# Automatic Estimation of Dynamic Lever Arms for a Position and Orientation System
## 3 Proposed SINS/GPS Integrated Navigation Model
### Observability Analysis for Lever Arms  
Usually, an observability analysis for an integrated navigation system is conducted to determine whether a specific error state is observable, and under what conditions it can be estimated [26]. According to global observability theory [27], a state is said to be observable if it can be determined uniquely from the measurements for a finite time and under a possible condition. Therefore, we investigate the observability of lever arms directly, depending on the measurement equation in this section. The position measurement in Equation (15) can be rewritten as an error model:  
\[\mathbf{z}_{p}=\delta\mathbf{p}_{I}-\mathbf{M}_{P}\mathbf{C}_{b_{c}}^{n}\Big{(}\mathbf{L}_{OG}^{b_ {c}}-\mathbf{C}_{b}^{b_{c}}\mathbf{L}_{OI}^{b}\Big{)}+\mathbf{w}_{gp}. \tag{18}\]  
Clearly, this is not a deterministic system, but a stochastic case. However, the related theoretical analysis is still useful for revealing the observable conditions and guiding the design of verification trajectories. Ignoring the driving noise and measurement noise temporarily, Equation (18) itself determines the observability of the associated errors \(\delta\mathbf{p}_{I}\), \(\mathbf{L}_{OI}^{b}\), and \(\mathbf{L}_{OG}^{b_{c}}\). It can be seen that the acceleration or deceleration excitations have little effect on lever arms \(\mathbf{L}_{OI}^{b}\) and \(\mathbf{L}_{OG}^{b_{c}}\), and their observabilities are determined by direction cosine matrices \(\mathbf{C}_{b_{c}}^{n}\) and \(\mathbf{C}_{b}^{n}\). If only the ISP rotates, \(\mathbf{C}_{b}^{b_{c}}\) changes while \(\mathbf{C}_{b_{c}}^{n}\) remains constant; thus, the lever arm \(\mathbf{L}_{OI}^{b}\) will be distinguished. In this situation, the SINS position error cannot be accurately estimated for the unobserved lever arm \(\mathbf{L}_{OG}^{b_{c}}\). Considering the construction of attitude matrix \(\mathbf{C}_{b}^{b_{c}}\), the rotation of one ISP gimbal can reveal two components of the lever arm \(\mathbf{L}_{OI}^{b}\) perpendicular to the rotating axis. Consequently, rotations of at least two gimbals are needed to estimate all three components. If only the attitude of the vehicle changes, \(\mathbf{C}_{b_{c}}^{n}\) varies while \(\mathbf{C}_{b}^{b_{c}}\) stays the same, and the combined lever arm \(\mathbf{L}_{IG}^{b_{c}}=\mathbf{L}_{OG}^{b_{c}}-\mathbf{C}_{b}^{b_{c}}\mathbf{L}_{OI}^{b}\) can be estimated. Thus, position measurement accuracy can be improved in spite of the unseparated lever arm \(\mathbf{L}_{OI}^{b}\). It can be seen that the change in \(\mathbf{C}_{b_{c}}^{n}\) is crucial to determining the final positioning accuracies. Similarly, improvement of all three dimensional position accuracies depends on attitude changing in at least two directions of the vehicle. In summary, the estimate of constant lever arms \(\mathbf{L}_{OI}^{b}\) and \(\mathbf{L}_{OG}^{b_{c}}\) requires the change of direction cosine matrices \(\mathbf{C}_{b}^{b_{c}}\) and \(\mathbf{C}_{b_{c}}^{n}\). The greater the change of the matrixes, the stronger the observability of the corresponding lever arms. Certainly, the final estimation accuracies of associated lever arms are limited by the measurement error noise \(\mathbf{w}_{gp}\). In aerial mapping missions, the aircraft usually maneuvers intentionally before entering the sensing area to enhance the observability of the POS [17; 20], especially to improve the heading accuracy. During the mapping process, long straight portions of the flight will cause heading error growth; thus, periodical aircraft maneuvers are required to separate the navigation errors from inertial sensor errors [18]. Common flight maneuvers include circling, coordinated turn, or \"8\" shaped flight, in which at least two directions of attitude change occur, such as in the heading and roll axes. Thus, in those cases without ISP, the lever arm \(\mathbf{L}_{IG}^{b_{c}}\) is usually treated as a constant to be estimated in these planned maneuver processes, and satisfactory results can be obtained [1; 17].