Identification of the Steering and Speed Systems of a Four-Wheel-Steering Tractor for Optimal Control

• Model-based control depends on a sufficiently accurate system model, while the tractor’s digital, mechatronic, and hydraulic components make first-principles modeling challenging or unnecessarily complex.Source: §1 · p. 1
• The study targets autonomous field operation of a four-wheel-steering agricultural tractor by investigating its steering and speed control systems.Source: §1 · p. 1
• The identified dynamics run from steering-angle setpoint to actual steering angle and are intended to help control design account for steering and speed response to inputs.Source: §1 · p. 1
• The paper uses data-driven gray-box system identification, balancing fit to observed input-output data against model complexity and using system insight to constrain the model set.Source: §1 · p. 1

• Indoor mobile-robot studies have modeled differential-drive and omnidirectional systems, including least-squares and instrumental-variable estimation and Gaussian-process regression [as cited in the paper].Ardiani et al. (2021) compare least-squares and instrumental variables; Eschmann et al. (2021) use Gaussian process regression.Source: §2 · p. 1
• For outdoor vehicles, Rödönyi et al. (2021) identify passenger-car lateral dynamics with a neural-network-based subspace-encoder method, while Vicente et al. (2021) compare data-driven and physical models.Vicente et al. report that their linear data-driven model outperformed linear and nonlinear physical models on their real-world driving dataset.Source: §2 · p. 2
• Soitinaho and Oksanen (2023) apply a kinematics-and-system-identification model in MPC path tracking field experiments, but their vehicle is front-wheel steered.Source: §2 · p. 2
• The paper focuses on a standard agricultural tractor with independently controlled front- and rear-wheel steering, which also enables crab steering.Source: §2 · p. 2
• The paper presents experiment design and system identification for steering and speed using field-collected data, aiming to support MPC-based path tracking for a 4WS tractor.The authors say validation-data fit suggests feasibility for field experiments in a model-predictive path-tracking controller.Source: §2 · p. 2

3. SYSTEM DESCRIPTION

• G-trac is a modified Lindner Lintrac 130 with front/rear steering, CAN-bus sensing and control, and autonomous operation through Simulink Real-Time.The front steering uses the built-in Danfoss system; the rear steering uses redeveloped controls with the original valves.Source: §3 · p. 2

3.1 Kinematic model

• The vehicle model uses a 4WS kinematic bicycle representation referenced at the rear-axle center, with position, heading, steering angles, and speed as state variables.The state includes the front and rear steering angles and the magnitude of current velocity.Source: §3.1 · p. 2

3.2 Steering actuation system

• Both steering actuators are modeled as first-order systems with unit steady-state gain, leaving their time constants and transport delays to estimate.A steady-state gain of 1 assumes each actuator can eventually reach its desired control value.Source: §3.2 · p. 3

3.3 Speed actuation system

• The speed actuator is modeled as a first-order system with unit steady-state gain, and identification estimates its time constant and transport delay.The assumed gain is set as known for system identification.Source: §3.3 · p. 3

3.4 Control system software

• The experiment design runs in Matlab/Simulink, while a Speedgoat target computer runs the Simulink Real-Time application.The target machine has an Intel Celeron 2 GHz four-core processor and 4 GB DDR3 RAM.Source: §3.4 · p. 3

4. EXPERIMENTAL PROCEDURE

• Experiments were conducted in autumn on a Southern German field, all on one day, with cloudy weather and no rain.The field’s maximum elevation difference was approximately 1.6 meters, and crop height was approximately 10 centimeters.Source: §4 · p. 3

4.1 Design of key experiments

• Square-wave excitation signals elicited step responses from each actuator while the other two actuators received constant-value control signals.For steering data collection, the tractor drove at constant speed; for speed data collection, it steered straight ahead.Source: §4.1 · p. 3

4.2 Excitation signals and data recording

• Table 1 describes the excitation data by experiment ID and signal, and lists the constant controls sent to the other actuators.The figures show excitation references as dashed lines and actuator-state sensor responses as solid green lines; rear-steering and speed traces were used for parameter estimation.Source: §4.2 · pp. 3–4, Table 1 · p. 4, Figure 3 · p. 4
• Steering tests used 10-second, 50%-duty-cycle bipolar square waves; speed tests used unipolar square waves, and each experiment was performed once.Steering amplitudes varied up to the maximum angle in both directions; speed excitation reached up to 2.34 m s with a slew rate of 0.84 m.Source: §4.2 · pp. 3–4

4.3 System identification method

• All three actuators were modeled with first-order SISO transfer functions, and iterative parameter estimation refined unknown parameters by minimizing prediction error.A combination of excitation-response data was used for identification and validation; the MATLAB toolbox used an adaptive subspace Gauss-Newton search.Source: §4.3 · p. 4, §4.2 · p. 4

5. RESULTS

• Time constants were τδf=0.73, τδr=0.62, and τv=1.13; delays were 430 ms (front), 390 ms (rear), and 390 ms (speed), with gains fixed at 1.The first-order actuator models use unit steady-state gain.Source: §5 · pp. 4–5
• Estimation fits were 91.0% (speed), 91.5% (front steering), and 95.1% (rear steering); validation fits were 84.9–93.2%, 66.5–89.4%, and 85.7–91.4%, respectively.Table 2 compares simulated model output with real-system data from recorded estimation experiments.Source: §5 · pp. 4–5, Table 2 · p. 5
• After 5% settling time, median tracking errors were -0.26° (1.7% below 15.0° input) for rear and -0.15° (0.4% below 34.0° input) for front steering.These medians were calculated to assess the unit steady-state gain assumption.Source: §5 · p. 5
• The final 4WS vehicle model combines the kinematic model with the identified actuator-system equations.Equation (6) gives the resulting system model.Source: §5 · p. 5

6. DISCUSSION

• The paper proposes a system identification procedure and resulting system model for a standard 4WS agricultural tractor.Source: §6 · p. 5
• Field-condition experiments were collected to estimate and validate models for speed actuation and the independently modeled front- and rear-wheel steering systems.The dataset covers the actuators’ relevant operating range, while additional excitation signals, such as a typical path-tracking turn, could further assess model response.Source: §6 · p. 5
• The model is intended for optimal-control path tracking on the research tractor G-trac, while the identification procedure can be adapted to similar 4WS or front-wheel-steering vehicles.Such models are needed, for example, in model-based control systems using model predictive control.Source: §6 · pp. 5–6
• The model omits slip, limiting its applicability to heavy work such as ploughing; the immediate use case is low-slip grassland operations such as mowing.A model that explicitly addresses slip, together with slip estimation, is left for future work.Source: §6 · p. 6

7. CONCLUSION

• A sufficient fit to validation data suggests that the kinematic 4WS model with estimated first-order steering and speed actuation parameters is feasible for field experiments in low-slip conditions.Source: §7 · p. 6