Modified Bryson-Frazier Smoother Research Articles

Estimation of wave-induced steady force using system identification, model tests, and numerical approach

This paper addresses the method for estimating maneuvering mathematical models in waves, and uses captive model tests to verify it. To obtain the wave force and moment, two kinds of captive model tests are performed. One is straight towing tests with different towing directions to consider various incident wave angles, while the other is circular motion tests at different regular wave frequency. These captive model tests provide added resistance and wave-induced steady force considering both different wave direction and ship speed, and different wave drift angles. The wave drift forces and moment obtained are validated by performing the maneuvering simulation in waves, and comparison with the results of free running model tests. This paper introduces the application of the Estimation-Before-Modeling (EBM) approach, one of the System Identification methods, to estimate mathematical models for ship maneuverability targeting KRISO Very Large Crude oil Carrier (KVLCC2). The most important force component for the prediction of the maneuverability of a ship is the accuracy of the wave-induced steady force acting on the ship. Here, the EBM approach is divided into two steps: First, the motion variables and forces are estimated by Extended Kalman Filter and Modified Bryson-Frazier smoother. Second, the wave-induced steady forces are determined by removing the hydrodynamic force acting on the ship during motion. Also, during the second step of the EBM approach, the database consisting of wave length and incident wave angles is constructed. All estimated variables are based on data obtained by free running model tests in regular waves. Finally, the estimated wave-induced steady force is interpolated, added to the hydrodynamic force component of the 3-DoF equations of motion, and the turning simulation of the ship in waves is then performed. Comparison of the results with the results of free running model tests confirms the validity of the established model.

Modified Bryson-Frazier Smoother Cross-Covariance

The Expectation Maximization algorithm can be used to estimate Kalman filter parameter matrices. This requires smoother results and typically the Rauch-Tung-Striebel smoother is used for this purpose. If either of the state transition or plant noise matrices require estimation, a sequence of single time-step smoother cross-covariance matrices, based on the Rauch-Tung-Striebel smoother formulation, are also required. The modified Bryson-Frazier smoother could be used as the smoother for the Expectation Maximization algorithm, but to estimate the state transition or plant noise matrix, this smoother would first need a method to generate the sequence of single time-step smoother cross-covariance matrices based on the modified Bryson-Frazier formulation. This paper develops an expression for these modified Bryson-Frazier cross-covariance matrices, thus allowing the modified Bryson-Frazier smoother to be substituted for the Rauch-Tung-Striebel smoother in the Expectation Maximization algorithm. The modified Bryson-Frazier smoother formulation requires inversion only of filter innovation matrices, which are generally much smaller than state error covariance matrices, which allows both greater computational speed and better numerical accuracy due to smaller matrix size. This formulation can be used even when covariance and transition matrices are singular.

Aerodynamic Modeling and Parameter Estimation from QAR Data of an Airplane Approaching a High-altitude Airport

Aerodynamic modeling and parameter estimation from quick accesses recorder (QAR) data is an important technical way to analyze the effects of highland weather conditions upon aerodynamic characteristics of airplane. It is also an essential content of flight accident analysis. The related techniques are developed in the present paper, including the geometric method for angle of attack and sideslip angle estimation, the extended Kalman filter associated with modified Bryson-Frazier smoother (EKF-MBF) method for aerodynamic coefficient identification, the radial basis function (RBF) neural network method for aerodynamic modeling, and the Delta method for stability/control derivative estimation. As an application example, the QAR data of a civil airplane approaching a high-altitude airport are processed and the aerodynamic coefficient and derivative estimates are obtained. The estimation results are reasonable, which shows that the developed techniques are feasible. The causes for the distribution of aerodynamic derivative estimates are analyzed. Accordingly, several measures to improve estimation accuracy are put forward.

Square Root Modified Bryson–Frazier Smoother

We derive here an algorithm for a complete square root implementation of the modified Bryson-Frazier (MBF) smoother. The MBF algorithm computes the smoothed covariance as the difference of two symmetric matrices. Numerical errors in this differencing can result in the covariance matrix not being positive semi-definite. Earlier algorithms implemented the computation of intermediate quantities in square root form but still computed the smoothed covariance as the difference of two matrices. We show how to compute the square root of the smoothed covariance by solving an equation in the form CCT=AAT-BBT using QR decomposition with hyperbolic Householder transformations.

