cuNRTO: GPU-Accelerated Nonlinear Robust Trajectory Optimization

Authors: Jiawei Wang, Arshiya Taj Abdul, Evangelos A. Theodorou · RSS 2026

Licences

Open Access
Paper: CC BY 4.0

Everything below draws on the full paper under its CC BY 4.0 licence; its figure is credited to the authors.

TL;DR

cuNRTO reworks the inner solver of nonlinear robust trajectory optimization so cone projections run in parallel on a GPU, cutting solve times by up to 139.6x versus interior-point.

Why it matters

Robust policies with feedback gains give guarantees against bounded disturbances, but their cost has limited them to small problems. Moving the inner loop onto the GPU makes the speed-up grow with the number of constraints, which brings robust trajectory optimization of a 7-DoF arm with obstacle constraints from about an hour of solve time down to a few minutes.

Problem

Robust trajectory optimization computes a control policy that satisfies safety constraints for every disturbance in a bounded set. The NRTO framework handles nonlinear dynamics and constraints, but its reformulated subproblems contain many second-order cone constraints, and solving them with interior-point methods is too slow for high-dimensional systems with many constraints.

Proposed solution

The authors replace the expensive inner subproblem solve of NRTO in two ways. NRTO-DR solves it with Douglas-Rachford splitting, factoring a fixed sparse KKT matrix once and running the cone projections in parallel. NRTO-FullADMM restructures the whole inner ADMM loop so cone projections are part of the ADMM updates and the loop runs entirely on the GPU, using custom CUDA kernels (one warp per constraint) for cone projections, cuBLAS GEMM chains for feedback-gain updates and preconditioned conjugate gradient for the QP block.

Key insights

The second-order cone projections in the robust subproblem are separable across constraints, so each can be mapped to its own GPU warp
Folding the cone projections directly into the inner ADMM updates removes the nested solver of NRTO-DR and avoids CPU-GPU data transfer inside the loop
Linear operators that stay constant within an outer iteration (KKT factorisation, preconditioners) can be computed once and reused across inner iterations
The GPU advantage grows with problem size: more obstacles mean more robust constraints and more parallel projections

Main results

Unicycle with 5 obstacles: NRTO-FullADMM cut solve time from 30,423.523 s to 217.932 s (139.60x); quadcopter with 5 obstacles: 13,374.375 s to 199.935 s (66.89x)
At the nominal horizon T = 30: unicycle 32.137 s to 6.231 s (5.16x) and quadcopter 1,327.390 s to 98.585 s (13.46x); at T = 45 the speed-ups were 6.94x and 13.21x
7-DoF Franka Panda with one obstacle: 3,948.523 s (NRTO) vs 1,342.455 s (NRTO-DR) vs 152.384 s (NRTO-FullADMM), a 25.9x speed-up
With the linearization-error extension (NRTO-LE) all methods reached 100% constraint satisfaction across every reported setting, checked with 1,000 random and 1,000 boundary disturbance rollouts
Without NRTO-LE, success fell as uncertainty grew (81-82% at tau = 0.075 for all three solvers); the GPU solvers stayed within a few points of the baseline, e.g. 72% and 69% vs 74% on the quadcopter with 5 obstacles

Limitations

The authors list extending cuNRTO to contact-rich manipulation and locomotion with larger conic programs, to multi-agent swarms and to heterogeneous forms of uncertainty as future work (no dedicated limitations section)
All main results are in simulation; the Franka model is a simplified joint-space model with per-joint damping and unit inertia, and the only real-robot test (a Robotarium unicycle, in the appendix) is qualitative and uses the NRTO policy
Even the fastest solves take from about 6 s to over 200 s, so the method speeds up offline planning rather than giving real-time control in the reported settings
The outer successive-linearization loop still runs on the CPU, and on the smallest unicycle problem (T = 15) NRTO-FullADMM was slower than NRTO-DR (5.983 s vs 5.324 s)
Without the NRTO-LE extension, constraint satisfaction under uniform and boundary disturbances dropped as low as 69% for NRTO-FullADMM on the quadcopter with 5 obstacles

Field impact

Finalist for the RSS 2026 Outstanding Paper Award; 1 citation so far. The paper does not list a code release or project page.

Abstract

Robust trajectory optimization enables autonomous systems to operate safely under uncertainty by computing control policies that satisfy the constraints for all bounded disturbances. However, these problems often lead to large Second Order Conic Programming (SOCP) constraints, which are computationally expensive. In this work, we propose the CUDA Nonlinear Robust Trajectory Optimization (cuNRTO) framework by introducing two dynamic optimization architectures that have direct application to robust decision-making and are implemented on CUDA. The first architecture, NRTO-DR, leverages the Douglas-Rachford (DR) splitting method to solve the SOCP inner subproblems of NRTO, thereby significantly reducing the computational burden through parallel SOCP projections and sparse direct solves. The second architecture, NRTO-FullADMM, is a novel variant that further exploits the problem structure to improve scalability using the Alternating Direction Method of Multipliers (ADMM). Finally, we provide GPU implementation of the proposed methodologies using custom CUDA kernels for SOC projection steps and cuBLAS GEMM chains for feedback gain updates. We validate the performance of cuNRTO through simulated experiments on unicycle, quadcopter, and Franka manipulator models, demonstrating speedup up to 139.6×.

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