Analysis of Motion Characteristics and Stability of Mobile Robot Based on a Transformable Wheel Mechanism

Abstract

In this research, we propose a novel wheel-legged mobile robot to address the problems of insufficient obstacle-crossing performance and poor motion flexibility of mobile robots in non-structural environments. Firstly, we designed the transformable wheel mechanism and tail adaptive mechanism. Secondly, the kinematic model of the robot is established and solved by analyzing the whole motion and wheel-legged switching motion for the operation requirements under different road conditions. By synthesizing the constraint relationships among the modules and analyzing the robot’s obstacle-crossing abilities, we systematically established the mechanical model of the robot when it encounters obstacles. Thirdly, we studied the stability of the robot based on the stable cone method in the case of slope and unilateral transformation wheel deployment and achieved the tipping condition in the critical state. Finally, we used ADAMS software to simulate and analyze the driving process of the robot in various types of terrain and obstacles in order to verify that it has superior performance through obstacles and motion flexibility. The analysis shows that the robot can passively adapt to various complex and variable obstacle-filled terrains with obstacle heights which are much higher than its center of gravity range. The results of the study can provide a reference for the structural optimization and the obstacle-crossing performance improvement of mobile robots.

1. Introduction

Currently, small mobile robots are widely used in unstructured environments. Mobile robots can assist or replace humans when performing actions, such as target detection and safe and efficient search and rescue [1]. Mobile robots with complex terrain adaptation capability is a current research hotspot in the field of mobile robotics [2,3]. To design robots with high terrain access performance and motion flexibility for complex terrain, most scholars and research institutions focus their research on new mobile mechanisms for robots [4]. Common mobility structures are broadly classified as wheeled, tracked, and legged [5]. By modeling and studying the body structure and movement mechanisms of living beings, walking structures for mobile robots, such as wheels, legs and feet, have been designed. It is worth mentioning that although they have been used successfully in some cases, the application of robots in rugged and complex terrain is still limited by obstacle traversal performance and control systems.

The emergence of multi-motion mode robots and transformable wheel structure robots has prompted innovative ideas [6]. The multi-motion mode robot mainly adopts the design method of the wheeled mobile mechanism mounted on the mobile platform of the leg-type robot [7]. Chao Liu designed a new type of mobile robot with transformable wheel legs in Figure 1a [8]. By combining the parallelogram mechanism with the Chebyshev mechanism, a single-loop closed chain 2RP3R metamorphic mechanism that can switch between multiple motion modes of the robot, such as four-legged, eight-legged, and wheeled, has been proposed. It has strong obstacle-crossing performance and anti-toppling ability. Hutter, M., developed a wheeled quadruped robot ANYMAL in Figure 1b [9,10], with excellent flexibility and dynamic movement ability. Using the 3D ZMP method, it can perform a variety of actions to overcome obstacles or stand up after falling. The design concept of transformable wheel legs is terrain adaptability. When moving on a flat surface or obstacle below the height of the wheel radius, the robot is in a round-wheeled movement mode. When the robot encounters an obstacle, the transformable wheel undergoes a structural transformation and the robot changes to wheel-legged movement mode [11]. Based on the walking gait of the cockroach, the “Whegs” series robot [12] adopts an open, three-spoke rotating wheel-leg design, and each spoke can be alternated periodically as support for contact with the ground when walking, showing a strong terrain-passing ability in Figure 1c. Combining the characteristics of the stable movement of insects in nature, Ni, Y. [13], designed a wheel-legged hexapod robot that can pass through complex terrain without any terrain sensing and with an independent, unconstrained, and flexible drive in Figure 1d. Using a unique switching mechanism, Chen, C. [14] designed a novel four-legged/quadruped mobile robot to switch the morphology of wheels and 2-degree-of-freedom legs under the same drive system in Figure 1e. With the help of external friction between the wheels and the ground, Kim S [15,16] designed a new wheel–leg hybrid mobile robot with a new triggering mechanism to improve the transition success rate, and the robot is able to climb over obstacles of 3.25 times its wheel radius in Figure 1f. Using a sea turtle as a bionic object, Rui, H. [17] designed a wheel–track–leg composite bionic robot and analyzed its gait planning and obstacle-crossing performance in Figure 1g. Combined with the passive deformation principle, She. Y., and his colleagues developed a robot that determines its mode of operation through physical contact force in Figure 1h [18].

