Stabilisation and Motion Control of Unstable Objects

by Alexander M. Formalskii

Overview

Systems with mechanical degrees of freedom containing unstable objects are analysed in this monograph and algorithms for their control are developed, discussed, and numerically tested. This is achieved by identifying unstable modes of motion and using all available resources to suppress them. By using this approach the region of states from which a stable regime can be reached is maximised.

The systems discussed in this book are models for pendula and vehicles and find applications in mechatronics, robotics as well as in mechanical and automotive engineering.

Product Details

ISBN-13:	9783110375824
Publisher:	De Gruyter
Publication date:	11/03/2015
Series:	De Gruyter Studies in Mathematical Physics , #33
Pages:	255
Product dimensions:	6.69(w) x 9.45(h) x 0.03(d)
Age Range:	18 Years

About the Author

Alexander Formalskii, Lomonosov State University, Moscow, Russia.

Table of Contents

Preface
Chapter 1. Devices Containing One-Link Pendulum
§ 1. Physical pendulum with fixed suspension point
1. Motion equations
2. Controllability domain
3. Maximization of attraction domain
4. Delay in feedback loop
5. Nonlinear control
6. Controllability domain for nonlinear model
§ 2. Pendulum with suspension point located at the center of a wheel
1. Motion equations
2. Controllability domain
3. Maximization of attraction domain
4. Nonlinear control
§ 3. Pendulum with flywheel
1. Design of the pendulum with flywheel
2. Motion equations
3. Local stabilization of the pendulum in the upper unstable equilibrium
4. Damping of the flywheel angular speed
5. Rocking and damping of the pendulum
6. Transferring the pendulum from the lower equilibrium to the upper one
7. Numerical study
8. Experiments
§ 4. Wheel rolling control using a pendulum
1. Mathematical model of the device
2. Steady motions
3. Stability of steady motions
§ 5. Optimal rocking and damping of the swing
1. On design of optimal feedback control for second-order systems
2. Mathematical model of the swing
3. Maximization of the swing amplitude
4. Minimization of the swing amplitude
5. Control of the swing with consideration of aerodynamic resistance and dry friction
§ 6. Pendulum control with minimal energy cost
1. Energy cost estimation
2. Transferring the pendulum to the unstable equilibrium
3. Transferring the pendulum to the stable equilibrium
Chapter 2. Two-link Physical Pendulum
§ 7. Local stabilization of inverted pendulum with one control torque
1. Mathematical model of the pendulum
2. Linearized model
3. Controllability domains
4. Feedback control design, maximization of the attraction domain
5. Numerical study
§ 8. Optimal control design of two-link pendulum rocking and damping
1. Mathematical model
2. Reduced angle
3. Optimal control for rocking the pendulum
4. Optimal control for damping the pendulum
5. On transferring the pendulum from the lower equilibrium to the upper one
§ 9. Global stabilization of inverted pendulum with control applied at the hinge between links
1. Mathematical model
2. Cascade form of dynamics equations
3. Control for rocking the pendulum
4. Tracking of desired inter-link angle variation
5. Local stabilization of the inverted pendulum
6. Numerical study
§ 10. Global stabilization of the inverted pendulum with control applied at the suspension point
1. Mathematical model
2. Pendulum rocking
3. Straightening of the pendulum
4. Linear model, local stabilization
5. Numerical study
§ 11. Multilink pendulum on a movable base
1. Multilink pendulum on a wheel
2. One-link pendulum on a wheel
3. Global stabilization of inverted pendulum
4. Controllability domain
5. Design of time-optimal trajectories
6. Pendulum on a cart
7. Decrease of frequencies under imposed constraint
Chapter 3. Ball on a Beam
§ 12. Stabilization of a ball on a rectilinear beam
1. Mathematical model of the system
2. Linearized model
3. Feedback control synthesis
4. Numerical study
§ 13. Stabilization of a ball on a circular beam
1. Mathematical model
2. Linearized model
3. Feedback control
4. Numerical study
Chapter 4. Gyroscopic Stabilization of Robot-Bicycle
§ 14. Designs of two bicycle types
1. Bicycle with one steering wheel
2. Bicycle with two steering wheels
3. Gyroscopic stabilizer
4. Equations of bicycle roll oscillations
§ 15. Control law for stabilizing bicycle in the upright position
1. Measurement of bicycle roll angle with accelerometers
2. Bicycle motion along a straight line
3. Bicycle motion along a circular line
4. Numerical and experimental study
References

Editorial Reviews

"The book can serve as an excellent basis for a graduate course on control and stabilization of multi-body mechanical systems. It provides a researcher with a comprehensive account of approaches and methods of designing controls stabilizing unstable equilibria. It will be a valuable reference to the classical and modern literature on the subject."
Oleg N. Kirillov in: Mathematical Reviews Clippings, July 2018, MR3726859

From the Publisher