2025.08.24 Astranis Interview Notes

Questions for Astranis hiring team

1. Do Astranis geo sats depend on GPS for position and velocity? If so how does that work with the GPS satellite lower in alttitude than the Astranis satellite.
2. What are the most exciting projects you are working on right now?
3. To the extent you have a regular day can you describe what that looks like? If your day is irregular can you describe how you spent yesterday?
4. How would you describe the Astranis culture?
5. What has been your favorite and least favorite part about working at Astranis?
6. What have been some unqiue challenges that emerged as a result of working with GEO-stat sats that you did not expect?
7. Do you guys allow your geo sats to drift in a figure 8 ground track? Or do you try to keep inclination pretty close to 0 degrees?

Notes about Geostaionary satellites

1. Cape Canaveral would require you to launch at 27 degrees. And then you would have to reduce the inclination to 0 degrees to be in a true Geo orbit. The delta v required for that inclination change is probably only about 1/2 the delta v required for the raise.

2. Geo sats can service pretty high lattitudes. Single sat can cover 1/3 of the earth.

3. Geo sats can use GPS for timing, and to a lesser extent positioning but weak signal strength and limited satellite visability pose challenges.

Relevant equations

Attitude Notes

1. Importance of attitude?

2. Main reference frames?

3. Describe three primary axes of rotation for a spacecraft and relationship to euler angles or quaternions?

   • ECI: z-axis aligns with Earth's rotation, x-axis points at vernal equinox (ascending node of celestial equator and place of solar orbit)
   • euler angles describe three angles in a specified order to describe rotation in terms of yaw, pitch, roll. Downside is gimbal lock.
   • Quaternions four-parameters (scalar + 3D vector ) avoid singularities and are more computationally efficient.

4. What is a TRIAD algorithm?

   • combines attitude from two vector observation (i.e. sun + earth's magnetic field)

5. What is a control law and how would you design a PD controller for attitude stabalization?

   • control law is an algorithm that computes the torque needed to acheive a desired attitude based on an error or residual between the current and the desired orientation.
   • A PD controller generates torque porportional to the attiude error and the rage of change to stabalize the space craft.

terms

1. Attitude matrix (always for complete rotation of one frame into another frame)
2. Rotation vector (magnitude is the degree of rotation, and axis is the direction of the vector)
3. Nadir refers to pointing directly down.

Rotation

Precision and nutation

• Precession -- occurs when external torque is applied perpendicular to angular momentum.

• Nutation -- occurs when the spinning axis is not aligned with the principle axis. Body nutation rate is the rate which the angular momentum vector rotates around the symetry axis. Can be damped with nutation dampners which are rings with viscous fluid.

Stability

• Spacecraft is stable about greatest or least rotational inertia access. Not stable about middle inertia access which is a saddle point of sorts.

Reaction wheels

• Euler's equations for rigid bodies modified to account for reaction wheels looks like:

$$\mathbf{h} = \mathbf{I} \boldsymbol{\omega} + \sum_{i} \mathbf{A}_i \mathbf{J}_{w,i} \boldsymbol{\omega}_{w,i}$$ where:

• $ \mathbf{I} $: Inertia tensor of the spacecraft (including wheels' contributions to the body inertia).
• $ \boldsymbol{\omega} $: Angular velocity of the spacecraft in the body frame.
• $ \mathbf{A}_i $: Direction cosine matrix or vector for the $ i $-th wheel's spin axis (maps wheel angular momentum to body frame).
• $ \mathbf{J}_{w,i} $: Moment of inertia of the $ i $-th reaction wheel about its spin axis.
• $ \boldsymbol{\omega}_{w,i} $: Angular velocity of the $ i $-th reaction wheel relative to the body.

