Implementation of a LQR in a 3 inverted pendulum (simulink model))

To implement the humanoid motion, the first step that need to achieved is its stabilization.
Stabilization is to keep quiet in a humanoid in a certain position.
In first approach, we will considerer that robot has the two legs parallels and the foot connect with the floor with 1DoF.
Thus, the control stabilization of the jacobian is similar like the stabilization of a 3 inverted pendulum under-actuated.
Before the tests with the real or the EZphisics model we perform the first tests with a MatLab/Simulink model, build with the SimMechanics blocks.
For detailed information about the construction of the 3 inverted link model and the implementations of the LQR in MatLab/Simulink see the report here.

Simulations results (continuous LQR):
We performed simulations of the system under vertical stabilization, corresponding to a period of 5 seconds. To analyze the robustness of the LQR we implemented an abrupt perturbation of 1N.m in the torque (along 1 second) of the 2nd actuator in t=1s.

Results for the follow  Ricatti equation parameters:
– Q = 0.1xI6
– R = 1xI2

Results for the follow  Ricatti equation parameters:
– Q = 10xI6
– R = 1xI2

Simulations results (discrete LQR):
Considering that the humanoid will be controlled through a PC with a sampling rate of 100Hz, it is necessary to design a digital controller for this condition.

To analyze the robustness of the LQR we implemented an abrupt perturbation of 0.25N.m in the torque (along 0.2 second) of the 2nd actuator in t=1s.

Results for the follow  Ricatti equation parameters:
– Q = 10xI6
– R = 1xI2