2.1. Mechanical design

The overall structure: The mechanical system includes a base, a two-wheeled legged mechanism, a spherical rolling component, a clutch, a housing, and electrical components, as shown in Figure 1. The base provides support for mounting components such as motors and controllers. The wheel-legged mechanism is a dual motor-driven parallel mechanism for the wheel-legged motion. The spherical rolling component consists of two meshing gears and a heavy pendulum. The clutch facilitates the multiplexing of the wheel motor drive and the sphere drive. The robot has a spherical housing. Electrical components mainly include drivers, controllers, and sensors.

Figure 1. Overall mechanical structure.

The wheel-legged structure: Each leg can be extended and retracted by a pair of motors installed on the spherical housing. As shown in Figure 2. The height of the body can be adjusted by extending the legs. The leg mechanism scheme is a five-link structure. The two leg drive motors are connected to the two upper links, respectively, and the wheel foot motors are connected to the two lower links. By controlling the two motors of the legs, the legs move in the direction of the center of mass of the system.

Figure 2. The wheel-legged structure.

The spherical structure: In the wheel-legged form, the robot’s legs can be retracted through the leg drive motor to switch to the ball form. While retracting the legs, the thigh can close the lower side protective cover by driving the protective cover link, so that the robot forms a complete and sealed spherical shape. As shown in Figure 3. Morphological transformation requires minimal clearances and high hub torque to provide smooth stability and counteract system noise interference. The protrusion on the wheelbase features a notch with guiding apertures located around its periphery. During the transformation from the wheelbase configuration to the spherical structure, the protrusion on the wheelbase applies pressure to the boundary plate of the boundary structure when the leg structure is folded. Additionally, the protruding notches form a secure connection with the I-beam drive shaft. The guiding edges of the grooves ensure smooth engagement between the protruding grooves and the drive shaft. To achieve this, we have customized a wheel assembly with frameless hub motors. This direct-drive configuration allows for almost gap-free motion and features a highly compact structure.

Figure 3. Spherical structure.

2.2. Hardware

Robot jumping motion needs to be able to provide large torque. Different motor types are used for leg motors and wheel motors. The overall electrical control scheme is shown in Figure 4.

Figure 4. Hardware design.

The leg motor uses the A1 motor, which can provide 33.5Nm of instantaneous torque and has built-in position and torque control. The wheel motor uses Ak60-6 motor with an instantaneous torque of 9Nm. Each wheel motor is equipped with an encoder for precise position and speed feedback. The four motors in the legs communicate with the controller through Universal Asynchronous Receiver/Transmitter (UART), and the wheel motors communicate with the controller through Controller Area Network (CAN) bus. As the main processing unit, Intel NUC using i7 processor is used as the main processing unit to control the system. The system is also equipped with an inertial measurement unit (IMU) and a microcontroller to allow communication between the IMU and computer. In order to power all motors, a lithium battery with a battery capacity of 5500mAh, a discharge rate of 75C, and a rated voltage of 22.2V is used.

2.3. Software

In order to implement a real-time controller, the control-related software needs to be computationally efficient, and all software is written in $\text{C}\texttt{++}$ . In addition, the Robot Operating System (ROS) framework is adopted for communication.

The controller module [Reference Di Vito, Di Lillo, Arrichiello, Ferone, Amico and Antonelli26] includes horizontal, jump, roll controls, and height position controllers. Thanks to optimized leg geometry, jumping, rolling, and traversing are decoupled and therefore controlled independently. The horizontal and height controllers calculate and send torque commands to drive the wheel hub motor. Similarly, the jump and roll controllers provide control of the leg and wheel motors.

The overall control architecture is shown in Figure 5, which illustrates the relationship between sensors, controllers, and ball wheel robots. It consists of two main parts. First, the terminal state is calculated based on the set horizontal speed, jump height, and other parameters. Then, secondly, the controller is designed to track the desired trajectory. The impedance tracking controller is used for jumping motion, the LQR controller is used for horizontal and vertical motion, and the ball is controlled by PD controller.

Figure 5. Software design.

3.1. Dynamics modeling of two-wheeled mode

A variety of modeling methods for two-wheeled robots are mentioned in the literature [Reference Chan, Stol and Halkyard27]. The wheeled motion of the wheel-legged robot can be equivalently simplified to a two-wheeled inverted pendulum model with variable pendulum length [Reference Wu and Zhang28], as shown in Figure 6. The main model parameters are shown in Table 1.

