The self-balancing non-coaxial two-wheeled robot achieves self-balancing based on the precession effect of dual gyro rotors. While the balancing effect of dual gyro rotors can be doubled, excessive precession angles and asynchronous precession are significant causes of system balance failure. To address these issues, this paper proposes a control strategy based on a gain scheduling algorithm and precession angle zero-compensation to eliminate precession angles. Simultaneously, mechanical methods are used to achieve hard synchronization of the gyro rotor precession, bringing both the system tilt angle and the gyro precession angle close to zero. The paper elaborates on the system's dynamic modeling, control simulation, and prototype experimental platform construction, and conducts related experiments to verify the system's balancing effect. Experimental results show that the system possesses excellent balancing performance and shock resistance, high real-time responsiveness, and small steady-state error, bringing the non-coaxial two-wheeled robot closer to practical application.
Two-wheeled balancing robots are a hot research topic in the field of manned mobile robots, and they are divided into two types: coaxial two-wheeled (left-right arrangement) and non-coaxial two-wheeled (front-back arrangement). Coaxial two-wheeled robots use a passive balancing method, that is, they rely on forward and backward motion to achieve dynamic balance (similar to an inverted pendulum system), and have achieved many research results and applications, such as the well-known Segway. Non-coaxial two-wheeled robots are underactuated, nonlinear, and laterally unstable systems, and their dynamics and control research is extremely challenging, currently still in the laboratory development stage. Compared with coaxial two-wheeled robots, non-coaxial two-wheeled robots have the advantages of flexible movement, good climbing and obstacle-crossing ability, and better safety, and are expected to be used in manned transportation, autonomous patrol, reconnaissance and other fields.
In 1998, Moscow State University in Russia designed a self-balancing bicycle by utilizing the precession torque generated by the gyroscopic effect to achieve lateral balance. In 2005, Murata Manufacturing Co., Ltd. of Japan launched the "Murata Boy," a bicycle that achieves left-right balance using a vertical inertial wheel mounted on its chest. In 2011, LIT Automotive Co., Ltd. of the United States applied for a patent for a manned motorcycle that uses the gyroscopic effect of two flywheels to maintain balance. In 2014, Ohio State University in the United States established the system dynamics equations using the Lagrange equations and designed a first-order sliding mode controller, which was experimentally tested on a bicycle and showed good robustness.
However, non-coaxial two-wheeled robotic vehicles still face technical challenges: poor balance robustness; weak impact resistance; and insufficient stability during high-speed driving, small-radius turning, and hill climbing.
This paper introduces a manned, non-coaxial, two-wheeled self-balancing robot that achieves self-balancing by relying on the gyroscopic torque generated by the synchronous precession of a pair of gyro rotors. This robot employs an active balancing method, exhibiting good balance robustness, shock resistance, and speed regulation. It can achieve balance while stationary or in motion, thus offering enhanced safety. The paper elaborates on the balancing principle, system structure and dynamic model, control system scheme, and experimental verification.
Principles and Structure
The gyroscopic torque τ generated by the precession of a single gyro rotor, as shown in Figure 1, satisfies the mathematical formula.
(1)
In the formula: ω<sub>inertia</sub> is the moment of inertia of the gyroscope rotor; ω<sub>rotation</sub> is the angular velocity of the gyroscope rotor's spin; ω<sub>precession</sub> is the angular velocity of the gyroscope rotor's precession. The direction of the gyroscopic torque follows the right-hand rule, pointing in the negative X-axis direction in the diagram. The direction of the gyroscopic torque changes with the precession angle.
Figure 1. Schematic diagram of the gyro rotor gyro effect.
Due to the precession angle, the gyro torque decomposes into rolling torque and yaw torque. The former is the effective torque that plays a balancing role, while the latter is a disturbing torque that affects the stability of the balance. Therefore, the balancing effect of a single gyro is very poor. The structural diagram of the dual-gyro balancing method used in this paper is shown in Figure 2. In order to ensure that the rolling torques of the two gyro rotors are superimposed and the yaw torques cancel each other out, the rotation and precession directions of the two gyro rotors should be kept opposite.
Figure 2. Analysis diagram of the gyroscope device's balance mechanism.
