Design, Analysis, and Experiment of a Wheel-Legged Mobile Robot

Abstract

In this paper, a wheel-legged mobile robot is proposed. The parameter design of the wheel-leg structure is completed using geometric analysis and statics modeling. The curling mechanism is chosen as the main part of the robot, expanding the robot’s application scenarios. Particle swarm optimization is used to determine the number of body sections and their length. After dynamic simulation, rubber is selected as the wheel-leg cover material. Based on the above results, a prototype is made and subject to experiments in different terrains. The experiment results verify that the robot is adaptable to multiple terrains with strong obstacle-crossing ability.

1. Introduction

As an important branch of robotics, mobile robots are an integrated system that incorporates environmental perception [1,2,3,4,5], dynamic decision-making and planning [6,7,8], behavior control and execution [9,10,11,12], and other functions. Owing to its good obstacle-crossing capability, mobility, and stability, it has broad application prospects in emergency rescuing, military reconnaissance, and scientific exploration [13]. According to the mode of movement, traditional mobile robots can be classified into wheeled mobile robots [14,15,16,17,18], legged mobile robots [19,20,21], and tracked mobile robots [22,23]. Wheeled mobile robots are characterized by a simple structure [14,15], reliable control [14,16,17,18], and faster moving speed [15]; however, their obstacle-crossing capability is poor in complex terrains such as stairs and rugged roads. Legged mobile robots have the advantages of strong obstacle-crossing capability and terrain adaptability [20,21]; however, they feature a complex leg structure, difficult controllability, and low movement speed. Tracked mobile robots demonstrate good obstacle-crossing capability [22] and reliable control [23]; however, their bulky structure results in low traveling speed, large power consumption, and damage to the ground.

Compared with the disadvantages of traditional mobile robots, hybrid mobile robots combine the characteristics of multiple mobile structures and thus have various advantages, which has attracted the wide attention of scholars. In addition to strong obstacle-crossing capability and stability, hybrid mobile robots can also balance indicators such as movement speed, comprehensive structure, and control complexity while showing satisfactory terrain adaptability. Hybrid mobile robots can be classified into the wheel-legged type [1,5,6,7,24,25,26,27,28,29,30,31,32], wheel-tracked type [33,34], track-legged type [35,36,37], wheel-track-legged type [13,38], etc., according to their structure, as shown in

Table 1. Summary of hybrid mobile robots.

Category	Previous Works		Features
Wheel-legged	MAMMOTH (Reid et al., 2016)		➢ Advantages: ✧ Great obstacle avoidance; ✧ Sound motion performance; ✧ Stable operation. ➢ Weakness: ✧ Higher control requirements.
	HyTRo-I (Lu et al., 2013)
	TurboQuad (Chen et al., 2017)
	ASGUARD (Eich et al., 2008)
Wheel-tracked	RHMBot (Luo et al., 2018)		➢ Advantages: ✧ Simple structure; ✧ Reliable control. ➢ Weakness: ✧ Poor terrains adaptation.
	NEZA-I Robot (Li et al., 2011)
Track-legged	TALBOT Robot (Guo et al., 2022)		➢ Advantages: ✧ Great obstacle avoidance; ✧ Reliable control. ➢ Weakness: ✧ Complex structure.
	Blade Walker (Servet et al., 2017)
Wheel-track-legged	WTL srobot (Zhu et al., 2016)		➢ Advantages: ✧ Great obstacle avoidance; ✧ Smooth movement. ➢ Weakness: ✧ Complex structure.
	WheTLHLoc Robot (Bruzzone et al., 2021)

. Among them, the wheel–leg structure is the most widely studied and applied for its advantages in multiple aspects.

The wheel-legged robot is a hybrid mobile robot with an excellent obstacle-crossing ability [5,6,9,25,26,27,28,29,30,31], great movement performance [1,5,6,7,25,26], efficient and stable operation [1,25,26,27], etc., and has attracted the attention of many researchers. Based on the operating mode and driving method during travel, wheel-legged robots can be roughly classified into four types: tandem type, separate type, transformable type, and hybrid type. The tandem wheel-legged robot [24,25,26,27] moves on flat ground with wheels and changes its posture using the leg structure when encountering obstacles. It has good mobility. A wheel-legged robot with a variable attitude robot mechanism was designed in [39], enabling the robot to conduct various configurations and omnidirectional motion. However, only its wheels come into contact with the ground, which results in slightly poor obstacle-crossing capability. The separate wheel-legged robot [28,29] switches walking modes according to different terrains. In the case of flat terrain, its wheels come into contact with the ground; in the case of steep terrain, its legs come into contact with the ground. It improves obstacle-crossing capability, but switching between the two modes causes complexity in control. The transformable wheel-legged robot [1,30] adapts to different environments by switching the wheel–leg state. The wheel-leg structure has a built-in motor, which allows for switching to the wheel or leg shape mode according to the terrain. Similarly, it can improve adaptability to the environment. However, the requisite wheel–leg conversion device increases the structural complexity and reduces the weight-bearing capacity. The hybrid wheel-legged robot [31] has its leg structure replaced with a wheel drive that rotates in a full circle and achieves different motion gaits by controlling the phase and rotation speed of the wheel-leg, which is convenient for achieving smooth obstacle-crossing movement and simple control. A hybrid wheel-legged robot was proposed in [40], and its great motion efficiency and obstacle-crossing ability were verified. However, there is space for structure optimization to realize better strength and rigidity. Based on these latest works, we explore a new hybrid wheel-legged mobile robot by designing a novel structure of the wheel-leg.

