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Manipulator Control

Introduction to ROBOTICS. Manipulator Control. Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu. Outline. Homework Highlights Robot Manipulator Control Control Theory Review Joint-level PD Control Computed Torque Method

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Manipulator Control

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  1. Introduction to ROBOTICS Manipulator Control Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu

  2. Outline • Homework Highlights • Robot Manipulator Control • Control Theory Review • Joint-level PD Control • Computed Torque Method • Non-linear Feedback Control • Midterm Exam Scope

  3. Homework 2 Find the forward kinematics, Roll-Pitch-Yaw representation of orientation Joint variables ? Why use atan2 function? Inverse trigonometric functions have multiple solutions: Limit x to [-180, 180] degree

  4. Homework 3 Find kinematics model of 2-link robot, Find the inverse kinematics solution Inverse: know position (Px,Py,Pz) and orientation (n, s, a), solve joint variables.

  5. Homework 4 Find the dynamic model of 2-link robot with mass equally distributed • Calculate D, H, C terms directly Physical meaning? Interaction effects of motion of joints j & k on link i

  6. Homework 4 Find the dynamic model of 2-link robot with mass equally distributed • Derivation of L-E Formula Erroneous answer Velocity of point Kinetic energy of link i point at link i

  7. Homework 4 Example: 1-link robot with point mass (m) concentrated at the end of the arm. Set up coordinate frame as in the figure According to physical meaning:

  8. Manipulator Control

  9. Manipulator Dynamics Revisit • Dynamics Model of n-link Arm The Acceleration-related Inertia term, Symmetric Matrix The Coriolis and Centrifugal terms The Gravity terms Driving torque applied on each link Non-linear, highly coupled , second order differential equation • Joint torque Robot motion

  10. Jacobian Matrix Revisit Forward Kinematics

  11. (x , y) 2 l2 l1 1 Jacobian Matrix Revisit • Example: 2-DOF planar robot arm • Given l1, l2 ,Find: Jacobian

  12. Robot Manipulator Control • Robot System: • Joint Level Controller Find a control input (tor), • Task Level Controller Find a control input (tor),

  13. Robot Manipulator Control • Control Methods • Conventional Joint PID Control • Widely used in industry • Advanced Control Approaches • Computed torque approach • Nonlinear feedback • Adaptive control • Variable structure control • ….

  14. d e dt Error signal e yd - ya actual ya V desired yd compute V using PID feedback - Motor actual ya Control Theory Review (I) PID controller: Proportional / Integral / Derivative control e= yd- ya V = Kp• e + Ki ∫ e dt + Kd ) Closed Loop Feedback Control Reference book: Modern Control Engineering, Katsuhiko Ogata, ISBN0-13-060907-2

  15. Evaluating the response overshoot steady-state error ss error -- difference from the system’s desired value settling time overshoot -- % of final value exceeded at first oscillation rise time -- time to span from 10% to 90% of the final value settling time -- time to reach within 2% of the final value How can we eliminate the steady-state error? rise time

  16. Control Performance, P-type Kp = 20 Kp = 50 Kp = 200 Kp = 500

  17. Control Performance, PI - type Kp = 100 Ki = 50 Ki = 200

  18. You’ve been integrated... Kp = 100 unstable & oscillation

  19. Control Performance, PID-type Kp = 100 Kd = 5 Kd = 2 Ki = 200 Kd = 10 Kd = 20

  20. PID final control

  21. Control Theory Review (II) • Linear Control System • State space equation of a system • Example: a system: • Eigenvalue of Aare the root of characteristic equation • Asymptotically stable all eigenvalues of A have negative real part (Equ. 1)

  22. Control Theory Review (II) • Find a state feedback control such that the closed loop system is asymptotically stable • Closed loop system becomes • Chose K, such that all eigenvalues of A’=(A-BK) have negative real parts (Equ. 2)

  23. Control Theory Review (III) • Feedback linearization • Nonlinear system • Example: Original system: Nonlinear feedback: Linear system:

  24. Robot Motion Control (I) • Joint level PID control • each joint is a servo-mechanism • adopted widely in industrial robot • neglect dynamic behavior of whole arm • degraded control performance especially in high speed • performance depends on configuration

  25. Robot Motion Control (II) • Computed torque method • Robot system: • Controller: How to chose Kp, Kv ? Error dynamics Advantage: compensated for the dynamic effects Condition: robot dynamic model is known

  26. Robot Motion Control (II) How to chose Kp, Kv to make the system stable? Error dynamics Define states: In matrix form: Characteristic equation: The eigenvalue of A matrix is: One of a selections: • Condition: have negative real part

  27. Robot Motion Control (III) • Non-linear Feedback Control Robot System: Jocobian:

  28. Robot Motion Control (III) • Non-linear Feedback Control Design the nonlinear feedback controller as: Then the linearized dynamic model: Design the linear controller: Error dynamic equation:

  29. Midterm Exam Scopes

  30. Thank you! HWK 5 posted on the web, Next Class: Midterm Exam Time: 6:30-9:00, Please on time!

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