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Introduction to ROBOTICS. Review for Midterm Exam. Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York Outline. Homework Highlights Course Review Midterm Exam Scope. Homework 2.

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review for midterm exam

Introduction to ROBOTICS

Review for Midterm Exam

Dr. John (Jizhong) Xiao

Department of Electrical Engineering

City College of New York

  • Homework Highlights
  • Course Review
  • Midterm Exam Scope
homework 2
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

homework 3
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.

homework 4
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

homework 41
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

homework 42
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:

course review
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
course coverage
Course Coverage
  • Robot Manipulator
    • Kinematics
    • Dynamics
    • Control
  • Mobile Robot
    • Kinematics/Control  
    • Sensing and Sensors
    • Motion planning
    • Mapping/Localization

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


composite rotation matrix
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
Homogeneous Representation
  • A frame in space (Geometric Interpretation)

Principal axis n w.r.t. the reference coordinate system

manipulator kinematics
Manipulator Kinematics


Jacobian Matrix



Jacobian Matrix: Relationship between joint

space velocity with task space velocity

Joint Space

Task Space

manipulator kinematics1
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
denavit hartenberg convention
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
denavit hartenberg convention1
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
link parameters


























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

jacobian matrix
Jacobian Matrix


Jacobian is a function of q, it is not a constant!

jacobian matrix revisit
Jacobian Matrix Revisit

Forward Kinematics

trajectory planning
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.
trajectory planning1
Trajectory planning
  • Path Profile
  • Velocity Profile
  • Acceleration Profile
trajectory planning2
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

manipulator dynamics
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
manipulator dynamics1
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


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 dynamics2
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

robot motion control
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
joint level controller
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

robot motion control1
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
task level controller
Task Level Controller
  • Non-linear Feedback Control

Robot System:


task level controller1
Task Level Controller
  • Non-linear Feedback Control

Nonlinear feedback controller:

Then the linearized dynamic model:

Linear Controller:

Error dynamic equation:

midterm exam scope
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
thank you
Thank you!

Next class: Oct. 23 (Tue): Midterm Exam

Time: 6:30-9:00