Two-wheeled mobile robots are strongly coupled, underactuated, and complex nonlinear systems, similar in principle to a walking inverted pendulum system. Such systems are highly typical in control theory and practical engineering, possessing significant academic research value. This paper studies the controller for a single-input fourth-order nonlinear dynamic system of a dual-flywheel mobile robot based on the gyro effect. First, the self-balancing principle of the dual-flywheel mobile robot is mechanically analyzed. The state variables of the fourth-order nonlinear system are embedded on a sliding mode surface, and the sliding mode control law is determined using Lyapunov's second method, achieving control of four state variables with a single control variable. Then, a control simulation platform is built using Matlab/Simulink to simulate the dynamic model of the mobile robot. Simulation results show that the sliding mode control law can achieve control of four state variables with a single control variable, and enables the dual-flywheel mobile robot model to stabilize to equilibrium even with an initial tilt angle of 30 degrees. This controller also achieves control of the flywheel deflection angle and angular velocity.
Working principle of dual-flywheel mobile robot
The gyroscopic effect of a flywheel is shown in Figure 1. A flywheel with moment of inertia rotates around an axis at an angular velocity. If the flywheel is deflected around the axis at an angular velocity of ω, then the flywheel will generate a gyroscopic torque M along the positive direction of the axis due to the gyroscopic effect. M = Jωω0. The mechanical model of the dual-flywheel mobile robot is shown in Figure 2. A rotary motor drives the flywheel to rotate around the Z-axis at an angular velocity ω. When the right side of the robot is impacted, the control system causes the deflection motor to drive the flywheel to deflect around the Y-axis at an angular velocity ω. The flywheel then generates a gyroscopic torque to counteract the impact force, maintaining the robot's lateral balance. This design adopts a dual-gyro linkage structure, the specific structure of which is shown in Figure 3. A gear transmission mechanism is used to control the synchronous deflection of the two deflection axes in opposite directions. Orthogonal decomposition of the gyroscopic torques generated by the front and rear flywheels in Figure 3 yields:
Figure 1
In equations (1) to (4), α is the deflection angle of the flywheel, JZ2 and JZ3 are the moments of inertia of the front and rear flywheels respectively, ωZ2 and ωZ3 are the angular velocities of the front and rear flywheels respectively, and ωY2 and ωY3 are the angular velocities of the front and rear flywheels respectively. MX2 and MX3 represent the gyroscopic torques generated by the front and rear flywheels, respectively, with MX2 representing the gyroscopic torque generated by the front flywheel. The component torque in the negative X1 direction of the robot's planar coordinate axis represents the component torque in the negative Z1 direction of the gyroscopic torque generated by the front flywheel, and the component torque in the positive Z1 direction of the gyroscopic torque generated by the rear flywheel represents the component torque in the positive Z1 direction of the robot's planar coordinate axis. Since ωZ2 and ωZ3 are equal in magnitude and opposite in direction, ωY2 and ωY3 are equal in magnitude and opposite in direction, and JZ2 and LZ3 are equal in magnitude, the correction torque and the vertical component torque for maintaining the balance of the mobile robot can be obtained as M2.
From equations (1)-(6), it can be seen that the gyroscopic torque generated by the dual flywheels only affects the tilting action of the robot.
Figure 2
Design of sliding mode controller
Based on this, the dynamic equations of the dual-flywheel mobile robot can be obtained as follows:
In equation (7), U is the voltage across the deflection motor, κ1 is a constant related to the deflection motor, τ is the control duty cycle, i.e., the control input of the system, κ2 is a constant related to the deflection of the flywheel, J3 is the moment of inertia of the flywheel about the deflection axis, ω is the angular velocity of the flywheel's rotation, J2 is the moment of inertia of the flywheel's rotation, G is the robot's gravity, h is the height of the robot's center of gravity above the ground, and J1 is the moment of inertia of the robot about the tilt axis.
Suppose that the dual-flywheel mobile robot moves in a straight line at a constant speed, with an inclination angle of X1, an inclination angular velocity of X2, a flywheel deflection angle of X3, and a deflection angular velocity of X4. Then equation (7) can be rewritten as:
Considering that the final deflection angle approaches 0, the deflection angle and angular velocity variables should be embedded in the sliding surface. Therefore, the switching function is selected as follows:
Figure 3 Dual gyroscope drive module
In equation (11), k>0 and ε>0. Substituting equation (11) into equation (10) yields:
Equation (12) shows that the designed sliding mode control system is asymptotically stable over a wide range.
Computer simulation
The structural parameters and correlation coefficients of the dual-flywheel mobile robot are shown in Table 1.
The initial conditions are x = (pi/6, 0, 0, 0), x1d, c1 = 5, c2 = 0.005, c3 = 1, k = 2, ε = 0.1, k1 = 0.43, U = 12, k2 = 1. The control algorithm and dynamic model are written using s-functions, and the switching function is replaced by a saturation function with a parameter of 0.1. The simulation block diagram built using Matlab/Simulink is shown in Figure 4.
Figure 5 shows the time-domain response diagrams of the tilt angle, tilt angular velocity, flywheel deflection angle, and deflection angular velocity of the dual-flywheel mobile robot obtained through Simulink simulation.
As shown in Figure 5, the designed sliding mode control law can simultaneously control the robot's tilt angle, tilt angular velocity, flywheel deflection angle, and deflection angular velocity, and enable the robot to stabilize to a balanced state even when the initial tilt angle is 30 degrees.
in conclusion
This paper designs a sliding mode controller for a single-input fourth-order nonlinear dynamic system of a dual-flywheel mobile robot. This controller achieves simultaneous control of the robot's tilt angle, tilt angular velocity, flywheel yaw angle, and yaw angular velocity from a single control input, exhibiting good dynamic response. A simulation platform for sliding mode control is built using MATLAB/SIMULINK to simulate the model. Simulation results show that this controller enables the robot to quickly recover to equilibrium from an initial tilt angle of 30 degrees.
