# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 2 Problem Formulation
### Dynamic Model  
The schematic diagram of the FLM is shown in Figure 2. The hub angle of the system is \(\theta\), the deformation angle of the flexible link is \(\alpha\), (X, Y) is the inertial coordinate system, and (x, y) is the local rotating coordinate system of the rotating center.  
Figure 1: Experimental platform FLM.  
In [24; 25], the flexible link is assumed to be a linear spring. The above simplification brings uncertainty to the system dynamics model. Applying the Euler-Lagrange equation, the lumped parameter model of the FLM is obtained as follows:  
\[\big{(}J_{eq}+J_{l}\big{)}\ddot{\theta}(t)+J_{l}\tilde{\alpha}(t)+B_{eq}\dot{ \theta}(t)=\tau(t), \tag{1}\]  
\[J_{l}\ddot{\theta}(t)+J_{l}\tilde{\alpha}(t)+K_{s}\alpha(t)=0, \tag{2}\]  
where \(\tau\) is input torque, \(B_{eq}\) is viscous friction coefficient, \(J_{eq}\) is equivalent moment of inertia, \(K_{s}\) is stiffness of the link, and \(J_{l}\) is link moment of inertia.  
The dynamic equation of the FLM in the joint space can be obtained by subtracting (2) from (3) as follows:  
\[\ddot{\theta}(t)=-\frac{B_{eq}}{J_{eq}}\dot{\theta}(t)+\frac{K_{s}}{J_{eq}} \alpha(t)+\frac{1}{J_{eq}}\tau(t). \tag{3}\]  
The input of the system is the voltage applied to the armature, so the relationship between the torque and the armature voltage is as follows [24; 25]:  
\[\tau=\frac{\eta_{m}\eta_{g}K_{t}K_{g}\Big{(}v-K_{g}K_{m}\dot{\theta}\Big{)}}{ R_{m}}, \tag{4}\]  
where \(\eta_{g}\) is gearbox efficiency, \(\eta_{m}\) is motor efficiency, \(K_{g}\) is gearbox ratio, \(K_{m}\) is back-emf constant, \(K_{t}\) is motor torque constant, \(R_{m}\) is motor armature resistance, and \(v\) is armature voltage.  
Combining (5) and (4), the system dynamics equation can be written as follows:  
\[\ddot{\theta}(t)=m\dot{\theta}(t)+n\alpha(t)+bv(t)+d(t), \tag{5}\]  
where \(m=-\frac{\eta_{g}\eta_{m}K_{t}K_{g}^{2}K_{m}+R_{m}B_{eq}}{R_{m}J_{eq}}\), \(n=\frac{K_{s}}{J_{eq}},b=\frac{\eta_{g}K_{g}\eta_{m}K_{t}}{R_{m}J_{eq}}\), \(d(t)\) represents system uncertainty and external disturbance.  
**Remark 1**.: _Adaptive ESOs are proposed to estimate the system input gain [22; 23], and these adaptive ESOs are based on pre-known partial model parameter information. In this paper, the rigid-flexible dynamic parameters and input gain in the system dynamics model are unknown, which means that the existing ESO design method provided by [22; 23] is not suitable for the problems proposed here. The primary research goal of this paper is to estimate the rigid-flexible coupling coefficient and input gain of the system by using partial measurement information of the system._  
Figure 2: The schematic diagram of the FLM.  
**Assumption 1:**_The system lumped disturbances \(d(t)\), including unmodeled dynamics and external disturbances, is continuously differentiable and bounded. Therefore, its derivative \(h=\dot{d}(t)\) is bounded, satisfying \(|h|\leq h_{up}\), where\({}_{up}\) is a positive constant._  
**Assumption 2:**_In the FLM system, only the flexible link servo angle \(\theta(t)\) and the link deflection angle \(\alpha(t)\) are measured through the encoder and the strain gauge._  
In practical applications, the exact or prior information of physical parameters \(m,n,b\) and \(d(t)\) cannot be obtained. In order to solve this problem, the parameter \(m,n,b\) is first estimated by an adaptive ESO, which is designed in the following section. Then, according to the estimated system parameters, the system uncertainties and external disturbances are estimated and compensated in real-time using ESO.