Given the above background, this paper proposes a hybrid wheel-legged mobile robot that incorporates the characteristics of both wheeled and legged structures. Based on the robot’s obstacle-crossing needs, the structural parameters are determined using modeling, theoretical analysis, etc. Then, the optimal wheel–leg cover material is determined using dynamic analysis, after which the prototype is made and subjected to experiments in various environments. As a hybrid mobile robot, the proposed wheel-legged mobile robot incorporates the structural advantages of wheeled and legged mobile robots. In addition to expanding the usage scenario, it realizes a relatively simple control and improves obstacle-crossing capability and movement stability, exhibiting satisfactory overall performance.

2. Structure Design

The wheel-leg structure is an essential part of the proposed mobile robot, playing a decisive role in determining whether the robot can complete obstacle-crossing. In this section, the obstacle-crossing environment is set up as steps that are difficult for traditional mobile robots. In terms of statics, the obstacle-crossing process of the wheel-legged mobile robot is analyzed to determine how the number of its spokes influences the obstacle-crossing capability. Based on the specific step parameters, an efficient step-climbing strategy is proposed using geometric analysis. According to the above results, the chord length of the spoke is optimized.

2.1. Statics Modeling

First, the force condition of a single wheel–leg during step climbing is analyzed. As shown in Figure 1, to meet practical needs, it is assumed that the width and height of a single step in a staircase are W = 25 mm and H = 15 mm, respectively. F and P are the component force of the robot body on the wheel-leg in horizontal and vertical directions; N and f are the support force and friction force of the steps on the wheel-leg; M_N and M_f are the torque generated by N and f at the geometric center of the wheel-leg; T_m is the driving torque generated by the motor on the wheel-leg; δ is the angle between the tangent of the wheel-leg at its contact point with the step and the horizontal line. It is assumed that the motor speed is slow and that the driving torque provided by the motor is sufficient. Then, Formula (1) can be obtained based on the force balance of the wheel-leg:

$$\left\{\begin{array}{c}P+\mu N\cos \delta -N\sin \delta =0\\ F-\mu N\sin \delta -N\cos \delta =0\\ T_m-M_{O_i\left(N\right)}-M_{O_i\left(f\right)}=0\end{array}\right.,$$

(1)

It is assumed that the maximum static friction force is approximately equal to the sliding friction force and that the dynamic friction coefficient between the wheel-leg and the step is μ. Then, when the wheel-leg is about to slip during step climbing, we have f = µN. Thus, Formula (2) is obtained:

$$\frac{F}{P}=\frac{\mu \tan \delta +1}{\tan \delta -\mu },$$

(2)

Next, the statics analysis of the entire robot during climbing steps is conducted. As shown in Figure 2, f₁, N₁, f₄, and N₄ are the friction and pressure of the robot when its front and rear wheel-legs come into contact with a step; P₁, F₁, P₄, and F₄ are the component force of the robot body on the wheel-leg in the horizontal and vertical directions; P₁′, F₁′, P₄′, and F₄′ are the reaction forces of P₁, F₁, P₄, and F₄, respectively; T_m1 and T_m4 are the driving torque of the front and rear drive motors on the wheel-leg, respectively; T_m1′ and T_m4′ are the reaction torques of T_m1 and T_m4, respectively; G is the gravity of the robot; M_O1(f1) and M_O1(N1) are the torque of f₁ and N₁ at the geometric center O₁ of the front wheel-leg, respectively; M_O4(f4) and M_O4(N4) are the torque of f₄ and N₄ at the geometric center O₄ of the rear wheel-leg, respectively; l₁ and l₂ are the distance from point O to O₁ and O₂, respectively; and γ is the angle between the robot body and the horizontal line. Ignoring the gravity of the wheel-leg, it is assumed that the force condition of the front and rear wheel-legs is the same. Then, Formula (3) can be obtained from the force balance of the robot:

$$\left\{\begin{array}{c}{f}_1\cos {\delta }_1-N_1\sin {\delta }_1+f_4\cos {\delta }_4-N_4\sin {\delta }_4=0\\ \frac{1}{2}G-f_1\sin {\delta }_1-N_1\cos {\delta }_1-f_4\sin {\delta }_4-N_4\cos {\delta }_4=0\\ M_{O\left(N_4\right)}-M_{O\left(f_4\right)}-M_{O\left(N_1\right)}-M_{O\left(f_1\right)}=0\end{array}\right.,$$

(3)

Formula (4) can be obtained from the force balance of the robot:

$$\left\{\begin{array}{c}{P_1}^{\prime }-{P_4}^{\prime }=0\\ \frac{1}{2}G-{F_1}^{\prime }-{F_4}^{\prime }=0\\ {T_{m1}}^{\prime }+{T_{m4}}^{\prime }+{P_1}^{\prime }{l}_1\sin \gamma +{P_4}^{\prime }{l}_2\sin \gamma +{F_1}^{\prime }{l}_1\cos \gamma -{F_4}^{\prime }{l}_2\cos \gamma =0\end{array}\right.,$$

(4)

Formula (5) can be obtained from the force balance of the robot’s front wheel-leg:

$$\left\{\begin{array}{c}{P}_1+f_1\cos {\delta }_1-N_1\sin {\delta }_1=0\\ F_1-f_1\sin {\delta }_1-N_1\cos {\delta }_1=0\\ T_{m1}-M_{O_1\left(N_1\right)}-M_{O_1\left(f_1\right)}=0\end{array}\right.,$$

(5)

The formula for the force balance of the robot’s rear wheel-leg is similar to the one above. In general conditions, δ₄ = 0 during the climbing process of the robot. According to Formulas (3)~(5), the maximum driving force that the driving wheel-leg could provide for the robot body is obtained as:

$$P_{4\max }=\frac{\mu }{\left(l_1+l_2\right)\left(1-\mu \tan \gamma \right)}(\frac{G{l}_1}{2}+\frac{T_{m1}+T_{m4}}{\cos \gamma }),$$

(6)

Then, the maximum of δ₁ is derived as:

$${\delta }_{1\max }=\arctan \left(\frac{{\mu }^{2}G}{\mu G-2(1+{\mu }^2)P_{4\max }}\right),$$

(7)

In other words, when the robot is driven with the rear wheel-leg and δ₄ = 0, the robot is capable of climbing a step if the angle between the front wheel-leg and the step meets the condition δ₁ ≤ δ_1max.

