As shown in Figure 3, the kinematic model of the standing posture of the wheel-legged robot is established. {W} is the world coordinate system, the Z axis is vertically upward, the X axis is perpendicular to the Z axis and points to the right end, and the Y axis direction is determined according to the right-hand rule. In order to simplify the kinematics problem, the base coordinate system {B} is established at the contact point between the wheel and the ground, the direction is parallel to the world coordinate system, and the coordinate system {0} is established at the center of the wheel. The coordinate system {1} is established at the connection between the Connecting rod 3 and the wheel, the coordinate system {2} is established at the connection between the Connecting rod 1 and the Connecting rod 3, and the coordinate system is established at the position shown in the Figure 3 in turn. The Z_i(i = 1∼5) axis of the link is along the joint. The positive direction of the axis is placed perpendicular to the surface of the paper, the positive direction of the X_i axis points to the common perpendicular of the i axis and the i+1 axis, and the direction of the Y_i axis is determined by the right-hand rule.

In the kinematic model of Figure 3, a_i–1 is the length of the connecting rod, α_i−1 is the rotation angle of the connecting rod, d_i is the offset distance of the connecting rod, θ_i is the joint angle, l_i is the distance between the origin of the coordinate system {i−1} and the origin of the coordinate system {i} in the Z-X plane. where i = 1∼5 is the rotational joint of the robot, d₁ = 0.026m, d₂ = 0.021m, d₃ = 0.016m, d₅ = 0.016m. l₂ = 0.14m, l₃ = 0.14m, l₄ = 0.09m, l₅ = 0.14m, l₆ = 0.09m, the robot wheel radius r = 0.095m.

According to the established kinematics model, the forward kinematics is solved, and the connecting rod transformation matrix is obtained as:

Where, C₁₂₃₄ is the meaning of cos(θ₁ + θ₂ + θ₃ + θ₄), S₁₂₃₄ is the meaning of sin(θ1+θ2+θ3+θ)4.

The robot is a parallelogram linkage mechanism, and the robot body is expected to be parallel to the ground while maintaining balance, so the constraint equation is attached:

Use UG software to analyze the position of the center of mass of each rod and the body of the robot, and obtain the coordinates of the center of mass c1 of the Connecting rod 3, the center of mass c2 of the Connecting rod 1, the center of mass c3 of the body, and the center of mass c4 of the Connecting rod 2 relative to the X-Y plane of the {1}, {2}, {3}, {4} coordinate system:

The angle between the line connecting the center of mass c2 and the origin of the coordinate system {2} and the positive direction of the X₂ axis is φ₂ = atan2(²Y_c2,²X_c2), the angle between the line of the center of mass c3 and the origin of the coordinate system of {3} and the positive direction of the X₃ axis is φ₃ = atan2(³Y_c3,³X_c3), the angle between the line connecting the center of mass c4 and the origin of the coordinate system of {4} and the positive direction of the X₄ axis is φ₄ = atan2(⁴Y_c4,⁴X_c4), the center of mass of the hip motor c5 is at the center of rotation. Let the lengths of the center of mass c1, c2, c3, and c4 from the origin of the {1}, {2}, {3}, and {4} coordinate systems be l_c1, l_c2, l_c3, and l_c4, respectively. Then the position of the center of mass of each rod ci relative to the world coordinate system is (X_ci, Z_ci), and the velocity is Xc⁢i2.+Zc⁢i2., i = 1∼5.

When the height of the obstacle is less than the radius of the wheel, the analysis is performed at the initial stage of the robot crossing the obstacle. At the beginning of obstacle crossing, keep the center of mass of the robot and the center of rotation of the wheel on the same vertical line, and perform force analysis on it.

Figure 6 shows the robot cross over the obstacle in wheel-leg mode, at the beginning of obstacle crossing, the robot is balanced by the forces and moments between the ground and the wheels and between the obstacles and the wheels. The balance formula of the robot’s two legs over obstacle is shown in Equation (21):

In Equation (21), M_w is the torque of the wheel motor, G is its own gravity, F_N1 is the support force of the ground facing the wheel, F_N2 is the support force of the obstacle contact point to the wheel, f₁ is the friction force of the ground facing the wheel, and f₂ is the frictional force on the wheel at the contact point of the obstacle, α is the angle between the supporting force of the contact point between the wheel and the obstacle and the ground, h is the height between the contact point of the obstacle and the ground, and r is the radius of the wheel.