The time derivative of the angular momentum, combined with Euler's rotational equation, gives: $$\dot{\mathbf{h}} = \mathbf{I} \dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times (\mathbf{I} \boldsymbol{\omega}) + \sum_{i} \mathbf{A}_i \mathbf{J}_{w,i} \dot{\boldsymbol{\omega}}_{w,i} + \boldsymbol{\omega} \times \sum_{i} \mathbf{A}_i \mathbf{J}_{w,i} \boldsymbol{\omega}_{w,i}$$ In the absence of external torques (or assuming they are known), the equation of motion is: $$\mathbf{I} \dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times (\mathbf{I} \boldsymbol{\omega} + \sum_{i} \mathbf{A}_i \mathbf{J}_{w,i} \boldsymbol{\omega}_{w,i}) = -\sum_{i} \mathbf{A}_i \mathbf{u}_{w,i}$$ where:

• $ \mathbf{u}_{w,i} = \mathbf{J}_{w,i} \dot{\boldsymbol{\omega}}_{w,i} $: Control torque applied by the $ i $-th reaction wheel (proportional to the wheel's angular acceleration).
• The term $ \boldsymbol{\omega} \times (\mathbf{I} \boldsymbol{\omega}) $ accounts for gyroscopic coupling due to the spacecraft's rotation.
• $ \boldsymbol{\omega} \times \sum_{i} \mathbf{A}_i \mathbf{J}_{w,i} \boldsymbol{\omega}_{w,i} $: Gyroscopic coupling from the wheels' angular momentum.

External torques

• gravity gradient
• magnetic torque
• aerodynamic torque

attitude determination

Triad method

• Compares two measured vectors from sensors in the body frame to their known counterpart in an inertial or earth-centered frame.

Astranis questions

1. Orbit determination

   • perterbations (J2, third bodies, solar)
   • also further earth spherical haramonic issues
   • drag neglibable in geo

2. J2 helpful for sun-sync but not geo?

   • Helpful for sun sync because at 98 degrees the RAAN precession matches sun, so Raan goes around 360 degrees in 360 days. For geo, so long as you are at 0 degrees inclination raan precession isn't well defined?
   • AHH in geo j2 leads to inclination drift

3. What equations govern relative motion between two nearby sats in Geo?

   • Hill equations

4. Station-keeping strat that controls both longitude drift + inclination in geo sats?

   • east/west control (corrects semi-major axis drift caused by solar/lunar perterbations)
   • north/south drift caused by solar/lunar perterbations

5. Bennefit of inclined GEO orbit?

   • Allowing inclination drift conserves fuel, daily figure 8.

6. Observability: means it is possible to estimate any state from a time history of measurments. Check column space of oberservability matrix.

7. Star tracker dropout? Can use gyroscope to maintain accurcacy for short timescale?

8. EKF instability?

   • Bad values for process/measurment covariance?
   • Bad sensor

9. Initialize with large state covariance matrix.

• TLEs -> convert to ECI position

10. Write a measurement model? What does measurment jacobian look like?

    • Identiy matrix?

11. How would you model the dynamics of a reaction wheel or thurster in a simulation?

    • differential equations (either linear or non-linear)

12. How do you design a quaternion-based attitude controller that avoids singularities and ensures smooth torque output?

    • error quaternion (MEKF)
    • apply porportional-derivative torque law.

13. Gimbal-locked sensor geometry?

    • How would you handle a situation where a sensor actually introduces singularities in certain attitude configuration?
    • Avoid euler angles

14. Attitude hold mode: Design a control mode that maintains nadir-pointing attitude?

    • Use qauternion feedback control.

15. Torque allocation?

    • Use a four wheel design
    • Probably possible to use thrusters to a degree

16. State machine archecture

17. GNC fault tolerant

    • Detect and record from stuck thruster?

18. Time synchonization important in GNC? How would you implement it?

    • Use a control loop. High frequency (50 Hz)

19. Sensor fusion pipeline? Outline data flow for raw IMU / GPS measurments to a fused position-velocity estimate.

    • Use EKF where you could have the state vector include both attitude and pos/vel.

20. Montecarlo to validate navigation filter or control algorithm?

    • Perterb truth and sensors paramaters across many runs.