Appropriate regulation of human arm mechanics is essential for completing the diverse range of tasks we accomplish each day. The steady state mechanical properties of the arm most relevant for postural tasks can be characterized by endpoint stiffness, the static forces generated by a limb in response to external perturbations of posture. Endpoint stiffness is directional, resisting perturbations in certain directions more than others. It has been shown that humans can voluntarily control the orientation of the maximum stiffness to meet specific task requirements, although the limits on this control are poorly understood. Both neural and biomechanical factors may limit endpoint stiffness control. The purpose of this work was to quantify the biomechanical constraints limiting the control of stiffness orientation. A realistic musculoskeletal model of the human arm coupled with a model of muscle stiffness was used to explore the range of endpoint stiffness orientations that could be achieved with changes in the feedforward control of muscle activation. We found that this range is constrained by the biomechanics of the neuromuscular system, and by the requirements of the specific task being performed by the subject. These constraints and the sensitivity to experimental conditions may account for some of the discrepancies in the literature regarding the ability to control endpoint stiffness orientation.