Nonlinear dynamics of yaw motion of surface vehicles

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.

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 chaoticlike 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.

Introduction
The dynamics of a surface vehicle-such as a boat or ship-is generally governed by six degrees of freedom, i.e., three translation motions (surge, sway and heave) and three rotational motions (roll, pitch, and yaw). General nonlinear dynamic analysis of ship motions is highly complex and has been investigated by many researchers, especially in pitch and roll, but the reported research is rather sparse in yaw. This paper focuses on the steering motion (and, hence, surge, sway, and heave) and hence we limit the literature review to only the relevant analyses.
Early studies on the steering dynamics of ships may be traced back to Davidson [1], Nomoto [2], and Abkowitz [3]. The stability of ship autopilots in yaw is discussed by Fossen and Lauvdal [4]. They analyzed the so-called Nomoto's model, a simplified linear second-order model, and discovered that the ship must move at a minimum speed to remain stable. Fossen and his colleagues at the Norwegian Institute of Technology have conducted more research in this area [5][6][7][8][9][10][11][12][13][14][15].
Hicks et al. [16] derived realistic hydrodynamic coefficients to analyze the nonlinear dynamics of planning hulls, and they discovered instabilities and bifurcations. Chen et al. [17] developed a systematic technique for modeling nonlinear ship dynamics in a very comprehensive and outstanding study. However, their focus is on roll, sway, and heave, which is of limited relevance for our current problem. Lefeber and colleagues [18] employed a nonlinear model to achieve tracking control and reported some fascinating findings. However, it is unclear if a simpler model would suffice, or whether a more complicated model is required to achieve their results.
Currently, the above ideas have not been fully demonstrated in naval studies, a field that is primarily dominated by linear concepts. As we know, an understanding of the nonlinear dynamics of surface vehicles enables the identification of the onset, evolution, and often unsafe responses beyond the linear regime that cannot be captured by conventional seakeeping techniques. When the relationship between stimulus and response in a dynamical system is nonlinear, multiple solutions can exist for certain values of the parameters. There are many well-known manifestations of nonlinear behavior, such as unexpected, violent motion that leads to vehicle escape, irregular behavior under regular excitation, subtle boundaries between domains of coexisting responses, and strong sensitivity to initial conditions. As a result, understanding the principles driving nonlinear behavior is imperative for assessing critical behavior.
In numerous cases, nonlinear phenomena have been discovered to underlie ship dynamics, much as they do for the dynamics of other engineering systems [19,20]. It is shown in [21] that a ship maneuvering problem can be modeled as a motion with three degrees of freedom. However, the roll induced by turning motion is not negligible for high-speed vessels. As a result, a four-degree-of-freedom description is required [22][23][24][25], which incorporates surge, sway, and roll modes.
Norrbin [26] proposed a nonlinear model to examine the surge, sway, and yaw motions for ship maneuvering in deep and confined waters, based on both experimental and analytical methods. Fang and Liao [27] demonstrated that nonlinear theory and time domain simulation can be utilized to investigate big ship motions induced by large waves. Spyrou [28] investigated the yaw motion of numerous ships in the presence of a steady wind. He discovered that heading winds can cause Hopf bifurcation, and concluded that most ships are dynamically unstable, requiring active steering to stay on course. Yasukawa et al. [29] concluded that steering control is necessary to reduce the instability of yaw motion of a ship in a steady wind. More recently, Nataraj et al. [30] proposed a tracking control algorithm for a three-degree-of-freedom steering model by deriving time varying coefficients for the linear error equations. As evidenced by some of the above papers, today's increased computer power does allow us to conduct research based on mathematical models with a high level of detail that can be interesting, but unfortunately these elaborate models offer limited understanding and insight.
The impetus for this study originates from the necessity to build robust and predictable automatic control of surface ships in order to implement autonomous behavior. As a result, a key question to consider at this point is whether the model we chose is acceptable for accurately representing the dynamics of real ships. Until now, the models employed in the literature to assess the yaw motion of autopilot ships have been either very sophisticated or very simple linear models. Complex models are more realistic in their predictions of real ship behavior, but they can be difficult to analyze and regulate. Linear models, on the other hand, such as the Nomoto's model, are insufficiently sophis-ticated to anticipate key occurrences, particularly nonlinear behavior. Therefore, there is a need for a simple model adequately representing the dynamics of real ships, which is the primary goal of this paper. Additionally, such a nonlinear model that is relatively easy to implement and design controllers for would go a long way toward accurate controller design for both manned and unmanned surface vehicles.
Note that what we really want is a model sophisticated enough to include all the critical phenomena, but nothing more complicated than that. That is, we would like to use the simplest model possible that still adequately captures the dynamics. It is hence the intention of this research (i) to derive a simple and yet nonlinear governing equation of yaw motion for an autopilot ship and (ii) to investigate its nonlinear behavior.
The manuscript is organized as follows: In Sect. 2, we derive a governing equation for the nonlinear yaw motion of an autopilot ship. In Sect. 3, we seek a solution using the harmonic balance method (HBM). Subsequently, we use the data available in the literature and investigate the nonlinear behavior of the ship analytically and numerically in Sect. 4. Finally, in Sect. 5, we summarize our findings. Figure 1 depicts a marine vehicle and typical coordinate reference frames. The coordinate system (x G , y G , z G ) is anchored to the ground with the z G -axis vertically pointing down. The body-fixed axes (x, y, z) are fixed at the vehicle's center of mass, O. The x-axis is always parallel to the vehicle's longitudinal symmetry axis (from aft to fore); the y-axis is perpendicular to the vehicle (directed to starboard); and the z-axis is defined by the cross product. The z-axis would be vertically down if the vehicle were horizontal [18].

