GABES-LSTM-Based Method for Predicting Draft Force in Tractor Rotary Tillage Operations

Abstract

During rotary tillage operations, the draft force is jointly affected by operating parameters and soil conditions, exhibiting pronounced nonlinearity, time-varying behavior, and historical dependence, which all impose higher requirements on tractor operating parameter matching and traction performance analysis. A draft force prediction method that is based on a long short-term memory (LSTM) neural network jointly optimized by a genetic algorithm (GA) and the bald eagle search (BES) algorithm, termed GABES-LSTM, is proposed to address the limited prediction accuracy and stability of traditional empirical models and single data-driven approaches under complex field conditions. First, on the basis of the mechanical characteristics of rotary tillage operations, a time-series mathematical description of draft force is established, and the prediction problem is formulated as a multi-input single-output nonlinear temporal mapping driven by operating parameters such as travel speed, rotary speed, and tillage depth. Subsequently, an LSTM-based draft force prediction model is constructed, in which GA is employed for global hyperparameter search and BES is integrated for local fine-grained optimization, thereby improving the effectiveness of model parameter optimization. Finally, a dataset is established using measured field rotary tillage data to train and test the proposed model, and comparative analyses are conducted against LSTM, GA-LSTM, and BES-LSTM models. Experimental results indicate that the GABES-LSTM model outperforms the comparison models in terms of mean absolute percentage error, mean relative error, relative analysis error, and coefficient of determination, effectively capturing the dynamic variation characteristics of draft force during rotary tillage operations while maintaining stable prediction performance under repeated experimental conditions. This method provides effective data support for draft force prediction analysis and operating parameter adjustment during rotary tillage operations.

1. Introduction

Rotary tillage is one of the most commonly used tillage methods in land preparation for dryland and paddy fields, and its operational quality and energy consumption level directly affect subsequent seeding quality and overall agricultural production efficiency [1,2,3]. Draft force, as a key dynamic indicator reflecting the interaction among the tractor–rotary tiller–soil system, is widely used for tractor power matching, operational performance evaluation, and energy consumption analysis [4]. Therefore, research on the modeling and prediction of draft force during rotary tillage can provide a quantitative basis for tractor power matching, operational performance assessment, and energy consumption analysis.

Although rotary tillage is an active tillage operation in which the dominant portion of engine power is transmitted through the Power Take-Off (PTO) to drive blade rotation, the total engine output is practically distributed between PTO power and traction power required to overcome the forward resistance of the implement [5]. The traction power component is directly determined by the draft force acting on the rotary tiller. Under typical operating conditions, effective soil fragmentation is primarily achieved by blade rotation, and the draft force remains relatively low [6]. However, unfavorable blade–soil interaction or inappropriate operating parameters may increase draft force, indicating inefficient energy allocation and undesired pushing effects rather than cutting-dominated behavior [7]. Moreover, abnormal draft force patterns, such as persistent negative values or pronounced temporal fluctuations, may provide valuable information for monitoring implement condition and operational quality, including blade wear, improper installation, or excessive soil disturbance. In this context, modeling and predicting the time-series behavior of draft force can support load condition assessment, operational state diagnosis, and operating parameter adjustment during rotary tillage, rather than acting as a direct surrogate for total energy consumption.

During rotary tillage operations, the draft force mainly originates from the cutting, fragmentation, and disturbance of soil by the rotary blades, and its magnitude is jointly influenced by multiple factors, including operating speed, tillage depth, blade shaft rotational speed, and the physicochemical properties of the soil [8]. The high-speed rotation of the blades significantly alters the soil stress state because rotary tillage is an active till-age method, causing the draft force to exhibit pronounced nonlinearity, time-varying behavior, and fluctuation characteristics over time [9]. Existing studies on draft force have primarily focused on the development of empirical and semi-empirical models [10,11,12]. These studies have established functional relationships between draft force and factors, such as operating speed, tillage depth, tool structure, and soil physical properties, by analyzing soil mechanical properties and the effects of operating parameters [13,14,15,16]. However, draft force models based on empirical or semi-empirical formulations typically rely on a limited number of operating parameters to construct simplified relationships, making it difficult to fully account for the cumulative effects of soil disturbance and the dynamic characteristics of the tillage process. As a result, their prediction accuracy and applicability under complex field conditions are inherently limited.