The above literature study demonstrates that different levels of research have been conducted on mobile robots’ ability to cross obstacles in unstructured terrain. Both multi-motion hybrid robots and transformable wheeled robots have increased the adaptability and obstacle-crossing capability of robots in complex unstructured environments to some extent, but at the same time, they have also caused problems, such as low motion efficiency and poor motion stability [19]. This paper designs a transformable wheel-legged mobile robot for application in unstructured environments. Under the action of the transformation mechanism, the wheels are changed from wheel mode to wheel-legged mode. It has terrain self-adaptation and obstacle-crossing capabilities. Compared to conventional mobile robots, the overall structure of the robot is simple, compact, and easy to install. The motion modules of the robot are independent of each other. Transformable wheel forces of the three legs are less constrained and more suitable for the analysis of motion characteristics and stability. It is able to meet the needs of working in different obstacle-filled environments, such as steps, ground of variable curvature, and slopes.

2. Structure and Model

2.1. Robot Concept Design

The whole design of the robot is shown in Figure 2.

The robot adopts a symmetrical structure, which mainly consists of three parts: a drive module, passive transformation wheels, and an adaptive tail mechanism. The drive module consists of the robot shell, internal controller, power supply, and gear motor. It is connected with two transformable independent drive wheels of identical dimensions to enable the robot’s movement and obstacle-crossing through the force between the ground and the obstacles. The transformable wheel can change from using a wheel motion to a wheel-legged motion. When moving on flat ground or ground with less curvature, the wheel shape remains as a complete geometric circle, and the robot is in wheeled motion mode. When encountering obstacle terrain that cannot be passed, the transformable wheels enter the passive switching state of wheel-leg, using the action of friction to drive the three curved legs to passively unfold, and the robot is then in wheel-legged motion mode. The adaptive tail mechanism adopts a segmented multi-stage design. The tail and drive module is fixed to the rear plate using pins. The robot tail section can be swung up and down in response to fluctuations in the terrain or obstacles and acts as a soft connection and auxiliary support when turning or crossing obstacles.

2.2. Transformable Wheel Design

The design of the transformable wheel is illustrated in Figure 3. It mainly consists of trigger legs, wheel hub, permanent magnets, resetting springs, limit screws, a triradial link, spindle, and extension springs. The triradial link and the spindle are fixed to each other. The resetting spring is connected to the wheel hub. The extension spring connects the triradial link to the trigger leg. The wheel hub is equipped with permanent magnets and a limiting screw, which are used to restrict the trigger leg from spreading too far when the wheel is transformed. The permanent magnets are used to attach the trigger leg to the wheel hub and prevent the trigger leg from spreading and transforming when the wheel is in the wheeled position. The transformation process is carried out using a motor that causes the wheel hubs to rotate relative to each other and the resetting spring is compressed. The triradial link drives the trigger leg to swing by the action of the extension spring, and the trigger leg touches the limit screw to stop opening. When the external torque of the motor is removed, the resetting spring returns to its reset state and the trigger leg returns to its original position, enabling the wheel to be closed.

2.3. Adaptive Tail Structure Design

This part mainly consists of a connecting rod, universal coupling, retaining ring, drive shaft, and wheels. The universal coupling completes the connection between the box and the tail section using a connection with the pin and the first-stage drive shaft and has a large angular compensation and cushioning shock absorption capacity. The wheels are connected to each stage by a set of connecting rods. A retaining ring is used to prevent the linkages from moving from side to side, thus ensuring stability during movement. The tail wheels are staggered in decreasing steps to ensure sufficient space between the connecting rods and to make the adaptive tail mechanism more flexible. The sketch of the tail mechanism is shown in Figure 4. The multi-stage mechanism can be used to reduce the impact of the external environment on the robot and prevent damage to the robot when it encounters small steps or obstacles with little change in curvature. After the robot has crossed the obstacle, due to the up and down movement and rotation characteristics, the degrees of freedom of the mechanism are not constrained. The tail wheel is always in contact with the ground due to gravity, and the motion attitude will be automatically adjusted to return the robot to the most reasonable position.

