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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- \(c\) :
-
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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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 26, 382–394 (2021). https://doi.org/10.1007/s00773-020-00743-4
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DOI: https://doi.org/10.1007/s00773-020-00743-4