With the advancement of sensor technologies and agricultural machinery operation data acquisition methods, data-driven modeling approaches based on measured data have gradually been introduced into studies on draft force and operational load prediction [17,18,19]. Some researchers have employed artificial neural networks, support vector regression, and related methods to model agricultural machinery operating resistance or energy consumption, achieving improved prediction accuracy to a certain extent com-pared with traditional empirical models [20,21]. However, the draft force during rotary tillage inherently exhibits pronounced time-series characteristics because its current state depends not only on the instantaneous values of operating parameters but also closely on the soil disturbance conditions from preceding stages [22,23]. Conventional feedforward models or static regression methods fail to adequately account for the cumulative effects of soil disturbance during rotary tillage operations and the temporal dependence of draft force.

LSTM networks selectively retain historical information through the introduction of gating mechanisms. They exhibit significant advantages in handling nonlinear time-series problems, and they have been widely applied in fields such as load prediction and energy consumption modeling [24,25]. Existing studies indicate that LSTM performs well in modeling nonlinear and time-varying load signals. However, its prediction performance is highly sensitive to network architecture and hyperparameter settings [26]. When parameters are improperly selected, the model is prone to becoming trapped in local optima. This degrades prediction accuracy and stability. Introducing intelligent optimization algorithms for LSTM hyperparameter tuning has emerged as a viable approach. Genetic algorithms demonstrate strong global search capability [27], whereas the bald eagle search algorithm performs well in local fine-grained optimization [28]. However, a single optimization strategy still exhibits inherent limitations when addressing complex non-convex problems.

In summary, existing studies on draft force prediction during rotary tillage still face several challenges. First, traditional empirical models show limited adaptability to com-plex operating conditions. Second, some data-driven methods fail to fully exploit the time-series characteristics of draft force. Third, there remains room for improvement in accuracy and stability when a single intelligent optimization strategy is applied to LSTM hyperparameter tuning. This study proposed an LSTM-based draft force prediction method jointly optimized by a GA and the BES algorithm to address these issues. A time-series prediction model for draft force during rotary tillage operations was developed, and GA and BES were introduced to jointly optimize key model hyperparameters. A dataset was constructed using measured field rotary tillage data to train and test the model, and its prediction performance was subsequently validated. The results provide data support for draft force prediction analysis and operating parameter adjustment during rotary tillage operations.

2. Materials and Methods

2.1. Mechanism Analysis and Mathematical Description of Draft Force in Rotary Tillage Operations

During tractor rotary tillage operations, the draft force mainly arises from the combined reaction forces generated when rotary blades cut, fracture, and throw soil. Its magnitude is jointly influenced by multiple factors, including operating parameters, soil conditions, and implement structure [13]. The rotation of the rotary shaft significantly alters the soil stress state because rotary tillage is an active tillage method. As a result, the draft force exhibits pronounced nonlinearity and fluctuation characteristics over time [29]. Based on the mechanical characteristics of rotary tillage operations, the draft force can be regarded as the combined effect of soil cutting resistance, resistance induced by soil disturbance and soil throwing, and the motion resistance of the implement.

$$F_t=F_c+F_s+F_m$$

(1)

where F_t denotes the draft force during rotary tillage operations; F_c represents the resistance required for the rotary blades to cut the soil; F_s refers to the additional resistance caused by soil disturbance and soil throwing induced by the blades; F_m denotes the frictional and structural resistance generated during the motion of the implement.

The volume of soil cut by the rotary blades per unit time is an important factor influencing cutting resistance. When the influence of blade structural differences is neglected, the soil volume cut per unit time can be approximated as being proportional to the rotary tillage depth and the tractor forward speed under ideal conditions [30]. The interaction between rotary blades and soil exhibits pronounced periodic characteristics. The rotational speed of the blade shaft determines the frequency of blade–soil contact within a given time interval, thereby exerting a significant influence on the instantaneous variation in draft force. Therefore, the draft force during rotary tillage operations can be abstractly expressed as a nonlinear function of the operating parameters as follows:

$$F_t=f(d,v,n)$$

(2)

where f(·) denotes a nonlinear function; d represents the rotary tillage depth; v denotes the tractor forward speed; n represents the rotational speed of the rotary tiller shaft.

At present, draft force is typically modeled using empirical or semi-empirical formulations [17,31]. The general expression can be written as follows:

$$F_t=\mu \cdot d^{\alpha }\cdot v^{\beta }\cdot n^{\gamma }$$

(3)

where μ is a composite coefficient, and α, β, and γ are empirical exponents.

