Balancing the Inverted Pendulum
Table of Contents

BEGIN HERE WITH SYNOPSIS

• EQUATIONS OF MOTION: The Inverted Pendulum
  1. Diagram
  2. Coordinates of point cg in terms of point p, angle , and length
  3. Sum of forces in X direction
  4. Sum of forces in Y direction
  5. Sum of moments about center of gravity
  6. Combine Equations
  7. Derivation of Transfer Function for the Inverted Pendulum
  
  
• EQUATIONS OF MOTION: The Free Hanging Pendulum
  1. Diagram
  2. Coordinates of point cg in terms of point p, angle , and length
  3. Sum of forces in X direction
  4. Sum of forces in Y direction
  5. Sum of moments about center of gravity
  6. Combine Equations
  7. Solution of characteristic equation of motion for the Free Hanging Pendulum
  8. Determination of Pendulum Parameters
  
  
• EQUATIONS OF MOTION: The Cart and Motor
  1. Diagram
  2. Basic Equations of Motor and Amplifier
  3. Combine Equations
  4. Calculate Torque
  5. Substitute Back in
  6. Observe following relations
  7. Substitute in
  8. Derivation of Transfer Function for the Cart and Motor
     
  9. Determination of parameters for the Cart and Motor
     • The Bode Plot: Equations
     • The Bode Plot: Diagram
     • Square Wave Input for 
     • Square Wave Input for 
  
  
• CONTINUOUS TIME TRANSFER FUNCTIONS
  1. Block Diagram
  2. DERIVATION OF STATE SPACE REPRESENTATION
     • State space equations for the cart and inverted pendulum
     • Calculation of Transfer Functions from State Space Equation
  
  
• DISCRETIZATION OF CONTINUOUS TIME STATE SPACE EQUATIONS
  1. Discretization Of Continuous Time State Space Equations
     • Calculation of Inverse Laplace Transforms
     • Row 1 Column 1
     • Row 1 Column 2
     • Row 2 Column 2
     • Row 3 Column 2
     • Row 4 Column 2
     • General Equation for Rows (3,3) (3,4) (4,3) (4,4)
     • Row 3 Column 3
     • Row 3 Column 4
     • Row 4 Column 3
     • Row 4 Column 4
  2. Calculation of G
  3. Calculation of G when = 0
  4. Calculation of H when = 0
  
  
• FEEDBACK
  1. Discussion of Observable Properties
  2. Discussion of Feedback
  3. Procedure to obtain
  4. Procedure to obtain
  5. Block Diagram of System with Discrete-Time Feedback
  6. Motor Dead Range
  7. Block Diagram of System with Discrete-Time Feedback and Motor Deadspace Cancelation
  
  
• CONTROLLING THE PENDULUM: AN EXAMPLE
  1. Pendulum Setup
  2. Determining Parameters for the Cart and Motor
  3. COMPLETE LIST OF PARAMETERS
  4. Determination of Feedback Control Parameters
  5. Control of the Pendulum
  6. Analysis of the results
  
  
• LINKS FOR FURTHER RESEARCH