## 4 Experimental Evaluation
### Mock-Up Tests
It is important to note that the friction force may fluctuate due to changes in the condition of the gripper's fingertips or the object's surface, such as when the fingertips become worn, leading to a reduction in frictional force. However, our proposed push manipulation method does not rely on achieving a precise amount of slip. Instead, it requires the desired force to exceed the friction force, but not excessively, to maintain stability. While fine-tuning of the desired force is necessary when the object changes, our approach is not overly sensitive to fluctuations in friction caused by environmental changes. Although not addressed in this paper, adaptive [39] or data-driven [40] control techniques for accurately estimating friction force during pushing manipulation could provide a robust solution to accommodate significant variations in the friction model. Figure 9: Hybrid position/force control input in the Y direction during the pushing action. The first row is the entire control force, which consists of the position and force control input and the nonlinear dynamics compensation term; the measured external force is included in the force control input. The control strategy is changed when the force control trigger arises. The rest of the direction is still controlled by position control, which maintains the pushing pose. The feedback term of the force control contributes to canceling the effect of the uneven frictional contact force. Note that the feedback term in Figure 10 includes vibrations caused by the dynamics of the SEAs as well as fluctuating forces to compensate for uneven friction. To finish the pushing action, the contact force should be diminished to move on to the next point without a sudden large feedback position control input, which may generate an undesirable recollision with the environment. Therefore, force smoothing is exploited at the end of the pushing period. The magnitude of the force reference is decreased from 13 N to 0 N during the smoothing period \(T_{D}\) by the first-order function that follows:  
\[\left\{ \begin{array}{cc}S_{f}(t)=1&for\ T_{P}-t>T_{D}\\ S_{f}(t)=T_{D}^{-1}\times(T_{p}-t)&for\ T_{P}-t\leqslant T_{D}\end{array} \right. \tag{13}\]  
where \(T_{P}\) is the pushing period and \(T_{D}\) is set to 0.2 s in this paper. The tracking control input is generated during this period. The time at which the control state change occurs is compared for the two cases of long and short grasping positions. The time at which this change occurs for the long grasping position is twice longer than that of the short grasping position. The time difference implies a different initial grasping position; in this case, the long grasping position is twice the length of the short grasping position. The end of the slip motion can be inferred from the end of fluctuation in the feedback term signal. Figure 10: The force control input (Y direction) consists of the measured external force and feedback control term. When the force control is triggered, the desired force is calibrated to induce slipping for the pushing action. The feedback term fluctuates when slip occurs because of uneven contact friction.