Mathematical model
The vehicle is governed by six rigid body degrees of freedom. The three linear velocities are surge (u) along the x-axis, sway (v) along the y-axis, and heave (w) along the z-axis; furthermore, the three angular velocities are roll ( p), pitch (q), and yaw (r ). Using angular orientation defined by a sequence of Euler angles (θ, φ, ψ), the motion of the vehicle can be specified in body-fixed coordinates or earth-fixed coordinates. The resultant equations are highly complex and have been published in a number of sources, including [30][31][32]. The following assumptions are made to decouple the steering dynamics from the surge mode in order to address a restricted problem: These assumptions are reasonable when one wants to consider only the yawing motion. Then, the kinematic equations simplify to the followinġ It is to be noted that these equations are nonlinear, despite being simpler than the three-dimensional case.
The dynamic equations become where ρ G is the distance of the origin from the center of gravity. X , Y , and N are the forces on the right-hand side which arise due to hydrodynamics, propulsion, and actuation.
In a series of pioneering papers and reports, Abkowitz [33] proposed a third-order Taylor series expansion of the forces about a steady operating condition of constant forward speed Although this model is not linearized, it is required that perturbations be small; however, note that the steady forward speed, u 0 can be-and is often-a large quantity. With certain restricted assumptions, the force equations of RHS in Eq. (3) can be described as follows where δ is the rudder angle. Note that the Taylor series expansion represented by Eq. (5) consists of up to thirdorder derivatives. It includes some non-negligible terms such as coupling between the velocity and acceleration, but some other terms are not included because their effect is insignificant in practice. As a matter of notation, note that the terms in the equation include the numerical factor of 1/n!, for example, 3) represents a three-DOF nonlinear model that is practically applicable to most surface vehicles when these forces are incorporated. The definition of various notations in Eq. (5) is given in Appendix A.