선박 조종미계수 식별 시 모델링 전 추정기법과 확장 Kalman 필터에 의한 계수추정법의 비교에 관한 연구

Two methods which estimate manoeuvring derivatives in the model of hydrodynamic force and moment acting on a manoeuvring ship using sea trial data were compared. One is the widely used parameter estimation method by using the Extended Kalman Filter (EKF), which estimates state variables of linearized state space model at every instant after dealing with the coefficients as the augmented state variables. The other one is the Estimation-Before-Modeling (EBM) technique, so called the two-step method. In the first step, hydrodynamic force of which dynamic model is assumed the third-order Gauss-Markov process is estimated along with motion variables by the EKF and the modified Bryson-Frazier smoother. Then, in the next step, manoeuvring derivatives are identified through the regression analysis. If the exact structure of hydrodynamic force could be known, which was an ideal case, the EKF method would be regarded as being more superior compared to the EBM technique. However the EBM technique was more robust than the EKF method from a realistic point of view where the assumed model structure was slightly different from the real one.

Identification of hydrodynamic coefficients in ship maneuvering equations of motion by Estimation-Before-Modeling technique

Estimation-Before-Modeling (EBM) technique (or the two-step method) is a system identification method that estimates parameters in a dynamic model. Given sea trial data, the extended Kalman filter and modified Bryson–Frazier smoother can be used to estimate motion variables, hydrodynamic force, and the speed and the direction of current. And using these estimated data, we can use the ridge regression method to estimate the hydrodynamic coefficients in a model. An identifiable state space model is constructed in case that current effect is included and the maneuvering characteristics of a ship are analyzed by correlation analysis. To better identify hydrodynamic coefficients, we suggest the sub-optimal input scenario that considers the D-optimal criterion. Finally, the algorithm is confirmed against real sea trial data of 113K tanker.

Determination of Nonlinear Aerodynamic Coefficients using the Estimation-Before-Modeling Methods

Aerodynamic coefficients of a Schweizer 2-32 sailplane have been estimated from flight test data for angles of attack (α) up to 30° and sideslip angles (β) to ±17°. The nonlinear aerodynamic model has been identified by applying the Estimation-Before-Modeling (EBM) technique to flight data derived from fifteen maneuvers including seven stalls and “post-stall gyrations”. An Extended Kalman-Bucy Filter and a Modified Bryson-Frazier smoother were used to estimate the time histories of the forces and moments from a fourteen-element discrete measurement vector. The optimal estimates and the measured control variables were sorted into 50 “subspaces”, and the aerodynamic modeling was performed using a multiple regression scheme in each subspace. The linear and nonlinear portions of the coefficients could be observed from the modeling results, providing insight into the stall dynamic behavior of the sailplane. The accuracy of the aerodynamic coefficients was assessed by comparing measured time histories of the sailplane with a six-degree-of-freedom simulation model response. A novel aspect of the research is the use of a quaternion-based mathematical model of aircraft dynamics. A microprocessor-based data acquisition system was used on board the sailplane.

An integrated multisensor aircraft track recovery system for remote sensing

This paper describes an application of the Kalman filter in a track recovery system (TRS) for postflight processing of aircraft navigation sensor data. The track recovery system has been successfully used as a key component of the Canadian aerial hydrography pilot project for mapping of shallow coastal waters. Recorded data from an inertial navigation system (INS) is combined with data obtained from a number of auxiliary sensors to construct a set of error measurements. The measurements are prefiltered to compress the data and are then processed using a U-D factorized Kalman filter and a modified Bryson-Frazier smoother to produce estimates of the time-correlated sensor errors. The flight profile is obtained by subtracting the computed error estimates from the recorded INS data. The residual errors observed in processing real data collected in a number of field tests are less than 1 m in position and less than 0.03 degrees in attitude.