==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 37394560 37916 10.1038/s41598-023-37916-1 Article Delayed feedback control for chaotic vibration in nonlinear impact dynamics of bouncing agricultural tractor Watanabe Masahisa masahisa-watanabe@go.tuat.ac.jp Sakai Kenshi grid.136594.c 0000 0001 0689 5974 Division of Environmental and Agricultural Engineering, Institute of Agriculture, Tokyo University of Agriculture and Technology, 3-5-8 Saiwai-cho, Fuchu, Tokyo 183-8509 Japan 2 7 2023 2 7 2023 2023 13 106959 1 2023 29 6 2023 © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Agricultural tractors often lose contact and recollide with the ground surface while driving on narrow paddy fields and bumpy farm roads owing to excessive vibrations. These nonlinear impact dynamics can cause chaotic vibrations during tractor operation. Chaotic vibrations are random complex motions that can deteriorate tractor stability and lead to tractor overturning accidents, causing damage to machinery and risk of injury to the operator. This study investigates the theoretical feasibility of chaos control to eliminate chaotic vibrations in tractor dynamics. Delayed feedback (DF) control is employed to eliminate complex vibrations in tractor dynamics. First, the frequency response, bifurcation diagram, and largest Lyapunov exponent are obtained to investigate the nonlinear dynamics of the tractor and identify the parametric region in which chaotic vibrations occur. Subsequently, the DF control is designed based on the trial-and-error method and applied to the tractor dynamics as the driving force control input. The numerical results demonstrate that the DF control can successfully eliminate chaotic vibration and reduce the vibration level. Therefore, this study is expected to contribute to improving the tractor safety by reducing the risk of overturning. Subject terms Mechanical engineering Engineering http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science 19H00959 issue-copyright-statement© Springer Nature Limited 2023 ==== Body pmcIntroduction An agricultural tractor is a self-propelled vehicle used for various farm operations, such as tillage, transportation, seeding, and towing. As tractors are mainly operated off-road, on farm fields and unpaved roads, they suffer from severe vibrations excited by undulating terrains. As the working speed of the tractor increases, vibrations become more problematic. Owing to severe vibrations, the operator may lose control of the tractor, and an overturning accident may occur. Tractor overturning is a major cause of farm fatalities worldwide1. In Japan, 53 out of 270 fatal farm accidents in 2020 were caused by tractor overturning2. Tractor dynamics and control have been investigated to predict vibrations and prevent overturning3–11. Therefore, active tractor control, including active steering control12,13, active driving force control14, and sliding mode control15, is intensively studied. In Japan, small agricultural tractors are generally used on narrow paddy fields and bumpy farm roads. Therefore, the vibration problem is more serious than that of a large tractor on relatively flat ground. In a small tractor, excessive and violent vibrations often cause the tractor tires to lose contact and recollide with the ground. These impact dynamics are similar to bouncing ball dynamics and are a strong nonlinear element in tractor dynamics16–19. Nonlinear tractor dynamics often lead to chaotic vibrations20–26. Chaotic vibrations or chaos are random-like motions, even though the motions are generated by deterministic dynamics such as motion equations. It is difficult to predict chaotic vibrations because of their inherent property of sensitive dependence on initial conditions and parameters. Thus, chaotic vibrations deteriorate tractor stability and lead to tractor overturning, causing damage to machinery and risk of injury to operators. In addition, chaotic vibrations cause undesirable peaks in the spectrum such as subharmonics, which are generally not considered in linear vehicle design theory. As well as agricultural tractor, chaotic vibration sometimes appears in automotive dynamics. Several studies on chaotic vibration in automotive dynamics have been conducted for safety and performance improvement27–31. Therefore, chaotic vibration is a significant research topic in both tractor and automotive engineering. Chaos is a generally undesirable phenomenon observed in various systems, such