MAE 506: Advanced System Modeling, Dynamics and Control

Lecture 6

Reading in Williams and Lawrence text:

Section 1.4

Spring Berman Fall 2014

Ballbot

Robotics Institute at Carnegie Mellon University, 2006

Tohoku Gakuin University, 2008

Lauwers, Kantor, and Hollis. A Dynamically Stable Single-Wheeled Mobile Robot with Inverse Mouse-Ball Drive. ICRA 2006

Simplifed Ballbot Model Ball wheel is a rigid sphere Body is rigid Control inputs: Torques applied

between the ball and the body

No slip between the wheel and the floor (no skidding)

Friction between wheel/floor and wheel/body is modeled as viscous damping

Can design a controller for full 3D system by designing independent controllers for 2 planar models

Simplifed Ballbot Model Use Lagranges equations to

derive equations of motion

Total kinetic energy: Total potential energy: Lagrangian:

Simplifed Ballbot Model Friction terms: Euler-Lagrange equations:

Mass matrix

Vector of Coriolis and centrifugal forces

Vector of gravitational forces

Friction terms

Component of torque applied between ball and body in direction normal to plane

Model in Nonlinear State-Space Form

Stabilizing Feedback Controller

Add a state variable:

LQR Control

Linearize Eqs of Motion, Apply LQR Control

References on Nonlinear Dynamics & Control

Slotine and Li, Applied Nonlinear Control, 1991 Sastry, Nonlinear Systems: Analysis, Stability, and

Control, 1999 Khalil, Nonlinear Systems, 3rd ed., 2001 Strogatz, Nonlinear Dynamics and Chaos, 2nd ed.,

2014