  We address the risk bounded trajectory optimization problem of stochastic
nonlinear robotic systems. More precisely, we consider the motion planning
problem in which the robot has stochastic nonlinear dynamics and uncertain
initial locations, and the environment contains multiple dynamic uncertain
obstacles with arbitrary probabilistic distributions. The goal is to plan a
sequence of control inputs for the robot to navigate to the target while
bounding the probability of colliding with obstacles. Existing approaches to
address risk bounded trajectory optimization problems are limited to particular
classes of models and uncertainties such as Gaussian linear problems. In this
paper, we deal with stochastic nonlinear models, nonlinear safety constraints,
and arbitrary probabilistic uncertainties, the most general setting ever
considered. To address the risk bounded trajectory optimization problem, we
first formulate the problem as an optimization problem with stochastic dynamics
equations and chance constraints. We then convert probabilistic constraints and
stochastic dynamics constraints on random variables into a set of deterministic
constraints on the moments of state probability distributions. Finally, we
solve the resulting deterministic optimization problem using nonlinear
optimization solvers and get a sequence of control inputs. To our best
knowledge, it is the first time that the motion planning problem to such a
general extent is considered and solved. To illustrate the performance of the
proposed method, we provide several robotics examples.
