Abstract
In this paper, we address the brachistochrone problem using Dynamic
Programming and an extension that accounts for forbidden
regions or obstacles. The brachistochrone problem seeks the curve
of fastest descent under gravity between two points, minimizing
travel time. While traditional approaches derive the cycloidal curve
as the solution using calculus of variations, this work employs a
discrete, grid-based Dynamic Programming formulation to approximate
the optimal trajectory. Computational experiments showcase
the method’s flexibility, particularly in adapting to constraints such
as forbidden regions, and its ability to dynamically recalculate paths.
Despite some limitations in angular resolution due to discretization,
the proposed approach demonstrates robustness and scalability in
addressing constrained trajectory optimization problems. Moreover,
this project lays the groundwork for extending the proposed
methodology to more complex scenarios, such as incorporating randomness
into the trajectory, where the adaptability of this approach
can be effectively leveraged.