Library / Guidance and trajectory following / GUI-04
Guidance and trajectory following
Lookahead (pure-pursuit) lateral acceleration
Lateral acceleration needed to fly a circular arc to a point ahead on the path, at an angle η from the current velocity.
- Ref
- GUI-04
- Rev
- v1
- Field
- Guidance and trajectory following
- Result
- a⊥ [m/s²]
If the vehicle follows a circular arc to a point a distance Ld ahead, at an angle η from its velocity, the arc has curvature κ = 2 sin η / Ld. The lateral acceleration needed is a⊥ = Vg²κ. This construction underlies the pure-pursuit path-following method. η is signed: positive when the point lies to the right of the velocity.
Lateral acceleration needed to fly a circular arc to a point ahead on the path, at an angle η from the current velocity. The distance to the point is Ld. This is the pure-pursuit and L1-style guidance law.
Calculator
- Path curvature, κ = 2 sin η / Ld0.006946 1/m
- Bank angle for a coordinated turn, arctan(a⊥/g)15.81 °
- Turn rate, a⊥/Vg7.959 °/s
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.
- Aiming at a point 50 m ahead and 10° off the velocity at 20 m/s needs a curvature of 0.006946 1/m and a lateral acceleration of 2.778 m/s², which a coordinated turn supplies with a bank of 15.8°.
- Acceleration rises with the square of speed: at 40 m/s the same geometry needs 11.11 m/s² (1.13 g), a bank of 48.6°. A guidance command must be limited to what the vehicle can do.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Vg Ground speed | 10 to 100 m/s | 0.6946 | 69.46 | 68.76 | +10% | +21 % |
| η Angle from velocity to the lookahead point | −40 to 40 ° | −10.28 | 10.28 | 20.57 | +1.6 ° | +15.8 % |
| Ld Distance to the lookahead point | 20 to 500 m | 0.2778 | 6.946 | 6.668 | +10% | −9.09 % |
Sweep of Ground speed
| Ground speed (m/s) | Lateral acceleration (m/s²) |
|---|---|
| 10 | 0.6946 |
| 21.25 | 3.137 |
| 32.5 | 7.337 |
| 43.75 | 13.29 |
| 55 | 21.01 |
| 66.25 | 30.49 |
| 77.5 | 41.72 |
| 88.75 | 54.71 |
| 100 | 69.46 |
Sweep one variable
Variables and how each one contributes
Lateral acceleration
What the equation gives you. Everything else feeds it.
Ground speed
The ground speed in m/s. The lateral acceleration grows with the square of speed, so a higher speed demands much more. Raising it by 10 % (to 22) moves the result by +21 %, from 2.778 to 3.362 m/s².
Distance to the lookahead point
The straight-line distance to the lookahead point, in m. Inversely proportional: a longer lookahead means a gentler arc and a smaller acceleration. Raising it by 10 % (to 55) moves the result by −9.09 %, from 2.778 to 2.526 m/s².
Angle from velocity to the lookahead point
The angle from the velocity direction to the lookahead point, in degrees. Positive means to the right. Roughly proportional for small angles: a bigger angle to the target needs a tighter arc. Raising it by 5° (to 15°) moves the result by +49 %, from 2.778 to 4.141 m/s².
Worked example with the values above
- Curvature: κ = 2 sin 10° / 50 = 2 × 0.1736 / 50 = 0.006946 1/m.
- Speed squared: Vg² = 20² = 400 m²/s².
- a⊥ = 400 × 0.006946 = 2.778 m/s².
Assumptions and limits
- A circular arc from the vehicle's current position and velocity direction to the lookahead point.
- A constant ground speed and ideal tracking of the commanded acceleration, with no wind in the turn.
Common mistakes
- Measuring η from north instead of from the current velocity direction.
- Ignoring the sign of η, so that the vehicle turns away from the path.
Statistics and uncertainty
Small errors in the angle η map to lateral acceleration with a factor of 2Vg²cos η/Ld. At the defaults a 1° error in η changes the command by about 0.275 m/s², 9.9 % of the command. Since the angle comes from position and heading estimates, their noise sets how smooth the commanded acceleration is.
Where it is used
- Path following for aircraft and ground vehicles, where the lateral acceleration is then turned into a bank or steering angle.
- Seeing how speed and lookahead distance trade off against the acceleration a vehicle needs.
Maths you need
- Circular motion: centripetal acceleration = V² × curvature
- Chord and arc geometry: a chord of length L at angle η subtends curvature 2 sin η / L