Analytical 3D rotation estimation using vector measurements with full variance-covariance matrix

Xu, Tianhe; Chang, Guobin; Wang, Qianxin; Hu, Chao

doi:10.1016/j.measurement.2016.11.037

Cited by 5 publications

(4 citation statements)

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“…Note that, in (18), (19), and (20), k=1 and 2. For an arbitrary non-singular 3×3 matrix X and an arbitrary 3×1 vectorx, the following can be validated:…”

Section: Degenerated Cases Withmentioning

confidence: 99%

“…The relaxation apparently introduces approximation or in other words costs the optimality. In fact, a mathematically feasible DCM can be extracted from an arbitrary 3×3 matrix by fulfilling certain optimality criterion; see, e.g., [18]. The problem can be transformed into Wahba's problem [19] to which analytical solutions exist; see, e.g., [20,21].…”

Section: Introductionmentioning

confidence: 99%

“…In this contribution, the analytical and the iterative methods in measurement domain are both studied, with the former providing an initial guess for the latter. In the analytical solution, similar methodology is adopted as in [17], but the orthogonality and +1 determinant properties of the estimated attitude matrix are strictly satisfied by the following [18]. Note here that only the attitude estimate is of interest as an intermediate quantity; its error analysis as presented in [18] is not necessary.…”

Section: Introductionmentioning

confidence: 99%

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Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Chen

Chang

et al. 2019

Mathematical Problems in Engineering

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Attitude determination using double-differenced GNSS carrier phase measurements is studied. A realistic stochastic model is employed to take the correlations among the double-differenced measurements into full consideration. Two important issues concerning iteratively solving the nonlinear least-squares attitude determination problem are treated, namely, the initial guess and the iteration scheme. An analytical and sub-optimal solution is employed to provide the initial guess. In this solution, the orthogonal and determinant constraints among the elements of the direction cosine matrix (DCM) of the attitude are firstly ignored, and hence a relaxed 3×3 matrix is estimated using the linear weighted least-squares method. Then a mathematically feasible DCM, i.e., orthogonal and with +1 determinant, is extracted from the relaxed matrix estimate, optimally in the sense of minimum Frobenius norm. This analytical initial guess estimation method can be used for all feasible cases, including some generated ones, e.g., the case with only 3 antennas and only 3 satellites, subject possibly to some necessary, yet minor modifications. In each iteration, an error attitude, whose DCM is parameterized using the Gibbs vector, is introduced to relate the previously estimated and the true DCM. By linearizing the measurement model at the zero Gibbs vector, the least-squares estimate of the Gibbs vector is obtained and then used to correct the previously estimated DCM. By repeating this process, the truly least-squares estimate of the attitude can be achieved progressively. These are in fact Gauss-Newton iterations. For the final estimate, the variance covariance matrix (VCM) of the attitude estimation error can be retained to evaluate or predict the estimation accuracy. The extraction of the widely used rollpitch-yaw angles and the VCM of their additive estimation errors from the final solution is also presented. Numerical experiments are conducted to check the performance of the developed theory. For the case with 3 2-meter long and orthogonally mounted baselines, 5 visible satellites, and 5-millimeter standard deviations of the carrier phase measurements, the root mean squared errors (RMSE) of the roll-pitch-yaw angles in the analytical solution are well below 0.5 degrees, and the estimates converge after only one iteration, with all three RMSEs below 0.2 degrees.

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“…Note that, in (18), (19), and (20), k=1 and 2. For an arbitrary non-singular 3×3 matrix X and an arbitrary 3×1 vectorx, the following can be validated:…”

Section: Degenerated Cases Withmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Chen

Chang

et al. 2019

Mathematical Problems in Engineering

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“…This approach is followed and expanded in that a mathematically feasible DCM is extracted from the estimated free 3 × 3 matrix in certain optimal sense. 32,33 In Cohen 34 and Bell, 35 the original problem was transformed into a Wahba’s problem, 36 which can be solved analytically. 37,38 However, this transformation is not an equivalent one, or in other words, approximation is introduced therein, because the implied weight matrix in the Wahba’s problem failed to be the inverse of the VCM of the measurement error.…”

Section: Introductionmentioning

confidence: 99%

A semi-analytical solution to the global navigation satellite system attitude determination problem

Jian-jun

Fan

et al. 2017

Proceedings of the Institution of Mechanical Engineers, Part G:

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Global navigation satellite system attitude determination based on carrier phase differencing technique is studied. A realistic stochastic model is followed to fully consider the correlations among different measurements in the least-squares problem formulation. A prior-free, computation-fixed, and averagely optimal solution is proposed suitable for real-time and high-dynamic situations. The prior-free property is achieved by developing an analytical sub-optimal solution. This solution follows first transforming the original problem into one with vector measurements and then further approximating it with a general Wahba’s problem. The fixed computation is guaranteed by performing only one or two rounds of correction of the analytical solution. Every single round follows a linearization-estimation-correction process. The process also provides an error or covariance analysis for the estimate. The average optimality in terms of the root mean squared errors is brought about by the relatively good quality of the analytical solution and the fast convergence of the correction processes. The numerical experiments, with three 4 m long baselines and 5 mm (standard deviation) carrier phase errors, show that the estimation errors (in magnitude) for all three channels are well below 0.4° for almost all epochs and within 0.2° for most epochs.

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Data snooping algorithm for universal 3D similarity transformation based on generalized EIV model

Wang

Liu

et al. 2018

Measurement

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Analytical 3D rotation estimation using vector measurements with full variance-covariance matrix

Cited by 5 publications

References 44 publications

Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

A semi-analytical solution to the global navigation satellite system attitude determination problem

Data snooping algorithm for universal 3D similarity transformation based on generalized EIV model

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