# A Safe Admittance Boundary Algorithm for Rehabilitation Robot Based on Space Classification Model
## 3 Experiments
### Results
#### 3.2.2 Out-of-Bounds Simulation Experiments  
For the purpose of evaluation, the output wrist force data collected from the five experimenters with the same intended trajectory of motion are imported into the simulation system. Depicted in Figure 12a-d are the output forces of wrists from experiment, 1 making four groups of reciprocating motion in the same direction. Figure 13a shows a small segment of the variation curve of the six-dimensional force FX after interception and filtering when experimenter 1 makes a motion along the specified intended trajectory. With this group of force data imported into the simulation system, the virtual mass m in the admittance model is consistently set to 0.01 kg and the virtual damping C is 1 Ns/m. The virtual stiffness k varies, its values being 0 N/m, 1 N/m, and 10 N/m, respectively. The velocity response curve output from the admittance model is obtained (as shown in Figure 13b). Figure 12: Six–dimensional force data from the wrist joint of experimenter 1: (**a**) six–dimensional force data 1; (**b**) six–dimensional force data 2; (**c**) six–dimensional force data 3; (**d**) six–dimensional force data 4. When the virtual stiffness is 0 N/m, the velocity response curve in the admittance model shows a variation trend consistent with that of the input interactive force. As the interactive force increases, the velocity of robotic motion increases as well. This indicates that the admittance controller can effectively recognize and track human motion intention. When the virtual stiffness increases from 0 N/m, the variation trend of the velocity response is not consistent with that of the input force anymore. This means that the admittance controller loses the ability to track human motion intention. The end-effector moves at an expected velocity and tends towards the expected position and pose. The above results demonstrate the effectiveness of the variable stiffness admittance strategy in defining the safe admittance boundary. Figure 14 shows the trajectories of the moving point in the end-effector with the input of the same six-dimensional force data with and without the application of the safe admittance boundary in the robotic simulation system. Along the trajectory from the starting point to the safe admittance boundary, the trajectories of the end-effector output from the simulation system with and without the application of the safe admittance boundary completely overlap. The above results indicate that in the control layer, the safe admittance boundary is generally accurate in judging the scope of the moving point. Twenty groups of data are analyzed in the same manner, and similar findings are made. The simulation system with the application of the safe admittance boundary also executes a change in the trajectory of the moving point in the end-effector at the safe admittance boundary so that the robotic motion is confined within the safe admittance boundary. This result proves the effectiveness of the applied safe admittance boundary. Figure 14: Visualization of the trajectories of the moving point in the end–effector with the input of the same six–dimensional force data with and without the application of the safe admittance boundary in the robotic simulation system. Figure 13: A segment of \(F_{x}\) variation curve and the velocity response curve of the moving point in the end–effector under different stiffness. (**a**) the six-dimensional force FX after interception and filtering for data 1; (**b**) the velocity response curves output from the admittance model with different virtual stiffness parameters. Finally, 20 groups of six-dimensional force data collected from the five experimenters are analyzed, as shown in Table 2. The above data are input into the simulation system. The average maximum response time for out-of-bounds behaviors is 165.62 ms. The kinesthetic reaction time of individuals without any known impairments typically falls within the range of 129.4 to 160.3 ms [38], and so the delay is almost imperceptible to the user.