Although empirical formulas can capture the overall trend of draft force variation with operating parameters, their coefficients and exponents are affected by factors such as soil moisture content, soil compaction, and tillage history. As a result, the draft force during rotary tillage exhibits pronounced time-varying characteristics, making it difficult for empirical models to accurately describe its dynamic behavior. When the cumulative effects of soil disturbance during rotary tillage are further considered, the draft force at the current time depends not only on the instantaneous values of operating parameters but also closely on the operating conditions over a preceding time interval. Accordingly, the draft force prediction problem in rotary tillage can be formulated as a multi-input single-output nonlinear time-series regression problem as follows:

$$\left\{\begin{array}{l}{F}_t=f(x(t),x(t-1),\cdots ,x(t-k))\\ x(t)=\left[d(t),v(t),n(t)\right]\end{array}\right.$$

(4)

where x(t) is the operating-parameter input vector at time t; d(t) is the rotary tillage depth; v(t) is the tractor forward speed; and n(t) is the rotary shaft speed.

The above mathematical formulation indicates that the draft force during rotary till-age exhibits pronounced nonlinearity, time-varying behavior, and historical dependence. For such problems, time-series prediction models capable of representing long-term dependencies are more appropriate. LSTM can effectively capture the dynamic mapping between operating parameters and draft force by selectively retaining historical information through gating mechanisms. However, the prediction performance of LSTM is highly sensitive to hyperparameter settings. When parameters are improperly chosen, the model is prone to becoming trapped in local optima, thereby degrading prediction accuracy. This study employed GA and BES to jointly optimize key LSTM hyperparameters to improve prediction accuracy and stability. On the basis of this strategy, a GABES-LSTM draft force prediction model for rotary tillage operations was developed, enabling high-accuracy prediction of draft force during rotary tillage.

2.2. Construction of GABES-LSTM Draft Force Prediction Model

2.2.1. Structure of LSTM-Based Draft Force Prediction Model for Rotary Tillage Operations

The draft force during rotary tillage exhibits clear continuity and cumulative effects over time. Its instantaneous value is influenced not only by the current operating parameters but also closely by the soil disturbance state induced by the blades over a preceding time interval. Therefore, in the LSTM unit, the forget gate is used to regulate the extent to which historical operating states contribute to the current draft force prediction. The corresponding calculation is expressed as

$$f_t=\sigma \left(w_f\left[h_{t-1},x_t\right]+b_f\right)$$

(5)

where $f_t$ denotes the forget gate vector at time t; σ is the sigmoid activation function; $w_f$ and $b_f$ are the weight matrix and bias vector of the forget gate, respectively; and $h_{t-1}$ is the previous hidden state reflecting the cumulative effect of past rotary tillage operating parameters on the draft force.

Through the forget gate, the model can automatically attenuate the influence of earlier operating states on the current draft force prediction when operating parameters change, thereby preventing historical information from interfering with the prediction results. The input gate is used to represent the instantaneous influence of the current operating parameters on the draft force. Its calculation is given as:

$$\left\{\begin{array}{l}{i}_t=\left(w_i\left[h_{t-1},x_t\right]+b_i\right)\\ {\tilde{c}}_t=\tanh \left(w_c\left[h_{t-1},x_t\right]+b_c\right)\end{array}\right.$$

(6)

where $i_t$ denotes the input gate vector of the LSTM unit at time t; $w_i$ is the weight matrix of the input gate, and $b_i$ is the corresponding bias vector; ${\tilde{c}}_t$ represents the candidate cell state generated from the current operating parameters; and $w_c$ and $b_c$ are the weight matrix and bias vector associated with the candidate cell state, respectively.

After the effects of historical operating states and the current operating parameters are integrated, the LSTM cell state can be updated as follows:

$$c_t=f_t\cdot c_{t-1}+i_t\cdot {\tilde{c}}_t$$

(7)

This state update process enables the modeling of the “memory effect” of draft force during rotary tillage operations. Specifically, $c_{t-1}$ represents the stress state of the soil formed under disturbance from the previous operating stage, and $c_t$ reflects the combined state under the current operating conditions.