Figure 6. The simplified model of two-wheeled inverted pendulum robot.

Table I. Main model parameters of sphere-wheel-legged robot.

The leg length can be independently adjusted to handle obstacles on various terrains. Since the leg length adjustment is not highly dynamic in real-time and is performed independently before jumping, it is not coupled with the jumping motion. As a result, our dynamics modeling does not account for changes in leg length, thereby simplifying the derivation and control of the dynamics. Experimental results demonstrate that this approach meets the requirements for dynamic control.

Dynamic analysis on the inverted pendulum model is performed by the Euler-Lagrange equation system. Mathematical model is expressed by Lagrange’s formula. The kinetic energy of the robot’s planar motion consists of translational kinetic energy and rotational kinetic energy. It can be expressed as:

(1) \begin{align} T_{\textit{trans}}=\frac{1}{2}m_{w}{v_{l}}^{2}+\frac{1}{2}m_{w}{v_{r}}^{2}+\frac{1}{2}m_{B}{v_{B}}^{2} \end{align}

(2) \begin{align} T_{\mathrm{rot}}=\frac{1}{2}J_{w}{\dot{\alpha }_{l}}^{2}+\frac{1}{2}J_{w}{\dot{\alpha }_{r}}^{2}+\frac{1}{2}J_{B}{\theta _{B}}^{2} \end{align}

where v _r and v _l denote the linear velocities of the right and left wheels, and J _w and J _B are the moments of inertia of the motor rotor about its axis and diameter, respectively.

Assume that the wheel is always in contact with the ground, so the potential energy of the wheel remains zero. The potential energy of the robot is expressed as:

(3) \begin{align} V=m_{B}gl\cos \theta _{B} \end{align}

where g is the gravitational acceleration constant.

The Lagrangian of the system is obtained from the difference between kinetic energy and potential energy, expressed as:

(4) \begin{align} L=T-V=T_{\textit{trans}}+T_{rot}-V \end{align}

The Lagrange equation is:

(5) \begin{align} \frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\partial L}{\partial \dot{\boldsymbol{q}}_{i}}\right)-\frac{\partial L}{\partial {\boldsymbol{q}}_{i}}= \boldsymbol{F}_{i} \end{align}

In the formula, q _i is the generalized coordinate and F _i generalized force vector. The generalized coordinate vector is $q\textbf{=}[\begin{array}{llllll} \dot{\boldsymbol{x}} & \ddot{\boldsymbol{x}} & \dot{\boldsymbol{\theta }}_{B} & \ddot{\boldsymbol{\theta }}_{B} & \dot{\boldsymbol{\delta }} & \ddot{\boldsymbol{\delta }} \end{array}]$ .

Based on Eq. 1, 2, 3, 4 to Eq. 5, the system of dynamic equations of the robot is:

(6) \begin{align} & \left(J_{z}+M_{B}l^{2}\right)\ddot{\theta }_{B}=Mgl\sin \theta _{B}-M\ddot{x}l\cos \theta _{B}-\left(T_{L}+T_{R}\right)\nonumber \\& \left(M+2m_{W}+\frac{2I}{r^{2}}\right)\ddot{x}=\frac{T_{L}+T_{R}}{r}-Ml\ddot{\theta }_{B}\cos \theta _{B}+Ml{\dot{\theta }_{B}}^{2}\sin \theta _{B} \nonumber\\ & \left(rmD+\frac{ID}{r}+\frac{2rJ_{y}}{D}\right)\ddot{\delta }=T_{L}-T_{R} \end{align}

3.2. Dynamics modeling of sphere mode

The robot’s rolling movement relies on a stepper reduction motor to drive the wheel hub to rotate through a gear. According to the force analysis, the robot ultimately relies on the friction between the wheel hub and the ground to move forward. Under ideal conditions, the force analysis of a spherical robot rolling at a constant speed is shown in Figure 7.

Figure 7. Force analysis of spherical robot rolling at constant speed.