3. Robot Motion Characteristics Analysis

3.1. Kinematic Model Analysis

3.1.1. Motion Analysis of the Entire Robot Mode

The complete motion of the robot is studied in a non-ideal state where the geometric center does not coincide with the center of mass in the wheeled movement mode. We assume that there is no relative sliding between the ground and the wheel and that the transformable wheel is in rigid contact with the ground without structural transformation. Figure 5 shows the kinematic model of the robot in the global coordinate system.

We define q = (x_D, y_D, θ)^T as the robot’s pose, (x_C, y_C) as the coordinate of the robot geometric center C in the global coordinate system, (x_D, y_D) as the coordinate of the robot center of mass D in the global coordinate system, $\stackrel{·}{x}$_r and $\stackrel{·}{y}$_r as the two components of the velocity in the local coordinate system, θ as the directional angle of the robot, d as the distance between the geometric center C and the center of mass D, R as the radius of the wheel, a as the width of the body, and $w_l$ and $w_r$ as the rotational speeds of the left and right wheels of the wheel [20,21].

Assuming that the robot velocity is $v$ and the robot component on O_xy is

$$\{\begin{array}{l}\stackrel{·}{x_C}=v\cos \theta \\ \stackrel{·}{y_C}=v\sin \theta \end{array}$$

(1)

then according to the geometric relationship, the positions of the geometric center C of the robot and the center of mass D are

$$\{\begin{array}{l}{x}_D=\stackrel{}{x_C}+d\cos \theta \\ y_D=\stackrel{}{y_C}+d\sin \theta \end{array}$$

(2)

Derivative of Equation (2):

$$\{\begin{array}{l}\stackrel{·}{x_D}=\stackrel{·}{x_C}-d\stackrel{·}{\theta }\sin \theta \\ \stackrel{·}{y_D}=\stackrel{·}{y_C}+d\stackrel{·}{\theta }\cos \theta \end{array}$$

(3)

The robot is then subjected to the non-holonomic constraint equation, which is obtained as:

$$\stackrel{·}{x_D}\sin \theta -\stackrel{·}{y_D}\cos \theta +d\stackrel{·}{\theta }=0$$

(4)

where $v=\frac{R}{2}(w_l+w_r)$ and $\stackrel{·}{\theta }=\frac{w_{r}R-w_{l}R}{a}$.

According to the robot motion conditions, we obtain the kinematic equation of the robot in its no-slip condition:

$$\left[\begin{array}{c}\stackrel{·}{x_D}\\ \stackrel{·}{y_D}\\ \stackrel{·}{\theta }\end{array}\right]=\left[\begin{array}{cc}\frac{R}{2}\cos \theta -\frac{Rd}{a}\sin \theta & \frac{R}{2}\cos \theta -\frac{Rd}{a}\sin \theta \\ \frac{R}{2}\sin \theta +\frac{Rd}{a}\cos \theta & \frac{R}{2}\sin \theta -\frac{Rd}{a}\cos \theta \\ \frac{R}{a}& -\frac{R}{a}\end{array}\right]\left[\begin{array}{c}{w}_r\\ w_l\end{array}\right]$$

(5)

3.1.2. Motion Analysis of the Robot Wheel-Legged Mode

Figure 6 shows the switching process of the transformed wheel from wheel to wheel-legged motion.

Without considering the slip, the robot achieves three cycles of 120° phase motion for each week of the rotation of the transformed wheel, as shown in Figure 7. At the beginning of one cycle, trigger leg 1 and trigger leg 3 directly touch the ground. As the motor rotates, trigger leg 1 is gradually lifted and trigger leg 3 begins to make a circular motion at its contact point with the ground. Until trigger leg 3 completes the change from the front leg to the rear leg of the wheel. At this point, trigger leg 3 makes a circular motion along the ground. Trigger leg 2 makes contact with the ground and becomes the front leg of the wheel [22].