The output gate is used to control the contribution of the current cell state to the draft force prediction. Its expression is given as

$$\left\{\begin{array}{l}{o}_t=\sigma \left(w_o\left[h_{t-1},x_t\right]+b_o\right)\\ h_t=o_t\cdot \tanh (c_t)\end{array}\right.$$

(8)

where $o_t$ denotes the output gate vector of the LSTM unit at time t; $w_o$ is the weight matrix of the output gate, and $b_o$ is the corresponding bias vector; and $h_t$ represents the hidden-layer output of the LSTM unit at time t. This output encodes the combined influence of current and historical operating parameters on the draft force.

After time-series feature extraction is completed, the hidden-layer output is mapped to the draft force prediction through a fully connected layer as follows:

$${\widehat{F}}_t=g\left(h_t\right)$$

(9)

where ${\widehat{F}}_t$ denotes the predicted draft force of rotary tillage at time t; and $g\left(\cdot \right)$ represents the linear mapping function.

2.2.2. Training Objective Function and Hyperparameter Representation

During the draft force prediction of rotary tillage operations, the objective function in the model training stage is primarily used to guide network parameter updates, with the aim of improving the consistency between predicted and measured draft force values. Considering that draft force is prone to large fluctuations when operating parameters change abruptly or soil conditions vary, the mean squared error (MSE) was selected as the objective function for model training [13,24]. Its expression is given as

$$J\left(\theta \right)={\displaystyle \frac{1}{N}}{\sum _{t=1}^N\left(F_t-{\widehat{F}}_t\right)}^2$$

(10)

where $J\left(\theta \right)$ denotes the objective function used for LSTM model training; N is the number of samples; θ represents the set of model hyperparameters; and $F_t$ denotes the measured draft force of rotary tillage at time t.

Using MSE as the training objective function imposes a higher penalty on time instants with large prediction errors, helping enhance model performance under operating conditions where draft force varies sharply. In addition, the MSE function is continuous and differentiable, making it suitable for gradient-based network parameter optimization. MSE improves the model’s fitting capability in regions with strong force variations be-cause draft force during rotary tillage is prone to instantaneous fluctuations when operating parameters change abruptly or soil conditions vary. This, in turn, enhances the overall reliability of the prediction results [32]. The prediction performance of the LSTM model largely depends on the selection of hyperparameters. Considering the characteristics of draft force prediction in rotary tillage operations, the number of hidden-layer neurons, the learning rate, and the number of training iterations were selected as the primary optimization variables. The corresponding hyperparameter vector is given as

$$\theta =\left[N_h,\eta ,E\right]$$

(11)

where N_h denotes the number of neurons, which determines the model’s capacity to represent the nonlinear characteristics of draft force; η represents the learning rate, which mainly affects the step size and convergence speed of model parameter updates; E denotes the number of training iterations.

2.2.3. GABES-Based Joint Hyperparameter Optimization Method

Genetic algorithms exhibit strong global search capability, whereas the BES algorithm demonstrates high convergence accuracy in the local search stage [27,28]. Therefore, combining GA and BES enables efficient optimization of LSTM hyperparameters over a large search space. During the joint optimization process, a GA is first employed to perform a global search over the hyperparameter vector. The hyperparameters are encoded as chromosomes, and a new generation of the population is produced through selection, crossover, and mutation operations. Individual fitness is then evaluated on the basis of the objective function value, allowing hyperparameter combinations with superior prediction performance to be identified. The optimization process at the GA stage effectively prevents the model training from becoming trapped in local optima.

Prior to optimization, the search ranges of key LSTM hyperparameters were predefined to ensure a well-constrained and computationally feasible optimization process. Specifically, the number of hidden neurons was searched within the range of [10, 100], the learning rate was constrained to [0.0005, 0.01], and the maximum number of training iterations was set within [50, 300]. These ranges were selected to cover commonly adopted configurations for LSTM-based time-series prediction while maintaining training stability and efficiency. After the GA obtains an initial set of preferred solutions, the BES algorithm is introduced to perform local fine-grained optimization. The BES algorithm achieves an adaptive balance between exploration and exploitation of the solution space by simulating the circling search and diving attack behaviors of bald eagles during predation. In the local optimization stage, the BES algorithm takes the preferred solutions generated by the GA as initial positions and continuously updates the hyperparameters, thereby further reducing the model prediction error.

