Areas Interest in Robotics

transcript

Industrial Automation

Industrial Engineering Department

Binghamton University

Outline• Introduction

– Historical Example– Mechanical Engineering and Robotics

• Review of Basic Kinematics and Dynamics• Transformation Matrices/Denavit-

Hartenberg• Dynamics and Controls• Example: Surgical Robot

Robot Configurations

Cartesian Cylindrical

SphericalSCARA

Phillip John McKerrow, Introduction to Robotics (1991)

Review of Basic Kinematics and Dynamics

• Case Study: Dynamic Analysis

• Software for Dynamic Analysis: ADAMS

• Rigid Body Kinematics

• Rigid Body Dynamics

Kinematics of Rigid Bodies

General Plane Motion: Translation plus Rotation

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

RotationIf 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

General Plane Motion can be analyzed as: A translation plus a rotation.

Chasle’s Theorem:1. 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 about an axis going through point A.

General Plane Motion: drA drB

General Plane Motion; (1) Translation

measured from originalPoint A

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

General Plane Motion = Translation + Rotation

Derivative of a Vector Fixed in a Moving Reference O

Two Reference Frames:XYZx'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.

AKinematics of Rigid Bodies (cont.)

AThe time rate of change of A as seen from x'y'z' is zero: R

AAs 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.)

2. Rotation. Rotation about an axis passing through O':

Establish a second stationary reference frame, X'Y'Z', such that the Z' axis coincides with the axis of rotation.

Locate a set of cylindrical coordinates at the end of A.

'' ZZrr AAAA

Because A is a fixed vector, the magnitudes Ar, A, and AZ' are constant. Therefore:

0' Zr AAA

Also, Z' is unchanging, therefore:

The time derivative as seen from the X'Y'Z' reference frame is: R

''''''''' ZYXZYX

Recall: rr

and Note:

The result for the time derivative as seen from the X'Y'Z' reference frame is: R

XYZZYX dt

Both the X'Y'Z' reference frame and the XYZ reference frame are stationary reference frames, therefore

ZYXXYZ

For: A

AAAAAAAA

000000

'' ZZrr AAAA

Ar 0 00 0 rA

For acceleration, differentiate:

By the product rule:

XYZXYZXYZdt

Summary of Equations: Kinematics of Rigid Bodies

B rraa AB

ABrvv AB

Degrees of Freedom

Degrees of Freedom (DOF) = df. The

number of independent parameters (measurements, coordinates) which are needed to uniquely define a system’s position in space at any point of time.

A rigid body in plane motion has three DOF.

Note: The three parameters are not unique.x, y, – is one set of three coordinates

r, , – is also a set of three coordinates

A rigid body in 3-D space has six DOF.

For example,x, y, z – three linear coordinates and – three angular coordinates

Links, Joints, and Kinematic Chains

Link = df. A rigid body which

possesses at least two nodes which are points for attachment to other links.

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.”

Type of contact between links

Lower pair: surface contact

Higher pair: line or point contactSix Lower Pairs

“Constrained Pin” “Screw”

“Slide” “Sliding Pin”

Kinematics of Rigid Bodies (cont.) CS 480A-34

Planar (F) Joint – 3 DOF

“Ball and Socket”

Open/Closed Kinematic Chain (Mechanism)Closed Kinematic Chain = df. A kinematic chain in

which there are no open attachment points or nodes.

Open Kinematic Chain = df. A kinematic chain in

which there is at least one open attachment point or node.

Dynamics of Rigid Bodies

Dynamic Equivalence

Lumped Parameter Dynamic Model

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

First Moment of Mass and Center of Gravity (CG)

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

Dynamics of Rigid Bodies (cont.)

Second Moment of Mass, Mass Moment of Inertia (I)

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.

m dmrI 2

where: r is the radius from the axis of interest to the increment of mass

Lumped Parameter Dynamic Models

The dynamic model of a mechanical system involves “lumping” the dynamic properties into three basic elements:

Mass (m or I)

Spring

Damper

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.)

Forward Kinematics:Denavit-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

Position Kinematics

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)

iiii xaxdzzi RotTransTransRotA ,,,,

Start hereStart here

End hereEnd here

iiii xaxdzzi RotTransTransRotA ,,,,

sascccs

casscsc

iiiiii

Example

1211221212121

sascscaccsssccs

cassccacsscsscc

12121 12

cacac sinsincoscoscos

sincoscossinsin 21 s

121 12 sasa

Given the angles, 1 and 2, 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:

2112121

sasacs

cacasc

Similarly for any robot configuration:

1000333231

232221

131211

Stanford manipulator configuration:

52452632

2155142541621321

5412515421621321

5254233

54152542123

54152542113

65264654232

646541652646542122

646541652646542112

65264654231

646541652646542121

646541652646542111

sscccddcd

sscssccsscddcdssd

ssssccscccddsdscd

ccscsr

ssccssccsr

ssscsscccr

ssccssccsr

ccscscsssssscccsr

ccscssssssssccccr

cscsscccsr

scccsccssssccccsr

scccsscsssscccccr

where:

Velocity Kinematics

JacobianThe Jacobian is a matrix valued function of derivatives.

coscoscos

sinsinsin

21221111

21221211

2qqlqqlql

qqlqqlql

Linear Velocities

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

In general the problem can be stated:Given the 4x4 D-H homogeneous transformation

Find one (or all) of the solutions of the equation

AAAqqqT

...,...,,

:where

,...,,

In other words, solve the system of equations:

hqqqT ijnij

,...,3 ,2 ,13 ,2 ,1

:where

,...,, 21

52452632

2155142541621321

5412515421621321

5254233

54152542123

54152542113

65264654232

646541652646542122

646541652646542112

65264654231

646541652646542121

646541652646542111

sscccddcd

sscssccsscddcdssd

ssssccscccddsdscd

ccscsr

ssccssccsr

ssscsscccr

ssccssccsr

ccscscsssssscccsr

ccscssssssssccccr

cscsscccsr

scccsccssssccccsr

scccsscsssscccccr

For example, the system of nonlinear trigonometric equations for the Stanford manipulator is:

Solve for: 1, 2, 4, 5, 6, d3

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.

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

Robot Dynamics

• 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.)

Robot Controls

Feedback Control System

DC Motor

Surgical Instrument

Good software cannot fix the problems caused by poor mechanical design. – Phillip John McKerrow, Introduction to Robotics

(1991)

Areas Interest in Robotics

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