Figure 8 shows the geometric analysis model of the wheel-legged mode. We define the transformed wheel motion process as T₀ = kT/3, T₁ = kT/3 + t₁, T₂ = (k + 1)T/3, where O₁ is the circle center of the transformed wheel, O₂ is the circle center of the circular arc segment of the wheel trigger leg, point B is the outermost endpoint of the wheel trigger leg 1, point C is the contact point between the wheel trigger leg 1 and the ground, and point D is the location point where the wheel trigger leg 2 is connected to the hub. The distance between O₁ and point B is the effective radius r_w of the wheel, and the distance between point O₁ and point O₂ is L₁, angleO₂BO₁ = α₁, angleO₁O₂P = α₂, angleO₂O₁B = α₃, and angleO₁O₂E = γ.

According to the geometric relationship in the figure, when t = T₀, the relationship between the transformed wheel centers S_t=0 and H_t=0 is

$$\{\begin{array}{c}{S}_{t=0}=0\\ H_{t=0}=R+L_1\cos \gamma \end{array}$$

(6)

where:

$${\phi }_{t=0}=\arcsin (H_{t=0}/r_w)$$

(7)

To solve H_t=0, it can be inferred that

$$\{\begin{array}{c}{L}_1=\sqrt{R^2+r_w{}^2-2r_{\mathrm{w}}\times R\times \cos {\alpha }_1}\\ \begin{array}{l}{\alpha }_1=\frac{\pi }{6}-\arccos [\frac{r_w{}^2+{(\sqrt{3}R)}^2-O_1{D}^2}{2\sqrt{3}R\times r_w}]\\ O_{1}D=\sqrt{L_1{}^2+R^2-2L_1\times R\times \cos {\alpha }_2}\\ {\alpha }_3=\arccos (\frac{r_w{}^2+L_1{}^2-R^2}{2L_1\times r_w})\end{array}\end{array}$$

(8)

The relationship between the height of the transformed wheel center H_t=0, as a function of the effective radius $r_w$ can be expressed as

$$H_{t=0}=H(r_w)$$

(9)

When t = T₁, trigger leg 3’s outer arc circle O₂′ and the ground contact point N of the line perpendicular to the ground, at the time T₁, can be expressed as

$$T_1=\frac{kT}{3}+t_1=\frac{kT}{3}+\frac{\pi /2-{\phi }_{t=0}+\angle {O_2}^{\prime }N{O_1}^{\prime }}{w}$$

(10)

According to the symmetric distribution of the transformed wheel, the relationship between $S_L$ and $H_L$ concerning $r_w$ and $w$ can be expressed as

$$H_L=\{\begin{array}{ll}{r}_w\times \sin [{\phi }_{t=T_0}+w(t-T_0)]& T_0\le t\le T_1\\ R+L_1\cos [{\alpha }_3+{\phi }_{t=T_1}-\frac{\mathsf{\Pi }}{2}+w(t-T_1)]& T_1\le t\le T_2\end{array}$$

(11)

$$S_L=\{\begin{array}{ll}{r}_w\left\{\cos {\phi }_{t=T_0}-\cos [{\phi }_{t=T_0}+w(t-T_0)]\right\}& T_0\le t\le T_1\\ S_{t=T_1}+wR(t-T_1)+L_1\sin [{\alpha }_3+{\phi }_{t=T_1}-\frac{\mathsf{\Pi }}{2}+w(t-T_1)]& T_1\le t\le T_2\end{array}$$

(12)

Figure 9 shows wheel center height and forward distance as a function of time. The mathematical simulation can prove that the motion law of the robot wheel center conforms to the expected variation. When the angular velocity and radius of wheel rotation are constants, the wheel center height and wheel advance distance are determined by the effective radius of the trigger leg spread. The different degrees of transformation of the transformed wheel affect the other structural parameters of the wheel, and the φ angle at the beginning of each motion cycle also affects the motion characteristics of the wheel.

3.2. Mechanical Analysis of the Process through an Obstacle

Figure 10 is a schematic diagram of the robot’s movement across a vertical obstacles of 160 mm heights. When the robot touches the vertical obstacle, the wheel reaches the position of transformation trigger from its general position and the wheel trigger leg and the contact points A and B form an effective support. As the motor continues to rotate, the transformed wheel continues to move in wheel-legged mode until one of the trigger legs moves over the obstacle and touches the ground. Under the influence of gravity and driving force, the transformed wheel makes a circular motion around contact point C, indirectly causing the whole machine to move up the height of the center of mass. With the action of traction, the adaptive tail mechanism maintains tangency with the contact surface of the obstacle until it steadily crosses the obstacle. Finally, the transformed wheel reverts from wheel-legged mode to wheeled mode by gravity and continues to advance, completing the pass of the vertical obstacle.