Linearized model
In this subsection, for comparison, we linearize the forces in Eq. (5) for the special case of constant speed forward motion assuming small perturbations. And the hydrodynamic forces then take the following form The LHS in Eq. (3) also simplifies as follows Applying steady-state conditions and reasonably assuming that the thrust of the propeller balances out the losses and hydrodynamic forces under constant speed condition, we can drop the constant terms in the RHS, as well as u terms to get Next, eliminating v from the above equations leads to the following equation often called Nomoto's second-order model [2] where T 1 , T 2 , T 3 , and K are constants given by Eqs. (27)- (33) in Appendix C which depend on parameters associated with hydrodynamics, propulsion, actuation, surge velocity, and dimensions of the ship. This is the simplest steering model one could employ [2] and is used for most autopilot analysis and design. Next, we design an autopilot with a proportional plus derivative (PD) controller for above model where the rudder angle, the control input, is given by δ = k pψ + k dψ by following the standard practice for controller design of linearized systems [32,34]. Substituting in Eq. (9) yields the following equation Equation (10) represents a generalized linearized steering model with PD controller which is applicable for ships and boats. In order to continue the numerical analysis, we consider the experimentally validated parameters of a container ship from the literature [22,35] are specified in Table 1 in Appendix B. Subsequently, we evaluate T 1 , T 2 , T 3 , and K using Eqs. (27)-(33) and parameters in Table 1 and are tabulated in Table 2 in Appendix C. For the above equation, we determine the controller gains using standard methods of control system design [36] to achieve a settling time of 90 seconds and an overshoot of 15%. This results in the following values for the controller gains: k p = 1.9 and k d = 0.1986. Note that Eq. (9) is obtained by linearizing the equations of motion around the zero rudder angle. As a result, the rudder angle should not exceed approximately 5 • ; otherwise, the model would be inaccurate. So, we need a model suitable for rudder angles that are larger than 5 • and can also describe the ship's behavior for a wide range of maneuvers. In order to derive such a model, we introduce a nonlinear function H (ψ) into Eq. (9). As a result, we substitute 1 where H (ψ) is a nonlinear function ofψ describing the maneuvering characteristic [37], which also ensures correct steady turn solutions. H (ψ) can be found from the relationship between δ andψ in steady state such that ... ψ =ψ =δ = 0. The expression for H (ψ) is obtained by fitting theψ −δ curve in Fig. 2 obtained by numerically solving the Abkowitz model [33] [Eqs. (2), (3), (5)] using parameters given in Table 1 in Appendix B. The expression for H (ψ) is as follows.
where a = 1.6, b = −0.052, c = 2.7 and d = −0.00057. Ship response to waves is generally of great importance in evaluating the sea keeping performance of the ship. The forces and moments exerted by the sea waves also greatly influence the steering dynamics. To account for the influence of the waves, in the simplest case, it may be assumed that the waves incident upon the body are plane progressive waves of small amplitude with sinusoidal time dependence. Hence we can represent the wave force N w as N w (t) = f cos(Ωt). (13) Note that the wave amplitude in Eq. (13) is proportional to the sea states. Sea state is the state of oscillation of the sea surface (waves) generated by wind energy and is defined as the mean wave height. The various sea states, the corresponding range of wave heights and the World Meteorological Organization (WMO) sea state codes are documented [38] in Table 3 in Appendix E. In this manuscript, we describe various sea states by the abbreviation SS followed by the WMO code. For example, the rough sea state is described as SS5.
Combining Eqs. (10)-(13) the nonlinear steering model of USVs in the yaw direction under external wave force is given by Substituting yaw rate, r forψ and rewriting Eq. (14), a 11r + a 12ṙ + a 13 r + a 14 r 2 + a 15 r 3 + a 16 = f cos(Ωt), where a 1i (i = 1, ..., 6) are given by Eq. (34) Finally, we nondimensionalize Eq. (16) using t = ωt where ω = √ b 11 and the resulting equation with nondimensional parameters is as follows: Finally, Eq. (17) represents a nonlinear steering model in the yaw direction and resembles a Duffing-like oscillator. In the next section, we seek an approximate solution using harmonic balance method (HBM) for above equation that importantly retains strong nonlinearity.

