Vine robots navigate cluttered environments by extending from their tip. Although their ability to operate in such environments has been extensively demonstrated, little work has addressed growth planning, i.e., finding optimal growth paths. Moreover, existing planners do not account for the growth pressure necessary to follow a given path, which can cause the robot to burst when it is too high. In this paper, we address the problem of finding minimum-pressure paths for vine robots growing around polytopic obstacles. We propose an efficient algorithm that is guaranteed to find globally optimal solutions in 2D and approximate solutions in 3D, with an error that vanishes as a discretization parameter approaches zero. First, we derive a growth pressure equation for vine robots of arbitrary shape, which we use to show that there always exists a minimum-pressure path that is piecewise-linear and can bend only at specific points on the obstacles. We then leverage this observation to reduce the growth-planning problem to a shortest-path problem with time-dependent weights, which we efficiently solve using a modified Dijkstra's algorithm. We demonstrate the speed and scalability of our approach through numerical simulations. We also validate our algorithm with hardware experiments and provide an open-source and high-performance implementation in the Python package, VinePlanner: https://github.com/Ahsoka/VinePlanner.

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