Quantitative analysis of aircraft aerodynamic derivatives using the least squares method in a six degrees of freedom flight simulation environment

Abstract

This thesis presents an in-depth analysis of aircraft aerodynamic derivatives using the Least Squares Method (LSM) within a six degrees of freedom (6-DOF) flight simulation environment. The primary objective is to evaluate and compare the performance of Ordinary Least Squares (OLS) and Recursive Least Squares (RLS) methods in estimating aerodynamic parameters under various flight conditions, including ideal, turbulent, and error-induced scenarios. A detailed 6-DOF flight simulation model was developed using data from the SIAI Marchetti S211 aircraft. This model integrates various subsystems, including equations of motion, aerodynamics, engine dynamics, and atmospheric conditions. The Newton-Raphson method was employed to maintain steady-state conditions, ensuring the aircraft's trim state was accurately represented. For solving the differential equations derived from the equations of motion, the Runge-Kutta method was chosen due to its robustness and accuracy in handling the nonlinearities associated with flight dynamics in the simulation model. The aerodynamic forces and moments were linearized using the small disturbance theorem, which simplifies the complex nonlinear equations into a more manageable linear form. This linearization allowed for the formulation of force and moment coefficients as functions of aerodynamic derivatives. These derivatives, critical for understanding the aircraft's behavior, were estimated using both OLS and RLS methods. Realistic flight data was simulated under various conditions, including ideal scenarios without any disturbances, scenarios with atmospheric turbulence, and scenarios with systematic sensor errors. The Dryden turbulence model was used to simulate realistic atmospheric disturbances, providing a continuous representation of turbulence that affects the aircraft during flight. Systematic sensor errors were introduced to understand their impact on the accuracy of parameter estimation. The OLS method provided single-step parameter estimates by processing all data points simultaneously, making it straightforward and computationally efficient. In contrast, the RLS method updated parameter estimates incrementally as new data became available. This dynamic approach allowed the RLS method to adapt to changes over time, making it particularly suitable for real-time applications where system characteristics may vary. Performance metrics such as the $R^2$ statistic and standard deviation were used to evaluate the estimation accuracy. These metrics provided quantitative measures of how well the estimated parameters matched the true values, with the $R^2$ statistic indicating the proportion of variance explained by the model and the standard deviation providing a measure of the estimation precision. The analysis revealed that both OLS and RLS methods produced accurate results under ideal and turbulent conditions. The presence of atmospheric turbulence did not significantly affect the estimation accuracy, as the average error introduced by the turbulence was zero. This robustness highlights the effectiveness of LSM in handling real-world flight data with environmental disturbances. However, when systematic sensor errors were introduced, both OLS and RLS methods showed biased estimation results. The bias was evident in the deviation of the estimated aerodynamic derivatives from their true values, underscoring the importance of accurate and error-free measurement data for reliable parameter estimation. Further analysis demonstrated that increasing the sampling frequency improved the performance of the RLS method. At higher frequencies, such as 50 kHz, the RLS estimates converged more closely to the true values, even in the presence of systematic sensor errors. This improvement is attributed to the reduced information loss in higher frequency sampling, which captures more details and variations in the data that might be missed at lower frequencies. This finding suggests that higher sampling rates can effectively mitigate the adverse effects of sensor errors on parameter estimation. The design of control surface inputs was identified as a crucial factor influencing the accuracy of aerodynamic parameter estimation. Optimal input design, which involves selecting appropriate control surface deflections, ensured accurate estimation results. Conversely, non-optimal inputs led to discrepancies between the estimated and true values. This emphasizes the need for carefully designed excitation maneuvers during flight tests to obtain reliable aerodynamic data. The RLS method demonstrated particular advantages in dynamic environments due to its ability to update estimates in real-time. This adaptive capability allowed it to maintain accuracy even when the system characteristics changed over time. However, the OLS method exhibited slightly better performance at lower frequencies, showing less sensitivity to variations in sampling rates. Both methods showed distinct strengths, with OLS excelling in stable, low-frequency scenarios and RLS proving superior in dynamic, high-frequency conditions. The theoretical expected value formulas for the parameter estimates were validated using the simulation model outputs. This validation confirmed the presence of bias when systematic errors were introduced and reinforced the high accuracy of estimates under both ideal and turbulent conditions. In conclusion, this thesis provides a comprehensive evaluation of OLS and RLS methods for estimating aerodynamic derivatives in a 6-DOF flight simulation environment. The findings demonstrate the robustness of these methods under various flight conditions, highlight the impact of systematic sensor errors, and underscore the importance of optimal input design and high-frequency data sampling under linear database.