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Article

Slip Estimation Model for Planetary Rover Using Gaussian Process Regression

1
School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China
2
Beijing Institute of Spacecraft System Engineering, China Academy of Space Technology, Beijing 100094, China
*
Author to whom correspondence should be addressed.
Submission received: 3 April 2022 / Revised: 1 May 2022 / Accepted: 6 May 2022 / Published: 9 May 2022

Abstract

:
Monitoring the rover slip is important; however, a certain level of estimation uncertainty is inevitable. In this paper, we establish slip estimation models for China’s Mars rover, Zhurong, using Gaussian process regression (GPR). The model was able to predict not only the average value of the longitudinal (slip_x) and lateral slip (slip_y), but also the maximum possible value that slip_x and slip_y could reach. The training data were collected on two simulated soils, TYII-2 and JLU Mars-2, and the GA-BP algorithm was applied as a comparison. The analysis results demonstrated that the soil type and dataset source had a direct impact on the applicability of the slip model on Mars conditions. The properties of the Martian soil near the Zhurong landing site were closer to the JLU Mars-2 simulated soil. The proposed GPR model had high estimation accuracy and estimation potential in slip value, and a 95% confidence interval that the rover could reach during motion. This work was part of a research effort aimed at ensuring the safety of Zhurong. The slip value may be used in subsequent path tracking research, and the slip confidence interval will be able to help guide path planning.

