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Obtaining precise attitude information is essential for aircraft navigation and control. This paper presents the results of the attitude determination using an in-house designed low-cost MEMS-based flight information measurement unit. This study proposes a quaternion-based extended Kalman filter to integrate the traditional quaternion and gravitational force decomposition methods for attitude determination algorithm. The proposed extended Kalman filter utilizes the evolution of the four elements in the quaternion method for attitude determination as the dynamic model, with the four elements as the states of the filter. The attitude angles obtained from the gravity computations and from the electronic magnetic sensors are regarded as the measurement of the filter. The immeasurable gravity accelerations are deduced from the outputs of the three axes accelerometers, the relative accelerations, and the accelerations due to body rotation. The constraint of the four elements of the quaternion method is treated as a perfect measurement and is integrated into the filter computation. Approximations of the time-varying noise variances of the measured signals are discussed and presented with details through Taylor series expansions. The algorithm is intuitive, easy to implement, and reliable for long-term high dynamic maneuvers. Moreover, a set of flight test data is utilized to demonstrate the success and practicality of the proposed algorithm and the filter design.

Obtaining an accurate vehicle attitude is essential for airplane navigation and control applications. The effectiveness of navigation and control is determined by the degree of precision of the navigation and control systems, including inertial measurement units. Traditional units—such as gimbaled gyroscopes, laser gyroscopes, and fiber optic gyroscopes and accelerometers—provide high-precision information for navigational calculations [

The attitude of the aero-vehicle can be determined by integrating the angular rates (pitch, roll, and yaw rates) of the vehicle. Nevertheless, accuracy requirements usually cannot be satisfied by using the inexpensive MEMS sensors. Therefore, some forms of Kalman filtering or complementary fusion algorithms are normally employed to provide more accurate and reliable attitude angles in the MEMS attitude determination systems. The pitch and roll angles can be estimated by using the gravity components from the measurement of the accelerometers [

This study considers the utilization of a low-cost MEMS-based flight information measurement unit to determine attitude. The main contribution of this paper is that an attitude estimation algorithm is proposed using MEMs sensors. In the proposed algorithm, an extended Kalman filter is used instead of the classical channel-wise complementary filters for attitude estimation. Specifically, the roll and pitch angles computed from the gravity force decomposition, and the heading angle determined from the electronic compass unit, are selected as the measurement of the Kalman filter. This is not commonly used in airplane attitude determination. The immeasurable gravity accelerations are deduced from the outputs of the three axes accelerometers, the relative accelerations, and the accelerations due to body rotation. The proposed attitude determination algorithm employed the evolution of the four elements of the quaternion method as the dynamical model for the Kalman filter. The constraint of the four elements of the quaternion method is thus treated as a perfect measurement and is integrated into the filter design. By adopting a coupled and time-varying approach, the proposed algorithm performs well under different dynamic maneuvers. The proposed attitude estimation algorithm is intuitive and easy to implement, and due to the vehicle dynamics has been considered in the gravity computation, the algorithm can handle the situation of the long-term high dynamic maneuvers. Detailed derivations establish the approximations of the time varying noise variances of the measured signals.

The proposed algorithm is demonstrated through a set of flight test data collected from the in-house designed MEMS-based attitude determination system. The duration of the flight covers a total of 3,620 s. Details of the evolutions of the attitude angles are examined and explored in the paper. The results show the long-term stability and practicality of the proposed attitude estimation algorithm.

The determination of attitude considered in this study is based on a set of signals measured from an in house-designed Flight Information Measurement Unit (FIMU) [

To avoid or reduce electromagnetic interference during the measurement process (because the magnetic sensor is extremely sensitive to its ambient disturbances), the FIMU is split into two separate units, namely, an inertial and air data measurement unit, and an electronic compass unit. On the one hand, the inertial and air data measurement unit provides the capability for three-axis accelerations, three-axis angular rates, airspeed, altitude, temperature, and angle of attack measurements. The compass system, on the other hand, measures the magnitude of the external magnetic fields. A Microchip PIC micro-controller is used in each measuring unit to perform data acquisition and signal processing, and to transfer the results to the host system for further analysis and processing. An image of the FIMU is shown in

Determination of flight attitude involves the computation of aircraft pitch angle, roll angle, and yaw angle. Pitch angle and roll angle can be computed through the measured aircraft accelerations and body rates from the accelerometers and rate gyros. The heading angle is determined by computing the magnetic heading: a magnetic sensor measures the magnetic field, and corrects for the magnetic declination. With the measured three axes acceleration signals and pitch, roll, and yaw rate information—the pitch and roll angles can be determined either by computing the gravitational acceleration components on the body axes, or by using the Euler quaternion method [

For example, assume that at a certain instant, the pitch angle of the aircraft is _{x}_{y}_{z}

Knowing _{x}_{y}_{z}

There are alternative ways to obtain _{x}_{y}_{z}_{x}_{y}_{z}_{x}_{y}_{z}_{x}_{y}_{z}_{x}_{y}_{z}

For magnetic heading computation, assume that the components of the Earth’s magnetic field along the X-, Y-, and Z-axis are _{X}_{Y}_{Z}_{X}_{Y}_{Z}_{1} and at right angles to the heading axis _{2} (as shown in

Thus, the magnetic heading of the aircraft _{M}

If the local magnetic declination is _{md}, the aircraft heading can be determined from _{M}_{md}.

