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

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

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

7. The City College of New York 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. The City College of New York 8 Manipulator Control

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

10. The City College of New York 10 Jacobian Matrix Revisit Forward Kinematics

11. The City College of New York 11 Jacobian Matrix Revisit Example: 2-DOF planar robot arm –Given l 1, l 2, Find: Jacobian 22 11 (x, y) l2l2 l1l1

12. The City College of New York 12 Robot Manipulator Control Joint Level Controller Task Level Controller Find a control input (tor), Robot System:

13. The City College of New York 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. The City College of New York 14 Control Theory Review (I) actual  a desired  d V Motor actual  a - compute V using PID feedback  d  a Error signal e PID controller : Proportional / Integral / Derivative control e=  d  a V = K p e + K i ∫ e dt + K d ) d e dt Closed Loop Feedback Control Reference book: Modern Control Engineering, Katsuhiko Ogata, ISBN0-13-060907-2

15. The City College of New York 15 Evaluating the response How can we eliminate the steady-state error? steady-state error settling time rise time overshoot 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 ss error -- difference from the system’s desired value

16. The City College of New York 16 Control Performance, P-type K p = 20 K p = 200 K p = 50 K p = 500

17. The City College of New York 17 Control Performance, PI - type K p = 100 K i = 50K i = 200

18. The City College of New York 18 You’ve been integrated... K p = 100 unstable & oscillation

19. The City College of New York 19 Control Performance, PID-type K p = 100 K i = 200K d = 2 K d = 10K d = 20 K d = 5

20. The City College of New York 20 PID final control

21. The City College of New York 21 Control Theory Review (II) Linear Control System –State space equation of a system –Example: a system: –Eigenvalue of A are the root of characteristic equation –Asymptotically stable all eigenvalues of A have negative real part (Equ. 1)

22. The City College of New York 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. The City College of New York 23 Control Theory Review (III) Feedback linearization –Nonlinear system –Example: Original system: Nonlinear feedback: Linear system:

24. The City College of New York 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. The City College of New York 25 Robot Motion Control (II) Computed torque method –Robot system: –Controller: Error dynamics How to chose Kp, Kv ? Advantage: compensated for the dynamic effects Condition: robot dynamic model is known

26. The City College of New York 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: Condition: have negative real part One of a selections:

27. The City College of New York 27 Robot Motion Control (III) Non-linear Feedback Control Jocobian: Robot System:

28. The City College of New York 28 Robot Motion Control (III) Design the nonlinear feedback controller as: Non-linear Feedback Control Then the linearized dynamic model: Design the linear controller: Error dynamic equation:

29. The City College of New York 29 Midterm Exam Scopes

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