# Trajectory Optimization of Chance-Constrained Nonlinear Stochastic Systems for Motion Planning Under Uncertainty

arXiv: https://arxiv.org/abs/2106.02801 · HTML: https://mlsanity.com/paper/2106.02801/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 67 references that are in the map.
- First posted on arXiv 5 Jun 2021.

## Builds on (most cited first, 15 of 15)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [Probabilistic model predictive safety certification for learning-based control](https://mlsanity.com/paper/1906.10417/) (2019)
- [Neural-Swarm2: Planning and Control of Heterogeneous Multirotor Swarms using Learned Interactions](https://mlsanity.com/paper/2012.05457/) (2020)
- [Asymptotically Optimal Sampling-based Kinodynamic Planning](https://mlsanity.com/paper/1407.2896/) (2014)
- [Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems](https://mlsanity.com/paper/2005.04374/) (2020)
- [Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty](https://mlsanity.com/paper/1504.08053/) (2015)
- [Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization](https://mlsanity.com/paper/2006.04359/) (2020)
- [A Real-Time Approach for Chance-Constrained Motion Planning with Dynamic Obstacles](https://mlsanity.com/paper/2001.08012/) (2020)
- [Asymptotically Optimal Planning by Feasible Kinodynamic Planning in State-Cost Space](https://mlsanity.com/paper/1505.04098/) (2015)
- [Nonlinear Uncertainty Control with Iterative Covariance Steering](https://mlsanity.com/paper/1903.10919/) (2019)
- [Limits of Probabilistic Safety Guarantees when Considering Human Uncertainty](https://mlsanity.com/paper/2103.03388/) (2021)
- [Stochastic Model Predictive Control for Linear Systems using Probabilistic Reachable Sets](https://mlsanity.com/paper/1805.07145/) (2018)
- [Comments on Truncation Errors for Polynomial Chaos Expansions](https://mlsanity.com/paper/1708.07655/) (2017)
- [Stochastic trajectory optimization for mechanical systems with parametric uncertainties](https://mlsanity.com/paper/1705.05506/) (2017)

## Built on by (most cited first, 4 of 4)

- [Conformal Predictive Programming for Chance Constrained Optimization](https://mlsanity.com/paper/2402.07407/) (2024)
- [Proactive Emergency Collision Avoidance for Automated Driving in Highway Scenarios](https://mlsanity.com/paper/2310.17381/) (2023)
- [Neural-Rendezvous: Provably Robust Guidance and Control to Encounter Interstellar Objects](https://mlsanity.com/paper/2208.04883/) (2022)
- [Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systems](https://mlsanity.com/paper/2603.07092/) (2026)

## Abstract (arXiv)

We present gPC-SCP: Generalized Polynomial Chaos-based Sequential Convex Programming to compute a sub-optimal solution for a continuous-time chance-constrained stochastic nonlinear optimal control (SNOC) problem. The approach enables motion planning for robotic systems under uncertainty. The gPC-SCP method involves two steps. The first step is to derive a surrogate problem of deterministic nonlinear optimal control (DNOC) with convex constraints by using gPC expansion and the distributionally-robust convex subset of the chance constraints. The second step is to solve the DNOC problem using sequential convex programming for trajectory generation and control. We prove that in the unconstrained case, the optimal value of the DNOC converges to that of SNOC asymptotically and that any feasible solution of the constrained DNOC is a feasible solution of the chance-constrained SNOC. We also present the predictor-corrector extension (gPC-SCPPC) for real-time motion trajectory generation in the presence of stochastic uncertainty. In the gPC-SCPPC method, we first predict the uncertainty using the gPC method and then optimize the motion plan to accommodate the uncertainty. We empirically demonstrate the efficacy of the gPC-SCP and the gPC-SCPPC methods for the following two test cases: 1) collision checking under uncertainty in actuation and physical parameters and 2) collision checking with stochastic obstacle model for 3DOF and 6DOF robotic systems. We validate the effectiveness of the gPC-SCP method on the 3DOF robotic spacecraft testbed.