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

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

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

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

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

Jacobian Matrix Revisit Forward Kinematics

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:

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

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

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

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