out by the Kalman filtering technique that necessitates a good initial attitude to obtain satisfying performance. Due to the airborne dynamics, the in-flight alignment is much difficult than alignment on the ground. This paper proposes an optimization-based coarse alignment approach using GPS position/velocity as input, founded on the newly-derived velocity/position integration formulae. Simulation and flight test results show that, with the GPS lever arm well handled, it is potentially able to yield the initial heading up to one degree accuracy in ten seconds. It can serve as a nice coarse in-flight alignment without any prior attitude information for the subsequent fine Kalman alignment. The approach can also be applied to other applications that require aligning the INS on the run.

I. INTRODUCTION

Initial SINS alignment must rapidly and accurately determine attitude, yet in-flight conditions make coarse alignment difficult and leave Kalman fine alignment dependent on a roughly known initial attitude. This paper extends an optimization-based approach with velocity/position integration formulae for alignment in motion using absolute velocity/position sensors.

  • Motivation: Initial alignment is critical because SINS performance largely depends on the alignment process’s accuracy and rapidness.
  • Challenge: Kalman-based fine alignment relies heavily on coarse alignment to provide a roughly known initial attitude for rapid and accurate results.
  • Challenge: In-flight alignment is difficult because GPS-derived velocity direction provides only rough pitch and heading information, which may be inadequate for reliable fine alignment.Carrier motion, water current, and air speed can degrade the usefulness of this information.
  • Contribution: The paper systematically extends prior work by devising velocity/position integration formulae for recursive in-flight alignment algorithms.The algorithms are evaluated using numerical and flight test data.
  • Contribution: Together with earlier work, the paper proposes a systematic approach to solve SINS alignment in motion using absolute velocity/position sensors.

II. PROBLEM STATEMENT · III. VELOCITY/POSITION INTEGRATION FORMULAE · A. Attitude Integration Formula

The paper formulates in-motion attitude determination from measured velocity and position, then derives an explicit attitude integration formula by combining body- and navigation-frame changes relative to an inertial frame.

  • II. PROBLEM STATEMENT: The navigation equations describe attitude, velocity, and position rates in the local-level navigation frame N.The formulation uses body frame B, inertial frame I, and Earth frame E, with quantities including angular rate, specific force, Earth rotation, gravity, and curvature.
  • II. PROBLEM STATEMENT: The alignment objective is to determine the attitude matrix n_b C while the system is moving, using measured velocity n v and position p over [0,t].The process is assumed to start at t_0.
  • A. Attitude Integration Formula: The attitude matrix n_b C depends on the body angular rate relative to the navigation frame, whose calculation is obtained from the body and navigation angular-rate relation.This dependence motivates deriving an analytic form for n_b C.
  • A. Attitude Integration Formula: The derivation separately integrates attitude changes of the body and navigation frames relative to a chosen inertial frame, then combines them through the attitude-matrix chain rule.The resulting relation expresses n_b C at any time using the corresponding frame transformations from the initial time.
  • A. Attitude Integration Formula: The body-frame and navigation-frame attitude changes encode their transformations from time 0 to t and satisfy their respective rate equations.These frame-specific changes are the components combined in the attitude integration relation.
  • A. Attitude Integration Formula: If the initial attitude matrix 0 n_b C is known, the attitude matrix n_b C at any time follows from the derived relations.The initial attitude matrix is treated as a constant quantity.
  • A. Attitude Integration Formula: For a fixed epoch t_0, the body and navigation frames relative to an inertial frame are inertially frozen after t_0, so their derivatives with respect to time t are zero.This distinguishes the explicit integration representation from the attitude differential equation.
  • A. Attitude Integration Formula: The explicit form of n_b C is termed the attitude integration formula, while its incremental version over one update interval has appeared in previous literature.The paper identifies the explicit form as distinct from the already familiar incremental formulation.

B. Velocity Integration Formula

The velocity integration formula is derived through integration transformations and reorganizing integrated velocity-rate terms. Equation (11) supports optimization-based in-flight alignment by relating initial attitude to measured inertial and aided-motion quantities, with unique determination possible under a practical independence condition.

  • Optimization basis: Equation (8) forms the basis for an optimization approach applicable to in-flight alignment.The approach requires velocity rate in addition to velocity and position information.
  • Derivation: Equation (11) is derived by integrating the velocity-rate equation and reorganizing the resulting terms.The derivation uses the attitude rate equation and the initial velocity.
  • Formula interpretation: The formula is an integration function of the initial attitude matrix and two defined vectors.The vectors are formed from gyroscope/accelerometer outputs and aided velocity/position information during alignment.
  • Attitude determination: Two linearly independent vectors theoretically determine the initial attitude matrix uniquely.The text states that this condition is easily fulfilled in practice.

C. Position Integration Formula

The position integration formula is derived in a local-level frame by integrating position dynamics and organizing the resulting terms. Its simplified form avoids aided velocity input and, under stated approximations, supports coarse in-flight alignment using position information only.

