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

City College of New York 2 Outline Homework Highlights Course Review Midterm Exam Scope

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

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.

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

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

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:

City College of New York 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

City College of New York 9 Course Coverage Robot Manipulator –Kinematics –Dynamics –Control Mobile Robot –Kinematics/Control –Sensing and Sensors –Motion planning –Mapping/Localization

City College of New York 10 Robot Manipulator

City College of New York 11 Position vector Rotation matrix Scaling 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 Homogeneous Transformation

City College of New York 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

City College of New York 13 Homogeneous Representation A frame in space (Geometric Interpretation) Principal axis n w.r.t. the reference coordinate system

City College of New York 14 Manipulator Kinematics Joint SpaceTask Space Forward Inverse Kinematics Jacobian Matrix: Relationship between joint space velocity with task space velocity Jacobian Matrix

City College of New York 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

City College of New York 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 Z i 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 Z i & Z i-1 or at the intersection of common normal between the Z i & Z i-1 axes and the Z i axis. Establish X i axis. Establish or along the common normal between the Z i-1 & Z i axes when they are parallel. Establish Y i axis. Assign to complete the right-handed coordinate system. Find the link and joint parameters

City College of New York 17 Denavit-Hartenberg Convention 1.Number the joints 2.Establish base frame 3.Establish joint axis Z i 4.Locate origin, (intersect. of Z i & Z i-1 ) OR (intersect of common normal & Z i ) 5.Establish X i,Y i

City College of New York 18 Link Parameters : angle from Z i-1 to Z i about X i : distance from intersection of Z i-1 & X i to O i along X i Joint distance : distance from O i-1 to intersection of Z i-1 & X i along Z i-1 : angle from X i-1 to X i about Z i-1 t J -l-l

City College of New York 19 Example: Puma 560

City College of New York 20 Jacobian Matrix Jacobian is a function of q, it is not a constant! Kinematics:

City College of New York 21 Jacobian Matrix Revisit Forward Kinematics

City College of New York 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.

City College of New York 23 Trajectory planning Path Profile Velocity Profile Acceleration Profile

City College of New York 24 Trajectory Planning n-th order polynomial, must satisfy 14 conditions, 13-th order polynomial trajectory trajectory t0  t1, 5 unknow t1  t2, 4 unknow t2  tf, 5 unknow

City College of New York 25 Manipulator Dynamics 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 Joint torques Robot motion, i.e. position velocity,

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

City College of New York 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:

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

City College of New York 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

City College of New York 30 Joint Level Controller 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 exactly

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

City College of New York 32 Non-linear Feedback Control Jocobian: Robot System: Task Level Controller

City College of New York 33 Task Level Controller Nonlinear feedback controller: Non-linear Feedback Control Then the linearized dynamic model: Linear Controller: Error dynamic equation:

City College of New York 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

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