# Low-Thrust Nonlinear Orbit Control for Very Low Lunar Orbits
## 3 Nonlinear Orbit Control
### Feedback Control Law  
The operational orbit conditions can be enforced as a nonlinear constraint,  
\[\mathbf{\
abla}(x,t)= \begin{bmatrix}\Psi_{1}\\ \Psi_{2}\\ \Psi_{3}\end{bmatrix} =\mathbf{0}. \tag{30}\]  
Specifically, the components of the constraint can be expressed as  
\[\Psi_{1}=(x_{1}-x_{1d})^{2}=0, \tag{31}\] \[\Psi_{2}=(x_{2}-x_{2d})^{2}+(x_{3}-x_{3d})^{2}=0,\] (32) \[\Psi_{3}=(x_{4}-x_{4d})^{2}+(x_{5}-x_{5d})^{2}=0. \tag{33}\]  
The first component \(\Psi_{1}\) corresponds to a constraint on the semilatus rectum,  
\[\Psi_{1}=(p-p_{d})^{2}. \tag{34}\]  
Instead, the second component \(\Psi_{2}\) pursues the desired eccentricity of the orbit,  
\[(x_{2}-x_{2d})^{2}+(x_{3}-x_{3d})^{2}=\] \[\left[e\cos{(\Omega+\omega)}-e_{d}\cos{(\Omega_{d}+\omega_{d})} \right]^{2}+\left[e\sin{(\Omega+\omega)}-e_{d}\sin{(\Omega_{d}+\omega_{d})} \right]^{2}= \tag{35}\] \[e^{2}+e_{d}^{2}-2 ee_{d}\cos{(\Omega-\Omega_{d}+\omega-\omega_{d })}.\]  
Finally, we can prove that component \(\Psi_{3}\) fixes the orientation of the orbital plane. Indeed, let us denote with \(\tilde{\mathbf{h}}\) and \(\tilde{\mathbf{h}}_{d}\) the actual and desired angular momentum unit vectors as expressed in the MCI-frame,  
\[\tilde{\mathbf{h}}= \begin{bmatrix}\sin{\Omega}\sin{i}\\ \cos{\Omega}\sin{i}\\ \cos{i}\end{bmatrix} ^{T} \begin{bmatrix}\hat{c}_{1}^{M}\\ \hat{c}_{2}^{M}\\ \hat{c}_{3}^{M}\end{bmatrix} = \begin{bmatrix}\frac{2x_{5}}{1+x_{4}^{2}+x_{5}^{ 2}}\\ \frac{2x_{4}}{1+x_{4}^{2}+x_{5}^{2}}\\ \frac{1-x_{4}^{2}-x_{5}^{2}}{1+x_{4}^{2}+x_{5}^{2}}\end{bmatrix} ^{T} \begin{bmatrix} \hat{c}_{1}^{M}\\ \hat{c}_{2}^{M}\\ \hat{c}_{3}^{M}\end{bmatrix} , \tag{36}\] \[\tilde{h}_{d}= \begin{bmatrix}\sin{\Omega}_{d}\sin{i}_{d}\\ \cos{\Omega}_{d}\sin{i}_{d}\\ \cos{i}_{d}\end{bmatrix} ^{T} \begin{bmatrix}\hat{c}_{1}^{M}\\ \hat{c}_{2}^{M}\\ \hat{c}_{3}^{M}\end{bmatrix} = \begin{bmatrix}\frac{2x_{5d}}{1+x_{4d}^{2}+x_{5d}^ {2}}\\ \frac{2x_{4d}}{1+x_{4d}^{2}+x_{5d}^{2}}\\ \frac{1-x_{4d}^{2}-x_{5d}^{2}}{1+x_{4d}^{2}+x_{5d}^{2}}\end{bmatrix} ^{T} \begin{bmatrix} \hat{c}_{1}^{M}\\ \hat{c}_{2}^{M}\\ \hat{c}_{3}^{M}\end{bmatrix} . \tag{37}\]  
Then, the alignment condition between the actual and the desired orbital plane can be expressed as a scalar product,  
\[\tilde{\mathbf{h}}\cdot\tilde{\mathbf{h}}_{d}=1. \tag{38}\]  
By expanding the scalar product, we obtain  
\[2x_{4}x_{4d}+2x_{5}x_{5d}+(1-x_{4}^{2}-x_{5}^{2})(1-x_{4d}^{2}- x_{5d}^{2})=(1+x_{4}^{2}+x_{5}^{2})(1+x_{4d}^{2}+x_{5d}^{2}). \tag{39}\]Equation (39) can be easily simplified into  
\[-2x_{4}x_{4d}-2x_{5}x_{5d}+x_{4}^{2}+x_{5}^{2}+x_{4d}^{2}+x_{5d}^{2}=(x_{4}-x_{4d })^{2}+(x_{5}-x_{5d})^{2}=0. \tag{40}\]  
Equations (31)-(33) define the target set. In order to achieve the desired final conditions, a nonlinear control law is identified through the definition of a meaningful Lyapunov function,  
\[V=\frac{1}{2}\mathbf{\Psi}^{T}\mathbf{K}\mathbf{\Psi}, \tag{41}\]  
where \(\mathbf{K}\) is a symmetric and positive definite matrix. It is straightforward to recognize that \(V>0\) unless \(\mathbf{\Psi}=\mathbf{0}\). Let us introduce the auxiliary vectors  
\[\mathbf{b}=\mathbf{G}^{T}\Big{(}\frac{\partial\mathbf{\Psi}}{ \partial x}\Big{)}^{T}\mathbf{K}\mathbf{\Psi}, \tag{42}\] \[\mathbf{d}=\mathbf{a}_{P}+\Big{(}\frac{\partial\mathbf{\Psi}}{\partial x }\mathbf{G}\Big{)}^{-1}\frac{\partial\mathbf{\Psi}}{\partial t}, \tag{43}\]  
where \(\mathbf{a}_{P}\) denotes the total perturbing acceleration. Then, as already proven in reference [23], the following nonlinear control law leads the dynamical system to converge to the target state, provided that \(\mathbf{\Psi}\) and \(\frac{\partial\mathbf{\Psi}}{\partial x}\) are continuous, \(\Big{[}\big{(}\partial\mathbf{\Psi}/\partial x\ \mathbf{G}\big{)}\Big{]}^{-1} (\partial\mathbf{\Psi}/\partial t)\) is finite, \(|\mathbf{b}|>0\) unless \(\mathbf{\Psi}=\mathbf{0}\), and \(u_{Tmax}>\gamma_{7}|\mathbf{d}|\):  
\[\mathbf{u}_{T}=-u_{T}\frac{\mathbf{b}+\mathbf{d}|}{|\mathbf{b}+\mathbf{d}|},\ \ \ \text{with} \tag{44}\] \[u_{T}=