# Development of an Adaptive Force Control Strategy for Soft Robotic Gripping
## 2 Materials and Methods
### Force Control  
During the initial testing of the normal force applied by the gripping, it was found that the gripper acted as a nonlinear spring with an approximately exponential stiffness within the gripper operating region. As the gripper deforms significantly, the layers of the fin ray begin to mesh together, creating an effect called layer jamming, which causes the stiffness of the mechanism to increase substantially. This increased stiffness results in nonlinear dynamics, particularly in the system's gain, and requires more advanced control schemes to regulate the force. The proposed controller is illustrated in the block diagram in Figure 8 [37]. As the object's shape and location along the finger were unknown before the gripping process, many system parameters were unknown, necessitating an online adaptive controller. The usual strategy for controlling uncertain systems is to use either robust control or adaptive methods. For this application, it was desirable that the controller performance would improve over time with more gripping cycles, making model reference adaptive control (MRAC) a good choice. Combining the MRAC with the virtual force sensing above creates the VRAC controller. The measurement of the system was performed using a camera and autoencoder system, which provided a fairly noisy measurement. To estimate the states of the system  
Figure 8: Model reference adaptive control for gripper force control. Figure 7: High-level cascaded control scheme. and deal with noise, an Extended Kalman Filter (EKF) was used [38]. The control output of the MRAC is given by Equation (2):  
\[u(t)=k_{r}r(t)+k_{x}x(t)-u_{ad} \tag{2}\]  
where \(u(t)\) is the control action, \(k_{r}\) is the feed-forward gain, \(r(t)\) is the reference signal, \(k_{x}\) is the feedback gain vector, \(x(t)\) is the estimated system states, and \(u_{ad}\) is the adaptive control action, derived from the disturbance model, show in Equation (3):  
\[u_{ad}=W^{T}\phi(x) \tag{3}\]  
The MRAC controller learns the controller gains to drive the closed-loop performance of the nonlinear system of the fin ray gripper to match the reference model. The parameter update equations are as follows:  
\[\dot{W} =\gamma_{w}\phi(t)e^{T}(t) \tag{4}\] \[\dot{k}_{x} =\gamma_{x}x(t)e^{T}(t)\] (5) \[\dot{k}_{r} =\gamma_{r}r(t)e^{T}(t) \tag{6}\]  
where \(\gamma_{w}\), \(\gamma_{x}\), and \(\gamma_{r}\) are the parameter learning rates; \(\phi(x)\) is a feature vector of known Gaussian functions, all having the same width, but with differing means; \(r(t)\) is the gripping force setpoint; \(x(t)\) is the estimated state vector for the gripper dynamics; \(e(t)\) is the error between the estimated plant states and the reference model states. The reference model is the state-space model shown in Equation (7):  
\[\dot{x}_{m}=A_{m}x_{m}(t)+B_{m}r(t) \tag{7}\]  
The model error is the difference between the observed and reference model states, as defined in Equation (8):  
\[e=x(t)-x_{m}(t) \tag{8}\]  
The gripper was programmed to operate in grip and release movements defined by a force setpoint and holding time. The gripping process is expected to perform poorly at the start of its life as the parameters are initialized with a guess. As more gripping cycles are performed, the performance should converge to the reference model dynamics. To improve the controller's performance at the start of the operation, the controller gains and the disturbance model can be initialized to values close to the converged solution. If a separate object needs to be gripped, the controller will adapt and find the new gains. In summary, a method for controlling the force applied by a soft robotic gripper was proposed, designing the complete workflow for the virtual reference adaptive controller, from building a force sensor to outlining the architecture of a control algorithm. A force sensor was proposed to extract the shape of a fin ray effect gripper and encode the image using an autoencoder into a latent representation of the gripper's features.