OpenSCvx: An Open-Source Modular and Extensible Nonlinear Trajectory Planning Package

Christopher R. Hayner; Griffin J. Norris; Fabio Spada; Samet Uzun; Avi Mittal; Behcet Acıkmese; Karen Leung

cs.ROAug 21, 2026

OpenSCvx: An Open-Source Modular and Extensible Nonlinear Trajectory Planning Package

Authors: Christopher R. Hayner, Griffin J. Norris, Fabio Spada, Samet Uzun, Avi Mittal, Behcet Acıkmese, Karen Leung

Organizations: Dept. of Aeronautics and Astronautics, University of Washington, Seattle, WA, USA. · General Robotics, Seattle, WA, USA. · Dept. of Mechanical Engineering, University of California, Berkeley, Berkley CA, USA. · Zoox, San Francisco, CA, USA. ·

Abstract

Trajectory optimization computes dynamically feasible motions that enable autonomous systems to accomplish complex tasks while satisfying operational and environmental constraints. This tutorial presents OpenSCvx, an open-source Python framework that bridges the gap between high-level problem specification and efficient numerical optimization. Rather than requiring users to derive solver-specific mathematical formulations, OpenSCvx provides a symbolic modeling interface that automatically constructs and solves trajectory optimization problems from modular descriptions of objectives, dynamics, and constraints. Beyond simplifying problem formulation, OpenSCvx supports (i) continuous-time constraint modeling, (ii) temporal and logical specifications, (iii) automatic vectorization for scalable and batched optimization, and (iv) a modular architecture that enables new algorithms, models, and solver backends to be incorporated with minimal effort. These capabilities allow researchers and practitioners to rapidly prototype, solve, and extend state-of-the-art trajectory optimization methods.

Figures & tables

Fig. 1 : Example trajectory generated with OpenSCvx : a quadrotor flies through a sequence of gates while controlling a laser pointer to trace the OpenSCvx logo on a target surface.

Panel of Figure 2 — Fig. 2 : A taxonomy of trajectory planning methods. The trajectory optimization categories reflect differences in problem formulation and optimization structure; numerical solvers used to solve intermediate subproblems such as IPM ’s or active set methods are not shown in this figure.

Framework	Transcription	Auto Trans.	Auto Diff	CTCS	STL / Logic	Mesh Adapt.	JIT / Compile	Batch / GPU	Language
General Modeling Frameworks
CasADi [ 7 ]	User-defined	–	✓	–	–	–	✓	–	C++ / Python / MATLAB
CusADi [ 42 ]	User-defined	–	✓	–	–	–	✓	✓	C++ / Python
Drake [ 76 ]	Collocation	✓	✓	–	–	–	✓	–	C++ / Python
CVXPY [ 22 ] / DNLP [ 19 ]	User-defined	–	✓	–	–	–	–	–	Python
JuMP [ 27 ]	User-defined	–	✓	–	–	–	✓	–	Julia

TABLE I : Representative trajectory optimization software and algorithmic paradigms. OpenSCvx combines the high-level problem specification of modern trajectory optimization frameworks with the structure-exploiting advantages of continuous-time successive convexification. Objective capability features replace subjective ratings: Auto Trans. indicates automatic continuous-to-discrete transcription; Auto Diff indicates automatic differentiation without manual Jacobians; CTCS indicates continuous-time path constraint satisfaction; STL / Logic indicates native support for Signal Temporal Logic or state-triggered constraints; Mesh Adapt. indicates adaptive mesh refinement; JIT / Compile indicates JIT/AOT compilation or C-code generation; and Batch / GPU indicates native vectorized batching or GPU acceleration.

Panel of Figure 4 — Fig. 3 : The constraint a @ (p - w) >= Norm(p - w) as authored (left) and after canonicalization (right). Leaf nodes are red; operator nodes and the edges to their children are blue. Because the subexpression p - w feeds both the inner product and the norm, it has two parents – the graph is a directed acyclic graph rather than a tree. Canonicalization rewrites the inequality into residual form, moving all terms to one side against a constant $0$ while reusing the shared subgraph.