If the angle through which the spherical shell rolls relative to the inertial coordinate system is $\alpha$ , then there is $x=\alpha R$ . Use the Lagrange equation to establish the dynamic model of the system. Choosing the plane where the center of the sphere is located as the zero potential energy surface, the motion of the spherical shell is decomposed into the linear motion of the center of the sphere along the x-axis and the rotation of the spherical shell around the center of the sphere. Then, the kinetic energy of the spherical shell can be expressed as

(7) \begin{align} T_{R}=\frac{1}{2}M_{d}\dot{x}^{2}+\frac{1}{2}J_{R}\dot{\alpha }^{2} \end{align}

where $\dot{x}$ is the horizontal movement speed of the center of the ball, and $\dot{x}=\dot{\alpha }R$ . $J_{R}$ is the moment of inertia of the spherical shell. M _d denotes the mass of body. In this article, the internal mechanism rolls with the spherical shell.

The components of the speed of a wheel with mass M _m on the plane are:

(8) \begin{align} \left\{\begin{array}{l} v_{x}=\dot{x}+r\theta \cos \theta \\ v_{y}=l\dot{\theta }\sin \theta \end{array}\right. \end{align}

where r is sphere radius, and θ is wheel offset angle. l is the distance from the center of mass of the wheel to the center of mass of the sphere.

The difference between the kinetic energy and potential energy of the eccentric wheel in the spherical state is:

(9) \begin{align} T_{m}-V_{m}& = \frac{1}{2}M_{m}\left[\left(\dot{x}+l\dot{\theta }\cos \theta \right){}^{2}{+}{}\left(l\dot{\theta }\sin \theta \right)^{2}\right]\\ & \quad + M_{m}gl\cos \theta \nonumber \end{align}

The Lagrangian function of the linear motion of the spherical robot is:

(10) \begin{align} L_{B}=T_{R}+T_{m}-V_{m} \end{align}

4.1. Balanced motion controller

LQR is an optimal control strategy that can regulate linear systems at the minimum cost. The LQR controller can achieve the robustness and reliability of a two-wheeled inverted pendulum [Reference Li, Yang and Fan29]. We design and implement an LQR controller to control and balance the robot. First, linearize the robot’s equilibrium point (θ = 0) and make the following small angle approximation as:

(11) \begin{align} \sin \theta \approx \theta, \cos \theta \approx 1 \end{align}

After linearizing the motion equation, the state space model of the system is defined according to the dynamic model $\dot{x}=f(x,u)$ .

(12) \begin{align} \dot{x} = Ax + Bu \end{align}

where $\boldsymbol{x}\boldsymbol{=}[\begin{array}{llllll} \dot{\boldsymbol{x}} & \ddot{\boldsymbol{x}} & \dot{\boldsymbol{\theta }} & \ddot{\boldsymbol{\theta }} & \dot{\boldsymbol{\delta }} & \ddot{\boldsymbol{\delta }} \end{array}]$ is the state vector, and $u=[\begin{array}{ll} \tau _{L} & \tau _{R} \end{array}]$ is the input of the system. A and B are the system dynamics matrix and input matrix. It is based on Eq. 7, where

\begin{equation*} A=\left[\begin{array}{l@{\quad}l@{\quad}c@{\quad}l@{\quad}l@{\quad}l} 0 & 1 & 0 & 0 & 0 & 0\\[3pt] 0 & 0 & \dfrac{{M_{B}}^{2}l^{2}gr}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r} & 0 & 0 & 0\\[3pt] 0 & 0 & 0 & 1 & 0 & 0\\[3pt] 0 & 0 & \dfrac{\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)M_{B}lg}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r} & 0 & 0 & 0\\[3pt] 0 & 0 & 0 & 0 & 0 & 1\\[3pt] 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]\qquad\qquad\qquad\qquad\end{equation*}

(13) \begin{align} B=\left[\begin{array}{c@{\quad}c} 0 & 0\\[3pt] \dfrac{M_{B}lr+J_{z}+Ml^{2}}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r} & \dfrac{M_{B}lr+J_{z}+Ml^{2}}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r}\\[3pt] 0 & 0\\[3pt] -\dfrac{\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)+M_{B}l}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r} & -\dfrac{\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)+M_{B}l}{\left(J_{z}+M_{B}l^{2}\right)\left(M_{B}r+2m_{w}r+\dfrac{2I}{r}\right)-{M_{B}}^{2}l^{2}r}\\[3pt] 0 & 0\\[3pt] \dfrac{1}{mrD+\dfrac{ID}{r}+\dfrac{2J_{y}}{D}} & -\dfrac{1}{mrD+\dfrac{ID}{r}+\dfrac{2J_{y}}{D}} \end{array}\right] \end{align}

The state feedback control law is expressed as :

(14) \begin{align} u=-k_{LQR}\cdot x \end{align}

The system optimizes the following cost function :

(15) \begin{align} J_{\boldsymbol{LQR}}=\int _{0}^{\infty }\left(x^{T}Qx+u^{T}Ru\right)dt \end{align}

Q and R serve as the weighting matrix of the system state and input, respectively.

