lecture 6 modeling and linearization(1)
DESCRIPTION
Linear Control SystemTRANSCRIPT
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MAE 506: Advanced System Modeling, Dynamics and Control
Lecture 6
Reading in Williams and Lawrence text:
Section 1.4
Spring Berman Fall 2014
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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
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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
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Simplifed Ballbot Model Use Lagranges equations to
derive equations of motion
Total kinetic energy: Total potential energy: Lagrangian:
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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
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Model in Nonlinear State-Space Form
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Stabilizing Feedback Controller
Add a state variable:
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LQR Control
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Linearize Eqs of Motion, Apply LQR Control
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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