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

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. How to chose Kp, Kv to make the system stable?
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