A. Driving Kinematics

Fig. 3(a) shows a diagram describing the side view of Ringbot. Note that the 2-D diagram does not include the legs but just the wheel and driving modules, as it only accounts for motions in the sagittal plane of the wheel. The two driving modules travel independently inside the wheel with the speed of the driving motor, $\omega _{m_{1}}$ and $\omega _{m_{2}}$, respectively. When the system is in a steady state that $\dot{\theta }$ is constant and $\omega _{m_{1}} = \omega _{m_{2}}$, a relationship between the wheel's rolling speed and the driving motor speed can be expressed as follows:

View Source \begin{align*} \dot{\theta } & = \omega _{r_{1}} = -N_{w} N_{d} \omega _{m_{1}}\\ &= \omega _{r_{2}} = -N_{w} N_{d} \omega _{m_{2}} \tag{1} \end{align*}

Fig. 3.

Diagrams for 2-D and 3-D modeling of Ringbot. (a) Side view 2-D model. (b) Simplified 3-D model.

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where $\omega _{r_{i}}$ is the driving gear speed of ith driving module. The driving gear speed $\omega _{r_{i}}$ can be calculated from the motor speed $\omega _{m_{i}}$ and the gear ratio, where $N_{d}$ represents the gear ratio of the transmission within the driving module, and $N_{w}$ represents the gear ratio between the wheel and the driving module, calculated as $N_{w} = \frac{r}{R_{i}}$, the ratio between the wheel's inner radius and the driving module gear pitch radius.

If the system is not in a steady state, the wheel and driving module have the following kinematic relationship:

View Source \begin{align*} \dot{\theta } &= \omega _{r_{1}} - \dot{\theta }_{1_{1}}\\ &= \omega _{r_{2}} - \dot{\theta }_{1_{2}}. \tag{2} \end{align*}

In Fig. 3(a), $CoM_{d}$ denotes the position of the combined CoM of the two driving modules. The movement of $CoM_{d}$ can be considered as a point-mass connected on a linkage $l_{d}$ that rotates with respect to the $y_{w}$ axis. The corresponding joint angle $\theta _{d}$ can be defined from the pitch angles of each driving module $\theta _{1_{1}}$ and $\theta _{1_{2}}$

View Source \begin{align*} \theta _{d} = \frac{\theta _{1_{1}} + \theta _{1_{2}}}{2}\cdot \tag{3} \end{align*}

Besides, as the monocycle is a nonholonomic system, the rolling constraint can be defined assuming an ideal rolling at a contact point without slipping

View Source \begin{align*} \begin{bmatrix}\dot{x}\\ \dot{y} \end{bmatrix} = R\dot{\theta }\begin{bmatrix}\cos {\psi }\\ \sin {\psi } \end{bmatrix} \tag{4} \end{align*}

B. Driving Dynamics

The equations of motion of Ringbot were derived using the simplified 3-D model as shown in Fig. 3(b). The simplified 3-D dynamics model includes a wheel and two-point mass attached at mass-less linkages. Each point mass represents the CoM of the two driving modules and the CoM of the legs, respectively. At the center of the wheel, a revolute joint connects the driving module CoM with the linkage $l_{d}$ that rotates along the $y_{w}$ axis. The leg CoM is attached at the linkage $l_{l}$, connected at the driving module CoM with a revolute joint rotates along the $x_{d}$ axis.

Using the simplified dynamic model, the design parameters of the prototype can be determined by considering the dynamics of the system. The equations of motion of the 3-D model were derived through the Newton–Euler method.

In Fig. 3(b), the world coordinate system is $[x, y, z]$. The yaw and roll angles of the wheel are represented with $\psi$ and $\phi$, respectively. Frame $[x_{w}, y_{w}, z_{w}]$ is a moving frame attached at the wheel's center. Note that the wheel's frame only reflects the yaw and roll angle; keep the pitch angle zero.

By applying $\psi$ and $\phi$ with a 3-1 Euler angle sequence, the angular velocity of the moving wheel frame can be represented as follows:

View Source \begin{align*} \omega _{x_{w}} &= \dot{\phi } \\ \omega _{y_{w}} &= \dot{\psi }s{\phi } \\ \omega _{z_{w}} &= \dot{\psi }c{\phi }. \tag{5} \end{align*}

Note that $s$ and $c$ are the abbreviations of $\sin$ and $\cos$, respectively. The abbreviations are used in the rest of the article. Adding the rolling angular velocity $\dot{\theta }$, the angular velocity of the rotating wheel, $\Omega$ can be defined as

View Source \begin{align*} \Omega _{x_{w}} &= \omega _{x_{w}} = \dot{\phi } \\ \Omega _{y_{w}} &= \omega _{y_{w}} + \dot{\theta } = \dot{\psi }s{\phi } + \dot{\theta } \\ \Omega _{z_{w}} &= \omega _{z_{w}} = \dot{\psi }c{\phi }. \tag{6} \end{align*}

As the rate of change of linear momentum is equal to the forces and the rate of change of angular momentum is equal to the torques

View Source \begin{align*} F &= \dot{p}_{s} = \dot{p}_{w} + \omega \times p_{w} \\ T &= \dot{L}_{s} = \dot{L}_{w} + \omega \times L_{w} \end{align*}

