Review for Midterm Exam

1. Introduction to ROBOTICS Review for Midterm Exam Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu

2. Outline • Homework Highlights • Course Review • 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. Course Review • What are Robots? • Machines with sensing, intelligence and mobility (NSF) • Why use Robots? • Perform 4A tasks in 4D environments 4A: Automation, Augmentation, Assistance, Autonomous • 4D: Dangerous, Dirty, Dull, Difficult

9. Course Coverage • Robot Manipulator • Kinematics • Dynamics • Control • Mobile Robot • Kinematics/Control   • Sensing and Sensors • Motion planning • Mapping/Localization

10. Robot Manipulator

11. Homogeneous Transformation Homogeneous Transformation Matrix • Composite Homogeneous Transformation Matrix • Rules: • Transformation (rotation/translation) w.r.t. (X,Y,Z) (OLD FRAME), using pre-multiplication • Transformation (rotation/translation) w.r.t. (U,V,W) (NEW FRAME), using post-multiplication Rotation matrix Position vector Scaling

12. Composite Rotation Matrix • A sequence of finite rotations • matrix multiplications do not commute • rules: • if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix • if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

13. Homogeneous Representation • A frame in space (Geometric Interpretation) Principal axis n w.r.t. the reference coordinate system

14. Manipulator Kinematics Forward Jacobian Matrix Kinematics Inverse Jacobian Matrix: Relationship between joint space velocity with task space velocity Joint Space Task Space

15. Manipulator Kinematics • Steps to derive kinematics model: • Assign D-H coordinates frames • Find link parameters • Transformation matrices of adjacent joints • Calculate kinematics model • chain product of successive coordinate transformation matrices • When necessary, Euler angle representation

16. Denavit-Hartenberg Convention • Number the joints from 1 to n starting with the base and ending with the end-effector. • Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1. • Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1. • Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. • Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel. • Establish Yi axis. Assign to complete the right-handed coordinate system. • Find the link and joint parameters

17. Denavit-Hartenberg Convention • Number the joints • Establish base frame • Establish joint axis Zi • Locate origin, (intersect. of Zi & Zi-1) OR (intersect of common normal & Zi ) • Establish Xi,Yi

18. J 1 -90 0 13 2 0 8 0 3 90 0 -l 4 -90 0 8 5 90 0 0 6 0 0 t Link Parameters : angle from Xi-1to Xi about Zi-1 : angle from Zi-1 to Zi about Xi : distance from intersection of Zi-1 & Xi to Oialong Xi Joint distance : distance from Oi-1 to intersection of Zi-1 & Xi along Zi-1

19. Example: Puma 560

20. Jacobian Matrix Kinematics: Jacobian is a function of q, it is not a constant!

21. Jacobian Matrix Revisit Forward Kinematics

22. Trajectory Planning • Motion Planning: • Path planning • Geometric path • Issues: obstacle avoidance, shortest path • Trajectory planning, • “interpolate” or “approximate” the desired path by a class of polynomial functions and generates a sequence of time-based “control set points” for the control of manipulator from the initial configuration to its destination.

23. Trajectory planning • Path Profile • Velocity Profile • Acceleration Profile

24. Trajectory Planning • n-th order polynomial, must satisfy 14 conditions, • 13-th order polynomial • 4-3-4 trajectory • 3-5-3 trajectory t0t1, 5 unknow t1t2, 4 unknow t2tf, 5 unknow

25. Manipulator Dynamics • Joint torques Robot motion, i.e. position velocity, • Lagrange-Euler Formulation • Lagrange function is defined • K: Total kinetic energy of robot • P: Total potential energy of robot • : Joint variable of i-th joint • : first time derivative of • : Generalized force (torque) at i-th joint

26. Manipulator Dynamics • Dynamics Model of n-link Arm The Acceleration-related Inertia matrix term, Symmetric The Coriolis and Centrifugal terms Driving torque applied on each link The Gravity terms

27. Example 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:

28. Manipulator Dynamics • Potential energy of link i : Center of mass w.r.t. base frame : Center of mass w.r.t. i-th frame : gravity row vector expressed in base frame • Potential energy of a robot arm Function of

29. Robot Motion Control • 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

30. Joint Level Controller • 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 exactly

31. Robot Motion Control 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

32. Task Level Controller • Non-linear Feedback Control Robot System: Jocobian:

33. Task Level Controller • Non-linear Feedback Control Nonlinear feedback controller: Then the linearized dynamic model: Linear Controller: Error dynamic equation:

34. Midterm Exam Scope • Study lecture notes • Understand homework and examples • Have clear concept • 2.5 hour exam • close book, close notes • But you can bring one-page cheat sheet

35. Thank you! Next class: Oct. 23 (Tue): Midterm Exam Time: 6:30-9:00