For a continuous time-invariant system, the LQR feedback gain is:

(16) \begin{align} k_{\boldsymbol{LQR}}=R^{-1}B^{T}P \end{align}

In the formula, P is obtained by solving the following algebraic Riccati equation:

(17) \begin{align} \mathrm{A}^{\boldsymbol{T}}P+PA-PBR^{\boldsymbol{-}\boldsymbol{1}}B^{\boldsymbol{T}}P+Q=\boldsymbol{0} \end{align}

Determine the A and B matrices according to the dynamic formulas 4, 5, and 6 and the robot parameters in Table 1, and select the corresponding Q and R matrices to calculate the feedback matrix K _LQR . Then the control variable u is calculated according to the formula. The change in the position of the robot’s overall center of mass is used as a trigger condition, and different K _LQR matrices are used for stable control.

4.2. Spherical motion controller

In this paper, a robot dynamics feed-forward controller is used to realize the spherical motion of the robot. In the spherical state of the robot, the controller is categorized into linear velocity control and angular velocity control, and the following controller is designed according to the dynamics equations in Section 3.2.

The linear velocity controller in the spherical state is:

(18) \begin{align} \tau _{1}=k_{P}\left(V_{ref}-V\right)+k_{i}\int \left(V_{ref}-V\right)+\tau _{v} \end{align}

where V _ref and V are the desired speed and actual speed of the sphere, and K _p and K _i are the Proportional Integral Derivative (PID) control coefficients. τ _v is the feedforward control rate.

The angular velocity controller in the spherical state is:

(19) \begin{align} \tau _{2}=\mathrm{D}_{P}\left({\phi} _{ref}-{\phi} \right)+D_{i}\int \left({\phi} _{ref}-{\phi} \right)+\tau _{{\phi} } \end{align}

${\phi} _{ref}$ and ${\phi}$ are the expected angular velocity and actual angular velocity of the sphere, and D _p and D _i are PID control coefficients. $\tau _{{\phi} }$ is the feedforward control rate.

4.3. Jumping motion controller

The virtual model control (VMC) algorithm has an efficient and intuitive control law that allows for the use of generalized variables and generalized force functions [Reference Pratt and Pratt30, Reference Pratt, Torres, Dilworth and Pratt31]. The robot leg motion control adopts the VMC method. The virtual force required for leg movement is established by adding a spring-damper virtual component, and then the end execution force of the leg link is obtained through the virtual force. Finally, the driving torque required for each joint motor is calculated from the end execution force.

For the virtual force Fy in the y direction, assuming that the elastic coefficient of the spring-damper virtual component is K _p1 and the damping coefficient is K _d1 , the equation of using the spring-damper virtual component to express the virtual force is:

(20) \begin{align} F_{y}=K_{p1}\left(y_{set}-y\right)+K_{d1}\left(\dot{y}_{set}-\dot{y}\right) \end{align}

where $y_{set}$ and $y$ are desired and actual positions along the y-direction.

The virtual forces P _L and P _R for the left and right legs were obtained from the dynamic equations:

(21) \begin{align} \left\{\begin{array}{l} P_{L}=\frac{K_{p1}\left(y_{set}-y\right)-K_{D1}\dot{y}-Mg}{2}\\ P_{R}=\frac{K_{p1}\left(y_{set}-y\right)-K_{D1}\dot{y}-Mg}{2} \end{array}\right. \end{align}

Map the virtual force to the end joint moment through the Jacobian matrix, that is:

(22) \begin{align} \left\{\begin{array}{l} T_{\boldsymbol{L}}=J^{\boldsymbol{T}}P_{\boldsymbol{L}}\\ T_{\boldsymbol{R}}=J^{\boldsymbol{T}}P_{\boldsymbol{R}} \end{array}\right. \end{align}

where J ^T is leg Jacobian matrix, and T _L and T _R are the calculated torques of the left and right wheels.

