More details and examples on robot arms and kinematics Denavit-Hartenberg Notation.

More details and examples on More details and examples on robot arms and kinematicsrobot arms and kinematics

Denavit-Hartenberg Denavit-Hartenberg NotationNotation

INTRODUCTIONINTRODUCTION

Forward Kinematics: to determine where the robot’s hand is? (If all joint variables are known)

Inverse Kinematics: to calculate what each joint variable is? (If we desire that the hand be located at a particular point)

Direct Kinematics

Direct Kinematics with no matriceswith no matrices

Where is my hand?

Direct Kinematics:HERE!

Direct Kinematics• Position of tip in (x,y) coordinates

Direct Kinematics Algorithm1) Draw sketch2) Number links. Base=0, Last link = n3) Identify and number robot joints4) Draw axis Zi for joint i5) Determine joint length ai-1 between Zi-1 and Zi

6) Draw axis Xi-1

7) Determine joint twist i-1 measured around Xi-1

8) Determine the joint offset di

9) Determine joint angle i around Zi

10+11) Write link transformation and concatenate

Often sufficient for

2D

Kinematic Problems for Manipulation

• Reliably position the tip - go from one position to another position

• Don’t hit anything, avoid obstacles

• Make smooth motions – at reasonable speeds and

– at reasonable accelerations

• Adjust to changing conditions - – i.e. when something is picked up respond to the change in weight

ROBOTS AS ROBOTS AS MECHANISMsMECHANISMs

Robot Kinematics: Robot Kinematics: ROBOTS AS MECHANISM

Fig. 2.1 A one-degree-of-freedom closed-loop four-bar mechanism

Multiple type robot have multiple DOF. (3 Dimensional, open loop, chain mechanisms)

Fig. 2.2 (a) Closed-loop versus (b) open-loop mechanism

Chapter 2Robot Kinematics: Position Analysis

Fig. 2.3 Representation of a point in space

A point P in space : 3 coordinates relative to a reference frame

^^^

kcjbiaP zyx

Representation of a Point Representation of a Point in Spacein Space


Fig. 2.4 Representation of a vector in space

A Vector P in space : 3 coordinates of its tail and of its head

^^^__

kcjbiaP zyx

w

z

y

x

P__

Representation of a Vector Representation of a Vector in Spacein Space


Fig. 2.5 Representation of a frame at the origin of the reference frame

Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

zzz

yyy

xxx

aon

aon

aon

F

Representation of a Frame Representation of a Frame at the at the OriginOrigin of a Fixed-Reference Frame of a Fixed-Reference Frame


Fig. 2.6 Representation of a frame in a frame

Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

1000

zzzz

yyyy

xxxx

Paon

Paon

Paon

F

Representation of a Frame Representation of a Frame in ain a Fixed Reference Frame Fixed Reference Frame

The same as last slide


Fig. 2.8 Representation of an object in space

An object can be represented in space by attaching a frame to it and representing the frame in space.

1000

zzzz

yyyy

xxxx

objectPaon

Paon

Paon

F

Representation of a Rigid Representation of a Rigid BodyBody


A transformation matrices must be in square form.

• It is much easier to calculate much easier to calculate the inverse of square matrices. • To multiply two matrices, their dimensions must match.

1000

zzzz

yyyy

xxxx

Paon

Paon

Paon

F

HOMOGENEOUS HOMOGENEOUS TRANSFORMATION MATRICESTRANSFORMATION MATRICES

Representation Representation of of

Transformations Transformations of rigid objects of rigid objects

in 3D space in 3D space


Fig. 2.9 Representation of an pure translation in space

A transformation is defined as making a movement in space.

• A pure translation.• A pure rotation about an axis.• A combination of translation or rotations.

1000

100

010

001

z

y

x

d

d

d

T

Representation of a Pure Representation of a Pure TranslationTranslation

identity

Same value a


Fig. 2.10 Coordinates of a point in a rotating frame before and after rotation around axis x.

Assumption : The frame is at the origin of the reference frame and parallel to it.

Fig. 2.11 Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis.