A GABES joint optimization mechanism was established by combining the global search capability of the GA with the local optimization capability of the BES algorithm. The workflow of the GABES-LSTM model is illustrated in Figure 1. An LSTM-based draft force prediction model was constructed using the hyperparameter configuration obtained through GABES optimization and trained with experimental data from rotary tillage operations. This enables high-accuracy prediction of draft force during rotary tillage operations.

2.3. Experimental Platform Setup and Data Processing Methods

As shown in Figure 2, a draft force prediction experimental platform for tractor rotary tillage operations was developed on the basis of an electric tractor. The main parameters of the platform are listed in

Table 1. Parameters of the field experimental platform for tractor rotary tillage draft force measurement.

Parameter	Value
Overall dimensions (length × width × height)/mm × mm × mm	3850 × 1240 × 2440
Wheelbase/mm	1585
Front track width/mm	1000
Rear track width/mm	1040
Main transmission shifting mode	Continuously variable motor speed control
Rated power/kW	18.4
Power source	Lithium iron phosphate (LiFePO4) battery
Rated voltage/V	76.8

. The tractor (YL-254ET, Jiangsu Yueda Intelligent Agricultural Equipment Co., Ltd., Yancheng city, China) operated in a dual-motor mode. The operating speed and rear power output were controlled via the CAN bus communication protocol. Through the tractor three-point hitch system, a rotary tiller (1GQN-125, Shandong Zhida Agricultural Machinery Co., Ltd., Jinan city, China) was coupled with the tractor to form a rotary tillage unit. A non-contact angular displacement sensor (GTCS33618, Fenner Electronics, Shenzhen city, China) was used to measure rotary tillage depth. A Hall-effect rotational speed sensor (CZ400, Shanghai Chuanzhen Co., Ltd., Shanghai city, China) was employed to acquire the rotary shaft speed. A pin-type force sensor (YGZX-2, Anhui Guangya Sensor Co., Ltd., Hefei city, China) was used to measure draft force. In addition, a ground speed radar sensor (USRR251, Ruida Technology Co., Ltd., Shenzhen city, China) was applied to obtain the tractor forward speed.

Field rotary tillage experiments were conducted at the experimental farm of Pukou Campus, Nanjing Agricultural University, Nanjing, Jiangsu Province, China (Figure 2b). These experiments were carried out to collect data required for developing the draft force prediction model for tractor rotary tillage operations. The data collection was performed during continuous rotary tillage operations covering an effective field area of approximately 1.2 ha, including multiple complete rotary tillage passes under typical field operating conditions. During the experimental period, weather conditions were stable, with no rainfall occurring during operations, and the meteorological conditions remained relatively consistent. The ambient temperature during the experiments ranged from 22 °C to 26 °C. The test field had an average soil moisture content of 18.7%, an average bulk density of 1.41 kg m⁻³, and an average soil compaction of 721 kPa. These conditions are representative of conventional rotary tillage operations in the local region and ensured stable soil–tool interaction during data acquisition. Some collected samples exhibited abnormal fluctuations and data inconsistencies owing to the field operating environment and the characteristics of the data acquisition system. To ensure the reliability of the training data, The raw data were preprocessed prior to modeling to ensure the reliability of the training data. When the deviation of a data sample from the mean exceeded 3σ, the sample was identified as an outlier and removed according to the 3σ criterion. This method was based on the assumption of a normal distribution and can effectively identify abnormal samples caused by random errors. Let the sample data be x₁, x₂, …, x_N, and the deviation can be defined as $v_i=x_i-\overline{x}$, where i = 1, 2, …, N. The standard deviation of the data samples is calculated as

$$\left\{\begin{array}{l}\sigma =\sqrt{{\displaystyle \sum _{i=1}^N{v}_i^2/\left(N-1\right)}}\\ \begin{array}{cc}{v}_i=x_i-\overline{x},& i=1,2,3,\dots ,N\end{array}\end{array}\right.$$

(12)

where σ denotes the standard deviation of the data samples; N is the number of samples; and $\overline{x}$ represents the mean value of the sample data.

After abnormal data were removed, the dataset for the draft force prediction model contained 975 time-series samples. These samples were extracted from steady-state rotary tillage segments characterized by stable forward speed, consistent tillage depth, and continuous soil engagement, thereby covering the dominant and repeatedly occurring operating states during rotary tillage. The preprocessed data were randomly divided into a training set and a test set to develop the draft force prediction model, with 90% of the samples used for model training and the remaining 10% used to validate the model’s pre-diction performance.