1. Introduction

In the field of planetary exploration, slip is one of the factors that can affect the safety of planetary rovers [1]. Slip is defined as the difference between the expected motion and the actual achieved motion [2]. Low slip can take the rover out of its planned path, which could cause the rover to collide with obstacles. High slip reflects a severe lack of traction, always accompanied by severe rover sinkage and even permanent immobility [3,4,5]. Almost all rovers that have conducted detection missions on Mars or the Moon have encountered slip-related hazards of varying degrees [6,7,8]. In one case, the rover Spirit was permanently immobilized and was repurposed as a stationary science platform [9]. Therefore, it is essential to monitor slip to ensure a rover’s safety.
According to information sources on sensors that predict slip, rover slip estimation research can be divided into two approaches: exteroceptive-based and proprioceptive-based [3]. Figure 1 displays common approaches used for rover slip estimation.
(1)
Exteroceptive-based approach
Exteroceptive sensors require input from the environment and the rover’s surroundings. The most widely used exteroceptive sensors are cameras. Cameras obtain image information for slip estimation, which can be subdivided into slip estimation based on natural environment images and slip estimation based on wheel–ground images.
Slip estimation based on natural environment images is usually realized by Visual Odometer (VO). The rover continuously obtains image data of the surrounding environment using cameras and calculates its own pose change according to the overlapping information of adjacent images. It uses these to obtain a motion estimation and then combines that with wheel speed information to calculate the slip rate [10,11]. VO is widely used in planetary exploration, and has been used with the Mars rovers Spirit, Opportunity and Curiosity, as well as the lunar rovers Tutu and Yutu-2. This method has high slip estimation accuracy, but it incurs high computational costs, and is limited for use in sparse-featured areas and shadowed areas [12,13].
The slip estimation based on wheel–ground images judges the slip using image information, such as ruts and wheels. Ding et al. [14] proposed to use monocular vision to judge the wheel slip rate under ideal conditions without side deflection, based on the ruts’ spacing. Li et al. [15,16] established the diffuse analysis model of trace imprint by analyzing the formation mechanism of the wheel imprint, and realized slip estimation based on the time domain features of the wheel imprint image. Li et al. [17] also proposed a high-brightness weak boundary rut extraction method to estimate wheel linear velocity and combined it with wheel angular velocity (estimated by tracking the wheel marking points) to estimate the wheel slip. Compared with the natural environment-based slip estimation method, the slip estimation method based on wheel–ground images has better robustness; it overcomes the acquisition of surface feature points and the dependence on lighting conditions. Additionally, the computational cost is significantly reduced. However, related research focused on wheel slip rate estimations has been performed under ideal conditions, e.g., no side deflection. These conditions do not conform to the complex situation of orbital operation of a planetary rover.
(2)
Proprioceptive-based approach
Proprioceptive sensors only require signals from the rover, without depending on surrounding environment information.
The lunar rovers, Lunokhod-1 and Lunokhod-2, added a non-driving wheel that could roll up and down freely at the rear of the rover. The actual forward distance of the lunar rover was measured by the number of revolutions of this wheel. This was combined with the number of revolutions to estimate the rover slip [18]. The Mars rover Curiosity determined whether it was in danger of slipping by comparing its current average to a safety threshold [19]. These methods are not suitable for uneven surfaces.
With the rise of data analytics and machine learning, proprioceptive-based data-driven approaches have been applied to rover slip estimation. In these methods, the selected signal data and the corresponding slip observations are used as input and output, respectively, and the model is trained through machine learning algorithms to obtain slip estimation. Gonzalez et al. [5,19] used linear acceleration, vertical acceleration, and pitch rate, measured by Inertial Measurement Unite (IMU), as input features, and realized the slip level classification through various machine learning algorithms, such as Artificial Neural Network (ANN), Support Vector Machines (SVM), Self-Organizing Mapping (SOM), and so on. The Lunar All-Terrain Utility Vehicle (LATUV) rover was used to validate the models, and a concrete-making factory and an abandoned mine were chosen to be test sites. Dimastrogiovanni et al. [20] took driving torque, longitudinal force, and friction coefficient as inputs, constructed a terrain classifier through SVM, and associated terrain types with slip level labels. Sand types corresponded to high slip and rock types corresponded to low slip. The research was tested using the rover SherpaTT, provided by German Research Center for Artificial Intelligence, and the test site was an outdoor area in soft sand dunes in Morocco. The proprioceptive-based data-driven approach is becoming a common trend in slip estimation research, as it can adapt to complex environment and overcome the dependence on images of the surrounding environment. However, few studies have taken into account the differences in gravitational acceleration and soil conditions between Earth and Mars or the Moon. The datasets used to train and validate these models were almost all obtained on completely natural Earth terrain, which cannot guarantee applicability of the trained models on Mars or the Moon.
China’s first Mars rover Zhurong successfully landed on the southern Utopia Planitia of Mars at 109.925° E, 25.066° N on 15 May 2021 [21]. Figure 2 shows the routing path of Zhurong during the three-month-long designed life and the typical pictures returned by Zhurong. After completing its primary exploration tasks, Zhurong continued to carry out an extended expedition toward an ancient coastal area of Utopia Planitia. Inevitably, it would have to pass through complex terrains with slip-prone areas in the exploration process.
The primary aim of our research was to establish a slip estimation model for the Mars rover Zhurong. In our previous work [22,23], we used the Back Propagation neural network (BP) and optimized BP based on a genetic algorithm (GA-BP) to train models, and GA-BP models were proven to have good slip estimation performance by model evaluation and validation. However, GA-BP models did not take uncertainty into consideration. As slip estimation models are expected to provide the basis for slip-based path planning and path tracking, obtaining uncertainty in slip estimation has significance for subsequent research. This paper was a continuation of previous work; we established models able to estimate slip values and confidence intervals, and further discussed models based on different soil types. All data were collected in a test site equipped with a low-gravity simulation device and simulated soil, such that we were able to simulate the conditions experienced by Zhurong on Mars as closely as possible.
The rest of this paper has been organized as follows: in Section 2, we illustrated the research framework of this paper. In Section 3, we introduced the data collection process of ground tests. In Section 4, we introduced the training methods. In Section 5, we showed the training results, obtained by ground tests, and the estimation results, compared to real data on Mars conditions. In Section 6, we provided our conclusions.
The main contributions of this paper are as follows:
(1)
Obtained a GPR slip estimation model, which can not only have high estimation accuracy but also provides uncertainty of estimated values. The confidence interval obtained by models can be used in subsequent robust path planning research.
(2)
Compared models based on two kinds of simulated soil data, the results demonstrate that the soil type and dataset source had a direct impact on the applicability of the slip model on Mars conditions. In contrast, the JLU Mars-2 simulated soil are more similar to the soil near the Zhurong landing site.

2. Research Framework

The research framework of this paper is shown in Figure 3. In general, it was consistent with our previous paper [22]; however, the variables considered in the study and the machine learning methods used were different. The specific research steps were as follows:
(1)
Collect test data from tests conducted at a site equipped with a low-gravity simulation device and simulated soil. In this paper, we collected data on two kinds of simulated soil, while our previous paper [22] only used the data collected on JLU Mars-2. As such, we were able, in this work, to compare and discuss the influence of soil type on the applicability of slip estimation models.
(2)
Train slip estimation models with machine learning algorithms. In order to establish slip estimation models, considering the slip uncertainty, we used Gaussian process regression (GPR) to train models. GPR can predict not only slip values, but also their confidence intervals. GA-BP has also been used to train models to compare the performance of the two algorithms.
(3)
Evaluate the trained GPR models and GA-BP models on ground test data. In this paper, mean absolute error (MAE) and root mean square error (RMSE) were chosen as evaluation indices to evaluate the performance of the models.
(4)
Validate the trained models on telemetry data from Zhurong. We combined the path planning information and slip estimation information to correct the planned path, and then compared the end of the slip path with the actual final point.

3. Data Collection

The ground tests were performed in an indoor test site at the Chinese Academy of Space Technology. The dataset D = { ( x i , s i ) | i = 1 n } for input feature vector x with the corresponding output, slip observation s , was collected to establish slip estimation models. In this section, we first introduce the ground testing site, and then detail the input features and output observations used in our data collection.
Some descriptions have been simplified in this paper; more details are available in our previous work [22].

