# Re-Optimized Design for the ADR-Based Dual-Loop Controller of 100 m Class Aperture Radio Telescope
## 2 Multiple-Motor-Driven Antenna Servo System Modeling  
Since it is a small ratio between weight and aperture, low cost, easy installation and maintenance, and high structural accuracy, a wheel-track azimuth-elevation-fully-steerable radio telescope is widely adopted in radio astronomy. Large-aperture radio telescopes in service, such as GBT, and Effisberg, and the under-construction QTT, employ this type of antenna. The azimuth and elevation axes of the azimuth-elevation antenna are geometrically orthogonal to each other, and their structures are similar. Since the azimuth axis subsystem is subject to the influence of the elevational pose changing, making its control performance susceptible, engineers usually undertake controller design and analysis on it. The azimuthal system of a wheel-track large-aperture antenna consists of a track and a mount with several driving arms and balancing arms, in which each driving arm is rotated by a truck with several motors with gearbos; for example, there are four driving arms, each employing four motors in the GBT azimuthal system [19]. Figure 1 shows a simplified azimuth structure of the multiple-motor-driven antenna of a radio telescope. It considers four principal degrees of freedom (4 DOF): the dish (major reflector) angular displacement \(\theta_{\rm d}\), the base angular displacement \(\theta_{\rm b}\), the truck system angular displacement \(\theta_{\rm t}\), and the gearbox angular displacement \(\theta_{\rm g}\). The azimuth system has n (n = 2 or 4) driving arms and n balancing arms. Each driving arm connects a truck consisting of k (k = 1, 2, 3, 4) wheels, and motors with gearboxes. The n drive arms have the same motor and structure parameters, so Figure 1 only shows one driving arm for the schematic drawing. In Figure 1, \(J_{\rm d}\), \(J_{\rm b}\), \(J_{\rm t}\), \(J_{\rm w}\), \(J_{\rm g}\), \(J_{\rm m}\) are the inertia moments of the dish, base, truck, wheel, gearbox, and motor, respectively; \(K_{s}\), \(K_{b}\), \(K_{g}\), \(K_{\rm m}\) and \(B_{s}\), \(B_{b}\), \(B_{g}\), \(B_{m}\) are the coefficients of the torsional stiffness and damping of the upper structure, one drive arm, gearbox, and motor shaft, respectively. Because of the relatively much lower damping and more significant stiffness of the gearbox, the motor shaft and gearbox are regarded as a whole. The equivalent inertia amount at the gearbox output of these two parts can be represented in Equation (1) under the condition of ignoring the gearbox damping \(B_{\rm g}\) and marking the reduction ratio as \(N\). \[\bar{J}_{\rm m}=N^{2}J_{\rm m}+J_{\rm g}, \tag{1}\]  
When the little \(J_{\rm g}\) is neglected, the equivalent inertia amount \(\bar{J}_{\rm m}\) can further be rewritten as \(\bar{J}_{\rm m}=N^{2}J_{\rm m}\). Let the radius ratio between the base and the wheel be \(R\), then \(\theta_{\rm w}=R\theta_{\rm t}\). The kinetic, potential, and dissipated energy equations of the 4 DOF azimuthal system shown in Figure 1 can be denoted as follows. \[E_{\rm k}=\frac{1}{2}J_{\rm d}\theta_{\rm d}^{2}+\frac{1}{2}J_{\rm b}\theta_{ \rm b}^{2}+n\big{(}\frac{1}{2}J_{\rm b}\theta_{\rm t}^{2}+\frac{k}{2}J_{\rm w} R^{2}\theta_{\rm t}^{2}+\frac{k}{2}\bar{J}_{\rm m}\theta_{\rm g}^{2}\big{)}, \tag{2}\]  
\[E_{\rm P}=\frac{1}{2}K_{\rm s}(\theta_{\rm d}-\theta_{\rm b})^{2}+n\big{(} \frac{1}{2}K_{\rm b}(\theta_{\rm b}-\theta_{\rm t})^{2}+\frac{k}{2}K_{\rm m}( R\theta_{\rm t}-\theta_{\rm g})^{2}\big{)}, \tag{3}\]  
\[D_{\rm b}=\frac{1}{2}B_{\rm s}(\theta_{\rm d}-\theta_{\rm b})^{2}+n\big{(} \frac{1}{2}B_{\rm b}(\theta_{\rm b}-\theta_{\rm t})^{2}+\frac{k}{2}B_{\rm m}( R\dot{\theta}_{\rm t}-\theta_{\rm g})^{2}\big{)}, \tag{4}\]  
Figure 1: Schematic of the multiple-motor-driven antenna’s azimuth system. Choosing the antenna's four displacement outputs as the generalized coordinate vector \(q=\left[ \begin{array}{ccc}\theta_{\rm d}&\theta_{\rm b}&\theta_{\rm t}&\theta_{ \rm g}\end{array} \right]^{T}\), the antenna system satisfies [PERSON]'s second law and the Euler-Lagrange equation as follows.