Observability of Strapdown INS Alignment: A Global Perspective

• Linearization-based analyses can require difficult rank tests, characterize only local nonlinear behavior, and rely on matrices that depend on the unknown state.Source: §I · p. 2
• The paper analyzes nonlinear strapdown INS alignment directly from observability’s basic definition, exploiting SO(3) attitude structure to construct a global account of tumbling effects.Source: §I · p. 2
• Global observability concerns estimating states over a finite interval, whereas instantaneous observability distinguishes neighboring states locally; nonlinear systems can be globally observable but instantaneously unobservable.Source: §I · p. 2
• The paper formulates alignment as solving continuous-time nonlinear equations and derives constructive global-observability theorems for static and maneuvering cases, including sufficient conditions for full observability.Source: §I · p. 2
• The global perspective exposes incompleteness and inconsistency in prior linearization-based studies and offers an efficient observability-analysis approach for other strapdown INS systems,, and.Source: §I · p. 2

• The formulation assumes known ground location, uses local-level frame N, models attitude, velocity, and constant gyro and accelerometer biases, and sets ground velocity to zero during alignment.Source: §II · pp. 2–3
• Previous linearization-based observability analysis approximates instantaneous observability but is not a consistent approximation.Source: §II · p. 2
• Deterministic observability means the initial state can be uniquely determined from known input and output over a finite time interval [12].The analysis excludes random noise, unlike stochastic observability.Source: §II · p. 3
• The SAOP asks whether known body angular rate and specific force, produced by attitude motion, uniquely determine initial attitude, gyroscope drift, and accelerometer bias.Source: §II · p. 3

Global perspective

• The global method decouples and solves the nonlinear system equations to analyze strapdown INS alignment observability.The stated non-polar location assumption applies unless explicitly noted otherwise.Source: §III · p. 3

Lemmas

• Two linearly independent vectors expressed in two frames determine the attitude matrix between those frames.Source: §III · p. 3
• An unknown point at a fixed distance from known points has a unique solution when those points are not coplanar.Source: §III · pp. 3–4
• An unknown constant vector in a(t) × m = b(t) is uniquely determined when the known vector a(t) changes direction over time.Source: §III · p. 4
• For a × m = b with nonzero a, knowing the magnitude of m yields explicit solutions for m.Source: §III · p. 4

A. Static Alignment

• Static alignment is unobservable, with infinitely many unobservable states because feasible sensor-bias pairs leave the initial body attitude indeterminate.The unobservable states are constrained by relations among the gyro bias, accelerometer bias, and attitude.Source: A. Static Alignment · p. 4
• A full-state estimator may converge to one unobservable state according to its settings, including its initial value, rather than determine the true state.The paper notes that practitioners commonly assume zero inertial sensor biases for static alignment; neglected-bias standard deviations limit the accuracy of the remaining states.Source: A. Static Alignment · p. 5
• Static segments yield observability if either their body angular-rate vectors or their specific-force vectors do not lie in any common plane.Under this condition, the sensor biases are uniquely determined, after which the initial body attitude can be solved.Source: A. Static Alignment · p. 5
• Two still positions with different headings do not guarantee unique sensor biases; at least four still positions are required, and the motion between positions matters.The cited two-position claim does not account for intervening rotation; later work and the authors’ note partly addressed the defect.Source: A. Static Alignment · p. 5
• The multiposition observability condition uses information from disconnected static stays alone, so it imposes tight conditions on the motion.The attitude motion connecting static positions can also contribute to observability, which the paper examines next.Source: A. Static Alignment · p. 5

B. Tumbling Alignment

• Under Theorem 3’s nonconstant-direction condition, single-direction rotation leaves two indistinguishable states unless the rotation direction is perpendicular to both gravity and Earth rate; one bias is observable when perpendicular to only one.The accelerometer bias is observable when the direction is perpendicular to gravity but not Earth rate; the gyroscope bias is observable when perpendicular to Earth rate but not gravity.Source: B. Tumbling Alignment · pp. 5–6
• Static periods contribute nothing to observability improvement; if the nonconstant-direction condition holds during a rotating interval, Theorem 3’s claims still apply.This also holds when static and rotating segments are interlaced.Source: B. Tumbling Alignment · p. 7
• The paper says earlier claims that rotating a strapdown INS around the vertical direction makes alignment observable, and hence globally observable, contradict Theorem 4 and Remark 6,, and.Source: B. Tumbling Alignment · p. 7
• Theorem 3’s constructive proof gives explicit analytical solutions for observable or unobservable states and enables an ideal observer, but requires exact first and second output derivatives.Source: B. Tumbling Alignment · p. 8
• The fixed-direction, fixed-magnitude assumption can be relaxed to fixed-direction, varying-magnitude rotation when the specific-force derivative is nonzero, yielding Theorem 3’s claims.Theorems 5 and 6 extend this result to an interval containing static periods and remove the smooth-tumbling-table requirement.Source: B. Tumbling Alignment · p. 8
• At least two nonzero constant-rate rotation intervals with linearly independent directions make the system observable.This establishes that successive tumbling about different directions can attain an observable alignment problem.Source: B. Tumbling Alignment · pp. 8–9
• Given linearly independent nb on the subintervals, bg and ba can be determined, making Cb n(0) unique.The passage attributes this result to (47).Source: B. Tumbling Alignment · p. 9
• At least two linearly independent subintervals where !b nb is nonzero, fixed-direction, and varying in magnitude, together with _fb ≠ 0, make the system observable.Theorem 8 gives these conditions for observability.Source: B. Tumbling Alignment · p. 9