  • Position integration setup: In a North-Up-East local-level frame, the curvature matrix is expressed as a function of current position using WGS-84 radii of curvature.The specific curvature expression changes with the local-level frame choice.
  • Position integration setup: The position in the N-frame is defined as an integral whose rate equation is n ṙ = v.This definition leads to the position integration formula after integrating over the interval 0 to t and organizing the terms.
  • Formula properties: Equations (20)–(22) are functions of the initial attitude matrix for any positive t, with terms determined by inertial-sensor outputs and aided velocity/position information.The vectors p_3 and p_4 are defined from gyroscope/accelerometer outputs and aided velocity/position information during alignment.
  • Formula simplification: Equation (22) has a simplified form that does not need aided velocity as an input.This distinguishes it from simply integrating the earlier velocity-based formula.
  • Position-only alignment: The double-integral velocity term can be omitted as relatively small, while the additive initial velocity can be eliminated by differencing time instants.Approximating n in the relevant term by n_ie supports coarse in-flight alignment using position information only.

IV. RECURSIVE IN-FLIGHT ALGORITHMS · A. Calculating Integrals in Velocity Integration Formula

The paper develops recursive numerical algorithms to calculate the integrals in the velocity/position integration formulae and solve alignment through attitude optimization. The method uses interval-based approximations, including slowly varying attitude and linearly varying velocity assumptions, while specially treating rapidly changing body rates.

  • IV. RECURSIVE IN-FLIGHT ALGORITHMS: The recursive in-flight algorithms accurately calculate the integrals in equations (11) and (20) before solving alignment through attitude optimization.These algorithms are designed for the current time M and updated intervals of duration T.
  • IV. RECURSIVE IN-FLIGHT ALGORITHMS: The update scheme evaluates quantities at integer multiples of the update interval, with sample times defined from the interval duration T.The passages define successive times using the current interval index and kT.
  • A. Calculating Integrals in Velocity Integration Formula: The last integral in equation (11) is approximated by treating the attitude matrix as slowly changing during each incremental interval.The relevant attitude and gravity-related quantities are approximately treated as constants at the lower integral limit.
  • A. Calculating Integrals in Velocity Integration Formula: The velocity n v is assumed to change linearly within each interval, enabling approximation of the first integral on the right side of equation (11).The approximation is combined with the treatment of the relevant specific-force quantity as constant over the interval.
  • A. Calculating Integrals in Velocity Integration Formula: Substituting the integral approximations into equation (11) yields the corresponding recursive expression for the velocity-integration formula.The derivation explicitly substitutes equations (25) and (27) into the right side of equation (11).
  • A. Calculating Integrals in Velocity Integration Formula: The vector v in equation (11) is approximated using an incremental integral evaluated with a two-sample correction.The samples are accelerometer-measured incremental velocities and gyroscope-measured incremental angles.
  • A. Calculating Integrals in Velocity Integration Formula: Unlike n in, the body rate b ib changes rapidly and therefore requires special treatment in the numerical integration.The paper identifies this handling as common practice in the inertial navigation community.

B. Calculating Integrals in Position Integration Formula

The position integration formula is evaluated by approximating its double and single integrals under a linear-change assumption for n v. The resulting expression also uses a two-sample correction for the second integral on the left-hand side.

  • The second double integral on the right side of (20) is approximated using (25).
  • Similarly, the first double integral on the right side of (20) is approximated using (26)-(27).
  • Assuming linearly changing n v as in (26), the single integral in (20) is approximated.
  • The second integral on the left side of (20) is approximated using a two-sample correction, with details given in Appendix B.

C. In-flight Alignment Algorithms Derived from Velocity/ Position Integration Formulae

The section constructs two optimization-based in-flight alignment algorithms from velocity and position integration formulae. Both solve for the initial attitude using aided measurements and inertial sensor data, with integration avoiding velocity differentiation and reducing measurement noise.

  • Two algorithms are constructed from the velocity integration formula (11) and the position integration formula (20).
  • The alignment solves for the initial attitude matrix using a known time-varying vector from aided velocity or position and another from gyroscope and accelerometer measurements.
  • A unit attitude quaternion parameter enables optimization-based solution of the attitude equation under a unit-norm constraint.
  • Integral equations avoid calculating the velocity derivative required by the differential equation in (6) in and can significantly depress noise in aided velocity/position measurements.
  • The velocity-derived algorithm, IFA-VIF, is listed in Table I, while the position-derived algorithm, IFA-PIF, is listed in Table II.

V. SIMULATION RESULTS

Simulation results validate both proposed algorithms under ideal sensing and characterize their behavior under Monte Carlo runs with GPS lever-arm effects. IFA-VIF stabilizes faster but exhibits roughly twice the scattering scope of IFA-PIF, while the lever arm causes error lumps, peaks, and biased estimates.

