IPEN2, IPEN3 – N-link inverted pendulum on cart

IPEN2, IPEN3 – N-link inverted pendulum on cart - Physical parameters

Block SymbolsLicensing group: MODEL

Function Description
The IPEN2 and IPEN3 blocks simulate the dynamics of double and triple inverted pendulums on a cart, respectively. These models enable users to conduct experiments with various control strategies, making them suitable for both educational and research purposes.

The primary input to the models is an analog signal u, interpreted based on the IACC parameter setting:

for IACC=on, the input u is assumed to be a force acting on the cart [N],
for IACC=off, the models assume the input represents speed [m/s].

The R1 signal is used to reset each model to its initial configuration.

Both models can be precisely configured with a series of parameters that reflect the system’s physical characteristics. These include the relative center of gravity positions a, moments of inertia J, lengths l, and masses of the pendulums m, as well as damping coefficients b and the initial state of the system (positions d_0, velocities dd_0). A schematic representation of the system with parameters is shown below. The parameters are intuitively defined. The relative position of the center of gravity for the i-th pendulum, $a_{i}$ , is determined by the equation

|A_{i - 1} T_{i}| = a_{i} l_{i},

where $A_{i - 1}$ is the position of the previous joint, $T_{i}$ is the position of the pendulum’s center of gravity, and $l_{i}$ is the length of the pendulum.

The computation of the models adheres to the mathematical model and physical parameters detailed in the literature [6]. The IPEN2pu and IPEN3pu blocks are used for simulating the inverted pendulum models with dynamic parameters.

This block does not propagates the signal quality. More information can be found in the 1.4 section.

Input

u	Analog input of the block	Double (F64)
R1	Block reset	Bool

Parameter

a1	Relative position of center of gravity $⊙$ 1.0	Double (F64)
a2	Relative position of center of gravity $⊙$ 1.0	Double (F64)
a3	Relative position of center of gravity $⊙$ 1.0	Double (F64)
J1	Moment of inertia of pendulum $⊙$ 1.0	Double (F64)
J2	Moment of inertia of pendulum $⊙$ 1.0	Double (F64)
J3	Moment of inertia of pendulum $⊙$ 1.0	Double (F64)
l1	Length of pendulum [m] $⊙$ 1.0	Double (F64)
l2	Length of pendulum [m] $⊙$ 1.0	Double (F64)
l3	Length of pendulum [m] $⊙$ 1.0	Double (F64)
m1	Mass of pendulum [kg] $⊙$ 1.0	Double (F64)
m2	Mass of pendulum [kg] $⊙$ 1.0	Double (F64)
m3	Mass of pendulum [kg] $⊙$ 1.0	Double (F64)
m0	Mass of cart [kg] $⊙$ 1.0	Double (F64)
b1	Damping coefficient of pendulum $⊙$ 1.0	Double (F64)
b2	Damping coefficient of pendulum $⊙$ 1.0	Double (F64)
b3	Damping coefficient of pendulum $⊙$ 1.0	Double (F64)
b0	Damping coefficient of cart $⊙$ 1.0	Double (F64)
d1_0	Initial angle of pendulum [rad] $⊙$ 1.0	Double (F64)
d2_0	Initial angle of pendulum [rad] $⊙$ 1.0	Double (F64)
d3_0	Initial angle of pendulum [rad] $⊙$ 1.0	Double (F64)
x_0	Initial position of cart [m] $⊙$ 1.0	Double (F64)
dd1_0	Initial angular velocity of pendulum [rad/s] $⊙$ 1.0	Double (F64)
dd2_0	Initial angular velocity of pendulum [rad/s] $⊙$ 1.0	Double (F64)
dd3_0	Initial angular velocity of pendulum [rad/s] $⊙$ 1.0	Double (F64)
dx_0	Initial velocity of cart [m/s] $⊙$ 1.0	Double (F64)
IACC	on=Input u is velocity, off=Input u is force	Bool

Output

d1	Angle of pendulum [rad]	Double (F64)
d2	Angle of pendulum [rad]	Double (F64)
d3	Angle of pendulum [rad]	Double (F64)
x	Position of cart [m]	Double (F64)
dd1	Angular velocity of pendulum [rad/s]	Double (F64)
dd2	Angular velocity of pendulum [rad/s]	Double (F64)
dd3	Angular velocity of pendulum [rad/s]	Double (F64)
dx	Velocity of cart [m/s]	Double (F64)
E	Error indicator	Bool

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