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

2. Grades Distribution 37 students taking Exam Minimum grade: 37 Maximum grade: 99

3. Q1 and Q2 • Q1 (a): 11/37 • Q1 (b): 14/37 • Q2: 1/37

4. 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

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

6. Jacobian Matrix Revisit Forward Kinematics

7. 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:

8. 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

9. Manipulator Dynamics • 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. Robot Motion Control • 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

11. 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