# Online Adaptive Parameter Estimation for Quadrotors
## 2 Adaptive Parameter Estimation  
In this section, an adaptive parameter estimation technology is proposed for quadrotors. The schematic diagram of a ground coordinate system and a body axes coordinate system is given in Figure 1. Consider the following typical quadrotor mathematical model [20]  
\[ \begin{split}\ddot{x}&=(\cos\psi\sin\theta\cos \phi+\sin\psi\sin\phi)U_{1}/m\\ \ddot{y}&=(\sin\psi\sin\theta\cos\phi+\sin\phi\sin \psi)U_{1}/m\\ \ddot{z}&=(\cos\theta\cos\phi)U_{1}/m-g\\ \ddot{\phi}&=[U_{2}+\dot{\theta}\dot{\psi}(J_{z}-J _{x})]/J_{y}\\ \ddot{\theta}&=[U_{3}+\dot{\phi}\dot{\psi}(J_{y}-J _{z})]/J_{x}\\ \ddot{\psi}&=[U_{4}+\dot{\phi}\dot{\theta}(J_{x}-J _{y})]/J_{z}\end{split}  \tag{1}\]  
where \(x\) is the displacement of the \(x\)-axis direction, \(y\) is the displacement of the \(y\)-axis direction, \(z\) is the displacement of the \(z\)-axis direction, \(\psi\) denotes the yaw angle, \(\phi\) means the rolling angle, and \(\theta\) defines the pitching angle. \(U_{1},U_{2},U_{3}\) and \(U_{4}\) denote the control actions of the vertical direction, roll direction, pitch direction, and the yaw direction, respectively, \(m\) denotes the total mass of the quadrotor, \(J\) is the moment inertia of the quadrotor, and \(g\) denotes the gravitational acceleration. In order to facilitate the parameter estimation, Equation (1) can be rewritten as  
\[ \begin{split}\ddot{x}&=a_{1}(\cos\psi\sin\theta\cos\phi +\sin\psi\sin\phi)U_{1}\\ \ddot{y}&=a_{1}(\sin\psi\sin\theta\cos\phi+\sin\phi \sin\psi)U_{1}\\ \ddot{z}&=a_{1}(\cos\theta\cos\phi)U_{1}-g\\ \ddot{\phi}&=a_{2}U_{2}+a_{3}\dot{\theta}\dot{\psi} \\ \ddot{\theta}&=a_{4}U_{3}+a_{5}\dot{\phi}\dot{\psi} \\ \ddot{\psi}&=a_{6}U_{4}+a_{7}\dot{\varphi}\dot{\theta} \end{split}  \tag{2}\]  
where \(a_{1}=1/m\), \(a_{2}=1/J_{y}\), \(a_{3}=(J_{2}-J_{x})/J_{y}\), \(a_{4}=1/J_{x}\), \(a_{5}=(J_{y}-J_{z})/J_{x}\), \(a_{6}=1/J_{z}\), and \(a_{7}=(J_{x}-J_{y})/J_{z}\). \(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\) and \(a_{7}\) are the unknown parameters to be estimated. The aim of this paper is to suggest a novel parameter estimation methodology and a novel adaptive law for estimating \(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\) and \(a_{7}\) of system (2). In this section, we will introduce a new adaptive parameter error framework, which is driven by the estimation error as [11; 12; 13]. In order to complete the parameter estimation, the filtered variables \(\dot{x}_{f},\dot{\phi}_{f},\dot{\theta}_{f},\dot{\psi}_{f}\) of \(\dot{x},\dot{\phi},\dot{\theta}\) and \(\dot{\psi}\) can be defined as  
\[\left\{ \begin{array}{l}k\ddot{x}_{f}+\dot{x}_{f}=\dot{x},\dot{x}_{f}(0)=0\\ k\ddot{\psi}_{f}+\dot{\phi}_{f}=\dot{\phi},\dot{\phi}_{f}(0)=0\\ k\ddot{\theta}_{f}+\dot{\theta}_{f}=\dot{\theta},\dot{\theta}_{f}(0)=0\\ k\ddot{\psi}_{f}+\dot{\psi}_{f}=\dot{\psi},\dot{\psi}_{f}(0)=0\end{array} \right. \tag{3}\]  
where \(k>0\) is the filtered parameter. Then, we can obtain the following equation from (2) and (3)  
\[\left\{ \begin{array}{l}\ddot{x}_{f}=\frac{\dot{x}-\dot{x}_{f}}{k}=a_{1}\cdot \mathrm{F}_{1f}\cdot U_{1}\\ \ddot{\phi}_{f}=\frac{\dot{\phi}-\dot{\phi}_{f}}{k}=a_{2}\cdot U_{2}+a_{3}F_{2 f}\\ \ddot{\theta}_{f}=\frac{\dot{\theta}-\dot{\theta}_{f}}{k}=a_{4}\cdot U_{3}+a_{ 5}\cdot F_{3f}\\ \ddot{\psi}_{f}=\frac{\dot{\psi}-\dot{\psi}_{f}}{k}=a_{6}\cdot U_{4}+a_{7} \cdot F_{4f}\end{array} \right. \tag{4}\]  
Figure 1: The schematic diagram of the ground coordinate system and the body axes coordinate system. where \(F_{1}=\cos\psi\sin\theta\cos\phi+\sin\psi\sin\phi\), \(F_{2}=\dot{\theta}\dot{\psi}\), \(F_{3}=\dot{\phi}\dot{\psi}\), and \(F_{4}=\dot{\phi}\dot{\theta}\). \(F_{1}\), \(F_{2}\), \(F_{3}\), and \(F_{4}\) are the defined functions, and \(F_{1f}\), \(F_{2f}\), \(F_{3f}\), and \(F_{4f}\) are the filtered variables of \(F_{1}\), \(F_{2}\), \(F_{3}\), and \(F_{4}\), respectively. In order to complete the parameters estimation, Equation (4) can be rewritten in a more compact form as  
\[X=YZ \tag{5}\]  
where  
\[X=\left[