Review for Midterm Exam

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

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Introduction to ROBOTICS

### Review for Midterm Exam

Dr. John (Jizhong) Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

Outline
• Homework Highlights
• Course Review
• Midterm Exam Scope
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

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

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 4

Find the dynamic model of 2-link robot with mass equally distributed

• Derivation of L-E Formula

Velocity of point

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
• 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
• 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

Scaling

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

Manipulator Kinematics

Forward

Jacobian Matrix

Kinematics

Inverse

Jacobian Matrix: Relationship between joint

space velocity with task space velocity

Joint Space

Manipulator Kinematics
• Steps to derive kinematics model:
• Assign D-H coordinates frames
• Transformation matrices of adjacent joints
• Calculate kinematics model
• chain product of successive coordinate transformation matrices
• When necessary, Euler angle representation
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 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

J

1

-90

0

13

2

0

8

0

3

90

0

-l

4

-90

0

8

5

90

0

0

6

0

0

t

: angle from Xi-1to Xi

: 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

Kinematics:

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

Jacobian Matrix Revisit

Forward Kinematics

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 planning
• Path Profile
• Velocity Profile
• Acceleration Profile
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
• 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 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

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

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
• 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 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
• Non-linear Feedback Control

Robot System:

Jocobian:

• Non-linear Feedback Control

Nonlinear feedback controller:

Then the linearized dynamic model:

Linear Controller:

Error dynamic equation:

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!

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

Time: 6:30-9:00