3.1. Ground Tests’ Site

Figure 4a displays the indoor test site used for ground tests. The rover used in the tests was the validator of Zhurong; as such, its design and functions are essentially the same as the Zhurong rover on Mars. The test sites were equipped with a low-gravity simulation device to provide a simulated Martian gravity environment for the rover. The vertical pressure of each wheel always stayed consistent with the values that the rover would experience on Mars. An indoor global positioning system (iGPS) was equipped for dynamic tracking and measurement of the position and attitude of the rover; 4 laser transmitters were arranged in the field, and a receiver was installed on the rover, and the measurement accuracy of position and attitude are better than 1 mm and 0.1°, respectively.
The tests site was covered with three kinds of simulated soil: TYII-2 simulated lunar soil, JLU Mars-2 simulated Martian soil, and JLU Mars-3 simulated soil. As shown in Figure 4b, the TYII-2 and JLU Mars-2 were laid in a horizontal area, the simulated soil was naturally laid to form gentle terrain undulations, and the slope variation range was small (0~5°). Research has demonstrated that the slope of most areas within 50 km of the Zhurong landing area are below 5° [24]; the data collected in the horizontal area were used in this paper. In addition, the JLU Mars-3 data will be used to analyze the influence of slope on slip data in our subsequent work. Table 1 presents the mechanical parameters of TYII-2-simulated lunar soil and JLU Mars-2-simulated Martian soil [25,26].

3.2. Input Features

Input features were expected to reflect the wheel–ground interaction state and were available from sensors on Zhurong. Four features were selected to form the input feature vector:
(1)
The rover’s pitch, θ .
The rover body frame is shown in Figure 5; as shown, the XB-axis and YB-axis were parallel to the ground, with the XB-axis pointing toward the driving direction and the YB-axis along the wheel axis. The ZB-axis was taken to be perpendicular to the XB-axis and YB-axis and pointed to the ground. θ represents the angle between the XB-axis of the rover body frame and the horizontal plane, which could be used to characterize the terrain slope. There was a highly nonlinear relationship between slip and slope [27], so it was selected as one of the feature inputs.
(2)
The rover’s roll φ .
φ represents the angle between the YB-axis of the rover body frame and the horizontal plane. It was selected for different rolling states corresponding to different wheel force distributions, which affect slip behavior.
(3)
The average current of six wheel-driving motors I ¯ .
According to the wheel–soil theory, the slip value directly affects the wheel drive torque T [28], so T can reflect the slip situation. Torque is known to be roughly proportional to motor current [29]; therefore, there was also a mapping relationship between the wheel drive motor current and the wheel slip. Since this paper focused on the rover slip, the slip conditions of the six wheels needed to be comprehensively considered. Therefore, the average current of six wheel-driving motors I ¯ was chosen as an input feature.
(4)
The average angular velocity w of six wheel-driving motors w ¯ .
Considering the nonideal behavior of the motor in the actual operation process, the speed of the wheel was not exactly the same as the command speed. Fluctuating within a certain range, wheel speed actually had some correspondence to wheel slip [30]. Therefore, the average rotational speed of the six wheel-drive motors was chosen as one of the input features.

3.3. Output Slip Observations

The slip of the rover is measured based on the rover body frame, as shown in Figure 5. While the rover was driving on natural terrain, the slip could be broken down into longitudinal slip (slip_x), lateral slip (slip_y), and rotational slip (slip yaw) [31]. This paper assumed that the slip yaw was trivial and could be canceled out by appropriate orientation control [32]; we mainly focused on slip_x and slip_y of the rover, defined as follows:
slip _ x = ( v c v x ) / v c
slip _ y = v y / v c
where v c is the expected velocity of rover, v x is the actual longitudinal velocity, and v y is the lateral slip velocity of the rover (Figure 6).
Slip cannot be directly observed. It needs to be calculated using position and attitude information from the rover in combination with wheel speed information. As shown in Figure 6, time n and time n + 1 were the adjacent sampling moments. The slip observations were calculated in the manner that follows.
(1)
First, we obtained the rover’s position, ( x C _ n , y C _ n , z C _ n ) and ( x C _ n + 1 , y C _ n + 1 , z C _ n + 1 ) at time of n and n + 1, respectively. Then, we calculated the moving distance of the rover in the site coordinate system (Figure 6).
( Δ x C , Δ y C , Δ z C ) = ( x C _ n + 1 x C _ n , y C _ n + 1 y C _ n , z C _ n + 1 z C _ n )
It should be noted that ( Δ x C , Δ y C , Δ z C ) was measured in the site coordinate system, but the slip was measured based on the rover body frame, so ( Δ x C , Δ y C , Δ z C ) must be converted to the rover body coordinate system ( Δ x B , Δ y B , Δ z B ) .
(2)
Next, we calculated the actual longitudinal velocity v x = Δ x B / Δ t and lateral slip velocity v y = Δ y B / Δ t of the rover. It was known that the sampling frequency of the rover’s position and attitude was about 20 Hz, so the time interval Δ t = 0.05   s .
(3)
We calculated the expected velocity of the rover. Due to the small time interval, we considered that the rover moved in a straight or near-straight path during the interval time Δ t , and the expected velocity of the rover was defined as follows:
v c = ( i = 1 6 ( w i r ) ) / ( 6 Δ t )
(4)
We calculated the slip_x and slip_y according to Equations (1) and (2), respectively.

