State Analysis and Emergency Control of Planetary Rover with Faulty Drive Wheel

Abstract

Wheel failure is one of the worst problems for a planetary rover working on Mars or the Moon, which may lead to the interruption of the exploration mission and even the loss of mobility. In this study, a driving test of a planetary rover prototype with a faulty drive wheel was conducted, and state analysis and dynamics modeling were carried out. The drag motion relationship between the faulty drive wheel and the normal wheels on the same suspension was established based on the targeted single wheel test (faulty wheel-soil bin). In order to maintain the subsequent basic detection capability of the planetary rover, an emergency control system is proposed that integrates the path planning strategy with faulty wheel priority and the motion control method of correcting heading and coordinating allocation. The experimental results and emergency strategies of this study on simulating Martian soil and terrain can provide researchers with ideas to solve such problems.

1. Introduction

Wheeled planetary rovers have been widely used in the exploration of Mars and the Moon, and they are responsible for many tasks, such as detection patrols and carrying instruments in current planetary exploration. However, the complex and unknown terrain environments of planetary surfaces pose great challenges to the hardware equipment of planetary rovers [1,2]. Especially, the wheels and driving equipment of planetary rovers are more prone to failure or damage as the components are in direct contact with the planetary surfaces [3,4]. Planetary rovers cannot be recovered and repaired in time when they fail as conventional field robots. Wheel failure will directly lead to the interruption of the exploration mission and even the loss of mobility [5]. Therefore, the treatment method of planetary rover wheel failure is an important research problem for the space agencies and researchers of various countries.

Planetary rover wheel failure problems can usually be categorized as wheel damage, wheel drive or steering actuator failure [6,7]. In the current study, researchers carried out a lot of research to optimize the wheel structure design in order to prevent wheel damage from affecting the driving of the planetary rover [8], and they combined terramechanics and structural mechanics to fully test and verify the stiffness and durability of the planetary rover wheel [9,10]. In order to reduce the impact of wheel steering actuator failure, the fault-safe motion planning method for a lunar rover based on deep reinforcement learning is proposed in [11] to solve the problem of serious mobility loss in the case of steering motor failure. Other emergency control methods for the steering function failure of ground wheeled vehicles can also provide references [12,13].

However, research related to the theoretical analysis and solution strategies for the wheel drive failure problems of planetary rovers is not widely carried out. In [14], the possibility and influence of various faults in the mobile system of the Curiosity Mars rover were evaluated. The possibility of hardware failure of the drive or steering actuator was assessed as medium, and the worst impact as medium. The failure probability of wheel damage that may affect driving was assessed as low, and the worst impact as high. However, the reason why wheel damage has a high degree of impact is because the researchers found that the damaged debris on the wheel surface can cut into the drive cable, causing the drive to become unusable. These all indicate that wheel drive failure is a serious accident during the exploration of the planetary rover. The Mars rover Spirit had a drive failure in the right front wheel before completing its exploration mission, making that wheel a bit like a dragged anchor, as shown in Figure 1a, and its driving ability was much reduced. This problem has attracted the attention of space agencies in various countries, and mobility tests of planetary rovers under wheel drive failures have become necessary. Figure 1b shows the moving test of the Zhurong Mars rover prototype under driving failure. The Zhu Rong Mars rover also uses an active suspension that lifts the faulty wheel to mitigate the impact [15]. In [16], the prototypes of partially failed propulsion wheels with different distributions were analyzed, and a control strategy of motion-drive topology transformation was proposed. In [17], the influence of different driving failure modes of mobile robots in moving over rough terrain was analyzed, and a driving configuration modification scheme that makes full use of the remaining driving forces was proposed.

Among the current related studies, there are few studies on the state of the whole vehicle when a planetary rover continues to function with a wheel-drive failure. Therefore, we hope to analyze the state of planetary rovers with a faulty drive wheel in advance and propose an emergency control method. This will enable timely and effective measures to be taken when similar accidents occur, or provide researchers with ideas to solve such problems.

The main work of this paper is as follows:

(1) A mobile performance test and state analysis of the rocker-bogie planetary rover prototype with a faulty drive wheel were carried out.

(2) A single wheel test (faulty wheel-soil bin) was carried out, and the drag motion relationship between the faulty drive wheel and the normal wheels on the same suspension was established.

(3) An emergency control system is proposed that integrates a path planning strategy with faulty wheel priority and the motion control method of correcting heading and coordinating allocation.

The rest of this paper is organized as follows. The results of the rover movement test and the faulty wheel test are described in Section 2. Section 3 carries out detailed state analysis and model establishment. The proposed design of an emergency control system is introduced in Section 4. Section 5 concludes the paper and outlines future work.

2. Experimental Study

In this section, we discuss the design of an experimental system built to restore the driving scene and performance test of the whole vehicle and a single wheel when a wheel drive fault occurs. The test phenomena and data are analyzed separately.

2.1. Experimental System

The vehicle movement test was carried out in the indoor soil-bin built by the terramechanics research group of Jilin University. During the test, the driving range of the rover was 3.5 m × 10.0 m, and the driving terrain was JLU Mars series simulated Martian soil (JLU Mars 1 and JLU Mars 3). The soil parameters are shown in

Table 1. Simulated soil parameters.

Simulated Soil	Deformation Index	Cohesion	Internal Friction Angle
JLU Mars 1	1.07	0.54 kPa	30.6°
JLU Mars 3	1.02	0.17 kPa	36.8°
JLU-2b	1.06	1.82 kPa	32.7°

, and the layer thickness was 0.6 m. The equipment layout of the vehicle mobile test system is shown in Figure 2a. Four ultra-wideband (UWB) fixed base stations were arranged in the four corners of the site; an UWB mobile base station and an inertial measurement unit were arranged in the center of the vehicle body to collect the motion information of the position and attitude of the planetary rover prototype. The image acquisition system was arranged at different angles of the site to record the test process.

The suspension structure and appearance of the planetary rover prototype used in this study are shown in Figure 2b. It adopts the rocker-bogie passive suspension, and has a steering motor at the front wheels and the rear wheels, which can realize the in situ steering of the planetary rover. During the test, any wheel could be locked by controlling, which can be used to restore the scene where the wheel drive of the planetary rover failed.