Similarly, when the robot is driven with the front wheel-leg and δ₁ = 0, the following calculations are derived:

$$\left\{\begin{array}{c}{P}_{1\max }=\frac{\mu }{\left({\mathrm{l}}_1+{\mathrm{l}}_2\right)\left(1-\mathsf{\mu }\tan \mathsf{\gamma }\right)}(\frac{G{l}_2}{2}-\frac{T_{m1}+T_{m4}}{\cos \mathsf{\gamma }})\\ {\delta }_{4\max }=\arctan \left(\frac{{\mu }^{2}G}{\mu G-2(1+{\mu }^2)P_{1\max }}\right)\end{array}\right.,$$

(8)

According to the above analysis, we attain the values of δ_1max and δ_4max with the value of the mass and center-of-mass coordinate of the robot, friction coefficient μ, step size, and other parameters of the robot.

2.2. Geometry Definition and Design

The wheel-leg structure is a key component of a wheel-legged mobile robot and directly determines the stability, mobility, and obstacle-crossing capability. A hybrid wheel-leg can achieve smooth movement with strong obstacle-crossing capability and simple control. Next, the structural design of the wheel-leg is proposed.

First, the number of wheel-leg spokes is determined. The wheel-leg structure of existing wheel-legged mobile robots can be roughly classified into four types according to the number of spokes: single-spoke, triple-spoke, four-spoke, and five-spoke. The single-spoke wheel-leg structure has poor stability and complex gait planning [30,41]. Therefore, the hybrid wheel-leg structure with three spokes or more is analyzed herein. The obstacle-crossing capability of the wheel-leg structure varies in forward and reverse rotation. During reverse rotation of the wheel-leg structure, for ease of analysis, it is assumed that the radius of curvature of spoke r = ∞, namely, it is assumed that spoke is linear. As shown in Figure 3, according to the geometric relationship, the expression for the crossing height H_s is obtained as:

$$H_s=R(\sin \frac{{\phi }_s}{2}+\cos {\theta }_s),$$

(9)

where s represents the number of spokes; R represents the cord length; ${\phi }_s$ represents the angle between spokes; and ${\theta }_s$ represents the angle between the wheel-leg spoke that contacts the ground and the vertical plane. The following results are derived: H₃ = 1.732R, H₄ = 1.414R, H₅ = 1.539R, H₆ = 1.5R, H₇ = 1.409R, and H₈ = 1.307R.

When the wheel-leg contacts obstacles while rotating forwardly, a similar obstacle-crossing result is desired for the analysis of the wheel-leg’s obstacle-crossing capability. It is assumed that the force of the robot body exerted on each wheel-leg is the same. Under the condition of the robot’s force balance, the relationship between the spoke’s radius of curvature r and chord length R is set as r = 0.67R, and the coefficient of friction is 0.5. The constraint δ_1max = 53.15° [42] is obtained. That is, under the corresponding conditions, as long as the angle between the tangent of the wheel-leg at the contact point with the step and the horizontal line is smaller than 53.15°, the wheel-leg structure is capable of crossing the obstacle, as shown in Figure 4. For a wheel-leg structure with s spokes, the forward crossing heights h_s are derived as h₃ = 1.44R, h₄ = 1.12R, h₅ = 1.17R, h₆ = 1.15R, h₇ = 1.10R, and h₈ = 1.02R.

Based on the above analysis, the forward and reverse obstacle-crossing capability of wheel-legs with different numbers of spokes can be obtained, as shown in Figure 5. It can be seen that the obstacle-crossing capability of a wheel-leg declines with an increase in the number of spokes s. Therefore, to achieve the strongest obstacle-crossing capability, we use s = 3.

After the number of spokes is determined, further optimization of the spoke shape is desired. The curved triple-spoke wheel-leg shown in Figure 6b evolved from the semicircular single-spoke wheel-leg shown in Figure 6a. Its spokes can touch the ground alternately, which improves the space utilization rate. However, during step climbing, there is a certain interference among spokes and the impact of curved structure on the stiffness of the wheel-leg. Since robots usually operate on flat ground or slopes, their structure is optimized while retaining the alternating movement of curved triple-spoke wheel-legs. When two spokes of the wheel-leg are in contact with flat ground, considering the contact point between the front spoke and the ground (that is, the tangent point between the front spoke and the ground) as the boundary point of the front spoke, the curved spokes from the center of the wheel-leg to this point are changed into straight-line spokes, and the other two spokes are also optimized in the same way. A schematic figure showing the optimized wheel-leg structure is shown in Figure 6c. The novel straight-curved, triple-spoke wheel-leg inherits the advantages of curved triple-spoke wheel-legs, whose spokes touch the ground alternatively; achieves improved stiffness after changing the spoke shape; and reduces interference between each spoke when climbing and crossing obstacles.