3.4. Dataset

Table 2 displays the raw data collected in ground tests. Due to the sampling frequency of the rover’s position/attitude (20 Hz) having inconsistency with the sampling frequency of the motor current/velocity (1 Hz), the current and angular velocity data needed to be interpolated by simple linear interpolation. The step time of the research was set to 0.05 s [22]. On TYII-2 and JLU Mars-2, approximately 7500 and 4600 data points were obtained, respectively. The hold-out method was used to randomly split the ground test data into the training set (66.7%) and testing set (33.3%). In this paper, each ground type was used to obtain a single trained model. Therefore, we would like to compare which model is more applicable to the Mars conditions, so that we can see which type of simulated soil is more similar to the soil in the Zhurong landing areas.

4. Training Methods

Under the same input conditions, slip also has distribution uncertainty. For the safety of the rover, it was critical to know not only the average value of the slip, but also the maximum possible value that the slip could reach [33]. GPR is a nonparametric model for regression analysis of data using the Gaussian process (GP) as a prior for Bayesian inference. It exhibits strong adaptability and generalization ability. Furthermore, GPR synthesizes both data and model uncertainty to estimate uncertainty [34,35]. Therefore, GPR was used to train models in this paper. GA-BP, proposed in our previous work, was also used to train models and compared with the performance of GPR, as fully detailed in the literature [22].
GP is defined as a collection of random variables, any finite number of which have consistent joint Gaussian distributions, and is fully specified by their mean function m(x) and covariance function k(x, x′), where x and x′ are random variables [36]. Treat the mean function m(x) as zero for the convenience of calculation, and there is no loss of generalization.
f ( x ) ~ G P ( 0 , k ( x , x ) )
Generally, the data were episodically affected by noise from various sources; the slippage observations can be represented as
s = f ( x ) + ε
where f ( ) is assumed as a zero-mean Gaussian process, is noise-dependent and follows ( 0 , σ 2 ) distribution. f ( ) is a nonlinear mapping relationship between inputs and outputs, and ε is noise. Assuming that they both fit zero-mean GP, then the prior distribution of sample observations, s, also fit GP.
s ~ G P ( 0 , K + σ n 2 I n )
The joint prior distribution of the sample observations, s, and the predicted value s * is:
[ s s * ] ~ N ( 0 ,   [ K + s 2 I K * T K * K * * ] )
where
K = k x 1 , x 1 k x 1 , x 2 k x 1 , x n k x 2 , x 1 k x 2 , x 2 k x 2 , x n k x n , x 1 k x n , x 2 k x n , x n
K * = [ k ( x * , x 1 ) k ( x * , x 2 ) k ( x * , x n ) ]
K * * = k ( x * , x * )
I n = [ 1 0 0 0 1 0 0 0 0 1 ]
where K is the covariance matrix for the training data, matrix elements k i j = k ( x i , x j ) are used to measure the correlation between x i and x j , K * * is the covariance matrix between the test points (prediction) and the training data, K * * is the covariance between the points in the test set, and I n is the identity matrix.
According to Bayesian formula
P ( s * | s ) = P ( s * , s ) P ( s )
We can obtain the posterior predictive distribution of s * at the location x * which is the obtained predicted result by GPR.
P ( s * | s ) ~ N ( K * ( K + σ 2 I ) 1 s , K * * K * ( K + σ 2 ) 1 K * T
Then
m ( s * ) = K * ( K + σ 2 I ) 1 s
c o v ( s * ) = K * * K * ( K + σ 2 ) 1 K * T
where m ( s * ) and c o v ( s * ) denote the prediction result and uncertainty, respectively.
GPR has several covariance functions, as shown in Table 1. The most commonly used is the Squared Exponential (SE),
k ( x , x ) = σ f 2 exp ( 1 2 l 2 | x x | 2 )
where σ f and l are hyper-parameters in need of identification, generally determined by maximum likelihood estimation.
L ( θ ) = l g   p ( s | x , θ ) = 1 2 s T ( K + σ n 2 I n ) 1 s 1 2 l g   | K + σ n 2 I n | n 2 l g   2 π
L ( θ ) θ i = 1 2 t r ( ( α α T ( K + σ n 2 I n ) 1 ) ( K + σ n 2 I n ) θ i )
where θ = { σ f , l } denotes the hyper-parameters and L ( θ ) is the negative log-likelihood function of the conditional probability of the training sample. After calculating the partial derivative with respect to θ , one can use a conjugate gradients algorithm to optimize the parameters. The flowchart of the model establishment process using GPR is shown in Figure 7.

