# Development of an Adaptive Force Control Strategy for Soft Robotic Gripping
## 3 Results and Discussion
### Force Control  
The force controller was tested by grasping a cylindrical object of approximately 10 mm in diameter. The controller gains and features weights were initially at zero, showing the ability of the system to converge to an appropriate set of gains and weights. It is important to note that the controller will converge faster if the gains start closer to the desired value. To reduce the control problem's complexity, the system's dynamics were simplified to a second-order approximation of the system. The reference model was designed to be defined by a damping ratio (\(\zeta\)) and settling time (\(t_{s}\)), with the system tested here using \(\zeta=1\) and \(t_{s}=0.75\). The system was tested with different reference models to observe the performance, and these values were chosen due to the reliability of the parameter convergence. However, the controller could converge at settling times up to 0.5 seconds and follow alternate damping ratios. The chosen reference model used in the experiments in this paper is shown in Equation (9):  
\[ \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\end{bmatrix} = \begin{bmatrix}0&1\\ -28.44&-10.67\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix} + \begin{bmatrix}0\\ 28.44\end{bmatrix} u \tag{9}\]  
Finding a set of learning rates to allow the gains to converge was a challenging problem. The chosen learning rates, denoted as \(\gamma\) in Equations (4)-(6) are shown in Table 9 for each parameter. \begin{table}
\begin{tabular}{c c c c} \hline
**Paper** & **MSE (N)** & **Max Force (N)** & **Normalized Error (\%)** \\ \hline Ours & 1.01 & 30 & 3.37 \\ \hline [PERSON] et al. [18] & 0.09 & 2 & 4.6 \\ \hline [PERSON] et al. [20] & 0.08 & 2 & 4 \\ \hline [PERSON] et al. [19] & 0.15 & 6 & 2.5 \\ \hline [PERSON] et al. [22] & 0.15 & 3 & 5 \\ \hline \end{tabular}
\end{table} 
Table 8: Comparison of other force-sensing methods. The experimental results of the gripping are shown in Figure 14. The test was conducted using a repeating grip and release pattern, with the first gripping setpoints being 30 N and 15 N, with a release between grasps. During the release, the controller parameters remained frozen and only updated when the gripper was in the gripping portion of the cycle. For the test, the object was gripped twenty times, ten at each setpoint, over three minutes. The convergence of the controller gains can be seen in Figure 15.  
As shown in the figure, by examining the plot for \(x_{1}\), the controller starts with abysmal performance, unable to reach the setpoint. After a few cycles, the gripping approaches the desired setpoint while still having a steady-state error, eventually disappearing with more gripping cycles. A similar trend appears for \(x_{2}\), where the real system approaches the dynamics of the reference model. Figure 14: Closed-loop force control performance of MRAC controller on fin ray fingers. Figure 15: Evolution of controller gains. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Learning Rate** \\ \hline \(r\) & 0.0001 \\ \hline \(x_{1}\) & 0.0001 \\ \hline \(x_{2}\) & 0.00001 \\ \hline \(W\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: MRAC learning rates. A practical issue in the MRAC controller is parameter drift, where noise in the measured signal causes the controller parameters to continue growing despite good performance, which leads to instability. This was resolved by placing a lower limit on the error signal, where the controller gains remain frozen if the error is below a set threshold. Saturating the error signal in this manner was an effective method of limiting parameter drift. After tuning the saturation limit for each parameter, the controller maintained good performance. The controller gains and weights were then saved to be loaded for a new gripping process, which can be seen in Figure 16, which shows a zoomed view of the gripping process. As can be seen in the zoomed plot, the gripping force can closely match the reference model for both of the system states.