A New Model to Predict the Slippage Coefficient of Tracked Vehicles During Steering

Tracked vehicles during steering are always subjected to track slippage relative to the ground. Studies show that an excessive slippage coefficient reduces the steering ability of tracked vehicles and increases the risk of soil shear failure. Accordingly, fast and accurate tests of the slippage coefficient of tracked vehicles under different ground conditions can be effectively referenced to select track parameters and reach optimal operation. Aiming to resolve shortcomings of the conventional testing method in determining the slippage coefficient of tracked vehicles, a novel method is proposed in the present study. Then the trajectory of an arbitrary point on the tracked vehicle during steering was analyzed. Moreover, the time of vehicles steering around and the speed of track sprocket were measured, and the slippage coefficient of tracked vehicles steering on the specific ground condition was determined. Meanwhile, the slippage coefficient of the tracked combined harvester steering under different ground conditions was explored using the conventional method. The obtained results were compared quantitatively, and the accuracy of the established model was verified. The performed analyses demonstrate that the proposed method has a simple structure and can be effectively applied to various complicated working conditions. This method is expected to lay a theoretical basis to establish an online testing system for the slippage coefficient of tracked vehicles during steering.


I. INTRODUCTION
In the steering process of tracked vehicles, the vehicle track is constantly accompanied by slippage relative to the ground. This phenomenon, which reduces the maneuverability of tracked vehicles compared to the theoretical maneuverability, can be classified by different degrees of slip and track skids on both sides under different ground conditions [1]- [5]. Studies show that excessive slippage coefficient directly affects the safety, steering stability, and operating efficiency of tracked vehicles [6]- [9]. Accordingly, the operation of vehicle tracks is limited to the optimal range of slippage coefficient to reach reasonable drive efficiency and steering stability. Furthermore, slippage of tracked vehicles in field operations may cause the compaction and shear of the tracked soil [10], destroy the original structure of the soil, and increase the risk of soil erosion [11], [12]. Accordingly, performing rapid and accurate tests on the slippage coefficient of tracked vehicles during steering is of significant importance for selecting the track parameters and optimal operating conditions under different working conditions.
The conventional methods to test the slippage coefficient of tracked vehicles are complicated [13]. Currently, the theoretical and actual moving speeds of tracked vehicles are tested respectively, and then the slippage coefficient is calculated. In this regard, the theoretical movement speed of tracks can be determined by measuring the sprocket rotation speed. The sprocket rotation speed is generally tested using a Hall sensor and photoelectric encoder. Moreover, the actual moving speed of a tracked vehicle can be calculated using GPS, radar, and other positioning devices. Zhong et al. [14] tested the slippage coefficient of a tractor. In this regard, the speed of the driver wheel of the tractor was measured using an encoder, while the actual speed of the vehicle was measured using GPS, radar method, and minimum wheel speed method. The obtained results revealed that the GPS method is highly affected by the weather condition, the radar method is applicable for good road conditions, and the minimum wheel speed method is applied to high-speed speeds. Lu et al. [15] designed a slippage coefficient prediction system of tractors based on simulations in the LabVIEW environment. To this end, a coded speed sensor, a low-speed radar velocimeter, and a signal acquisition and processing system were used, and the system accuracy was then verified through numerous field tests.
A series of methods have been proposed by some researchers to predict the track slippage coefficient using models. Song et al. [16] built a model relating to the theoretical moving speed, actual steering radius, steering angular speed of tracked vehicles, as well as the slippage coefficient of tracks. As indicated by their study, the slippage coefficient of both sides of tracks could be obtained by measuring the parameters of the moving track. Lu et al. [17] developed a soft-switching slippage mode observer to measure the track slip parameters of the tracked robot, and the observations were compensated into the adaptive backstepping controller to reduce the effects arising from track slippage. Song et al. [18] developed a slippage mode observer to estimate track slippage using the kinematic model of a skid-steering tracked vehicle and the measurement results achieved by on-board sensors. The accuracy of the slippage mode observer was verified in accordance with both simulation and experimental results. Yamauchi et al. [19] built a slip estimation method using a slippage model for tracked vehicles and applied it to slipcompensated odometry on Loose Slope. Their method was validated through an indoor slope-traveling experiment. Dar et al. [20] presented a method to estimate the slip parameters, orientation and trajectory of tracked vehicles using extended Kalman filter (EKF). The present method was validated through experiments using small-scale robotic tracked vehicles.
As indicated by the review of the relevant literature, the actual moving speed of vehicles is generally required for the calculation of track slippage coefficient. However, the actual moving speed of tracked vehicles is difficult to measure directly, the test methods are complicated, and rigorous operating environment requirements should be satisfied. Thus far, existing methods have been not sufficiently fast and accurate to calculate the slippage coefficient of tracked vehicles during steering.
In this paper, to build an accurate and simple method of calculating the slippage coefficient of tracked vehicles during steering, a model of predicting the track slippage coefficient without detecting the actual moving speed of vehicles was built. A hypothesis that proposed method could effectively reduce the measuring errors as compared with the conventional method was put forward. After the steering trajectory of tracked vehicles at different track slippage coefficients was theoretically analyzed, the correlation between the vehicle steering time and slippage coefficient of tracked vehicles was investigated. Subsequently, the track slippage coefficient was obtained by examining the steering around time of vehicles under the given ground conditions. Afterward, a tracked combined harvester was adopted to calculate the track slippage coefficient under different ground conditions, so as to verify the accuracy of the model.