A common method used to compute the attitude angle is the Euler quaternion method [

The four parameters are subjected to the following constraint equation:

The dynamics of the four parameters regarding the aircraft body rate (

Therefore, the attitude angle in

As discussed in the previous section, the attitude computed from the gravitational force components provides long-term accuracy with high noise contents. The Euler quaternion method, however, provides less noisy results but suffers from a long-term drift problem. Either method alone may be inadequate for attitude computation. Hence, in this study, the extended Kalman filter [

The Kalman filter is a model-based estimation technique. The dynamics that characterize the relationship between the aircraft body rate (pitch, roll, and yaw rates) and the four parameters of the quaternion method described in _{0} _{1} _{2} _{3}]^{T}^{T}

The measurements of pitch angle _{x}_{y}_{z}

Except the measurement of accelerations (_{x}_{y}_{z}_{T}_{N}_{E}_{D}_{N}_{E}_{D}

The velocities’ rates (_{N}_{E}_{D}_{N}_{E}_{D}_{N}_{E}_{D}_{N}_{N}_{N}_{E}_{E}_{E}_{D}_{D}_{D}

For real time computation, the dynamical system in ^{A}^{Δ}^{t}_{n}_{n}_{n}

With all the necessary information available, the last step of the iteration loop is to perform the measurement update of the state estimate

In the Kalman filter design, we use the covariances of the process noise and measurement noise as design parameters to achieve minimum variance of the estimation error. Hence, we must determine the variances of the process and measurement noises as precisely as possible. For the attitude determination considered in this study, the process noises are mainly derived from the outputs of the gyros (

The second term on the right hand side is considered as the process noise

For a discrete time system, the process noise

It is simple to show that the variance of the process noise is
_{x}_{y}_{z}_{x}_{y}_{z}_{x}_{y}_{z}

The roll angle

Expanding the function

To determine

Similarly, the deviation of

Thus, the deviation of

Defining:

Then:

Thus, the variance of pitch and roll angles,

The variance of the heading angle
_{1} and _{2} are computed based on the filtered roll and pitch angles. Thus, the deviations of _{1} and _{2} can be expressed as:

Hence the deviation of

Therefore, the deviation of the magnetic heading

Thus, the variance of the heading angle
_{X}_{Y}_{Z}_{X}_{Y}_{Z}_{M}

To evaluate the design of the flight information measurement unit and the extended Kalman filtering for attitude determination, the system is tested on an ultralight aircraft. The test data—including the three axes body rates (_{x}_{y}_{z}_{X}_{Y}_{Z}

The results for nearly level flight with wing wiggling (intentionally performed) are shown in

From the above examinations, we demonstrated that the proposed attitude determination algorithm estimates the attitude angles successfully. The algorithm is also reliable for long-term and high-dynamic maneuvers.

To further investigate the performance of the proposed attitude estimation method, the innovation around ±3 standard deviations (extracted from the innovation covariance) and the quaternion norm constraint in

Because there is no primary standard, the proposed attitude estimation algorithm is also tested using a motion platform under a controlled environment. An image of the test arrangement is shown in

The motion platform is controlled to perform uniform circular motion with an angular velocity of 0.873 rad/s. Under this test condition, the velocities (^{2}. Using these velocity and acceleration information together with the accelerations measured from the accelerometers and the body rates from the gyro outputs, the pitch angle and roll angle are computed by using each of the gravity force components, the Euler quaternion method (which can be considered as a direct 6-DOF mechanization for attitude determination [

This study presented attitude determination using a low cost MEMS-based flight information measurement unit. The proposed attitude estimation algorithm is intuitive, easy to implement, and reliable for long-term high dynamic maneuvers. The algorithm utilized the evolution of the four elements of the quaternion method as the dynamical model for the Kalman filter. The roll and pitch angles (computed from the gravity force decomposition) and the heading angle (determined from the electronic compass unit) are selected as the measurements of the Kalman filter. The immeasurable gravity accelerations are deduced from the outputs of the three axes accelerometers, the relative accelerations, and the accelerations due to body rotation. The constraint of the four elements of the quaternion method is treated as perfect measurement and is integrated into the filter design. Approximations of the time varying noise variances of the measured signals are established with detailed derivations. The proposed algorithm is successfully demonstrated through a set of flight test data collected from the in house-designed MEMS based attitude determination system. To improve the performance of the attitude estimation, estimation of the gyro drift and velocity derivatives can be included into the Kalman filter formulation with the expense of computation demands.

This research is supported by the National Science Council, Taiwan, Republic of China, under Grant No. NSC99-2212-E-032-012.

System Structure of the Flight Information Measurement Unit.

The Flight Information Measurement Unit.

Gravitational force decomposition.

Vector change in velocity components due to angular rotation.

Magnetic force decomposition.

Computation process for attitude determination.

Extended Kalman Filter.

Trajectory of the complete flight.

Evolutions of the attitude angles.

Nearly level flight with wing wiggling.

Attitude for climbing-up condition.

Attitude during the descent flight.

Attitude evolutions during the time period 1,106–1,130 s (making a left turn). The arrow in (a) shows the direction of flight.

Attitude evolutions during the time period 2,600–2,675 s (making a right turn). The arrow in (a) shows the direction of flight.

Attitude evolutions during an S-turn maneuver. The arrow in (a) shows the direction of flight.

Innovation around ±3 standard deviations. Green and red lines are the boundaries of the ±3 standard deviations.

Evolution of the norm of the quaternion vector.

The motion platform and the test arrangement.

Results from lab test.