Based on the geometric conditions of the transformed wheel and the robot model, the key processes of crossing obstacles are mechanically analyzed, and the mathematical symbols and meanings used in the analysis are described in

Table 1. Parameters and meaning of each parameter in the process of crossing the obstacle.

Symbol	Meaning
li (i = 1, 2, 3)	The length between the center of the transformed wheel and the center of the tail wheel 1, the center of the tail wheel 1 and the center of the tail wheel 2, and the center of the tail wheel 2 and the center of the tail wheel 3.
li (i = 4, 5, 6)	The length between the trigger leg and the contact points A, B, and C.
ri (i = 1, 2, 3)	The radius of tail wheels 1, 2, and 3
h	Vertical obstacle height
ϴi (i = 1, 2, 3, 4)	The angle between the line segment li (i = 1, 2, 3, 4) and the horizontal direction of the ground
ϴ5	The acute angle between contact B and the center of rotation of the transformed wheel in the horizontal direction
ϴ6	The acute angle between point C and the line connecting the center of the transformed wheel in the vertical direction
fni (i = 1, 3)	The pressure generated by points A and C with the ground or obstacle
fsi (i = 1, 2, 3)	The friction generated by points A, B, and C and the obstacle or the ground
fni (i = 4, 5, 6)	The pressure generated by the contact point between the tail wheel and the ground
fsi (i = 4, 5, 6)	The friction generated by the contact point between the tail wheel and the ground
Gi (i = 1, 2, 3, 4)	The gravity of the transformed wheel and tail wheel

.

Figure 11 shows the mechanical analysis of the robot moving through the vertical obstacle 1 stage, taking the point O₁ as the reference point, and the following relationship is obtained.

$$\begin{array}{ll}{M}_1=& l_1\cos {\theta }_2\left({\displaystyle \sum _{i=4}^6{f}_{ni}-}{\displaystyle \sum _{i=2}^4{G}_i}\right)+(r_1+l_1\sin {\theta }_2){\displaystyle \sum _{i=4}^6{f}_{ni}}+l_2\cos {\theta }_3\left({\displaystyle \sum _{i=3}^6{f}_{ni}-}{\displaystyle \sum _{i=3}^4{G}_i}\right)+l_3\cos {\theta }_4\left(f_{n6}-G_4\right)\\ & +l_4\left(f_{\mathrm{s}1}\sin {\theta }_1-f_{\mathrm{n}1}\cos {\theta }_1\right)+l_5\sin {\theta }_5{f}_{\mathrm{n}2}-l_6\left(f_{\mathrm{s}3}\sin {\theta }_6+f_{\mathrm{n}3}\cos {\theta }_6\right)\end{array}$$

(13)

When the robot is in the position shown in Figure 12a, the following relationship is obtained:

$$\begin{array}{ll}{M}_2=& l_1\cos {\theta }_2\left({\displaystyle \sum _{i=5}^6{f}_{ni}-}{\displaystyle \sum _{i=2}^4{G}_i}\right)+l_2\cos {\theta }_3\left({\displaystyle \sum _{i=5}^6{f}_{ni}-}{\displaystyle \sum _{i=3}^4{G}_i}\right)+l_3\cos {\theta }_4\left(f_{n6}-G_4\right)-l_4(f_{\mathrm{s}1}\sin {\theta }_1\\ & +f_{\mathrm{n}1}\cos \theta )-(\mathrm{h}+l_4\sin {\theta }_1)\left(f_{\mathrm{s}5}-f_{\mathrm{s}6}\right)\end{array}$$

(14)

