Title: Bendable Cuboid Robot Path Planning with Collision Avoidance using Generalized $L_p$ Norms,
Abstract: Optimal path planning problems for rigid and deformable (bendable) cuboid
robots are considered by providing an analytic safety constraint using
generalized $L_p$ norms. For regular cuboid robots, level sets of weighted
$L_p$ norms generate implicit approximations of their surfaces. For bendable
cuboid robots a weighted $L_p$ norm in polar coordinates implicitly
approximates the surface boundary through a specified level set. Obstacle
volumes, in the environment to navigate within, are presumed to be
approximately described as sub-level sets of weighted $L_p$ norms. Using these
approximate surface models, the optimal safe path planning problem is
reformulated as a two stage optimization problem, where the safety constraint
depends on a point on the robot which is closest to the obstacle in the
obstacle's distance metric. A set of equality and inequality constraints are
derived to replace the closest point problem, which is then defines additional
analytic constraints on the original path planning problem. Combining all the
analytic constraints with logical AND operations leads to a general optimal
safe path planning problem. Numerically solving the problem involve conversion
to a nonlinear programing problem. Simulations for rigid and bendable cuboid
robot verify the proposed method.