$$\begin{align*} F_{R_{x_{w}}} &= \dot{p}_{x_{w}} + \omega _{z_{w}} p_{y_{w}} - \omega _{y_{w}} p_{z_{w}}\\ F_{R_{y_{w}}} - Mgs{\phi } &= \dot{p}_{y_{w}} + \omega _{x_{w}} p_{z_{w}} - \omega _{z_{w}} p_{x_{w}}\\ F_{R_{z_{w}}} - Mgc{\phi } &= \dot{p}_{z_{w}} + \omega _{y_{w}} p_{x_{w}} - \omega _{x_{w}} p_{y_{w}}\\ &\quad F_{R_{y_{w}}}R + \tau _{l_{x}} + \tau _{d_{x}} + \tau _{R_{l}} c{\theta _{d}} \\ &= I_{x_{w}}\dot{\Omega }_{x_{w}} + I_{y_{w}} \Omega _{y_{w}} \omega _{z_{w}} - I_{z_{w}} \Omega _{z_{w}} \omega _{y_{w}} \\ &\quad -F_{R_{x_{w}}}R + \tau _{l_{y}} + \tau _{d_{y}} + \tau _{R_{d}} \\ &= I_{y_{w}}\dot{\Omega }_{y_{w}} + I_{z_{w}} \Omega _{z_{w}} \omega _{x_{w}} - I_{x_{w}} \Omega _{x_{w}} \omega _{z_{w}} \\ &\quad\tau _{l_{z}} + \tau _{d_{z}} + \tau _{R_{l}} s{\theta _{d}} \\ &= I_{z_{w}}\dot{\Omega }_{z_{w}} + I_{x_{w}} \Omega _{x_{w}} \omega _{y_{w}} - I_{y_{w}} \Omega _{y_{w}} \omega _{x_{w}} \tag{7} \end{align*}$$ View Source \begin{align*} F_{R_{x_{w}}} &= \dot{p}_{x_{w}} + \omega _{z_{w}} p_{y_{w}} - \omega _{y_{w}} p_{z_{w}}\\ F_{R_{y_{w}}} - Mgs{\phi } &= \dot{p}_{y_{w}} + \omega _{x_{w}} p_{z_{w}} - \omega _{z_{w}} p_{x_{w}}\\ F_{R_{z_{w}}} - Mgc{\phi } &= \dot{p}_{z_{w}} + \omega _{y_{w}} p_{x_{w}} - \omega _{x_{w}} p_{y_{w}}\\ &\quad F_{R_{y_{w}}}R + \tau _{l_{x}} + \tau _{d_{x}} + \tau _{R_{l}} c{\theta _{d}} \\ &= I_{x_{w}}\dot{\Omega }_{x_{w}} + I_{y_{w}} \Omega _{y_{w}} \omega _{z_{w}} - I_{z_{w}} \Omega _{z_{w}} \omega _{y_{w}} \\ &\quad -F_{R_{x_{w}}}R + \tau _{l_{y}} + \tau _{d_{y}} + \tau _{R_{d}} \\ &= I_{y_{w}}\dot{\Omega }_{y_{w}} + I_{z_{w}} \Omega _{z_{w}} \omega _{x_{w}} - I_{x_{w}} \Omega _{x_{w}} \omega _{z_{w}} \\ &\quad\tau _{l_{z}} + \tau _{d_{z}} + \tau _{R_{l}} s{\theta _{d}} \\ &= I_{z_{w}}\dot{\Omega }_{z_{w}} + I_{x_{w}} \Omega _{x_{w}} \omega _{y_{w}} - I_{y_{w}} \Omega _{y_{w}} \omega _{x_{w}} \tag{7} \end{align*}

$$\begin{align*} \tau _{d_{x}} &= l_{d} m_{d} g c{\theta _{d}} s{\phi } \\ \tau _{d_{y}} &= -l_{d} m_{d} g c{\phi } s{\theta _{d}} \\ \tau _{d_{z}} &= -l_{d} m_{d} g s {\phi } s{\theta _{d}} \\ \tau _{l_{x}} &= m_{l} g s{\phi } (l_{d} c{\theta _{d}} - l_{l} c{\phi _{l}} c{\theta _{d}}) + l_{l} m_{1}\,g c{\phi } s{\phi _{l}} \\ \tau _{l_{y}} &= -m_{l} g c{\phi } s{\theta _{d}}(l_{d} - l_{l} c{\phi _{l}}) \\ \tau _{l_{z}} &= -m_{l} g s{\phi } s{\theta _{d}}(l_{d} - l_{l} c{\phi _{l}}). \tag{8} \end{align*}$$ View Source \begin{align*} \tau _{d_{x}} &= l_{d} m_{d} g c{\theta _{d}} s{\phi } \\ \tau _{d_{y}} &= -l_{d} m_{d} g c{\phi } s{\theta _{d}} \\ \tau _{d_{z}} &= -l_{d} m_{d} g s {\phi } s{\theta _{d}} \\ \tau _{l_{x}} &= m_{l} g s{\phi } (l_{d} c{\theta _{d}} - l_{l} c{\phi _{l}} c{\theta _{d}}) + l_{l} m_{1}\,g c{\phi } s{\phi _{l}} \\ \tau _{l_{y}} &= -m_{l} g c{\phi } s{\theta _{d}}(l_{d} - l_{l} c{\phi _{l}}) \\ \tau _{l_{z}} &= -m_{l} g s{\phi } s{\theta _{d}}(l_{d} - l_{l} c{\phi _{l}}). \tag{8} \end{align*}

The legs are modeled as an inverted pendulum with a massless rod and a point mass at its end, representing the center of mass of the legs $CoM_{l}$, which is connected to the center of mass of the two driving modules $CoM_{d}$. It is designed to rotate only around the $y_{w}$ axis, with the angle of rotation denoted by $\phi _{l}$.

As the wheel is rolling on the ground, the linear momentum of the moving frame has relations with the angular momentum with the following constraints derived from (4):

View Source \begin{align*} p_{x_{w}} - MR \Omega _{y_{w}} &= 0\\ P_{y_{w}} + MR \Omega _{x_{w}} &= 0\\ P_{z_{w}} &= 0. \tag{9} \end{align*}

Combining (5) to (9), the equations of motion expressed in the absolute orientation of the wheel $[\phi, \theta, \psi ]^{T}$. can be obtained.