When the robot is in the position shown Figure 12b, the tail wheels are raised and pass the obstacle under the traction force, and the following relationship can be obtained:

$$\begin{array}{ll}{M}_3=& l_1\cos {\theta }_2\left(f_{n6}-{\displaystyle \sum _{i=2}^4{G}_i}\right)++l_2\cos {\theta }_3\left(f_{n6}-{\displaystyle \sum _{i=3}^4{G}_i}\right)+l_3\cos {\theta }_4\left(f_{n6}-G_4\right)-f_{\mathrm{s}6}(h+{\mathrm{l}}_4\sin {\theta }_1)\\ & -l_4\left(f_{\mathrm{s}1}\sin {\theta }_1+f_{\mathrm{n}1}\cos {\theta }_1\right)\end{array}$$

(15)

By analyzing the forces on the robot throughout its movement, increasing the magnitude of the support force at the contact point can effectively improve the success rate of the robot moving past the vertical obstacle. The crossing performance is positively related to the wheel radius, trigger leg length, and tail length. The minimum output torque M_min of the drive module is greater than the torque at each stage of passing the obstacle.

$$M_{min}=\{M_1,M_2,M_3\}$$

(16)

4. Robot Stability Analysis

4.1. Slope Stability Analysis

In unstructured environments, the theoretical analysis of the robot’s stability helps the robot to pass smoothly across different types of terrain and avoid tipping phenomena that can further lead to loss of control of the system and damage to the components [23,24]. Figure 13 shows the stability model of the robot ramp motion based on the stable cone approach.

We establish a right-angle coordinate system with the center of mass of the robot as the coordinate origin. The forward direction of the robot is the positive Y-axis direction, the right side of the forward direction is the positive X-axis direction, and the height direction is the positive Z-axis direction. We define the coordinates of contact point N_i as {x_i, y_i, z_i}, the vector as h_i = {x₀, y₀, z₀}, and the gravity vector as f_g = (0, 0, −1). According to the principle of stable cone analysis, the relationship between stability angle ${\epsilon }_i$ and ${{\epsilon }_i}^{\prime }$ under the conditions of sideline tipping and angular tipping can be introduced.

$$\{\begin{array}{l}{\epsilon }_i={\sigma }_i{\cos }^{-1}\left({\widehat{f}}_g\cdot {\widehat{h}}_i\right)={\sigma }_i\arccos \frac{-z_0}{\sqrt{x_0{}^2+y_0{}^2+z_0{}^2}}(i=1,2\dots 8)\\ {{\epsilon }_i}^{\prime }={\xi }_i{\cos }^{-1}\left({\widehat{f}}_g\cdot {\widehat{q}}_i\right)={\xi }_i\arccos \frac{-z_i}{\sqrt{x_i{}^2+y_i{}^2+z_i{}^2}}(i=1,2\dots 8)\end{array}$$

(17)

${\epsilon }_i$ and ${{\epsilon }_i}^{\prime }$ take values in the range [−π, π]. ${\sigma }_i$ and ${\xi }_i$ are used to determine the positive or negative stability angle, and the unit vector defined can be expressed as

$${\sigma }_i=\{\begin{array}{cc}+1\hfill & \left({\widehat{f}}_g\times {\widehat{h}}_i\right)\cdot {\widehat{a}}_i<0\\ -1\hfill & otherwise\end{array}$$

(18)

$${\xi }_i=\{\begin{array}{ll}+1& \left({\widehat{f}}_g\times {\widehat{h}}_i\right)\cdot {\widehat{a}}_i<0\mathrm{or}\left({\widehat{f}}_g\times {\widehat{h}}_{i+1}\right)\cdot {\widehat{a}}_{i+1}<0\\ -1& \hspace{1em}\hspace{1em}\hspace{1em}otherwise\end{array}$$

(19)

Figure 14a shows the stable cone model for the lateral motion of the robot on the slope. When point O′ coincides with point N₃, the robot is in a critical angle tipping state. Figure 14b shows the stable cone model for the longitudinal motion of the robot on the slope, where f_g gradually points to point N₈ as the slope increases. When O′ coincides with point N₈, the robot is in a critical angle tipping state. When the structural parameters of the robots are certain, the stability of the robot can be improved by increasing the coefficient of friction between the robot and the ground and by reducing the height of the robot’s center of gravity.

