Abstract
This paper demonstrates the effectiveness of using global optimal rudder control with the covariance matrix adaptation evolution strategy, applied to a surgeswayyawroll coupled numerical model, in preventing broachingto, 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 historyoptimized approach. With the help of stability analysis of surfriding equilibria, two different mechanisms to successfully prevent broachingto were identified: one is to tend to a periodic orbit with additional slight rudder actions and the other is to stay in surfriding on a position of unstable equilibrium in regular waves for a longer duration with a larger differential gain. These outcomes could facilitate realtime prevention of broachingto 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
 \({X}_{w}\) :

Waveinduced surge force
 \({Y}_{r}\) :

Derivative of sway force with respect to yaw rate
 \({Y}_{v}\) :

Derivative of sway force with respect to sway velocity
 \({Y}_{W}\) :

Waveinduced 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 GrantinAid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant Number 19H02360).
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Maniyappan, S., Umeda, N., Maki, A. et al. Effectiveness and mechanism of broachingto prevention using global optimal control with evolution strategy (CMAES). J Mar Sci Technol (2020). https://doi.org/10.1007/s00773020007434
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Keywords
 CMAES
 Broachingto
 Dynamical system approach
 Unstable surfriding
 Periodic orbit