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Online Gaussian processes (GPs), typically used for learning models from time-series data, are more flexible and robust than offline GPs. Both local and sparse approximations of GPs can efficiently learn complex models online. Yet, these approaches assume that all signals are relatively accurate and that all data are available for learning without misleading data. Besides, the online learning capacity of GPs is limited for high-dimension problems and long-term tasks in practice. This paper proposes a sparse online GP (SOGP) with a forgetting mechanism to forget distant model information at a specific rate. The proposed approach combines two general data deletion schemes for the basis vector set of SOGP: The position information-based scheme and the oldest points-based scheme. We apply our approach to learn the inverse dynamics of a collaborative robot with 7 degrees of freedom under a two-segment trajectory tracking problem with task switching. Both simulations and experiments have shown that the proposed approach achieves better tracking accuracy and predictive smoothness compared with the two general data deletion schemes.