1) SLIP AND SKID OF TRACKED VEHICLES DURING STEERING
To simplify the model, the following assumptions were made.
(1)The tracked vehicle is operating in a horizontal plane and has a steady-state steer.
(2)The longitudinal and lateral load transfers caused by the centrifugal force during the vehicle's steer are absent.
(3)The steering speed of tracked vehicle is constant. (4)The friction forces between the moving parts of the track are absent.
When the tracked vehicle steers, the actual steering center of the low-speed and the high-speed side tracks shifts by Y1 and Y2, respectively. Fig. 1 shows that this phenomenon mainly originates from the braking of the low-speed side track and the driving action of the high-speed side track. Accordingly, the theoretical speed U (the winding speed of the track relative to the vehicle body) of both sides differs from the actual speed V (the rotation speed of the tracked vehicle around the steering center).
The theoretical velocity of the high-speed side track U2 is greater than the actual velocity V2, resulting in the track slip. Moreover, the theoretical velocity of the low-speed side track U1 is less than the actual velocity V1, resulting in the track skid. Considering the slip and skid of the track, the actual steering radius RS of the vehicle is always greater than its theoretical steering radius RL. Similarly, the actual steering angular velocity ωS is always less than the theoretical steering angular velocity ωL. The skid coefficient σ1 of the low-speed side track and the slip coefficient σ2 of the highspeed side track can be expressed in the form below: where SL1 and SL2 are the theoretical driving distances of the low-speed and high-speed tracks, respectively. Furthermore, SS1 and SS2 denote the actual running distance of the low-VOLUME XX, 2017 9 speed and high-speed tracks, respectively. These parameters can be calculated using the following expressions: where n1 and n2 are the sprocket rotation speeds of the lowspeed and high-speed tracks, respectively. Moreover, α, r and B denote the vehicle steering angle, radius of the track sprocket, and the track gauge of the tracked vehicle, respectively. t is the running time of tracked vehicle under the vehicle steering angle α. Under the slip and skid conditions of tracks, the actual steering radius RS and the actual steering angular velocity ωS of the vehicle can be expressed in the form below:

2) TRAJECTORY OF THE TRACKED VEHICLE DURING STEERING
Given the slippage factor of tracked vehicles during steering, the trajectory of an arbitrary point on the track during steering can be analyzed. Fig. 2 shows a ground-based static coordinate system XOY and a dynamic coordinate system x1o1y1 rotating with the track chassis. The subscript P=1 and P=2 represent the low-speed and high-speed side track, respectively. Considering the correlation between the unit vector in the moving coordinate system and the unit vector in the static coordinate system, the velocity at an arbitrary point on the track in the static coordinate system can be obtained indirectly. Subsequently, integrating the velocity with respect to time t results in the trajectory equation on the track.   shows that at the initial state t=0, the static coordinate system XOY overlaps with the dynamic coordinate system x1o1y1. In this case, the coordinate of a point M on the track of the P side in the moving coordinate system x1o1y1 can be expressed as (xp0, yp0). As the test progresses, point M constantly moves relative to the ground. When the test time reaches 0<t, the steering angle of the tracked vehicle reaches φ, and the coordinate of the point M in the coordinate system x1o1y1 can be expressed as (xp, yp), where components are as follows: indicates that the implicated motion velocity of the point, which overlaps with the point M on the body can be calculated from the following expression: The relative movement speed of the track relative to the vehicle is: where i1 and j1 are two unit vectors in the moving coordinate system x1o1y1. Velocity of the point M relative to the ground is: Substituting Eqs. (6), (7), and (8) into Eq. (9) yields: On the other hand, the unit vectors i and j in the static coordinate system XOY can be expressed as: In the static coordinate system XOY, the velocity Vap can be expressed in the form below: Comparing Eqs. (12) and (13) yields the following system of equations:

3) CALCULATING THE SLIPPAGE COEFFICIENT OF TRACKED VEHICLES DURING STEERING.
In this section, it is intended to establish a mathematical model of the relationship between vehicle steering time t and slippage coefficient σp based on the steering trajectory equation of tracked vehicle during slippage. Fig. 3 shows the displacement of an arbitrary track grouser on the P-side track that contacts the ground with a slippage coefficient of σp. When the track is at the initial position t=0, the track grouser A0C0 at the front end of the track is in contact with the ground, and the track grouser A20C20 is at the last end of the track. When the vehicle steering time reaches t=t1, the corresponding track steering angle reaches α. Meanwhile, track grouser A0C0 leaves the ground, while track grouser A20C20 touches the ground. Under this circumstance, the angle α of track steering in the period from t=0 to t=t1 can be expressed as follows: The required time T for steering around of the tracked vehicle is: Substituting Eqs. (4), (5), (14) and (15)  As revealed by the preceding analysis, the track slippage coefficients refer to a function of the vehicle motion parameters and the structure parameters. The input of the proposed model consists of track width B, sprocket radius r, vehicle steering around time T, and track sprocket rotation speed np; the output of the model refers to track slippage coefficient σp. The track slippage coefficient can be calculated by measuring the input parameters under the given soil conditions.

1) TEST VEHICLE AND LOCATION
In the present study, a tracked vehicle (Ruilong 4LZ-5.0E, China) combined with a harvester (Jiangsu Word Agricultural Machinery Co., Model, China) was used in the experiments, as shown in Fig. 4. The vehicle was unloaded during the test. The main parameters of the vehicle are presented in Table 1. It is worth noting that steering tests were performed on three different surfaces, including cement ground, sand ground, and soft ground. The steering test on soft ground was carried out in paddy soil, and the soil parameters are listed in Table 2.
To evaluate the influence of the soil water content on the slippage coefficient of tracks, experiments were performed in VOLUME XX, 2017 9 the paddy soil with different water contents. To this end, tests were performed before and after rainfall. To facilitate the test, the combined harvester adopted the steering mode of unilateral braking, low-speed track braking, and high-speed track driving to complete the steering. During steering, the moving velocity was kept constant, and high-, medium-and low-speed gears were selected, respectively. Moreover, three groups of repeated tests were performed under each condition. Before the test, the surface soil was sampled to determine the soil moisture content.

2) TESTING PROCESS
The slippage coefficient of the combined harvester under different ground conditions was calculated using the conventional [21] and the proposed test method. In the conventional method, the sprocket rotation speed np and the actual steering radius RS are initially measured, and the track slippage coefficient can be calculated through Eqs. (1), (2), and (3). In the proposed method, the sprocket rotation speed np and the time T of the steering around were measured, then the slippage coefficient of the track was calculated by the proposed model in the MATLAB environment. The actual steering radius RS of the combined harvester was calculated by measuring the real trajectory of a certain point when the tracked vehicle was being steered. Fig. 5 shows the hourglass device installed at point B at the rear of the vehicle. Transverse and longitudinal distances of the hourglass device relative to the geometric center OV of the vehicle were measured. Then the coordinate value (xv,yv) of point B relative to the geometric center OV of the vehicle was determined [22].
When the vehicle entered the stable steering state, the hourglass device was used to record the steering trajectory of the vehicle. Then the starting point A and the ending point B were determined using the path remaining on the sand, and the midpoint C was marked. The steering trajectory was reduced to a certain proportion and was plotted on the graph. The center OS of the arc AB was determined by the vertical bisector of chord length AC and BC. The radius Rd of the arc AB can be determined by connecting AOS or BOS, and the actual steering radius RS of tracked vehicles can be calculated from the following expression: Track sprocket rotation speed np was tested using a wireless telemetry analysis system (Jiangsu Donghua Testing Co. DH5905, China), and the test system were illustrated in Fig. 6, a Hall sensor (Company name CHE18-15N11-HZF710, China) was used to measure the sprocket rotation speed. The main parameters of Hall sensor are listed in Table  3. Subsequently, the acquired data was transmitted to the computer acquisition system through a wireless router to realize real-time monitoring of track sprocket rotation speed np. The time T for steering around was measured using the manual timing method [21], [23].