3. Results and Discussion

3.1. Evaluation Metrics for Prediction Models

The mean absolute percentage error (MAPE), mean relative error (MRE), relative percent deviation (RPD), and coefficient of determination (R²) were selected as evaluation metrics to quantitatively evaluate the prediction performance of the draft force prediction model for rotary tillage operations. These indicators were used to analyze the model from multiple aspects, including prediction error magnitude, error distribution characteristics, and goodness of fit.

MAPE is used to quantify the relative deviation between predicted values and measured values. Its calculation is given as

$$MAPE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|\frac{{\widehat{F}}_i-F_i}{F_i}\right|\times 100\%$$

(13)

MRE is used to describe the average relative magnitude of prediction errors. It reflects the overall deviation direction and magnitude of the predicted results relative to the measured values. Its expression is given as

$$MRE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\frac{{\widehat{F}}_i-F_i}{F_i}}$$

(14)

RPD is used to evaluate the relationship between the dispersion of the predicted results and that of the measured data. A larger RPD value indicates better predictive capability. Its expression is given as

$$\left\{\begin{array}{l}RPD={\displaystyle \frac{SD}{RMSE}}\\ SD=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(F_i-{\overline{F}}_i\right)}^2}}\\ RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(F_i-{\widehat{F}}_i\right)}^2}}\end{array}\right.$$

(15)

where SD denotes the standard deviation of the measured draft force data; RMSE represents the root mean square error between the predicted and measured values; and $\overline{F}$ is the mean value of the measured draft force.

The coefficient of determination is used to evaluate the degree to which the predicted results fit the variation trend of the measured data. Its expression is given as follows:

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N{\left({\widehat{F}}_i-F_i\right)}^2}}{{\displaystyle \sum _{i=1}^N{\left(F_i-{\overline{F}}_i\right)}^2}}}$$

(16)

3.2. Comparison of Prediction Performance Among Different Models

LSTM, GA-LSTM, BES-LSTM, and GABES-LSTM models were established and tested using the same dataset to comparatively analyze the performance of the proposed GABES-LSTM draft force prediction model. All models were trained under identical training conditions and applied to predict draft force during rotary tillage operations on the test set. The prediction results of the different models on the test set are shown in Figure 3, Figure 4, Figure 5 and Figure 6.

On the basis of the evaluation metrics of the prediction models, the predictive performance of each model on the test set was quantitatively assessed, and the results are presented in

Table 2. Performance evaluation results of the models.

Model	MAPE/%	MRE/%	RPD	R2
LSTM	7.7869	6.6105	6.8302	0.9483
GA-LSTM	7.3032	4.7643	7.2556	0.9560
BES-LSTM	3.4794	1.8923	11.1563	0.9826
GABES-LSTM	2.4482	0.4805	13.3711	0.9902

. As shown in Table 2, clear differences existed among the models across all evaluation metrics, reflecting the influence of different hyperparameter optimization strategies on prediction performance. As the baseline model, LSTM achieved MAPE, MRE, RPD, and R² values of 7.7869%, 6.6105%, 6.8302, and 0.9483, respectively. These results indicate that, without hyperparameter optimization, the model can learn the basic variation trend of draft force. However, deficiencies remained in terms of error magnitude and systematic bias. This limitation is mainly attributed to the fact that draft force during rotary tillage is jointly influenced by multiple operating parameters and exhibits nonlinear fluctuations over time. Under such conditions, an LSTM with a fixed architecture and parameter configuration has difficulty maintaining consistently high prediction performance across different operating regimes.

After a GA was introduced to optimize the LSTM hyperparameters, the GA-LSTM model achieved MAPE and MRE values of 7.3032% and 4.7643%, respectively. Compared with the baseline LSTM model, these values decreased by 6.21% and 27.93%. At the same time, the RPD and R² values increased to 7.2556 and 0.9560, respectively. These results indicate that adjusting network hyperparameters through a global search strategy can alleviate systematic bias to a certain extent. However, the ability of GA to capture local, sharp fluctuations in draft force remains limited because GA tend to suffer from reduced convergence efficiency in the later stages of the search. This limitation is reflected in the relatively large local deviations observed in certain intervals of the GA-LSTM prediction curves.

