Formation control simplifies minimizing multi-robot cost functions by encoding a cost function as a shape the robots maintain. However, by reducing complex cost functions to formations, discrepancies arise between maintaining the shape and minimizing the original cost function. For example, a Diamond or Box formation shape is often used for protecting all members of the formation. When more information about the surrounding environment becomes available, a static shape often no longer minimizes the original protection cost. We propose a formation planner to reduce mismatch between a formation and the cost function while still leveraging efficient formation controllers. Our formation planner is a two-step optimization problem that identifies desired relative robot positions. We first solve a constrained problem to estimate non-linear and non-differentiable costs with a weighted sum of surrogate cost functions. We theoretically analyze this problem and identify situations where weights do not need to be updated. The weighted, surrogate cost function is then minimized using relative positions between robots. The desired relative positions are realized using a non-cooperative formation controller derived from Lyapunov's direct approach. We then demonstrate the efficacy of this approach for military-like costs such as protection and obstacle avoidance. In simulations, we show a formation planner can reduce a single cost by over 75%. When minimizing a variety of cost functions simultaneously, using a formation planner with adaptive weights can reduce the cost by 20-40%. Formation planning provides better performance by minimizing a surrogate cost function that closely approximates the original cost function instead of relying on a shape abstraction.

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