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Optimal path planning problems for rigid and deformable (bendable) cuboidrobots are considered by providing an analytic safety constraint usinggeneralized $L_p$ norms. For regular cuboid robots, level sets of weighted$L_p$ norms generate implicit approximations of their surfaces. For bendablecuboid robots a weighted $L_p$ norm in polar coordinates implicitlyapproximates the surface boundary through a specified level set. Obstaclevolumes, in the environment to navigate within, are presumed to beapproximately described as sub-level sets of weighted $L_p$ norms. Using theseapproximate surface models, the optimal safe path planning problem isreformulated as a two stage optimization problem, where the safety constraintdepends on a point on the robot which is closest to the obstacle in theobstacle's distance metric. A set of equality and inequality constraints arederived to replace the closest point problem, which is then defines additionalanalytic constraints on the original path planning problem. Combining all theanalytic constraints with logical AND operations leads to a general optimalsafe path planning problem. Numerically solving the problem involve conversionto a nonlinear programing problem. Simulations for rigid and bendable cuboidrobot verify the proposed method.

Bendable Cuboid Robot Path Planning with Collision Avoidance using Generalized $L_p$ Norms