# Input Shaping Control of a Flexible Structure for Rest-to-Rest and Non-Rest-to-Rest Maneuvers
## 2 Dynamic System Model  
A single-mode flexible system is modeled as two rigid bodies connected by a torsional spring without damping. The first body, \(\mathcal{L}_{1}\), is assumed to be a cube, while the second body, \(\mathcal{L}_{2}\), is a thin plate. We assume planar motion, where the system is free-floating on a plane and the axis of the torsional spring is perpendicular to the plane of motion. This model is intended to represent a general two-body spring-mass system, such as the in-plane motion of a spacecraft with a rigid solar panel or a robotic vehicle with a flexible appendage. Zero damping is used as a \"worst case\" assumption, where the system will not naturally come to rest without an external control. This assumption has been used in the literature on flexible spacecraft systems [15; 16]. However, note that the proposed approach can be used in exactly the same way as described in this article for a system with damping. Figure 1 shows the system model in the plane of motion, where \(\mathcal{L}_{1}\) appears as a square with sides of length \(L_{1}\), and the edge of \(\mathcal{L}_{2}\) appears as a thin rectangle of length \(L_{2}\). We assume both bodies are homogeneous with centers of mass \(C_{1}\) and \(C_{2}\) at the geometric centers of the bodies. Point \(C\) indicates the center of mass of the entire two-body system. Point \(A_{12}\) is the articulation point between the two bodies; we assume there is a torsional spring with spring constant \(k\) located at this point. The generalized coordinates of the system include the absolute position \((X,Y)\) of point \(C\), relative to vector base \(\hat{e}^{N}\) in the inertial reference frame, and the internal configuration of the system \((\theta_{1},\theta_{2})\), where \(\theta_{1}\) is the absolute orientation of body \(\mathcal{L}_{1}\) with respect to \(\hat{e}^{N}\), and \(\theta_{2}\) is the relative orientation of \(\mathcal{L}_{2}\) with respect to \(\mathcal{L}_{1}\). We define the generalized coordinate vector  
\[\mathbf{q}=[X,Y,\theta_{1},\theta_{2}] \tag{1}\]  
The equation of motion of the system can be written as  
\[\mathbf{H}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=\mathbf{G} \tag{2}\]  
where \(\mathbf{H}\), \(\mathbf{C}\), and \(\mathbf{G}\) represent the mass matrix, centrifugal terms matrix, and force matrix. These matrices are given in Appendix B. As a free-floating system, linear momentum is conserved, so point \(C\) will conserve its initial velocity \((\dot{X}(0),\dot{Y}(0))\). Thus, the motion of the center of mass of the system \((X,Y)\) is decoupled from the internal configuration of the system \((\theta_{1},\theta_{2})\). Henceforth, we focus our analysis on the dynamics of the angular states \((\theta_{1},\theta_{2})\). Using small angle approximations, the equation of motion can be linearized:  
\[\dot{\mathbf{\theta}}(t)=A\mathbf{\theta}(t)+\mathbf{bu}(t) \tag{3}\]  
Figure 1: Schematic of a planar free-floating two-rigid-body, torsional-spring-joint system, in flat space. where  
\[\theta(t) = [\theta_{1}(t)\quad\dot{\theta}_{1}(t)\quad\theta_{2}(t)\quad\dot{ \theta}_{2}(t)]^{T}\] \[A = \begin{bmatrix}0&1&0&0\\ 0&0&c_{1}&0\\ 0&0&0&1\\ 0&0&-c_{3}^{2}&0\\ \end{bmatrix}  \tag{4}\] \[b = \begin{bmatrix}0&b_{2}&0&b_{4}\\ \end{bmatrix} ^{T} \tag{5}\]  
Note that \(c_{3}\) is the natural frequency of the system, \(c_{1}>0\) is the coupling parameter, and \(u(t)\) is the input torque applied to body \(\mathcal{L}_{1}\). For the system pictured in Figure 1, \(c_{1}\), \(c_{3}\), \(b_{2}\), and \(b_{4}\) are given in Appendix B. For a rest-to-rest problem, the initial state is \(\theta(t_{0})=[\theta_{0},0,0,0]\), and the desired final state is \(\theta(t_{f})=[0,0,0,0]\). The objective is to find the control \(u(t)\) such that the final rest state can be obtained at final time \(t_{f}\). This article considers an example system to illustrate the effects of various control strategies.