The soil bin test bench for single fault wheel performance test is shown in Figure 2c. The size of the soil bin was 1.5 m × 0.8 m, and the interior was a JLU-2b series simulated planetary soil with a thickness of 0.3 m. By collecting the sensor data of the test bench, the wheel torque, horizontal displacement, normal load, and drawbar pull of the tested wheel were obtained. During the test, the scene of wheel drive failure could be restored by locking the drive motor, and the wheel drag speed could be controlled using the drag motor.

Figure 2d shows the tested wheel of planetary rover in this study. The wheel was a rigid wheel (diameter 250 mm, wheel width 200 mm), and there was no wheel lug on the wheel surface.

2.2. Vehicle Test of Fault Planetary Rover

2.2.1. Mobile Test

In order to fully test the mobility of the planetary rover, the vehicle state of the prototype under each fault condition (front wheel, middle wheel, rear wheel) continued to be tested in the conventional mode (straight, in situ steering). The specific test was set as follows: in the straight mode, the forward speed of the prototype body $v_x$ was 30 mm/s, and the expected driving distance was 3 m (wheel drive running 100 s). In the in situ steering mode, the body rotation angular velocity of the planetary rover prototype $w_z$ was 0.0233 rad/s, and the desired steering angle was 180° (wheel drive operation 95 s).

Figure 3 shows the position and heading deviation of the faulty planetary rover during the straight driving test. It can be seen that the actual trajectory of the planetary rover deviated to the side with the driving fault wheel under each fault condition. The deviation degree of the right front wheel drive fault was the most serious. At the end of the test, the lateral position deviation (x direction) reached −42 cm, and the longitudinal position deviation (y direction) reached −50 cm. The deviation of the right rear wheel drive fault was the smallest, and the lateral and longitudinal position deviations were −11 cm and −13 cm, respectively. Figure 4 shows the straight test process of the faulty planetary rover. It can be seen that the body of the planetary rover yawed to the faulty side during the driving process. The heading angle deviations of the right front, right middle and right rear wheel drive faults were −31.0°, −17.7°, −13.7°.

The observation and analysis of the prototype test process effectively found the difference between the driving state of the fault driving wheel and the normal wheel on the soft ground. Figure 5 shows the state comparison of the planetary rover after the mobile test was carried out with the driving fault of the right rear wheel. It can be seen that in the straight-line driving test, the faulty wheel relied on the drag or push of other normal wheels to move, and left a sunken rut on the driving path. This was also the main reason for the large deviation between the planetary rover trajectory and the expected path.

The in situ steering mode is the most important course adjustment mode of the existing rocker-bogie planetary rover. In the experiment of this study, Figure 6 shows the in situ steering test process of the planetary rover prototype under different wheel drive faults. It can be seen that the driving fault state of the planetary rover will also affect the in situ steering motion process. For example, Figure 7 shows the in situ steering state of the planetary rover under different driving states of the right front wheel. Under normal conditions, the planetary rover prototype can be more stable around the center of the body for heading adjustment. The ruts of the middle wheel and the front and rear wheels are in the shape of concentric rings. When the wheel of the planetary rover has a driving fault, the rut of the planetary rover is more chaotic. The planetary rover cannot perform stable in situ steering, and sideslips or obstructions will occur.

Figure 8 shows the position and heading deviation of the faulty planetary rover during the in situ steering test. As shown in Figure 8a, the actual steering center of the rover deviated from the desired point and changed continuously in the presence of wheel drive failure. When right front and right rear wheel drive failure occurred, the maximum deviation of the steering center position reached 34.5 cm and 32.6 cm, respectively. When the right middle wheel failed, the deviation was small, and the maximum position deviation was 25.2 cm. Meanwhile, it can be seen in Figure 8b that the body of the planetary rover did not change its course according to the desired rotational angular velocity during the steering process, and the preset steering angle was not completed at the end of the wheel drive running time. The body rotation angle only reached 155.8° and 151.4° when the right front and right rear wheel drive failed. The influence of the middle wheel failure was small, and the body could turn to 163.9° at the end of the test. This phenomenon was also caused by driving the faulty wheel hindering the movement of the planetary rover.

2.2.2. Obstacle Crossing Test

In this study, the obstacle surmounting ability under each driving fault condition was also tested in order to evaluate the mobility of the faulty planetary rover more comprehensively. The height of the obstacle stone slab used was 3 cm (the appearance is shown in Figure 2b) and the driving speed was 30 mm/s. Figure 9 shows the obstacle-crossing test process of the planetary rover prototype. Figure 9a shows that the right front wheel of the planetary rover in normal state crossed the three-stone slab obstacle, and took 25 s to cross the obstacle successfully. Figure 9b shows a stone slab obstacle and the right front wheel of the planetary rover with right front wheel drive failure as it tried unsuccessfully to surmount it.

Table 2. Obstacle crossing ability of each wheel under fault condition.

Faulty Drive Wheel	Number of Obstacle Slabs
	RF	RM	RR	LF	LM	LR
Right front wheel	0	—	—	5	4	3
Right middle wheel	1	0	—	5	4	3
Right rear wheel	4	3	0	5	5	4

shows the maximum number of slabs that each wheel could cross in each scene of the test. The obstacle-crossing ability of each wheel of the planetary rover in normal state is about seven stone plates. It can be seen that the wheel with driving failure almost lost its obstacle-crossing ability, and the obstacle-crossing ability of the wheel on the same side suspension was greatly reduced. Although the wheel of the suspension on the other side of the rover was also affected, it still retained a certain obstacle crossing ability. For example, one obstacle slab for the right rear wheel that had a driving fault could not be crossed, but it did cross seven blocks under normal conditions. At this time, the right front and right middle wheels can cross four and three slabs respectively, which is less than the obstacle-surmounting ability of the right suspension (average 5 slabs).