as electronic and mechanical systems, and should be eliminated via the control input. Thus far, many studies have been conducted on chaos control methods to eliminate chaos in nonlinear dynamics. Ott, Grebogi, and York32 developed a chaos control method that can eliminate chaos in dynamics by making a small time-dependent perturbation (OGY control). OGY control is a time-discrete control method that requires a permanent computer analysis of the state of the system. However, because of the abovementioned properties, OGY control can lead to an occasional burst of the system when a large noise occurs in the system. Pyragas33,34 developed the delayed feedback control (DF control), a time-continuous control method, to eliminate chaos. The DF control method involves reference-free control that is calculated based on the difference between the current state of the system and the time-delayed state. As DF control is easy to implement, it is applied to various experimental and theoretical systems such as electronic oscillators35, mechanical pendulums36, helicopter rotor blades37, and car-following traffic38. Even though these practical applications suggest that DF control has a potential to apply to tractor dynamics, DF control for agricultural tractor has not been intensively investigated yet. The objective of this study is to investigate the theoretical feasibility of DF control in eliminating chaotic vibrations in tractor impact dynamics. First, a nonlinear tractor dynamic model with three degrees of freedom is considered. Subsequently, the DF control for a tractor is introduced as an active driving force control. The frequency response, bifurcation diagram, and largest Lyapunov exponent are obtained to identify the parametric region in which chaotic vibrations occur. DF control is applied to the chaotic vibration in the tractor. The remainder of this paper is organized as follows. In “Nonlinear impact dynamic model for agricultural tractor” and “Delayed feedback control design” sections introduce the nonlinear tractor dynamic model and the DF control for the tractor, respectively. The numerical results are described in “Numerical results” section. Finally, in “Conclusion” section presents the conclusions. The innovative point of the present study is to demonstrate numerical feasibility of DF control application to agricultural tractor and to provide basic policy for tuning DF control parameters specific to agricultural tractor applications. Nonlinear impact dynamic model for agricultural tractor This section explains the nonlinear impact dynamic model of the agricultural tractors used in this study. Figure 1 shows a schematic of the tractor model with three degrees of freedom.Figure 1 Schematic of the tractor dynamic model with three degrees of freedom; vertical motion z, longitudinal motion x; pitch motion γ. The tractor model has three degrees of freedom: vertical motion z, longitudinal motion x, and pitch motion γ. According to the force balance method, the motion equations of the developed model are derived according to Eqs. (1)–(3), as follows:1 MT+MIz¨=fv,1+fv,2-MT+MIg, 2 (Iyy+MIl32)γ¨=l1fv,1-l2fv,2+Fd-μr,1fv,1-μr,2fv,2+fl,2hCG, 3 MT+MIx¨=Fd-μr,1fv,1-μr,2fv,2+fl,2, where MT is the mass of the tractor body; MI is the mass of the implement; g is the gravitational acceleration; hCG is the height of the center of gravity; fv,1 and fv,2 are the vertical forces acting on the front and rear tires, respectively; fl,2 is the longitudinal force acting on the rear tire; Fd is the driving force on the rear tire. In this study, a rear wheel drive tractor was assumed, and thus, no driving force was exerted on the front wheel. In vertical and longitudinal tire force modeling, tires are expressed by the Kelvin–Voigt model (linear spring and damper) when the tire makes contact with the ground. The tire forces become zero when the tire loses contact with the ground. These impact nonlinearities are modeled using switching equations, involving bouncing ball dynamics. The vertical and longitudinal forces are calculated using Eqs. (4)–(6), as follows:4 fv,1=-kv,1z1-d1-cv,1z˙1-d˙1,ifthefrontwheelcontactstheground0,ifthewheellosescontactwiththeground, 5 fv,2=-kv,2z2-d2-cv,2z˙2-d˙2,iftherearwheelcontactstheground0,iftherearwheellosescontactwiththeground, 6 fl,2=-kl,2x-cl,2x˙,iftherearwheelcontactstheground0,iftherearwheellosescontactwiththeground, where kv,1, kv,2, and k l, 2 are the stiffnesses of the front vertical, rear vertical, and