With the advent of force control in legged robots, there is an increasing demand in research on controlling contact forces that can ensure stable interaction and balance of the system. This article aims to solve the force distribution problem by an analytic solution to regulate the contact forces particularly in a computationally efficient manner. To this end, compliance models, consisting of a virtual model of the torso and impedance models of supporting feet, are developed for a quadruped robot. The linear constraints are formulated for the analytic method based on the compliance models, and the minimization of foot slippage and the internal forces within the closed chain are also taken into account. Moreover, given the compliance models, the postural compensation of the torso can be achieved by modifying the trajectories of supporting feet in order to generate desired forces. The comparisons between the proposed analytic and numerical methods show that the analytic one is advantageous for embedded controllers due to its high computational efficiency. Finally, the effectiveness of the proposed method is first validated in simulations and then in experiments on a physical quadruped robot, and the data are presented and analyzed.