Meanwhile, the vehicle current of the planetary rover prototype during the obstacle crossing test was also detected. The black curve in Figure 10 shows the current of the rover as the right front wheel crossed the three-stone obstacle under the normal state of the right rear wheel drive. The average current of the process was 0.98 A and the voltage was 23.9 V. The green curve in Figure 10 is the current of the planetary rover during the right front wheel obstacle crossing under the condition of right rear wheel drive failure. The average current of the process was 1.05 A and the voltage was 23.9 V. It can be seen that the wheel drive failure caused the energy consumption of the planetary rover to increase during subsequent movement and obstacle crossing. In addition, when the wheel was in contact with the obstacle in the test, the drag force caused the planetary rover with a faulty drive wheel to stop and then cross the obstacle. This is the reason why the current growth trend of the faulty planetary rover occurred slightly later than the normal state in Figure 10.

2.3. Single Wheel Test of Faulty Drive Wheel

The single wheel soil bin test is a direct and effective method to obtain the motion characteristics and performance of the special state of planetary rover wheels [18,19]. In this study, the driving state of a faulty wheel and a normal wheel were tested in the case of drag and drop. The specific test was set as follows: the wheel motion test distance was about 0.8 m, and the normal load was 8.5 kg. The speed of dragging or releasing the tested wheel was 0 mm/s, 10 mm/s, 20 mm/s, 30 mm/s, 40 mm/s, 50 mm/s, and the normal angular velocity of the tested wheel was 0.4 rad/s (the slip rate s_w was 1, 0.8, 0.6, 0.4, 0.2, 0). It should be noted that when the slip rate s_w was 1, the wheel of the planetary rover was in a pure slip state, and the test duration of the group was set to 15 s. The wheel test under each test condition was repeated three times.

Figure 11a and Figure 12a show the hook traction and driving torque at different slip rates during the normal wheel dragging motion in the test. As the wheel slip rate increased, the force generated for drag (drawbar pull) also gradually increased. When the driving was stable, the hook traction force of the normal wheel under the set slip rate was 10.4 N, 23.2 N, 36.4 N, 43.8 N, and reached 45.0 N in the case of pure slip. Figure 11b and Figure 12b are the drag and stop torque of the faulty wheel in the test. It can be seen that the drag force required to maintain a certain speed increased first and then stabilized during the drag test of the fault wheel in the soil bin. It shows that the drive fault wheel could be dragged at a relatively stable speed when providing a suitable drag force. At the same time, it can be seen in the diagram that with the increase of the drag speed of the driving fault wheel, the resistance generated gradually increased, but the increase of the resistance value was not significant. When the driving was stable, the drag resistance of the driving fault wheel at the set speed was 42.4 N, 45.4 N, 50.8 N, 54.9 N, and 56.4 N, and the maximum difference was only 14.0 N.

Meanwhile, the dragging process of the fault wheel and the deformation of the soil were observed and compared with those of the normal wheel. With the increase of drag speed and slip rate, the sinkage of the tested wheel increased, and the sinkage of the driving fault wheel was higher than that of the normal wheel. For example, when the drag and release speed was 30 mm/s, the sinkage of the normal wheel was 19 mm, and the sinkage of the driving fault wheel was 28 mm. It can be seen from the motion scene and soil state of the tested wheel shown in Figure 13 that a serious soil accumulation was formed in front of the faulty wheel during the dragging process, and it was difficult to transfer the soil in front of the wheel to the back of the wheel like a normal wheel. This bulldozing effect is the main reason for the drag of the faulty wheel during dragging motion. With the increase of slip rate of normal wheel, the interaction between wheel and soil is more intense resulting in greater traction, but the wheel subsidence caused by slip is also increased and the traction provided is limited.

3. State Analysis

In this section, the state analysis of the planetary rover is carried out combined with the phenomena and results of the test of the planetary rover fault scene reduction. The drag motion relationship between the faulty drive wheel and the normal wheel between the same side suspension is analyzed.

3.1. Planetary Rover Model with Faulty Drive Wheel

${\left[{\begin{array}{ccc}{x}_D& y_D& \theta \end{array}}_R\right]}^T$ is defined as the center D position and orientation of the planetary rover in the world coordinate system. In the planetary rover model with a faulty drive wheel (left rear wheel) shown in Figure 14, the relationship between the speed in the world coordinate system and the speed in the rover reference system is:

$$\left[\begin{array}{c}{x}_D\\ y_D\\ {\theta }_R\end{array}\right]=\left[\begin{array}{ll}\cos {\theta }_R\hfill & 0\\ \sin {\theta }_R\hfill & 0\\ 0\hfill & 1\end{array}\right]\left[\begin{array}{c}{v}_x\\ w_z\end{array}\right]+\left[\begin{array}{c}-\sin {\theta }_R\\ \cos {\theta }_R\\ 0\end{array}\right]v_y+{\Delta }_{dist}$$

(1)

where $v_x,v_y,w_z$ are the longitudinal speed, lateral speed, and angular velocity of the body rotation of the roaming car. ${\Delta }_{dist}$ is the combined term of the fault wheel and other external factors on the state disturbance of the planetary rover.

Without considering the change of the suspension attitude of the planetary rover, the drag speed driving the faulty wheel n is:

$${\mathit{v}}_{\mathit{n}}={\mathit{v}}_{\mathit{x}}+{\mathit{v}}_{\mathit{y}}+{\mathit{w}}_{\mathit{z}}\times \overrightarrow{\mathit{D}{\mathit{A}}_{\mathit{n}}}$$

(2)

Under the influence of wheel failure and terrain factors, the slip of the remaining normal wheel i is $s_i$ (the value range of i is $\left\{\left.i\in Z\right|i\in \left[1,6\right],i\ne n\right\}$):

$$s_i=\left\{\begin{array}{ll}\left(r_w{w}_i-v_{wi}^{\phi }\right)/r_w{w}_i\hfill & \left(r_w{w}_i\ge v_{wi}^{\phi },s_i\in \left[0,1\right]\right)\hfill \\ \left(r_w{w}_i-v_{wi}^{\phi }\right)/v_{wi}^{\phi }\hfill & \left(r_w{w}_i<v_{wi}^{\phi },s_i\in \left[-1,0\right]\right)\hfill \end{array}\right.$$

(3)

where $v_{wi}^{\phi }={\mathit{v}}_{\mathit{w}\mathit{i}}\cdot {\mathit{a}}_{\mathit{i}}$ is the projection of the speed of wheel i in the longitudinal direction. ${\mathit{a}}_{\mathit{i}}$ is the unit vector along the longitudinal direction of wheel i. $r_w$ is the wheel radius of the planetary rover.

