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Adaptive control can address model uncertainty in control systems. However, it is preliminarily designed for tracking control. Recent advancements in the control of quadruped robots show that force control can effectively realize agile and robust locomotion. In this paper, we present a novel adaptive force-based control framework for legged robots. We introduce a new architecture in our proposed approach to incorporate adaptive control into quadratic programming (QP) force control. Since our approach is based on force control, it also retains the advantages of the baseline framework, such as robustness to uneven terrain, controllable friction constraints, or soft impacts. Our method is successfully validated in both simulation and hardware experiments. While the baseline QP control has shown a significant degradation in the body tracking error with a small load, our proposed adaptive force-based control can enable the 12-kg Unitree A1 robot to walk on rough terrains while carrying a heavy load of up to 6 kg (50% of the robot weight). When standing with four legs, our proposed adaptive control can even allow the robot to carry up to 11 kg of load (92% of the robot weight) with less than 5-cm tracking error in the robot height.