Research → Mechatronic systems → Inverted pendulum driven by pneumatics → Model of the inverted pendulum system



Nonlinear mathematical model of the dynamics of the inverted pendulum is given by:


           

Below is given the state space description of a linearized system, including dynamics of pressure changes in the pneumatic cylinder. The linearization is obtained near the equilibrium point, assuming:

System states are:

In the above mathematical model the dynamics of pneumatics is simplified, and given as a first-order lag system. Here, the input is voltage signal u (0-10 V) on the solenoid of the pneumatic valve, and the output is a pressure difference Δp between two chambers of the rodless pneumatic cylinder:

where τ is a time constant (0.12 s), and K is a gain (0.11 MPa/V). These parameters were experimentally determined from transient responses.

The symbols and numerical values of physical system parameters in above equations are as follows:

A – area of the piston, 0.00018 m2
b– coefficient of viscous friction, 65 Ns/m
F– applied force [N]
g– gravity acceleration, 9.81 m/s2
K– pneumatic gain, 0.11 MPa/V
L– length of pendulum's gravity center, 0.18 m
m1– mass of the slider, 1.5 kg
m2– mass of the pendulum, 0.06 kg
D p– pressure difference between two chambers [Pa]
q1– slider position [m]
q2– pendulum angle [rad]
u– input voltage applied on the valve [V]
t – pneumatic time constant, 0.12 s