A wheel clamp attaches to a vehicle wheel for performing a wheel alignment on the vehicle. The wheel clamp has a body for supporting a target or measuring head, and three elongated arms mounted to the body. Each arm has a gripping portion for gripping the tire's tread surface. A self-centering linkage has enmeshed gears that, when rotated, cause the elongated arms to simultaneously proportionally move relative to the clamp body. An attachment interface mechanically connected to the self-centering linkage is configured to engage with a rotational actuator. Alternatively, the wheel clamp includes an actuator mechanically coupled to the self-centering mechanism. Activation of the actuator causes the gripping portions of the arms to engage and grip the tire tread surface and tighten the clamp onto the tire.