To absorb the vibrations and shocks caused by road obstacles effectively in any direction within the wheel rotation plane, a planar suspension system (PSS), in which there are spring-damper struts in both the vertical and longitudinal directions, is proposed to improve the ride quality of a vehicle with such novel suspension systems. The longitudinal spring-damper strut in a PSS is considerably soft compared with the longitudinal connection in a conventional suspension. Consequently, the wheels in a vehicle with PSS can move forth and back with respect to the body. The dynamic behaviours of a PSS vehicle under some special conditions, such as a differential braking in which the braking torque applied to the wheels at two sides of an axle are uneven, may exhibit special characteristics. The directional stability of the PSS vehicle in such a case may be one of the major concerns. The dynamic performance of the PSS vehicle in the differential braking condition is thus necessary to be investigated. In this study, an 18 DOF full-car vehicle model is developed to study the dynamic performance of a vehicle with PSSs. The model is validated with the Adams/car models for both a PSS vehicle and a conventional vehicle in a braking and a turning manoeuvre. This model, together with a dynamic 2D friction model, can account for the effect of the longitudinal elasticity of a suspension system on the overall dynamic behaviour. The possible relative longitudinal motions between the body and tyres, the normal wheel load undulation, and the 2D frictions are taken into account. In this study, the dynamic response of a PSS vehicle is predicted and analysed in a straight line and an in-turn differential braking manoeuvres. The analysis is conducted by a comparison with the response of a conventional vehicle in the same conditions. The study results show that the implementation of PSSs will not hurt the roll, pitch and yaw performance of a vehicle while it improves the ability of the longitudinal vibration isolation. In some cases such as in differential braking-in-turn, the PSS vehicle exhibits, although very slight, a more understeer character. Actually there is no surprise to the investigation results because the 'soft' longitudinal strut in a PSS is only soft when the deflection under a threshold value (d_r). When the longitudinal deflection exceeds this threshold value, the longitudinal spring becomes as stiff as the longitudinal connection in the conventional vehicle. The possible relative longitudinal motion between the body and wheels is constrained by this property and thus cannot be too large even in a differential braking operation. The small relative longitudinal motion cannot change the vehicle dynamic performance significantly.