# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 3 Controller Design  
### Control Objective  
The objective is to develop a control approach able to perform full control of the spacecraft exclusively using the DMD system. Uncertainties in the spacecraft physical parameters as well as in the environmental parameters are considered. Specifically, the attitude and relative orbit controllers are required to compensate for uncertainties in the atmospheric density \(\rho\), the magnitude of \(\mathbf{V_{r}}\) opposite to the direction of motion \(\mathbf{\Delta\hat{y}}\), the spacecraft inertia matrix \(J\), CoM location \(\mathbf{r_{c}}\), drag \(C_{D,j}\) and lift \(C_{L,j}\) coefficients. Several models for estimating the atmospheric density are available; however, even the more complex ones (e.g., NRLMSISE-00 [26]) have significant level of uncertainty [27], and may be difficult to compute on-board due to the need of solar and geomagnetic indices forecasts. The use of simpler linearly parameterizable models to represent the behavior of this parameter in the short term (i.e., few days) has demonstrated to be sufficient for completing attitude and relative orbit maneuvers and even estimating the time-varying behavior of the atmospheric density online. **Assumption 3**. : _The atmospheric density can be modeled using linearly parameterizable models such as those proposed in [28; 29; 30], which can be generalized in the form_  
\[\rho\triangleq\mathbf{Y_{\rho}}\mathbf{\Theta_{\rho}} \tag{20}\]  
_where the \(\mathbf{Y_{\rho}}^{T}\in\mathbb{R}^{n}\) is a measurable regression vector containing all the known parameters and states, and the vector \(\Theta_{\rho}\in\mathbb{R}^{n}\) contains \(n\) uncertain calibration constants._  
An approximate model to represent the variation of the inertia matrix \(J\) and CoM location \(\mathbf{r_{c}}\) can be obtained by representing the spacecraft as a collection of multiple geometric shapes. The CubeSat body can be represented as a rectangular box, while the deployed and rolled portions of the DMD surfaces can be represented as flat plates and thick walled cylinders, respectively, see Figure 3. However, such approximations introduce uncertainties in \(J\) and \(\mathbf{r_{c}}\), which are critical quantities to determine the applied environmental torques. In this work, the components of \(\mathbf{r_{c}}\) are considered uncertain and the inertia matrix is assumed partially known. **Assumption 4**. : _The spacecraft is capable of computing an approximate \(J_{m}\in\mathbb{R}^{3\times 3}\) of its time-varying inertia matrix, provided an on-board simplified analytical model. The real inertia matrix \(J\) can be expressed as_  
\[J\triangleq J_{m}+\Delta J, \tag{21}\]  
_where \(\Delta J\in\mathbb{R}^{3\times 3}\) is the mismatch between \(J_{m}\) and \(J\), the parameters in \(\Delta J,\Delta J\) and \(J\) are assumed bounded by known constants._  
The differential drag acceleration is the control input considered for the relative maneuver and is defined as  
\[\mathbf{u}=\frac{F_{D}^{t}}{m_{t}}-\frac{F_{D}^{i}}{m_{t}}, \tag{22}\]  
where \(t\), and \(i\) denotes the target and \(i\)th chaser, respectively. From (22), the direction of the differential drag is approximately opposite to the direction of motion \(\mathbf{\Delta\hat{y}}\), but not entirely, due to the rotation of the atmosphere. Therefore, for control design purposes, the differential drag is assumed to act opposite to \(\mathbf{\Delta\hat{y}}\) and the time-varying magnitudes of \(\mathbf{V_{r,t}}\) and \(\mathbf{V_{r,i}}\) are included among the uncertain parameters. The purpose for considering uncertainties in \(\|\mathbf{V_{r,t}}\|\) and \(\|\mathbf{V_{r,i}}\|\) is to account for the deviations of \(\mathbf{V_{r}}\) with respect to \(\mathbf{\Delta\hat{y}}\) due to the rotation of the atmosphere and winds. **Assumption 5**.