As shown in Figure 2, the three coordinate systems—the vehicle body coordinate system X1Y1Z1, the gyroscope rotor coordinate system X2Y2Z2, and the gyroscope rotor coordinate system X3Y3Z3—are in the same plane (i.e., the directions of the Y1, Y2, and Y3 axes are the same, facing outwards from the paper), with the X1 axis representing the vehicle's forward direction. For ease of analysis, the precession angular velocities of the two gyroscopes are assumed to be equal in magnitude and opposite in direction. Assuming the precession angle of the two gyroscopes at a certain moment is , the tumbling torque generated along the X1 axis during the precession of the dual gyroscope rotors satisfies the mathematical formula...
(2)
In the formula: 2 and 3 are the moments of inertia of the gyroscope rotor; 2 and 3 are the angular velocities of the gyroscope rotor's spin; 1 is the precession angular velocity of the gyroscope rotor. The direction of the tumbling torque is the negative direction of the X1 axis.
The yaw moment generated by the two gyroscopes along the Z1 axis satisfies the mathematical formula.
(3)
As shown in the above equation, when the structural parameters of the two gyroscopes are identical and their precession velocities are the same, the yaw torque is zero. Therefore, the advantage of a dual-gyroscope device is that it can counteract the disturbance torque generated by the precession of a single gyroscope, enabling the system to achieve balance and stability. To ensure the synchronicity of the precession process of the two gyroscope rotors, the balancing device in this paper uses a mechanical method to achieve synchronous motion, namely, a synchronous belt drive mechanism. Thus, the precession angular velocity and precession angle are key factors in the balance of the control system.
To further simplify the analysis, it is assumed that the contact points of the robot's front and rear wheels with the ground lie on the X-axis of the geographic coordinate system XYZ, and that the robot's body coordinate X1 is parallel to the geographic X-axis. Its dynamic model is shown in Figure 3. Ignoring air resistance and the torque between the tires and the ground, when the tilt angle of the robot body relative to the Z-axis is θ, the dynamic equation of the self-balancing vehicle can be obtained as follows:
(4)
In the formula: is the moment of inertia of the vehicle body relative to the X-axis; is the angular acceleration of the vehicle body tilting left and right (i.e., roll acceleration); G is the gravity acting on the robot; h is the distance between the center of gravity and point O.
Figure 3 Simplified mechanical model of the system
Figure 4 shows the mechanical structure of a non-coaxial two-wheeled self-balancing robot. Its overall structure consists of three parts: the front, the rear, and the body. The front part includes rubber wheels, servo motors, and a front wheel steering mechanism; the rear part mainly consists of hub motors and shock absorption mechanisms; the body includes a dual-gyro balancing device and an electronic control system. The dual-gyro balancing device consists of front and rear gyro rotors, a brushless motor, a gyro rotor synchronization mechanism, and a gyro precession motor. The electronic control system consists of a main control board, motor drivers and sensors, and a lithium battery pack.
Figure 4. Non-coaxial two-wheeled self-balancing robot
Balance control scheme
Figure 5 shows the hardware composition of the control system for a non-coaxial two-wheeled self-balancing robot. The control system is functionally divided into five units: host computer, main control unit, driver, sensor detection, and power management. The motors include four types: gyro-rotating motor, gyro-precessing motor, front wheel steering servo motor, and rear wheel forward drive motor.
Figure 5. Hardware composition of the control system
The most important parameter of a self-balancing robot system is the vehicle tilt angle θ, which should be zero to achieve balance. Simple PD control can achieve temporary balance, but it cannot maintain stability, and the system is prone to oscillations and even collapse. The reason for this is that a self-balancing robot is a nonlinear, coupled system, and in order to resist the external torque (mainly the gravitational torque) on the vehicle body, the gyroscope will always precess in one direction, resulting in an excessively large precession angle, which causes the gyro effect to fail in its balancing role.
During the balance control process, a gyroscope precession angle zeroing compensation θα needs to be added to make the gyroscope precession angle tend to zero; otherwise, the robot body will tip over due to gyroscopic failure. This paper employs fuzzy PD and gain scheduling to control the robot's self-balancing process, as shown in Figure 6. The balance controller reduces the left-right oscillation of the robot body through gain scheduling, and these parameter values are all obtained through experimental testing.
Figure 6. Block diagram of balance control for a non-coaxial two-wheeled robot.
As we know from daily experience, when turning a bicycle or motorcycle at a certain speed, the vehicle needs to tilt at a certain angle to ensure it doesn't tip over. This is because the vehicle needs to provide a centripetal force during the turn. Considering the turning factor, the left and right dynamic equations of the vehicle change from (4) to
(5)
In the formula: FC is the centrifugal force generated when the vehicle body turns.