  • Ideal case: With perfect sensors and no GPS lever arm, IFA-VIF and IFA-PIF produce correct attitude angles with negligible errors.This validates the preceding analysis and the implemented algorithms.
  • Monte Carlo evaluation: IFA-VIF has shorter stabilizing time but about one time larger scattering scope than IFA-PIF.At 300s, IFA-VIF angle errors (3σ) are -0.006 0.004 0.021 0.227 0.003 0.005 T ( ( ( in degree, versus IFA-PIF’s -0.006 0.003 -0.012 0.129 0.005 0.003 T ( ( ( in degree.
  • Lever-arm effect: For IFA-VIF, the GPS lever arm produces obvious lump peaks at about 30s and biased angle estimates.The comparison uses GPS-error-contaminated SINS velocity and position inputs with and without the lever arm.
  • Lever-arm effect: IFA-PIF shows a similar lever-arm-induced phenomenon, with error effects visible in its mean alignment angle errors.The corresponding behavior is reported in Fig. 6.

VI. FIELD TEST RESULTS

Flight tests on a Cessna 208 show that IFA-VIF outperforms IFA-PIF, especially when the SINS-GPS lever arm is eliminated. The lever arm must be seriously considered because it can markedly degrade alignment and is difficult to compensate.

  • Field-test setup: Flight data from a Cessna 208 were evaluated over three 100-second maneuvering segments: ascending S1, turning S2, and descending S3.GPS raw measurements at 2 Hz were linearly interpolated to provide velocity and position at both ends of each 0.02T-second update interval.
  • Raw GPS measurements: 5 seconds: both IFA-VIF level angle errors reduced to within 1 degree, while IFA-PIF level angle errors reduced to within 2 degree.For S1, IFA-VIF heading error reached about 5 degree in 10 seconds, compared with about 13 degree for IFA-PIF.
  • Raw GPS measurements: 20 seconds: S2-S3 yaw angle errors were below 3 degree for IFA-VIF and below 5 degree for IFA-PIF.Both methods had level angle errors below 2 degree in 5 seconds.
  • Lever-arm-compensated measurements: With the lever arm eliminated, IFA-VIF errors were less than 1 degree in 10s and 0.3 degree in 20s for heading, with level errors below 0.1 degree in 10 seconds.IFA-PIF heading errors were less than 5 degree in 10s and 2 degree in 20s, with level errors below 0.25 degree in 10 seconds.
  • Lever-arm effect: Over ten times: IFA-VIF showed remarkable improvement after lever-arm elimination, demonstrating that the lever arm should be seriously considered or avoided.Its effect can be minimized by placing the GPS antenna close to the SINS or avoiding quick turns, but precise compensation requires the attitude being solved.

VII. CONCLUSIONS

The paper presents an optimization-based in-flight coarse alignment approach using velocity/position integration formulae and recursive algorithms to address the difficulty of obtaining an initial attitude during motion. With the GPS lever arm handled, simulations and flight tests indicate heading alignment up to one degree accuracy in ten seconds, while future work targets low-quality SINS and sensor bias.

  • VII. CONCLUSIONS: Good attitude initialization is important for rapid, accurate SINS fine alignment, but obtaining a rough initial angle during motion or maneuvering is difficult.The conclusion frames moving-carrier alignment as the motivating problem.
  • VII. CONCLUSIONS: The paper proposes an optimization-based in-flight coarse alignment approach founded on velocity/position integration formulae derived from the traditional navigation rate equation.The formulae are obtained through integration manipulations.
  • VII. CONCLUSIONS: Two recursive alignment algorithms are designed after carefully handling the integrals to reduce calculation errors caused by maneuvers, and simulations and flight tests evaluate them.The integral treatment is intended to reduce maneuver-induced calculation errors as much as possible.
  • VII. CONCLUSIONS: One degree accuracy in ten seconds is potentially achievable for SINS heading alignment when the GPS lever arm is well handled.The result comes from simulations and flight test data and suggests usefulness for applications requiring alignment on the run.
  • VII. CONCLUSIONS: Future work will further test the algorithms on low quality SINS and address sensor bias using the proposed optimization approach.The paper also identifies on-the-run INS alignment as a broader application area.

APPENDIX · A. Two-sample Approximation of Integral in (29) · B. Two-sample Approximation of Integral in (35)

The appendix develops two-sample approximations for the integrals in (29) and (35). It assumes approximately linear body angular velocity and specific force, then derives coefficient vectors and incremental angle and velocity approximations.

  • A. Two-sample Approximation of Integral in (29): Appendix A assumes body angular velocity and specific force follow approximate linear forms.This assumption underpins the two-sample approximation in subsection A.
  • A. Two-sample Approximation of Integral in (29): The incremental-angle approximation is obtained from the assumed linear representations.The passage introduces the appropriate coefficient vectors used in this derivation.
  • A. Two-sample Approximation of Integral in (29): The coefficient vectors are solved from the incremental-angle relations.The appendix proceeds from the angle expression to coefficient-vector solutions.
  • A. Two-sample Approximation of Integral in (29): The same two-sample procedure is applied to derive the incremental-velocity expression.This extends the approximation from angular increments to velocity increments.
  • A. Two-sample Approximation of Integral in (29): The integral in (29) is then replaced by its two-sample approximation.The visible passage states the approximation step without exposing the formula text.
  • B. Two-sample Approximation of Integral in (35): Appendix B presents further displayed derivation fragments and identifies a derivation of b_5.The subsection also states that the body rotation vector can be approximated, but the accompanying formula is not recoverable in the supplied text.
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