Harmonic balance method
In this section, we apply the harmonic balance method (HBM) by following the standard procedure [39] to Eq. (17) to get approximate solutions. We assume a solution with one harmonic as Substituting Eq. (18) into Eq. (17), using the trigonometric identities for cos(Ωt) 3 , sin(Ωt) 3 , cos(Ωt) 2 and sin(Ωt) 2 and collecting the harmonics results in where hh(3) denotes third and higher harmonics. Equating to zero the coefficients of cos Ωt and sin Ωt in Eq. (19), respectively, yields We choose A c = a cos θ and A s = a sin θ such that A 2 c + A 2 s = a 2 and substitute in Eq. (20) which results in α 1 − Ω 2 a sin θ − γ Ωa cos θ + 3 4 α 2 a 3 sin θ 3 + a sin θa 2 cos θ 2 = 0.
Squaring and adding Eqs. (23) and (24) yields amplitude frequency relationship as Solving Eq. (25) for Ω results in the following frequency response equation

Results and discussion
In this section, we investigate the nonlinear behavior of the yaw motion of a ship derived above. As explained in the previous section, this paper considers an example of a container ship and the various experimentally determined nondimensional parameters for the container ship as documented in [22,35] are given in Table 1 in Appendix B. These parameters depend on the hydrodynamics, propulsion, actuation, surge velocity and dimensions of the ship. The constants of Nomoto's second-order model for the container ship are calculated using Eqs. (27)-(33) for parameters in Table 1 and are presented in Table 2 in Appendix C. The nondimensional amplitudes F of the sea wave calculated for different sea states are in Table 3 in Appendix E. We evaluate the nondimensional parameters in Eq. (17) γ = 0.0228, α 1 = 0.6457, α 2 = 0.0717, Δ = 0.01 and F = 2.66 f . Note that any changes in α 1 , α 2 , and Δ result from changes in parameters mentioned above as well as controller gains whereas changes in F account for changes in amplitudes of the external wave force for different sea states.

Frequency response using HBM
We solve Eq. (25) (using MATCONT [40]) or Eq. (26) for the frequency response for various parameters.  Table 3 in Appendix E). To analyze the stability of the solution, we compute the Jacobian matrix and eigenvalues associated with Eq. (25). The equilibrium solution Eq. (25) is stable when Re(λ) < 0; otherwise, the solution becomes unstable. Figure 3b corresponds to eigenvalues versus Ω for parameters γ = 0.0228, α 1 = 0.6457, α 2 = 0.0717 and F = 0.0358. In both the figures, the blue solid lines indicate stable solutions (Re(λ) < 0 in Fig. 3b) and the red dotted lines indicate unstable solutions (Re(λ) > 0 in Fig. 3b). The red circles in Fig. 3a are solutions from numerical simulation of Eq. (17) which are in good agreement with the solutions of HBM. It is evident from Fig. 3a that the amplitude of frequency response has increased with higher sea states for same linear (α 1 ) and nonlinear (α 2 ) stiffness.
Frequency response for the same external wave force (F = 0.08, 0.2) corresponding to SS4 and SS6 sea states and damping (γ = 0.05) and for different values of α 1 and α 2 are shown in Fig. 4a (for α 1 = 0.1 and α 2 = 0.5) and Fig. 4b (for α 1 = 0.5 and α 2 = 0.8). It is observed that for the same external wave force and for the same damping, with an increase in α 1 from 0.1 to 0.5 and α 2 from 0.5 to 0.8, the amplitude of frequency response reduces considerably. This is because with the increase in α 1 and α 2 , the ship becomes stiffer and overcomes the external wave force.

