## 2 System Identification Technique by Parameter Estimator  
### Modeling of a 2-Axes Gimbal Structure Loading Device  
The inner frame of the 2-axes gimbal structure has much smaller properties like mass and moment of inertia than the outer frame. Also, the tilt of the loading box is mainly caused by the stair climbing of the delivery robot, and the tilt of the inner frame is insignificant. The 2-axes gimbal structure loading device is interpreted as two independent pendulum systems. This simplification is the main assumption of this paper. Each pendulum system maintains the balance by a motor located at the center of its rotation axis. The center of gravity of the loading device can change with loaded luggage. Figure 2 is the free body diagram, and Equation (1) is the dynamic equation of a pendulum structure loading device.  
\[J\;\ddot{\varnothing}=-M\;g\;\sqrt{C_{P}{}^{2}+C_{H}{}^{2}}\;\sin\!\left( \varnothing+\tan^{-1}\!\left(\frac{C_{H}}{C_{P}}\right)\right)+D_{m}\;\dot{ \alpha}+T_{m} \tag{1}\]  
* \(\varnothing\): the tilt of the loading device (degree)
* \(J\): the moment of inertia of the loading device (kg m\({}^{2}\))
* \(M\): the mass of the loading device (kg)
* \(C_{P}\): the vertical position of the center of gravity from the rotate axis (m)
* \(C_{H}\): the horizontal position of the center of gravity from the rotate axis (m)
* \(g\): gravitational acceleration (m/s\({}^{2}\))
* \(\dot{\alpha}\): angular velocity of motor shaft (m/s)
* \(D_{m}\): damping coefficient of the motor (N s/m)
* \(T_{m}\): motor torque (N m)  
Figure 2: Free body diagram of a pendulum structure loading device.  
In Equation (1), the \(\varnothing,\ \dot{a}\), and \(T_{m}\) are known values that can be measured by an IMU sensor and an encoder of the motor, and the \(D_{m}\) is determined as a fixed value by experimental data for simplifying. The \(J,\ M,\ C_{P}\), and \(C_{H}\) are unknown values that change depending on the loaded luggage. For system identification by a parameter estimator, the dynamic equation is expressed by dividing it into the known parameters and unknown parameters in Equations (2)-(4).  
\[\ddot{\varnothing}=-\frac{M\ g\ \sqrt{C_{P}{}^{2}+C_{H}{}^{2}}}{J}\left\{\frac{C_{P} \ \sin(\varnothing)}{\sqrt{C_{P}{}^{2}+C_{H}{}^{2}}}+\frac{C_{H}\ \cos(\varnothing)}{\sqrt{C_{P}{}^{2}+C_{H}{}^{2}}}\right\}+\frac{1}{J}\left\{D _{m}\ \dot{\alpha}+T_{m}\right\} \tag{2}\]  
\[\ddot{\varnothing}=-\frac{M\ g\ C_{P}}{J}\ \sin(\varnothing)-\frac{M\ g\ C_{H}}{J}\ \cos( \varnothing)+\frac{1}{J}\left\{D_{m}\dot{\alpha}+T_{m}\right\} \tag{3}\]  
\[\ddot{\varnothing}=\left[\sin(\varnothing)\ \ \ \cos(\varnothing)\ \ \ D_{m}\ \dot{\alpha}+T_{m}\ \right]\left[-\frac{M\ g\ C_{P}}{J}\atop-\frac{M\ g\ C_{H}}{J}\atop\frac{1}{J} \right]=N^{T}\ X=N^{T}\ \left(X_{\text{empty}}+\Delta X\right)=N^{T}\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\end{bmatrix} \tag{4}\]  
In Equation (4), the \(N\) vector is the known parameters vector with measurable values, and the \(X\) is the unknown parameters vector. The \(X\) cannot be determined because the loaded luggage is unspecific, and it makes it difficult to design an appropriate controller. The \(X\) is divided into the \(X_{\text{empty}}\) of the empty loading device and the \(\Delta X\) which represents the difference between the empty loading device by luggage. The system identification process aims to estimate the \(\Delta X\) to calculate the \(X\) in real-time.