3) STATISTICAL ANALYSIS
In the previous sections, the effects of the sprocket rotation speed and soil water content on the measured track slippage coefficient were explored. It is found that none of the studied variables follow a normal distribution. Accordingly, a oneway analysis of variance on ranks (Kruskal-Wallis test) with Post Hoc analysis (Dunn test) was applied. To this end, Origin 8.0 data analysis software was used to perform the calculations.

A. COMPARISON OF SLIPPAGE COEFFICIENT BETWEEN TWO TEST METHODS
Since the combined harvester adopts the steering mode of unilateral braking, the low-speed track is completely braked during steering. Accordingly, the skid coefficient remains almost constant [24], [25]. The high-speed track's slip coefficient of the combined harvester under different ground conditions at specified sprocket rotation speed is shown in Table 4. It is observed that the relative error between the average slip coefficient of the track calculated by the proposed model and the average measured using the conventional method is 5.7%-11.9%, which is within a reasonable range. Accordingly, it is inferred that the proposed model for the slip coefficient of tracked vehicles during steering is accurate and reliable. The deviation between the results of the two methods can be explained as follows: First, the measurement of the time T for steering around is not accurate enough. Moreover, since the steering time is measured manually, human errors are unavoidable. Second, the measurement of the actual steering radius RS is not accurate enough. Considering large vibrations during the test, deviations may occur when using the hourglass method to record the trajectory of the vehicle steering. Moreover, the centrifugal force of vehicle steering increases the actual steering radius [26], [27].
Compared with the conventional testing method, the proposed new method is simpler and easier to operate so that it can be used in complicated ground conditions. When the conventional testing method is applied to test the actual moving speed V of the track, vehicle vibration originating from uneven ground and the sensitivity of the measuring tool decreases. On the other hand, bad weather conditions which occasionally occur, adversely affect the testing accuracy [28]. However, the proposed testing method can predict track slip coefficient without detecting vehicle actual moving speed. The time T for steering around was used to replace the actual track speed V, thereby simplifying the testing process regardless of ground conditions and environmental factors. Table 4 reveals that the largest slip coefficient can be achieved from the tracked vehicles being steered on soft ground, followed by the coefficient on the sand and cement ground. Analyzing the obtained results demonstrate that the slip coefficient of tracked vehicles varies from 0.18-0.32 on hard ground (i.e. cement ground and sand ground) and 0.3-0.6 on soft ground. This may be interpreted as follows: (1) the friction coefficient between tracked vehicles and soft ground is smaller than that on hard ground; (2) Relative movement occurs among particles in the soil. Fig. 7 illustrates the mean values of the measured track slippage coefficient obtained using the conventional testing method and the proposed method under different soil water contents at a specified sprocket rotation speed. With the increase in the soil water content, the corresponding slippage coefficient of the track tended to increase based on both two methods. There was a significant difference in the track slippage coefficient between soil with relatively low water content (23.8% and 28.2%) and that high water content (38.9%) (P< 0.05). Thus, the track slippage was highly impacted by the soil water content. The reasons for the high slippage coefficient on soil with high water content include: (1) when the moisture content of soil increases, the friction coefficient between the track and soil decreases; (2) As the water content of soil increases, the corresponding soil strength decreases; (3) The increase of the soil moisture content reduces the relative movement resistance between soil particles, making the track more prone to slip.  Previous Studies [29]- [31] show that as the slip coefficient of tracks increases, the tractive force of tracked vehicles and the imbalance of track resistance moment on both sides decreases, thereby decreasing the steering performance of tracked vehicles. Meanwhile, the amount of track subsidence increases, thereby increasing the steering instability of the tracked vehicle [32], [33]. This may be attributed to the slipsinkage effect of the tracked vehicle. It is inferred that tracked vehicles should not be recommend to used in the field with high water content during the harvest time (e.g. in or after rainy days) to reduce the risk of reducing the production efficiency and safety accidents caused by unstable steering. Table 4 lists the mean values of the calculated track slippage coefficient using the conventional and proposed testing methods at a specified sprocket rotation speed. In our field test, three vehicle velocities (0.3, 0.8, and 1.4 m · s -1 ) were selected. Normally, the harvesting velocity of the combine harvester is between 0.8 m·s -1 and 1.4 m·s -1 , which depends on the type of cereals. Moreover, the traffic velocity at specific operation conditions (i.e. traffic in the field with high water content) is lower than normal speed. We choose these three traffic velocities because we would like to look at the track slip coefficient at different working conditions of combine harvester. As indicated by this table, the track slippage coefficient obtained using the two test methods did not change significantly with the increase in the track sprocket rotation speed (P > 0.05), indicating that the slip coefficient of tracks is independent of the vehicle steering speed and is mainly affected by the ground conditions (e.g. soil water content). The influence of other properties of the soil, including the soil texture and the soil strength on the slip coefficient of tracks should be further studied. Fig. 8 presents the change of mean measured track slippage coefficient with sprocket rotation speed obtained using two methods under different ground conditions. According to the figure, with the increase in the sprocket rotation speed of the vehicle, the corresponding slippage coefficient tended to decrease when it was calculated using the conventional method (Fig. 8a), while it remained almost constant when calculated using the proposed method (Fig.  8b). The different results by two methods may due to the increase of steering centrifugal force when increasing the steering speed of the tracked vehicle, which results in the excessive measured value of the actual steering radius Rs of the vehicle. It is concluded that the proposed test method can avoid the calculation error of slip coefficient caused by inaccurate measurement of the actual steering radius Rs. The performed analyses demonstrate that the established model to calculate the slippage coefficient of the tracked vehicle during steering has high accuracy in different ground conditions and moving speeds, which gives a potential for the establishment of an online prediction system of tracked vehicles' slippage coefficient [34]- [38]. Although the proposed model is simpler and faster than the conventional testing method, motion parameters of tracked vehicles still need to be tested. In order to resolve this shortcoming, more investigations are required to find a more rapid and convenient method to obtain track slippage coefficient during steering (e.g. prediction of track slippage coefficient according to soil mechanical parameters).

