# Review for Midterm Exam

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30 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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City College of New York

1

Dr. John (Jizhong) Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

Review for Midterm Exam

Introduction to ROBOTICS

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Outline

Homework Highlights

Course Review

Midterm Exam Scope

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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

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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.

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Homework 4

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

Calculate D, H, C terms directly

Physical meaning?

Interaction effects of motion of joints j & k on link i

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Homework 4

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

Derivation of L
-
E Formula

i

Velocity of point

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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:

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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

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Course Coverage

Robot
Manipulator

Kinematics

Dynamics

Control

Mobile Robot

Kinematics/Control

Sensing and Sensors

Motion planning

Mapping/Localization

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Robot Manipulator

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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

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Composite Rotation Matrix

A sequence of finite rotations

matrix multiplications do not commute

rules:

if rotating coordinate O
-
U
-
V
-
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

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Homogeneous Representation

A frame in space (Geometric
Interpretation)

Principal axis
n

w.r.t. the reference coordinate system

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Manipulator Kinematics

Joint Space

Forward

Inverse

Kinematics

Jacobian Matrix: Relationship between joint

space velocity with task space velocity

Jacobian
Matrix

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Manipulator Kinematics

Steps to derive kinematics model:

Assign D
-
H coordinates frames

Calculate kinematics model

chain product of successive coordinate transformation
matrices

When necessary, Euler angle representation

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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

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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

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: angle from Z
i
-
1

to Z
i
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

i
-
1

t

0

0

6

0

0

90

5

8

0

-
90

4

0

0

90

3

8

0

2

13

0

-
90

1

J

-
l

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Example:
Puma 560

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Jacobian Matrix

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

Kinematics:

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Jacobian Matrix Revisit

Forward Kinematics

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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.

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Trajectory planning

Path Profile

Velocity Profile

Acceleration Profile

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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

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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,

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Manipulator Dynamics

Dynamics Model of n
-

The Acceleration
-
related Inertia
matrix term, Symmetric

The Coriolis and Centrifugal terms

The Gravity terms

Driving torque

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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:

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Manipulator Dynamics

: 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

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Robot Motion Control

Joint level PID control

each joint is a servo
-
mechanism

neglect dynamic behavior of whole arm

high speed

performance depends on configuration

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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

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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:

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Non
-
linear Feedback Control

Jocobian:

Robot System:

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Nonlinear feedback controller:

Non
-
linear Feedback Control

Then the linearized dynamic model:

Linear Controller:

Error dynamic equation:

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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

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Thank you!

Next class:
Oct. 23

(Tue): Midterm Exam

Time: 6:30
-
9:00