## 3 Incremental Nonlinear Dynamic Inversion Control Law Design  
### INDI Derivation  
The nonlinear control-affine dynamic system can be expressed as  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{g}(\mathbf{x})\mathbf{u} \tag{5}\]  
where \(\mathbf{x}\in\mathbf{R}^{n}\) is the state vector, \(\mathbf{u}\in\mathbf{R}^{m}\) is the control input vector, \(\mathbf{f}\) is the mapping function \(\mathbf{R}^{n}\to\mathbf{R}^{n}\), and \(\mathbf{g}\) is the mapping function \(\mathbf{R}^{n}\to\mathbf{R}^{n\times m}\), whose columns are smooth and nonsingular vector fields [13]. In practice, \(\mathbf{f}\) is a nonlinear dynamic function related mainly to the state variables, and \(\mathbf{g}\) is the control distribution function. It is assumed that \(m=n\) and the output vector \(\mathbf{y}=\mathbf{x}\). A first-order Taylor series expansion for the system in Equation (5) in the neighborhood of the point \([x_{0},\mathbf{u}_{0}]\) can be expressed as  
\[\dot{\mathbf{x}}\cong\dot{\mathbf{x}}_{0}+\mathbf{F}_{0}(\mathbf{x}-\mathbf{x}_{0})+\mathbf{G}_{0}(\bm {u}-\mathbf{u}_{0})+\mathbf{o}\left(\Delta\mathbf{x}^{2}\right) \tag{6}\]  
where  
\[ \begin{array}{l}\mathbf{F}_{0}=\left.\frac{\partial}{\partial\mathbf{x}}\left[\mathbf{f }(\mathbf{x})+\mathbf{g}(\mathbf{x})\mathbf{u}\right]\right|_{\mathbf{x}=x_{0},\mathbf{u}=\mathbf{u}_{0}} \\ \mathbf{G}_{0}=\left.\frac{\partial}{\partial\mathbf{u}}\left[\mathbf{f}(\mathbf{x})+\mathbf{g}( \mathbf{x})\mathbf{u}\right]\right|_{\mathbf{x}=x_{0},\mathbf{u}=\mathbf{u}_{0}}\end{array}  \tag{7}\]  
where \(x_{0}\) and \(\mathbf{u}_{0}\) are the current state and control variables, respectively. \(\mathbf{F}_{0}\) and \(\mathbf{G}_{0}\), indicated by the subscript '\(0\)', are the current state matrix and control matrix, respectively; \(\mathbf{o}\left(\Delta\mathbf{x}^{2}\right)\) denotes higher-order terms (HOT). Note that the control matrix \(\mathbf{G}_{0}\) needs to be inverted. The sampling rate is assumed small enough that the state variable \(\mathbf{x}\) changes significantly more slowly than the control variable \(\mathbf{u}\). Thus, \(\mathbf{x}\approx\mathbf{x}_{0}\) can be assumed to be comparable to the increment of \(\mathbf{u}\) in one sampling time step. Without considering the higher-order part and the increment of the state, Equation (6) can be simplified as  
\[\dot{\mathbf{x}}\cong\dot{\mathbf{x}}_{0}+\mathbf{G}_{0}\Delta\mathbf{u} \tag{8}\]  
where \(\Delta\mathbf{u}=\mathbf{u}-\mathbf{u}_{0}\), which represents the control increment in one sampling time step \(\Delta t\). For the inner-loop angular rate control, \(\dot{\mathbf{x}}_{0}\) can be estimated using a virtual angular accelerometer [16], calculated from an onboard model (OBM) with the derivative of the angular rates [22] or measured directly by an angular accelerometer [23]. In addition, the input-output relative control degree of the inner-loop angular rate control is equal to one, which means there are no internal dynamics [24]. The model information \(\mathbf{f}\) in Equation (8) is replaced with sensor measurements; however, in a regular explicit nonlinear dynamic inversion (NDI) controller, the nonlinear feedback term \(\mathbf{f}\) is needed [25]. Thus, the INDI control approach is less sensitive to model mismatch [13] and improves the robustness of NDI to model uncertainties [26].