The structure and specific geometric parameters of the wheel-leg are shown in Figure 7. Based on the chord length R, the remaining geometric parameters are obtained using the geometric relationship of wheel-leg: r = $\sqrt{3}$R/3, Ψ = 86.9°, ζ = 33.1°, φ = 30°, and l = 0.795R [43].

2.3. Further Optimization

Steps are common obstacles in daily life and represent a complex irregular terrain that is difficult for various mobile robots to cross. The height of steps is generally 10 mm~20 mm. Owing to its structural advantages, the proposed hybrid wheel-legged mobile robot can quickly climb steps using a relatively simple control strategy. The following statics modeling is used to analyze the force conditions and requirements that enable step climbing.

The straight-line part and circular-arc part of the straight-curved wheel-leg have different contact situations with steps during climbing and should be classified and analyzed to obtain the range of geometric parameters that allow step climbing. According to the statics analysis of wheel-leg step climbing and based on the structural characteristics of the wheel-leg, the main circumstances of the robot’s step crossing are divided into several types, as shown in Figure 8:

When the straight-line part of the wheel-leg comes into contact with a step, Formula (10) is obtained according to the geometric relationship between the wheel-leg and the step:

$$\left\{\begin{array}{c}l\sin \delta +R\cos \alpha =H\\ d=l\cos \alpha \end{array}\right.,$$

(10)

Based on the different positions and the geometric parameters of the wheel-leg, different δ can be obtained and compared with δ_max to determine whether the steps can be crossed.

When the junction between the straight-line and circular-arc part of the wheel-leg comes into contact with a step, since the position of the wheel-leg is determined, Formula (11) is obtained according to the positional relationship between the wheel-leg and the step:

$$\left\{\begin{array}{c}l\sin \delta +R\cos \alpha =H\\ \eta +\delta +\alpha =\frac{\mathsf{\pi }}{6}\\ l=0.1464R\\ \eta =0.2889\end{array}\right.,$$

(11)

The following expression is derived:

$$\delta =\arcsin \left(\frac{H}{1.0438R}\right)-1.1992,$$

(12)

When the geometric parameters of the wheel-leg are determined and substituted into the above expression, the corresponding δ is obtained. When δ ≤ δ_max, the wheel-leg can cross the step; when δ ≥ δ_max, the wheel-leg will slip.

When the circular-arc part of the wheel-leg comes into contact with a step, Formula (13) is obtained according to the position between the wheel-leg and the step:

$$\left\{\begin{array}{c}R\cos \alpha +2r\cos \left(\frac{\mathsf{\pi }}{6}-\alpha -\beta +\phi \right)\sin \beta =H\\ R=\sqrt{3}r\\ \delta =\frac{\mathsf{\pi }}{3}+\alpha +2\beta -\phi \\ \phi =\frac{\mathsf{\pi }}{6}\end{array}\right.,$$

(13)

When the geometric parameters and initial position of the wheel-leg are determined and substituted into the above formula, the corresponding δ is obtained. When δ ≤ δ_max, the wheel-leg can cross the step; when δ > δ_max, the wheel-leg will slip at the step.

When the wheel-leg climbs steps, the extreme condition of the contact between the straight part and a step is shown in Figure 9. The geometric parameters of the wheel-leg under such conditions are analyzed. According to the geometric relationship between the wheel-leg and step, Formula (14) is obtained:

$$\left\{\begin{array}{c}R\cos \alpha +l\sin \delta =H\\ R\sin \alpha =l\cos \delta \\ \alpha =\eta +\delta -\frac{\mathsf{\pi }}{6}\end{array}\right.,$$

(14)

The following expression is derived:

$$\delta =\arccos \frac{1.0282H}{R},$$

(15)

Under this extreme condition, to ensure that the wheel-leg crosses the step, the following requirement should be met:

$$\delta =\arccos \frac{1.0282H}{R}\le {\delta }_{\max },$$

(16)

The following expression is derived:

$$\delta =\arcsin \left(\frac{H}{1.0438R}\right)-1.1992 \le {\delta }_{\max },$$

(17)

Therefore, the chord length of the wheel-leg is 13.27 mm ≤ R ≤ 20.43 mm. When the step height is 14 mm, the chord length of the wheel-leg is 17 mm. The obstacle crossing height is 1.13 times the chord length. For steps with common size, the wheel-leg can finish obstacle crossing.

3. Curling Mechanism Design

3.1. Key Parameters

To further expand the application scenarios of the robot, a curling mechanism is selected as the robot’s main body to make the robot move more smoothly. The robot’s curling structure is shown in Figure 10. According to the configuration of the robot’s curling mechanism and the requirements for stability of the curling procedure, two indicators are proposed from the evaluation of kinematics and dynamics: the roundness R of the robot’s curling at the end state and the torque fluctuation TF of the curling procedure. R represents the sum of the difference between each joint angle and π. The smaller the value of R, the greater the roundness. TF reflects the motion stability of the robot during the curling procedure. The smaller the TF, the better the motion stability of the robot. The specific expressions for R and TF are as follows:

$$\left\{\begin{array}{c}R=\sum _{i=1}^n\sum _{k=1}^2\left|{\theta }_{\mathrm{ik}-f}-\mathsf{\pi }\right|\\ TF=\sum _{i=1}^n\sum _{k=1}^2\sum _{p=1}^m\frac{\left|B_{\mathrm{ik}}{t}_p-{\tau }_{\mathrm{ik}-p}+A_{\mathrm{ik}}\right|}{\sqrt{B_{ik}^2+1}}\end{array}\right.,$$

(18)

where:

$$\left[\begin{array}{c}{B}_{\mathrm{ik}}\\ A_{\mathrm{ik}}\end{array}\right]={\left[\begin{array}{cc}{\sum }_{p=1}^m{t}_p& m\\ {\sum }_{p=1}^m{t}_p^2& {\sum }_{p=1}^m{t}_p\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{\sum }_{p=1}^m{t}_p{\tau }_{\mathrm{ik}-p}\\ {\sum }_{p=1}^m{\tau }_{\mathrm{ik}-p}\end{array}\right],$$

(19)

and (t_p, τ_ik−p) is the coordinates of the p_th point on the curling acceleration curve.

