# Drag and Attitude Control for the Next Generation Gravity Mission
## 3 Drag and Attitude Control Role in the Measurement of the Non-Gravitational Accelerations
### The Measurement of the Non-Gravitational Accelerations and the Main Error Terms
Taking into account these terms, the acceleration actually measured by the accelerometer (\(\mathbf{a}_{\rm meas}\)) can be expressed as a function of the \"true\" acceleration \(\mathbf{a}\) as follows (see also illustration in Figure 7):  
\[\mathbf{a}_{\rm meas}=\mathbf{a}+[\rm dM]\mathbf{a}+[\rm K2]\mathbf{a}^{2}+[\rm C]\hat{\mathbf{ \Omega}}+\mathbf{b}+\mathbf{n}=\mathbf{a}+\delta\mathbf{a}\]  
Notably, the error \(\delta\mathbf{a}\) on the measured acceleration is a function, besides other factors, of the acceleration itself through the coupling factors [dM] and [K2]. Therefore, the reduction of the error \(\delta\mathbf{a}\) implies the reduction of the acceleration \(\mathbf{a}\) (and in particular of the drag acceleration \(\mathbf{a}_{\rm D}\)) through the implementation of a \"drag-free control\". Additionally, the misalignment between the accelerometer reference frame and the satellite-to-satellite reference frame (SSRF) will also induce a projection error in the measured acceleration along the direction joining the CoMs of the two satellites (Figure 8). \[\mathbf{a}_{\rm D}=\mathbf{a}_{\rm meas,X}+\mathbf{\theta}_{Z}\;\mathbf{a}_{\rm meas,Y}+\mathbf{ \theta}_{\rm Y}\;\mathbf{a}_{\rm meas,Z}\]  
\[\rightarrow\delta\mathbf{a}_{\rm D}=\delta\mathbf{a}_{\rm meas,X}+\mathbf{\theta}_{Z}\; \delta\mathbf{a}_{\rm meas,Y}+\delta\mathbf{\theta}_{Z}\;\mathbf{a}_{\rm meas,Y}+\mathbf{ \theta}_{\rm Y}\;\delta\mathbf{a}_{\rm meas,Z}+\delta\mathbf{\theta}_{\rm Y}\;\mathbf{a}_ {\rm meas,Z}\]  
Figure 8: Accelerometer reference frame (\(\rm X_{A}\), \(\rm Y_{A}\), \(\rm Z_{A}\)) to SSRF (\(\rm X_{SS}\), \(\rm Y_{SS}\), \(\rm Z_{SS}\)) misalignment giving rise to the pitch/yaw-dependent error term \(\delta\mathbf{a}_{\rm D}\). Figure 7: Illustration of the errors affecting \(\mathbf{a}_{\rm meas}\). The reduction of the error \(\delta a_{\text{D}}\) implies a fine pointing and pointing stability of the accelerometer X-axis along the CoM-to-CoM direction, i.e., the tight control of the pitch and yaw angles \(\theta_{\text{Y}}\), \(\theta_{\text{Z}}\). From the above discussion, the errors affecting the measurement of the non-gravitational differential acceleration between the CoMs of the satellites (\(\Delta a_{\text{D}}\)) can be classified in the following classes (illustrated in the error tree of Figure 9):  
* Accelerometer Intrinsic Errors (I): sensor intrinsic noise and bias, parasitic forces on the PM originated internally to the accelerometer (gold-wire stiffness, thermo-molecular forces, etc.). * Accelerometer-Satellite Coupling Errors (C): errors originated from the interactions of the accelerometer scale factor, internal misalignments, quadratic factors with residual non-gravitational accelerations, and the attitude dynamics of the spacecraft. * Satellite Generated Errors (S): errors produced by sources dependent only on the spacecraft (self-gravity forces, stability of the accelerometer-CoM relative position, etc.). * Transformation Errors (T): errors originated by the projection of the measured acceleration along the CoM-to-CoM direction.