## 4 Simulation and Experiment
### Experiment  
The existing simulation verifies the ability of the proposed control method to track the joint position trajectory in a virtual environment. To further illustrate the performance of the proposed controller applied in the upper-limb exoskeleton robot, an experiment was designed to mimic the robot-assisted passive rehabilitation task on the human elbow joint. First, the experimental setups were introduced. The whole equipment involved in the experiment is depicted in Figure 11. The proposed control method was programmed in a host with an Intel Core i7-4790 processor and 16 GB RAM. The execution rate can take up 1 KHz by running the Simulink real-time programs (Matlab2016b, Mathworks Inc., Natick, MA, USA). For safety, a manikin arm was installed on the exoskeleton instead of on the actual arm during the experiment. In the sensor system, a force sensor (BJM-3, JN sensor Inc., Bengbu, Anhui, China) was installed on the exoskeleton to detect the actual joint drive torque determined by the joint position, which achieves the torque-loop of the exoskeleton system (with a programmed PI controller). Subsequently, the two inertial measurement units (WT61 PC, Witmotion Inc, Beijing, China) were adopted to measure the actual joint position. The mentioned signals were acquired with a microcontroller (Arduino Mega2560, Hesai Technology Inc., Shenzhen, China) and then transmitted to the host at a communication frequency of 100 Hz. In the meantime, another microcontroller is exploited to send the control signal to the printed circuit board to drive the two high-speed valves (MHE2-MS1H-3/20-M7, FESTO AG & Co. KG, Esslingen Berkheim, Germany). Then, gas from the air source flows through the air filter regulator via the high-speed valves and finally passes into the PAM. The experimental process is illustrated in Figure 12. A mimic robot-assisted passive rehabilitation training environment is set with a certain sense of reality. By complying with the proposed control method, the upper-limb exoskeleton performs a periodic flexion and extension movement (with 25 degrees of amplitude) on the elbow joint (as shown in Figure 13). The training trajectory was set as a Fourier function, as expressed in Equation (43), which applies to the human joint training trajectory [14]:  
\[q_{d}(k) =15.36-12.67\text{cos}(0.5 kT)-1.23\text{sin}(0.5 kT)-2.49\text{cos}( kT)-0.48\text{sin}(kT) \tag{43}\]  
The parameters of the proposed controller were selected as \(\eta=30\), \(\mu=0.6\), \(\Phi_{1}(0)=0.95\), \(\Phi_{2}(0)=0.95\), \(t_{in}=5\) s, \(\rho=2\) degrees, \(K=1.5\), \(\Gamma_{1}=0.05\), and \(\Gamma_{2}=0.001\). Then, the initial elbow position remained distant from the reference value with several instances of human intervention (i.e., \(y(0)=10\)) using the proposed controller. Moreover, an original MFAC-SMC-PP controller was used to track the same reference trajectory, except for \(y(0)=0\)  
Figure 10: Tracking errors of the simulation in group 2 (proposed method). The parameters of the prescribed performance function were selected as \(v=0.05\), \(\rho(0)=5\), and \(\rho=2.5\). Other parameters are as same as the proposed controller. The experimental results are presented as follows. According to Figure 14, by using the proposed method, the actual joint angular position is capable of effectively tracking the reference trajectory in the rehabilitation experiment. Meanwhile, the actual joint position error \(q-q_{d}\) is depicted. From Figure 15, the error curve can fall into two parts by \(t=k_{in}\), i.e., the curve in the gray area and the white area. In the first part, the error curve is progressively regulated and finally enters the predesigned boundary as impacted by the tuning function. At \(t=0.9\) s, the joint position error of the exoskeleton is equal to \(-2.53\) degrees (i.e., the maximal value of the negative error). As indicated in the second part, the joint position error of the exoskeleton never escapes the boundary of \((-\rho,\rho)\).