# How to calculate thruster force calculation during maneuvers

I'm working in a satellite operator company, operating a GEO cubic shaped satellite, with 16 chemical thrusters and 3 reaction wheel for attitude control maneuvering. We have vendor provided flight dynamics software. How the flight dynamics software calculates the required delta-V and force/pulse needed in each thruster for station keeping? And how does it calculate wheel unloading situations?

• What the research you did yourself? If you add it to the question (edit), it can help. – Heopps Jul 8 '18 at 11:31
• Surely the vendor can give more specific info than Random Internet People. – Organic Marble Jul 8 '18 at 14:16
• Well. The vendor is not helping, saying the algorithm is closed source – Bruno Jul 8 '18 at 14:32
• Actually my plan is to learn basic equations at first and compare the result with the software's output. Afterwards try to reverse engineer step by step. So, any good source of regarding thruster force/pulse calculation is appreciated. – Bruno Jul 8 '18 at 15:52
• Sounds like a good plan. I hope someone knowledgable answers. – Organic Marble Jul 8 '18 at 17:55

How does the flight dynamics software calculate the required delta-V and force/pulse needed in each thruster for station keeping?

To compute d-V's for station keeping, the deviations of the orbital elements from nominal must be known. For a GEO satellite, there are two primary perturbations: an inclination perturbation caused by the earth and moon of about 0.85 degrees per year, and an eccentricity perturbation caused by the ellipticity of the earth's polar symmetry.

For a circular orbit, the d-V required for an inclination change is given by the following equation:

$\Delta v_i = 2v*sin(\frac{\Delta i}{2})$

where $v$ is the orbital velocity, and $\Delta i$ is the inclination change. The velocity must be imparted in the normal direction at the descending node or in the anti-normal direction at the ascending node. The velocity at geostationary orbit is approximately 3.07 km/s, which gives a d-V of about 46 m/s. This could be a single burn performed annually (e.g. GOES), or several smaller burns over the course of the year. The location of the line of nodes can be computed using the specific relative angular momentum vector.

Adjusting for the eccentricity can be approximated by assuming a simple Hohmann style maneuver to raise or lower the orbit extrema back to circular. Correcting this at GEO requires a small 2 m/s per year. That being said, a more complex burn is typically used. For a detailed description of tangential, radial, and orthoganal d-V components required, see this PDF.

De-saturating a reaction wheel requires thrusters to impart a torque to keep the s/c from spinning as the wheel is spun-down. The torque imparted on the s/c by a reaction wheel is defined as:

$T = I_w * \alpha_w$

where $I_w$ is the wheels moment of inertia, and $\alpha_w$ is the rate at which the wheel is decelerated. The thrusters must maintain this torque throughout the de-saturation. With a non-symmetric bank of thrusters, as seen on the infamous Mars Climate Orbiter, performing these maneuvers will impart a d-V and change the satellites trajectory. Assuming that the s/c in question has thrusters facing opposite directions, this can be avoided. The torque imparted by a thruster is given as:

$T = F_{thr}*d$

where $F_{thr}$ is the force provided by the thruster, and $d$ is the orthogonal distance, or "moment arm", of the thruster from the axis that is parallel with the wheel spin axis and intersects the s/c center of mass. As touched on before, this torque can be provided by a single thruster or a sum of smaller torques provided by multiple thrusters, but in both cases, the total torque must equal the the torque exerted by the wheel.

• The RAAN drift will also be significant if I'm not mistaken. IIRC, the RAAN drift decreases as your inclination approaches 90 degrees, so near zero degrees of inclination is where you RAAN drifts a lot. That said, if you are in a near-circular orbit, it does not "seem" to matter much from the ground. – ChrisR Jul 10 '18 at 2:40
• @AMcKelvy, thanks for the equation of delta-V. How can I calculate needed force/impulse in each thruster, which are at different corners of the body? I'm not very fluent with 3-dimensional vector mechanics. – Bruno Jul 10 '18 at 15:17
• @Bruno, The needed force for station-keeping is going to be dependent on the mass of the spacecraft. I'm not familiar with your system, but I would assume that the s/c has the ability to impart d-V's in a given direction without needing to use multiple non-parallel thrustsers. As far as the de-sat maneuver goes, the complexity of the problem is highly dependent on the configuration of the wheels and the thrusters. I think it's a bit too broad to answer how to generally compute the needed force. – A McKelvy Jul 10 '18 at 15:31
• @ChrisR, I was mistaken; GEO does not imply GSO. So your point is entirely valid. I will delete my previous comment. – A McKelvy Jul 10 '18 at 15:32
• @AMcKelvy, OK. Let me narrow down the problem a bit. How to calculate force needed in different thrusters, for various configurations? – Bruno Jul 10 '18 at 15:54