The balance formula of the robot’s one leg over obstacle is shown in Equation (22):

In Equation (22), M_h and F_h are, respectively, the torque of the hip motor in the adaptive contraction state of the robot when crossing the obstacle with one leg, and the force transmitted by the torque of the hip motor to the wheel through the connecting rod of the wheel leg. The meanings of other parameters are consistent with Equation (21).

In Equations (21, 22), f₁=μF_N1, f₂=μF_N2, α=a⁢r⁢c⁢s⁢i⁢n⁢r-hr. When α > 0, the wheel can advance over the obstacle by its own rotational motion driven by the motor.

In the analysis of the robot jumping over obstacles, due to the symmetric design of the robot, the center of mass in the process of movement can be kept only in the X and Z direction displacement in the world coordinate system, and the jumping process is analyzed according to the plane robot analysis method. Assume that the take-off phase time of the robot is [t₀,t₁) and the flight phase time is [t₁,t₄). If the robot wants to jump off the ground after energy storage, it needs to meet the following conditions:

Therefore, the following conditions should be met in the take-off phase time [t₀,t₁):

When the speed of the robot in the horizontal and vertical directions is greater than zero, it will make an oblique throw motion with only the initial speed in the flight phase. It is assumed that the robot makes uniform linear motion in the horizontal direction before jumping, and the velocity of the center of mass in the horizontal and vertical directions is v_x and v_y, respectively. Then, the robot’s flight time is t=2⁢vyg, the highest jumping height is Hm⁢a⁢x=vy22⁢g, and the horizontal displacement distance is S=2⁢vx⁢vyg.

The obstacle crossing process is shown in Figure 7, it’s required that when the horizontal displacement is S_t2 = D₁, the jump height H_t2 > H_o. When the horizontal displacement is S_t3 = D₁ + D_o, the jump height H_t3 > H_o. Then, the conditions under which the robot can overcome obstacles are:

When considering the robot perform jumps has higher request to the speed and acceleration, vertical direction of the robot center of mass in accordance with the planned five times polynomial interpolation method, the initial and the end of a robot position, velocity and acceleration, respectively, y_init, v_init, a_init and y_end, v_end, a_end, using the method of undetermined coefficients can get desired trajectory:

Among them,

In general, the wheel-leg movement mode of the robot proposed in this paper can be divided into two forms: wheel movement and jumping over obstacles, which are self-balancing mode and jumping mode, respectively. The robot can switch between the two modes to complete the task, and it is necessary to design controllers for the two motion modes, respectively. The balance and speed control of the robot are realized by the Linear Quadratic Regulator (LQR) method. The state equation of the linear time-invariant system is:

For the balance system of the robot, the speed, tilt angle and tilt angle velocity are taken as the state variables, and the state variable is x=[x.b,ϕ,ϕ.]. Then the equilibrium system can be expressed as:

If the robot takes x.r⁢e⁢f as the reference speed and keeps balance, then ϕ.=0, x.=x.r⁢e⁢f, the system stable state variable xs=f⁢(x.r⁢e⁢f), the stability control input us=g⁢(x.r⁢e⁢f), take the new state variable Δx = x−x_s, the new system input Δu = T_ϕ−u_s, then the new system state equation is:

The optimal control input is obtained as follows:

Where, f⁢(x.r⁢e⁢f)=[x.r⁢e⁢fA22⁢B2-A42⁢B1A43⁢B1-A23⁢B20], g⁢(x.r⁢e⁢f)=A42⁢A23-A22⁢A43A43⁢B1-A23⁢B2.

In this paper, the jump control of the robot adopts fuzzy Proportion Differentiation (PD) control. According to the dynamic equation of the jump stage obtained in the dynamic analysis:

The control scheme for calculating torque is set as:

Where, e = q_d−q, e=q.d-q., q_d and q are the ideal angle and the actual angle, respectively.

The fuzzy PD controller is designed to calculate the torque control, e and e. are taken as the input of the fuzzy controller. Mamdan rule is used for fuzzy inference, and min-max-center of gravity method is used for fuzzy resolution. The output is ΔK_p and ΔK_d, let the self-tuning parameters K_p = K_p0 + ΔK_p, K_d = K_d0 + ΔK_d. K_p0 and K_d0 are initial values, KP and KD are final values. The block diagram of the control scheme is shown in Figure 8.

For the dynamic model of the robot in the take-off phase, assuming that the angle tracking command of the robot hip joint is q_d=2sin(πt)rad, the dynamic equation is written as s-function by the Simulink module in Matlab to test the simulation effect. Fuzzy PD control is used to design the control law, and the initial values of PD parameters are set as K_p0=20 and K_d0=20. The simulation results are shown in Figure 9.

