Optimal LQG controller design for self-balancing robot

LQG controller design process with MATLAB & Simulink®

Control requirements: follow a sinusoidal position reference with an 0.5m amplitude and 15s period remaining stable. The tolerance allowed is 1s in time and 10cm in position.

Formulation of plant model:

• Euler-Lagrange equations
• Non-linear model state-space representation
• Non-linear model linearization around a fixed point
• LTI model state-space representation
• Sensor modeling: Quadrature encoder and IMU (Ref. MPU6050) with gyroscope and accelerometer

Full-state feedback design:

• Controllability analysis
• Manual pole placement
• Linear Quadratic Regulator (LQR) design
• Non-minimal phase system analysis

Full-state estimator design:

• Observability analysis
• Manual estimator's pole placement
• Linear Quadratic Estimator (Kalman filter) design

Mechanical model with Simscape Multibody™

In order to have a more reliable simulation, a mechanical model of the robot was implemented with Simscape Multibody™ (formerly SimMechanics™) which provides a multibody simulation environment for 3D mechanical systems.

An automatically generated 3D animation provides a visualization of the system's dynamic.

Following reference (LQR)	Disturbance rejection (LQR)

An interesting feature of this model is the wheel-floor contact model by using the external Simscape Multibody Contact Forces Library. The contact is modeled as a sphere to plane force and, as a small disclaimer, it's working but with a different behavior from the previous analysis. It's neccesary to tune the different parameters to make it works as supposed: Contact stiffness, contact damping, static and kinetic friction, etc. (If you find the parameters to make it works please make a pull request!).

Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Author and institution

Gonzalo Gabriel Fernández. Rodrigo Gonzalez (advisor). Facultad de Ingeniería, Universidad Nacional de Cuyo