In this paper, the Global Positioning System (GPS) interferometer provides the preliminarily computed quaternions, which are then employed as the measurement of the extended Kalman filter (EKF) for the attitude determination system. The estimated quaternion elements from the EKF output with noticeably improved precision can be converted to the Euler angles for navigation applications. The aim of the study is twofold. Firstly, the GPS-based computed quaternion vector is utilized to avoid the singularity problem. Secondly, the quaternion estimator based on the EKF is adopted to improve the estimation accuracy. Determination of the unknown baseline vector between the antennas sits at the heart of GPS-based attitude determination. Although utilization of the carrier phase observables enables the relative positioning to achieve centimeter level accuracy, however, the quaternion elements derived from the GPS interferometer are inherently noisy. This is due to the fact that the baseline vectors estimated by the least-squares method are based on the raw double-differenced measurements. Construction of the transformation matrix is accessible according to the estimate of baseline vectors and thereafter the computed quaternion elements can be derived. Using the Euler angle method, the process becomes meaningless when the angles are at

The Global Positioning System (GPS) [

Traditionally used in precise static relative positioning, and thereafter in kinematic positioning, the GPS offers the interferometer [

The rotation angles that relate a coordinate system fixed in the body (body frame) to a coordinate system fixed in space are referred to as the attitudes. The space coordinate system is typically defined to be a local level NED (north-east-down) frame, also referred to as the navigation frame. The purpose of attitude determination essentially involves calculation of the three Euler angles, namely roll, pitch, and yaw. The quaternion method [

The purpose of the Kalman filter (KF) [

The remainder of this paper is organized as follows. In Section 2, preliminary background on the GPS carrier phase observation model is reviewed; the computed quaternion vectors based on the GPS interferometer is introduced. Section 3 presents the modelling of quaternion dynamics for the extended Kalman filter. In Section 4, simulation experiments are carried out to evaluate the performance on estimation accuracy using the proposed method as compared to the conventional one. Two numerical examples are presented for illustration. Conclusions are given in Section 5.

In a GPS interferometer, the receiver-satellite double-differenced carrier phase observable has been commonly utilized to solve for the antenna baseline vector. The GPS carrier phase observables representing sum of range, an unknown integer ambiguity and some ranging errors -can be represented by:

where the parameters involved in

The receiver-satellite double differencing operator is defined as

where the effects of errors associated with the satellites are greatly reduced. Similarly, for satellite

where

which eliminates or greatly reduces the satellite and receiver timing errors.

The signals received from

which can be expressed in matrix form

Referring to

where

In certain applications, the body angular velocities:

Once the Euler angle rates are available with sufficiently good accuracy, the body angular velocity can be obtained. Taking the inverse transformation for

The direction cosine matrix (DCM)

where the subscripts

In

Using the Euler angle method, the singularity problem occurs when the angles are at

A quaternion is a four-dimensional extension to complex numbers, containing four real parameters. The first is considered a scalar and the other three vector components in three-dimensional space:

where a constant equation exists of the form

where

Using the transformation matrix to obtain quaternion parameters can be done through

By comparing

See [

The interferometer offers the preliminarily computed quaternion vector using the GPS double-differenced carrier phase observables. The implementation procedure is highlighted as the following steps: (1) Determining the baseline vector using the receiver-satellite double-differenced carrier phase observables:

Consider a dynamical system whose state is described by a nonlinear, vector differential equation. The process model and measurement model are represented as

It is assumed that

where

where the state vector

where

The nonlinear process model can be linearized along the currently estimated trajectory where the influence of the perturbations on the trajectory can be represented by a Taylor series expansion about the nominal trajectory.

The actual trajectory is the sum of the estimated trajectory and the small increment:

When working with incremental state variables, the linearized measurement equation presented to the EKF is

in which the measurement residual is given by:

Adding

which shows that in an EKF it is accessible to keep track of the total estimates rather than the incremental ones. Once

- Initialization: Initialize state vector |

- Time update |

(1) Evaluate the predicted state vector through the process model |

(2) Propagate the predicted error covariance matrix |

- Measurement update |

(3) Evaluate the Kalman gain |

(4) Perform update for state vector |

(5) Perform update for error covariance matrix |

where |

It can be shown that the quaternion elements are propagated according to the differential equation:

The differential equations for describing the propagation of quaternion elements are given by

where the product terms in the parentheses are introduced by quaternion product between the angular rate and the quaternion, which can be written in matrix form

Therefore, the quaternion vector is propagated according to the differential equation

where

The differential equations for describing the quaternion vector can be represented by

where

and

Augmented by the propagation of the body angular rates, described by the random walk models, i.e.,

