# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 3 Controller Design
### Control Development
The magnitude of the differential drag acting along \(\mathbf{\Delta\hat{y}}\) (i.e., \(u_{y}\)), can be expressed as  
\[u_{y}\triangleq\rho_{i}C_{D}^{i}\|\mathbf{V_{r}}_{i}\|^{2}\frac{S_{i}}{2m_{i}}-\rho_ {t}\|\mathbf{V_{r,t}}\|^{2}\frac{C_{D}^{t}S_{t}}{2m_{t}}, \tag{29}\]  
where \(i\) denotes the \(i\)th chaser. The expression in (29) can be linearly parameterized as  
\[u_{y}=Y_{2}\mathbf{\Theta_{2}}, \tag{30}\]  
with the regression vector \(\mathbf{Y_{2}}^{T}\in\mathbb{R}^{6}\) and the vector of uncertain parameters \(\mathbf{\Theta_{2}}\in\mathbb{R}^{6}\) defined as  
\[\mathbf{Y_{2}} \triangleq\bigg{[}S_{i}\;\;S_{i}\sin(\Omega t)\;\;S_{i}\cos( \Omega t)\;\;-\frac{S_{t}}{2m_{t}}\;-\frac{S_{t}}{2m_{t}}\sin(\Omega t)\;\;- \frac{S_{t}}{2m^{t}}\cos(\Omega t)\bigg{]}, \tag{31}\] \[\mathbf{\Theta_{2}} \triangleq \begin{bmatrix}D_{1,t}C_{D}^{i}\|V_{r,t}\|^{2}\\ D_{2,t}C_{D}^{i}\|V_{r,t}\|^{2}\\ D_{3,t}C_{D}^{i}\|V_{r,t}\|^{2}\\ D_{2,t}C_{D}^{i}\|V_{r,t}\|^{2}\\ D_{3,t}C_{D}^{i}\|V_{r,t}\|^{2}\end{bmatrix} . \tag{32}\]  
In (32), the parameters \(D_{1,k},D_{2,k},D_{3,k}\in\mathbb{R}\) with \(k=i,t\), are uncertain constants used under Assumption 3 to calibrate the atmospheric density at the position of the \(i\)th chaser and target, respectively, see [29] for details. The control and adaptive update laws, assuming that the target broadcasts its own area to mass ratio \(S_{t}/m_{t}\) to all chasers, are proposed as  
\[S_{d,i} \triangleq 2m_{i}\Bigg{[}\bigg{(}\hat{\rho_{i}}\widehat{C_{D}^{i}} \widehat{\|V_{r,t}\|^{2}}\bigg{)}^{-1}\bigg{(}\frac{\widehat{C_{D}^{t}}S_{t}} {2m_{t}}\hat{\rho_{t}}\widehat{\|V_{r,t}\|^{2}}-\mathbf{K_{LQR}}\mathbf{X}\bigg{)} \Bigg{]}, \tag{33}\] \[\hat{\mathbf{\Theta_{2}}} \triangleq \text{proj}\Big{(}2\Gamma_{2}\mathbf{Y_{2}}^{T}\mathbf{B}^{T}\mathbf{X}\Big{)}, \tag{34}\]  
where \(\widehat{(\cdot)}\) represents the estimate of the uncertain parameter \((\cdot)\), \(\mathbf{K_{LQR}^{T}}\in\mathbb{R}^{4}\) is a constant control gain vector, \(P\in\mathbb{R}^{4\times 4}\) is a symmetric positive-definite matrix, \(\Gamma_{2}\in\mathbb{R}^{6\times 6}\) is a symmetric positive-definite adaptation gain, \(\mathbf{B}\triangleq[0\;0\;0\;1]^{T}\in\mathbb{R}^{4}\) and \(C_{D}^{k}\triangleq\sum_{j=1}^{4}C_{D,j}\), with \(\;k=t,i\). To integrate the attitude and relative orbit controllers, the minimization problem in (27) is modified to account for relative maneuvering requirements by introducing the additional constraint of a required total cross-sectional area, which is the parameter that can be varied to modify the differential drag. The minimization problem is modified for each spacecraft as  
\[\min_{\{L_{1},L_{2},L_{3},L_{4}\}}\Big{\{}W_{att}\|\mathbf{\bar{u}}- \mathbf{\bar{u}_{d}}\|+W_{orb}(S-S_{d})^{2}\Big{\}} \tag{35}\] \[\text{subject to }\;\Big{\{}\;0\leq L_{j}\leq 3.7m\;\;j=1,2,3,4,\]  
where \(S\in\mathbb{R}_{>0}\) is the total cross-sectional area of the spacecraft, and \(\;S_{d}\in\mathbb{R}_{>0}\) is the desired cross-sectional area. Note that the parameters \(\mathbf{\bar{u}}\) and \(S\) depend on the spacecraft attitude and orbital states as well as the DMD lengths, while \(\mathbf{\bar{u}_{d}}\) and \(S_{d}\) are computed from the corresponding adaptive control laws. By using (35), a set of DMD lengths can be computed accounting for both control requirements; however, residual errors due to the use of a numerical algorithm must be considered in the stability analysis. In addition to the residual error, the drag coefficients of the DMD surfaces may change over time due to the spacecraft changing orientation. **Assumption 6**. : _The time derivative of the vector of uncertain parameters (i.e., \(\dot{\mathbf{\Theta_{2}}}\)), can be bounded by a known constant._