rear longitudinal wheels, respectively; cv,1, cv.2, and cl, 2 are the damping coefficients of the front vertical, rear vertical, and rear longitudinal wheels, respectively; z1 and z2 are the vertical displacements of the front and rear wheels, respectively; x is the longitudinal displacement of the rear wheel; d1 and d2 denote the front and rear road elevation, respectively. The bouncing process of an agricultural tractor was explained by Watanabe and Sakai23. z1 and z2 are calculated using Eqs. (7) and (8), respectively:7 z1=z+l1+l3MIMT+MIγ, 8 z2=z-(l2-l3MIMT+MI)γ. In this study, the road excitation functions are defined as sine functions, as expressed by Eqs. (9) and (10):9 d1=d0sin2πftt, 10 d2=d0sin2πftt-l1+l2V, where d0, V, and ft is the road amplitude, the travel velocity of the tractor, the forcing frequency respectively. The road amplitude d0 was set as 0.015 m in the numerical simulations. Delayed feedback control design This section explains the DF control of driving forces. The driving forces on the rear wheel are calculated from Eq. (11), as follows:11 Fd=μr,1fv,1+μr,2fv,2+Fc,iftherearwheelcontactstheground0,iftherearwheellosescontactwiththeground, where the term μr,1 fv,1 + μr,1 fv,1 denotes the force necessary for overcoming motion resistance, and Fc is a DF control force term. The driving force Fd should be zero when the wheel loses contact with the ground. The DF control force Fc is calculated based on the method described by Pyragas33. DF control can stabilize chaotic vibrations by converting them into periodic vibrations. Figure 2 shows a block diagram of the DF control implemented in tractor dynamics.Figure 2 Block diagram of delayed feedback control based on Pyragas33. In DF control, the feedback term is calculated based on the signal difference γt-γt-τ between the current state of the system and the state of the system delayed τ. As the pitch angle γ(t) can be easily measured in a practical situation, it is used as the representative signal of tractor dynamics in this study. The delayed feedback control forces are proportional to the signal difference, as shown in Eq. (12):12 Fc=Kdfγt-γt-τ, where Kdf is the DF gain, and τ is the time delay for DF control. In DF control, unstable periodic orbits embedded in a strange attractor are stabilized by input perturbations A general determination methodology for the DF parameters has not been established thus far. The time delay τ and gain Kdf are tuned based on try-and-error. Table 1 shows the parameter specifications of the small farm tractor used in the present study.Table 1 Nomenclature and parameter specifications of the small tractor for the numerical experiments39. Parameters Symbol Value Unit Vertical stiffness of front wheel k1, v 304 kN m-1 Vertical stiffness of rear wheel k2, v 372 kN m-1 Longitudinal stiffness of rear wheel k2, l 316 kN m-1 Vertical damping coefficient of front wheel c1, v 5500 N s m-1 Vertical damping coefficient of rear wheel c2, v 6690 N s m-1 Longitudinal damping coefficient of rear wheel c2, l 6400 N s m-1 Mass of tractor body MT 874 kg Mass of implement MI 149 kg Pitch moment of inertia Iyy 320 kg m-2 Distance between the center of gravity of the tractor body and the front wheel l1 0.78 m Distance between the center of gravity of the tractor body and rear wheel l2 0.587 m Distance between the center of gravity of the tractor body and the center of gravity of the implement l3 1.549 m The wheelbase of the tractor WB 1.35 m The height of the center of gravity from the ground hCG 0.586 m Motion resistance coefficient of front wheel μr,1 0.12 – Motion resistance coefficient of rear wheel μr,2 0.12 – Sinus road function amplitude d0 0.015 m Vertical motion z – m Vertical velocity vz – m s-1 Pitch motion γ – rad Longitudinal motion x – m Vertical force on front wheel fv,1 – kN Vertical force on rear wheel fv,2 – kN Longitudinal force on rear wheel fl,2 – kN Driving force Fd – kN Delayed feedback control force Fc – kN Vertical motion of front wheel z1 – m Vertical motion of rear wheel z2 – m Gravitational acceleration g – g Travel velocity V – m s-1 Largest Lyapunov exponent λ1 – – Forcing frequency ft – Hz Primary frequency in the spectrum fp – Hz Time delay for DF control τ – s Gain for DF control Kdf – kN rad-1 Numerical results Chaotic vibrations in tractor dynamics In this section, the observations on chaotic vibration in tractor dynamics are presented. In this study, delayed feedback control is applied to eliminate