The reaction wheel pendulum is a pendulum with a rotating wheel at the end, which is free to spin about an axis parallel to the axis of rotation of the pendulum (see Figure 3.1). The wheel is actuated by a DCcmotor, while the pendulum is unactuated. The coupling torque generated by the angular acceleration of the disk can be used to actively control the system.

The control objective here will be to swing the pendulum up and balance it about its unstable inverted position. We will focus our study on the swinging-up control law. The non-linear swinging-up controller will be based on the total energy of the system. The control design will exploit the passivity property of the complete Lagrangian system dynamics. We make use of LaSalle's theorem to prove that the system trajectories asymptotically converge to a homoc1inic orbit. Therefore, asymptotically, after every swing of the pendulum, the system state gets successively closer to the origin.

In Section 3.2, we develop the equations of motion of the reaction wheel pendulum.

In Sections 3.3 control algorithm are presented.

Simulation results are given in Section 3.4.

Principle of operation

A project to stabilize a reaction wheel pendulum. It is a type of inverted pendulum but unlike moving cart type this pendulum has a reaction wheel mounted on the top. Torque is produced by the change of the angular momentum of the reaction wheel. This is a nonlinear system. To stabilize this system there are three control algorithms involved:

1. Stabilizing pendulum about the upright position using PID, LQR LQG etc (Linear Control);

2. Swinging the pendulum from the stable to the unstable equilibrium (Nonlinear Control);

3. Switching between the two controllers at a proper angular position.

Vertical position stabiliziruemost due to the Encoder and PID library. In PID control value is set to 180 degrees, then the algorithm introduced error conditions, and was found the benefits. Now, when the pendulum is in a vertical position, it is able to stabilize.

Now we must swing the pendulum from the unstable position to a vertical position, and then balancing it. It's not easy, because the dynamics is nonlinear. Controller Bang controller based on energy we will be able to swing it with appropriate speed. Then, switching between stabilizing controller in a vertical position, balance.

The bang-bang controller is basically a controller on-off and off. It is planned to implement only the drum hit, and then integrate it with the controller on the basis of energy. To swing the pendulum slider to strike, we need to find the rate of change of the angle of the pendulum. Therefore, in order to swing the pendulum applied torque in the direction of the swing. If the reaction wheel moves clockwise, the pendulum is moving counterclockwise and Vice versa. We assume that the angle clockwise to be positive. Therefore, when the pendulum moves in a clockwise direction, the rate of change of the angle is positive and negative when counterclockwise. Thus, the motor driving the jet impeller can be set for voltage and the direction of the Motor Driver and Arduino respectively.

In the normal state, the pendulum will be at θ = 0, i.e. a stable equilibrium. The pendulum was vertical, i.e. θ = 180 (a predetermined value). Since the engine torque is not very high to reach a predetermined value in one revolution, we need to swing, and then when it reaches around θ = 180, we switch to stabilizing control, and then stabilize it. Use PID controller to stabilize it. When the pendulum rides at 175-185 degrees, it can balance itself. Now we need to swing the pendulum in this position and then switch between the stabilizing controller. But waving is difficult because the dynamics is not linear. To make the case easier to use the controller of the explosion. Controller explosion - just the controller enable. We found a way to find the rate of change of the angle of the pendulum. We will use this concept to control the controller of the explosion.

Pendulum angle is measured by the optical encoder 1000 PPR. When the pendulum moves in a clockwise direction, the rate of change of the angle is positive and negative when counterclockwise. So let's say at θ = 0, when we move the wheel reactions (mounted on the shaft of a DC motor) counter-clockwise for a short time, then the pendulum will move in a clockwise direction, reaches a certain maximum position and then begin to fall. This means that in the fall of the rate of change of the angle is negative. Then we move the reaction wheel in a clockwise direction, then the pendulum is moving counterclockwise. This means that we help the pendulum to move in a counterclockwise direction as it falls. Then, after it reaches the maximum, it again starts moving in the other direction, then the rate of change of the angle is positive, then we move the reaction wheel counter-clockwise, so it starts moving clockwise, and you see that the process is iterative until it reaches the vertical.

Thus,

1. When the rate of change of the angle is greater than or equal to zero, move the reaction wheel counter-clockwise;

2. When the rate of change of the angle is negative, move the reaction wheel clockwise.