Representation of a Pure Representation of a Pure RotationRotation about an Axis about an Axis

Projections as seen from x axis

x,y,z n, o, a

Fig. 2.13 Effects of three three successive successive transformations

A number of successive translations and rotations….

Representation of Combined Representation of Combined Transformations Transformations

Order is Order is importantimportant

x,y,z n, o, a

nioi

ai T1

T2

T3

Fig. 2.14 Changing the order of transformations will change the final result

Order of Order of TransformationTransformations is important s is important

x,y,z n, o, a

translation


Fig. 2.15 Transformations relative to the current frames.

Example 2.8

Transformations Relative Transformations Relative to the Rotating Frameto the Rotating Frame

translation

rotation

MATRICES FORMATRICES FORFORWARDFORWARD AND AND

INVERSEINVERSE KINEMATICS OF KINEMATICS OF

ROBOTSROBOTS

• For position• For orientation


Fig. 2.17 The hand frame of the robot relative to the reference frame.

Forward Kinematics Analysis: • Calculating the position and orientation of the hand of the robot.

If all robot joint variables are known, one can calculate where the robot is at any instant. .

FORWARDFORWARD AND AND INVERSEINVERSE KINEMATICS KINEMATICS OF ROBOTSOF ROBOTS


Forward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates.

Forward and Inverse KinematicsForward and Inverse Kinematics Equations for Equations for PositionPosition


IBM 7565 robot • All actuator is linear. • A gantry robot is a Cartesian robot.

Fig. 2.18 Cartesian Coordinates.

1000

100

010

001

z

y

x

cartPR

P

P

P

TT

Forward and Inverse KinematicsForward and Inverse Kinematics Equations for PositionEquations for Position (a) (a) CartesianCartesian (Gantry, (Gantry,

Rectangular) CoordinatesRectangular) Coordinates


2 Linear translations and 1 rotation • translation of r along the x-axis • rotation of about the z-axis • translation of l along the z-axis

Fig. 2.19 Cylindrical Coordinates.

1000

100

0

0

l

rSCS

rCSC

TT cylPR

,0,0))Trans(,)Rot(Trans(0,0,),,( rzllrTT cylPR

Forward and Inverse KinematicsForward and Inverse Kinematics Equations Equations for Position:for Position:

Cylindrical CoordinatesCylindrical Coordinates

cosine

sine


2 Linear translations and 1 rotation • translation of r along the z-axis • rotation of about the y-axis • rotation of along the z-axis

Fig. 2.20 Spherical Coordinates.

1000

0

rCCS

SrSSSCSC

CrSCSSCC

TT sphPR

))Trans()Rot(Rot()( 0,0,,,,, yzlrsphPR TT

Forward and Inverse KinematicsForward and Inverse Kinematics Equations Equations for Positionfor Position

(c) (c) Spherical CoordinatesSpherical Coordinates


3 rotations -> Denavit-Hartenberg representation

Fig. 2.21 Articulated Coordinates.

Forward and Inverse KinematicsForward and Inverse Kinematics Equations Equations for Positionfor Position

(d) (d) Articulated CoordinatesArticulated Coordinates


Roll, Pitch, Yaw (RPY) angles

Euler angles

Articulated joints

Forward and Inverse Kinematics Equations for

Orientation


Roll: Rotation of about -axis (z-axis of the moving frame)

Pitch: Rotation of about -axis (y-axis of the moving frame)

Yaw: Rotation of about -axis (x-axis of the moving frame)

aaon

on

Fig. 2.22 RPY rotations about the current axes.

Forward and Inverse Kinematics Equations for Orientation

(a) Roll, Pitch, Yaw(RPY) Angles


Fig. 2.24 Euler rotations about the current axes.

Rotation of about -axis (z-axis of the moving frame) followed by

Rotation of about -axis (y-axis of the moving frame) followed by

Rotation of about -axis (z-axis of the moving frame).

a

oa

Forward and Inverse Kinematics Equations for Orientation

(b) Euler Angles


)()( ,,,, noazyxcartHR RPYPPPTT

)()( ,,,, EulerTT rsphHR

Assumption : Robot is made of a Cartesian and an RPY set of joints.

