The present paper discusses a planning and control strategy for robotic whole-arm manipulation of a slippery polygonal object by taking into account the estimated bounds of the frictional coefficient. In this study, randomized planning methods are first proposed in order to generate contact state transitions. Second, a novel control strategy that can switch manipulation between quasi-static, dynamic, and caging manipulation modes is proposed. Solving the forward dynamics problem, which is formulated as a complementarity problem, generates the desired trajectory and thus ensures that undesirable contact modes do not occur within the estimated bounds of the frictional coefficient. The validity of the proposed methods is then demonstrated through simulations and experiments.