# Automatic Estimation of Dynamic Lever Arms for a Position and Orientation System
## 2 System Description  
As shown in Figure 1, the IMU and surveying sensors are usually fixed on the inner gimbal of the ISP. Thus, the mounting angle and lever arm between the two are constant, reducing the accuracy requirements for the ISP. The GPS antenna is mounted on the roof of the plane for better satellite visibility. The related coordinate systems are defined as follows:  
\(n\)-frame, the navigation frame, chosen as the local level east-north-up (ENU) coordinate;  
\(b\)-frame, the IMU body frame, implicitly predefined by the calibrated sensitive axes of the inertial sensors, with the origin located at the sensitive center of IMU (point \(\mathbf{O}_{I}\)), and axes pointing right, forward, and upward;\(b_{c}\)-frame, the carrier frame, fixed to the flight vehicle and originating at the rotating center of the ISP (point \(\mathbf{O}_{O}\)). The axis directions of \(b_{c}\)-frame do not rotate with the gimbals, and coincide with the ISP only if the encoder data of three gimbals are all zeros;  
\(b_{s}\)-frame, the surveying sensor frame, rigidly fixed to the surveying sensor with the origin located at the surveying center (point \(\mathbf{O}_{S}\)). Based on the established frames, the related lever arms are defined as follows:  
\(\mathbf{L}_{OI}^{b}\), the lever arm from the rotating center of the ISP (point \(\mathbf{O}_{O}\)) to the sensitive center of IMU (point \(\mathbf{O}_{I}\)) projected in the \(b\)-frame;  
\(\mathbf{L}_{OG}^{b_{c}}\), the lever arm from the rotating center of the ISP (point \(\mathbf{O}_{O}\)) to the phase center of the GPS antenna (point \(\mathbf{O}_{C}\)) projected in the \(b_{c}\)-frame. Whether the ISP rotates or not, there is no relative motion between the point \(\mathbf{O}_{O}\) and point \(\mathbf{O}_{G}\) in the \(b_{c}\)-frame; thus, the lever arm \(\mathbf{L}_{OG}^{b_{c}}\) is a constant;  
\(\mathbf{L}_{IG}^{b_{c}}\), the lever arm from the sensitive center of the IMU (point \(\mathbf{O}_{I}\)) to the phase center of the GPS antenna (point \(\mathbf{O}_{G}\)) projected in the \(b_{c}\)-frame;  
\(\mathbf{L}_{IS}^{b}\), the lever arm from the sensitive center of the IMU (point \(\mathbf{O}_{I}\)) to the measuring center of the surveying center (point \(\mathbf{O}_{S}\)), projected in the \(b\)-frame. Three lever arms--\(\mathbf{L}_{OI}^{b}\), \(\mathbf{L}_{OG}^{b_{c}}\), and \(\mathbf{L}_{IS}^{b}\)--are constant, while lever arm \(\mathbf{L}_{IG}^{b_{c}}\) changes with the ISP rotation. Because the lever arm \(\mathbf{L}_{IG}^{b_{c}}\) is uncertain and time-varying, it is referred to as a dynamic lever arm. Since the attitude matrix \(\mathbf{C}_{b}^{n}\) can be obtained from the SINS solution, the transformation matrix \(\mathbf{C}_{b_{c}}^{n}\) from \(b_{c}\)-frame to the \(n\)-frame can be obtained as follows:  
\[\mathbf{C}_{b_{c}}^{n}=\mathbf{C}_{b}^{n}\mathbf{C}_{b_{c}}^{b}, \tag{1}\]  
in which the direction cosine matrix \(\mathbf{C}_{b_{c}}^{b}\) is formed with the encoder data of the ISP [7]:  
\[\mathbf{C}_{b_{c}}^{b}=\mathbf{C}_{\#T}\mathbf{C}_{\theta_{T}}\mathbf{C}_{\gamma T}, \tag{2}\]  
where:  
\[\mathbf{C}_{\phi_{T}}=\left[ \begin{array}{ccc}\cos\psi_{T}&\sin\psi_{T}&0\\ -\sin\psi_{T}&\cos\psi_{T}&0\\ 0&0&1\end{array} \right],\ \mathbf{C}_{\theta_{T}}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\theta_{T}&\sin\theta_{T}\\ 0&-\sin\theta_{T}&\cos\theta_{T}\end{array} \right],\ \mathbf{C}_{\gamma T}=\left[ \begin{array}{ccc}\cos\gamma_{T}&0&-\sin\gamma_{T}\\ 0&1&0\\ \sin\gamma_{T}&0&\cos\gamma_{T}\end{array} \right],\]  
Figure 1: The installation relationship of the position and orientation system (POS) and surveying system. the angles \(\psi_{T}\), \(\theta_{T}\), and \(\gamma_{T}\) are encoder data obtained from the inner, middle, and outer gimbals of the ISP, respectively. Referring to Equation (2), the angular rate of the \(b\)-frame with respect to the \(b_{c}\)-frame, denoted as \(\mathbf{\omega}_{b_{c}b}^{b}\), can be calculated as follows [7]:  
\[\mathbf{\omega}_{b_{c}b}^{b}=\left[ \begin{array}{c}0\\ 0\\ \dot{\psi}_{T}\end{array}