# Automatic Estimation of Dynamic Lever Arms for a Position and Orientation System
## 3 Proposed SINS/GPS Integrated Navigation Model
### Observability Analysis for Lever Arms
As long as any two gimbals of ISP are controlled to rotate during these auxiliary maneuvering processes, the two constant -level arms \(L^{b}_{OI}\) and \(L^{b_{c}}_{OG}\) are solvable. Therefore, the observable conditions for the dynamic lever arm estimation are easy to satisfy in practical applications. It should be noted that the auxiliary matrix \(\mathcal{C}^{b_{c}}_{b}\) is constructed from the encoder data of the ISP, as shown in Equation (2). Since there are some unavoidable errors in the encoder angle measurements, their influences should be evaluated. Denoting the angle measurement errors as \(\mathbf{\mu}^{b_{c}}\), the introduced position errors are deduced as follows:  
\[\delta\mathbf{p}_{\mu}=-\mathbf{M}_{P}\mathcal{C}^{b}_{b}\mathcal{C}^{b}_{b_{c}} \mathcal{C}^{b}_{b_{c}}\mathbf{L}^{b}_{OG}=\mathbf{M}_{P}\mathcal{C}^{n}_{b_{c}}\Big{(} \mathbf{L}^{b_{c}}_{OG}\times\mathbf{\mu}^{b_{c}}\Big{)}. \tag{19}\]  
The term \((\mathbf{L}^{b_{c}}_{OG}\times\mathbf{\mu}^{b_{c}})\) determines the magnitude of the position errors that are introduced by encoder errors. Assuming that the lever arm \(\mathbf{L}^{b_{c}}_{OG}\) is 2 m and the encoder angle errors of the ISP are 3 arcminutes, the related position errors will be at the 2 mm level. The commonly used ISP PAV80 [22] and GSM3000 [23] can provide such angular accuracy. In this situation, the influence of encoder angle errors is limited and much smaller than the positioning accuracy of typical CDGPS. However, if the lever arm \(\mathbf{L}^{b_{c}}_{OG}\) is too large, the interrelated influence will not be negligible. The residual encoder error of the ISP can be modelled as a zero mean white noises process, which has the same effect on automatic estimation method and manual measurement method of lever arm. Thus, the GPS antenna and the ISP should be installed as close as possible, otherwise higher performance ISP is needed. The time delay of the GPS information relative to SINS is another major error source. Since the GPS receiver and the SINS unit are two separate subsystems, the frequency difference between the 1 PPS and SINS sampling clock could cause the time delay to be a variable (usually a ramp or triangular wave). Thus, the method of taking the time delay as a constant error state to be estimated will not work. The effective way is to discipline the SINS sampling clock with 1 PPS [28], or to record the delayed time with a counter for software compensation [11]. Even so, it is difficult for the time delay to be perfectly compensated. In addition, although the same sampling clock is used in the IMU, the time-asynchrony between each inertial sensor is inevitable, due to the differences of physical characteristics [29], which makes the time synchronization more complicated. Denoting the time-asynchrony error as \(\delta t\), its influence on position measurement is:  
\[\delta\mathbf{p}_{dt}=M_{P}\mathbf{v}^{n}\delta t. \tag{20}\]  
Assuming that \(\delta t\) is 0.1 ms and the normal flight velocity is 50 m/s, the introduced position error is about 5 mm. Clearly, larger time-asynchrony error is intolerable at this speed. After time compensation as mentioned earlier, the residual time-asynchrony error can be modelled as a zero mean white noises process for analysis.