A preliminary analysis shows that the parameters that affect the robot’s performance include the number of robot sections n and the length of each section l. In this paper, the optimal solutions for the parameters n and l are determined with the goal of obtaining the ideal R and TF. Therefore, the optimization goal can be expressed as:

$$f={\eta }_1\frac{R}{\max R}+{\eta }_2\frac{TF}{\max \mathrm{TF}},$$

(20)

where η_i (i = 1~2) is the optimization coefficient and η₁ + η₂ = 1.

Based on the constraint conditions, the optimization model of the curling mechanism is:

$$\left\{\begin{array}{c}\min f\\ s.t.\hspace{1em}\left|{\tau }_{\mathrm{ik}-p}\right|\le {\tau }_{\max }\\ \&\hspace{1em}\hspace{1em}{n}_1\le n\le n_2\\ \&\hspace{1em}\hspace{1em}0<l\le l_2\end{array}\right.,$$

(21)

where τ_max is the maximum value of the motor torque of each robot joint and is set to 0.025 N·m. [n₁, n₂] is the range of the number of sections in the curling mechanism. According to the concept of bionics, the range can be determined as [5, 8].

Particle swarm optimization is used to determine the parameters n and l. The principle is to find the global optimal solution to the problem through collaboration within the swarm population. A flow chart depicting particle swarm optimization is shown in Figure 11.

First, the model for the velocity and position of particle i in n-dimensional space is established, as shown below:

$$V_{in}^{k+1}=\omega V_{in}^k+c_1{r}_1\left(p_{\mathit{in}}^k-X_{\mathit{in}}^k\right)+c_2{r}_2(p_{\mathit{gn}}^k-X_{\mathit{in}}^k),$$

(22)

$$X_{in}^{k+1}=X_{\mathit{in}}^k+V_{in}^{k+1}$$

(23)

where c₁ and c₂ are learning factors, namely, the acceleration coefficients with a range of [0, 2]. r₁ and r₂ are pseudo-random numbers with a range of [0, 1]. ω is the inertial weight. $V_{in}^k$ is the n-dimensional vector of particle i’s velocity vector after the k_th iteration. $X_{\mathit{in}}^k$ is the n-dimensional vector of particle i’s position vector after the k_th iteration. $p_{\mathit{in}}^k$ is the n-dimensional vector of the optimal position of the particle in history after the k_th iteration. $p_{\mathit{gn}}^k$ is the n-dimensional vector of the global optimal position of the population after the k_th iteration.

Next, the fitness function is established. The penalty function is an effective treatment method. The basic idea of the penalty function is to construct an adaptation value function f₀ [44] with a penalty term by adding a penalty term m to the objective function f and to punish particles that fail to meet the requirements. The penalty function is as follows:

$$m=h·{\sum }_{i=1}^8\max (0, g_i),$$

(24)

where h is the penalty coefficient whose value is 10⁶. Finally, the fitness function f₀ = min(f + m) is obtained.

The parameters of particle swarm optimization are initialized. The inertial weight ω is set to 0.7, and the learning factors c₁ and c₂ are set to the traditional fixed value of 2.0. After 150 iterations, the final optimization results n = 5 and l = 16 mm are derived.

3.2. Model Construction

The following assumptions are made during the modeling process: each segment of the body is symmetrically distributed concerning the center of mass and has the same mass. The gravity at the contact point between the wheel legs and the ground is uniformly distributed. There is no longitudinal slip when the wheel legs are in contact with the ground. To avoid unnecessary abundance, there are simplifications in the process and expressions. The detailed derivative process is shown in Appendix A.

Since the mass of the robot is symmetrically distributed, the structure of the robot can be derived as shown in Figure 12. The coordinate O₁-XY is established. The origin is located in the middle of the rear wheel leg. The X-axis is horizontal to the right, and the Y-axis is vertical up.

$${\phi }_2={\phi }_2^1+{\phi }_2^2, {\phi }_4={\phi }_4^1+{\phi }_4^2+{\phi }_4^3,$$

(25)

where φ_i is the angle between rod A_i−1A_i and rod A_iA_i+1 (i = 2~5). φ₂ⁱ (i = 1~2) is the angle between rod A_iA_i+1 and rod A₂A₄. φ₄ⁱ (i = 1~3) is the angle between rod A₃A₄ and rod A₂A₄, rod A₂A₄ and rod A₄A₆, and rod A₄A₅ and rod A₄A₆, respectively. The lengths of rod A_iA_i+1 (i = 1~5) are all L. A₇ and A₈ are the axes of the wheel-legs.

As shown in Figure 13, a closed loop can be found starting from point A₂.

$$A_2{A}_1+A_1{A}_6=A_2{A}_4+A_4{A}_6,$$

(26)

After projecting the above equation onto the X-axis and Y-axis and performing geometric analysis, the solutions for ${\phi }_1$ and $L_3$ are derived. Therefore, we have:

$${\phi }_1^2=\arcsin \left(\frac{Z_{A8}-Z_{A7}}{L_3}\right),$$

(27)

where ${\phi }_1^2$ is the angle between A₁A₆ and A₁B. Z_A8 is the motion of the front wheel-leg on the Z-axis. Z_A7 is the motion of the rear wheel-leg on the Z-axis.

