Kinematics

Time to Derive Kinematics Model of the Robotic Arm

Amirkabir University of TechnologyComputer Engineering & Information Technology Department

Direct Kinematics

Where is my hand?

Direct Kinematics:HERE!

Kinematics of Manipulators

Objective:

To drive a method to compute the position and orientation of the manipulator’s end-effector relative to the base of the manipulator as a function of the joint variables.

Degrees of Freedom

The number of :

• Independent position variables needed to locate all parts of the mechanism,

• Different ways in which a robot arm can move,

• Joints

The The degrees of freedomdegrees of freedom of a rigid body is defined as of a rigid body is defined as the number of independent movements it has.the number of independent movements it has.

DOF of a Rigid Body

In a planeIn a plane

In spaceIn space

Degrees of Freedom

As DOF

3 position3 orientation

3D Space = 6 DOF

In robotics:DOF = number of independently driven joints

computational complexitycostflexibilitypower transmission is more difficult

positioning accuracy

Robot Links and Joints

In open kinematics chains (i.e. Industrial Manipulators):

{No of D.O.F. = No of Joints}

A manipulator may be thought of as a set of bodies (links) connected in a chain by joints.

Lower Pair

The connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another

The Six Possible Lower Pair Joints

Higher PairA higher pair joint is one which contact occurs only at isolated points or along a line segments

Robot Joints

Spherical Joint3 DOF ( Variables - 1, 2, 3)

Revolute Joint1 DOF ( Variable - )

Prismatic Joint1 DOF (linear) (Variables - d)

Robot Specifications

Number of axes

Major axes, (1-3) => position the wrist

Minor axes, (4-6) => orient the tool

Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration

An Example - The PUMA 560

The PUMA 560 has SIX revolute joints.A revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle.

1

23

4

There are two more joints on the end-effector (the gripper)

Note on Joints

Without loss of generality, we will consider only manipulators which have joints with a single degree of freedom. A joint having n degrees of freedom can be modeled as n joints of one degree of freedom connected with n-1 links of zero length.

Link

A link is considered as a rigid body which defines the relationship between two neighboring joint axes of a manipulator.

Link n

n+1a n

n

Joint n+1

Joint n

z n

x n

x n+1

z n+1

x n

z n

The Kinematics Function of a Link

The kinematics function of a link is to maintain a fixed relationship between the two joint axes it supports.This relationship can be described with two parameters: the link length a, the link twist

Link Length

Is measured along a line which is mutually perpendicular to both axes.

The mutually perpendicular always exists and is unique except when both axes are parallel.

Link twist

Project both axes i-1 and i onto the plane whose normal is the mutually perpendicular line, and measure the angle between them

Right-hand sense

Link Length and Twist Axis i

Axis i-1

a i-1

i-1

Joint Parameters

A joint axis is established at the connection of two links. This joint will have two normals connected to it one for each of the links.

The relative position of two links is called link offset dn whish is the distance between the links (the displacement, along the joint axes between the links).

The joint angle n between the normals is measured in a plane normal to the joint axis.

Link and Joint Parameters

a i-1

Axis i-1

Axis i

dii

i-1

ai-1

Link and Joint Parameters4 parameters are associated with each link. You

can align the two axis using these parameters.

Link parameters:

a0 the length of the link.

n the twist angle between the joint axes.

Joint parameters:

n the angle between the links.

dn the distance between the links

Link Connection Description:

For Revolute Joints: a, , and d. are all fixed, then “i” is the. Joint Variable.

For Prismatic Joints: a, , and . are all fixed, then “di” is the. Joint Variable.

These four parameters: (Link-Length ai-1), (Link-Twist i-1(, (Link-Offset di), (Joint-Angle i) are known as the Denavit-Hartenberg Link Parameters.

A 3-DOF Manipulator Arm

0

1

23

Links Numbering Convention

Base of the arm:Link-01st moving link:Link-1

. .

. .

. .

Last moving linkLink-n

Link 0

Link 1

Link 2Link 3

First and Last Links in the Chain

a0= n=0.0

0= n=0.0

If joint 1 is revolute: d0= and 1 is arbitrary

If joint 1 is prismatic: d0= arbitraryand 1 =

Affixing Frames to LinksIn order to describe the location of each link

relative to its neighbors we define a frame attached to each link.

The Z axis is coincident with the joint axis i.

The origin of frame is located where ai perpendicular intersects the joint i axis.

The X axis points along ai( from i to i+1).If ai = 0 (i.E. The axes intersect) then Xi is perpendicular to axes i and i+1.

The Y axis is formed by right hand rule.

Affixing Frames to LinksFirst and last links

Base frame (0) is arbitrary Make life easy Coincides with frame {1} when joint parameter is 0

Frame {n} (last link) Revolute joint n:

Xn = Xn-1 when n = 0 Origin {n} such that dn=0

Prismatic joint n: Xn such that n = 0 Origin {n} at intersection of joint axis n and Xn when dn=0

Link n-1

Link n

zn-1 yn-1

xn-1

zn

xn

yn

zn+1

xn+1

yn+1

dnn

Joint n+1

an

Joint n-1Joint n

an-1

Affixing Frames to Links

Affixing Frames to LinksNote: assign link frames so as to cause as many

link parameters as possible to become zero!