It can be seen that the joint angle tracking curve of the fuzzy PD controller is close to the expected tracking angle curve in Figure 9, and has a fast response speed, no obvious overshoot, and has a good control effect.

The 3D model of the robot was imported into Adams simulation software, and the co-simulation of the self-balance and obstacle crossing of the robot is carried out by using Simulink module of Matlab. The hip motor rotates at different angles, which can adjust the height of the robot when standing. First, the simulation of the robot shifting from the wheel mode to the wheel-leg mode is carried out. When the hip motors rotate in wheel-leg mode, the robot switches to wheel movement mode, and then reverts to the original posture through motors reversals. The simulation process is shown in Figure 10A. In order to realize the function of jumping over obstacles, the hip motors need to rotate with different steering to complete the contraction and extension of the wheel leg connecting rod, so as to storage and release the energy needed for jumping. Taking the single jump process of the robot as an example, its screenshot is shown in Figure 10B. When the robot moves at a constant speed of 1 m/s, the robot reaches the highest jump height of 0.16 m at 0.8 s. The take-off phase of the robot is 0.5–0.6 s, the flight phase is 0.6–0.95 s, and the landing phase is 0.95–1.2 s.

In the process of mode switching, the position and velocity change curve of the robot’s center of mass in the vertical direction are shown in Figure 11. It can be seen that the velocity change curve is smooth and continuous, which can ensure the stable mode switching. According to different expected jumping heights, the jumping trajectories of the robot are different. Jump simulation experiments are conducted for the expected heights H_d=0.16m and H_d=0.11m, and the longitudinal trajectories of the bottom of the robot wheel and the robot center of mass are obtained as shown in Figure 12A. It can be seen from Figure 12B, the motion of the robot after take-off is oblique throwing motion, and the planned motion wants to achieve a higher jumping height, so a larger longitudinal velocity is needed at the take-off point to prolong the time of flight. In the take-off phase, the wheel-leg linkage mechanisms successively contract and extend to realize the energy storage and release of the machine mechanisms. When the condition of leaving the ground is reached, the end of the robot’s wheels leave the ground and enter the phase of flight. In the flight phase, the robot system is in the state of momentum conservation, so the method of contracting the wheel-legs to increase the jumping height causes the overall centroid velocity to fluctuate. In order to reduce the impact of landing, a method is used to keep the mechanism retraction until the wheels touch the ground. After the robot lands, the wheel-legs extend back to the position before the jump.

In Figure 13, the curve of ground support reaction force on the wheel in the process of jumping is measured, and it’s found that the support reaction force gradually increases in the take-off phase, reaches the maximum value at 0.6 s at the end of wheel-leg contraction, and decreases rapidly at the moment of take-off. When the wheels leave the ground, the support reaction force is 0. At the moment of landing, the support reaction force will change due to impact. The change of the support reaction force corresponds to the height above the ground in each stage of the robot jumping process, which verifies the feasibility of the jumping.

Figure 14A shows the angle variation of each joint when the jump height is 0.16 m. Since the wheel-leg of the robot is a parallelogram linkage mechanism, q₄ = q₅. Among them, the angles q₂, q₃, q₄, and q₅ represent are the angle between Connecting rod 3 and Connecting rod 1, the angle between Connecting rod 3 and Connecting rod 2 at the hinge point, The angle between Connecting rod 2 and the motor rotation center and the Connecting rod 2 hinge, and the angle between Connecting rod 2 and Connecting rod 3. Robot hip motor torque curves is shown in Figure 14B, it can be seen two hip motors’ output torques are basically consistent, the motors’ torques of the wheel-legs contraction in the take-off phase are shown in the curves from 0.5 to 0.6 s, which are the torques curves of the wheel-legs extension from 0.6 to 0.65 s, and the torque generated near 0.97 s is generated when the robot lands and touches the ground.

When the robot passes over the pavement with pothole (major diameter: 15 cm, minor diameter: 10 cm, depth: 5 cm) as shown in Figure 15, the robot is required to keep moving smoothly. When the left leg of the robot passes over the pothole, the linkage mechanism of the right leg contracts to keep the axes of the motor rotation of the hip joint of the two legs coincide as much as possible. Figure 16 shows the height of the rotation axes of the left and right wheel-leg hip motors in this process. It can be seen that the axes of the two motors basically coincide. At time of 1.2 s, the deviation of the axis of the two legs is relatively large, but as a whole, the deviation is small, only about 2 mm. The robot can smoothly pass through the pothole terrain.

