Effectiveness and mechanism of broaching-to prevention using global optimal control with evolution strategy (CMA-ES)

Abstract

This paper demonstrates the effectiveness of using global optimal rudder control with the covariance matrix adaptation evolution strategy, applied to a surge-sway-yaw-roll coupled numerical model, in preventing broaching-to, occurring under conventional PD rudder controls in stern quartering waves. In particular, the PD control with optimized PD parameters is more efficient than the time history-optimized approach. With the help of stability analysis of surf-riding equilibria, two different mechanisms to successfully prevent broaching-to were identified: one is to tend to a periodic orbit with additional slight rudder actions and the other is to stay in surf-riding on a position of unstable equilibrium in regular waves for a longer duration with a larger differential gain. These outcomes could facilitate real-time prevention of broaching-to in actual seas.

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Abbreviations

\(c\) :

Wave celerity

F n :

Nominal Froude number

\(g\) :

Acceleration due to gravity

\(\text{GZ}\) :

Righting arm

H :

Wave height

\({I}_{xx}\) :

Moment of inertia in roll

\({I}_{zz}\) :

Yaw moment of inertia

\(J\) :

Objective function

\({J}_{xx}\) :

Added roll moment of inertia

\({J}_{zz}\) :

Added yaw moment of inertia

\(K\) :

Penalty value for capsizing

\({K}_{\rm{D}}\) :

Rudder differential gain

\({K}_{p}\) :

Derivative of roll moment with respect to roll rate

\({K}_{P}\) :

Rudder proportional gain

\({K}_{r}\) :

Derivative of roll moment with respect to yaw rate

\({K}_{v}\) :

Derivative of roll moment with respect to sway velocity

\({K}_{w}\) :

Wave induced roll moment

\({K}_{\delta }\) :

Derivative of roll moment with respect to rudder angle

\({K}_{\phi }\) :

Derivative of roll moment with respect to roll angle

\(m\) :

Ship mass

\({m}_{x}\) :

Added mass in surge

\({m}_{y}\) :

Added mass in sway

\(n\) :

Propeller revolution number

\({N}_{r}\) :

Derivative of yaw moment with respect to yaw rate

\({N}_{v}\) :

Derivative of yaw moment with respect to sway velocity

\({N}_{w}\) :

Wave induced yaw moment

\({N}_{\delta }\) :

Derivative of yaw moment with respect to rudder angle

\({N}_{\phi }\) :

Derivative of yaw moment with respect to roll angle.

\(p\) :

Roll rate

\(\mathbf{p}\) :

Control vector

\(R\) :

Ship resistance

\(r\) :

Yaw rate

\({t}_{f}\) :

Final time

\(t\) :

Time

\(T\) :

Propeller thrust

\({T}_{E}\) :

Time constant of steering gear

\(u\) :

Surge velocity

\(\mathbf{u}\) :

Control vector

\(v\) :

Sway velocity

\({\mathrm{v}_i}\) :

Eigenvector

\(\mathbf{x}\) :

State vector

\({X}_{w}\) :

Wave-induced surge force

\({Y}_{r}\) :

Derivative of sway force with respect to yaw rate

\({Y}_{v}\) :

Derivative of sway force with respect to sway velocity

\({Y}_{W}\) :

Wave-induced sway force

\({Y}_{\delta }\) :

Derivative of sway force with respect to rudder angle

\({Y}_{\phi }\) :

Derivative of sway force with respect to roll angle

\({Z}_{H}\) :

Height of center of sway force due to lateral motions

\(\delta\) :

Rudder angle

\(\upepsilon\) :

Small perturbation

\(\lambda\) :

Wave length

\(\phi\) :

Roll Angle

\({\phi }_{v}\) :

Angle of vanishing stability

\(\upchi\) :

Heading angle

\({\chi }_{c}\) :

Desired heading angle for autopilot

\({\upxi }_{\mathrm{G}}\) :

Longitudinal position of center of gravity from wave trough

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Acknowledgements

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant Number 19H02360).

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Correspondence to Naoya Umeda.

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Maniyappan, S., Umeda, N., Maki, A. et al. Effectiveness and mechanism of broaching-to prevention using global optimal control with evolution strategy (CMA-ES). J Mar Sci Technol (2020). https://doi.org/10.1007/s00773-020-00743-4

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Keywords

  • CMA-ES
  • Broaching-to
  • Dynamical system approach
  • Unstable surf-riding
  • Periodic orbit