D. LIMITATION OF RESEARCH.
In our model, the tracked vehicle was assumed to turn under steady-state conditions, and the longitudinal and lateral load transfers caused by the centrifugal force were absent, this means that the mass center of the vehicle is at the same point as the geometric center of the vehicle. In general, the centrifugal force may cause the fore-aft offset of vehicle's mass center, resulting in increase of vehicle's actual steering radius RS, which may affect the accuracy of the model. Our future study will focus on developing more accurate models by accounting for additional effects from the vehicle centrifugal force.
Moreover, the vehicle steering speed (i.e. sprocket angular velocity) was assume to be constant in the model. In general, the steering speed of tracked vehicle may not be constant due to different working conditions. Thus, more situations (e.g. vehicle steering at variable speed) should be taken into account to improve the applicability of model in the future studies.

IV. CONCLUSION
1) A novel model to predict the slippage coefficient of tracked vehicles steering under different ground conditions without detecting the actual moving speed of vehicles was built. As demonstrated by the analyses, the proposed model was accurate and could effectively reduce the measuring errors as compared with the conventional testing method.
2) The slippage coefficient of tracked vehicles steering on the soft ground was higher than that on the hard ground (i.e., gravel ground and cement ground). The soil water content had a significant effect on the track slippage coefficient. Vehicles steering on the soil with high water content had larger slippage coefficients the on the soil with low water content.
3) The steering speed of tracked vehicles had an insignificant effect on the track slippage coefficient under different ground conditions.

ACKNOWLEDGMENT
This work was supported by the Jinhua Science and Technology Research Project (Grant no. 2021-2-020). Key Laboratory of Crop Harvesting Equipment and Technology of Zhejiang Province (Grant no. 2021E10018). Author thanks Dongmin Bai for helpful discussions, and thanks Dongmei Liu, Jie Wang and Lidong Ren for their work and assistance in the field. The authors would like to thank all the reviewers who participated in the review and MJEditor for its linguistic assistance during the preparation of this manuscript.