  Existing motion planning methods often have two drawbacks: 1) goal
configurations need to be specified by a user, and 2) only a single solution is
generated under a given condition. In practice, multiple possible goal
configurations exist to achieve a task. Although the choice of the goal
configuration significantly affects the quality of the resulting trajectory, it
is not trivial for a user to specify the optimal goal configuration. In
addition, the objective function used in the trajectory optimization is often
non-convex, and it can have multiple solutions that achieve comparable costs.
In this study, we propose a framework that determines multiple trajectories
that correspond to the different modes of the cost function. We reduce the
problem of identifying the modes of the cost function to that of estimating the
density induced by a distribution based on the cost function. The proposed
framework enables users to select a preferable solution from multiple candidate
trajectories, thereby making it easier to tune the cost function and obtain a
satisfactory solution. We evaluated our proposed method with motion planning
tasks in 2D and 3D space. Our experiments show that the proposed algorithm is
capable of determining multiple solutions for those tasks.
