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Background and Theory
The inverted pendulum is a classic controls problem with that can be described and stabilized using feedback control. In the figure below, the inverted pendulum setup is shown. A uniform weight rod pendulum is attached to a movable cart of negligible friction. The system can be classified as single input multiple output (SIMO), in which a single voltage command controls both the cart location and pendulum angle with respect to the normal. Background and Theory
Governing Mechanical System Equations
I will use the following variable names to describe the dynamics of the system:Governing Mechanical System Equations
By first considering a force balance equation on the moment arm, we can determine the force, F1, exerted on the pendulum in the x direction by the cart, as well as the weight force, F2. We can then determine the net torque on the pendulum which yields the first equation below. The second equation is derived by doing a force balance on the total mass and pendulum system.
The two equations above are then substituted into the cart dynamics, described by the equation below:
Cart Positional Control
Cart Positional Control
Prior to the control of the full inverted pendulum, a working control design of the cart position needed to be established. For a given voltage as input and a position of the cart as output, the follow transfer function represents the cart system dynamics:
Full Derivation Here
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