Rehabilitation robots have proven their usefulness in assisting with physical therapy. This article presents a trajectory tracking controller for a wrist rehabilitation robot with three degrees of freedom. The nonlinearity of the human-robot interaction dynamics has been defined as the Koopman linear system in terms of nonlinear observable functions of the state variables. Koopman operators are learned using linear regression to encode the states into object-centric embedding space for a linear approximation of a nonlinear dynamical system. The learned Koopman operators ascertain the system dynamics applied to design the wrist robot's trajectory tracking task controller. This is a data-driven approach that yields an explicit control-oriented model. The efficiency and feasibility of the controller were evaluated through experiments with three healthy human subjects. The experiments demonstrated the ability of the controller to guide the subject's wrist along the reference trajectory.