Nonlinear Stochastic Attitude Filters on the Special Orthogonal Group 3: Ito and Stratonovich

This paper formulates the attitude filtering problem as a nonlinear stochastic filter problem evolved directly on the Special Orthogonal Group 3 (<inline-formula> <tex-math notation="LaTeX">${\mathbb {SO}}(3)$ </tex-math></inline-formula>). One of the traditional potential functions for nonlinear deterministic complimentary filters is studied and examined against angular velocity measurements corrupted with noise. This paper demonstrates that the careful selection of the attitude potential function allows to attenuate the noise associated with the angular velocity measurements and results into superior convergence properties of estimator and correction factor. The problem is formulated as a stochastic problem through mapping <inline-formula> <tex-math notation="LaTeX">$ {\mathbb {SO}}(3)$ </tex-math></inline-formula> to Rodriguez vector parameterization. Two nonlinear stochastic complimentary filters are developed on <inline-formula> <tex-math notation="LaTeX">$ {\mathbb {SO}}(3)$ </tex-math></inline-formula>. The first stochastic filter is driven in the sense of Ito and the second one considers Stratonovich. The two proposed filters guarantee that errors in the Rodriguez vector and estimates are semi-globally uniformly ultimately bounded in mean square. Simulation results are presented to illustrate the effectiveness of the proposed filters considering high level of uncertainties in angular velocity as well as body-frame vector measurements.

[1]  Robert E. Mahony,et al.  Nonlinear Complementary Filters on the Special Orthogonal Group , 2008, IEEE Transactions on Automatic Control.

[2]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[3]  Frank L. Lewis,et al.  Neuro-adaptive cooperative tracking control with prescribed performance of unknown higher-order nonlinear multi-agent systems , 2018, Int. J. Control.

[4]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[5]  Miroslav Krstic,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 2001, IEEE Trans. Autom. Control..

[6]  Shaocheng Tong,et al.  Observer-Based Adaptive Fuzzy Backstepping Control for a Class of Stochastic Nonlinear Strict-Feedback Systems , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[7]  Henrique Marra Menegaz,et al.  A Systematization of the Unscented Kalman Filter Theory , 2015, IEEE Transactions on Automatic Control.

[8]  E. J. Lefferts,et al.  Kalman Filtering for Spacecraft Attitude Estimation , 1982 .

[9]  R. G. Medhurst,et al.  Topics in the Theory of Random Noise , 1969 .

[10]  Philippe Martin,et al.  Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[11]  Sami El-Ferik,et al.  Optimal Tuning of Fuzzy Feedback filter for L1 Adaptive Controller Using Multi-Objective Particle Swarm Optimization for Uncertain Nonlinear MIMO Systems , 2017, 1710.05423.

[12]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[13]  S. Haykin.  Kalman Filtering and Neural Networks , 2001 .

[14]  Hai-Bo Ji,et al.  Adaptive output-feedback tracking of stochastic nonlinear systems , 2006, IEEE Trans. Autom. Control..

[15]  Robert E. Mahony,et al.  Minimum-Energy Filtering for Attitude Estimation , 2013, IEEE Transactions on Automatic Control.

[16]  M. Shuster,et al.  Three-axis attitude determination from vector observations , 1981 .

[17]  Derong Liu,et al.  An Approximate Optimal Control Approach for Robust Stabilization of a Class of Discrete-Time Nonlinear Systems With Uncertainties , 2016, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[18]  Taeyoung Lee,et al.  Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers , 2012, Syst. Control. Lett..

[19]  I. Bar-Itzhack,et al.  Novel quaternion Kalman filter , 2002, IEEE Transactions on Aerospace and Electronic Systems.

[20]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[21]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[22]  F. Markley.  Attitude determination using vector observations and the singular value decomposition , 1988 .

[23]  J. Crassidis,et al.  Particle Filtering for Sequential Spacecraft Attitude Estimation , 2004 .

[24]  M. Shuster.  A survey of attitude representation , 1993 .

[25]  Jianbin Qiu,et al.  Adaptive Neural Control of Stochastic Nonlinear Time-Delay Systems With Multiple Constraints , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[26]  Guang-Hong Yang,et al.  Observer-Based Output Feedback Control for Discrete-Time T-S Fuzzy Systems With Partly Immeasurable Premise Variables , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[27]  M. Krstić,et al.  Stochastic nonlinear stabilization—I: a backstepping design , 1997 .

[28]  Frank L. Lewis,et al.  Adaptive synchronisation of unknown nonlinear networked systems with prescribed performance , 2017, Int. J. Syst. Sci..

[29]  James Richard Forbes,et al.  Nonlinear Estimator Design on the Special Orthogonal Group Using Vector Measurements Directly , 2017, IEEE Transactions on Automatic Control.

[30]  G. Wahba.  A Least Squares Estimate of Satellite Attitude , 1965 .

[31]  F. Markley.  Attitude Error Representations for Kalman Filtering , 2003 .

[32]  R. Mortensen.  Maximum-likelihood recursive nonlinear filtering , 1968 .

[33]  H. D. Black,et al.  A passive system for determining the attitude of a satellite , 1964 .

[34]  Yonggui Kao,et al.  A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems , 2015, Autom..

[35]  John L. Crassidis,et al.  Survey of nonlinear attitude estimation methods , 2007 .

[36]  Malcolm D. Shuster.  Survey of attitude representations , 1993 .

[37]  Xiaoming Hu,et al.  Drift-free attitude estimation for accelerated rigid bodies , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[38]  F. Markley,et al.  Unscented Filtering for Spacecraft Attitude Estimation , 2003 .

[39]  Tarek Hamel,et al.  Attitude and gyro bias estimation for a VTOL UAV , 2006 .

[40]  H. Mohamed.  Improved Robust Adaptive Control of High-order Nonlinear Systems with Guaranteed Performance , 2018, 1807.00876.

[41]  R. L. Stratonovich,et al.  Topics in the theory of random noise , 1967 .

[42]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.