Contraction-Based Trajectory Tracking Control for AUVs on SE(3) with Hierarchical Gain Certification

Authors

Jinjun Jia, Kang An, Yuchen Liao, Xun Yan, Tiedong Zhang, Dapeng Jiang

Abstract

This paper develops a contraction-certified trajectory-tracking and gain-selection framework for fully actuated autonomous underwater vehicles on SE(3). The vehicle dynamics are represented in port-Hamiltonian form with a Rayleigh-type dissipation potential, and a dual potential shaping controller provides an energy-structured rotational–translational cascade. Regional contraction certificates are derived separately for the rotational and translational subsystems. The rotational analysis uses fixed left-trivialised momentum coordinates and retains anisotropic-inertia effects and the complete off-diagonal differential coupling. The translational analysis applies to a general known symmetric positive-definite inertia matrix through an attitude-cover semidefinite programme, with an exact endpoint reduction for isotropic inertia. A scaled composite metric combines the subsystem certificates and guarantees every strict complete-cascade rate below the slower subsystem rate. Large initial attitude errors are handled by an energy-entry phase followed by contraction within a prescribed tube, without controller switching. The four-dimensional gain-selection problem is decomposed into two independent two-dimensional offline searches using bisection and SDP/LMI feasibility tests. Numerical studies on the ODIN AUV quantify the region–gain–rate trade-off and examine small-angle, large-angle, and near-antipodal manoeuvres. The framework certifies complete-cascade rates of 0.042096s−1 and 0.008524s−1 for the 60∘/60∘ and 150∘/80∘ regions, respectively.

Citation

• Journal: Journal of Marine Science and Engineering
• Year: 2026
• Volume: 14
• Issue: 16
• Pages: 1465
• Publisher: MDPI AG
• DOI: 10.3390/jmse14161465

BibTeX

@article{Jia_2026,
  title={{Contraction-Based Trajectory Tracking Control for AUVs on SE(3) with Hierarchical Gain Certification}},
  volume={14},
  ISSN={2077-1312},
  DOI={10.3390/jmse14161465},
  number={16},
  journal={Journal of Marine Science and Engineering},
  publisher={MDPI AG},
  author={Jia, Jinjun and An, Kang and Liao, Yuchen and Yan, Xun and Zhang, Tiedong and Jiang, Dapeng},
  year={2026},
  pages={1465}
}