View Source \begin{align*} &(I_{x_{w}} + MR^{2})\ddot{\phi } + (I_{y_{w}} + MR^{2})(\dot{\psi }s{\phi } + \dot{\theta })\dot{\psi }c{\phi } \\ &\quad -I_{z_{w}}\dot{\psi }^{2}c{\phi }s{\phi } = MgRs{\phi } + \tau _{l_{x}} + \tau _{d_{x}} + \tau _{R_{l}} c{\theta _{d}} \tag{10}\\ & (I_{y_{w}} + MR^{2})\left(\ddot{\psi }s{\phi } + \dot{\psi }\dot{\phi }c{\phi }+\ddot{\theta }\right)-MR^{2}\dot{\psi }\dot{\phi }c{\phi } \\ &\quad = \tau _{l_{y}} + \tau _{d_{y}} + \tau _{R_{d}} \tag{11}\\ &\quad I_{z_{w}}\ddot{\psi }c{\phi } - I_{y_{w}}\dot{\phi }\left(\dot{\psi }s{\phi }+\dot{\theta }\right) = \tau _{l_{z}} + \tau _{d_{z}} + \tau _{R_{l}} s{\theta _{d}}. \tag{12} \end{align*}

From the equations of motion, the critical velocity can be derived, which is the minimum velocity of the wheel such that the gyroscopic forces stabilize the wheel from a minor disturbance. Assume the wheel is rolling upright with a constant speed ($\phi _{l} = \theta _{d} = \tau _{R} = 0$, $\dot{\theta }$ = constant) with a little disturbance at the side, causing a small tilt angle $\phi = \alpha$.

By assuming $\dot{\psi }$ and $\dot{\alpha }$ are very small, higher order terms can be neglected. Then, (10) can be expanded to

View Source \begin{align*} &(I_{x_{w}} + MR^{2})\ddot{\alpha } + (I_{y_{w}} + MR^{2})\dot{\theta }\dot{\psi }c{\alpha } \\ &\quad= (MR + m_{l} (l_{d} - l_{l}) + m_{d} l_{d}) g\alpha \tag{13}\\ &I_{z_{w}}\ddot{\psi } c{\alpha } -I_{y_{w}} \dot{\alpha }\dot{\theta } = 0. \tag{14} \end{align*}

$$\begin{align*} \dot{\psi } = \frac{I_{y_{w}}}{I_{z_{w}}c{\alpha }}\dot{\theta }\alpha. \tag{15} \end{align*}$$ View Source \begin{align*} \dot{\psi } = \frac{I_{y_{w}}}{I_{z_{w}}c{\alpha }}\dot{\theta }\alpha. \tag{15} \end{align*}

Substitute $\dot{\psi }$ in (13) to get the following simplified equation:

View Source \begin{align*} &(I_{x_{w}} + MR^{2})\ddot{\alpha } = \\ &- \left[ (I_{y_{w}} + MR^{2}) \frac{I_{y_{w}}}{I_{z_{w}}}\dot{\theta }^{2} - (MR+m_{l}(l_{d} - l_{l}) + m_{d} l_{d}) \right] \alpha. \tag{16} \end{align*}

To ensure that (16) yields a stable solution, indicating that the roll angle will converge to zero and the wheel will maintain balance, the value within the square brackets must be positive.

View Source \begin{align*} \dot{\theta } > \sqrt{\frac{I_{z_{w}} (MR+m_{l}(l_{d} -l_{l}) + m_{d} l_{d})}{I_{y_{w}} (I_{y_{w}} + MR^{2})g}}\cdot \tag{17} \end{align*}

If we consider the wheel as a thin hoop, $I_{x_{z}} = \frac{1}{2} MR^{2}$ and $I_{y_{w}} = MR^{2}$. By substituting the inertia terms, we can obtain the critical angular velocity as follows:

View Source \begin{align*} \dot{\theta } > \sqrt{\frac{(MR+m_{l}(l_{d} -l_{l}) + m_{d} l_{d})g}{4MR^{2}}}\cdot \tag{18} \end{align*}

C. Simplified Lateral Dynamics

The roll angle control is important for steering and balancing Ringbot, which can be achieved by shifting the legs laterally. As shown in Fig. 4, the lateral dynamic model of Ringbot can be simplified to a 2-DoF planar serial linkage by modeling the contact between the wheel and the ground as a 1-DoF revolute joint. In this model, the wheel and its two driving modules are treated as a single body, denoted as Link 1, while the two legs are combined into Link 2. It is important to note that leg 1 and leg 2 are symmetrical, each maintaining a constant angular offset from the midplane while in driving mode, except in cases where they reach the limits of their range of motion or make contact with the environment. When the robot is static, there is no torque acting on the first joint as it is just a ground contact point. In this case, the simplified model is equivalent to an Acrobot model, an underactuated 2-DoF inverted pendulum with an actuator at the second revolute joint [23]. By applying the Acrobot model, the state variables of the simplified model are defined as

View Source \begin{align*} X = \begin{bmatrix}\phi \\ \phi _{l} \end{bmatrix}. \tag{19} \end{align*}

Fig. 4.

Diagrams for lateral dynamics modeling of ringbot. (a) Frontal view of Ringbot. (b) Simplified 2-D model.

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Again, $\phi$ is the roll angle corresponding to the first joint angle which represents the ground contact point, and $\phi _{l}$ denotes the leg CoM roll joint angle as shown in Figs. 3(b) and 4(b).

Then, the manipulator equation is expressed as

View Source \begin{align*} \begin{bmatrix}M_{11} & M_{12} \\ M_{21} &M_{22} \end{bmatrix} \ddot{X} + \begin{bmatrix}C_{11} & C_{12} \\ C_{21} & 0 \end{bmatrix}\dot{X} = \begin{bmatrix}\tau _{g_{1}} \\ \tau _{g_{2}} \end{bmatrix} + B u \tag{20} \end{align*}

