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This paper presents the construction of a realistic linear parameter-varying (LPV) model of a robotic manipulator using parameter set mapping, for the purpose of synthesizing an LPV gain-scheduled controller. A nonlinear dynamic model of the manipulator is obtained and a quasi-LPV model is derived. Since the quasi-LPV model has a large number of affine scheduling parameters and a large overbounding, parameter set mapping is used to reduce conservatism and complexity in controller design by finding tighter parameter regions with fewer scheduling parameters. Then, a polytopic LPV gain-scheduled controller is synthesized and implemented experimentally on an industrial robot for a trajectory tracking task. Comparison of results with a decentralized PD controller illustrates that the designed LPV controller improves the tracking error significantly. Moreover, it achieves a slightly better accuracy than a model-based inverse dynamics controller while being of lower complexity.