Library / Terramechanics and off-road mobility / TER-17
Terramechanics and off-road mobility
Ideal differential steering radius
Turning radius of a tracked vehicle steered by running its two tracks at different speeds, assuming no slip.
- Ref
- TER-17
- Rev
- v1
- Field
- Terramechanics and off-road mobility
- Result
- R [m]
vR and vL are the ground speeds of the right and left tracks and B is the distance between the track centres. The vehicle speed is the mean U = (vR + vL)/2 and the yaw rate is r = (vR − vL)/B, so the radius of the path of the vehicle centre is R = U/r = (B/2)(vR + vL)/(vR − vL). The model assumes no slip and no sideways scrub. Equal speeds give a straight line (infinite radius) and equal and opposite speeds give a pivot turn on the spot (R = 0).
Turning radius of a tracked vehicle steered by running its two tracks at different speeds, assuming no slip. Equal speeds give a straight line and equal opposite speeds give a pivot turn on the spot.
Calculator
- Vehicle forward speed, (vR + vL)/22 m/s
- Yaw rate, (vR − vL)/B1 rad/s
- Radius in track spacings, R / B1
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.
- With the right track at 3 m/s and the left at 1 m/s the vehicle moves at 2 m/s, turns at 1 rad/s and follows a 2 m radius, one track spacing. Stopping the left track entirely gives a pivot about it, with R = 1 m (half the spacing).
- Running the left track backwards at 3 m/s gives R = 0 m: the vehicle pivots on the spot. At the other end, speeds of 3 and 2.9 m/s give 59 m, a gentle bend.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| vL Left track speed | 0 to 10 m/s | −19 | 17 | 36 | +10% | +7.89 % |
| vR Right track speed | 0 to 10 m/s | −5 | 7 | 12 | +10% | −6.52 % |
| B Distance between track centres | 1 to 4 m | 1 | 4 | 3 | +10% | +10 % |
Sweep of Left track speed
| Left track speed (m/s) | Turning radius (m) |
|---|---|
| 0 | 1 |
| 1.25 | 2.429 |
| 2.5 | 11 |
| 3.75 | −9 |
| 5 | −4 |
| 6.25 | −2.846 |
| 7.5 | −2.333 |
| 8.75 | −2.043 |
| 10 | −1.857 |
Sweep one variable
Variables and how each one contributes
Turning radius
What the equation gives you. Everything else feeds it.
Distance between track centres
The distance between the centres of the two tracks, in m. Directly proportional: a wider vehicle takes a larger circle for the same track speeds. Raising it by 10 % (to 2.2) moves the result by +10 %, from 2 to 2.2 m.
Right track speed
The ground speed of the right track, in m/s (negative for reverse). The vehicle turns left when it is faster than the left track. Widens the speed difference, so the radius falls: a faster outer track tightens the turn. Raising it by 10 % (to 3.3) moves the result by −6.52 %, from 2 to 1.87 m.
Left track speed
The ground speed of the left track, in m/s (negative for reverse). Brings the two speeds closer, so the vehicle goes more nearly straight and the radius increases a lot. Raising it by 10 % (to 1.1) moves the result by +7.89 %, from 2 to 2.158 m.
Worked example with the values above
- Mean speed: U = (3 + 1) / 2 = 2 m/s. Yaw rate: r = (3 − 1) / 2 = 1 rad/s.
- Radius: R = U / r = 2 / 1.
- R = 2 m.
Assumptions and limits
- Ideal kinematics: no longitudinal slip and no sideways scrub of the tracks, on firm level ground.
- Track speeds are the actual ground speeds, constant over the turn.
Common mistakes
- Using the track belt speed as the ground speed. On soil the difference is the slip.
- Expecting a pivot turn to have the radius given here. In practice the tracks scrub sideways and the effective pivot point shifts.
Statistics and uncertainty
The radius is a ratio of a sum and a difference of speeds, so it is very sensitive when the speeds are close: with vR = 3 and vL = 2.9 m/s, a 1 % error in either one changes R by 25 to 40 %. In practice the measured turning radius of a tracked vehicle differs from this ideal one by 20 to 100 % because of longitudinal slip and lateral scrub of the tracks, which grow as the radius gets smaller.
Where it is used
- Predicting the turns of tracked vehicles and robots, and the track speeds a steering command requires.
- Skid-steer loaders and wheeled vehicles that steer the same way.
Maths you need
- Relative velocity: speeds of the two sides
- Circular motion: R = U / r