$$\begin{align*} M_{11} &= m_{2} l_{1}^{2} + 2m_{2} l_{1}l_{c_{2}} c{\phi _{l}} + m_{2} l_{c_{2}}^{2} + I_{1} + I_{2} \\ M_{12} &= m_{2} l_{c_{2}}^{2} + l_{1} m_{2} l_{c_{2}} c{\phi _{l}} + I_{2}\\ M_{22} &= m_{2} l_{c_{2}}^{2} + I_{2}\\ C_{11} &= -2l_{1}l_{c_{2}}m_{2}\dot{\phi }_{l} s{\phi _{l}} \\ C_{12} &= -l_{1} l_{c_{2}} m_{2} \dot{\phi }_{l} s{\phi _{l}} \\ C_{21} &= l_{1} l_{c_{2}} m_{2} \dot{\phi } s{\phi _{l}} \\ \tau _{g_{1}} &= m_{2}\,g (l_{1} s{\phi } + l_{c_{2}} s{(\phi + \phi _{l})}) + m_{1}\,g l_{c_{1}} s{\phi }\\ \tau _{g_{2}} &= m_{2}\,g l_{c_{2}} s{(\phi + \phi _{l})}\\ B &= \begin{bmatrix}0 \\ 1 \end{bmatrix}. \tag{21} \end{align*}$$ View Source \begin{align*} M_{11} &= m_{2} l_{1}^{2} + 2m_{2} l_{1}l_{c_{2}} c{\phi _{l}} + m_{2} l_{c_{2}}^{2} + I_{1} + I_{2} \\ M_{12} &= m_{2} l_{c_{2}}^{2} + l_{1} m_{2} l_{c_{2}} c{\phi _{l}} + I_{2}\\ M_{22} &= m_{2} l_{c_{2}}^{2} + I_{2}\\ C_{11} &= -2l_{1}l_{c_{2}}m_{2}\dot{\phi }_{l} s{\phi _{l}} \\ C_{12} &= -l_{1} l_{c_{2}} m_{2} \dot{\phi }_{l} s{\phi _{l}} \\ C_{21} &= l_{1} l_{c_{2}} m_{2} \dot{\phi } s{\phi _{l}} \\ \tau _{g_{1}} &= m_{2}\,g (l_{1} s{\phi } + l_{c_{2}} s{(\phi + \phi _{l})}) + m_{1}\,g l_{c_{1}} s{\phi }\\ \tau _{g_{2}} &= m_{2}\,g l_{c_{2}} s{(\phi + \phi _{l})}\\ B &= \begin{bmatrix}0 \\ 1 \end{bmatrix}. \tag{21} \end{align*}

When the wheel is rolling with a nonzero angular velocity, the gyroscopic effect caused by the angular momentum acts in the opposite direction to gravity, preventing the wheel from tippling over. In (10), the gyroscopic effect is expressed as taking the term acting opposite to the gravity torque

View Source \begin{align*} \tau _{gyro} = (I_{y_{w}} + MR^{2})\dot{\theta }\dot{\psi }c{\phi }. \tag{22} \end{align*}

A. Driving Module

The two driving modules have identical configurations, except only one train has a single-board computer to operate the entire robot system. The main role of the driving modules is to shift the CoM in the longitudinal direction to accelerate/decelerate the wheel. Also, the driving modules become platforms for computers, sensors, and batteries needed to operate the robot system. Furthermore, the legs are attached to the top of the driving modules.

The selection of a motor to actuate the driving module is one of the most important design decisions. The primary design objective was to surpass a max speed of 4 km/h, faster than human walking speed. At the same time, a compact driving module size was pursued to keep the CoM lower as well as to provide more space for the leg linkages. An integrated servo motor package with a controller and sensors was preferred for concise electronics in limited space. To meet these demands, a compact-size servo motor was selected that has a built-in microcontroller that processes the internal feedback control as well as provides a serial network connection between all the connected motors.

As the single motor does not have sufficient power to generate enough torque and speed to reach the target speed, a single driving module employs two twin servo motors and synchronizes its actuation. By using two motors within one driving module, the driving torque can be enhanced. Consequently, the internal gearbox of the servo motors had custom modifications to increase the speed by 3.7 times compared to the original product in order to reach faster driving speed. The driving motors are powered at 7.5 V and operated in the pulse width modulation (PWM) control mode.

In addition, 3-D printed gears are attached to each servo motor output with the 2.61:1 gear ratio to amplify the speed of the motors. For efficient transmission, a herringbone gear design is applied to the 3-D printed gears.

Each driving module is equipped with a 9-DoF inertial measurement unit (IMU) that is capable of measuring absolute orientation, angular velocity, and linear accelerations. Combining the independent IMU measurements from each driving module makes it possible to estimate the current states of both the driving modules and wheels more accurately.

To facilitate tetherless, stand-alone operation, a single-board computer is installed on one of the driving modules and connected to all the motors and sensors in the Ringbot system. All the motors, including the leg motors, are connected through Transistor–Transistor Logic (TTL) serial communication. In addition, the IMUs are serially connected via I2C communication. Each driving module is powered by a 300 mAh three-cell Lithium Ion Polymer (LiPo) battery mounted at the driving module body and an 850 mAh three-cell LiPo battery at the leg linkage, respectively. A coil cable, which is elastic and capable of being stretched beyond the wheel's inner diameter, connects the two driving modules. This ensures that the distance between the two modules is not restricted by the cable. The motor communication lines and I2C lines are combined via the cable, enabling a single driving module's computer to manage the entire system.

As seen in Fig. 5(d), the driving module is mounted on the rail by six ball bearings that only allow for movement along the longitudinal direction of the rail. Two bearings roll along grooves at the top of the rail, while the other four bearings are angled at 45$^\circ$ and support the side of the rail to prevent any lateral movement. Gear 3 engages with the internal gear at the wheel and has thrust bearings inside that contact the wheel body's side wall to reduce slack between the wheel and driving modules. To further prevent deformation of the structural frame, auxiliary rollers are installed at the axis of Gear 2 to distribute stress all over the wheel structure. It is crucial to ensure a secure fit between the driving module and wheel body while minimizing friction, as any looseness or distortion can adversely impact driving performance and balance stability.

B. Wheel

The wheel is a crucial component of Ringbot, which allows the robot to move by rolling on the ground and houses all other elements. The wheel comprises three parts: 1) the wheel body; 2) rim; and the 3) tire. The wheel body is a structural frame for the wheel, providing a rail structure for the driving modules to travel on. In addition, it contains an internal gear that engages with the pinion gears of the driving modules. The rim serves as an intermediary piece between the wheel body and the tire. The tire was fabricated with an elastic material that directly contacts the ground. With a total width of 50 mm, the wheel has a 22-mm flat surface, providing more contact area for a greater stability margin.