There is a complex ground mechanical relationship between the wheel and the soft ground. The force of the normal working planetary rover wheel can be recorded as ${\left[\begin{array}{ccc}{F}_{DPi}& F_{Si}& F_{Ni}\end{array}\right]}^T$, and the force of the driving fault wheel is different as ${\left[\begin{array}{ccc}{F}_{Rn}& F_{Sn}& F_{Nn}\end{array}\right]}^T$. Among them, $F_{DPi},F_{Si},.F_{Ni}$ are the drawbar pull, lateral force, and support force provided by wheel i to the planetary rover, and $F_{Rn}$ is the resistance of the fault wheel n when it moves passively.

Therefore, the dynamic model of the planetary rover with a faulty drive wheel can be described as:

$$m{\dot{v}}_x={\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}(F_{DPi}}\cos {\phi }_i+F_{Si}\sin {\phi }_i)-F_{Rn}\cos {\phi }_n+F_{Sn}\sin {\phi }_n+F_{xdist}$$

(4)

$$m{\dot{y}}_x={\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}(F_{Si}}\cos {\phi }_i-F_{DPi}\sin {\phi }_i)+F_{Sn}\cos {\phi }_n+F_{Rn}\sin {\phi }_n+F_{ydist}$$

(5)

$$I_{bz}{\dot{\mathit{w}}}_z={\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}[\mathit{D}{\mathit{A}}_{\mathit{i}}\times ({\mathit{F}}_{\mathit{D}\mathit{P}\mathit{i}}}+{\mathit{F}}_{\mathit{S}\mathit{i}})]+[\mathit{D}{\mathit{A}}_{\mathit{n}}\times ({\mathit{F}}_{\mathit{R}\mathit{n}}+{\mathit{F}}_{\mathit{S}\mathit{n}})]+{\mathit{M}}_{\mathit{z}\mathit{d}\mathit{i}\mathit{s}\mathit{t}}$$

(6)

where m is the mass of the rover. ${\phi }_i$ is the steering angle of wheel i, which is positive when turning left. $I_{bz}$ is the moment of inertia of the planetary rover in the vertical direction. $F_{xdist}, F_{ydist}, M_{zdist}$ are the longitudinal, lateral, and heading disturbance of the planetary rover.

3.2. Drag Motion Relationship of Faulty Drive Wheel

Figure 15 shows the flow law of deformable soil under the wheel when the planetary rover with a faulty drive wheel moves. In addition to extreme fault conditions, there are generally two flow zones under normal wheels with slip. In the front area $A_i{C}_i{B}_i$ soil flows forward, and in the rear area $A_i{E}_i{D}_i$ soil flows backward. During the drag motion, a soil wedge $A_n{C}_n{D}_n$ is generated in front of the locked drive failure wheel and pushes the soil in the front area $A_n{C}_n{B}_n$ forward. This effect is similar to treating the soil plane at $A_n{C}_n$ as a virtual moving bulldozing plate.

The force of the wheel and the stress distribution (normal stress $\sigma$, shear stress $\tau$) between the wheel and soil are shown in Figure 16. In order to simplify the stress analysis, the soil wedge and the fault wheel are regarded as a moving body, and the influence of the lateral force of the wheel. The repeated passing of the wheel and the quality of the soil wedge are ignored. Under this setting, the force and torque applied to the wheel can be obtained by integrating the normal stress and shear stress.

The normal wheel:

$$F_{Ni}=r_w{b}_w\left\{{\displaystyle {\int }_{{\theta }_{im}}^{{\theta }_{i1}}{\sigma }_{i1}\left(\theta \right)\cos \theta d\theta +{\displaystyle {\int }_{{\theta }_{i2}}^{{\theta }_{im}}{\sigma }_{i2}\left(\theta \right)\cos \theta d\theta }+{\displaystyle {\int }_{{\theta }_{i2}}^{{\theta }_{i1}}{\tau }_i\left(\theta \right)\sin \theta d\theta }}\right\}$$

(7)

$$F_{DPi}=r_w{b}_w\left\{{\displaystyle {\int }_{{\theta }_{i2}}^{{\theta }_{i1}}{\tau }_i\left(\theta \right)\cos \theta d\theta -{\displaystyle {\int }_{{\theta }_{im}}^{{\theta }_{i1}}{\sigma }_{i1}\left(\theta \right)\sin \theta d\theta }-{\displaystyle {\int }_{{\theta }_{i2}}^{{\theta }_{im}}{\sigma }_{i2}\left(\theta \right)\sin \theta d\theta }}\right\}$$

(8)

$$T_i={r_w}^2{b}_w{\displaystyle {\int }_{{\theta }_{i2}}^{{\theta }_{i1}}{\tau }_i\left(\theta \right)d\theta }$$

(9)

The faulty drive wheel:

$$F_{Nn}=r_w{b}_w\left\{{\displaystyle {\int }_0^{{\theta }_{nb}}{\sigma }_{n2}\left(\theta \right)d\theta +{\displaystyle {\int }_{{\theta }_{n2}}^0{\sigma }_{n3}\left(\theta \right)\cos \theta d\theta }}\right\}$$

(10)

$$F_{Rn}=r_w{b}_w\left\{{\displaystyle {\int }_0^{{\theta }_{nb}}{\tau }_n\left(\theta \right)d\theta +{\displaystyle {\int }_{{\theta }_{inb}}^{{\theta }_{n1}}{\sigma }_{n1}\left(\theta \right)d\theta }-{\displaystyle {\int }_{{\theta }_{n2}}^0{\sigma }_{n3}\left(\theta \right)\sin \theta d\theta }}\right\}$$

(11)

$$T_n={r_w}^2{b}_w\left\{{\displaystyle {\int }_0^{{\theta }_{nb}}{\tau }_n\left(\theta \right)d\theta +{\displaystyle {\int }_{{\theta }_{nb}}^{{\theta }_{n1}}{\sigma }_{n1}\left(\theta \right)\sin {\theta }_{n1}\cot \theta d\theta -}{\displaystyle {\int }_0^{{\theta }_{nb}}{\sigma }_{n2}\left(\theta \right)\tan \theta d\theta }}\right\}$$