5. Ground Tests Results and Discussion

To ensure better stability and reliability of the model, we repeated ten iterations of hold-out validation on the models of each type. A total of 40 slip models were acquired based on the data collected on two kinds of simulated soil and two training methods. We used mean absolute error (MAE) and root mean square error (RMSE) as evaluation indices in this paper.
MAE _ x = 1 n i = 1 n | s ^ x i s x i |  
MAE _ y = 1 n i = 1 n | s ^ y i s y i |  
RMSE _ x = i = 1 n ( s ^ x i s x i ) 2 n  
RMSE _ y = i = 1 n ( s ^ y i s y i ) 2 n
where MAE_x and RMSE_x are evaluation indices for slip_x, MAE_y and RMSE_y are evaluation indices for slip_y; s ^ x i and s ^ y i are predicted values for slip_x and slip_y, respectively; s x i and s y i are real observations for slip_x and slip_y, respectively; n is the amount of testing sets.
Table 3 and Table 4 detail the models’ performance results for slip_x and slip_y, respectively. Figure 8 and Figure 9 present the results visually using line charts and box plots. For longitudinal slip, Figure 8a,c shows that the estimation results of GPR models fluctuated greatly, and the box plots displayed in Figure 8b,d also show that GA-BP estimation was more concentrated. These all demonstrated that the estimation performance of the GPR models was unstable compared to the GA-BP models. Take MAE_x results on TYII-2 dataset as an example: the MAE_x values of the GA-BP models were between 2.57~2.70%, with an average of 2.65%, and a maximum difference of 0.13%. The MAE_x of the GPR models were between 0.29~4.35%, with an average of 1.94% and a maximum difference of 4.06%. Although GPR models are unstable, the optimal performance of GPR models is beyond the reach of GA-BP models. MAE_x = 0.29% signified that the estimated slip_x values closely approximated the true values, which demonstrated the estimation potential of the GPR models. For lateral slip, the performance results of slip_y were roughly similar to those of slip_x. In conclusion, both GPR models and GA-BP models could complete the estimation with good accuracy. The estimation performance of the GPR models was unstable compared to the GA-BP models, but GPR models also demonstrated estimation potential, which GA-BP models cannot reach.
To further explore the estimation effects of GPR models, Figure 10 displays part of the GPR estimation results in the TYII-2 group (1) and JLU Mars-2 group (1). Since the test set data were randomly selected, there was no continuous trend between adjacent training data. In order to exhibit the results more clearly, Figure 10 only shows the estimation of 100 data points to avoid excessively dense data points. The gray band area in the figure represents the 95% confidence interval corresponding to the estimation result. As shown, no matter what kind of soil dataset utilized, and whether it was slip_x or slip_y, the estimation results were consistent with the trends of actual values, and almost all real points were located within the 95% confidence interval, which means GPR models can provide a reasonable estimation of slip value uncertainty.

