Library / Guidance and trajectory following / GUI-05
Guidance and trajectory following
Bank angle for a commanded lateral acceleration
Bank angle that supplies a required horizontal acceleration in a level coordinated turn, together with the resulting turn rate.
- Ref
- GUI-05
- Rev
- v1
- Field
- Guidance and trajectory following
- Result
- φcmd [°]
In a level coordinated turn the horizontal part of the lift supplies the lateral acceleration and the vertical part carries the weight, so tan φ = a⊥/g and the bank command is φcmd = arctan(a⊥/g). The heading changes at the rate a⊥/V. The load factor, the lift as a multiple of the weight, is √(1 + (a⊥/g)²) = 1/cos φ, so a steeper bank raises the load and the stall speed. Limit the commanded bank to the permitted envelope.
Bank angle that supplies a required horizontal acceleration in a level coordinated turn, together with the resulting turn rate. The lift must also carry the weight, so the load factor rises as 1/cos φ.
Calculator
- Load factor, √(1 + (a⊥/g)²)1.039
- Turn rate, a⊥/V7.964 °/s
- Stall-speed increase, √(load factor)1.019
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A lateral acceleration of 2.78 m/s² needs a bank of 15.82° and turns the aircraft at 7.96 °/s at 20 m/s. The load factor is 1.039, hardly above straight and level flight.
- A bank limit of 45° corresponds to a⊥ = g tan 45° = 9.81 m/s², a load factor of 1.41 and a stall speed about 19 % higher. A guidance law that asks for more acceleration than the limit must be saturated.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a⊥ Commanded lateral acceleration | 0.5 to 8 m/s² | 2.918 | 39.2 | 36.28 | +10% | +1.49 ° |
| V True airspeed | 10 to 100 m/s | 15.82 | 15.82 | 0 | +10% | 0 ° |
Sweep of Commanded lateral acceleration
| Commanded lateral acceleration (m/s²) | Bank angle (°) |
|---|---|
| 0.5 | 2.918 |
| 1.438 | 8.336 |
| 2.375 | 13.61 |
| 3.313 | 18.66 |
| 4.25 | 23.42 |
| 5.188 | 27.87 |
| 6.125 | 31.98 |
| 7.063 | 35.75 |
| 8 | 39.2 |
Sweep one variable
Variables and how each one contributes
Bank angle
What the equation gives you. Everything else feeds it.
Commanded lateral acceleration
The commanded lateral acceleration in m/s², for example from a pure-pursuit law. Roughly proportional for small angles. At larger bank the arctangent flattens the rise. Raising it by 10 % (to 3.058) moves the result by +9.43 %, from 15.82 to 17.31 °.
True airspeed
The true airspeed in m/s. No effect on the bank angle, which depends on the acceleration alone. It sets the turn rate: a⊥/V = 7.96 °/s at the defaults.
Worked example with the values above
- Ratio to gravity: a⊥/g = 2.78 / 9.81 = 0.2834.
- Bank angle: arctan(0.2834) = 15.82°.
- Turn rate: a⊥/V = 2.78 / 20 = 0.139 rad/s = 7.964 °/s.
Assumptions and limits
- Level coordinated flight, negligible sideslip and no wind. Climb and large flight-path changes need a fuller model.
- The aircraft's speed is the true airspeed, equal to the ground speed.
Common mistakes
- Using the command without a bank limit. Large lateral accelerations demand banks beyond the flight envelope.
- Using it in strong wind with the ground speed, when the aircraft's airspeed and the wind both matter.
Statistics and uncertainty
Near small angles the bank is almost proportional to the acceleration, so a 5 % error in a⊥ is nearly a 5 % error in φ. At steeper banks the arctangent flattens, so errors in the command matter less in angle but more in load factor, since the load rises sharply. The result also depends on g, which varies by about 0.5 % over the Earth.
Where it is used
- Converting a guidance acceleration command into a roll command, with a bank limit set by the flight envelope.
- Checking the load factor and stall speed implied by a manoeuvre.
Maths you need
- Forces in a banked turn: lift resolved vertically and horizontally
- Load factor: the lift divided by the weight