Fig. 5

Dynamic response of a FREE. Numerical solution for simulated open-loop response based on the original and the linearized model (left). The simulation results suggest insignificant changes in radius r and winding angle \(\gamma\) of the FREE before and after pressurization. Closed-loop response to a PID controller with step and trajectory following inputs (right). Experimental data is compared with the simulation results, and the trajectory-based controller produces a smoother response. All results correspond to a FREE with constant geometric properties \(\Gamma = 40^{\circ }\), \(L = 11\) cm, and \(R = 0.7\) cm, external forces and moments \(F_{l} = F_{gravity}\) and \(M_l = 0\), stiffness \(k_e = 10110\) N/m, and \(k_t = 0.086\) N/rad, and damping constants \(c_e = 5\) N-s/m, and \(c_t = 0.005\) N-m-s/rad that was controlled with a PD controller (\(K_p = 32000\) and \(K_d = 1200000\) Pa/rad-s). See our project repository for the codes used in our simulations and experiments: https://github.com/soheil-h/FREE.git