The wheel diameter is one of the most important design parameters of the monocycle that determines driving performance and dynamics, as well as the space constraints for the mechanisms inside. To ensure a metacentric weight distribution and avoid collisions between the legs, we have established a minimum wheel size to accommodate four driving modules in the lower half of the wheel. As the wheel's inner diameter gets larger, it also necessitates a higher driving motor speed to achieve the angular speed that generates a gyroscopic effect to stabilize the wheel. Therefore, it is important to limit the wheel size to ensure the motors at the driving module can generate sufficient speed to exceed the critical velocity, which is the minimum angular velocity needed to stabilize a rolling wheel. The constraint for the maximum wheel radius can be determined by the equation for critical velocity derived from the equations of motion of the simplified dynamics model. For determining the wheel size upper limit, we neglect $m_{l}$ and $m_{d}$ and only consider the wheel's mass in (18). Combining (1) and (18), we can get the upper limit of the inner radius, which is equivalent to the pitch diameter of the ring gear inside of the wheel

View Source \begin{align*} R_{i} < \frac{4r^{2}N_{d}^{2}\omega _{m_{\text{max}}}^{2}}{g}\cdot \end{align*}

For the selected driving motor, the allowable range for the wheel diameter is between 230 mm and 1126 mm. After taking into account the safety factor and the fabrication feasibility, we have chosen an inner wheel diameter of 500 mm and the resulting outer diameter of 515 mm.

In pursuit of a high degree of freedom in part geometry and a quick manufacturing time, the wheel components are fabricated using 3-D printing technology. As the wheel is designed as a large ring-shape, it is crucial to make the wheel structure manage stress and impacts effectively despite the absence of spokes. As depicted in Fig. 5(a), the wheel body parts are printed using micro carbon fiber-filled nylon with continuous carbon fiber reinforcement layers in pursuit of enhanced stiffness and strength. The rim parts are printed with polylactic acid (PLA), and the tire parts are printed with thermoplastic polyurethane. Since the printing volume is limited, the wheel body, rim, and tire are divided into eight segments and assembled to form a single wheel. The assembly is constructed to prevent separation between parts by interlocking the joints and firmly connecting the tire and rim to the wheel body using screws. In addition, four carbon fiber tubes are inserted inside the wheel body and rim to reinforce the assembly and prevent separation due to impacts as well as reinforce the wheel structure.

C. Leg

Ringbot has two legs, each of which has 3-DoF and is mounted on top of each driving module. In Fig. 5(c), the workspace of a leg is marked with a green shaded area, taking into account the joint angle limit of each joint, but without considering internal collisions. The link length and joint configuration of each leg are designed to allow the tip of the leg is able to touch the ground while the wheel is standing upright, enabling the robot to perform various motions using the legs to assist the wheeled driving.

Three position-controlled servo motors actuate the leg, arranged in a Yaw-Roll-Roll configuration. The joint configuration of entire legs is presented in Fig. 6. The servo motors in the legs are controlled using the same serial communication as the driving module motors, which simplifies the electronic peripherals and the software framework while enabling coordinated control of both the leg and driving module.

Fig. 6.

Joint configuration of Ringbot. (a) Leg joint diagram overlapped on the CAD drawing. (b) Name assignment for the leg joint angles.

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In order for the robot to balance or steer using its legs as inverted pendulums, it is crucial that the legs have an effective mass capable of influencing the wheel's dynamics. As the robot's structures are 3-D printed using PLA, it is challenging to achieve an effective mass without adding extra weights on the leg linkages. To address this, a LiPo battery is inserted into the first linkage of each leg, increasing the mass by 160% heavier compared to printing the same volume with solid infill. Moreover, having additional batteries greatly extends the robot's operational time compared to relying solely on the driving module batteries.

The presence of legs allows Ringbot to perform diverse tasks that distinguish it from conventional monocycle robots. In order to control the leg for various legged motions, the forward and inverse kinematics of the robot's legs are solved analytically. Fig. 6 illustrates how a driving module and its corresponding leg can be modeled as a 4-DoF robot arm mounted at the center of the wheel. The first joint represents the driving module's movement, which travels along the wheel's inner diameter and can be considered a revolute joint located at the wheel's center, with a linkage length of $l_{1}$. Table II displays the Denavit–Hartenberg (DH) parameter table for the leg, constructed using the modified DH parameter convention. Note that as the DH parameters of the two legs are identical, the subscript used to indicate the specific driving module has been omitted. This table enables the solution of forward kinematics, which calculates the foot's position based on joint angles.

TABLE II DH Parameter Table for the Leg

An analytical solution for inverse kinematics has been obtained, enabling task-space control of the legs. The solution finds joint angles that satisfy a given foot position. Note that the first joint angle $\theta _{1_{1}}$ and $\theta _{1_{2}}$ are controlled by the driving controller. Therefore, these angles are not found by the inverse kinematics. The analytical solution finds the leg motor angles $[\theta _{2_{i}}, \theta _{3_{i}}, \theta _{4_{i}}]$ for the given foot position of $i$th leg, $P_{F_{i}}$.

The inverse kinematics solution for an arbitrary foot position $P_{F_{i}} = [P_{x_{i}}, P_{y_{i}}, P_{z_{i}}]$, can be obtained as follows:

View Source \begin{align*} \theta _{2_{i}} &= \text{atan}2{(P_{x_{i}} c{\theta _{1_{i}}} + P_{z_{i}} s\theta _{1_{i}},P_{y_{i}})}\\ \theta _{3_{i}} &= -{\rm{sign}}{(P_{y_{i}})} \frac{\pi }{2} - (\gamma _{i} + \beta _{i})\\ \theta _{4_{i}} &= -\pi + \text{acos}\left(\frac{l_{2}^{2} + l_{3}^{2} - a_{i}^{2}}{2 l_{2} l_{3}}\right) \tag{23} \end{align*}