chaotic vibration. First, the frequency response, bifurcation diagram, and largest Lyapunov exponent λ1 are obtained to identify the parametric region in which chaotic vibration occurs. Frequency response is typically used to analyze the response of a dynamical system. In this study, the forcing frequency ft is varied from 1.0 to 7.0 Hz, and the maximum value of vertical displacement z is plotted at each forcing frequency. In nonlinear dynamics, bifurcation is a structural change that describes the stability of a periodic orbit40. The bifurcation diagram is widely used to visualize orbit stability and easily identify the parametric region in which chaotic vibration occurs. In this study, a bifurcation diagram is obtained by plotting the Poincaré point of the vertical displacement z concerning the forcing frequency ft. In the bifurcation diagram, a single bifurcation plot indicates one periodic vibration, whereas multiple complex plots indicate multiple complex vibrations. The largest Lyapunov exponent λ1 quantifies the divergence or convergence rate of the trajectory in the phase space and is generally used to identify chaos in dynamics41. The negative, zero, and positive largest Lyapunov exponents indicate a fixed point (no vibration), normal periodic vibration, and chaotic vibration, respectively. In this study, the largest Lyapunov exponent λ1 is calculated based on vertical displacement time series data42,43. Based on the time delay embedding technique, the attractor is reconstructed, and the Jacobian G(t) is estimated at each time point t. Given a set of unit vectors that are orthogonal to each other, denoted as ui(t) (where i = 1, 2, …, m), each unit vector is transformed to ei(t + 1) using the estimated Jacobian G(t). Here, m represents the embedding dimension. The calculation formula for the transformation is as follows:13 eit+1=Gtuit, In each iteration, Eq. (13) is orthonormalized by Gram-Schmit method.14 ei′t+1=eit+1-∑j=1i-1eit+1·ujt+1ujt+1 15 uit+1=ei′t+1ei′t+1 Lyapunov exponent is calculated based on e’i(t) as follows:16 λi=limN→∞∑t=0N-1lnei′tN, where N is data size of the time series. Figure 3a–c show the frequency response, bifurcation diagram, and largest Lyapunov exponent λ1, respectively.Figure 3 (a) Frequency response, (b) Bifurcation diagram, (c) Largest Lyapunov Exponent λ1. Figure 3 shows that the frequency response is continuous, and the bifurcation diagram indicates periodic vibration from 1.0 to 4.1 Hz. In addition, the largest Lyapunov exponent λ1 is zero in the forcing frequency range of 1.0–4.1 Hz, indicating that periodic vibration occurs from 1.0 to 4.1 Hz. At the point of 4.1 Hz, the frequency response increases discontinuously. This discontinuous response is a typical feature of nonlinear dynamics. The bifurcation plot shows multiple complex plots, and the largest Lyapunov exponent λ1 is positive from 4.1 to 5.4 Hz, indicating that chaotic vibration occurs in the range of 4.1–5.4 Hz. In the frequency region above 5.4 Hz, the largest Lyapunov exponent λ1 is zero, indicating that periodic vibrations occur in this frequency band. The bifurcation plot becomes complex from 5.8 to 6.0 Hz with λ1 = 0, indicating that quasi-periodic vibration occurs from 5.8 to 6.0 Hz. Typical examples of periodic, quasiperiodic, and chaotic vibrations are presented as follows. Figure 4 shows a typical example of periodic vibration at 1.5 Hz.Figure 4 Periodic vibration at 1.5 Hz. (a) Time-series of vertical displacement z, (b) Fourier spectrum of vertical displacement z, (c) Phase plane spanned by vertical displacement z and vertical velocity vz, (d) Poincaré section spanned by vertical displacement z and vertical velocity vz. Figure 4a shows the time series of vertical displacement is periodic. Figure 4b shows the Fourier spectrum of the vertical displacement z. In the spectrum, the primary frequency fp is equal to a forcing frequency of 1.5 Hz. Figure 4c shows the motion trajectory in the phase plane spanned by vertical displacement and vertical velocity. The trajectory is a limit cycle corresponding to periodic motion. Figure 4d shows the Poincaré section of the trajectory in the phase plane. The Poincaré section is typically employed to study the trajectory structure in the phase plane; it can be derived via stroboscopic monitoring of the point in the phase plane. Therefore, one point in the Poincaré section corresponds to one specific period. In Fig. 4d, the Poincaré section is derived at different