The relationship between the vehicle tilt angle θ and the turning angle β when turning.
(6)
In the formula: v is the forward speed of the vehicle body; d is the distance between the contact points of the front and rear tires; g is the acceleration due to gravity; m is the mass of the vehicle body; M is the total mass (including load); L is the height of the vehicle body's center of gravity.
To address the balance issue during cornering, a tilt compensation based on the centrifugal force during cornering needs to be added to the PD control (its magnitude is related to the front wheel cornering angle and the rear wheel speed).
The total compensation for the vehicle body tilt angle is given by the formula, which gives the vehicle body tilt angle deviation as follows:
(7)
In the formula: is the desired tilt angle; is the total compensation for the tilt angle.
Therefore, the tumbling torque generated by the two gyroscopes is controlled by a fuzzy PD controller.
(8)
In the formula: represents the total tumbling torque generated by the front and rear gyroscope rotors; and represents the output control gain of the fuzzy PD controller.
Based on fuzzy theory, the fuzzy subsets for tilt angle and tilt angular velocity are {NB, NM, NS, Z, PS, PM, PB}. According to the fuzzy subsets and the simplified dynamic model of the self-balancing robot system, the fuzzy universe of discourse for the fuzzy input parameters tilt angle error and angular velocity error is [-1, 1], with quantization levels of {-1, 0.5, -0.25, 0, 0.25, 0.5, 1} and quantization factors of 0.3 rad and 1 rad/s, respectively. The fuzzy output control quantity has a universe of discourse of [-1, 1], quantization levels of {-1, 0.5, -0.25, 0, 0.25, 0.5, 1}, and a scaling factor of 10 rad/s. The membership functions of the input and output are shown in Figure 7.
Figure 7 Input-output membership functions
The fuzzy control rules are shown in Table 1.
Table 1 Fuzzy Control Rules
The dynamic data time response curve of the balancing robot obtained by MATLAB simulation is shown in Figure 8. The simulation results show that the system can quickly enter a stable equilibrium state.
Figure 8. Attitude parameter response curves during vehicle self-balancing.
Experiment and Results Analysis
This paper presents experimental verifications of the balance performance of a non-coaxial two-wheeled robot, comparing and testing the prototype using both PD and fuzzy PD gain scheduling methods. The specific specifications of the experimental prototype are shown in Table 2. The main control unit chip is an ARM processor, and the data sampling rate is 100Hz.
Table 2 System Specifications
Figure 9 shows the curves of the vehicle's tilt angle and angular velocity during the process from a stationary tilt to automatic recovery of balance. As can be seen from the figure, the vehicle initially tilts to the left, and the transition from a tilted to a balanced state takes only 1-2 seconds. Using only PD control can only maintain the vehicle in a near-balanced state for a few seconds before high-frequency oscillations quickly occur, eventually causing the system to collapse and tilt to the right. In contrast, the fuzzy PD gain scheduling control method can consistently maintain the vehicle in a balanced state, with small overshoot and a stable tilt angle θ between -0.02 rad and +0.02 rad (i.e., a deviation of less than 1.5°).
Figure 9. Tilt angle and angular velocity during vehicle self-balancing process
Figure 10 shows the curves of the gyroscope's precession angle and angular velocity during the vehicle's self-balancing process. Similarly, it can be seen that when using pure PD control, the gyroscope rotor precession angle begins to oscillate at high frequency at 30s until the gyroscope torque fails. However, when using fuzzy PD gain scheduling, the final deviation of the gyroscope precession angle is between -0.2rad and -0.1rad. As can be seen from the figure, the center of deviation of the gyroscope precession angle at the final state is not on the 0 line. This is due to the prototype's center of gravity being biased to the right, creating an eccentric torque. The gyroscope needs to continuously precess in one direction to generate a counter-torque.
Figure 10. Gyroscope precession angle and angular velocity
Conclusion
The prototype of the non-coaxial two-wheeled manned vehicle developed in this paper can achieve automatic balancing. Experimental results show that the balancing effect of the non-coaxial two-wheeled self-balancing robot using pure PD control is not ideal, while the balancing performance using fuzzy PD gain scheduling is very good, with the vehicle tilt angle and gyroscope precession angle both within a relatively stable and reasonable range.