Download the bib file

References

• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/260000000 – 10.1561/2600000002
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278- – 10.1016/s0005-1098(01)00278-3
• Guerrero-Sánchez ME, Hernández-González O, Valencia-Palomo G, Mercado-Ravell DA, López-Estrada FR, Hoyo-Montaño JA (2021) Robust IDA-PBC for under-actuated systems with inertia matrix dependent of the unactuated coordinates: application to a UAV carrying a load. Nonlinear Dyn 105(4):3225–3238. https://doi.org/10.1007/s11071-021-06776- – 10.1007/s11071-021-06776-7
• Guerrero-Sánchez ME, Montoya-Morales JR, Valencia-Palomo G, Hernández-González O (2024) Robust IDA-PBC for non-separable PCH systems under time-varying external disturbances. Nonlinear Dyn 113(4):3499–3510. https://doi.org/10.1007/s11071-024-10380- – 10.1007/s11071-024-10380-w
• Donaire A, Perez T (2012) Dynamic positioning of marine craft using a port-Hamiltonian framework. Automatica 48(5):851–856. https://doi.org/10.1016/j.automatica.2012.02.02 – 10.1016/j.automatica.2012.02.022
• Desai RP, Manjarekar NS (2024) Interconnection and damping assignment passivity‐based control for dynamic steering position stabilization of an underactuated AUV. Adv Control Appl 6(3). https://doi.org/10.1002/adc2.22 – 10.1002/adc2.225
• Fujimoto K, Sakurama K, Sugie T (2003) Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica 39(12):2059–2069. https://doi.org/10.1016/j.automatica.2003.07.00 – 10.1016/j.automatica.2003.07.005
• Donaire A, Romero JG, Perez T (2017) Trajectory tracking passivity-based control for marine vehicles subject to disturbances. Journal of the Franklin Institute 354(5):2167–2182. https://doi.org/10.1016/j.jfranklin.2017.01.01 – 10.1016/j.jfranklin.2017.01.012
• Donaire A, Guadalupe Romero J, Perez T (2015) Passivity-based Trajectory-tracking for Marine Craft with Disturbance Rejection. IFAC-PapersOnLine 48(16):19–24. https://doi.org/10.1016/j.ifacol.2015.10.25 – 10.1016/j.ifacol.2015.10.252
• Lv C, Yu H, Chen J, Zhao N, Chi J (2022) Trajectory tracking control for unmanned surface vessel with input saturation and disturbances via robust state error IDA-PBC approach. Journal of the Franklin Institute 359(5):1899–1924. https://doi.org/10.1016/j.jfranklin.2022.01.03 – 10.1016/j.jfranklin.2022.01.036
• Bullo F, Murray RM (1999) Tracking for fully actuated mechanical systems: a geometric framework. Automatica 35(1):17–34. https://doi.org/10.1016/s0005-1098(98)00119- – 10.1016/s0005-1098(98)00119-8
• Sanyal A, Nordkvist N, Chyba M (2011) An Almost Global Tracking Control Scheme for Maneuverable Autonomous Vehicles and its Discretization. IEEE Trans Automat Contr 56(2):457–462. https://doi.org/10.1109/tac.2010.209019 – 10.1109/tac.2010.2090190
• Maithripala DHS, Berg JM (2015) An intrinsic PID controller for mechanical systems on Lie groups. Automatica 54:189–200. https://doi.org/10.1016/j.automatica.2015.01.00 – 10.1016/j.automatica.2015.01.005
• Liao Y, Yan X, An K, Wang Z, Zhang T, Wu S, Jiang D (2024) Fixed-time geometric tracking control of autonomous underwater vehicles on SE(3). Ocean Engineering 311:118757. https://doi.org/10.1016/j.oceaneng.2024.11875 – 10.1016/j.oceaneng.2024.118757
• Yaghmaei A, Yazdanpanah MJ (2017) Trajectory tracking for a class of contractive port Hamiltonian systems. Automatica 83:331–336. https://doi.org/10.1016/j.automatica.2017.06.03 – 10.1016/j.automatica.2017.06.039
• Barabanov N, Ortega R, Pyrkin A (2019) On contraction of time-varying port-Hamiltonian systems. Systems & Control Letters 133:104545. https://doi.org/10.1016/j.sysconle.2019.10454 – 10.1016/j.sysconle.2019.104545
• Yaghmaei A, Yazdanpanah MJ (2024) On Contractive Port-Hamiltonian Systems With State-Modulated Interconnection and Damping Matrices. IEEE Trans Automat Contr 69(1):622–628. https://doi.org/10.1109/tac.2023.327339 – 10.1109/tac.2023.3273394
• Manchester IR, Slotine J-JE (2017) Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design. IEEE Trans Automat Contr 62(6):3046–3053. https://doi.org/10.1109/tac.2017.266838 – 10.1109/tac.2017.2668380
• Wu D, Yi B, Manchester IR (2024) Control Contraction Metrics on Submanifolds. 2024 IEEE 63rd Conference on Decision and Control (CDC) 3735–374 – 10.1109/cdc56724.2024.10886040
• Vang B, Tron R (2019) Geometric Attitude Control via Contraction on Manifolds with Automatic Gain Selection. 2019 IEEE 58th Conference on Decision and Control (CDC) 6138–614 – 10.1109/cdc40024.2019.9029723
• Vang B, Tron R (2020) Global Attitude Control via Contraction on Manifolds with Reference Trajectory and Optimization. 2020 59th IEEE Conference on Decision and Control (CDC) 2006–201 – 10.1109/cdc42340.2020.9303862
• Yaghmaei A, Yazdanpanah MJ (2015) Trajectory tracking of a class of port Hamiltonian systems using Timed IDA-PBC technique. 2015 54th IEEE Conference on Decision and Control (CDC) 5037–504 – 10.1109/cdc.2015.7403007
• LOHMILLER W, SLOTINE J-JE (1998) On Contraction Analysis for Non-linear Systems. Automatica 34(6):683–696. https://doi.org/10.1016/s0005-1098(98)00019- – 10.1016/s0005-1098(98)00019-3
• Simpson-Porco JW, Bullo F (2014) Contraction theory on Riemannian manifolds. Systems & Control Letters 65:74–80. https://doi.org/10.1016/j.sysconle.2013.12.01 – 10.1016/j.sysconle.2013.12.016
• Slotine JE (2003) Modular stability tools for distributed computation and control. Adaptive Control & Signal 17(6):397–416. https://doi.org/10.1002/acs.75 – 10.1002/acs.754
• Fossen TI (2011) Handbook of Marine Craft Hydrodynamics and Motion Contro – 10.1002/9781119994138
• Duong T, Altawaitan A, Stanley J, Atanasov N (2024) Port-Hamiltonian Neural ODE Networks on Lie Groups for Robot Dynamics Learning and Control. IEEE Trans Robot 40:3695–3715. https://doi.org/10.1109/tro.2024.342843 – 10.1109/tro.2024.3428433
• Lee T (2012) Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers. Systems & Control Letters 61(1):231–237. https://doi.org/10.1016/j.sysconle.2011.10.01 – 10.1016/j.sysconle.2011.10.017
• Lee JM (2018) Introduction to Riemannian Manifolds. Springer International Publishin – 10.1007/978-3-319-91755-9