(12)

where $r_w, b_w$ are the radius and width of the planetary rover wheels, and ${\theta }_1, {\theta }_2$ are the entry angle and exit angle on soft ground, respectively. ${\theta }_{im}$ is the angle position where the normal wheel–ground contact stress reaches the maximum value, and its value can be considered as a function of slip rate [20]. ${\theta }_{nb}$ is the angular position at the junction of the vertical plane $A_n{C}_n$ and the horizontal plane $A_n{C}_n$ of the soil wedge in front of the driving fault wheel. It can be obtained as:

$${\theta }_{im}=\left(c_1+c_2{s}_i\right){\theta }_{i1}$$

(13)

$${\theta }_{nb}={\tan }^{-1}\left(\sin {\theta }_{n1}\right)$$

(14)

The normal stress and shear stress of normal wheels under slip conditions during dragging motion can be expressed as [20,21]:

$$\left\{\begin{array}{ll}{\sigma }_{i1}\left(\theta \right)=\left({\displaystyle \frac{k_c}{b_w}}+k_{\phi }\right){r_w}^n{\left(\cos \theta -\cos {\theta }_{i1}\right)}^n\hfill & \left({\theta }_{im}\le \theta \le {\theta }_{i1}\right)\\ {\sigma }_{i2}\left(\theta \right)=\left({\displaystyle \frac{k_c}{b_w}}+k_{\phi }\right){r_w}^n{\left[\cos \left({\theta }_{i1}-\left({\displaystyle \frac{\theta -{\theta }_{i2}}{{\theta }_{im}-{\theta }_{i2}}}\right)\left({\theta }_{i1}-{\theta }_{im}\right)\right)-\cos {\theta }_{i1}\right]}^n\hfill & \left({\theta }_{i2}\le \theta \le {\theta }_{i1}\right)\end{array}\right.$$

(15)

$${\tau }_i\left(\theta \right)=\left[c+{\sigma }_i\left(\theta \right)\tan \phi \right]\left[1-\exp \left({\displaystyle \frac{-r_w\left[\left({\theta }_{i1}-\theta \right)-\left(1-s_i\right)\left(\sin {\theta }_{i1}-\sin \theta \right)\right]}{K}}\right)\right]$$

(16)

When the faulty drive wheel is dragged, the state of the faulty wheel can be described by analyzing the bulldozing effect of the soil wedge in front of the wheel and its stress distribution and shear displacement. The normal stress at $A_n{C}_n$ can be expressed as [22]:

$${\theta }_{n1}\left(\theta \right)={\gamma }_s{r}_w{K}_{p\gamma }\cos {\theta }_{n1}\left(\tan \theta -1\right)+q{K}_{pq}+c{K}_{pc}$$

(17)

where $K_{p\gamma }, K_{pq}$, and $K_{pc}$ are the correlation constants of soil parameters, and $q$ is the external stress. The normal stress under the wheel can be described by Bekker’s pressure-settlement model in the same form as Equation (15):

$$\left\{\begin{array}{ll}{\sigma }_{n2}\left(\theta \right)=\left({\displaystyle \frac{k_c}{b_w}}+k_{\phi }\right){r_w}^n{\left(1-\cos {\theta }_{n1}\right)}^n\hfill & \left(0\le \theta \le {\theta }_{nb}\right)\\ {\sigma }_{n3}\left(\theta \right)=\left({\displaystyle \frac{k_c}{b_w}}+k_{\phi }\right){r_w}^n{\left[\cos \left({\theta }_{n1}-\left({\displaystyle \frac{\theta -{\theta }_{n2}}{{\theta }_{nb}-{\theta }_{n2}}}\right)\left({\theta }_{n1}-{\theta }_{nb}\right)\right)-\cos {\theta }_{n1}\right]}^n\hfill & \left({\theta }_{n2}\le \theta \le 0\right)\end{array}\right.$$

(18)

The shear stress under the soil wedge can be described by the Coulomb’s failure criterion:

$${\tau }_n\left(\theta \right)=\left[c+{\sigma }_{n2}\left(\theta \right)\tan \phi \right]\left[1-\exp \left({\displaystyle \frac{-r_w\left(\sin {\theta }_{i1}-\tan \theta \right)}{K}}\right)\right]$$

(19)

A faulty drive wheel’s dragging motion is a special driving state of the planetary rover, and it is necessary to summarize its motion characteristics. Combined with the above analysis, the motion relationship and force of the faulty drive wheel and the normal wheel on the same side suspension mainly meet the following constraints without considering the change of suspension attitude:

$$\left\{\begin{array}{l}{F}_{Ni}=F_{Nn}>0\hfill \\ Q_i{F}_{DPi}\ge F_{Rn}\hfill \\ T_i\le T_{i\max }\hfill \\ {\displaystyle \frac{\left|{\mathit{v}}_{\mathit{i}}-{\mathit{v}}_{\mathit{D}}\right|}{\left|{\mathit{v}}_{\mathit{n}}-{\mathit{v}}_{\mathit{D}}\right|}}={\displaystyle \frac{\left|\overrightarrow{\mathit{D}{\mathit{A}}_{\mathit{i}}}\right|}{\left|\overrightarrow{\mathit{D}{\mathit{A}}_{\mathit{n}}}\right|}}\hfill \\ 0<s_i<1\hfill \end{array}\right.$$

(20)

where $Q_i$ is the number of normal wheels on that side. $T_{i\max }$ is the maximum torque that the normal wheel drive motor can provide. It should be noted that Equation (20) and the simultaneous Equations (7)–(19) can be transformed or simplified according to the control requirements in the construction of the actual emergency control system.

4. Emergency Control

In this section, the emergency control strategy and control algorithm are proposed for the prototype test and state analysis of a planetary rover with a faulty drive wheel.

According to the different functions of the software layer in the planetary rover control system, it can be divided into an advanced control layer, a motion control layer, and a bottom execution layer [23]. By combining the state analysis in Section 2, it can be found that a faulty drive wheel will directly change the overall driving state of the planetary rover. It will not be able to properly match the original control system. Therefore, emergency control needs to consider control strategy adjustment and a targeted control algorithm deployment of the advanced control layer and the motion control layer at the same time. The specific adjustment method is shown in Figure 17.

