## 3 Simulation Results and Discussion
### Deployment Strategy  
Smooth deployment of the tether is the key to TB-TSS operation. However, due to the Coriolis force induced by the relative motion of libration and the motion of tether deployment, it is not easy to quickly and stably deploy the tether at the expected length. Therefore, an effective deployment strategy is important to achieving stable and rapid deployment of the TB-TSS. A common open-loop deployment strategy is to control the tether length rate [26; 30]. Because of its simplicity, it is widely used to verify the deployment dynamics of tethered satellite systems. There are two common tether length-rate control laws: uniform speed development and exponential development. In the actual development process, the deployment speed of the uniform speed control strategy will change abruptly in the initial and end stages of tether deployment, which will result in rebounding and even winding of the tether. The exponential deployment strategy will increase infinitely or approach zero with time. There is a risk of tether entanglement occurring when the speed of the tether is up to 20 m/s [22]. In addition, the exponential deployment speed should approach zero when the deployment task is complete. Therefore, the development dynamics of the TB-TSS are surveyed with a predesigned deployment strategy in this section. Since a TB-TSS can be deployed with the gravity gradient or tension, the state of the system may be far from the ideal target goals and objectives in the final phase. The deployment phase is an unstable process; the deployment task can only be completed without any slack or winding of the tether [22]. This means that libration motion is inevitable, and the tension should always be positive to keep the tether straight. A deployment strategy needs to be designed to ensure that the tension is positive and the libration angle does not exceed 90\({}^{\circ}\). The results in the existing literature show that a too fast deployment speed and sudden changes in the tether speed will lead to system instability. Therefore, a predesigned deployment strategy was developed. The deployment strategy involves predesigning a constrained piecewise function that regulates the deployed velocity \(V_{i}\) as three phases. During the initial phase, the deployment velocity accelerates exponentially from rest to cruise velocity. In the second phase, the deployment velocity maintains this state. Finally, in the last phase, the deployment velocity is restrained exponentially to rest as it reaches the desired length. The specific form is given as follows:  
\[V_{i}=\left\{ \begin{array}{ll}c_{i1}e^{c_{i}t}&if\ 0\leq t\leq t_{1}\\ v_{c}&if\ t_{1}\leq t\leq t_{2}\\ c_{i2}e^{c_{i}(t_{2}-t)}&if\ t_{2}\leq t\leq t_{f}\end{array} \right.\;for\;i= 1,2 \tag{35}\]  
where the arguments of deployed acceleration of two tethers is designed as \(c_{i}=7.7\times 10^{-4}\), \(c_{11}=1.54\times 10^{-3}\), and \(c_{21}=1.54\times 10^{-2}\); \(v_{1c}=0.077\) m/s and \(v_{2c}=0.77\) m/s are the values of constantly deploying speeds in the second phase; \(c_{12}=7.7\times 10^{-2}\) and \(c_{22}=0.77\) represent the deployed acceleration of two tethers in the final phase. According to the total deployed tether length and deployed velocity, timing \(t_{1}\), \(t_{2}\), and \(t_{3}\) can be computed as \(t_{1}=5080.5\), \(t_{2}=16,768.9\) s, and \(t_{3}=21,849.4\) s. These parameters are designed according to the simulation requirement that the maximum deployment velocity is insufficient to cause system instability and reaches the desired length when the deployment velocity converges to zero in the final phase. The methods of applying two tether lengths and deploying velocity are plotted in Figure 10.  
To investigate the effect of this deployment strategy, we performed a simulation with the predesigned motion. It should be noted that the specified deployment rate is assumed to be driven solely by tension, and the tension can be obtained with Equations (9) and (12).