13:53:20 UTC] = [CA −2′, CA + 1′] when the unexpected acceleration spike, described in the previous paragraph, was detected. The misalignment is visible on the three axes, but is clearly highlighted on the MCS + X axis. This circumstance let the ISA team argue that exactly at the time of the spike in acceleration detected by ISA, BepiColombo AOCS compensated a torque to keep the spacecraft at the commanded attitude, mainly around the S/C+ˆxsc axis:

[Figure: Figure 8: ISA measured calibrated accelerations during the second Venus flyby.]

direction fully compatible with a force that pushed the spacecraft mainly along +ˆysc direction.
AOCS is caused by the acceleration detected by ISA, the angle between the force direction and the torque direction must be as close as possible to 90 degrees. Indeed, the relationship between these quantities can be written as:
Tdis = mS/CrA × as
(4)
ISA
where Tdis = [Tx,dis, Ty,dis, Tz,dis] is the difference between the total torque exerted on the spacecraft by the reaction wheel TRW and the expected torque due to the gravity gradients TGG expressed in the body fixed reference frame. The quantity rA = [xA, yA, zA] is the vector from the S/C COM to the force application point expressed in the body fixed reference frame and as ISA = [as x,ISA, as y,ISA, as
z,ISA] is the spike acceleration measured by ISA computed as the difference between the solid and the dotted lines of figure 8. The misplacement from the orthogonality between Tdis and as ISA can be computed as the quantity β:

[Figure: Figure 9: Comparison between reaction wheel executed torque and computed signal due to gravity gradient disturbance torque.]

β = arcsin( Tdis · as
ISA
|Tdis||as ISA|)
If the disturbance torque detected by the .

From the figure 10 it can be observed that when the disturbance torque starts to act on the spacecraft, the β angle reaches the expected value of 0 deg, proving that the AOCS and ISA observations are related to the same physical phenomenon.
with the MCS body model in which markers color are based on the β value, the greener are the dots, the closer is the angle to zero. Despite the estimation solution is not constrained in any direction, it can be noted that green points are all placed in nearby of MPO radiator: the average of the estimated application point, in the interval at low beta angle, is:
[0.0235 −2.1841 −1.7233][m] in the MPO reference frame and the X and Z coordinates variation is about few centimeters, when the modulus of β angle is under the 10 degrees. However, Y coordinates shows bigger oscillation in time, moving the application point also outside the MPO body.

[Figure: Figure 10: Angle between the measured disturbance torque and the measured ISA acceleration.]

The fact that two independent sensors measured the same event gave ISA team the opportunity to use the equation 4, linking the two quantities, to resolve for the unknown force application point vector rA in the period in which β angle is close to 0 degrees. ISA team used a non-linear least-square estimation method to minimise the function:
(a) (b)
(c) (d)
N
N
�
�
e2 i =
Tdis|C i −Tdis|i �2 , (5)
�

[Figure: Figure 11: Estimated force actuation points in the MPO reference frame. Dots colors are related to the colormap showing the |β| angle.]

ε =
i=1
i=1
where the i −th residual ei at at time ti is calculated as the difference between Tdis|C i , computed following the equation 4, and Tdis|i effectively measured from AOCS at time ti but cleaned from the Gravity Gradient effects. The estimation is performed every N acceleration samples to follow the evolution of the application point rA during the time. In figures 11 are depicted the reconstructed application point as evolving in time, compared
The reason of that is clearly related to the geometry of the reconstructed force direction: when the absolute value of β angle is under 10 degrees, the force is almost directed towards the MPO +Y , keeping the equation 5 bad conditioned for that coordinate. Using the Jacobian matrix, we evaluated from the numerical solution of the non linear least square fit for the equation 5, the standard deviation respect to the converged solution for the three component of the vector.