# Modeling and Control of an Underactuated System for Dynamic Body Weight Support
## 4 Control System Modeling  
The actual purpose of the control system of a support load system is to maintain constant load support. Constant rope tension relates to constant deflection of the rope. Therefore, an assumption may be made that the issue related to keeping a constant support load is synonymous to follow-up motion related to vertical movement of the patient. According to the presented model, these movements directly impact the movement of the passive part, namely generalized coordinate \(x\). Further calculations were based on an adjusted trajectory of the passive part \(x_{d}(t)\) featuring continuous derivatives, and an assumption was made regarding the existence of the adjusted trajectory of the active part \(\varphi_{d}(t)\) with continuous derivatives [41]. Next, the trajectory error was determined:  
\[\left\{ \begin{array}{l}e_{a}=\varphi_{d}-\varphi\\ e_{p}=x_{d}-x\end{array} \right. \tag{2}\]  
where \(e_{a}\) is the follow-up movement error of the active part and \(e_{p}\) is the follow-up movement error of the passive part. The generalized follow-up movement error was determined using:  
\[\left\{ \begin{array}{l}s_{a}=\dot{e}_{a}+\lambda_{a}e_{a}\\ s_{p}=\dot{e}_{p}+\lambda_{p}e_{p}\end{array} \right. \tag{3}\]  
where \(\lambda_{a}\) and \(\lambda_{p}\) indicate positive design constants related to the ratio of proportional and differential gains in the PD controller. Figure 3: Physical model of the dynamic body weight support system. Equation (3) considers Equation (2). Through conversion and derivation, the acceleration was determined:  
\[\left\{ \begin{array}{l}\dot{\varphi}=-\dot{s}_{a}+\ddot{\varphi}_{d}+\lambda_{a }\dot{e}_{a}=-\dot{s}_{a}+\dot{v}_{a}\\ \dot{x}=-\dot{s}_{p}+\ddot{x}_{d}+\lambda_{p}\dot{e}_{p}=-\dot{s}_{p}+\dot{v}_{p }\end{array} \right. \tag{4}\]  
where \(v_{a}=\dot{\varphi}_{d}+\lambda_{a}e_{a}\) and \(v_{p}=\dot{x}_{d}+\lambda_{p}e_{p}\) are additional variables applied to simplify recording. The application of (4) to (1) resulted in a description of the generalized follow-up movement error function:  
\[\left\{ \begin{array}{l}I_{Z}\dot{s}_{a}=I_{Z}\dot{v}_{a}+\left(b_{Z}+b_{s} \frac{h^{2}}{4\pi^{2}}\right)\dot{\varphi}-b_{s}\frac{h}{2\pi}\dot{x}+k_{s} \frac{h^{2}}{4\pi^{2}}\varphi-k_{s}\frac{h}{2\pi}x-M_{Z}\\ m_{Z}\dot{s}_{p}=m_{Z}\dot{v}_{p}+b_{s}\dot{x}-b_{s}\frac{h}{2\pi}\dot{ \varphi}+k_{s}x-k_{s}\frac{h}{2\pi}\varphi-2F\end{array} \right. \tag{5}\]  
Expressions related to the system's condition were determined with the following functions:  
\[\left\{ \begin{array}{l}f_{a}=I_{Z}\dot{v}_{a}+\left(b_{Z}+b_{s}\frac{h^{2}}{ 4\pi^{2}}\right)\dot{\varphi}-b_{s}\frac{h}{2\pi}\dot{x}+k_{s}\frac{h^{2}}{4\pi ^{2}}\varphi-k_{s}\frac{h}{2\pi}x\\ f_{p}=m_{Z}\dot{v}_{p}+b_{s}\dot{x}-b_{s}\frac{h}{2\pi}\dot{\varphi}+k_{s}x-k_{ s}\frac{h}{2\pi}\varphi-2F\end{array} \right. \tag{6}\]  
Finally, the following set of Equation (5) was determined:  
\[\left\{ \begin{array}{l}I_{Z}\dot{s}_{a}=f_{a}-M_{Z}\\ m_{Z}\dot{s}_{p}=f_{p}\end{array} \right.. \tag{7}\]  
To obtain the adjusted trajectory of motion, an assumption was made regarding the control inputs of the proportional-derivative controller with compensation:  
\[\left\{ \begin{array}{l}u_{a}=k_{a}s_{a}+f_{a}\\ u_{p}=k_{p}s_{p}+f_{p}\end{array} \right. \tag{8}\]  
where \(u_{a}\) is the control of the active part equal to \(u_{a}=M_{Z}\) and \(u_{p}\) is the fictional control of the passive part equal to \(u_{p}=0\)[42]. Here, \(k_{a}s_{a}=k_{a}\big{(}\dot{e}_{a}+\lambda_{a}e_{a}\big{)}\) and \(k_{p}s_{p}=k_{p}\big{(}\dot{e}_{p}+\lambda_{p}e_{p}\big{)}\) describes the PD controller for the active and passive parts, respectively, where \(k_{a}\) and \(k_{p}\) improve the derivation. The assumed control method in (8) includes two components, whereas the second equals zero as the passive part is not controlled. This component is called fictional control, as it does not have a direct impact on the movement of the second part of the system.