MANEUVERING AND CONTROL OF MARINE VEHICLES

1. MANEUVERING AND CONTROL OF MARINE VEHICLES Michael S. Triantafyllou Franz S. Hover 指導教授：曾慶耀　博士姓名：鄭敦仁學號：10167027

2. Momentum of a Particle 質點系統的運動方程式 內力的合力為零 因此質點系統的動力方程式變為 N個質點的系統 (system of particles)

3. Linear Momentum in a Moving Frame

4. Linear Momentum in a Moving Frame 定義重心向量 利用向量三重積特性化簡

5. Linear Momentum in a Moving Frame 速度 質心 角速度 外力

6. Linear Momentum in a Moving Frame 縱移 橫移 起伏 橫搖 俯仰 平擺

7. Example: Mass on a String The vector equation at the start is Consider a mass on a string, being swung around around in a circle at speed U, with radius r.

8. Moving Frame Affixed to Mass

9. Rotating Frame Attached to Pivot Point

10. NONLINEAR COEFFICIENTS IN DETAIL We employ the following basic facts and assumptions to derive the fluid forces acting on a ship, submarine or vehicle: 流體的慣性項是線性取決於加速度 預計單獨依靠加速度慣性力 消除一定數量零或很小的係數 在很廣的參數範圍內，導出方程是足夠精確的。 1.We retain only first order acceleration terms. 2.We do not include terms coupling velocities and accelerations. 3.We consider port/starboard symmetry. 4.We retain up to third order terms.

11. Helpful Facts To exploit symmetries, we consider the following simple facts 方程對所有實數x都成立，則f為偶函數 n為奇數 方程對所有實數x都成立，則f為奇函數 n為偶數

12. Nonlinear Equations in the Horizontal Plane To demonstrate the methodology, we will derive the governing nonlinear equations of motion in the horizontal plane (surge, sway and yaw), employing the assumptions above.

13. Fluid Force X we derive the following expression for the fluid force X, valid up to third order: This is a result of port/starboard symmetry and is expressed as: In summary, the symmetries provide the following zero coefficients:

14. Fluid Force Y Finally, we derive the following expansion for Y :

15. Fluid Moment N

16. Surface Vessel Linear Model Letting u=U+u，where U>>uand eliminating higher-order terms, this set is :

17. a state-space representation of the sway/yaw system is 矩陣M是質量或慣性矩陣，這是可逆的。最後形成的方程式是一個標準之一。 A代表內部動態系統。 B是一個增益矩陣的控制和干擾輸入。

18. Stability of the Sway/Yaw System A necessary and sufficient condition for stability of this ODE system is that each coefficient must be greater than zero:

19. Stability of the Sway/Yaw System The denominator for A’s components reduces to Hence the first condition for stability is met:

20. Stability of the Sway/Yaw System C被稱為船隻穩定性參數。 For the second condition, since the denominators of the Aij are identical, we have only to look at the numerators. For stability, we require When only the largest terms are considered for a vessel, a simpler form is common: The terms of C compete, and yaw/sway stability depends closely on the magnitude and sign of Nv. Adding more surface area aft drives Nv more positive, increasing stability as expected.

21. Basic Rudder Action in the Sway/Yaw Model Adding Yaw Damping through Feedback Heading Control in the Sway/Yaw Model Employing the control law

22. Response of the Vessel to Step Rudder Input Phase1:Accelerations Dominate When the rudder first moves, acceleration terms dominate, since the velocities are zero. The equation looks like this:

23. Phase3: Steady State When the transients have decayed, the vessel is in a steady turning condition, and the accelerations are zero. The system equations reduce to The steady turning rate is thus approximated by

24. With C> 0, the steady-state yaw rate is negative. If the vessel is unstable (C< 0), it turns in the opposite direction than expected. This turning rate equation can also be used to estimate turning radius R:

25. Thankyou for your attention