6. Mars Condition Validation Results and Discussion

Zhurong operated in a highly efficient detection mode within its design life; the most commonly used movement scheme is shown Figure 11. When Zhurong arrived at Station N, its surrounding environment was imaged with navigation cameras located on the mast. Ground flight controllers performed visual positioning, target point selection (intermediate transition Station X and target Station N + 1), and path planning (the path from Station N to Station X) according to images obtained at Station N. From station N to station X, the rover drove in blind movement mode; that is, Zhurong drove completely according to the path planned by the ground controllers without autonomous obstacle avoidance. Meanwhile, visual positioning was performed at Station X to obtain the relative position of Station X to Station N. From station X to station N + 1, Zhurong drove in blind autonomous obstacle avoidance movement mode; that is, the rover autonomously planned a path to drive toward the target, Station N + 1. It should be noted that the rover controller is not equipped with the slip estimation model at present, thus the actual rover arrival point is reached without slip compensation.
From the above-described process, the ground cannot obtain real-time rover coordinates when the rover is driving, thus it is impossible to directly verify the accuracy of the slip estimation value during the driving process. However, we knew the planning path and the actual end point/arrival after blind movement. In this paper, the predicted slip_x and slip_y were converted into the longitudinal slip offset ∆x and the lateral offset ∆y, respectively, and the planned path was corrected according to the longitudinal and lateral offset values. Thus, we were able to obtain the slip estimation corrected path. The end point of the slip-corrected path was compared with the actual points the rover reached (accurate coordinates obtained by visual positioning) to evaluate the estimation effects of the slip estimation models against the telemetry data. The required telemetry information is shown in Table 5, and more details of the specific implementation steps are available in [22]. It should be noted that the frequency of telemetry data for the rover’s attitude and wheel driving motor is about 0.2~0.3 Hz, which is lower than the sampling frequency of the ground tests. It means that it needs to use a estimated slip value to represent the mean slip within 3 to 5 s, which may causes errors in slip offset calculations where the terrain was extremely rugged. Since the terrain of the landing area of Zhurong is relatively flat, the data of Zhurong does not change drastically when driving; we can consider this frequency to be acceptable.
In this paper, validation was conducted based on data obtained from mid-June 2021 (sol.27–sol.40). Table 6 shows the planning path information and actual end points of arrival. As shown, there was always an offset ∆L between the actual arrival point and the planned target point. The ∆L varied from 0.267 m to 0.678 m, and the largest offset occurred at sol.34. Due to the limited sample, there was no uniform linear relationship between the ∆L of these five movements, the planned distance, and the planned curvature. According to the planned path information shown in Table 6 and the models’ estimation results, Figure 12 displays the slip-corrected path and planning path (left figure of each subfigure). In order to quantify and compare the estimation accuracy of the GA-BP and GPR models, the offsets, ∆L_GA-BP and ∆L_GPR, between the actual arrival point and the planned target point, are summarized in Table 7. Figure 12 visualizes the offset results of each movement (right figure of each subfigure); the points in all subfigures represent the linear deviation distance between the actual arrival point of the current move and the planned path point, while the horizontal dotted lines in each subfigure represent the offset ∆L between the actual arrival point and the planned target point.
Applsci 12 04789 i001 and Applsci 12 04789 i002 represent the estimation points of the GA-BP models based on TYII-2 and JLU Mars-2 datasets, respectively. In all subfigures, Applsci 12 04789 i001 path deviated the furthest from the planned path, and the end point of Applsci 12 04789 i001 path was always farther than the end point of Applsci 12 04789 i002. Notably, on sol.40, ∆L = 0.562 m, the ∆L_GA-BP (TYII-2) was 1.129 m, while ∆L_GA-BP of JLU Mars-2 was just 0.242 m, which demonstrated that the GA-BP models established based on TYII-2 data performed poorly in comparison to the others. The GPR models based on two soil datasets also showed the same regularity. The slip-corrected Applsci 12 04789 i003 path based on the JLU Mars-2 dataset exhibited better performance than the Applsci 12 04789 i004 path based on the TYII-2 dataset. The end point of the Applsci 12 04789 i003 path was closer to the actual arrival point, and the actual arrival point was always within the Applsci 12 04789 i005 95% confidence interval. ∆L_GPR1 and ∆L_GPR2 represent the end-point deviation from the actual arrival point of the 95% confidence interval boundary predicted by the GPR model. The maximum possible point the rover could reach was always over 0.5 m away from the real arrival point. Comparing the GA-BP model and the GPR model, based on the same soil dataset, the GPR model always achieved a better estimation effect. ∆L_GA-BP of TYII-2 were always larger than or roughly equal to ∆L, while most ∆L_GPR of JLU Mars-2 were smaller than ∆L. The best es-timation points all appeared on the GPR model based on JLU Mars-2; on sol.27, ∆L_GPR even reached 0.044 m, meaning that the predicted point was very close to the real arrival point.
In conclusion, the model established based on the data of the JLU Mars-2 dataset had the best adaptability to the Martian environment near Zhurong’s landing site, which means the properties of the Martian soil in this region may be more similar to the simulated soil, JLU Mars-2. The GPR model had high estimation accuracy and estimation potential, and at the same time offered the maximum deviation that the rover could achieve, which is very useful for robust path planning.

7. Conclusions

This paper developed slip estimation models for the Mars rover Zhurong using GPR and GA-BP. The training data came from ground tests on two simulated soils, TYII and JLU Mars-2. The data also had higher accuracy and sampling frequency. The models were validated using Zhurong data on Mars conditions, and the results demonstrated that the soil type and dataset source had a direct impact on the applicability of the slip model on Mars conditions. The properties of the Martian soil near the Zhurong landing site were most similar to the JLU Mars-2 simulated soil. This demonstrated the necessity of using simulated Martian soil for ground tests, which has not been not considered in the existing literature. The GPR models had high estimation accuracy, and were able to show the slip behavior of the rover. These models offered a 95% confidence interval. Thus, this work offered a basic research direction for subsequent slip-based path planning and path tracking. There are some limitations of our work; only two kinds of simulated soils were used in the test, thus it is hard to deeply explore the correlation between soil properties and slip behavior. Moreover, the training of the Gaussian process regression model takes a long time, and the algorithm needs to be improved.