Assumption : Robot is made of a Spherical Coordinate and an Euler angle.

Another Combination can be possible……

Denavit-Hartenberg Representation

Forward and Inverse Forward and Inverse KinematicsKinematics Equations for OrientationEquations for Orientation

Roll, Pitch, Yaw(RPY) Angles

Forward and Forward and Inverse Inverse

Transformations Transformations for robot armsfor robot arms

Fig. 2.16 The Universe, robot, hand, part, and end effecter frames.

Steps of calculation of an Inverse matrix: 1. Calculate the determinant of the matrix.2. Transpose the matrix.3. Replace each element of the transposed matrix by its own minor

(adjoint matrix).4. Divide the converted matrix by the determinant.

INVERSE OF TRANSFORMATION MATRICES

Identity Identity TransformationsTransformations

1. We often need to calculate

INVERSE MATRICES

2.2. It is good to reduce the number It is good to reduce the number of such operationsof such operations

3.3. We need to do these calculations We need to do these calculations fastfast

How to find an Inverse Matrix B of How to find an Inverse Matrix B of matrix A?matrix A?

Inverse Homogeneous Inverse Homogeneous TransformationTransformation

Homogeneous Coordinates• Homogeneous coordinates: embed 3D

vectors into 4D by adding a “1”• More generally, the transformation matrix

T has the form:

FactorScalingTrans. Perspect.

Vector Trans. MatrixRot.T

a11 a12 a13 b1

a21 a22 a23 b2

a31 a32 a33 b3

c1 c2 c3 sf

It is presented in more detail on the WWW!

For various types of robots For various types of robots we have different we have different maneuvering spacesmaneuvering spaces

For various types of robots we calculate For various types of robots we calculate different forward and inverse different forward and inverse

transformationstransformations

For various types of robots we solve different forward and inverse For various types of robots we solve different forward and inverse kinematic problemskinematic problems

Forward and Forward and Inverse Kinematics: Inverse Kinematics: Single Link ExampleSingle Link Example

Forward and Inverse Kinematics: Forward and Inverse Kinematics: Single Link ExampleSingle Link Example

easy

Denavit – Denavit – Hartenberg Hartenberg

ideaidea

Denavit-Hartenberg Representation :

Fig. 2.25 A D-H representation of a general-purpose joint-link combination

@ Simple way of modeling robot links and joints for any robot configurationany robot configuration, regardless of its sequence or complexity.

@ Transformations in any coordinatesin any coordinates is possible.

@ Any possible combinations of joints and links and all-revolute articulated robots can be represented.

DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT


⊙ : A rotation angle between two links, about the z-axis

(revolute).

⊙ d : The distance (offsetoffset) on the z-axis, between links

(prismatic).

⊙ a : The length of each common normal (Joint offset).

⊙ : The “twist” angle between two successive z-axes (Joint

twist) (revolute)

Only and d are joint variables.

DENAVIT-HARTENBERG REPRESENTATIONDENAVIT-HARTENBERG REPRESENTATION

Symbol Terminologies :

Links are in 3D, any shape

associated with Zi always

Only rotation Only translation Only offset

Only offset Only rotation Axis alignment

DENAVIT-DENAVIT-HARTENBERG HARTENBERG

REPRESENTATIOREPRESENTATION for N for each linkeach link

4 link parameters4 link parameters


⊙ : A rotation angle between two links, about the z-axis

(revolute).

⊙ d : The distance (offset) on the z-axis, between links

(prismatic).

⊙ a : The length of each common normal (Joint offset).