IV. SIMULATION STUDY

• Simulations use IO-NCR, IO-NFVR, and an EKF to verify the analytical results.Sensor noise is omitted to make observability changes apparent.Source: §IV · p. 9
• The simulated INS has gyroscope drift of 0.01 deg/h, accelerometer bias of 50 ¹g, 100 Hz output bandwidth, and is located at L = 28:2204 deg and h = 60 m.The stated Euler-angle order is yaw about y, pitch about z, then roll about x.Source: §IV · p. 9
• The 12-state indirect EKF models angle, velocity, gyroscope-bias, and accelerometer-bias errors, with 20 s of coarse alignment and an added 1 deg random alignment error.Initial sensor biases are zero unless otherwise stated.Source: §IV · p. 9

A. Static Alignment

• The EKF converges quickly in both 300-s runs, but produces quite different estimates for different initial attitudes.The bias constraints in (9)–(11) govern the unobservable sensor-bias estimates, and (13) likewise constrains the attitude estimate.Source: A. Static Alignment · p. 9
• Gyroscope and accelerometer bias estimates lie on unobservable circles formed by intersecting a sphere and a cone.For each bias, the cone’s axis is the negative of the corresponding measured vector, and its half-angle is π/2 − L.Source: A. Static Alignment · p. 9
• In 1000 Monte Carlo EKF runs, gyroscope bias estimates form a visible circular segment, while accelerometer estimates are too close to distinguish.The bias spaces follow approximately from (11) because sensor biases are small relative to gyroscope and accelerometer outputs.Source: A. Static Alignment · p. 9

Single-axis results

• Up-down and north-south rotations admitted two bias solutions, whereas east-west rotation yielded a unique solution.For east-west rotation, the passage identifies the system as observable and reports zero separation between the alternative bias solutions.Source: B. Tumbling Alignment by Single-Axis… · pp. 10, 12

Up-Down Direction

• During up-down rotation, EKF bias estimates converged to the true values, but an alternate initial accelerometer bias led to an unobservable state with roll at 180 degrees.After rotation stopped, gyroscope estimates approached truth faster while accelerometer estimates departed from it; the static segment did not improve observability but appeared to mitigate numerical errors.Source: B. Tumbling Alignment by Single-Axis… · pp. 10–11, Figure 9 · p. 13
• Varying-magnitude up-down rotation produced the same IO-NFVR bias result as IO-NCR, while EKF convergence to the correct or wrong solution depended on the initial value.The varying rate was 6 + 4sin(0.04π(t − 100)) deg/s.Source: B. Tumbling Alignment by Single-Axis… · p. 11

North-South Direction

• For north-south rotation, EKF estimates converged to the true values despite oscillations, but a different initial gyroscope bias led to the wrong solution.The gyroscope bias along the rotation direction had two solutions, while the other reported sensor biases had one.Source: B. Tumbling Alignment by Single-Axis… · p. 12

East-West Direction

• For east-west rotation, EKF attitude estimates showed severe jumps, and the gyroscope bias in the rotation axis converged extremely slowly; nonzero initial attitude improved convergence speed.The passage also reports strong oscillations in the z-axis accelerometer bias.Source: B. Tumbling Alignment by Single-Axis… · pp. 12–13