Next, θ₁ is calculated:

$${\theta }_1={\phi }_1+{\phi }_1^2,$$

(28)

The structure of the wheel-leg is shown in Figure 14. Since the structure of all wheel-legs is the same, we consider one of them for motion analysis. The center point A₇₍₈₎ of the wheel-leg is equivalent to C₅. The inferior arc C₁C₂ has C₃ as the center and r as the radius. The angle between each wheel leg is 2/3π. The length of C₂C₅ is R_max, the length of C₁C₅ is R, the angle between C₁C₃ and C₂C₃ is φ, the angle between C₂C₃ and C₂C₄ is β, the angle between C₂C₅ and C₂C₄ is δ₁, and the angle between C₄C₅ and C₂C₄ is δ₂.

The rotation of the wheel-leg can be divided into three repetitive cycles. When the wheel leg rotates clockwise, the point C₅ in the X-axis and Y-axis displacement can be expressed, thus:

$${\theta }_7=f_2\left({\theta }_2, {\theta }_3, {\theta }_4, {\theta }_5, {\theta }_8\right),$$

(29)

where θ_i (i = 7, 8) is the turned angle of the wheel-leg.

Appendix A shows the detailed derivative process and specific expressions.

3.3. Prototype Manufacturing

The appearance of the robot prototype is shown in Figure 15a. The shell was made of aluminum alloy and the bottom plate was made of resin, so as to reduce the weight of the robot. The motor shaft’s fixing parts and wheel-leg were all made of carbon steel, which can improve the strength of the robot. There are threaded fixing holes on the sides of the two parts to fix the parts and the motor’s D-shaft. Non-slip materials were added to the wheel-leg to increase the friction of the robot when crossing obstacles and climbing slopes, which improves the efficiency of the robot’s operation.

The internal structure of the robot is shown in Figure 15b. Gear motors other than transmission components were used at all joints of the robot, so as to increase the strength of the robot to resist the impact of landing. The layout of the internal parts of the robot is also shown. The spacer and connecting shaft were 3D printed with resin.

The structure of the robot’s wheel-leg is shown in Figure 16. Its accessories were made of carbon steel. The connecting screws were made of alloy steel. The fit between the wheel-leg rack and the motor’s shaft neck was a clearance fit. Since the wheel-leg suffers impact force during operation, the fit will become slightly loosened after a long time of operation. Therefore, threaded holes were added to the sides of the wheel-leg rack to strengthen the tightness between the wheel-leg and the motor shaft. This helps prevent the wheel-leg from falling off the motor shaft and also extends the service life of the robot. There is a through hole at the connection between the wheel-leg rod and the wheel-leg rack. The wheel-leg rack clamps the wheel-leg rod, which effectively increases the tightness of the connection. Bolts and nuts were used for the connection.

4. Results

To better protect the robot, a wheel-leg cover is necessary. The cover can prevent direct collision between the wheel-leg and the ground and provide certain cushioning and damping. A careful selection of wheel-leg cover materials helps reduce the requirements for the wheel-leg drive motor and curling motor, which enables more reliable operation of the robot. ADAMS is used for the analysis of the dynamics. Upon comparison of the motor torque under different wheel-leg cover materials, the best wheel-leg cover material can be selected.

4.1. Simulation Settings

As shown in Figure 17a, a virtual prototype and terrain were created. The simulation environment was composed of two types of terrain: flatland and slope. The flatland width was 800 mm, the slope angle was 30 degrees, and the slope height was 240 mm. The length of the prototype was 110.5 mm, and the width was 84 mm. The movement process of the robot is shown in Figure 17b–d. The movement includes movement on the flatland and rolling on the slope. Given these two situations, ADAMS was used for dynamics analysis.

The dynamic equations were established using the widely used Lagrange’s equation with ADAMS. The three coordinates of the barycenter of each rigid body in the inertial reference system and the three Euler angles of the rigid body were selected to determine the Cartesian generalized coordinates. Lagrange’s equation with multipliers was used to process the complete or incomplete constrained system of multiple coordinates. To avoid over-complexity of the calculations, default values were used for some of the parameters without affecting the results [39]. Then, a kinematics equation with Cartesian generalized coordinates as variables was derived. The calculation program of the software uses the gear rigid integration algorithm and sparse matrix, which greatly improves the calculation efficiency.

The purpose of the simulation is to analyze the influence of wheel-leg stiffnesses, ground friction, and wheel-leg rotation speed on the torque of the driving motor. Based on the analysis result, the appropriate wheel-leg cover material was determined. Nylon, rubber, and acrylic were used as alternatives. The ground stiffness was set to that of steel. The remaining parameters are shown in

Table 2. Parameters of the alternative materials.

Material	Collision Rigidness N/mm	Resistance Ns/mm	Force Index	Static Translational Velocity mm/s	Frictional Translational Velocity mm/s
Nylon	3800	1.52	2.0	0.1	10
Rubber	2855	0.57	1.1	0.1	10
Acrylic	1150	0.68	2.0	0.1	10

.

4.2. Comparison and Analysis

For different wheel-leg materials, wheel-leg speeds, and friction coefficients, curves for the torque of each motor over time during movement were drawn. Among them, motor 1 to motor 4 were the driving motors of wheel-legs, and motor 5 to motor 8 were the motors for curling. It is expected that a suitable material will be selected to achieve smaller torque requirements and fewer torque mutations.