$$\begin{align*} a_{i} &= \sqrt{l_{1}^{2} - 2s{\theta _{1_{i}} l_{1}} P_{x_{i}} + 2c{\theta _{1_{i}}} l_{1} P_{z_{i}} + P_{x_{i}}^{2} + P_{y_{i}}^{2} + P_{z_{i}}^{2}}\\ \beta _{i} &= \text{acos} \left(\frac{l_{2}^{2} + a_{i}^{2} - l_{3}^{2}}{2 l_{2} a_{i}}\right)\\ \gamma _{i} &=-\text{atan} \left(\frac{l_{1} + P_{z_{i}} c{\theta _{1_{i}}} - P_{x_{i}} s{\theta _{1_{i}}}}{P_{y_{i}} c{\theta _{2_{i}}} + P_{x_{i}} c{\theta _{1_{i}}}s{\theta _{2_{i}}} + P_{z_{i}} s{\theta _{1_{i}}}s{\theta _{2_{i}}+}}\right)\cdot \tag{24} \end{align*}$$ View Source \begin{align*} a_{i} &= \sqrt{l_{1}^{2} - 2s{\theta _{1_{i}} l_{1}} P_{x_{i}} + 2c{\theta _{1_{i}}} l_{1} P_{z_{i}} + P_{x_{i}}^{2} + P_{y_{i}}^{2} + P_{z_{i}}^{2}}\\ \beta _{i} &= \text{acos} \left(\frac{l_{2}^{2} + a_{i}^{2} - l_{3}^{2}}{2 l_{2} a_{i}}\right)\\ \gamma _{i} &=-\text{atan} \left(\frac{l_{1} + P_{z_{i}} c{\theta _{1_{i}}} - P_{x_{i}} s{\theta _{1_{i}}}}{P_{y_{i}} c{\theta _{2_{i}}} + P_{x_{i}} c{\theta _{1_{i}}}s{\theta _{2_{i}}} + P_{z_{i}} s{\theta _{1_{i}}}s{\theta _{2_{i}}+}}\right)\cdot \tag{24} \end{align*}

Note that the inverse kinematic solution finds the elbow-up solution all the time.

A. State Estimation

In order to control the robot system, it is crucial to have knowledge of its current state, describing the driving modules as well as the entire robot system. To estimate these states, two IMUs are utilized in each driving module, along with encoder information from the driving motors.

There is an IMU in each driving module measuring the angular velocity, linear acceleration, and absolution orientation of each module. The IMU estimates the absolute orientation with a 100 Hz rate using an internal sensor fusion algorithm to combine data from the gyroscope, accelerometer, and magnetometer.

The yaw and roll orientation of the wheel are estimated from the IMU values. The IMU value output for the absolute orientation is obtained in a quaternion. By converting the quaternion into a 3-1-2 Euler angle, the roll and yaw orientation of the wheel frame can be estimated, and then the pitch orientation for describing the driving module's movement can be obtained. This is because the wheel frame undergoes a 3-1 Euler angle sequence transformation from the world coordinate, as defined in Section III.

The acquisition of absolute roll and yaw angles from the IMUs on each driving module provides a redundant measure of the wheel's roll and yaw orientation. Leveraging this redundancy enhances the accuracy of estimating the wheel's yaw and roll angles. This is achieved by averaging the measurements, which reduces noise and drift that might occur with a single IMU. In addition, this method helps to minimize bias arising from hardware-related errors, such as inaccuracies during installation or slack in the rails.

Consequently, when the quaternion value $[w_{i}, x_{i}, y_{i}, z_{i}]$ represents the absolute orientation measured by the IMU at driving module $i$, the estimated roll and yaw angles can be derived as follows:

View Source \begin{align*} \hat{\psi } &= \frac{1}{2}\left(\text{atan}\left(\frac{-2(xy_{1} - wz_{1})}{1-2(x_{1}^{2} + z_{1}^{2})}\right) + \text{atan}\left(\frac{-2(xy_{2} - wz_{2})}{1-2(x_{2}^{2} + z_{2}^{2})}\right) \right) \\ \hat{\phi } &= \frac{1}{2}\left(\text{asin}(2(yz_{1} + wx_{1})) + \text{asin}(2(yz_{2} + wx_{2})) \right). \end{align*}

Since the driving modules only rotate along the $y$ axis of the wheel, the pitch orientation of IMU can be directly considered as $\theta _{1_{1}}$ and $\theta _{1_{2}}$ in Fig. 6(b).

View Source \begin{align*} \hat{\theta }_{1_{i}} = \frac{\text{asin}(-2(xz_{i}-wy_{i}))}{\cos {\phi _{i}}} \end{align*}

The wheel's angular velocity is also estimated using the IMUs, which are used for the roll angle control, driving velocity control, and steering control. Since each driving module only has a pitch angle rotation relative to the $y$-axis of the wheel frame, the wheel's angular velocity can be obtained by projecting the measured angular velocity vectors $[\omega _{x_{i}}, \omega _{y_{i}}, \omega _{z_{i}}]^{T}$to the wheel frame axes. As the wheel's angular velocity can be obtained from each IMU, the two values are averaged to estimate the angular velocity of the wheel frame less affected by sensor noise and bias

View Source \begin{align*} \begin{bmatrix}\hat{\omega }_{x_{w}} \\ \hat{\omega }_{y_{w}} \\ \hat{\omega }_{z_{w}} \end{bmatrix} =& \frac{1}{2}\begin{bmatrix}\omega _{x_{1}}c{\hat{\theta }_{1_{1}}} - \omega _{z_{1}} s{\hat{\theta }_{1_{1}}}\\ 0\\ \omega _{z_{1}}c{\hat{\theta }_{1_{1}}} + \omega _{x_{1}} s{\hat{\theta }_{1_{1}}} \end{bmatrix} \\ &+\frac{1}{2} \begin{bmatrix}\omega _{x_{1}}c{\hat{\theta }_{1_{1}}} - \omega _{z_{1}} s{\hat{\theta }_{1_{1}}}\\ 0\\ \omega _{z_{1}}c{\hat{\theta }_{1_{1}}} + \omega _{x_{1}} s{\hat{\theta }_{1_{1}}} \end{bmatrix}. \end{align*}

Similar to the orientation estimation, the y-axis angular velocity of the wheel is zero. Instead, we consider $\omega _{y_{1}}$ and $\omega _{y_{2}}$ as the angular velocities corresponding to each driving module's respective joint, represented by $\dot{\theta }_{1_{1}}$ and $\dot{\theta }_{1_{2}}$.

