Areas Interest in Robotics

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Areas Interest in Robotics

• Industrial Engineering Department
• Binghamton University

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Outline

• Introduction
• Historical Example
• Mechanical Engineering and Robotics
• Review of Basic Kinematics and Dynamics
• Transformation Matrices/Denavit-Hartenberg
• Dynamics and Controls
• Example Surgical Robot

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Robot Configurations
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Cylindrical
Cartesian
Spherical
SCARA
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Phillip John McKerrow, Introduction to Robotics
(1991)
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Review of Basic Kinematics and Dynamics

• Case Study Dynamic Analysis
• Software for Dynamic Analysis ADAMS
• Rigid Body Kinematics
• Rigid Body Dynamics

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Kinematics of Rigid Bodies

• General Plane Motion Translation plus Rotation

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Kinematics of Rigid Bodies (cont.)

• Translation
• If a body moves so that all the particles have at
  time t the same velocity relative to some
  reference, the body is said to be in translation
  relative to this reference.

Rectilinear Translation
Curvilinear Translation
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Kinematics of Rigid Bodies (cont.)

• Rotation
• If a rigid body moves so that along some straight
  line all the particles of the body, or a
  hypothetical extension of the body, have zero
  velocity relative to some reference, the body is
  said to be in rotation relative to this
  reference.
• The line of stationary particles is called the
  axis of rotation.

Motion
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Kinematics of Rigid Bodies (cont.)

• General Plane Motion can be analyzed as
• A translation plus a rotation.
• Chasles Theorem
• Select any point A in the body. Assume that all
  particles of the body have at the same time t a
  velocity equal to vA, the actual velocity of the
  point A.
• 2. Superpose a pure rotational velocity w about
  an axis going through point A.

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Kinematics of Rigid Bodies (cont.)

• General Plane Motion drA ? drB

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Kinematics of Rigid Bodies (cont.)

• General Plane Motion
• (1) Translation
• measured
• from original
• Point A

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Kinematics of Rigid Bodies (cont.)

• General Plane Motion
• (2) Rotation about axis through Point A

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Kinematics of Rigid Bodies (cont.)

• General Plane Motion Translation Rotation

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Kinematics of Rigid Bodies (cont.)
Derivative of a Vector Fixed in a Moving Reference
A
P
Two Reference Frames
XYZ
x'y'z'
Let R be the vector that establishes the relative
position between XYZ and x'y'z'.
Let A be the fixed vector that establishes the
position between A and P.
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Kinematics of Rigid Bodies (cont.)
The time rate of change of A as seen from x'y'z'
is zero
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Kinematics of Rigid Bodies (cont.)
As seen from XYZ, the time rate of change of A
will not necessarily be zero. Determine the time
derivative by applying Chasles Theorem. 1.
Translation. Translational motion of R will not
alter the magnitude or direction of A. (The line
of action will change but the direction will
not.)
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Kinematics of Rigid Bodies (cont.)
2. Rotation. Rotation about an axis passing
through O'w
Establish a second stationary reference frame,
X'Y'Z', such that the Z' axis coincides with the
axis of rotation.
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Kinematics of Rigid Bodies (cont.)
Locate a set of cylindrical coordinates at the
end of A.
Because A is a fixed vector, the magnitudes Ar,
Aq, and AZ' are constant. Therefore
Also, eZ' is unchanging, therefore
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Kinematics of Rigid Bodies (cont.)
The time derivative as seen from the X'Y'Z'
reference frame is
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Kinematics of Rigid Bodies (cont.)
The result for the time derivative as seen from
the X'Y'Z' reference frame is
Both the X'Y'Z' reference frame and the XYZ
reference frame are stationary reference frames,
therefore
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Kinematics of Rigid Bodies (cont.)
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Kinematics of Rigid Bodies (cont.)
For acceleration, differentiate
By the product rule
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Kinematics of Rigid Bodies (cont.)
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Kinematics of Rigid Bodies (cont.)
Summary of Equations Kinematics of Rigid Bodies
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Degrees of Freedom
Kinematics of Rigid Bodies (cont.)
Degrees of Freedom (DOF) df. The number of
independent parameters (measurements,
coordinates) which are needed to uniquely define
a systems position in space at any point of time.
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Kinematics of Rigid Bodies (cont.)
A rigid body in plane motion has three DOF.
Note The three parameters are not unique. x, y,
q is one set of three coordinates
O
r, f, q is also a set of three coordinates
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Kinematics of Rigid Bodies (cont.)
y
A rigid body in 3-D space has six DOF.
For example, x, y, z three linear coordinates
and f, q, y three angular coordinates
O
X
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Links, Joints, and Kinematic Chains
Kinematics of Rigid Bodies (cont.)
Link df. A rigid body which possesses at least
two nodes which are points for attachment to
other links.
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Kinematics of Rigid Bodies (cont.)
Joint df. A connection between two or more
links (at their nodes) which allows some motion,
or potential motion, between the connected
links. Also called kinematic pairs.
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Kinematics of Rigid Bodies (cont.)
Type of contact between links Lower pair
surface contact Higher pair line or point
contact
Six Lower Pairs
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Kinematics of Rigid Bodies (cont.)
CS 480A-34
Constrained Pin
Screw
Slide
Sliding Pin
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Kinematics of Rigid Bodies (cont.)
Ball and Socket
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Kinematics of Rigid Bodies (cont.)
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Open/Closed Kinematic Chain (Mechanism)
Kinematics of Rigid Bodies (cont.)
Closed Kinematic Chain df. A kinematic chain in
which there are no open attachment points or
nodes.
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Kinematics of Rigid Bodies (cont.)
Open Kinematic Chain df. A kinematic chain in
which there is at least one open attachment point
or node.
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Dynamics of Rigid Bodies

