Deployment Kinematic Analysis and Control of a New Hoop Truss Deployable Antenna

Firstly, based on the structural characteristics of a new type of hoop truss deployable antenna, this paper derives the motion transformation relation between two hoop modules by using the method of coordinate transformation, and establishes the general model for deployment kinematic analysis, which can be applied to analyze the position, velocity and acceleration of any point on the structure. Secondly, according to the relation between the driving cable and the hoop module, the motion planning of the hoop module is transformed into the motion control of the driving cable, which can realize the deploying position control of the antenna. Finally, numerical simulations show the control method can make the antenna smoothly deploy following the specified deployable motion.


Introduction
In the modern satellite communications, wireless broadcasting systems, earth observation and deep space exploration, large deployable antennas are one of the key equipment [1][2][3]. The hoop truss deployable antenna is getting more and more attention due to its superior performance. Its structure is mainly composed of deployable truss, tension supporting net, metal mesh and driving device [4]. Compared with other antenna structures, it has the advantages of large flexibility, good space thermal stability, large storage ratio, simple structure, and the increase of the aperture in a certain range does not change the structural form of the antenna, and the quality does not increase proportionally, so it is the ideal structural form for a large deployable antenna at present. However, the traditional hoop truss deployable antenna restricts its development to super-large-diameter antennas because the storage ratio is difficult to increase. Based on this, Cao, a master student of Xidian University, proposed a new structure about the hoop truss deployable antenna [5], which significantly improved the storage ratio and received great attention.
The deployable antenna has two states of folding and unfolding. In the process of satellite launching, it is in a collapsed state. After entering the orbit, the antenna automatically deploys into working state. The deploying process is one of the most brittle parts for the deployable antenna. Therefore, it is necessary to carry out the simulation of the deploying process and the prediction of the performance of the antenna [6,7]. The paper carried out detailed kinematics and controlling analysis by studying the new hoop truss deployable structure proposed by Cao of Xidian University.

Location analysis of deployment
A schematic view of the deploying process of hoop truss unit is shown in Fig. 1. The points A i to N i are the associate points of truss unit. The angle of θ is the deploying angle, which is the angle between the bar i i A H and i x axis. In the process of deployment, point 1 A is fixed at ground and point 1 B move down along 1 y axis. Fig.1. Truss unit coordinate system description.
In order to analyze this structure, we defined that the body coordinate system 1 Any point P i can be described as in the body coordinate system Denote the length of AH as 1 l , the length of AD and FH as 2 l , the length of BD and FG as 3 l and the length of CD and EF as l 4 . The coordinate of any point on the truss could be obtained by the following four steps.
Step. Step.3. The conversion process from the body coordinate system Step.4. The truss is divided into n segments and n must be even. The coordinate value of point

Speed and acceleration analysis
On both sides of Eq. (7), the first order derivative with respect to the time t is obtained as On both sides of Eq. (7), the second order derivative with respect to the time t is obtained as (9) Based on Eq. (8) and Eq. (9), the velocity and acceleration values of any point on the hoop truss structure can be analyzed.

Deployment and control analysis
In the deployment of antenna, whether the deployment of angle  is smooth or not directly determines whether there will be large vibrations. Therefore, it is necessary to plan the motion of the truss unit. The rate of change of the deploying angle can be defined as and the integral function is obtained as Meanwhile, it is also necessary to establish the relationship between the changeable length of the driving cable and the angle in deployment of the hoop truss unit since the deployment of antenna is achieved by contracting the driving cable with a motor.
As shown in Fig. 1, the effect of the pulley radius on the change in cable length is ignored here, and considering that the driving cable generally has a good stiffness, we also ignore the effects of its force deformation. So the driving cable is similarly arranged in the mechanism at the diagonal position of the hinged four-bar units ( i i i i H F J I and . The relationship between the length of the driving cable and the angle of deployment can be obtained from the geometric relationship (This relationship describes only kinematic relation and does not involve mechanical properties).
On both sides of Eq. (12), the first order derivative with respect to the time t is obtained as 2 2 sin So in the view of the perspective of mechanism kinematic, the corresponding relationship between the change of the length of the driving cable of the hinged four-bar units and each deploying time is obtained during the entire deploying process. When the position controlling of the deployment of antenna is performed, the linear displacement of the motor output corresponds to the frequency of the pulse input, so that the deploying process of antenna can be smoothly controlled according to the planned curve of the deploying angle.

Simulation verification
As illustrated in Fig. 2, the deployment of antenna can be generally designed as a motion which consists of acceleration stage, constant speed stage and deceleration stage. time T of 500s is exampled here whose simulation results are shown in Figs. 3-6. It can be seen from Fig. 3 which shows the trajectory of the vertices of each rigid truss during the deployment of the antenna that the vertices on the truss are deployed along the trajectory shown by the dashed line according to the angle of deployment of planning, and the feasibility of the structure is preliminarily verified. Figs. 4-5 show the motion trajectory of unit. The trajectory is smooth, which indicates the unit can smoothly deploy. Fig.6 shows the structural state at different times during the deploying process. It can be seen that the antenna is smoothly deployed according to the planned angular velocity variation.

Conclusion
A general motion analysis model for the deployment of a new type of hoop truss deployable antenna is established. For such antennas with different structural parameters, the position, velocity and acceleration of any point on the structure can be analyzed during the deployment by using this model, so it has strong practicability. At the same time, the control strategy of deployment based on position controls is studied. On the one hand, according to the requirement of deployment, the movement planning of the unit is carried out. On the other hand, according to the structural geometry, the motion of unit is transformed into the displacement plan of the deploying device. By implementing effective motion control of the driving cable, the antenna can be smoothly deployed according to the planned variation of the angle.