A Total Least-Squares Estimate for Attitude Determination

2019 ◽  
Author(s):  
Yang Cheng ◽  
John L. Crassidis
Keyword(s):  
Least Squares ◽  
Attitude Determination ◽  
Total Least Squares ◽  
Least Squares Estimate

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

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Baowei Chen ◽  
Guobin Chang ◽  
Shengquan Li ◽  
Kailiang Deng
Keyword(s):  
Least Squares ◽  
Carrier Phase ◽  
Attitude Determination ◽  
Estimation Error ◽  
Direction Cosine ◽  
Initial Guess ◽  
Frobenius Norm ◽  
Phase Measurements ◽  
Least Squares Estimate ◽  
Gibbs Vector

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 roll-pitch-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.

Total Least-Squares Determination of Body Segment Attitude

2021 ◽  
Author(s):  
John Challis
Keyword(s):  
Rigid Body ◽  
Least Squares ◽  
Total Least Squares ◽  
Body Segment ◽  
Noise Levels ◽  
Least Squares Estimate ◽  
Body Attitude ◽  
Transformation Parameters ◽  
Marker Cluster ◽  
Rigid Body Transformation

Abstract To examine segment and joint attitudes when using image based motion capture it is necessary to determine the rigid body transformation parameters from an inertial reference frame to a reference frame fixed in a body segment. Determine the rigid body transformation parameters must account for errors in the coordinates measured in both reference frames, a total least-squares problem. This study presents a new derivation that shows that a singular value decomposition based method provides a total least-squares estimate of rigid body transformation parameters. The total least-squares method was compared with an algebraic method for determining rigid body attitude (TRIAD method). Two cases were examined: Case 1 where the positions of a marker cluster contained noise after the transformation, and Case 2 where the positions of a marker cluster contained noise both before and after the transformation. The white noise added to position data had a standard deviation from zero to 0.002 m, with 101 noise levels examined. For each noise level 10000 criterion attitude matrices were generated. Errors in estimating rigid body attitude were quantified by computing the angle, error angle, required to align the estimated rigid body attitude with the actual rigid body attitude. For both methods and cases as the noise level increased the error angle increased, with errors larger for Case 2 compared with Case 1. The SVD based method was superior to the TRIAD algorithm for all noise levels and both cases, and provided a total least-squares estimate of body attitude.

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