4.2. Stability Analysis of Unilateral Unfolding of Transformed Wheels

According to the previous analysis, ${\epsilon }_{min}$ should be greater than 0. The larger ${\epsilon }_{min}$ is, the more stable the robot is. When crossing a larger unilateral obstacle, the transformed wheel is passively transformed at the moment of contacting the obstacle, at this time the transformed wheel A moves in the wheeled mode, and the transformed wheel B moves in the wheel-legged mode. The state of crossing the obstacle is shown in Figure 15, where one side of the transformed wheel is subjected to a larger load acting on the edge, and the height of the center of gravity changes, causing the body to tilt, and the robot may move with an unstable motion [25,26,27].

The height of the transformed wheel centers on the upper and lower sides are H_a and H_b, and Δδ is the cross-roll angle of the body. Δθ is the adjustment angle of the robot relative to the slope, and a_s is the equivalent slope of the ramp. The following equations are derived:

$$\{\begin{array}{l}\Delta \delta =a_s-\Delta \theta \\ a_s=\arcsin \frac{\Delta h\cos \Delta \delta }{a}\\ \Delta \theta =\arctan \frac{H_b-H_a}{a}\end{array}$$

(20)

The wheel center height H_a is varied with the side wheel spread radius r_wa, H_b is varied with the lower side wheel spread radius r_wb, and the robot’s body cross-roll angle is

$$\Delta \delta =a_s-\arctan \frac{H(r_{wb})-H(r_{wa})}{a}$$

(21)

The following equations are derived:

$$\{\begin{array}{l}{\epsilon }_1=\arctan \frac{a}{H(r_{wb})}-\Delta \delta \\ {\epsilon }_2=\arctan \frac{a}{H(r_{wa})}+\Delta \delta \end{array}$$

(22)

When $w$ = 1 rad/s and r_wa = 75 mm, the fluctuation of the robot’s body traverse angle during obstacle crossing is solved through the stability criterion. Figure 16a shows the relationship between the height of the obstacle or the inclined surface and the effective radius of the transformation wheel. Figure 16b shows the relationship between the maximum traverse angle of the robot, the effective radius of the transformation wheel, and the stability criterion.

The following conclusions were obtained.

• When the terrain height difference is small, the excessive wheel diameter difference affects the significant increase of the fuselage cross-roll angle, the stability criterion decreases, and the fuselage tilts backward.
• When the terrain height difference is greater than 100 mm, the fluctuation of the fuselage cross-roll angle decreases with the increase of the wheel diameter difference. The cross-roll angle decreases by more than 25% in the wheel–leg maximum spread mode and the wheeled mode.
• Due to the small change in wheel diameter of the lower transformed wheels, the effect of fuselage adjustment affected by the radius difference cannot offset the decrease in the stability angle caused by the increase in its center of gravity. There is a phenomenon where the stability angle first decreases and then increases as the effective radius of the wheels becomes larger.
• When the terrain height difference is 185 mm, the stability angle appears less than 0, and the fuselage tilts, increasing the possibility of tipping.

5. Simulation Experiment

5.1. Slope and Variable Curvature Pavement Simulation Experiment

Slopes and variable curvature road surfaces are common obstacle terrains affecting robot motion. Slope and terrain curvature radius are the main parameters that affect robots when crossing obstacles. Figure 17 shows the simulation of robot motion for the slope and the variable curvature pavement. The simulation of the robot is available at https://ibb.co/gDppV99 (accessed on 31 October 2022) and https://ibb.co/VYmc59W (accessed on 31 October 2022).

Figure 18a shows the displacement and the velocity variation curves of the center of mass during the robot’s ramp movement. The center of mass undergoes a brief sudden change in velocity under the action of traction during the two time periods when the robot initially touches the ramp and then passes over the ramp. It remains smooth throughout the slope movement. Figure 18b shows the acceleration the change curve over the slope, the maximum change value in the z-direction is 97 mm/s², and the robot remains stable on the ground without tipping or backing up.

Figure 19a shows the displacement and velocity variation curves of the robot’s centroid during the motion on the variable curvature surface. The robot centroid height changes with the change of slope. The trajectory of robot motion is basically the same as on the terrain of the variable curvature road. Figure 19b shows the robot centroid acceleration variation curve. During the advance, as the radius of curvature becomes larger, the robot centroid acceleration also increases. When encountering continuous uphill and downhill terrain, the robot’s motion is more supple, which proves that the robot has terrain adaptive ability and motion flexibility.

