Abstract
The paper presents a methodology to design an electric controller for marine cycloidal propeller. The controller is designed considering the torque and the rotational speed limit of the motor. The influence of manoeuvring dynamics of the ship, rotational speed of the disc, eccentricity ratios and torque, pitching speed and pitch angle of the blades on the controller design are investigated. Feedback signals are used for the controller and combined with multiple PID control logic for controlling the motion of disc and blades. The proposed PID controller helps to stabilize the rotational speed of propeller blades and disc when requirement of torque exceeds the maximum limit of motor torque. The proposed control algorithm enhances the chances of optimizing propulsion efficiency of the blade. This is achieved due to decoupling of the motion of individual blades. Simulation results of different manoeuvring and straight run cruising conditions demonstrate the application of proposed control scheme. Finally, the simulated results are validated with the experimental results of mechanically controlled cycloidal propeller.
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Abbreviations
 \(\left\{ \begin{gathered} S \hfill \\ P \hfill \\ \end{gathered} \right\}\) :

Starboard or port propeller
 a (m):

Chord length of blade
 C _{D} :

Coefficient of drag
 CF (N):

Centrifugal force
 CG:

Centre of gravity of blade aerofoil section
 C _{L} :

Coefficient of lift
 C _{M} :

Coefficient of moment
 CCW:

Counterclockwise direction
 CW:

Clockwise direction
 dt (s):

Time step
 D (m):

Diameter of disc
 \({e_{\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) :

Eccentricity ratio
 \({e_{1\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (m):

The distance of eccentricity point along yaxis from disc centre in disc coordinate system
 \({e_{2\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (m):

The distance of eccentricity point along xaxis from disc centre in disc coordinate system
 F_{D} (N):

Drag force on propeller blade
 \({F_{X\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (N):

Component of thrust on the port/starboard disc along the xaxis in disc coordinate system
 \({F_{Y\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (N):

Component of thrust on the port/starboard disc along the yaxis in disc coordinate system
 I_{b} (kgm^{2}):

Mass moment of inertia of the blade about zaxis
 I_{d} (kgm^{2}):

Mass moment of inertia of the disc about the zaxis
 K _{db} :

Derivative gain of the blade controller
 k _{dd} :

Derivative gain of the disc controller
 K _{Ib} :

Integral gain due to blade pitching of the blade controller
 K _{Id} :

Integral gain due to disc rotation
 K _{Pb} :

Proportional gain of the blade controller
 k _{Pd} :

Proportional gain of the disc controller
 L (N):

Lift force on blade
 L_{S} (m):

Length of ship
 M_{B} (kg):

Mass of the propeller blade
 M_{D} (kg):

Mass of the propeller disc with all accessories including blade and machinery
 N_{B} (rpm):

Rotational speed of propeller blade
 \({N_{D\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (rpm):

Rotational speed of propeller disc
 O :

Centre of propeller disc
 P :

Steering centre
 P_{b} (KW):

Powers consumed by blade actuator
 P_{d} (KW):

Powers consumed by disc actuator
 \({Q_{BF\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (Nm):

Torque due to bearing friction on propeller disc
 \({Q_{BL\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Torque due to fluid friction on the propeller disc
 \({Q_{TH\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Torque on propeller disc due to resultant thrust on vertical bearing
 \({Q_{D\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Hydrodynamic torque on propeller disc
 r (m):

Radius of the propeller blade shaft
 r_{b} (m):

Average radius of propeller blade bearing
 R (m):

Radius of propeller disc
 R_{d} (m):

Average radius of propeller disc bearing
 R_{TS} (N):

Resistance force on ship
 Rn:

Reynolds number of propeller disc
 t (s):

Time
 \({T_{x\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (N):

Thrust acting on the blade stock due to hydrodynamic action along the xaxis of disc coordinate system
 \({T_{y\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (N):

Thrust acting on the blade stock due to hydrodynamic action along the yaxis of disc coordinate system
 u (m/s):

Velocity of ship
 V_{R} (m/s):

Resultant inflow velocity on blade
 V_{T} (m/s):

Tangential velocity on propeller disc
 v_{x} (m/s):

x component of velocity of ship
 v_{y} (m/s):

y component of velocity of ship
 Z :

Total number of blades in a propeller unit
 \({\alpha _{\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (deg):

Angle of attack
 \({\theta _{\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (deg):

Blade orbit angle
 \({\dot {\theta }_{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\)(rad/s):

Angular velocity of propeller disc
 \({\ddot {\theta }_{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (rad/s^{2}):

Angular acceleration of propeller disc
 \({\delta _{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (deg):

Pitch angle of propeller blade
 \({\dot {\delta }_{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (rad/s):

Angular velocity of propeller blade
 \({\ddot {\delta }_{\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (rad/s^{2}):

Angular acceleration of propeller blade
 \(\lambda\) :

Advance coefficient of propeller disc
 \({\tau _{\left\{ {\begin{array}{*{20}{c}} S \\ P \end{array}} \right\}}}\) (Nm):

Torque on the stock of a single blade
 \({\tau _{{\text{BF}}\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Torque due to bearing friction on propeller blade
 \({\tau _{TH\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Torque on single blade due to friction force in vertical bearing
 \({\tau _{HY\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (Nm):

Torque on single blade due to hydrodynamic lift and drag force
 \({\phi _{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) (deg):

Angle of the connected line of eccentricity point and blade stock with the positive \({x_2}\)axis of disc coordinate system
 \({\eta _{O\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\) :

Openwater efficiency
 \({\eta _{\text{D}}}\) :

Propulsive efficiency
 \({\mu _{\left. h \rightb}}\) :

The rolling friction coefficient of the horizontal support bearing of propeller blade
 \({\mu _{\left. h \rightd}}\) :

The rolling friction coefficient of the horizontal support bearing of the disc
 \({\mu _{\left. v \rightb}}\) :

The vertical bearing friction coefficient of propeller blade
 \({\mu _{\left. v \rightd}}\) :

The vertical bearing friction coefficient of propeller disc
 \(\nu\) :

Kinematic viscosity
 \({\xi _{\left\{ \begin{subarray}{l} S \\ P \end{subarray} \right\}}}\)(deg):

Angle between resultant flow and thrust direction
 CP:

Centre of pressure
 ECMCP:

Electronically controlled marine cycloidal propeller
 MCP:

Marine cycloidal propeller
 PPS:

Pulses per second
 SP:

Stock position of the blade
 VSP:

Voith Schneider Propeller
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Acknowledgements
The first author is awarded by “High value PhD Scholarship” under “Prof R P Gokarn Innovation Grant” by Tiara Charitable Foundation for experimentation. First author would like to express deepest gratitude and hearted acknowledgment of thankfulness to coauthors for their full support, engagement, expert guidance, encouragement and valuable comments and suggestions during the research work.
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Nandy, S., Prabhu J, J., Nagarajan, V. et al. PIDtype controller for marine cycloidal propeller: a simulation study. J Mar Sci Technol 25, 111–137 (2020). https://doi.org/10.1007/s00773019006352
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Keywords
 PID control
 Marine cycloidal propeller
 manoeuvring
 Hydrodynamics
 Pitch angle