angles, that is, 0°, 90°, 180°, and 270°. All the Poincaré sections show a single point in the phase plane. This implies that the vibration is periodic. Figure 5 shows a typical example of quasi-periodic vibration at 5.8 Hz.Figure 5 Quasi-periodic vibration at 5.8 Hz. (a) Time-series of vertical displacement z, (b) Fourier spectrum of vertical displacement z, (c) Phase plane spanned by vertical displacement z and vertical velocity vz, (d) Poincaré section spanned by vertical displacement z and vertical velocity vz. In Fig. 5a, the time series of vertical displacement demonstrates multiple periodic vibrations. Multiple peaks are observed in the Fourier spectrum, corresponding to the primary frequency fp = 5.8 Hz, subharmonics fp/3 = 1.93 Hz, and ultra-subharmonics of 2fp/3 = 3.86 Hz, 4fp/3 = 7.72 Hz, and 5fp/3 = 9.65 Hz. In Fig. 5c, the trajectory is a torus in the phase plane. A closed loop is observed in the Poincaré section in Fig. 5d. This serves as evidence for the existence of quasi-periodic vibration. Figure 6a shows that the vertical displacement vibration is complex and random. The noise level is significantly high in the Fourier spectrum shown in Fig. 6b, even though the motion equations do not include any stochastic noise terms. Fourier spectrum also has multiple peaks, including primary frequency ft = 4.2 Hz, super-harmonics 2ft = 8.4 Hz, subharmonics ft/3 = 1.4 Hz, and ultra-subharmonics 2fp/3 = 2.8 Hz and 4fp/3 = 5.6 Hz. Figure 6c shows the complex trajectory of chaotic motion in the phase plane. Figure 6d shows the Poincaré section of chaotic vibration. A stretching and folding structure, corresponding to chaos characteristics, is observed. These features serve as typical evidence for the existence of chaotic vibrations.Figure 6 Chaotic vibration at 4.2 Hz. (a) Time series of vertical displacement z, (b) Fourier spectrum of vertical displacement z, (c) Phase plane spanned by vertical displacement z and vertical velocity vz, (d) Poincaré section spanned by vertical displacement z and vertical velocity vz. Delayed feedback control DF control is designed and applied to eliminate chaotic vibrations. Figure 7 shows the numerical results for the forcing frequency of ft = 4.2 Hz with Kdf = 55,000, and τ = 0.238 tuned by the trial-and-error. Figure 7a–c show the vertical motion z, pitch angle γ, and DF control force Fc, respectively.Figure 7 Delayed feedback control application results for forcing frequency ft = 4.2 Hz; (a) Vertical displacement z; (b) Pitch angle γ; (c) Delayed feedback control force Fc. The DF control force Fc is applied at 35.0 s. After applying the control force, the complex motion gradually stabilizes as it is converted into periodic motion. The DF control force Fc decreases as the motion becomes periodic. The vibration level is significantly decreased, and chaotic vibration is eliminated. Figure 8a–d show the time series of vertical displacement, Fourier spectrum of vertical displacement, phase space spanned by vertical displacement, velocity, and pitch angle, and Poincaré section, respectively.Figure 8 Delayed feedback control application results for forcing frequency ft = 4.2 Hz; (a) Time series of vertical displacement, (b) Fourier spectrum of vertical displacement, (c) Phase plane spanned by vertical displacement z and vertical velocity vz, velocity, (d) Poincaré section spanned by vertical displacement z and vertical velocity vz. Figure 8a shows that the vertical acceleration is stabilized as it is converted into periodic motion owing to DF control. Furthermore, the vibration level in Fig. 8a is smaller than that in Fig. 6a. A comparison of the Fourier spectra shown in Figs. 6b and 8b indicates that the DF control eliminates chaotic motion and reduces the noise level. Moreover, Figs. 8c and d indicate periodic motion. The period of the stabilized vibration corresponds to the inverse of the forcing frequency, which is 1/4.2 = 0.238 s in this case. To investigate the relationship between dynamic stability and the parameters of delayed feedback (DF) control, the values of Kdf and τ are varied within the ranges of 0 to 85,000 and 0 to 0.28, respectively. Figure 9 illustrates the maximum vertical displacement value z obtained from each DF control simulation.Figure 9 The maximum value of vertical displacement z at different control gain Kdf and time delay τ. In Fig. 9, as gain Kdf increases, the vibration level is reduced, and chaos is stabilized into periodic vibration. Regarding