⊙ : The “twist” angle between two successive z-axes (Joint

twist) (revolute)



Symbol Terminologies :

Example with three Revolute Joints

i (i-1) a(i-1) di

i

0 0 0 0 0

1 0 a0 0 1

2 -90 a1 d2 2

Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2

d2

a0 a1

Denavit-Hartenberg Link Denavit-Hartenberg Link Parameter TableParameter Table

The DH The DH Parameter Parameter

TableTable

Apply first Apply last

Denavit-Hartenberg Representation of Denavit-Hartenberg Representation of Joint-Link-Joint TransformationJoint-Link-Joint Transformation

NotationNotation for Denavit-Hartenberg for Denavit-Hartenberg Representation of Joint-Link-Joint Representation of Joint-Link-Joint

TransformationTransformation

Alpha applied first

Four Transformations from one Joint to the NextFour Transformations from one Joint to the Next

• Order of multiplication of matrices is inverse of order of applying them• Here we show order of matrices

Joint-Link-JointJoint-Link-Joint

Denavit-Hartenberg

Representation of Joint-Link-

Joint Transformation

• Alpha is applied first

How to How to create a create a single single matrix A matrix A nn

EXAMPLE: Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Type 1 LinkType 1 Link

Final matrix from previous slide

substitute

substitute

Numeric or symbolic matrices

The Denavit-The Denavit-Hartenberg Matrix for Hartenberg Matrix for another link typeanother link type

1000

cosαcosαsinαcosθsinαsinθ

sinαsinαcosαcosθcosαsinθ

0sinθcosθ

i1)(i1)(i1)(ii1)(ii

i1)(i1)(i1)(ii1)(ii

1)(iii

d

d

a

• Similarity to HomegeneousSimilarity to Homegeneous: Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. from one coordinate frame to the next.

• Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.

Z(i - 1)

X(i -1)

Y(i -1)

( i -

1)

a(i - 1 )

Z i Y

i X

i

a

i

d

i

i

Put the transformation here for every link

1.1. InIn DENAVIT- DENAVIT-HARTENBERG HARTENBERG REPRESENTATION REPRESENTATION we we must be able to find must be able to find parameters for parameters for each linkeach link

2.2. So we must know link typesSo we must know link types

Links between Links between revoluterevolute joints joints

llnn=0=0

Type 3 Type 3 LinkLink

Joint n+1Joint n+1

Joint nJoint n

ddnn=0=0

Link nLink n

xxn-1n-1

xxnn

ln=0dn=0

Type 4 Link

Origins coincide

n-1n-1

Joint n+1Joint n+1

Joint nJoint n

Part of Part of ddn-1n-1

Link nLink n

xxn-1n-1 yyn-1n-1

xxnn

nn

Links between Links between prismaticprismatic

jointsjoints

ForwardForward and and InverseInverse Transformations Transformations on on

MatricesMatrices

Start point:

• Assign joint number n to the first shown joint.

• Assign a local reference frame for each and

every joint before or

after these joints.

• Y-axis is not used in D-H representation.

DENAVIT-HARTENBERG DENAVIT-HARTENBERG REPRESENTATION REPRESENTATION

PROCEDURESPROCEDURES

1. .All joints are represented by a z-axis ٭

• (right-hand rule for rotational joint, linear movement for

prismatic joint)

2. The common normal is one line mutually perpendicular to

any two skew lines.

3. Parallel z-axes joints make a infinite number of common

normal.

4. Intersecting z-axes of two successive joints make no

common normal between them(Length is 0.).

DENAVIT-HARTENBERG REPRESENTATION

Procedures for assigning a local reference frame to each joint:


⊙ : A rotation about the z-axis.

⊙ dd : The distance on the z-axis.

⊙ aa : The length of each common normal (Joint

offset).

⊙ : The angle between two successive z-axes

(Joint twist)



Symbol Terminologies ReminderReminder:


(I) Rotate about the zn-axis an able of n+1. (Coplanar)

(II) Translate along zn-axis a distance of dn+1 to make xn and xn+1

colinear.

(III) Translate along the xn-axis a distance of an+1 to bring the

origins

of xn+1 together.

(IV) Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis

with zn+1-axis.

DENAVIT-HARTENBERG REPRESENTATION

The necessary motions to transform from one reference frame from one reference frame to the

next.

More details and examples on robot arms and kinematics Denavit-Hartenberg Notation.

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