• Dynamic Equivalence
• Lumped Parameter Dynamic Model

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Dynamic System Model
For a model to be dynamically equivalent to the
original body, three conditions must be satisfied
1. The mass (m) used in the model must equal the
mass of the original body.
2. The Center of Gravity (CG) in the model must
be in the same location as on the original body.
3. The mass moment of inertia (I) used in the
model must equal the mass moment of inertia of
the original body.
m, CG, I
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First Moment of Mass and Center of Gravity (CG)
Dynamics of Rigid Bodies (cont.)
The first moment of mass, or mass moment (M),
about an axis is the product of the mass and the
distance from the axis of interest.
where r is the radius from the axis of interest
to the increment of mass
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Second Moment of Mass, Mass Moment of Inertia (I)
Dynamics of Rigid Bodies (cont.)
The second moment of mass, or mass moment of
inertia (I), about an axis is the product of the
mass and the distance squared from the axis of
interest.
where r is the radius from the axis of interest
to the increment of mass
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Lumped Parameter Dynamic Models
Dynamics of Rigid Bodies (cont.)
The dynamic model of a mechanical system involves
lumping the dynamic properties into three basic
elements
Mass (m or I)
m
Spring
Damper
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Manipulator Dynamics and Control

• Forward Kinematics Given the angles and/or
  extensions of the arm, determine the position of
  the end of the manipulator
• Inverse Kinematics Given the position of the
  end of the manipulator, determine the angles
  and/or extensions of the arm needed to get there
• Dynamics Determine the forces and torques
  required for or resulting from the given
  kinematic motions.
• Control Given the block diagram model of the
  dynamic system, determine the feedback loops and
  gains needed to accomplish the desired
  performance (overshoot, settling time, etc.)

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Forward KinematicsDenavit-Hartenberg (D-H)
Transformation Matrix

• Forward Kinematics Given the angles and/or
  extensions of the arm, determine the position of
  the end of the manipulator

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Position Kinematics
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While the kinematic analysis of a robot
manipulator can be carried out using any
arbitrary reference frame, a systematic approach
using a convention known as the
Denavit-Hartenberg (D-H) convention is commonly
used. Any homogeneous transformation is
represented as the product of four 'basic"
transformations
Mark W. Spong and M. Vidyasagar, Robot Dynamics
and Control (1989)
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Mark W. Spong and M. Vidyasagar, Robot Dynamics
and Control (1989)
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Example
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• Given the angles, q1 and q2, along with the link
  lengths, a1 and a2, the position of the end point
  of the two-link planar manipulator with respect
  to the base of the manipulator can be found using
  the D-H transformation matrix

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• Similarly for any robot configuration

Stanford manipulator configuration
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where
d6
d3
d2
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Velocity Kinematics
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• Jacobian
• The Jacobian is a matrix valued function of
  derivatives.

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Linear Velocities
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Inverse Kinematics

• Inverse Kinematics Given the position of the
  end of the manipulator, determine the angles
  and/or extensions of the arm needed to get there

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In general the problem can be stated Given the
4x4 D-H homogeneous transformation
Find one (or all) of the solutions of the equation
In other words, solve the system of equations
Mark W. Spong and M. Vidyasagar, Robot Dynamics
and Control (1989)
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For example, the system of nonlinear
trigonometric equations for the Stanford
manipulator is
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There is no simple, universal method to solve
inverse kinematic problems. A common technique
used for a 6 DOF robot with a 3 DOF end-effector
(roll, pitch, yaw) is "kinematic decoupling"
find a location for the robot wrist and then
determine the orientation of the end-effector.
Also, in general, there is no unique solution to
the inverse kinematic problem.
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Robot Dynamics

• Dynamics Determine the forces and torques
  required for or resulting from the given
  kinematic motions.

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Robot Controls

• Control Given the block diagram model of the
  dynamic system, determine the feedback loops and
  gains needed to accomplish the desired
  performance (overshoot, settling time, etc.)

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Feedback Control System
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DC Motor
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Surgical Instrument
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Good software cannot fix the problems caused by
poor mechanical design. Phillip John
McKerrow, Introduction to Robotics (1991)