This paper presents a proposal of an iterative learning control method for a musculoskeletal arm to acquire adequate internal force to realize human-like natural movements. Additionally, a dynamic damping ellipsoid at the end-point is introduced to evaluate internal forces obtained through the iterative learning. In our previous works, we have presented that a human-like smooth reaching movement using a musculoskeletal redundant arm model can be obtained by introducing a nonlinear muscle model and "the Virtual spring-damper hypothesis". However, the internal forces have been determined heuristically, so far. In this paper, an iterative learning control method is used for acquisition of an adequate dynamic damping ellipsoid according to a given task, in order to determine internal forces more systematically. It is presented that the learning control scheme can perform effectively to realize given desired tasks, even under the existence of strong nonlinear characteristics of the muscles. After acquiring a given task, the dynamic damping ellipsoid is introduced to evaluate the relation between a damping effect generated by the acquired internal forces and a trajectory of the end-point. Some numerical simulations are performed and the usefulness of the learning control strategy, despite strong nonlinearity of the muscles, is demonstrated through these results.