Author Contributions

Conceptualization, Y.J. and T.Z.; software, S.P. and J.S.; data curation, H.T. and J.S.; writing—original draft preparation, T.Z.; supervision, C.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors are grateful to the Chinese Academy of Space Technology for test and data supporting.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. Most common slip estimation approaches for rovers.
Figure 1. Most common slip estimation approaches for rovers.
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Figure 2. The routing path of Zhurong during its designed life.
Figure 2. The routing path of Zhurong during its designed life.
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Figure 3. Research framework.
Figure 3. Research framework.
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Figure 4. (a) The indoor test site of ground tests; (b) the distribution of simulated soil.
Figure 4. (a) The indoor test site of ground tests; (b) the distribution of simulated soil.
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Figure 5. The body frame of Zhurong.
Figure 5. The body frame of Zhurong.
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Figure 6. The velocity decomposition.
Figure 6. The velocity decomposition.
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Figure 7. Flowchart of the model establishment process using GPR.
Figure 7. Flowchart of the model establishment process using GPR.
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Figure 8. The estimation results of five test groups for slip_x using GA-BP and GPR: (a) the line charts of MAE_x; (b) the box plots of MAE_x; (c) the line charts of RMSE_x; (d) the line charts of RMSE_x.
Figure 8. The estimation results of five test groups for slip_x using GA-BP and GPR: (a) the line charts of MAE_x; (b) the box plots of MAE_x; (c) the line charts of RMSE_x; (d) the line charts of RMSE_x.
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Figure 9. The estimation results of five test groups for slip_y using GA-BP and GPR: (a) the line charts of MAE_y; (b) the box plots of MAE_y; (c) the line charts of RMSE_y; (d) the line charts of RMSE_y.
Figure 9. The estimation results of five test groups for slip_y using GA-BP and GPR: (a) the line charts of MAE_y; (b) the box plots of MAE_y; (c) the line charts of RMSE_y; (d) the line charts of RMSE_y.
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Figure 10. GPR estimation results (in part): (a) slip_x on TYII−2; (b) slip_x on JLU Mars−2; (c) slip_y on TYII−2; (d) slip_y on JLU Mars−2.
Figure 10. GPR estimation results (in part): (a) slip_x on TYII−2; (b) slip_x on JLU Mars−2; (c) slip_y on TYII−2; (d) slip_y on JLU Mars−2.
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Figure 11. The Schematic diagram of Zhurong station.
Figure 11. The Schematic diagram of Zhurong station.
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Figure 12. Planning path, predicted path, and deviation from sol.27–sol.36: (a) sol.27; (b) sol.28; (c) sol.34; (d) sol.36; (e) sol.40.
Figure 12. Planning path, predicted path, and deviation from sol.27–sol.36: (a) sol.27; (b) sol.28; (c) sol.34; (d) sol.36; (e) sol.40.
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Table 1. The mechanical parameters of TYII-2 and JLU Mars-2.
Table 1. The mechanical parameters of TYII-2 and JLU Mars-2.
c (kPa) φ (°)K (m) k c   ( kPa m ( n + 1 ) ) k φ   ( kPa m ( n + 2 ) ) n
TYII-20.03530.490.0163.711.101.12
JLU Mars-20.60937.200.01915.69983.361.02
c: cohesion; φ : internal friction angle; k: shear deformation modulus; k c : the cohesive modulus of the soil; k φ : the frictional modulus; n: slip-sinkage.
Table 2. The data collected in ground tests.
Table 2. The data collected in ground tests.
Raw DataSampling
Frequency
Rover’s pitch20 Hz
Rover’s roll20 Hz
Rover’s yaw20 Hz
Current of each wheel driving motor1 Hz
Angular velocity of each wheel driving motor1 Hz
Position of rover20 Hz
