We address the problem of distributed control of a network of cooperative and competitive robot manipulators in end-effector coordinates. We propose a distributed bipartite formation controller that guarantees collision avoidance. In the considered setting two groups are formed and reach inter-group bipartite consensus or disagreement. On the other hand, the end-effectors achieve intra-group formation. To ensure that the end-effectors do not collide, we design gradient-based control laws using barrier-Lyapunov functions. In addition, the proposed controller ensures that the closed-loop system is robust to external disturbances. The latter are assumed to be generated by an exosystem, so they are effectively rejected by an internal-model-based compensator. More precisely, we establish asymptotic stability of the bipartite formation manifold. Finally, we illustrate our theoretical results via numerical simulations.