Fig. 4 : Monte Carlo analysis of 6DoF powered descent guidance (PDG): 200 random initial positions denoted by the blue points. The coloring of the trajectories denotes the 2-norm of velocity, $\|v\|_{2}$ .

Panel of Figure 6 — Fig. 5 : Continuous-time successive convexification (CT-SCvx) . The red blocks are problem parsing steps, the blue blocks are the iterative steps.

Panel of Figure 7 — Fig. 6 : Nodal feasibility does not imply continuous-time feasibility. All nodes (green) lie outside the obstacle (blue), yet the trajectory connecting them cuts through the obstacle (hatched red), motivating continuous enforcement.

Panel of Figure 8 — Fig. 7 : Control Parameterization for ZOH and FOH

Panel of Figure 9 — Fig. 8 : Multiple shooting transcription. Early iterations (light gray) exhibit large defect gaps between integrated states (red) and the next state variables (yellow). As the algorithm converges, these defects are reduced, yielding a dynamically feasible trajectory (black).

	$\displaystyle\nu_{\mathrm{vc}}$	$\displaystyle=\|\mathbf{x}_{k+1}-\mathbf{x}_{\mathrm{prop}}\|,$
	$\displaystyle\nu_{G}$	$\displaystyle=\max(0,G(\mathbf{z}_{k},\theta)),$
	$\displaystyle\nu_{H}$	$\displaystyle=\|H(\mathbf{z}_{k},\theta)\|$

Algorithm 2 Augmented Lagrangian-inspired Autotuning

Problem	Rel. error (%)	Pos. RMSE	$t_{\mathrm{init}}$ (s)	$t_{\mathrm{solve}}$ (s)
Brachistochrone	$6.65\times 10^{-6}$	$1.01\times 10^{-4}$ m	$1.614$	$0.053$
Hohmann transfer	$5.04\times 10^{-7}$	$4.49\times 10^{-2}$ km	$1.813$	$0.132$

TABLE II : Verification of OpenSCvx against analytical optima for two canonical nonlinear optimal control problems.

Fig. 9 : Simulated optimal state – HS

Solver	Avg CPU (s) $\downarrow$	Std CPU (s) $\downarrow$
OpenSCvx	1.208	0.266
GPOPS-II	0.751	0.014
ICLOCS2	1.898	0.025
MAPTOR	10.540	0.136
PSOPT	17.138	0.807

TABLE III : Computational performance – HS

Solver	Reported $J$ $\downarrow$	Simulated $J$ $\downarrow$	% error $\downarrow$
OpenSCvx	3.890	3.890	$\approx$ 0.00 %
GPOPS-II	3.362	3.351	0.33%
ICLOCS2	3.362	3.277	2.53%
MAPTOR	3.362	4.556	35.52%
PSOPT	3.362	3.361	0.03%

TABLE IV : Solver vs. simulated objective – HS

Fig. 10 : Path constraint violations – HS

Fig. 11 : Simulated optimal altitude profile – ALV

Solver	Avg CPU (s) $\downarrow$	Std CPU (s) $\downarrow$
OpenSCvx	2.125	0.379
GPOPS-II ( IPOPT )	115.483	1.323
GPOPS-II ( SNOPT )	8.442	0.161
MAPTOR	3.844	0.152
PSOPT	1.681	0.302

TABLE V : Computational performance – ALV

Solver	Final $r$ err. (km) $\downarrow$	Final $v$ err. (m/s) $\downarrow$
OpenSCvx	1.67e-03	6.26e-03
GPOPS-II ( IPOPT )	7.03e-04	4.27e-01
GPOPS-II ( SNOPT )	2.20e-02	5.76e-02
MAPTOR	3.77e-03	1.16e+00
PSOPT	1.01e-03	1.95e-03

