Performance Evaluation by Measurement Combination of Loosely Coupled GPS/INS Integration

This paper presents an extended Kalman filter derivation for loosely coupled GPS (Global Positioning System)/INS (Inertial Navigation System) integration based on quaternion attitude representation using the Earth-Centered Earth (ECEF) Frame. In this loosely coupling integration, both the position and velocity estimates from GPS receiver are used as the measurements to extended Kalman filter, and then they are integrated with inertial measurements from inertial measurement units (IMU) to estimate the attitude, position and velocity of an air vehicle. The velocity estimates which have centimeter level estimation error from the GPS receiver are used to improve the filter performance. For attitude estimation, the global attitude parameterization is given by a quaternion and a multiplicative quaternion-error approach is used to guarantee a normalization constraint of quaternion in the filter. Simulation results are shown to obtain the estimation of the attitude, position, velocity, biases and scale factors and to evaluate the performance of the EKF with the measurement combination composed of the two different types of GPS navigation solutions (the position only, and both the position and the velocity in the ECEF frame).

In this paper, an EKF filter using the Earth-Centered Earth-Fixed (ECEF) frame formulation is derived for loosely coupled GPS/INS integration to determine both the position and the attitude of an unmanned aerial vehicle (UAV). The GPS navigation solutions (position and velocity in the ECEF frame) are directly integrated with inertial measurements from gyros and accelerometers without coordinate transformation for loosely coupled GPS\INSS integration in the ECEF frame. This paper was motivated by the advantage of the ECEF frame for loosely coupled GPS/INS integration so that GPS navigation solutions are directly used as the EKF filter measurements in the ECEF-frame navigation equations without coordinate transformation.
Furthermore, the global attitude parameterization expressed by a quaternion is used while a generalized three-dimensional attitude representation is used to define the local attitude error. A loosely coupled GPS/INS integration algorithm is easy to implement and needs to be derived with the following characteristics [7]. 1) Firstly, loosely coupled GPS/INS integration algorithm should be described using the ECEF frame so that the EKF can estimate the position, velocity and attitude of an UAV with respect to the ECEF frame using the GPS navigation solutions. 2) Secondly, centimeter per second-level of accuracy GPS velocity solutions are assumed to be available to the EKF as measurements as well as standalone GPS position solutions since recently a commercial GPS receiver can provide very precise velocity estimates [14]. 3) Thirdly, the EKF uses a multiplicative error quaternion after neglecting higher-order terms for attitude estimation since the attitude kinematics here is based on the quaternion, which must obey a normalization constraint that can be violated by the linear measurement updates associated with the standard EKF [4,13]. Then the four-component quaternion can effectively be replaced by a three-component error vector [15]. As a simulation result, the three different Thus, the main contribution of this paper is the derivation of the EKF for the loosely coupled GPS/INS integration algorithm so that the benefits of the ECEF frame, highly precise GPS velocity estimates and the quaternion parameterization can be used simultaneously with the measurement combination. Since the GPS navigation solutions are given in the ECEF frame, they are directly used to update the INS-derived trajectory with no coordinate transformation in the filter. In addition, the derived EKF based on the ECEF frame navigation equations has the benefit of faster computation compared to the filter based on the local navigation frame equations [10]. The derived EKF can use two different kinds of GPS navigation solutions composed of the position, and both the position and the velocity as the measurements obtained from the GPS receiver to the EKF. The loosely coupled GPS/INS integration algorithm can attain improvement in the filter performance when centimeter per second-level of accuracy GPS velocity solutions are available under the normal visibility of the GPS.

Attitude Kinematics
In this section the basic properties of attitude kinematics is summarized. One of the most useful attitude parameterizations is given by the quaternion, which is a four-dimensional vector. Like the Euler axis/angle parameterization, the quaternion is also four-dimensional vector, defined as where ê is the axis of rotation and  is the angle of rotation [15]. The quaternion components are dependent each other since a four-dimensional vector is used to describe three dimensions. The quaternion satisfies a single constraint given by 1.
To describe the attitude in different reference frames, superscripts and subscripts i, e, n and b which denotes the where 33 I  is a 33  identity matrix and []  is a notation for a cross product matrix [11].
For small angles the vector part of the quaternion is equal to half angles so that /2  ρ α and 4 , q where α is a vector of the roll, pitch and yaw angles. The attitude matrix can then be approximated by which is valid to within first-order in the angles.
The attitude kinematics equation is given by where e eb ω is the angular velocity of the B frame relative to the E frame expressed in B coordinates. A major benefit of using quaternions is that the kinematics equation is linear in the quaternion and is also free of singularities. Another advantage of quaternion is that successive rotations can be achieved using quaternion multiplication. Here the convention of Lefferts et al. [16] who multiply the quaternions in the same order as the attitude matrix multiplication (in contrast to the general convention established by Hamilton [16]) is adopted. This is written by The composition of the quaternion is bilinear, with Also, the inverse quaternion is defined by Note that which is the identity quaternion.