By comparison, the BES-LSTM model exhibited a further improvement in prediction performance. Its MAPE and MRE decreased to 3.4794% and 1.8923%, respectively, representing reductions of 55.33% and 71.38% relative to the baseline LSTM model. Meanwhile, the RPD and R² values increased to 11.1563 and 0.9826, respectively. These results demonstrate that the BES algorithm has strong capability in local hyperparameter search and fine-tuning. This capability enhances the model’s ability to fit nonlinear variations and abrupt changes in draft force time-series data, thereby significantly reducing prediction error magnitude and improving overall goodness of fit.

On this basis, the GABES-LSTM model achieved the best performance across all evaluation metrics on the test set. Its MAPE and MRE decreased to 2.4482% and 0.4805%, respectively, representing reductions of 68.56% and 92.73% compared with the baseline LSTM model. Meanwhile, the RPD increased to 13.3711, which is 95.76% higher than that of the LSTM model, and the R² value improved to 0.9902, corresponding to an increase of 4.42%. These results indicate that the joint optimization strategy combining GA and BES balances global exploration and local fine-tuning during hyperparameter search. As a result, the LSTM model structure becomes better aligned with the time-series characteristics of draft force during rotary tillage operations. This leads to further improvements in reducing prediction errors, mitigating systematic bias, and enhancing fitting stability.

As shown in Figure 7, the predictions of the GABES-LSTM model were distributed closely around the regression line (y = x), and the coefficient of determination reached 0.9902. This finding a high consistency between the predicted results and the measured tractor draft force in terms of variation trends, demonstrating good predictive performance. Considering that intelligent optimization algorithms are influenced by random initialization and search paths during the optimization process, repeated experiments were conducted for each model under identical experimental conditions to further examine prediction stability. The corresponding results are summarized in

Table 3. Results of repeated experiments for the prediction models.

Model	Parameter	Number of Repeated Tests			Mean Value
		No. 1	No. 2	No. 3
LSTM	MAPE	7.3975	7.8676	7.3975	7.5542
	MRE	6.8081	6.8085	6.5439	6.7202
	RPD	8.6539	9.0604	8.7418	8.8187
	R2	0.9372	0.9570	0.9418	0.9453
GA-LSTM	MAPE	7.5953	7.0112	6.9380	7.1815
	MRE	4.7167	4.8119	4.6217	4.7168
	RPD	6.8928	7.3286	7.0379	7.0864
	R2	0.9462	0.9650	0.9560	0.9557
BES-LSTM	MAPE	3.4794	3.5837	3.5489	3.5373
	MRE	1.8737	1.9123	1.9301	1.9054
	RPD	11.2678	10.7108	11.4909	11.1565
	R2	0.9783	0.9689	0.9891	0.9787
GABES-LSTM	MAPE	2.3527	2.3257	2.4971	2.3918
	MRE	0.4685	0.4992	0.4915	0.4864
	RPD	13.3711	13.2379	13.2373	13.2821
	R2	0.9843	0.9914	0.9886	0.9881

. The results show that the GABES-LSTM model maintained relatively stable MAPE, MRE, RPD, and R² values across multiple trials. Compared with the LSTM model’s values, the average values improved by 68.33%, 92.76%, 50.60%, and 4.54%, respectively, indicating a substantial performance enhancement. Figure 8 presents a comprehensive performance radar chart of the different prediction models on the basis of the evaluation metrics. The radar chart further demonstrates that the GABES-LSTM model exhibited a well-balanced overall performance in terms of error suppression and fitting capability. These findings indicate that the joint optimization strategy reduces the model’s sensitivity to random factors to a certain extent. As a result, it enables reliable prediction of tractor draft force during rotary tillage operations. This provides robust data support for real-time draft force estimation and adaptive operational control and lays a foundation for subsequent operating parameter optimization and power matching analysis.

4. Discussion

4.1. Model Limitations

Soil physical properties, including soil moisture content, bulk density, and soil compaction, are widely recognized as important factors influencing draft force during rotary tillage operations. Variations in these parameters affect soil shear strength, soil–tool interaction behavior, and the resistance encountered during soil cutting, fragmentation, and displacement processes, which in turn influence the magnitude and fluctuation characteristics of draft force. In the present study, field experiments were conducted under relatively stable soil conditions, and soil parameters were treated as fixed background variables. Accordingly, the model inputs were limited to operating parameters, namely forward speed, rotary shaft speed, and tillage depth. Under this experimental scope, the proposed GABES-LSTM model is intended to characterize the nonlinear temporal dependency between operating parameters and draft force under representative field conditions.