TABLE VI : Position/velocity integration accuracy – ALV

	$\|\Delta a\|\,\downarrow$	$\|\Delta e\|\,\downarrow$	$\|\Delta i\|\,\downarrow$	$\|\Delta\Omega\|\,\downarrow$	$\|\Delta\omega\|\,\downarrow$
Solver	(km)	( $10^{-6}$ )	( $10^{-6}\,{}^{\circ}$ )	( $10^{-6}\,{}^{\circ}$ )	( $10^{-3}\,{}^{\circ}$ )
OpenSCvx	$0.212$	$2.42$	0.0022	0.029	$0.0402$
GPOPS-II (I.)	$0.810$	$10.3$	$0.532$	$1.04$	$5.60$
GPOPS-II (S.)	$0.213$	$2.71$	$22.1$	$94.7$	$0.601$
MAPTOR	$2.030$	$25.9$	$0.883$	$1.68$	$15.4$
PSOPT	0.013	0.113	$0.289$	$2.57$	0.0219

TABLE VII: Terminal orbital-element absolute errors against the target orbit (

a=24361140

e=0.7308

i=28.5^{\circ}

\Omega=269.8^{\circ}

\omega=130.5^{\circ}

) – ALV

Solver	$J$ (kg) $\uparrow$
OpenSCvx	7514.45
GPOPS-II ( IPOPT )	7529.71
GPOPS-II ( SNOPT )	7529.67
MAPTOR	7529.71
PSOPT	7529.71

TABLE VIII : Solver/simulated objective – ALV

Fig. 12 : Simulated optimal altitude and latitude profiles - RRV

Solver	Reported $J$ ( ∘ ) $\uparrow$	Simulated $J$ ( ∘ ) $\uparrow$	% error $\downarrow$
OpenSCvx (f.)	34.1163	34.1160	0.00 %
OpenSCvx (s.)	34.1544	34.1544	0.00 %
GPOPS-II	34.1641	34.1641	0.00 %
ICLOCS2	34.1422	34.1554	0.04%
MAPTOR	34.1641	34.1641	0.00 %
PSOPT	34.1412	34.1521	0.03%

TABLE IX : Solver vs. simulated objective – RRV

Solver	Avg CPU (s) $\downarrow$	Std CPU (s) $\downarrow$
OpenSCvx (f.)	1.869	0.027
OpenSCvx (s.)	2.314	0.114
GPOPS-II	1.025	0.129
ICLOCS2	4.052	0.313
MAPTOR	4.197	0.278
PSOPT	1.643	0.418

TABLE X : Computational performance – RRV

Panel of Figure 24 — Fig. 13 : UR5e pen-tracing experiment. Left: sequence composite of the hardware execution. Right: optimized simulation.

Fig. 14 : Time-optimal 6-DoF quadrotor racing trajectory.

Panel of Figure 26 — Fig. 15 : Demonstration of implication-based compound state-triggered constraints in a 6-DoF powered descent guidance problem. The shaded areas denote the feasible regions. Note the intersample constraint satisfaction in fig. 15(a) .

Fig. 16 : Real-world experiment of the 6-DoF compound-STC problem. Left: NASA SENSS 6-DoF tendon robot executing the trajectory. Right: aligned nonlinear single-shot simulation. Both panels are sequence composites.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Arithmetic		JAX	CVXPY
$+,-,*,/$	Addition, subtraction, multiplication, division	✓	✓
$\leq$ , ==	Inequality, equality	✓	✓
@	Matrix multiplication	✓	✓
	Power	✓	✓
Sqrt()	Element-wise square root	✓	✓
Exp()	Element-wise exponential	✓	✓

Appendix

TABLE XI : Basic Mathematical Operators

Logic & Signal Temporal Logic
All(), Any()	Logical AND / OR over all predicates
And(), Or(), Not()	Conjunction, disjunction, negation
Always()	Temporal operator, formula holds over a node or time interval
IfThen()	Implication
Cond()	Conditional, selects a branch by predicate
IntegerVariable()	Discrete/integer variable

Appendix

TABLE XII : Specialized Operators ( JAX -only)

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Open paper AskPDF arXiv

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 implementations 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 speedups of up to 139.6

\times

. More details are available at https://cunrto.github.io.

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