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We consider the motion planning problem for stochastic nonlinear systems inuncertain environments. More precisely, in this problem the robot hasstochastic nonlinear dynamics and uncertain initial locations, and theenvironment contains multiple dynamic uncertain obstacles. Obstacles can be ofarbitrary shape, can deform, and can move. All uncertainties do not necessarilyhave Gaussian distribution. This general setting has been considered and solvedin [1]. In addition to the assumptions above, in this paper, we considerlong-term tasks, where the planning method in [1] would fail, as theuncertainty of the system states grows too large over a long time horizon.Unlike [1], we present a real-time online motion planning algorithm. We builddiscrete-time motion primitives and their corresponding continuous-time tubesoffline, so that almost all system states of each motion primitive areguaranteed to stay inside the corresponding tube. We convert probabilisticsafety constraints into a set of deterministic constraints called riskcontours. During online execution, we verify the safety of the tubes againstdeterministic risk contours using sum-of-squares (SOS) programming. Theprovided SOS-based method verifies the safety of the tube in the presence ofuncertain obstacles without the need for uncertainty samples and timediscretization in real-time. By bounding the probability the system statesstaying inside the tube and bounding the probability of the tube colliding withobstacles, our approach guarantees bounded probability of system statescolliding with obstacles. We demonstrate our approach on several long-termrobotics tasks.

Real-Time Tube-Based Non-Gaussian Risk Bounded Motion Planning for Stochastic Nonlinear Systems in Uncertain Environments via Motion Primitives