In this paper, the analysis of manipulability of robotic systems comprised of multiple cooperating arms is considered. Given bounds on the capabilities of joint actuators for each robot, the purpose of this study is to derive the bounds for task velocity achievable by the system. Since bounds on each joint velocity form a polytope in joint-velocity space and the task space velocity is connected with joint velocity through Jacobian matrices of each robot, the allowable task velocity space, i.e. velocity workspace, for multiple cooperating robot system is also represented as a polytope which is called manipulability polytope throughout this paper. Based on the fact that the boundaries of the manipulability polytope are mapped from the boundaries of allowable joint-velocity space, slack variables are introduced in order to transform given inequality constraint given on joint velocities into a set of normal linear equalities in which the unknowns of the equation are composed of the vertices of manipulability polytope, vectors spanning the null space of the Jacobian matrix, and the slack variables. Either redundant or nonredundant cooperating robot systems can be handled with the proposed technique. Several different application examples including simple SCARA-type robots as well as complex articulated robot manipulators are included, and, under the assumption of firm grip, it will be shown that the calculated manipulability polytope for cooperating robot system is actually the intersection of all the manipulability polytopes of every single robot which is hard to be derived through geometrical manipulation.