Cartesian robots have position-dependent dynamics that should be accounted for in high performance applications. Traditional methods design Linear Time Invariant (LTI) controllers which are robustly stable with respect to position variations, but require a trade-off in performance to account for the changing dynamics. Advanced methods require Linear Parameter Varying (LPV) models and LPV controller design methods that are not well established in industry. On the other hand, classical model-based gain-scheduled technique requires paramet- ric identification, design of high performance controllers for each position, interpolation of the controller parameters and real-time validation of the gain-scheduled controller, which takes costly engineering time. We propose a new approach, using the frequency response of a system at different operating points, to design a Linear Parameter Varying (LPV) controller. The controller parameters are optimised by a convex optimisation algorithm based on second-order cone programming. The approach is applied to an industrial 3-axis Cartesian robot, showing significant improvements over state-of-the-art control design strategies. Data acquisition and controller design can be performed automatically, reducing significantly the engineering costs for controller synthesis