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In multi-point contact systems, precise force control is crucial for achieving stable and safe interactions between robots and their environment. Thus, we demonstrate an admittance controller with auto-tuning that can be applied for these systems. The controller's objective is to track the target wrench profiles of each contact point while considering the additional torque due to rotational friction. Our admittance controller is adaptive during online operation by using an auto-tuning method that tunes the gains of the controller while following user-specified training objectives. These objectives include facilitating controller stability, such as tracking the wrench profiles as closely as possible, ensuring control outputs are within force limits that minimize slippage, and avoiding configurations that induce kinematic singularity. We demonstrate the robustness of our controller on hardware for both manipulation and locomotion tasks using a multi-limbed climbing robot.