time delay τ, any time delay above 0.01 is able to suppress chaotic vibration. However, combination of larger Kdf and τ can lead to unbounded or unstable results, as indicated by the white blank in Fig. 9. Therefore, for practical application, starting with smaller values of Kdf and τ as the initial control tuning point and gradually increasing them to larger values can effectively suppress chaotic vibration. The period of the stabilized vibration is not affected by the DF control parameters. All stabilized periodic motion has a period equal to the inverse of the forcing frequency, which is 0.238 s for a forcing frequency of 4.2 Hz. To show the DF control effect in other frequency points, frequency response is conducted with DF control with Kdf = 55,000, and τ = 0.238. Figure 10 compares the uncontrol and control frequency responses to investigate the vibration reduction effect of DF control. Figure 10a–c show the frequency responses of the vertical displacement z, pitch angle γ, and longitudinal displacement x, respectively.Figure 10 Comparison between uncontrol and DF control frequency response; the solid and hollow lines denote the uncontrol and control responses, respectively; (a) Frequency response of vertical displacement z; (b) Frequency response of pitch angle γ; (c) Frequency response of longitudinal displacement x. DF control is applied in the range of 4.1–6.0 Hz to eliminate chaotic vibrations. The vibration level of the controlled frequency response curve is smaller than that of the uncontrolled frequency response curve. The results indicate that DF control is effective in reducing vibration levels and eliminating chaotic vibration. Conclusion This study investigated the theoretical feasibility of DF control in eliminating chaotic vibrations in nonlinear impact dynamics of agricultural tractors. The DF control force was input as a longitudinal driving force on the rear wheel; it was designed based on the trial-and-error method. The frequency response, bifurcation diagram, and largest Lyapunov exponent λ1 were obtained to identify the parametric region in which chaotic vibration occurred. Delayed feedback control was applied to the agricultural tractor dynamics during the numerical simulation. The delayed feedback successfully eliminated chaotic vibrations and reduced the vibration level in the frequency response curve. The numerical results demonstrated the theoretical feasibility of delayed feedback control for agricultural tractors. This feasibility study is expected to contribute to improving the safety aspects of agricultural tractor operators and preventing damage to the tractors by reducing the risk of overturning. Although the feasibility of applying DF control to agricultural tractors is clarified numerically in this research, there are several topics that should be explored in future research. In order to apply DF control to real agricultural tractors, it is important to consider factors such as road randomness and driver behavior models within the tractor model. In addition, it is necessary to validate the proposed DF control through field experiments. For a deeper understanding of the complex dynamics of agricultural tractors with DF control, simpler modeling approaches should be explored and advanced dynamic analyses should be conducted, including multistability and basin analysis. Considering the influence of random road surfaces, the equations of motion for the tractor become nonlinear differential equations with stochastic components, requiring intensive theoretical analysis for control design. Furthermore, conducting a comparison of different nonlinear control methods would be beneficial for selecting the most suitable control method for agricultural tractor dynamics. Acknowledgements We thank Professor Awadhesh Prasad for his kind support to this study. This work was supported by JSPS Grant-in-Aid (Nos. 19H00959, 22H00388 and 23K14043). Author contributions M.W. and K.S. conceived the research. M.W. made MATLAB program for numerical simulation. M.W. wrote whole manuscript text and prepared all figures and tables. M.W. and K.S. reviewed the manuscript. Data availability The datasets that support this study are available with the corresponding author and can be accessed upon reasonable request. Competing interests The authors declare no competing interests. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ==== Refs References 1. 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