## 2 Problem Formulation  
### Model Building  
In this article, we designed a set of mechanisms composed of three identical parts. Its function was to apply the set value of vertical upward force to the given three points in the space through a steel wire rope, and it could adapt to the three-dimensional space movement of the given three points maintaining the vertical direction and tension of the steel wire rope at the set value. The three components of this mechanism operated independently, and each part could apply force to a given point. We called this part the \"lifting point\". The structure of a single lifting point and the definition of the coordinate system used in the article are shown in Figure 1.  
The Y direction of the coordinate system is the vertical upward direction, and the XZ direction is in the horizontal plane. There were three pulleys in the lifting point named ABC. Pulley A was a fixed pulley with a sensor above it to measure the tension of the steel wire rope. Pulley B was a movable pulley that could actively move in the Y direction as an actuator, causing the retraction and release of the steel wire rope in the Y direction. The C pulley was an actuator that could wind the steel wire rope, which could also control the retraction and release of the steel wire rope in the Y direction. Between points N and M there was a part where the steel wire rope extended out of the lifting point. Point M was connected with the point to which we applied the force in the space so that the force could be applied to point M through the steel wire. There was an inclination measuring mechanism at point N, which could measure the inclination of the steel wire rope along the X and Z directions. These two inclination angles were \(\alpha\) and \(\gamma\). Maintaining these two inclination angles at zero, it could be considered that the steel wire rope was vertical and  
Figure 1: Free body diagram for defining the coordinate system and constructing a single lifting point. the force applied to point M was also vertical. A spring was connected in series to the steel wire rope between points M and N, and the total distance between points M and N was l.  
To describe this system, the coordinates of the given point in space are expressed as \(\left(x_{2},y_{2},z_{2}\right)^{\mathrm{T}}\), the coordinates of M in the natural state are \(\left(x_{1}(t-\tau),y_{1}(t-\tau),z_{1}(t-\tau)\right)^{\mathrm{T}}\), \(\left(x_{1},y_{1},z_{1}\right)^{\mathrm{T}}\) is the input of the system, and \(\left(\alpha,\mathrm{F},\gamma\right)^{\mathrm{T}}\) is the output of the system. The relationship between them can be expressed by Equation (1):  
\[\left\{ \begin{array}{l}1\cos\alpha(t)\cos\gamma(t)=y_{\mathrm{max}}-y_{2}(t- \tau)\\ 1\sin\alpha(t)=x_{2}(t)-x_{1}(t-\tau)\\ \frac{\mathrm{F}(t)}{\mathrm{k}}=-y_{2}(t)+y_{1}(t-\tau)\\ 1\cos\alpha(t)\sin\gamma(t)=z_{2}(t)-z_{1}(t-\tau)\end{array} \right. \tag{1}\]  
where k is the coefficient in [PERSON]'s law, which is the ratio of force to displacement. In this model, this is the inherent parameter of the system. We considered carrying out linearization when the wire rope was nearly in a vertical state with the condition of small inclination. Under this condition, \(\sin(\alpha)\) approximately equals \(\alpha\), and \(\cos(\alpha)\) approximately equals 1. We linearized Equation (1) and sorted it out to obtain Equation (2):  
\[\left\{ \begin{array}{l}\alpha(t)=\frac{1}{\mathrm{x}_{2}}(t)-\frac{1}{ \mathrm{x}_{1}}(t-\tau)\\ \mathrm{F}(t)=-ky_{2}(t)+ky_{1}(t-\tau)\\ \gamma(t)=\frac{1}{\mathrm{i}}z_{2}(t)-\frac{1}{\mathrm{i}}z_{1}(t-\tau)\end{array} \right. \tag{2}\]  
Equation (2) is the mathematical model of an independent lifting point. In the modern control theory, we usually use the letter \"x\" to represent the state of the system. In this article, the letter \"x\" had already been used to represent the coordinates of points.