4.2.1. Material: Nylon

In this section, the results of the nylon wheel-leg cover material are discussed. As shown in Figure 18a–c (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7054, accessed on 30 August 2023), when the wheel-leg speed was 10 rpm, when the friction coefficient was increased, the driving torque and curling torque of the wheel-leg became slightly larger. At some moments, the torque far exceeded the torque required for normal movement, which was due to the collision between the robot and the ground during movement. When the wheel-leg moved at a uniform speed, the contact between the wheel-leg and the ground was intermittent. Therefore, during each cycle of the wheel-leg’s movement, the wheel-leg strings suffered elastic collisions, and the peak torque was shown in the simulation. Although the torque increased instantly, the duration was for milliseconds. A small amount of abnormal peak torque did not affect the movement of the robot.

As shown in Figure 18d–f (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7055, accessed on 30 August 2023), when the wheel-leg speed was 20 rpm, the driving torque and curling torque of the wheel-leg increased with the increase in the friction coefficient. It was noticed that the robot lifted due to the collision force of the wheel-leg at this speed, and as a result, the torque of one wheel-leg decreased due to insufficient contact with the ground.

As shown in Figure 18g–i (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7056, accessed on 30 August 2023), when the wheel-leg speed was 30 rpm, the driving torque and curling torque of the wheel-leg also increased with the increase in the friction coefficient. The torque of one wheel-leg decreased due to insufficient contact with the ground.

4.2.2. Material: Rubber

In this section, the results of the rubber wheel-leg cover are discussed. As shown in Figure 19a–c (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7057, accessed on 30 August 2023), when the wheel-leg speed was 10 rpm, compared with nylon, the peak torque generated by the collision was smaller, but the torque in operation was larger. This is because rubber is elastic and has a certain cushioning effect on large collision forces, and the torque of the motor increases when the collision force is small. From the comparison, it was determined that when the static friction coefficient was 0.35, and the driving and curling torques of the wheel-leg were the smallest among the three data sets.

As shown in Figure 19d–f (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7058, accessed on 30 August 2023), when the wheel-leg speed was 20 rpm, the driving torque and curling torque of the wheel-leg were the smallest when the static friction coefficient was 0.35, which was the same as when the wheel-leg speed was 10 rpm. However, the motor’s peak torque increased slightly at a speed of 20 rpm.

As shown in Figure 19g–i (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7059, accessed on 30 August 2023), when the wheel-leg speed was 30 rpm, the driving torque and curling torque of the wheel-leg were the smallest when the coefficient of friction was 0.6, which was different from when the speed was 10 rpm and 20 rpm. In addition, the peak torque of the motor due to the collision was further increased.

When a rubber wheel-leg cover is selected, the motor torque required for the rotation of curling joints is uncertain. Therefore, it is necessary to further determine the wheel-leg’s speed in the actual operating environment of the prototype using experiments. In addition, due to the great elasticity of rubber, the driving torque and curling torque of the wheel-leg become larger.

4.2.3. Material: Acrylic

As shown in Figure 20a–c (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7060, accessed on 30 August 2023), when the wheel-leg speed was 10 rpm, the driving torque and curling torque of the wheel-leg increased with the increase in the friction coefficient. The torque became smaller compared with the use of the other two materials.

As shown in Figure 20d–f (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7061, accessed on 30 August 2023), when the wheel-leg speed was 20 rpm, the driving torque of the wheel-leg increased with the increase in the friction coefficient, but the curling torque first increased and then decreased with the increase in the friction coefficient. Due to the large amount of abnormal torque generated by the collision, the possibility of using acrylic as the wheel-leg cover material at a wheel-leg speed of 20 rpm was not considered further.

As shown in Figure 20g–i (high-res figures available at http://me.tju.edu.cn/nbxx_content.action?cla=3&news.id=7062, accessed on 30 August 2023), when the wheel-leg speed was 30rpm, the torque during rolling was greater than that under normal movement, indicating that the force of the rolling collision was greater. Since electronic components such as circuit boards are usually fragile and subject to a certain degree of interference after being launched at high altitude, to protect the electronic components inside the motor from damage, the use of acrylic as the wheel-leg cover material at a wheel-leg speed of 30 rpm was not considered further.

4.2.4. Material Selection

The simulation results were compared and analyzed. The operating torque required for the nylon wheel-leg cover is relatively small and regular, and the peak torque lasts for milliseconds. However, further structural design is required to meet the climbing requirements, which increases the design difficulty and reduces its effectiveness. The peak torque also lasts for milliseconds when rubber is used for the wheel-leg cover, but it easily reaches a large friction coefficient to complete the climbing requirements. Although the operating torque is relatively large, it is within the acceptable range. For the acrylic wheel-leg cover, in addition to the large abnormal torque and long duration, the operating torque value is irregular. Therefore, this material is not favorable. The performance of different wheel-leg cover materials was evaluated in all aspects, as listed in

Table 3. Evaluation of different wheel-leg cover materials.

Material	Operating Torque	Peak Torque	Climbing Capacity
Nylon	☆☆☆	☆☆☆	☆
Rubber	☆☆	☆☆☆	☆☆☆
Acrylic	☆☆	☆	☆☆

.

In summary, with priority given to climbing ability, the rubber wheel-leg cover was finally determined to increase friction based on the comprehensive torque and the motor model.

5. Experiments

After constructing the prototype and selecting wheel-leg cover materials, experiments were conducted to determine the robot’s movement in different terrains. Before the experiments, relevant physical parameters of the robot prototype were measured. The length of the prototype was 110.5 mm, and the width was 84 mm; the height was 58 mm in the curled state and 27.5 mm in the stretched state. The prototype weighed 153 g. The experiments were carried out in terrains that are difficult for traditional mobile robots to cross, such as flat ground with certain ups and downs, gravel obstacles, and slopes. The obstacle-crossing capability of the robot can be determined based on its performance.