4.1. Path Planning

With the development of wheeled mobile robot control technology and planning algorithms, some planetary vehicles have deployed and applied autonomous driving algorithms for autonomous environment recognition and path planning [14]. However, most of the path planning functions do not include emergency handling and subsequent algorithm adjustment for the sudden failure of the planetary rover. This study proposes an emergency control strategy for planetary rover path planning in a special scenario where a wheel drive fails and cannot be repaired. The main adjustments include the following aspects:

(1) Transfer of responsibility for path planning

The original automatic or semi-automatic path planning task of the planetary rover is transferred from the planetary rover controller to the ground expert system. It can avoid further deterioration of the planetary rover situation due to improper analysis of the surrounding environment or mobile planning of the planetary rover in the subsequent movement. In the usual solution, an expert system is usually devised by experienced geological experts, engineers who are proficient in planetary rover state analysis and prediction, and ground planetary rover driving operators.

(2) Simplification of the path form of the planetary rover

The composition of the planetary rover’s driving path is simplified. In the subsequent path planning, the desired trajectory is set to be a combination of two basic driving modes: straight-line driving and in situ steering; that is, point turning-line-point turning (PLP) path. This will reduce the mobility performance requirements of the faulty planetary rover and avoid unexpected loss of control during curve driving or real-time course change. Meanwhile, this broken-line driving path will be conducive to real-time monitoring of the driving state of the planetary rover and facilitate the construction and execution of the motion controller.

(3) Correction of path selection basis

According to the particularity of the faulty wheel, the basis of path selection is modified, and the strategy of faulty wheel priority is adopted to improve the proportion of the suspension on the side of the faulty driving wheel. We should avoid regarding the planetary rover as a whole in the conventional path evaluation, but comprehensively evaluate the passing effect when synthesizing the respective driving trajectories of the suspensions on both sides. This is mainly because the heading control and obstacle crossing ability of the planetary rover on the side of the faulty wheel are seriously reduced, which is reflected in Section 2. This targeted treatment is also necessary to improve the follow-up motion control effect and reduce the energy consumption of the vehicle.

Specifically, the application object in the scene is defined as a faulty planetary rover with state $\psi \left(n,q_t,\varsigma \right)$ to be planned, where n is the number of the faulty wheel and $q_t$ is the attitude and position state of the planetary rover when it moves. It is necessary to plan subsequent exploration missions within the range of the surrounding environment and the terrain state of $\varsigma \left(m\right)$. The specific steps can be described as follows:

(1) Establishment of trajectory library

The expert system on the ground needs to combine the comprehensive influence of factors such as the current state of the planetary rover, the surrounding planetary surface environment, and the requirements of the exploration mission. A trajectory library $\mathrm{{\rm K}}$ consisting of N more reasonable driving path trajectories is initially proposed:

$$\mathrm{{\rm K}}=\left\{{\zeta }_j\right\},j=1:N$$

According to the simplified driving path form, each trajectory can be expressed as a combination of straight and in situ steering:

$${\zeta }_j:{\psi }_0\mapsto {\kappa }_1\mapsto {\lambda }_1\mapsto \cdots \mapsto {\kappa }_{N^{\prime }}\mapsto {\lambda }_{N^{″}}\mapsto {\psi }_j,j=1:N$$

where ${\psi }_0, {\psi }_j$ are the starting point and the final point of the path trajectory of the j path, respectively, and ${\kappa }_{N^{\prime }}, {\lambda }_{N^{″}}$ are the straight line of the $N^{\prime }$ part and the in situ turning of the $N^{″}$ part in the path composition.

(2) Selection of initial optimal path

The subsequent further selection of the path in the trajectory library essentially consists of the prediction and evaluation of the state of the faulty planetary rover when it travels along each path trajectory. The evaluation of this quality can be defined as the cost and cost function $C\left(\cdot \right)$ when traversing the trajectory. The preferential process and the optimal path can be expressed as:

$$\stackrel{*}{\zeta }=\underset{\zeta \in {\rm K}}{\arg \min }C\left({\zeta }_j\right)$$

(21)

$$C\left({\zeta }_j\right)=\alpha \cdot C_{goal}\left({\zeta }_j,{\psi }_0\mapsto {\psi }_j\right)+\beta \cdot {\displaystyle \sum _{t=1}^T{C}_{block}}\left({\zeta }_j,\psi \left(n,q_t,\varsigma \right)\right)$$

(22)

where $C_{goal}$ is the basic theoretical cost of implementing this path, such as the time cost required to complete this drive, the minimum energy consumption and the complexity of the path. $C_{block}$ is the actual obstacle effect of factors such as terrain conditions along the trajectory on the driving of the planetary rover, such as the probability of collision with the obstacle stone on the path, and the degree of wheel subsidence caused by the soft ground.

Aiming at the change of the driving path of the faulty planetary rover, the driving cost of the straight and the in situ steering is calculated respectively, and the cost function expression of Equation (22) is further written as:

$$C\left({\zeta }_j\right)=\alpha \cdot C_{goal}\left({\zeta }_j,{\psi }_0\mapsto {\psi }_j\right)+{\beta }_{\kappa }\cdot {\displaystyle \sum _{t=1}^{N^{\prime }}{C}_{block}}\left({\zeta }_j,{\kappa }_t\right)+{\beta }_{\lambda }\cdot {\displaystyle \sum _{t=1}^{N^{″}}{C}_{block}}\left({\zeta }_j,{\lambda }_t\right)$$

(23)

(3) Path adjustment for fault conditions

Because the cost function evaluation in the obtained preliminary optimal path $\stackrel{*}{\zeta }$ is designed to evaluate the moving cost of the planetary vehicle as a whole, the particularity of the suspension on the side of the fault wheel is not considered. Further path adjustment is needed in emergency path control:

$$\widehat{\zeta }=\underset{\zeta \in \Xi }{\arg \min }\stackrel{\ast }{C}\left({\zeta }_k\right)$$

