Reading a Spacecraft Rate Budget When Half of It Is in Radians
An attitude determination and control subsystem produces body rates in radians per second, because that is what the estimator, the wheel controller and the simulation all run on. The requirements around it rarely agree. A launch provider states the maximum tip-off rate imparted at separation in degrees per second, a payload states its stability requirement in arcseconds per second, and an operations team asking whether the vehicle has finished detumbling wants a figure it can picture.
Why One Subsystem Carries Both Units
Flight Software Integrates, Requirements Communicate
Body Rate and Wheel Speed Are Different Numbers
Detumbling Is Judged on Rate, Pointing on Angle
Jitter Sits Three Decimal Places Below a Slew
Checking a Rate Before It Goes Into the Budget
Four passes that turn a raw telemetry figure into something a review board can argue about.
Enter the rate exactly as the subsystem reports it
Take the body rate per axis rather than a magnitude when the requirement is written per axis. The field opens at 1 and recomputes on every keystroke, so working down three axes of a telemetry snapshot is quick.
Compare against the separation specification
Tip-off limits are quoted in degrees per second by the deployer, so a measured rate has to be restated before the comparison means anything. The same applies to the detumble exit threshold the controller checks internally.
Sanity-check a slew against its peak rate
For an accelerate-then-decelerate profile the peak rate is half the total angle divided by half the duration. Converting that peak shows whether the wheels can supply it, or whether the manoeuvre needs a longer, gentler ramp.
Reverse it when a limit has to reach the flight code
A threshold agreed in degrees must become radians before it is compiled in. The swap arrows reverse the pair, typing into the second field does the same since both are editable, and copying returns digits with no unit attached.
From a Post-Separation Tumble Down to Imaging Stability
These rates span five orders of magnitude within one mission. The values are representative of a small Earth-observation satellite in low orbit, and each conversion uses the exact factor rather than a rounded 57.3.
| Mission phase or requirement | Body rate (rad/s) | Degrees per second | What it means in practice |
|---|---|---|---|
| Worst-case tip-off from a deployer | 0.175 | 10.027 °/s | A full turn every 36 seconds; sun and star sensors both struggle |
| Typical rate shortly after separation | 0.05 | 2.865 °/s | Solar arrays flicker in and out of sunlight several times a minute |
| Detumble exit threshold | 0.0175 | 1.003 °/s | Slow enough for a star tracker to lock and acquire attitude |
| Peak rate during a 30° slew | 0.0302 | 1.730 °/s | Reached mid-manoeuvre at 0.1 °/s², with the slew taking 34.6 s |
| Nadir tracking in a 90-minute orbit | 0.0011636 | 0.06667 °/s | The steady pitch rate that keeps the camera facing down |
| Coarse pointing residual | 0.001 | 0.05730 °/s | Roughly 206 arcseconds of drift per second of exposure |
| Stability during a science exposure | 0.0001 | 0.005730 °/s | About 20.6 arcseconds per second, the level fine sensors chase |
The nadir-tracking row is worth dwelling on, because it is a rate the vehicle must hold continuously rather than one it passes through. Circling once every 90 minutes means pitching at exactly one revolution per orbit to keep a camera on the ground beneath, and holding a constant rate steadily is a different control problem from settling on a fixed attitude.
Telemetry and Interface Documents on One Scale
A measured body rate and a contractual tip-off limit become directly comparable rather than sitting in two notations nobody wants to reconcile by hand.
Stability Terms Kept Beside Slew Peaks
Values a thousandfold apart run through the same step, so exposure stability and a manoeuvre peak share one budget without a lost decimal.
Thresholds Copied Into a Parameter Table
Copying gives the number with no unit or spacing, which is what a controller configuration file or an operations checklist expects.
Questions from the Attitude-Control Desk
Why do attitude-control documents mix radians and degrees at all?
Because the two halves of the subsystem answer to different audiences. Everything computational lives in radians: quaternion kinematics, inertia-matrix products, gain calculations and the simulation verifying them. Everything contractual lives in degrees, because a provider describing tip-off, a payload team describing smear and a reviewer asking how fast the vehicle turns all think in a visible fraction of a turn. Arcseconds sit on top for the finest requirements.
How fast does a satellite tumble after separation, and what counts as detumbled?
Deployers usually guarantee an upper bound rather than a value, commonly around 10 °/s per axis for a small satellite ejected by a spring mechanism, while the rate actually seen is often a few degrees per second. That is enough to make sun sensors ambiguous, stop a star tracker locking, and swing the arrays repeatedly through shadow. The first job is normally a B-dot law driving magnetorquers against the changing measured field. Detumbled generally means residual rate below roughly a degree per second.
How long does a slew of a given angle actually take?
For a short manoeuvre with no coast phase, the vehicle accelerates for half the angle and decelerates for the rest, giving twice the square root of the angle divided by the angular acceleration. At 0.1 °/s², a 30° target change takes 34.6 seconds and peaks at 1.73 °/s halfway. A 90° slew at a gentler 0.05 °/s² takes 84.9 seconds and peaks at 2.12 °/s. Settling time is added afterwards, since pointing is not met the instant rate crosses zero.
What does reaction-wheel saturation mean and why is momentum dumping needed?
A wheel turns the vehicle by taking on the opposite angular momentum, so every manoeuvre and disturbance leaves it spinning a little faster. Aerodynamic drag, solar radiation pressure and gravity-gradient torque push consistently over an orbit, so wheel speed creeps towards its mechanical ceiling. A saturated wheel cannot accelerate further and so gives no control torque in that direction — a genuine loss of authority. Dumping restores headroom with an external torque, from magnetorquers or thrusters.
How does a pointing stability figure in arcseconds per second relate to these units?
An arcsecond is one three-thousand-six-hundredth of a degree, so degrees per second becomes arcseconds per second by multiplying by 3 600, and one radian per second is 206 265 arcsec/s. Payloads use the smaller unit because smear is what degrades a product: multiply residual rate by exposure time and compare with the angular size of one pixel. A rate of 0.001 rad/s is 206 arcsec/s, smearing over 400 arcseconds across a two-second exposure — which is why stability is specified apart from accuracy.
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