5.1. Flatland Experiment

5.1.1. Grassland Experiment

The height of the customized lawn was 12 mm, which is the undulation level of the lawn. The prototype was tested using this lawn undulation. As shown in Figure 21, the robot moved a distance that was 5 times its body length in 10 s. The movement was fast and stable. During this process, the bottom shell of the robot provided adequate protection by preventing motor failure that would have resulted from the tangling of leaves in the gear of the motor.

5.1.2. Gritty Land Experiment

Gritty land is land covered with gravel grains. The undulation level was about 0~18mm, and the surface was soft. The movement of the robot on stony and sandy land in stretched mode is shown in Figure 22 and Figure 23, respectively. The robot’s travel speed was reduced to some extent due to the uneven ground. Nonetheless, the travel speed was still enough to ensure the effective operation of the robot on the gritty land.

The operation of the robot in curled mode on the gritty land is shown in Figure 24. It can be seen that the operation was satisfactory, and the curling was of some help when crossing obstacles. Harsh gritty land is a major challenge for traditional mobile robots, but the proposed wheel-legged robot completes the crossing well.

5.2. Climbing Experiment

As shown in Figure 25, when the slope gradient was 30°, the robot passed it smoothly in 13 s. During the entire climb, the robot did not distort, which suggests that the coefficient of friction between the robot and the ground and the stability of the robot meet the requirements.

As shown in Figure 26, when the slope gradient was 45°, the robot climbed in a relatively smooth manner. However, the robot’s head distorted to a certain extent during climbing. The friction between the wheel-leg and the slope confined the robot’s climbing gradient.

For slope climbing, the robot designed herein demonstrates satisfactory climbing capability on high-gradient slopes while ensuring a certain speed.

5.3. Curling Experiment

The curling and stretching processes of the robot are shown in Figure 27. The curling and stretching processes both occur within 2 s. The curling process of the robot is shown in Figure 27a. The curling of the robot was achieved with the cooperation of slight wheel-leg rotation. The stretching process of the robot is shown in Figure 27b. It can be seen that stretching took less time than curling because the robot is subject to gravity. The motor should overcome gravity during curling, but during stretching, gravity can act as part of the driving force.

The downhill rolling of the robot after recognition and autonomous curling is shown in Figure 28. The robot completed the switch between stretching and curling modes well. It can be seen that at t = 3.0 s, the robot recognized the downhill slope; at t = 3.3 s, the robot started curling; and at t = 4.3 s, the robot completed curling and started rolling. The entire curling process lasted for about a second.

Compared with traditional mobile robots, the robot designed herein rolls downhill after curling. While the safety of the robot is guaranteed, the downhill time is significantly shortened under this scheme, which effectively improves the movement efficiency of the robot and makes up for the poor speed of traditional-legged robots.

5.4. Summary

After the experiments, the operation of the robot was summarized. It can be seen that the robot operates well with relatively satisfactory indicators and can achieve fast and stable crossing on flat ground with a high undulation level. In the case of a steep slope, it can move downhill efficiently and safely using rolling after curling. The experiments verified that the robot is adaptable to multiple terrains with strong obstacle-crossing capability. In terrains that are difficult for traditional mobile robots to cross, such as flat ground with a high undulation level or steep slope, the hybrid wheel-legged mobile robot crosses them safely while ensuring movement speed. The performance of the prototype is shown in

Table 4. Performance of the prototype.

Requirements	Prototype Performance
Grassland	Undulation: 0~0.66R
Gritty land	Undulation: 0~1.06R
Slope angle	45°
Transformation mode	Two modes
Transformation time	Less than 2 s
Operation	Smooth

. The experimental results prove that the robot can be used in rescue, reconnaissance, and detection of complex terrain and complete tasks that are difficult for existing robots.

6. Conclusions

As hybrid mobile robots have the advantages of various mobile structures after combining their characteristics, they attract wide attention from many scholars. A hybrid wheel-legged mobile robot that combines the structural characteristics of wheeled and legged structures is proposed within the scope of hybrid mobile robots. The contents are as follows:

• After statics modeling and analysis, it is determined that the number of spokes of the robot’s wheel-legs is three. Based on the specific step parameters, a geometric analysis is used to determine the chord length of the spokes in the robot’s wheel-leg. The design of the parameters of the wheel-legged structure ensures the satisfactory obstacle-crossing capability of the robot.
• To further expand the application scenarios of the robot, the curling mechanism is selected as the main robot body. Particle swarm optimization is used to determine the number of robot sections n = 5 and the length of each section l = 16 mm. Based on this result, a prototype robot is made taking into account the strength of the components, the lightweight of the robot body, etc., and the materials for each key part are also determined.
• To improve the working conditions of the drive motor and the curling motor, the material of the wheel-leg in contact with the ground, namely, the wheel-leg cover material is selected after dynamic analysis. After considering the climbing ability and values of torque, etc., rubber is used for the wheel-leg cover.
• After prototype production and wheel-leg cover material selection, experiments are carried out on different terrains, including relatively complex terrains such as grassland, gritty land, and slopes. The robot demonstrates good performance.

In conclusion, the proposed wheel-legged mobile robot is adaptable to multiple terrains with strong obstacle-crossing capability, which makes up for the drawbacks of traditional mobile robots in these aspects. To further improve the performance of the robot, future research can be carried out on control strategies and intelligence in the future. This will help give full play to wheel-legged mobile robots’ advantages, enabling them to assist humans when completing tasks in scientific exploration and rescue.