The relative position between two driving modules, $\theta _{dist}$, is estimated through the driving motor encoder measurements, which are more reliable than IMUs measurements. Since the distance between the driving modules is strictly regulated by a feedback controller with a large feedback gain, it is crucial to minimize sensor errors like noise or delay.

The driving motor has an absolute encoder that can measure a motor output angle with 0.088$^\circ$ resolution, with up to 256 revolutions in both clockwise and counterclockwise rotation. Note that upon rebooting the motor system, the stored encoder value is erased and initialized in a range of 0$^\circ$ to 360$^\circ$, even if it previously surpassed this range before the reboot.

Due to the encoder values reset, an initial calibration is necessary using the pitch angle obtained from the IMU measurement. Once the initial IMU measurement and initial motor position sensor values are captured, the relative angle between the two modules, $\theta _{dist}$, is estimated as follows:

View Source \begin{align*} \hat{\theta }_{dist} =& \left(\hat{\theta }_{1_{1_{0}}}-\hat{\theta }_{1_{2_{0}}}\right) + N_{w} N_{d} \left((\theta _{m_{1}} - \theta _{m_{1_{0}}})\right.\\ & -\left. (\theta _{m_{2}} - \theta _{m_{2_{0}}})\right). \end{align*}

B. Speed Controller

The speed controller's design is based on the intuitive principle that an offset in the CoM produces torque, propelling the wheel toward the biased direction to recenter the CoM. If the lumped CoM of both driving modules shifts forward, the wheel accelerates, and conversely, it decelerates if the CoM shifts backward. Consequently, speed control can be accomplished by simply increasing or decreasing the driving module motors' speed.

Nevertheless, controlling the relative position between the two driving modules during driving is also crucial, as an internal collision between them could lead to hardware issues. Moreover, if the gap continues to widen, the connecting wire between the modules could become overstretched and disconnected. Asynchronized movement also hinders the efficiency of shifting the CoM. To maintain the distance between the driving modules while concurrently managing wheel speed, the controller depicted in Fig. 8(a) is employed.

Fig. 8.

Decoupled controller of Ringbot. (a) Wheel speed controller. (b) Steering angle controller.

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This controller features an outer angular speed feedback loop, an inner orientation feedback loop, and an additional train distance feedback loop. The design of the controller is designed on a very intuitive behavior of the monocycle: To accelerate the wheel's rotation, increased torque is necessary. To generate more torque, the driving modules need to climb up the ring, thereby utilizing their mass more effectively for the driving torque. For the first part, there is a speed feedback controller, and the output of this controller then feeds into the pitch orientation controller, integrating these two aspects. As the driving modules move within the wheel along a circular rail, their ascent corresponds to larger pitch orientations. To ensure that a driving module does not surpass the wheel's peak point, we cap the reference pitch angle at 180$^\circ$, preventing the modules from exceeding the peak.

As shown in Fig. 6, the output from the outer speed control loop, comprising a PD controller and a command feedforward, is utilized to set the pitch orientation for each module, corresponding to $\theta _{1_{1}}$ and $\theta _{1_{2}}$. This approach is preferred over directly manipulating motor PWM with the speed controller output.

The output from the speed controller is added to the initial pitch position of each driving module and is then used as an input for the pitch orientation controller of each respective driving module. The pitch orientation controller output is converted to the driving module motors' PWM signal, controlling the speed of the motors. In addition, a PI controller regulates the distance between the two modules at a desired distance by superimposing its output onto the pitch orientation controller output.

Consequently, the proposed speed controller can control the speed of the wheel while regulating the distance between two driving modules at the same time.

C. Steering Angle Controller

In a unicycle configuration with a single wheel, it is not possible to change the heading direction when the wheel is stationary. Steering of the wheel can result from gyroscopic precession, which occurs when the wheel rolls with a nonzero angular velocity and an external force, such as gravity, causes a change in the rotation axis perpendicular to both the angular momentum and the external torque. Therefore, a roll angle change can cause steering of the rolling wheel. As the direction of the precession is determined by the right-hand rule, it matches with the direction of the cross product of the angular momentum $L$ and the torque $\tau$.

Fig. 9 illustrates the turning direction based on the interplay between the angular momentum and gravitational torque. It is clear that the wheel's turning direction is towards the roll angle deviation, regardless of whether it is moving forward or backward.

Fig. 9.

Turning direction of the rolling wheel caused by the procession effect depends on the rolling velocity and roll angle.

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Using the correlations between the precession rate and roll angle, a steering angle controller is implemented that consists of an outer loop for the yaw angle control and an inner loop for the roll angle loop. The outer loop includes a PI controller to track the desired yaw angle of the wheel. It is challenging to achieve effective yaw angle control solely with the feedback controller as the yaw angle dynamics as complexly coupled with the other states of the robot, particularly the rolling angular velocity. The equation for a precession rate is given as follows:

View Source \begin{align*} \Omega = \frac{\tau }{L}\cdot \tag{26} \end{align*}

The equation illustrates that the precession rate is inversely related to the angular momentum, denoted by $L = I\dot{\theta }$. Therefore, the same feedback controller gain might not work well for different driving speeds. In order to address this problem, a compensation term $K_{y}aw\dot{\theta }$ is applied to compensate for the influence of the angular velocity.

The steering angle controller's output serves as an input for the roll angle controller. Roll angle control of the wheel is accomplished by shifting the legs laterally as a 2-DoF inverted pendulum. As described in Section III-C, the simplified lateral dynamic model represents the wheel as the first linkage connected to the ground with a revolute joint at the contact point. The legs are then considered as the second linkage connected. When the wheel is not rolling, there is no torque on the first joint, which makes the model equivalent to the Acrobot model. The Acrobot, an underactuated 2-DoF system, has been a prominent subject of nonlinear control research [24], [25], [26], [27]. A widely recognized method for Acrobot balancing control is employing a state feedback control law $U = -KX$ for the second joint torque with linear quadratic regulator (LQR) gains [23]. Ringbot adopts a similar state feedback control law to control the wheel's roll angle. The control states are defined as

View Source \begin{align*} X = [\phi \quad \phi _{l} \quad \dot{\phi } \quad \dot{\phi _{l}}]^{T}. \tag{27} \end{align*}

It is important to recognize that $\phi _{l}$ corresponds to the joint angle of the second link in the simplified lateral dynamic model depicted in Fig. 4, representing the lumped model of the two legs. When the robot is in driving mode, the legs only make lateral movements through the third joint ($\theta _{3_{i}}$) of the leg, consistently maintaining a configuration where each leg is the mirror image of the other, therefore the legs can behave as one inverted pendulum with a fixed length. This also prevents internal collisions between the legs. Unlike the conventional Acrobot controller, Ringbot's roll angle controller regulates the joint velocity $\dot{\phi }_{l}$ instead of the joint torque, as the leg servo motors are operated in a position control mode. As the legs are also involved in diverse legged motion tasks, it is more beneficial to use the motor in position control mode and take advantage of the strong position control capabilities offered by the servo motor's built-in controller. Therefore, the control effort needed to be converted into a position command to the motors.