(24)

where $\Xi$ is the space for further path adjustment centered on path $\stackrel{*}{\zeta }$. $\stackrel{\ast }{C}\left({\zeta }_k\right)$ is the cost function evaluation of the ${\zeta }_k$ in the adjusted determined path in the space to be used again for adjustment. The specific form of the cost function is:

$$\stackrel{\ast }{C}\left({\zeta }_k\right)={\beta }_{L\kappa }\cdot {\displaystyle \sum _{t=1}^{N^{\prime }}{C}_{block}}\left({\zeta }_k,{\kappa }_t\right)+{\beta }_{R\kappa }\cdot {\displaystyle \sum _{t=1}^{N^{\prime }}{C}_{block}}\left({\zeta }_k,{\kappa }_t\right)+{\beta }_{L\lambda }\cdot {\displaystyle \sum _{t=1}^{N^{″}}{C}_{block}}\left({\zeta }_k,{\lambda }_t\right)+{\beta }_{L\lambda }\cdot {\displaystyle \sum _{t=1}^{N^{″}}{C}_{block}}\left({\zeta }_k,{\lambda }_t\right)$$

(25)

${\beta }_{L\kappa }, {\beta }_{L\lambda }, {\beta }_{R\kappa }, {\beta }_{R\lambda }$ are the weight coefficients of the cost function of the left suspension and the right suspension in the straight and in situ steering process, respectively. In the hypothetical scenario, since the faulty wheel is the left front wheel, there is a relationship of ${\beta }_{L\kappa }>{\beta }_{R\kappa }, {\beta }_{L\lambda }>{\beta }_{R\lambda }$ between the coefficients. It should be noted that the whole process of the whole path planning emergency control strategy needs to be carried out under the supervision of the ground expert system personnel, and the variable parameters are specifically adjusted by experience and theoretical analysis. The final path results also need to be manually checked to ensure the safety of the subsequent movement of the planetary rover.

4.2. Motion Control

Through the scene restoration test in Section 2, it can be seen that a planetary rover with a faulty driving wheel will experience serious yaw and slip when it continues to use the original motion control wheel speed distribution method. Therefore, motion control correction is indispensable for a planetary rover with faulty driving wheels. However, a faulty planetary rover is a multivariable, nonlinear, and strong interference system and we need to take into account the driving efficiency and energy consumption. The slip problem can usually be compensated for by appropriately extending the driving time and is relatively simple. Therefore, this study mainly designed an active disturbance rejection heading correction and coordinated allocation motion control method.

The active disturbance rejection controller for the course correction of the planetary rover body is composed of three parts: tracking differentiator, extended state observer, and nonlinear state error feedback law. It does not depend on the precise mathematical model of the controlled faulty planetary rover and has the ability to suppress system disturbances.

From the mathematical model of the planetary rover with driving faults described by Equations (4)–(6) in the state analysis, the dynamic equation of the body heading control can be written as:

$$I_{bz}{\dot{\mathit{w}}}_z={\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}[\mathit{D}{\mathit{A}}_{\mathit{i}}\times {\mathit{F}}_{\mathit{D}\mathit{P}\mathit{i}}}]+f_{Mzdist}\left(n,{\mathit{F}}_{\mathit{S}},{\mathit{F}}_{\mathit{R}\mathit{n}},{\mathit{M}}_{\mathit{z}\mathit{d}\mathit{i}\mathit{s}\mathit{t}}\right)$$

(26)

$${\ddot{\theta }}_z=Bu+f_{Mzdist}\left(n,{\mathit{F}}_{\mathit{S}},{\mathit{F}}_{\mathit{R}\mathit{n}},{\mathit{M}}_{\mathit{z}\mathit{d}\mathit{i}\mathit{s}\mathit{t}}\right)$$

(27)

where $B=1/I_{bz}$ is a constant coefficient. $u={\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}[\mathit{D}{\mathit{A}}_{\mathit{i}}\times {\mathit{F}}_{\mathit{D}\mathit{P}\mathit{i}}}]$ is the virtual output of the active disturbance rejection control part, which represents the sum of the torque provided to the body heading angle change when the remaining normal wheels are drive. $f_{Mzdist}$ is the merging term of the disturbance.

Therefore, the discrete-time ADRC heading correction controller can be summarized as follows:

$$\left\{\begin{array}{l}\left.\begin{array}{l}{\eta }_{z1}\left(k+1\right)={\eta }_{z1}\left(k\right)+h{\eta }_{z2}\left(k\right)\hfill \\ {\eta }_{z2}\left(k+1\right)={\eta }_{z2}\left(k\right)+hfhan\left({\eta }_{z1}\left(k\right)-{\eta }_{zd}\left(k\right),{\eta }_{z2}\left(k\right),\delta ,h_0\right)\hfill \end{array}\right\}TD\hfill \\ \left.\begin{array}{l}{e}_{\eta }={\xi }_{\eta 1}\left(k\right)-{\eta }_z\left(k\right)\hfill \\ {\xi }_{\eta 1}\left(k+1\right)={\xi }_{\eta 1}\left(k\right)+\left[{\xi }_{\eta 2}\left(k\right)-l_{\eta 1}{e}_{\eta }\right]\hfill \\ {\xi }_{\eta 2}\left(k+1\right)={\xi }_{\eta 2}\left(k\right)+\left[{\xi }_{\eta 3}\left(k\right)-l_{\eta 2}{e}_{\eta }+b_{0}u\right]\hfill \\ {\xi }_{\eta 3}\left(k+1\right)={\xi }_{\eta 3}\left(k\right)+\left(-l_{\eta 3}{e}_{\eta }\right)\hfill \end{array}\right\}ESO\hfill \\ \left.\begin{array}{l}{e}_1={\eta }_{z1}\left(k+1\right)-{\xi }_{\eta 1}\left(k+1\right)\hfill \\ e_2={\eta }_{z2}\left(k+1\right)-{\xi }_{\eta 2}\left(k+1\right)\hfill \\ u_0=k_{\eta p}{e}_1+k_{\eta d}{e}_2\hfill \end{array}\right\}LSEF\hfill \\ u=\left(u_0-{\xi }_{\eta 3}\left(k+1\right)\right)/b_0\hfill \end{array}\right.$$

(28)

In the formula, h is the control period of the active disturbance rejection controller, $\delta ,h_0$ are the controller parameters, ${\eta }_{zd}$ is the expected heading of the advanced control layer input, and ${\eta }_{z1},{\eta }_{z2}$ are the expected input and its differential signal, respectively. ${\xi }_{\eta 1},{\xi }_{\eta 2}$ are the heading state estimation, and ${\xi }_{\eta 3}$ estimates the extended state of the total disturbance of the whole system. $L={\left[\begin{array}{ccc}{l}_{\eta 1}& l_{\eta 2}& l_{\eta 3}\end{array}\right]}^T={\left[\begin{array}{ccc}3w_0& 3w_0^2& w_0^3\end{array}\right]}^T$ is the ESO gain and $w_0$ is the bandwidth of the observer.

