# Modeling and Control of an Underactuated System for Dynamic Body Weight Support
## 8 Summary and Conclusions  
This paper includes the results of studies regarding the methodology used for tests on the stability of a control system for a dynamic body weight support system. It also describes the object of tests for which equations were formulated regarding a motion and control  
Figure 21: Follow-up movement errors: (**a**) position error of the active part \(e_{\varphi}\); (**b**) position error of the passive part \(e_{x}\); (**c**) speed error of the active part \(e_{dp/dt}\); (**d**) speed error of the passive part \(e_{dx/dt}\). Figure 22: Value of the trainee’s unloading force as a function of time. method. The device has a control system based on a PD controller with compensation, which is essential considering its classification as an underactuated system. This work presents the complete algorithm for calculations used to describe the control system and to analyze the stability. The Lyapunov theory of stability function was also determined. The presented methodology of calculations may be employed to analyze the stability of similar devices. However, knowledge of the mathematical model is required to determine the dynamic performance of a device and its parameters, as well as its control method. The proposed control system features the trajectory of the active part, which is determined on the basis of movement of the passive part. During the performed tests, the trajectory of the passive part related to the adjustable function is formulated on the basis of previous studies of the gait kinematic parameters. It is also possible to modify the control system and use the measurement of the position of the passive part as the set trajectory. Alternatively, the force sensor, which is equivalent to an approach applied previously in similar devices, can be used [21; 25; 34]. In this case, the device performs a movement that follows the vertical movement of the patient only. During the conducted tests, it was the body weight support system that forced patient gait thanks to the passive part movement. The use of such a control system in a rehabilitation device for the re-education of human gait is a new approach. Therefore, it is not possible to directly compare the obtained results with other works and the quality of control systems used so far in similar devices, e.g., with the control system of the ZeroG device, where only force controller occurs [21]. However, we know from the literature review that for underactuated mechanical systems, the use of the PD controller with compensation ensures better operation than the feedback loop with the PD controller itself; this has been shown inter alia on underactuated systems such as inverted pendulums, ball and beam systems, overhead cranes, or robots with flexible joints [54; 55; 56; 57; 58; 59; 60]. However, it is necessary to demonstrate the stability of such a control system so that the term associated with the compensation of the dynamics of the passive part does not cause destabilization [61; 62; 63]. The unstable control algorithm may result in dangerous movements of the drive in the body weight support system, which is particularly important due to the cooperation of the device with patients during rehabilitation. In the presented approach, the stability of the developed system has been demonstrated, which has not been proven so far in other studies on body weight support systems. However, in future studies it will be possible to compare the proposed algorithm with more advanced controllers, such as active disturbance rejection control, sliding mode control, or with adaptive algorithms. On the basis of the results of empirical studies and the subjective opinion of the person performing the training, it is concluded that the use of this control system in a dynamic body weight support system forces the patient to adjust their walking cadence to the rope movement. Consequently, this control system has a significant impact on the person's gait kinematics.