Numerical analysis
Next, we study the nonlinear behavior of the container ship by numerically integrating Eq. (17) and investigate the bifurcation of the system under the effect of various parameters leading to various patterns of behavior including chaos. Figure 5a depicts the frequency response for the parameters γ = 0.0228, α 1 = 0.6457, α 2 = 0.071, Δ = 0.0106, F = 0.0358 (for SS3 sea state, see Table 3 in Appendix E). Figure 5b-d shows the time history, phase plane with Poincaré point, and power spectra of a periodic motion for ω = 0.85. Under this condition, the system remains periodic throughout the frequency range of Ω = 0.5 to 1.1 as shown by time history, phase plane, and power spectra in Fig. 5b-d. Therefore, under this operating condition and for a slight sea state (SS3), we can conclude that the container ship operates safely.
The bifurcation diagram for the parameters γ = 0.05, α 1 = 0.1, α 2 = 0.5, Δ = 0.01, and F = 0.2 is shown in Fig. 6a. For this case of SS6 sea state, the bifurcation diagram shows the transition of periodic motions to period doubled and period tripled motions. Time history, phase plane with Poincaré points, and power spectra of period doubling and period tripling motions for Ω = 0.86 and Ω = 1.2 corresponding to Fig. 6a are, respectively, shown in Fig. 7a-f.
The dynamic behavior becomes more complex for SS8 sea state for the parameters γ = 0.05, α 1 = 0.1, α 2 = 0.5, Δ = 0.01, F = 0.5 and the bifurcation diagram is depicted in Fig. 6b. It is observed from Fig. 6a that bifurcation diagram illustrates the transition of chaotic (Ω = 0.3255) to period tripling (Ω = 0.633), period doubling (Ω = 1.1) and periodic solutions (Ω = 1.8). Time history, phase plane with Poincaré points and power spectra for chaotic (Ω = 0.3255), period Note that, for low linear stiffness, considerably high damping, and high nonlinear stiffness, the steering dynamics of the container ship becomes quite complex for SS6 sea state leading to even more complex dynamics for the case of SS8 sea state. Figure 11 shows the bifurcation diagram of the container ship for (a) γ = 0.03, α 1 = 0.05, α 2 = 0.7, Δ = 0.01, F = 0.3, (b) γ = 0.05, α 1 = 0.05, α 2 = 0.7, Δ = 0.01, F = 0.6. It is observed from Fig. 11a that the steering dynamics of the container ship is very complex with lower values of damping, and linear stiffness and with slightly higher value of nonlinear stiffness for the case of SS7 sea state. This increase in complexity is also evident for SS9 sea state even with higher values of damping as depicted in Fig. 11b.

Closing remarks
Surface vehicles-such as boats and ships-have been designed and used for a very long time in human history. However, their physics and complexity of behavior are still not quite understood. This is because their dynamics even when modeled as a rigid body is strongly nonlinear and would normally include six coupled degrees of freedom. Rolling and pitching motion has received a lot of research attention over the years. While all of the dynamics is interesting and important, we are particularly interested in the steering dynamics when approximately decoupled from pitching and rolling. This is because steering becomes most important for path plan-  ning of boats which has been a focus 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. Hence, in our quest for an analytically manageable model we have derived a nonlinear single-degree-offreedom governing equation with a cubic nonlinearity to represent the yaw motion of an autopilot ship integrated with a PD controller under the influence of an external wave force. The model is conditionally equivalent to the Abkowitz model, a three-degree-of-freedom nonlinear steering model. Our model provides an accessible nonlinear model for steering motion of the ship adequately representing the dynamics of real ships. The nonlinearity in the model is introduced by a nonlinear function of the yaw rate into Nomoto's second-order model which is obtained by comparing the Abkowitz model.
Using the parameters of an example container ship from the literature [22,35], we investigated the nonlinear dynamics analytically and numerically. We employed the harmonic balance method (HBM) to study the nonlinear frequency response under the influence of external wave force, linear, nonlinear stiffness, and damping. From our analysis, we found that the stiffer ships with large linear and nonlinear stiffness balance out large external wave forces.
Further, a comprehensive numerical analysis of the container ship reveals bifurcation structures that comprise multiple distinct regions with different behaviors. The governing parameters of the bifurcation structure are linear and nonlinear stiffness, damping, and excitation due to external wave force. For a certain set of parameters, the system responses are periodic. This would be a well-behaved region which helps the designer to predict the trends accurately and without ambiguity. However, analyzing systematically for other parametric values, such as for lower values of linear stiffness, higher values of nonlinear stiffness, and higher values of external wave force corresponding to very rough (SS6), high (SS7), and very high (SS8) sea states, the bifurcation structure reveals a mixed dynamic response in which periodic solutions evolve into period doubled, period tripled, and chaotic solu-tions. This complex behavior persists for extremely high sea states (SS9, technically termed 'phenomenal') even when the damping is considerably higher.
Thus, in conclusion, the nonlinear single-degree-offreedom steering model derived in this paper is quite simple, robust, and efficient. This model can be further utilized to gain insights into the nonlinear dynamics of yaw motion of real ships and to design and implement smarter autopilots with much improved controllers for both non-autonomous and autonomous surface vehicles.