The following simulation experiments are done for the robot to cross the slope obstacle with one leg. The obstacle set by the simulation is about 6.5 cm in height and 65 cm in length. When the robot crosses the obstacle without adaptive contraction wheel-legs, the side view of the obstacle crossing process of the robot is shown in Figure 17. In this case, crossing the obstacle may cause damage to the connecting rod structure of the robot wheel-leg or the motors of the hip joints. Therefore, when the vertical height of the contact between the left and right wheels and the ground is different, the wheel-leg linkage mechanism with relatively high vertical position of the contact point needs to be properly contracted to ensure the stability of the robot body. For this purpose, the one-leg obstacle crossing simulation experiment was carried out, and the whole process of obstacle crossing was captured, as shown in Figure 18.

Figure 19A shows the height variation curve of the rotation axis of the left and right hip joint motors in the vertical direction. It can be seen from the figure that the left leg of the robot starts to cross the obstacle when it contacts the obstacle at about 2 s, and the robot starts to leave the obstacle at about 7.15 s. In the process of obstacle crossing, the highest position deviation of the motor axis of the hip joint of the two legs is generated at 7.9 s, and the maximum deviation between the motor position of the left leg and the motor position of the right leg is about 6.6 mm. It can be seen that in the whole process of obstacle crossing, the position height deviation of the two hip motors is low, and the performance of obstacle crossing is better. And because of gravity, the time for going up and down is different, and the time for going down is less than that for going up. Figure 19B is the robot’s left leg hip motor torque figure, can be seen from the figure when the robot starts uphill because access to the obstacles, will jump a torque value, and in the process of uphill, with the contraction of the wheel-leg, it gradually attenuates to the torque value of maintaining the motor position locking, and also produces a sudden torque when downhill.

On the basis of simulation, in order to verify the feasibility of jumping over obstacles designed in this paper and the correctness of the simulation results, real robot tests are carried out. The experiments are based on the DDT robot platform of Direct Drive Technology Ltd. The total mass of the robot is about 30 kg, the height of the robot in the wheeled mode is about 21 cm, the length of the robot is about 35 cm, and the width is about 53 cm. The height of the robot can be adjusted under the wheel-leg mode. The robot moves in wheel mode and wheel-leg mode at different height of center of mass on flat ground, the motion posture is shown in Figure 20.

The real tests are carried out in different terrain, first of all, the jump height and jump feasibility will be analyzed and verified under flat ground. Control the robot to jump and compare the jump height with the scale of the preset whiteboard (see Figure 21). The energy needed for the robot to jump is stored by the contraction of the leg mechanisms, the energy release is completed by the extension of the leg mechanism and the jump is carried out, the wheel-legs are contracted in the flight phase to improve robot’s height off the ground, and the purpose of jumping over obstacles is realized. After comparing with the scale of the white board, it is found that when the robot jumps at a low speed, the jump height can be 0.16 m. The experimental results are basically consistent with the simulation results, which verifies the feasibility of the jump action of the wheel-legged robot designed in this paper.

The adaptive contraction of the wheel leg is actually tested on the pothole ground. In Figure 16, it can be seen that in the simulation experiment of pothole terrain, the robot passes through the pothole in about 1.2 s, corresponding to the robot state shown in serial number ① and serial number ② of Figure 22. At this time, the robot walks on the road with potholes. The axes of the two hip motors are always in the same vertical position by contracting the wheel-leg which is at a higher position, it ensures the stability of the robot when walking on the pothole ground.

For the test of single-leg obstacle crossing, a single block, two blocks and three blocks of wood are set in front of the right leg as obstacles. The height of a single plank is 2 cm, the height of the corresponding obstacles is 2, 4, and 6 cm, respectively, and the radius of the wheel is 9.5 cm. The obstacle crossing process is shown in Figure 23. The process from contacting the obstacle to adaptively adjusting the expansion of the wheel-leg is shown. It can be seen that the degree of contraction of the wheel-leg is different when the height of the obstacle is different. The robot can easily cross the obstacles with height lower than the radius of its own wheels, and keep moving smoothly through obstacles by adjusting the wheel-leg adaptively.

In addition to jumping on the flat and open ground, jumping over obstacles on the uneven ground is an important embodiment of the robot’s ability to jump over obstacles. The jumping ability was further tested on the uneven grass outside, and the test results in Figure 24 shows that the robot still had good jumping ability on the uneven ground.