The corresponding Jacobian matrix can be shown to be

The corresponding discrete state transition matrix is given by

Improved accuracy is accessible for the attitude solutions based on the EKF using the preliminarily computed quaternion vector from the GPS interferometer as the measurement. The measurement equations in this case are linear and are much simpler, and they are the four quaternion components

i.e.,

and the covariance matrix for the measurement noise is given by

respectively. The measurement model

or the first-order Gauss-Markov process

Simulation study has been carried out to evaluate the performance of the proposed approach in comparison with the conventional method for GPS-based attitude determination. Two illustrative examples were implemented through numerical experiments. Computer codes were developed using the Matlab® software. The commercial software Satellite Navigation (SatNav) Toolbox by GPSoft LLC (2003) [

The error sources corrupting the GPS carrier phase observables include ionospheric error, tropospheric delay, receiver thermal noise and multipath errors. The variances of the receiver noise are assumed to be 1 m^{2}. Most of the receiver-independent common errors can be corrected through the differential correction while the multipath and receiver thermal noise cannot be eliminated. The antenna geometry is set up as in

with initial Euler attitude angles

where

The proposed method in which Kalman filtering is used provides estimate of the quaternion vector with noticeable accuracy improvement. It should be noticed that it is risky to set the process noise variance zero to avoid filter divergence. As a result, when the system reaches steady state, the steady-state gain will not approach zero and, subsequently, the filter is able to catch the time-varying attitude dynamics. One still needs to find the suitable values to meet the specific design/mission requirement. The estimation accuracy of the quaternion vectors are essential and directly influence the resulting accuracy of Euler attitude angles. To confirm the correctness of the solutions, the estimated quaternion vectors were examined, as shown in

Axis | Roll | Pitch | Yaw | ||
---|---|---|---|---|---|

Methods | Conventional | Mean (deg) | −0.05610 | 0.00819 | −0.024156 |

Variance (deg^{2}) |
1.92910 | 2.23251 | 0.38579 | ||

Proposed | Mean (deg) | −0.05924 | 0.00758 | −0.01660 | |

Variance (deg^{2}) |
0.13366 | 0.15787 | 0.027261 |

Segment number | Description of the motion |
---|---|

1 | Level acceleration for 10 seconds |

2 | Pitch up transition |

3 | 15 second climb |

4 | Level off |

5 | Roll into a turn |

6 | 90 degree turn |

7 | Roll back to straight and level |

8 | 10 second straight segment to the West |

Axis | Roll | Pitch | Yaw | ||
---|---|---|---|---|---|

Methods | Conventional | Mean (deg) | 0.0234 | −0.06765 | −0.03536 |

Variance (deg^{2}) |
1.9301 | 1.57445 | 0.36217 | ||

Proposed | Mean (deg) | 0.0225 | −0.06585 | −0.03999 | |

Variance (deg^{2}) |
0.13530 | 0.134683 | 0.091516 |

The standard EKF is sensitive to the changes of the process and measurement models, thus yielding poor performance. Furthermore, the EKF framework does not possess capability to deal with the non-Gaussian measurement errors/outliers. The EKF associates the contaminated measurements with an increase in the measurement covariance, causing the reformulated error covariance. This modified covariance inflation is known to cause an increase of error in the state estimation. Performance based on the EKF will degrade when the noise strength is varying and/or the actual distribution deviates from the assumed Gaussian model. To further improve the performance of the EKF, an adaptive mechanism or a robust technique can be incorporated for performance improvement. Due to appropriate tuning, the adaptive EKF (AEKF) exhibits robust behavior and therefore outperforms the standard EKF when the time-varying dynamic process and measurement models are involved.

A novel GPS-based attitude determination method has been presented. The quaternion vector derived from GPS interferometer has been used as the measurement of the EKF in the attitude determination system. Although utilization of the carrier phase observables enables the relative positioning to achieve centimeter level accuracy, nevertheless, the quaternion elements derived from the GPS interferometer are inherently noisy. This is essentially due to the fact that the baseline vectors estimated by the least-squares method are based on the noisy double-differenced measurements.

The preliminarily computed quaternion elements from the GPS interferometer is employed as the measurement of the EKF based quaternion estimator. The model based approach using the EKF is adopted for estimating the quaternion elements, which can then be converted to the Euler angles. Results show that, by incorporating the EKF into the GPS interferometer, the errors in the solutions of the baseline vectors, and thereafter the quaternion elements have been remarkably mitigated and the estimation accuracy of the attitude solutions has been noticeably improved. The proposed method provides several advantages, such as accuracy improvement, reliability enhancement, and real-time characteristics.

Since the EKF is sensitive to the changes of the process and measurement models, when implementing the EKF approach, poor knowledge of the noise statistics may seriously degrade the estimation performance, and even provoke the filter divergence. For achieving improved estimation accuracy, the designers are required to possess the complete