This paper deals with the problem of output regulation for left invariant systems defined on general matrix Lie-Groups. Reference trajectories to be tracked are generated by a right invariant exosystem defined on the same Lie-Group and driven by a linear oscillator defined on the Lie-algebra. Only partial relative geometrical information of the exosystem with respect to actual system is available in the form of invariant measurements in a homogeneous space, a quotient of the Lie-group. In the spirit of the internal model principle, the proposed control structure contains an embedded copy of the exosystem kinematics. The dynamics associated with the mechanical systems can be handled by dynamic extension of the proposed control approach. We demonstrate the approachby providing the details of this extension for the specific case of rotational dynamics on SO(3). The problem formulation considered is motivated by a wide range of real applications in robotics, aerospace, and computer vision, where the state space of systems such as mobile robots, aerial robots, unmanned drones, and camera homographies are naturally Lie-groups associated with inherent symmetry properties of the underlying physical system. (C) 2017 Elsevier Ltd. All rights reserved.