1. Thrust Allocation Ole Jakob Sørdalen, PhD Counsellor Science & Technology The Royal Norwegian Embassy, Singapore

2. Controller architecture • Sensor signal processing • Signal QA • Filtering and weighting • Vessel Model • Separate LF/WF model • Kalman filter estimator • Mooring model • Optimal Control • Positioning and damping • Reduce fuel, tear and wear • Mooring line break compensation • Feedforward control • Wind load compensation • Reference model tracking • Optimal thrust allocation • Adaptive control

3. Thruster configuration

4. Problem statement Given desired forces and moment from the controller, tc =[txc, tyc, tyc]T. Determine thrusts T=[T1, T2,..., Tn]T and azimuth angles a=[a1, a2,..., an]T so that • ||A(a) T - tc|| is minimal to minimize the error • ||T|| is minimal to minimize fuel consumption • ai(t) is slowly varying to reduce wear and tear Assumption here: Thrusters are bi-directional

5. Challenges • Singularities: the singular values of A(a) can be small; A(a) T = t ,  simple pseudo inversion can give high gains and high thrust • An azimuth thruster cannot be considered as two independent perpendicular thrusters since the rotation velocity is limited • If the thruster is not symmetric, how should the azimuth respond to 180o changes of desired thrust directions? • Forbidden zones

6. T 3 t t T y x 1 T t 2 y Singularities There is an azimuth angle where det A(ais) = 0 A(ais) cannot be inverted Example of a singular configuration:

7. Singular Value Decomposition Any m x n matrix A can be factored into A = U S VT Where U snd V are orthogonal matrices. S is given by

8. About SVD ... • Coloumns of U: orthonormal eigen vectors of AAT • Coloumns of V: orthonormal eigen vectors of ATA • si = sqrt (eig(ATA) i) • Pseudo inverse of A: A+ = V S+ UT • The least square solution to Ax = y is x = A+y i.e. either min ||Ax – y||2 or min ||x|| 2 Can use weighted LS.

9. Example: plot of smallest singular value Bow azimuth fixed 90o. Aft azimuts rotate

10. Area where s < 0.05

11. Area where s < 0.015

12. Fixed angle between aft azimuths s < 0.05

13. Forces and torque as function of azimuth angles

14. How to determine angles a? • Consider azimuth thrusters as two perpendicar fixed thrusters • New (expanded) relation: AeTe = t • desired ”expanded” thrust vector Ted: Ted = A+etc

15. How to determine thrust T? • Note: T = A+(af)tc large T close to singular configurations! • Modified pseudo inverse: Ad+ = V Sd+ UT  T = V Sd+ UTtc

16. Geomtrical interpretation • Commanded thrust in directions representing small singular values are neglected • This is GOOD • Azimuth angles are always oriented towards the mean environment forces & torques • Other commanded forces typically due to noise  efficient ”geometrical” filtering of this noise

17. Simulation results

22. Fixed azimuth angles:

23. Features • Automatic azimuth control • Automatic avoidance of forbidden sectors: not shown here • Optimal direction control • Smooth turning • Optimal singularity handling • Avoidance of unnecessary use of thrust • Reduced wear and tear of propulsion devices • Optimal priority handling • Among thruster devices • Among surge, sway, yaw