This paper considers linear parameter-varying (LPV) control within a passive systems framework. A very strictly passive (VSP) controller is presented in state-space form where the state-space matrices have an affine dependence on plant parameters. The resulting controller is a VSP-LPV controller guaranteed to stabilize nonlinear passive systems via the passivity theorem. An H-infinity-inspired control synthesis method is presented along with an experimental comparison between the proposed VSP-LPV controller and a self-scheduled LPV controller. Experimental results demonstrate effective joint angle and rate tracking of a two-link flexible-joint manipulator when the proposed VSP-LPV controller is used.