Nonlinear dynamics of yaw motion of surface vehicles

Abstract

The dynamics of surface vehicles such as boats and ships, when modeled as a rigid body, is complex as it is strongly nonlinear and involves six degrees of freedom. We are particularly interested in the steering dynamics decoupled from pitching and rolling as the basis of our research on unmanned boats and autonomy. Steering dynamics has been traditionally modeled using a linear Nomoto’s model, which however does a poor job of capturing real nonlinear phenomena. On the other hand, there exists a somewhat simplified three-degree-of-freedom nonlinear model derived by Abkowitz, which, although better than the Nomoto’s model, is still too intractable for analytical methods. In this paper, we derive a nonlinear single-degree-of-freedom model with a cubic nonlinearity that is conditionally equivalent to the Abkowitz model. Using this model, we analyze the yaw motion of an autopilot ship with a PD controller under the influence of an external wave force. Analytical and numerical investigations of the nonlinear dynamics are performed using parameters of an example container ship for various sea states. We employ the harmonic balance method to investigate the nonlinear frequency response of the ship under the influence of different parameters and demonstrate that the external wave force is balanced by stiffer ships. We also numerically investigate the nonlinear dynamic behavior to understand the different mechanisms involved in transitions from periodic to chaotic behavior under the influence of various parameters and different sea state conditions. The analysis reveals periodic response of the ship for a calm sea state. It is noteworthy that for lower values of linear stiffness, higher values of nonlinear stiffness, and higher values of external wave force corresponding to higher sea states, the bifurcation structure reveals mixed dynamic response in which periodic solutions evolve into period doubled, period tripled, and chaotic-like solutions even with high levels of damping. This work aims to demonstrate the utility of a simple nonlinear model and nonlinear behavior of steering motion of the ships to gain better insight into the nonlinear dynamics of autonomous surface vehicles. Further, the model stands out as an accessible nonlinear model for steering motion of the ship adequately representing the dynamics of real ships. It hence provides a basis for much improved controller design to implement smarter autopilots on manned ships but also aids in the development of robust autonomy in surface vehicles.

Appendices

Appendix A: Notations and parameters

m :

Mass

p :

Rolling rate

q :

Pitching rate

r :

Yawing rate

u :

Forward velocity (surge)

v :

Lateral velocity (sway)

w :

Vertical velocity (heave)

x :

Displacement along the x-axis in body-fixed coordinates

y :

Displacement along the y-axis in body-fixed coordinates

z :

Displacement along the z-axis in body-fixed coordinates

I :

Component of the moment of inertia tensor

K :

Moment component about the x-axis in body-fixed coordinates

M :

Moment component about the y-axis in body-fixed coordinates

N :

Moment component about the z-axis in body-fixed coordinates

X :

Force component along the x-axis in body-fixed coordinates

Y :

Force component along the y-axis in body-fixed coordinates

Z :

Force component along the z-axis in body-fixed coordinates

$\delta$ :

Rudder angle

$\psi , \theta , \phi$ :

Euler angles

Appendix B: Data for a container ship

Appendix C: Coefficients of Nomoto’s second-order model [Eq. (9)]

$$\begin{aligned} K= & {} \frac{-\Big (N(2,1)b(1,1)-N(1,1)b(2,1)\Big )}{\textrm{det}(N)} \end{aligned}$$

(27)

$$\begin{aligned} T_{3}= & {} \frac{(M(2,1)b(1,1)-M(1,1)b(2,1))}{(K\textrm{det}(N))}, \end{aligned}$$

(28)

$$\begin{aligned} A= & {} \frac{\textrm{det}(M)}{\textrm{det}(N)}, \end{aligned}$$

(29)

$$\begin{aligned} B= & {} \frac{\Big ( N(1,1)M(2,2)+N(2,2)M(1,1)-N(1,2)M(2,1)-N(2,1)M(1,2)\Big )}{\textrm{det}(N)}, \nonumber \\ \end{aligned}$$

(30)

$$\begin{aligned} C= & {} \sqrt{(B^2-4A)}, \end{aligned}$$

(31)

$$\begin{aligned} T_{1}= & {} \frac{(B+C)}{2}, \end{aligned}$$

(32)

$$\begin{aligned} T_{2}= & {} B-T_{1},\ \end{aligned}$$

(33)

where

$$\begin{aligned}{} & {} M= \left[ \begin{array}{cc} m-Y_{{\dot{v}}} &{} mx_{G}-Y_{{\dot{r}}} \\ mx_{G}-N_{{\dot{v}}} &{} I_{z}-N_{{\dot{r}}} \end{array} \right] , \quad \\{} & {} N_{u_{0}}=\left[ \begin{array}{cc} -Y_{{v}} &{} mu_{0}-Y_{{r}} \\ -N_{{v}} &{} mx_{G}u_{0}-N_{{r}} \end{array} \right] ,\quad b=\left[ \begin{array}{cc} Y_{\delta } \\ N_{\delta } \end{array} \right] \end{aligned}$$

Table 2 Constants of second-order Nomoto’s model using parameters in Table 1 and Eqs. (27)–(33)

Table 3 Sea states, corresponding range of wave heights and sea state codes as specified by the World Meteorological Organization (WMO) [38], and nondimensional wave amplitude

Appendix D: Coefficients of Eq. (15)

$$\begin{aligned} a_{11}= & {} \Big (1-{\frac{K}{T_{{1}}T_{{2}}}}k_{{d}}T_{{3}}\Big ),\nonumber \\ a_{12}= & {} \Bigg [\Big (\frac{1}{T_{{1}}}+\frac{1}{T_{{2}}}\Big )+{\frac{K}{T_{{1}}T_{{2}}}}\left( k_{{p}}T_{{3}}-k_{{d}} \right) \Bigg ], \nonumber \\ a_{13}= & {} {\frac{K}{T_{{1}}T_{{2}}}}\left( a+k_{{p}} \right) ,\quad a_{14}=b{\frac{K}{T_{{1}}T_{{2}}}},\,\nonumber \\ a_{15}= & {} c{\frac{K}{T_{{1}}T_{{2}}}},\quad a_{16}=d {\frac{K}{T_{{1}}T_{{2}}}}. \end{aligned}$$

(34)

Appendix E: Sea states