Table 3. Estimation results of GA-BP and GPR on TYII-2 and JLU Mars-2 (slip_x).
Table 3. Estimation results of GA-BP and GPR on TYII-2 and JLU Mars-2 (slip_x).
TYII-2JLU Mars-2
Group NumberGA-BP GPRNoGA-BP GPR
MAE
(%)
RMSE
(%)
MAE
(%)
RMSE
(%)
MAE
(%)
RMSE
(%)
MAE
(%)
RMSE
(%)
(1)2.69 3.39 0.841.09(1)3.16 4.02 0.540.63
(2)2.69 3.41 0.921.18(2)3.08 4.05 4.194.72
(3)2.70 3.36 0.931.13(3)3.14 4.09 1.821.91
(4)2.65 3.38 1.742.31(4)3.06 4.06 3.453.46
(5)2.66 3.31 3.293.65(5)3.12 4.06 1.251.27
(6)2.57 3.29 2.693.24(6)2.99 3.96 4.945.61
(7)2.57 3.29 4.355.43(7)3.19 4.10 1.802.18
(8)2.67 3.32 1.401.50(8)3.15 4.14 0.580.68
(9)2.69 3.35 0.290.36(9)3.18 4.09 0.580.79
(10)2.62 3.29 2.943.98(10)3.02 4.00 0.200.21
mean2.653.341.942.39mean3.11 4.06 1.942.15
Table 4. Estimation results of GPR on TYII-2 and JLU Mars-2 (slip_y).
Table 4. Estimation results of GPR on TYII-2 and JLU Mars-2 (slip_y).
TYII-2JLU Mars-2
Group NumberGA-BPGPRNoGA-BPGPR
MAE_y
(%)
RMSE_y
(%)
MAE_y
(%)
RMSE_y
(%)
MAE_y
(%)
RMSE_y
(%)
MAE_y
(%)
RMSE_y
(%)
(1)1.15 1.50 0.570.7311.150 1.577 0.860.87
(2)1.16 1.51 1.912.1621.184 1.523 3.383.46
(3)1.15 1.47 1.972.5031.127 1.487 1.501.95
(4)1.14 1.48 4.255.0041.190 1.537 0.840.85
(5)1.16 1.53 1.802.4151.117 1.450 5.406.97
(6)1.16 1.51 1.361.3661.131 1.474 3.393.41
(7)1.16 1.51 3.214.2271.058 1.414 3.183.60
(8)1.15 1.48 3.844.2581.194 1.529 7.489.89
(9)1.15 1.50 0.891.1991.115 1.470 3.364.71
(10)1.21 1.54 1.661.77101.131 1.522 0.150.19
mean1.161.502.1462.56mean1.140 1.498 2.953.59
Table 5. Planning end points and actual end points of arrival.
Table 5. Planning end points and actual end points of arrival.
NoRequired InformationUsageInformation Type
1Planning path curvatureDraw planning pathGround planning information
2Planning path distance
3Planning target point
4Rover’s yaw angle at start point and target point
5Rover’s pitchInput features of modelsTelemetry information
6Rover’s roll
7The current of each wheel driving motor
8The angular velocity of each wheel driving motor
9Visual positioning of end pointJudge the deviation between predicted point and real pointGround visual positioning information
Table 6. Planning end points and actual end points of arrival.
Table 6. Planning end points and actual end points of arrival.
SolPlanning Curvature (1/m)Planning Distance (m)Rover’s Yaw(o)Planning PointsActual PointsDeviation
l
(m)
Start PointTarget PointX_Plan
(m)
Y_Plan
(m)
X_Actual
(m)
Y_Actual (m)
270.0388.459−135.508−116.999−4.9805−6.7917−4.9408−7.17530.386
28−0.0719.196−129.597−167.031−7.6869−4.7449−7.8924−4.91530.267
34−0.0399.23−162.377176.774−9.107−1.1502−9.7244−1.43130.678
36−0.05949.644−106.654−139.298−5.1775−7.9800−5.5369−8.38480.541
400.00969.516171.782177.010−9.46700.9290−10.00640.77070.562
Table 7.L_GA-BP and ΔL_GPR.
Table 7.L_GA-BP and ΔL_GPR.
SolTYII-2JLU Mars-2
GA-BPGPRGA-BPGPR
L_GA-BP (m)L_GPR (m)L_GPR1 (m)L_GPR2 (m)L_GA-BP (m)L_GPR (m)L_GPR1 (m)L_GPR2 (m)
270.4340.4200.4260.8510.1200.0440.4720.551
280.3120.2410.7390.8960.2220.1440.6880.880
340.6330.1411.2691.5490.2400.0841.2171.380
360.5480.1650.9381.2670.3020.1430.5180.700
401.1290.1330.5310.7120.2420.0750.6530.731
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Zhang, T.; Peng, S.; Jia, Y.; Sun, J.; Tian, H.; Yan, C. Slip Estimation Model for Planetary Rover Using Gaussian Process Regression. Appl. Sci. 2022, 12, 4789. https://0-doi-org.brum.beds.ac.uk/10.3390/app12094789

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Zhang T, Peng S, Jia Y, Sun J, Tian H, Yan C. Slip Estimation Model for Planetary Rover Using Gaussian Process Regression. Applied Sciences. 2022; 12(9):4789. https://0-doi-org.brum.beds.ac.uk/10.3390/app12094789

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Zhang, Tianyi, Song Peng, Yang Jia, Junkai Sun, He Tian, and Chuliang Yan. 2022. "Slip Estimation Model for Planetary Rover Using Gaussian Process Regression" Applied Sciences 12, no. 9: 4789. https://0-doi-org.brum.beds.ac.uk/10.3390/app12094789

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