C. Tumbling Alignment by Multiple-Axis Rotation

• The second rotation, starting at 800 s, removes the remaining solution ambiguity and reduces attitude and bias errors, especially accelerometer y- and z-axis bias errors.The EKF estimates converge satisfactorily to their true values. The rotations proceed east-west at 100–700 s, north-south at 800–1400 s, and vertically at 1500–2100 s, with static periods otherwise.Source: C. Tumbling Alignment by Multiple-Axis… · pp. 13–14
• With an initial accelerometer bias of [0,18 m=s2,0]T, the EKF stabilizes at a wrong solution during the first up-down rotation but diverges from it when the second rotation begins.The second rotation eliminates the solution ambiguity, so the wrong solution is no longer a stabilizing state for the EKF.Source: C. Tumbling Alignment by Multiple-Axis… · p. 14

V. CONCLUSIONS AND FUTURE WORKS

• Successive rotation about two axes makes the strapdown INS alignment problem fully observable.The analysis frames observability as whether the initial state can be determined from continuous-time nonlinear equations under known body angular rate and specific force.Source: §V · p. 15
• Single-axis rotation yields near observability, with no more than two unobservable states whose explicit solutions are derived analytically.Source: §V · p. 15
• For observability, rotating motion matters, not static positions.The global observability problem is formulated as SAOP: whether the initial state can be determined by solving infinite nonlinear equations over a continuous time interval.Source: §V · p. 15
• The constructive global analysis clarifies strapdown INS alignment and highlights incompleteness and inconsistency in previous results, prompting review of linearization-based observability studies.Source: §V · p. 15
• Simulations with constructed ideal observers and an approximate nonlinear EKF support the theorems; EKF results show static segments mitigate numerical errors in rotating and interlaced simulations.Source: §V · p. 15
• The conclusions can guide practical tumbling strategy and observer design, but lever-arm effects on zero-velocity measurement accuracy and estimation performance remain future work.The lever arm between the table rotation center and INS origin may remarkably decay zero-velocity measurement accuracy and thus estimation performance.Source: §V · p. 16

A. Proof of Lemma 2

• The equality |a_k − x| = r constrains x to a sphere of radius r centered at each a_k.The proof subtracts the equation for k = 1 from those for the remaining points to obtain linear constraints.Source: A. Proof of Lemma 2 · p. 16
• With full-rank constraints, x is determined when the points a_k do not lie in one plane, which requires at least four points.Any three points lie in a common plane.Source: A. Proof of Lemma 2 · p. 17
• When rank(A) = 2 and the a_k lie in a plane but not a line, feasible x values form a line that intersects the sphere in one or two solutions.The number depends on how the line meets the sphere.Source: A. Proof of Lemma 2 · p. 17
• When rank(A) = 1 and the a_k lie on a common line, possible solutions lie at the intersection of the sphere and a plane normal to that line.The solution space has dimension 2.Source: A. Proof of Lemma 2 · p. 18
• When rank(A) = 0 and all a_k are equal, the solution set is the sphere centered at a_1.This is the sphere defined by equation (49).Source: A. Proof of Lemma 2 · p. 18

B. Proof of Lemma 3

• Because a(t) has nonconstant directions, two instants with different directions can be chosen, and m then has a unique solution.The proof identifies t1 and t2 for which a(t1) and a(t2) have different directions.Source: B. Proof of Lemma 3 · p. 18

C. Proof of Lemma 4

• The unknown vector m lies in the plane spanned by a and a × b.The proof then expresses m using real coefficients along those two vectors.Source: C. Proof of Lemma 4 · p. 19
• With a^T b = 0, the proof uses a × (a × b) = −|a|^2 b to constrain the coefficient of a × b.Substitution gives −β|a|^2 b = b.Source: C. Proof of Lemma 4 · p. 19

D. Why Are Only Two of Four Pairs Valid?

• The angular relation allows only one of the two sign choices for the bias pair to satisfy the equality.The “+” case means the accelerometer and gyroscope biases have the same sign; otherwise they have different signs.Source: D. Why Are Only Two of Four Pairs Valid? · pp. 20, 22
• If (ω^b_nb × g^b) < 0, the feasible pairs have matching signs; otherwise, they have opposite signs.The matching pairs are (b_a+, b_g+) and (b_a−, b_g−); the opposite-sign pairs are (b_a+, b_g−) and (b_a−, b_g+).Source: D. Why Are Only Two of Four Pairs Valid? · p. 22

E. Added Linear Acceleration Helps Resolve the Ambiguity

• An added linear-acceleration maneuver after the first rotation uses a nonzero reference velocity v^n while acceleration is present.The passage introduces this condition to obtain an additional relation from the navigation equation.Source: E. Added Linear Acceleration Helps… · p. 22
• Differentiating the relation yields the accelerometer bias when ḟ^b is not parallel to ω^b_nb, thereby removing the gyroscope-bias ambiguity.The passage connects this recovery condition to Lemma 3.Source: E. Added Linear Acceleration Helps… · p. 22