How to Calculate Roll Center (Double Wishbone)

The SAE defines the roll center as the point in the front-view plane where a lateral force can be applied to the body without making it roll (summarized here). The version you can calculate from the suspension geometry alone is called the kinematic roll center, and that is the one this guide finds. Its height is one of the first numbers anyone checks on a double-wishbone suspension, and you can find it with four points, a ruler and some patience. This guide shows the method, works one example through with numbers, and says plainly what the answer leaves out.

If you'd rather skip the arithmetic, the free roll center calculator does the same calculation as you type or drag the points. Working one by hand first is the best way to understand what its numbers mean.

What you need

Four hardpoints in the front view: the upper arm's chassis pivot and ball joint, and the lower arm's chassis pivot and ball joint. Each is a pair of numbers: how far outboard from the car's centerline, and how high above the ground.
The tire contact patch: on the ground, half the track width out from the centerline.
Static ride height: measure the heights with the car at its normal loaded ride height, not hanging from a jack.

Each arm pivots about an axis. For this front-view method, use the front-view projection of each pivot point, and remember that if the axes are tilted in side view the result is only an approximation.

The four steps

1. Draw the two arm lines. Draw a line through the upper arm's two points and extend it well past the ball joint toward the car's centerline. Do the same for the lower arm.
2. Find the instant center. The two extended lines meet at one point. That is the instant center (IC), the point the wheel is momentarily rotating about as the suspension moves.
3. Draw the force line. Draw a straight line from the tire contact patch through the instant center and continue it to the car's centerline.
4. Read the roll center. The height where the force line crosses the centerline is the roll center height, measured up from the ground.

Two special cases are worth knowing. If the arms are parallel, the lines never meet, the instant center is infinitely far away, and the force line from the contact patch simply runs parallel to the arms. If the arms are also level and equal in length, that line is flat on the ground, so the roll center is on the ground. If the lines meet on the opposite side of the car, the instant center lies past the centerline, which is normal and not an error.

The arithmetic

Drawing is fine for a sketch, but a number is better. Write each arm as a line through two points and find where the lines cross with the standard two-line intersection formula. Then take the line from the contact patch (x = half the track, y = 0) to the instant center and find its height at x = 0. In one line: if the contact patch is at (c, 0) and the instant center is at (x, y), the roll center height is y multiplied by c and divided by (c minus x).

A worked example

Take a generic FSAE-style front axle with a 1220 mm track, so the contact patch is 610 mm from the centerline. Coordinates are millimetres, with x measured outboard from the centerline and y up from the ground.

Upper arm: chassis pivot at (284, 332), ball joint at (562, 345).
Lower arm: chassis pivot at (167, 142), ball joint at (580, 136).
Contact patch: (610, 0).

The upper arm slopes up slightly outboard and the lower arm slopes down slightly outboard, so the two lines converge toward the inboard side. Extending them gives an instant center at about x = −2844 mm and y = 186 mm. That's far past the centerline, about 3.5 metres from the contact patch, which is the front-view swing arm length. A long swing arm like that means the wheel changes camber slowly as it moves, because camber gain depends on the swing arm length (Suspension Secrets on camber gain).

The force line from the contact patch (610, 0) to that point crosses the centerline at a height of about 33 mm. So this axle has a roll center roughly 33 mm above the ground. Check it yourself: 186 × 610 ÷ (610 + 2844) is about 33.

We checked these numbers against our own solver, and the roll center calculator loads this same axle as its example car so you can move the points and watch the result change.

The roll center moves

The number you just calculated is the static roll center, with the car sitting still. As the suspension moves, the arms rotate, the instant center shifts, and the roll center moves with it. For the same example axle, moving the wheel 25 mm into bump takes the roll center from about 33 mm down to about 8 mm, and 25 mm of droop takes it up to about 58 mm. The wheel also gains about half a degree of negative camber in that bump travel.

That is why a single static number can mislead. A roll center that looks right at ride height but swings 50 mm over the travel gives a car that feels different mid-corner than it does on a straight. Plotting the roll center through bump and droop is a more useful check than the starting value alone.

What the number means

Height above ground: a higher roll center reduces body roll, because the springs and bars have less force to deal with, but it adds jacking force: part of the lateral force goes through the links as a vertical push that lifts the body in a corner. A roll center below the ground pulls the body down instead (see the SAE paper on roll centres and jacking forces).
Height relative to the center of gravity: the distance from the roll center to the center of gravity is the roll moment arm. The farther apart they are, the more the body rolls in a corner (Suspension Secrets on roll moment).
Front versus rear: the line joining the front and rear roll centers is the roll axis, so the two ends have to be considered together.

There isn't one right height, and we haven't found a published typical range. One Formula Student design paper settled on 50 mm above the ground for its front axle (Procedia Engineering, 2016), but that is one team's choice, not a rule. Published design reports from teams in your competition are a better guide than any single number.

What this method leaves out

It's a front view only. Side-view geometry and arm pivot angles are ignored.
It assumes a symmetric, rigid axle. In a corner the body rolls and the two sides stop being mirror images, so the roll center shifts further.
It treats the tire as a point. Real tires have width, flex, and a contact patch that moves across the tread as the car leans.
It is a kinematic roll center. Force-based definitions depend on how the tires share the lateral force, so they can give a different height from the geometric one.
It describes geometry, not behaviour. Springs, bars, damping and tire properties decide how the car actually rolls.

Use it to compare options and to catch mistakes early, then confirm the design with a full model of the car or testing before you commit to hardware.

To see every number update as you move a point, try the free roll center calculator. For FSAE and Formula Student teams, our Vehicle Dynamics Workbench builds the full double-wishbone model, and the free suspension tools cover spring rates and weight transfer too.