The control loop rate of the system is limited to the relatively low rate of 100 Hz due to the motor communication speed and the IMU sampling rate. If the control effort is treated as the joint acceleration and integrated twice to calculate corresponding the joint position, it causes significant delay and lagging due to the slow discrete loop time. Therefore, we decided to integrate the control effort only once to control the joint velocity instead.

As a result, the state feedback control law in the $i$th discrete step is

View Source \begin{align*} U = \dot{\theta }_{l_{r_{i}}} = -KX_{i}. \end{align*}

The control effort is then numerically integrated using the trapezoidal rule to compute the corresponding joint angle.

View Source \begin{align*} \theta _{l_{r_{i}}} = \theta _{l_{r_{i-1}}} + 0.5\left(\dot{\theta }_{l_{r_{i}}} + \dot{\theta }_{l_{r_{i-1}}}\right)\Delta T. \end{align*}

Here, $\Delta T$ stands for the loop time. Given that the legs move laterally in unison, functioning as a single inverted pendulum with a fixed length, the position of each foot can be determined in a way that aligns with the CoM position defined by $\theta _{l}$ and the pendulum length $l_{c_{2}}$. Once the foot position for each leg is established, the joint angles of the legs can be calculated using inverse kinematics.

When the wheel is stationary, the robot can be considered as an underactuated system due to the absence of torque at the first joint in the simplified model. Consequently, the control state can only be manipulated around the fixed upright equilibrium point ($\phi = \phi _{l} = 0$.). Although the wheel features a flat surface, allowing for a substantial margin to maintain an upright stance without control, balancing control is still essential to maintain balance for uneven ground surfaces and external disturbance.

As the wheel moves with adequate rolling speed, the gyroscopic torque stabilizes the robot by counteracting the torque induced by gravity. Hence, the same controller's capability extends beyond the fixed point, allowing it to manage the wheel's nonzero roll angle. Since the state feedback gains are tuned for the static balancing control case, a feedforward term is incorporated to track the desired roll angle more effectively without excessively increasing the gain, which could potentially make the system unstable.

D. Legged Motion

Ringbot utilizes its legs for multiple purposes, not only for balancing and steering, but also actively engaging with its surroundings to assist the wheeled driving. A finite state machine (FSM) has been developed to facilitate the transition between driving mode and various legged motion modes, as illustrated in Fig. 11. Transitions between states are prompted by specific conditions indicated alongside the arrows. Detailed explanations for each state are provided below as follows.

Fig. 10.

Snapshots of the implemented legged motions. The legged motions are also demonstrated in the supplementary video. (a) Fall recovery motion. (b) Standing-up motion. (c) Holonomic turning motion.

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Fig. 11.

Visual representation of the leg motion FSM.

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A. Steering Rate

Through the simulation, we assessed Ringbot's steering capacity. In the simulation scenario, Ringbot maintained a steady linear velocity while its legs were shifted to one side until the maximum range of motion was reached for getting the maximum turning rate. Simulations were conducted at velocities from 3.5 km/h, the slowest linear velocity, where the simulation model was able to sustain balance with the weight shift, to 5 km/h, the robot's fastest linear velocity. The results are shown in Fig. 12.

Fig. 12.

Simulation results for identifying the maximum turning rate.

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As indicated by (26), the precession rate is proportional to the external torque and inversely related to the angular momentum. The robot has more substantial angular momentum when the angular velocity is higher. Meanwhile, the external torque is decreased, as the greater gyroscopic torque in (22) has a greater effect, reducing the roll angle deviation caused by the same leg shifting. Thus, the wheel's turning rate decreases as the linear speed rises. The simulation outcomes correspond with the previously mentioned phenomenon, where turning occurs at a slower rate when the speed is higher. At 5 km/h, the simulation model successfully executed a turn with a 0.217-m turning radius while it made the sharpest turning with 0.037 m at 3.5 km/h.

B. Driving Performance

The decoupled controllers for speed and steering were validated through a simulation in which the model simultaneously tracked both speed and steering angle commands. The controllers proposed in Section V were used to control the simulation model. The plots in Fig. 13 present the results of the simulation tracking varying speed commands on a straight trajectory without steering. The speed commands increment by 1 km/h every 10 s, reaching a peak at 5 km/h, the maximum speed. This speed is maintained for 20 s, followed by a rapid decrease in velocity to 0, simulating an aggressive braking scenario. To assess mechanical configurations, such as motor selection, gear ratio, and wheel size, both the speed and torque of the driving motor were monitored. The results illustrate the simulation model's adeptness in adhering to diverse velocity commands and efficiently transitioning from maximum velocity to a complete halt in just 2.6 s, all without exceeding the operational limits of the driving motor.

Fig. 13.

Simulation result for varying speed commands and braking.

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An additional simulation was carried out to evaluate the driving performance during steering, wherein the robot maintained a constant speed of 5 km/h while following arbitrary desired yaw angle commands set by a pilot. Motor torque and speed saturation were also incorporated into the simulation model to confirm the hardware design and specifications through the simulation. As depicted in Fig. 14, the robot successfully maintained a constant speed while steering to follow the arbitrary yaw angle inputs.

Fig. 14.

Simulation result for tracking arbitrary steering angle inputs at a constant speed command.

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