Find the ideal wheel angles for a multi-axle vehicle whose wheels share one instantaneous turning center. Specify front and rear reference steering angles or enter the turning center directly.

The rear reference is x = 0 and the front reference is x = wheelbase. Axle rows contain x, track width and steer or fixed; enter all rows in the selected coordinate unit.

Ideal low-speed, no-slip rigid-body geometry. Positive steering turns left; x is forward and y is left from the rear reference. This does not validate hardware travel, tire slip, stability or road clearance.

One row per axle, for example: 0, 2, fixed then 3, 2, steer. No header. Use a decimal point and separate columns with commas or spaces.

Optional settings

Coordinate system and formulas

Place the rear reference at x = 0 and the front reference at x = L. Positive x points forward, positive y points left and positive steering turns left. For front/rear centerline angles δf and δr, yc = L / (tan δf − tan δr) and xc = −yc tan δr.

A wheel at (x,y) needs δ = atan((x − xc) / (yc − y)). For track width t, the left wheel is at y = t/2 and the right wheel at y = −t/2. Enter one axle per line as x, track width, steer or fixed. Every fixed axle must have x = xc to roll without lateral scrub.

The reported radius is √(xc² + yc²) from the rear reference origin. It is not the vehicle-body swept radius. This model requires the turning center outside every wheel track.

Worked example

With a turning center at (0,5) m, a fixed axle at x = 0 and a steering axle at x = 3 m, both with a 2 m track, the steering axle’s left angle is atan(3/4) = 36.8699°. Its right angle is atan(3/6) = 26.5651°. The rear-reference radius is 5 m.

Frequently asked questions

What happens with equal front and rear angles?

The ideal motion is straight crab motion with no finite turning center. A fixed axle cannot follow a nonzero crab angle without scrub.

Can two fixed axles occupy different x positions?

Not in an exact finite no-slip turn. Real vehicles can turn by tire slip or compliance, which this geometry omits.

Does this prove a steering design is safe?

No. Hardware travel, loads, tire behavior, speed and stability require separate engineering analysis.

Further kinematic context: Lim and colleagues: four-wheel steering and path tracking. The wheel-angle calculation above follows rigid-body tangent geometry.