## 2 System Identification Technique by Parameter Estimator
### Simulation of the System Identification Techniques  
The proposed system identification techniques are verified by simulation using MATLAB and Simulink (MathWorks, Natick, MA, USA). To verify the system identification, a dynamic equation assumed as a real loading device is defined in advance. The \(X\) of the predefined dynamic equation is expressed as \(X_{\mathrm{loaded}}\). The \(X\) estimated by the least-squaresmethod-based system identification is expressed as \(X_{\text{estimated}}\). \(X_{\text{estimated}}\)\({}_{\text{NSS}}\) is the estimated \(X\) by the improved system identification using a null-space solution.  
Figure 3 and Table 1 show the simulation results. Figure 3a shows that the pendulum motions of the dynamic equation of the \(X_{\text{loaded}}\), \(X_{\text{estimated}}\), and \(X_{\text{estimated}}\)\({}_{\text{NSS}}\) are equal. In Figure 3c, the \(X_{\text{estimated}}\) and the \(X_{\text{estimated}}\)\({}_{\text{NSS}}\) are checked how close they are to the \(X_{\text{loaded}}\). Figure 3b is the torque of the motor that is related to the relative size of the terms of the \(N\). Table 1 is a table that summarizes \(X_{\text{empty}}\), \(X_{\text{loaded}}\), \(X_{\text{estimated}}\) (final), and \(X_{\text{estimated}}\)\({}_{\text{NSS}}\)(final). The \(X_{\text{estimated}}\) (final) and \(X_{\text{estimated}}\)\({}_{\text{NSS}}\)(final) are the converged \(X_{\text{estimated}}\) and \(X_{\text  
In the simulation, the movement of the stair-climbing delivery robot is a step function of 15 degrees to simulate the tilting of the frame of the roll axis which is the inner frame of the gimbal structure. A controller of the loading device is a fixed PID controller. The fixed values are as follows:  
* \(D_{m}=0.02\) N s/m
* \(g=9.81\) m/s\({}^{2}\)
* Gains of the PID controller: \(P_{\text{gain}}=1.2\), \(I_{\text{gain}}=0.2\), \(D_{\text{gain}}=0.5\)
* The weights: \(w_{1}=-1.5\), \(w_{2}=-0.5\), \(w_{3}=-6\)  
In Figure 2(a), the root-mean-square errors of the estimated system and the estimated (NSS) system are \(4.378\times 10^{-13}\) and \(8.438\times 10^{-13}\). It shows that the proposed system identification techniques estimate the dynamic equation that moves in equal motion with the real system.  
In Figure 2(c) and Table 1, the \(X_{1}\) of the \(X_{\text{estimated}}\) stays at the \(X_{\text{empty}}\). The cause is that \(\sin(\varnothing)\), the first term of the known parameter vector \(N\), that is multiplied by \(X_{1}\), is almost 0 because the \(\varnothing\) converges to 0. In the calculation of the \(X\) by minimizing the error function of the \(\bar{\varnothing}\), the adjustment of the \(X_{1}\) has a negligible effect, and it causes that the \(\Delta X_{1}\) is almost 0. This problem is also shown for the \(X_{2}\) and the \(X_{3}\) depending on the relative size between \(\cos(\varnothing)=1\), which is the second term of \(N\) and motor torque in Figure 2(b) affecting the third term of \(N\). In Table 1, the error rates of each term of the \(X_{\text{estimated}}\) are 31.960%, 399.427%, and 399.479%. On the other hand, the improved system identification technique using a null-space solution can adjust the \(X\). The error rates of each term of the \(X_{\text{estimated NSS}}\) are 0.928%, 0.925%, and 0.925%. The accuracy of the parameter estimation can be improved by using a null-space solution.