The time optimal control synthesis function fhan (·) defined by Han [24] is:

$$\left\{\begin{array}{l}d=h_0{\delta }_0^2,a_0=h_0{\eta }_{z2},y={\eta }_{z1}+a_0\hfill \\ a_1=\sqrt{d\left(d+8\left|y\right|\right)}\hfill \\ a_2=a_0+sign\left(a_1-d\right)/2\hfill \\ fsg\left(x,d\right)=\left(sign\left(x+d\right)-sign\left(x+d\right)\right)/2\hfill \\ a=\left(a_0+y-a_2\right)fsg\left(y,d\right)+a_2\hfill \\ fhan=-{\delta }_0\left[a/b-sign\left(a\right)\right]fsg\left(a,d\right)-\delta sign\left(a\right)\hfill \end{array}\right.$$

(29)

After obtaining the output of the ADRC part, the coordinated distribution of the slip rate of the remaining normal planetary rover wheels based on both driving efficiency and energy consumption is performed. The process needs to be carried out based on the drag-and-drop motion relationship of the faulty drive wheel established in Section 3.

In order to simplify the allocation operation, in this study, the method of making the slip state of each wheel as close as possible to the set optimal slip rate $s_{osr}$ was used to achieve coordinated allocation. The specific setting method is as follows: through the overall analysis of the wheel on the driving fault side, the traction coefficient PC and the driving efficiency TE of the wheel on this side are taken as the movement and energy consumption indexes under the fault [25].

$$PC=Q_i{F}_{DPi}/\left(Q_i{F}_{Ni}+F_{Nn}\right)$$

(30)

$$TE={\displaystyle \frac{\left(Q_i{F}_{DPi}-F_{Rn}\right)v_i}{Q_i{T}_i{\omega }_i}}={\displaystyle \frac{\left(Q_i{F}_{DPi}-F_{Rn}\right)}{Q_i{T}_i}}\cdot \left(1-s_i\right)$$

(31)

According to the numerical change of the single-wheel test data with the slip rate, the optimal slip rates under each index are recorded as $s_T, s_E$, respectively. At the same time, in order to prevent the excessive pursuit of mobility in the control, in which the wheels are in a large slip rate and cause excessive slip subsidence, the upper limit value $s_{ul}$ of slip rate distribution is set. The distribution of slip ratio needs to consider the constraints of the motion relationship and force condition between the faulty driving wheel and the normal wheel on the same side of the suspension established in Section 3. Here, the second line in Equation (20) is simplified by writing $F_{DPi}$ as $s_i$ fitting function [26] and using the maximum value $F_{Rn\max }$ of fault wheel resistance in the test.

$$F_{DPi}=\left(a_1{s_i}^{b_1}-c_1\right)F_{Ni}$$

(32)

Therefore, according to Equations (20) and (28)–(32), the coordinated distribution process of the slip ratio can be described as:

$$\min J\left(s_i\right)=\min {\displaystyle \mathbf{\sum }_{i=\mathbf{1}}^{\mathbf{6}}\left|s_i-s_{osr}\right|}$$

(33)

The constraints are as follows:

$$s.t.\left\{\begin{array}{l}{s}_{osr}=K_s{s}_E+\left(1-K_s\right)· \min \left(s_T,s_{ul}\right)\hfill \\ {\displaystyle \mathbf{\sum }_{\mathbf{1}}^{\mathbf{6}}[\mathit{D}{\mathit{A}}_{\mathit{i}}\times {\mathit{F}}_{\mathit{D}\mathit{P}\mathit{i}}}]=u\hfill \\ {s_{i1}}^{b_1}+{s_{i2}}^{b_1}\ge {\displaystyle \frac{F_{Rn\max }/F_{Ni}+2c_1}{a_1}}\hfill \\ 0<s_i<1\hfill \end{array}\right.$$

(34)

where $K_s$ is the weight constant set when weighing the optimal slip rate $s_{osr}$. $s_{i1},s_{i2}$ are the slip rates of the normal wheel of the side suspension. In the controller, according to the expected motion state $v_{xd},w_{zd}$ of the vehicle in the planning, the expected speed $v_{id}$ of each wheel is obtained. Finally, the angular velocity command $w_{id}=v_{id}/\left[r_w\left(1-s_i\right)\right]$ of each vehicle wheel is calculated and output to the bottom execution layer for the execution of each wheel drive motor.

5. Conclusions and Future Work

In this paper, the wheel drive failure of a planetary rover that performs the detection task on the planetary surface is systematically studied. The research is based on the phenomena and data of vehicle prototypes or single wheel performance tests, so that the theoretical model can fully restore and analyze the actual fault scenes and states. A planetary rover model with a faulty drive wheel and the dragging motion relationship for the faulty drive wheel are established. Meanwhile, when combining the state analysis to design the emergency control system, a path planning strategy based on faulty wheel priority is proposed to simplify and adjust the driving path of the faulty planetary rover. In order to maintain the basic movement of the faulty planetary rover, an emergency motion controller was built with active disturbance rejection body heading correction control and coordinated distribution of wheel slip rate.

This study focused only on theoretical research and experimental testing of a serious accident of wheel drive failure. Future research will focus on two areas: other fault types, such as planetary rover steering motor failure, or wheel breakage. We will develop a planetary rover active suspension system that can reduce the impact of single-wheel failure based on our previous work.