## 2 Dynamic Model of Stewart-Gough Flight Simulation Platform  
In this paper, the dynamic model of Stewart-Gough flight simulation platform is built based on the virtual working principle. The experimental platform is shown in Figure 2. The virtual work equation, according to the authors of [27], is:  
\[\delta q^{T}\tau+\delta x_{p}^{T}\hat{F}_{p}+\sum_{i}^{6}\left(\delta x_{i_{1} }^{T}\hat{F}_{i_{1}}+\delta x_{i_{2}}^{T}\hat{F}_{i_{2}}\right)=0, \tag{1}\]  
where \(\delta q=[\delta q_{1}\dots\delta q_{6}]^{T}\) denotes the virtual displacement vector of actuated joints, \(\tau=[d\tau_{1}\dots\,d\tau_{6}]^{T}\) denotes the force and torque vector acting on the actuated joints, and \(\delta x_{p}=\left[\delta x_{p},\delta y_{p},\delta z_{p},\delta\theta_{x_{p} },\delta\theta_{y_{p}},\delta\theta_{z_{p}}\right]^{T}\) denotes the virtual variables of a contacting point of the moving platform. Furthermore, \(\delta x_{i_{1}}=\left[\delta x_{i_{1}},\delta y_{i_{1}},\delta z_{i_{1}}, \delta\theta_{x_{i_{1}}},\delta\theta_{y_{i_{1}}},\delta\theta_{z_{i_{1}}} \right]^{T}\) represents the virtual variables vector of the upper legs center of gravity (c.o.g.). Similarly, \(\delta x_{i_{2}}\) denotes the virtual displacement vector c.o.g. of down legs, \(\hat{F}_{p}\) denotes the inertial forces at the c.o.g. of the moving platform, and \(\hat{F}_{i_{1}}\) and \(\hat{F}_{i_{2}}\) are the inertial forces at the c.o.g. of up legs and at the c.o.g. of the down legs, respectively, which are defined from the relationship, according the authors of [27], as:  
\[\hat{F}_{p}=\left[ \begin{array}{c}f_{d}+m_{p}g-m_{p}\ddot{x}_{p}\\ n_{d}-I_{p}\dot{\omega}_{p}-\omega_{p}\times I_{p}\omega_{p}\end{array} \right]= \tau_{d,p}-M_{p}\ddot{x}_{p}-C_{p}\dot{x}_{p}-G_{p}, \tag{2}\]  
Figure 1: Structure diagram of the \(\mathcal{L}_{1}\) adaptive flight simulation control system. \[\hat{F}_{i_{1}} =\left[ \begin{array}{c}m_{i_{1}}g-m_{i_{1}}\ddot{x}_{i_{1}}\\ -I_{i_{1}}\dot{\omega}_{i_{1}}-\omega_{i_{1}}\times I_{i_{1}}\omega_{i_{1}}\\ \end{array} \right]=-M_{i_{1}}\ddot{x}_{i_{1}}-C_{i_{1}}\dot{x}_{i_{1}}-G_{i_{1}}, \tag{3}\] \[\hat{F}_{i_{2}} =\left[ \begin{array}{c}m_{i_{2}}g-m_{i_{2}}\ddot{x}_{i_{2}}\\ -I_{i_{2}}\dot{\omega}_{i_{2}}-\omega_{i_{2}}\times I_{i_{2}}\omega_{i_{2}}\\ \end{array} \right]=-M_{i_{2}}\ddot{x}_{i_{2}}-C_{i_{2}}\dot{x}_{i_{2}}-G_{i_{2}}, \tag{4}\]  
where \(M_{p}\), \(M_{i_{1}}\), and \(M_{i_{2}}\) denote the mass matrix of the moving platform, of the up legs, and of the down legs given in Equation (5); and \(C_{p}\), \(C_{i_{1}}\), and \(C_{i_{2}}\) denote the Coriolis and centrifugal matrix of the moving platform, of the up legs, and of the down legs given in Equation (6). Furthermore, \(G_{p}\), \(G_{i_{1}}\), and \(G_{i_{2}}\) denote the gravity vector of the moving platform, of the up legs, and of the down legs given in Equation (7). \[M_{p}=\left[ \begin{array}{cc}m_{p}1&0\\ 0&I_{p}\\ \end{array} \right]M_{i_{1}}=\left[ \begin{array}{cc}m_{i_{1}}1&0\\ 0&I_{i_{1}}\\ \end{array} \right]M_{i_{2}}=\left[ \begin{array}{cc}m_{i_{2}}1&0\\ 0&I_{i_{2}}\\ \end{array} \right], \tag{5}\]  
\[C_{i_{1}}=\left[ \begin{array}{cc}0&0\\ 0&\omega_{i_{1}}\times I_{i_{1}}\\ \end{array} \right]C_{i_{2}}=\left[ \begin{array}{cc}0&0\\ 0&\omega_{i_{1}}\times I_{i_{1}}\\ \end{array} \right]C_{p}=\left[ \begin{array}{cc}0&0\\ 0&\omega_{p}\times I_{p}\\ \end{array} \right], \tag{6}\]  
\[G_{p}=\left[ \begin{array}{c}-m_{p}g\\ 0\\ \end{array} \right]G_{i_{1}}=\left[ \begin{array}{c}-m_{i_{1}}g\\ 0\\ \end{array} \right]G_{i_{2}}=\left[ \begin{array}{c}-m_{i_{2}}g\\ 0\\ \end{array} \right], \tag{7}\]  
The virtual displacement \(\delta q\) of the actuated joints and the virtual displacement of the center of gravity \(\delta x_{i_{1}}\) and \(\delta x_{i_{2}}\) of the legs, however, must be compatible with the kinematic constraints imposed by the virtual displacement of the moving platform.