From a data-driven perspective, the influence of soil properties is implicitly embedded in the collected time-series data, while explicit variation in soil parameters is beyond the scope of the current experimental design. As a result, model performance is primarily evaluated with respect to its ability to capture temporal dynamics and nonlinear responses of draft force to operating parameters under the tested conditions. Future studies may further explore the application of the proposed framework under a wider range of field conditions to enhance its adaptability in practical rotary tillage operations.

In addition to soil conditions, the working width of the rotary tillage implement affects the overall magnitude of draft force by determining the scale of soil engagement and tool arrangement. In the present study, all experiments were conducted using a rotary tiller with a fixed working width, corresponding to a consistent implement configuration throughout data collection.

The primary objective of this study is to evaluate the effectiveness of a hybrid optimization–LSTM framework for time-series prediction of draft force under a given implement configuration and operating conditions. Under this setting, the proposed GABES-LSTM model focuses on learning the nonlinear temporal relationship between operating parameters and draft force for the tested implement configuration. Variations in working width, which would involve changes in implement geometry and soil engagement scale, were not considered in the current experimental design and are therefore beyond the scope of the present validation. From an application-oriented perspective, extending the proposed framework to rotary tillage operations with different implement configurations represents a meaningful direction for future research.

4.2. Comparison with Existing Approaches

In recent years, hybrid optimization–LSTM frameworks have been increasingly adopted to improve prediction accuracy and stability in agricultural machinery load and energy-related modeling. Commonly used optimization strategies include GA, particle swarm optimization, and other swarm intelligence methods, which are primarily employed to tune LSTM hyperparameters such as network structure and learning rate. Existing hybrid approaches typically rely on a single optimization algorithm, which may exhibit inherent limitations when dealing with complex, non-convex hyperparameter spaces. Global optimization-oriented algorithms, such as GA, are effective in exploring a wide solution space but may converge slowly or lack fine-grained local search capability. In contrast, local search-oriented strategies often demonstrate fast convergence but are more sensitive to initial solutions and may become trapped in local optima.

In this study, GA and BES algorithm are coupled to exploit their complementary strengths. GA is first employed to perform global exploration and identify promising regions in the hyperparameter space, while BES is subsequently used to refine the solution through local exploitation. This cooperative optimization strategy aims to balance exploration and exploitation, thereby enhancing convergence stability and reducing the risk of premature convergence. Compared with single-optimizer hybrid LSTM approaches reported in the literature, the proposed GA–BES–LSTM framework demonstrates improved robustness and prediction stability under field operating conditions, particularly for draft force signals characterized by strong nonlinearity and temporal variability. These results suggest that coupling multiple optimization strategies can be a practical and effective way to enhance LSTM-based time-series modeling performance for agricultural machinery applications.

5. Conclusions

This study addresses the challenges of strong nonlinearity, pronounced time-varying behavior, and historical dependence of draft force during tractor rotary tillage operations. A draft force prediction method based on an LSTM network jointly optimized by GA and BES is proposed. The model performance is validated and comparatively analyzed using field experimental data. The main conclusions are summarized as follows:

(1) A time-series mathematical description of draft force was established on the basis of the mechanical characteristics of rotary tillage operations and the variation patterns of operational parameters. The nonlinear multi-input single-output relationship between draft force and key operational parameters was clearly identified. These parameters include forward speed, rotary tiller shaft speed, and tillage depth. This modeling framework provides a solid foundation for the development of data-driven prediction models.

(2) A draft force prediction model based on an LSTM neural network was developed. GA and BES algorithms were introduced to jointly optimize the key hyperparameters of the model. The proposed optimization strategy enabled effective searching of the model parameter space by combining global exploration with a local fine-grained search. As a result, the model’s capability to capture the temporal variation characteristics of draft force during rotary tillage operations was significantly enhanced.

(3) Field rotary tillage experimental data were used to comparatively evaluate the proposed GABES-LSTM model against the LSTM, GA-LSTM, and BES-LSTM models. The results show that the GABES-LSTM model achieved higher prediction accuracy across multiple evaluation metrics. In addition, it maintained good stability under repeated experimental conditions. These findings indicate that the joint optimization strategy effectively reduces the model’s sensitivity to random factors.