5.2. Unilateral Steps and Vertical Obstacle Terrain

Typical obstacles, such as unilateral bosses and vertical terrain, can reflect the obstacle, overcoming the performance of the robot. Unilateral steps are obstacles that the robot can overcome in its wheeled state and where the height of the step does not trigger the passive transformation of the transformed wheel. Vertical obstacles are generally present on road surfaces that are difficult to cross. The robot needs to passively switch to a wheel-legged state in order to pass. Figure 20 shows the motion simulation of a unilateral step with vertical terrain. The simulation of the robot is available at https://ibb.co/F4vq1zX (accessed on 31 October 2022) and https://ibb.co/TY1c0Tw (accessed on 31 October 2022).

Figure 21 shows the variation curve of the center of mass displacement versus velocity during the unilateral step of the robot. From 1.2 s to 5.4 s, the z-directional displacement and velocity of the center of mass fluctuate as the robot passes over the unilateral step. The speed of the robot tends to level off when the single wheel falls on the step and then continues to move steadily after the whole robot has completely passed the step.

Figure 22 shows the variation curve of displacement versus the acceleration of the robot over vertical terrain. The simulation results demonstrate that the robot can effectively and quickly move over a vertical obstacle of 150 mm height within 0 to 10 s. Due to the traction force (green line part), the robot’s Z-directional acceleration of the center of mass increases during the overturning process until it passes the obstacle, and then the acceleration gradually decreases. After the whole machine completely passed the obstacle, the robot recovered smoothly and no overturning phenomenon occurred in the process, which is the motion law of the overturning process. The black arrow represents the movement trend and change process of the robot’s center of mass. This proved that the robot has the flexibility to move past the obstacle and has motion stability.

6. Specification and Conclusions

Based on the wheel-legged transformable mechanism and the control system, this robot can adapt to the working environment by changing the diameter and morphology of the wheel-legs. The essential parameters of the robot are summarized in

Table 2. Robot specifications.

	Parameters	Numerical Value
Robot	Max overall dimension under wheeled mode	600 × 400 × 150 mm
	Max overall dimension under wheel-legged mode	600 × 400 × 255 mm
	Weight	5 kg
	Minimum climbing angle	15°
	Minimum height through vertical barriers	160 mm
	Mode switching time	<3 s
Transformable wheel	Wheel radius (unfolded)	170 mm
	Wheel radius (folded)	75 mm
	Transformation ratio	2.405
	Material	Aluminum
Actuator	Motion Actuators Type	DC motors
	Power	Li-Po battery
	Main controller	Arduino Mega

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In summary, this paper proposes a structured design and simulation experimental result for a new transformable wheel-legged mobile robot. According to the design, analysis, and experiments, the conclusions can be drawn as follows:

• The robot can switch between wheeled and wheel-legged modes by changing its wheels to address the problems of inadequate obstacle-crossing performance and inflexible movement of mobile robots in unstructured environments. The robot has fewer degrees of freedom than traditional wheel-legged robots, is easier to control, and can adapt to terrain and cross obstacles softly.
• A kinematic model of the robot was developed to analyze the trajectory of the transformed wheel. The relationship between the radius of the transformed wheel and time concerning z-height and forward displacement was obtained. At the same time, the mechanics model when passing vertical obstacle was established, and the ability of the robot to pass obstacles in wheel-legged mode was analyzed by combining force and dimensional constraints.
• A stab cone approach was used to study the stability conditions of robots in different terrains, based on the need for stable robot movement and the supple crossing of obstacles. Robot tipping conditions in the unilateral wheel-legged spread state were analyzed.
• Using ADAMS software, the robot was simulated as moving across slopes, complex terrain, vertical obstacles, and unilateral steps. The experimental results are in good agreement with the theoretical scores, verifying the correctness of the kinematic analysis of the wheeled and wheel-legged movement mode and the rationality of the robot solution design.
• This new wheeled-legged mobile robot has good survivability in the face of unstructured environments. As a mobile platform for carrying other functional modules, this robot will have a wide range of applications in areas such as counter-terrorism, detection, and reconnaissance.