On the Convergence of Iterative Orthogonalization Processes

Bar-Itzhack, Itzhack Y.; Meyer, Jenny

doi:10.1109/taes.1976.308289

Cited by 38 publications

(23 citation statements)

References 6 publications

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“…A Newton method. Consider the iteration (the real matrix version of which is discussed in [3], [4], [5], [6], [28] To analyse the rate of convergence we write (3.6) in the form…”

mentioning

confidence: 99%

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Computing the Polar Decomposition—with Applications

Higham¹

1986

SIAM J. Sci. and Stat. Comput.

379

281

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Abstract. A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice.To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A and 1/2(A / H) is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived.Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.

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“…A Newton method. Consider the iteration (the real matrix version of which is discussed in [3], [4], [5], [6], [28] To analyse the rate of convergence we write (3.6) in the form…”

mentioning

confidence: 99%

“…It is shown in [3], [5], [28] [8], [10], [20], [21], [25]. [10], [21], [25] for the case where A is symmetric positive definite.…”

mentioning

confidence: 99%

Computing the Polar Decomposition—with Applications

Higham¹

1986

SIAM J. Sci. and Stat. Comput.

379

281

View full text Add to dashboard Cite

show abstract

“…see [18]. Nevertheless, additional orthogonalization cycles may be employed should it be required in order to achieve solutions closer to SO(3).…”

Section: Solution Refinements 1) Full Cascade Observermentioning

confidence: 99%

Computationally efficient GES cascade observer for attitude estimation

Batista

Silvestre

Oliveira

2011

Proceedings of the 2011 American Control Conference

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This paper presents the design, analysis, and performance evaluation of a novel cascade observer for attitude estimation. First, a sensor-based observer, which resorts to rate gyro readings and a set of vector observations, estimates the rate gyro bias. Afterwards, a second observer explicitly estimates the attitude in the form of a rotation matrix based on the rate gyro measurements, the vector observations, and the estimate of the rate gyro bias provided by the first observer. The error dynamics of the overall cascade estimation system are globally exponentially stable (GES) and do not suffer from drawbacks common to attitude estimation solutions such as singularities, unwinding phenomena, or topological limitations for achieving global asymptotic stability (GAS). In addition, the proposed system is computationally efficient and hence it is easily implementable with low computational capabilities. The fact that the observer does not evolve explicitly on SO (3), providing in fact estimates that converge asymptotically to SO(3), is also addressed and an effective and efficient solution is proposed. Finally, the resulting estimator is evaluated, where a Motion Rate Table (MRT) that provides ground truth data is employed for performance evaluation purposes.

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“…This algorithm is suboptimal but is less computationally intensive than the OBF procedure, which involves the SVD ofD k=k . When applied recursively, this algorithm produces a sequence of estimates that converges to the optimal solution of (70) [19]. That technique was succesfully applied in a previous work on DCM identification [11].…”

Section: Iterative Brute-forcementioning

confidence: 99%

Direction Cosine Matrix Estimation from Vector Observations using a Matrix Kalman Filter

Choukroun

Weiss

Bar-Itzhack

et al. 2010

IEEE Trans. Aerosp. Electron. Syst.

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This work presents several algorithms that use vector observations in order to estimate the direction cosine matrix (DCM) as well as three constant biases and three time-varying drifts in body-mounted gyro output errors. All the algorithms use the matrix Kalman filter (MKF) paradigm, which preserves the natural formulation of the DCM state-space model equations. Focusing on the DCM estimation problem, the assumption of white noise in the gyro and in the vector observations errors yields reduced and efficient filter covariance computations. The orthogonality constraint on the DCM is handled via the technique of pseudomeasurement, which is naturally embedded in the MKF. Two additional known "brute-force" procedures are implemented for the sake of comparison. Extensive Monte-Carlo simulations illustrate the performances of the different estimators. When estimating only the DCM, it is shown that all the proposed orthogonalization procedures accelerate the estimation convergence. Nevertheless, the pseudomeasurement technique shows a smoother and shorter transient than the brute-force procedures, which on the other hand yield more accurate steady-states. The reduced covariance computations yield a more accurate steady-state than the full covariance computations but show a slower transient. When estimating the DCM as well as the gyro biases and drifts, enforcing orthogonalization seems to penalize the DCM estimation as long as the biases are not correctly identified. For the sake of computation savings during long duration missions, a mixed estimator, switching between long periods of DCM-only estimation and short periods of DCM-biases estimation, appears to be a promising strategy. Manuscript

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On the Convergence of Iterative Orthogonalization Processes

Cited by 38 publications

References 6 publications

Computing the Polar Decomposition—with Applications

Computing the Polar Decomposition—with Applications

Computationally efficient GES cascade observer for attitude estimation

Direction Cosine Matrix Estimation from Vector Observations using a Matrix Kalman Filter

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