Estimation of UAV Position with Use of Smoothing Algorithms

Abstract

The paper presents methods of on-line and off-line estimation of UAV position on the basis of measurements from its integrated navigation system. The navigation system installed on board UAV contains an INS and a GNSS receiver. The UAV position, as well as its velocity and orientation are estimated with the use of smoothing algorithms. For off-line estimation, a fixed-interval smoothing algorithm has been applied. On-line estimation has been accomplished with the use of a fixed-lag smoothing algorithm. The paper includes chosen results of simulations demonstrating improvements of accuracy of UAV position estimation with the use of smoothing algorithms in comparison with the use of a Kalman filter.

Motion Compensation for Radar Terrain Imaging Based on INS/GPS System

“…The dynamics model describes how the system state vector $\mathrm{bold-italicx}\left(k\right)$ changes in time [13]:$boldnormalx\left(k\right)=boldnormal\mathsf{\Phi }\left(k,k-1\right)boldnormalx\left(k-1\right)+boldnormalw\left(k-1\right)$ where $k$ is an index of discrete time, $boldnormal\mathsf{\Phi }\left(k,k-1\right)$ is a state transition matrix from $k-1$ to $k$ moment, and $boldnormalw\left(k\right)$ is the process noise vector. The observation model shows a relationship between the state vector and the measurement vector $boldnormalz\left(k\right)$ [15]:$boldnormalz\left(k\right)=boldnormalH\left(k\right)boldnormalx\left(k\right)+boldnormalv\left(k\right)$ where $boldnormalv\left(k\right)$ is the measurement noise vector and $boldnormalH\left(k\right)$ is the observation matrix. In the proposed system, the state vector consists of INS errors: $\mathrm{bold-italicx}={\left[\begin{array}{ccccccccc}\mathrm{sans-serif\delta }n& \mathrm{sans-serif\delta }{v}_{is1,N}^n& {\mathsf{\varphi }}_E& \mathrm{sans-serif\delta }e& \mathrm{sans-serif\delta }{v}_{is1,E}^n& {\mathsf{\varphi }}_N& \mathrm{sans-serif\delta }d& \mathrm{sans-serif\delta }{v}_{is1,D}^n& {\mathsf{\varphi }}_D\end{array}\right]}^T$ where $...$…”

“…The dynamics model was determined using equations described in [15,17]. Finally, the transition matrix $boldnormal\mathsf{\Phi }\left(k,k-1\right)$ has the following form:$\mathrm{bold-sans-serif\Phi }=[\begin{array}{ccccccccc}1& T_p& -\frac{f_{is1,D}^n{T}_p^2}{2}& 0& 0& 0& 0& 0& {\mathrm{sans-serif\Phi }}_{19}\\ 0& {\mathrm{sans-serif\Phi }}_{22}& -f_{is1,D}^n{T}_p& 0& 0& 0& 0& 0& {\mathrm{sans-serif\Phi }}_{29}\\ 0& -\frac{T_p}{R}& {\mathrm{sans-serif\Phi }}_{33}& 0& 0& 0& 0& 0& {\mathrm{sans-serif\Phi }}_{39}\\ 0& 0& 0& 1& T_p& \frac{f_{is1,D}^n{T}_p^2}{2}& 0& 0& {\mathrm{sans-serif\Phi }}_{49}\\ 0& 0& 0& 0& {\mathrm{sans-serif\Phi }}_{55}& f_{is1,D}^n{T}_p& 0& 0& {\mathrm{sans-serif\Phi }}_{59}\\ 0& 0& 0& 0& \frac{T_p}{R}& {\mathrm{sans-serif\Phi }}_{66}& 0& 0& {\mathrm{sans-serif\Phi }}_{69}\\ 0& 0& 0& 0& 0& 0& 1+\frac{g_{b,D}^n{T}_p^2}{R}& 1& 0\\ 0& 0& 0& 0& 0& 0& \frac{2g_{b,D}^n{T}_p}{R}& 1+\frac{g_{b,D}^n{T}_p^2}{R}& 0\\ 0& 0& 0& 0\end{array}$…”

“…The process noise covariance matrix $boldnormalQ\left(k-1\right)$, which must be known for implementing a Kalman filter, was determined using modified equations from [15,17]:$\mathrm{boldQ}=[\begin{array}{ccccccccc}{Q}_{11}& Q_{12}& Q_{13}& Q_{14}& Q_{15}& Q_{16}& 0& 0& Q_{19}\\ Q_{12}& Q_{22}& Q_{23}& Q_{24}& Q_{25}& Q_{26}& 0& 0& Q_{29}\\ Q_{13}& Q_{23}& Q_{33}& Q_{34}& Q_{35}& Q_{36}& 0& 0& Q_{39}\\ Q_{14}& Q_{24}& Q_{34}& Q_{44}& Q_{45}& Q_{46}& 0& 0& Q_{49}\\ Q_{15}& Q_{25}& Q_{35}& Q_{45}& Q_{55}& Q_{56}& 0& 0& Q_{59}\\ Q_{16}& Q_{26}& Q_{36}& Q_{46}& Q_{56}& Q_{66}& 0& 0& Q_{69}\\ 0& 0& 0& 0& 0& 0& \frac{S_{vD}{T}_p^3}{3}& \frac{S_{vD}{T}_p^2}{2}& 0\\ 0& 0& 0& 0& 0& 0& \frac{S_{vD}{T}_p^2}{2}& \end{array}$…”

“…In the proposed system, the INS and the GPS data integration was implemented using a Kalman filter [13,14,15,16]. The algorithm proposed by the authors was modified in comparison to the solutions used thus far in MOCO procedures [1,2,3].…”

Fusion of identification information from ELINT-ESM sensors

“…Methods of estimating the position of electronic entities have been presented, among others, in papers [15,16,17]. Assuming that sensors send all reports on the tracked electronic entities to the superior operation center in the electronic recognition system, such a center can perform the fusion function of the identification information.…”