The required virtual force is converted into joint torque and applied to the motors of each joint of the robot. By adjusting parameters, the robot can exhibit different impedance characteristics in different situations. The robot needs higher damping during the landing phase to smooth out the kinetic energy and ensure a smooth landing.

5.1. Basic motion modes test

In order to verify the performance of SWLR, basic motion experiments are conducted as shown in Figure 10. The stability performance of the robot in the standing state is tested in the wheel-legged mode. In the slope-climbing test, we tested the robot’s performance on a 30° in the wheel-legged mode as shown in Figure 10 a. In the spherical mode, the robot’s steering and moving on gravel roads are tested as shown in Figure 10 b. Finally, we tested the robot’s jumping ability as shown in Figure 10c. From the experimental results, the performance of the robot is excellent and fully meets the design expectations.

5.2. Stability of wheel-legged balancing motion

Balance and stability are the precursors of robot movement. In order to test the robot’s ability to resist interference, we conducted experiments on the robot in the case of external interference. Figure 11 shows the distance from the equilibrium position, the θ angle output of the robot in the state of external interference, and the output torques of the robot’s left and right wheels. It can be seen that in the case of external interference, the robot can respond quickly to return to the horizontal position.

Figure 11. The distance from the equilibrium position, the θ angle output of the robot, and the output torques of the robot’s left and right wheels.

5.3. Jumping motion performance

In order to obtain the jumping performance of the robot, the robot’s jumping experiment is carried out. As shown in Figure 10 c, the robot’s in-situ jumping height and running jumping ability are tested, respectively. In the experiment, the robot jumped over a 20 cm high platform to verify its overall jumping ability. Figure 12 shows the curves of leg motor torque changing with time during robot jumping.

Figure 12. Torque curves of leg motor changing with time during robot jumping.

Figure 13. The change-of-state landing cushion planning schematic.

Figure 14. The change-of-state landing cushion planning process.

As can be seen from Figure 12, at the instant of the robot jump, the torque of the motors reaches 4.7 Nm in 0.25 s, providing an instantaneous force for the robot to jump. The completion time of the entire robot jumping process is controlled within 0.5 s. From the performance of the leg motors, the robot and the control method we designed can realize the robot’s jumping motion with a small torque output.

5.4. Robot landing buffer strategy

Robots need energy cushioning when falling to ensure a smooth landing. Inspired by human jumps from high platforms, we capitalized on the properties of robots to design a robot landing strategy that transitions from the wheeled-legged mode to the spherical mode. Experiments have shown that this approach effectively absorbs energy.

As shown in the Figure 13, the whole process is divided into four stages. First, the robot accelerates the traveling stage as the first stage. The second stage is the process of the robot falling from the high platform. The critical value of the output torque of the robot leg motors is detected to determine whether the robot is in the flight phase or falling to the ground, and buffering is performed at the instant of contact with the ground (judged by the torque of the motors). By buffering the height of the fall, it is judged that the spherical shell touches the ground in the third stage, at which time the rowing pendulum strategy starts to take effect. When the leg retracts to a certain position (using the position as a threshold), it switches to the fourth stage of the spherical mode and rolls forward.

The robot landing experiment is shown in Figure 14. When the robot detects a landing, it first performs leg cushioning. When the legs are retracted to approach the spherical shell, the robot’s operating state switches to the spherical operating state and rolls forward.

In the process of landing cushion, we make full use of the characteristics of the designed mechanism and propose a variable structure cushioning method. Figure 15 shows the output curve of the leg motor torque in the cushioning stage. Figure 15 a shows the results of the traditional cushioning method. The curve of the motor torque fluctuates a lot, and then it returns to a flat cross after 0.26 s. The results using the method in this paper are shown in Figure 15 b. It can be seen that the robot has less impact and faster convergence time. In our experiments, we found that the traditional cushioning method cannot effectively eliminate the impact of the robot falling, and the robot will fly up again under the reaction force, which is unfavorable to the subsequent tasks. The rowing method proposed in this paper can effectively decompose the impact force into the horizontal direction, and with the help of its own structural characteristics, it can effectively discharge the force and prevent the robot from taking off again.

Figure 15. Comparison of motor torque change curves during planning for variable state grounding.