ECEF Frame Navigation Equations
In this section the E frame navigation equations with the quaternion parameterization are derived.
Beginning from Newton's 2 nd law of motion in the gravitational field of the Earth, which relates specific force and gravitation, the total acceleration is given by where eb r is the position vector of the body relative to the E frame, ib f is the specific force vector sensed with respect to the I frame, and ib γ is the gravitation which does not incorporate with any centripetal components.  (13) where e eb v is the velocity of the body relative to the E frame described in the E frame.
Equation (12)   (20) Figure 1 shows the algorithmic flowchart of the mechanization equation in the E frame. In this algorithm, the numerical integration of the IMU raw data is computed in two steps. In the first step, the quaternion kinematics including the angular velocity b ib ω is integrated numerically to obtain the transformation matrix from the E to the B frame. In the second step, the specific force b ib f is transformed into the E frame, and then the velocity and position of the body are obtained by integrating the acceleration in the E frame consecutively.
To represent the attitude in the N frame the rotation matrix b n C from the N frame to the B frame is obtained through the successive rotation matrices: with sin cos sin sin cos sin cos 0 cos cos cos sin sin where n e C is the rotation matrix that transforms the E frame to the N frame,  and  are geodetic latitude and longitude which can be computed using the position vector e eb r with WGS-84 parameters [14,15]. Here the gyro measurement model is given by [4] ib ω is the true angular velocity of the B frame relative to I frame expressed in the B coordinates, b ib ω is the measured angular velocity of the gyro,   respectively. The accelerometer measurement model is given by [4] where b ib f is the true specific force of the B frame relative to I frame expressed in the B coordinates, b ib f is the measured specific force of the accelerometer,   C q of Eq. (28) is coupled into the position vector as shown in Eq. (28), which permits to estimate the attitude from the position measurements.

Extended Kalman Filter equations for loosely coupled GPS/INS integration
We now derive the attitude error equations, which are used in the EKF covariance propagation. The linearized model error kinematics follows directly from that the quaternion is linearized using a multiplicative approach [3,13]. The error quaternion and its derivatives are given by where the quaternion multiplication is defined by Eq. (9). The equivalent attitude error matrix is given by If the error quaternion is small enough to within first order, the vector part of the quaternion error can be approximated by /2   ρ α and 4 1, q   where α is a small error-angle rotation vector. Also, the quaternion inverse is defined by Eq. (10). The linearized model error-kinematics then takes the form: ω is constant. Note that the fourth error-quaternion component is constant. The first-order approximation, which assumes that the true quaternion is close to the estimated quaternion, yields 4 1. q   The order of the system in the EKF is then reduced to one state using this property. From the angular velocity model of Eq. (23a), the error of b ib ω to within first-order approximation in the EKF can be written using the approximation This yields The state, state-error vector, process noise vector and covariance used in the EKF are defined as where 33 0  is a 3×3 null matrix, the state vector x is a 22-dimensional vector, and the state-error vector x is a 21-dimensional vector since a 4-dimensional quaternion is used to describe three dimensions. The error-dynamics used in the EKF are given by 11  3 3  3 3  14  3 3  16  3 3   3 3  3 3  3 3  3 3  3 3  3 3  3 3   31  32  33  34  35  36  37   3 3  3 3  3 3  3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3  3 3  3 3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3   3 3  3 3  3 3  3 3   3 3  3 3  3        The state error covariance follows the standard EKF update. The position, velocity, and bias states also follow the standard EKF additive correction while the attitude error state update is computed using a multiplicative update [3,15]. The updated quaternion is renormalized by brute force. Finally, the propagation equations follow the standard EKF model. To reduce the computational load a discrete-time propagation of the covariance matrix is used where k  is the discrete-time state transition matrix and k Q is the discrete-time process noise covariance matrix. Though k Q can be obtained using numerical solution given by van Loan [18], the first-order approximation of the solution is given by where t  is the constant sampling time.

Simulation results
In this section simulation results are shown that estimate a flying vehicle's attitude, position, and velocity, as well as the gyro and accelerometer biases and scale factors. All measurements are assumed to be sampled The resulting EKF using both the EKF position and velocity measurements are shown in Figure 2.   Table 2 shows the averaged Ja and Jr values for different measurements using both the position and velocity, and the velocity only, respectively. It shows that smaller averaged Ja and Jr are obtained when both the GPS position and velocity are used together. Table 2. Averaged Ja and Jr values for different measurements Figure 3 shows the norms of the position and attitude estimation errors using two different measurement combinations of the GPS navigation solutions, respectively. The norms of the attitude estimation errors are very similar and converge to less than 0.1 degrees in 3600 s, as shown in Fig. 2 (a). On the other hand, the norms of the position estimation errors converge to smaller than 1.  Table 3 shows  Table 3 is still smaller than, Jr in Table 2 for the position measurement only. More accurate EKF estimation results can be obtained when both the position and velocity measurements are used together with more accurate velocity measurement. Therefore, the velocity information from the GPS receiver may improve the EKF estimation result when it is used with the ECEF position. Furthermore, as the velocity accuracy is higher, more improved EKF result can be attained. Table 3. Averaged Ja and Jr values using both the position and velocity     51  3 3  52  3 3  53  3 3  54  3 3  55  3 3   56  3 3  57  3 61  3 3  62  3 3  63  3 3  64  3 3  65  3 3   66  3 3  67  3 71  3 3  72  3 3  73  3 3  74  3 3  75  3 3   76  3 3  77  3