# State Observer Based on an Accelerometer for an Elastic Joint with Nonlinear Friction
## 4 Experiment Results  
Even though our goal is to observe all states of the elastic joint expressed as (1), we carry out experiments for a two-inertia system shown in Figure 2 to verify the effectiveness of the proposed observer in the previous section because it represents quite well the dynamic properties of the manipulator having a flexible joint [18]. As shown in Figure 3, a two-motor system consisting of a driving motor and a load motor is illustrated in this experiment. The two motors connected with an elastic shaft and a friction adjustment on the driving motor side are similar to [19]. The proposed observer has been implemented in a high-performance embedded motion controller (manufactured by National Instruments Corporation) with servo motor drivers (manufactured by Panasonic Industry Corporation).  
The parameters of the two-inertia system, including the nonlinear stiffness and friction, are listed in Table 1. In addition, the link acceleration \(\dot{\omega}_{l}\) of (6) is obtained from the encoder value of the load motor in Figure 3. In other words, the angle acceleration is obtained by differentiating the position value of the encoder twice. Figure 4 shows the block diagram of the overall system with the proposed state observer. Here, the block \(\Omega(\hat{x})\) means the nonlinear terms of the friction and stiffness in the elastic joint model. The difference in design approach between the proposed method and the linear observer is the addition of the block \(\Omega(\hat{x})\) and the design method of the observer gain \(L\) that depends on the Lipschitz constant \(\gamma\) of \(\Omega(x)\), as shown in the Equations (7) and (9). The design parameters, including the observer gain \(L\) of (8), are chosen as depicted in Table 2.  
Figure 3: Experiment Setup.  
Figure 2: The two-inertia system.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline damping (\(D\)) & 600 & Nm s/rad \\ motor inertia (\(J_{m}\)) & 0.001027 & kg \(\cdot\) m\({}^{2}\) \\ linear stiffness coefficient (\(k_{1}\)) & 1.5 \(\times\) 10\({}^{6}\) & Nm/rad \\ nonlinear stiffness coefficient (\(k_{2}\)) & 9.85 \(\times\) 10\({}^{11}\) & Nm/rad\({}^{3}\) \\ breakpoint deflection (\(\theta_{B}\)) & 2 & arcmin \\ viscous friction coefficient (\(f_{v}\)) & 0.006 & Nm s/rad \\ coulomb friction coefficient (\(f_{c}\)) & 1.5 & Nm \\ \(\mu_{k}\) & 0.5 & \\ \(\beta\) & 0.5 & \\ \(\alpha\) & 5 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the experimental system.  
Figure 4: Block Diagram of the Overall System.  
position estimation error for the conventional method and proposed approach, respectively, as follows.  
\[\text{RMSE}(\hat{\theta}_{l},\theta_{l})=\left\{\begin{array}{ll}124.8400& \text{for conventional method,}\\ 7.8889&\text{for proposed method.}\end{array}\right.\]  
The comparison of the observation performance can also be confirmed from the estimation error of the actual link trajectory, as shown in Figure 7.  
Figure 5: Observer performance comparison (link position).  
Figure 6: Enlarged version of Figure 5.