The reference vector z of a link-frame is always on a joint axis.

The parameter di is algebraic and may be negative. It is constant if joint i is revolute and variable when joint i is prismatic.

The parameter ai is always constant and positive.

a i is always chosen positive with the smallest possible magnitude.

The robot can now be kinematically modeled by using the link transforms ie: 

Where 0

nT is the pose of the end-effector relative to base; Ti is the link transform for the ith joint; and n is the number of links.

The Kinematics Model

nin TTTTTT 3210

The Denavit-Hartenberg (D-H) Representation

In the robotics literature, the Denavit-Hartenberg (D-H) representation has been used, almost universally, to derive the kinematic description of robotic manipulators.

The Denavit-Hartenberg (D-H) Representation

The appeal of the D-H representation lies in its algorithmic approach. The method begins with a systematic approach to assigning and labeling an orthonormal (x,y,z) coordinate system to each robot joint. It is then possible to relate one joint to the next and ultimately to assemble a complete representation of a robot's geometry.

Denavit-Hartenberg Parameters

Axis i-1

a i-1

i-1

Axis i

Link i

di

i

The Link Parametersai = the distance from zi to zi+1.

measured along xi.

i = the angle between zi and zi+1.

measured about xi.

di = the distance from xi-1 to xi.

measured along z i.

i = the angle between xi-1 to xi.

measured about z i

General Transformation Between Two Bodies

In D-H convention, a general transformation between two bodies is defined as the product of four basic transformations:

A translation along the initial z axis by d,

A rotation about the initial z axis by ,

A translation along the new x axis by a, and.

A rotation about the new x axis by .

A General Transformation in D-h Convention

D-H transformation for adjacent coordinate frames:

44,,,,

1

ITTTT xaxdzz

i

iT

Denavit-Hartenberg Convention

D1. 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.D2. Initialize and loop Steps D3 to D6 for I=1,2,….n-1D3. Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1.

D4. 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.

D5. Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.

D6. Establish Yi axis. Assign to complete the right-handed coordinate system.

),,( 000 ZYX

iiiii ZZZZX 11 /)(

iiiii XZXZY /)(

0Z

Denavit-Hartenberg Convention

D7. Establish the hand coordinate systemD8. Find the link and joint parameters : d,a,,

D-H transformation for adjacent coordinate frames:

44,,,,

1

ITTTT xaxdzz

i

iT

Example

Joint i i ai di

i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

Joint 1

Joint 2

Joint 3

Example

))()(( 2103213

0 TTTT

1000

01iii

iiiiiii

iiiiiii

ii dCS

SaCSCCS

CaSSSCC

T

Example

Joint i i ai di

i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

1000

0100

0cosθsinθ

0sinθcosθ

00

00

000

00

1 sin

cos

a

a

T

1000

000

sinθ

cosθ

1

1

1 1

sincos0

cossin0

111

111

2

a

a

T

1000

0sinθcosθ 22

22

223

100

00cossin

0

dT

Example (3.3):

Link Frame Assignments

Example (3.3):

1000

0100

00

0

1000

0100

00

0

1000

0100

00

00

1000

0

33

233

23

22

122

12

11

11

1000101

1000101

011

01

cs

Lsc

Tcs

Lsc

T

cs

sc

dccscss

dsscccs

asc

T

Example (3.3):

.

1000

0.00.10.00.0

0.0

0.0

12211123123

12211123123

03

slslcs

clclsc

TTBW

Example:SCARA Robot

The location of the sliding axis Z2 is arbitrary, since it is a free vector. For simplicity, we make it coincident with Z3 . thus 2 and d2 are arbitrarily set.

The placement of O3 and X3 along Z3 is arbitrary, since Z2 and Z3 are coincident. Once we choose O3, however, then the joint displacement d3 is defined.

We have also placed the end link frame in a convenient manner, with the Z4 axis coincident with the Z3 axis and the origin O4 displaced down into the gripper by d4.

Example:SCARA Robot

Example: Puma 560

Joint i i

i ai(mm) di(mm)

1 1 0 0 0

2 2 -90 0 d2

3 3 0 a2 d3

4 4 90 a3 d4

5 5 -90 0 0

6 6 0 0 0

Example: Puma 560

Forearm of a PUMA

a3

x5

y5

x6

z6

x3

y3

x4

z4

d4

Spherical joint

Example: Puma 560Different Configuration

Link Coordinate Parameters

Joint i i i ai(mm) di(mm)

1 1 -90 0 0

2 2 0 431.8 149.09

3 3 90 -20.32 0

4 4 -90 0 433.07

5 5 90 0 0

6 6 0 0 56.25

PUMA 560 robot arm link coordinate parameters

Example: Puma 560

Example: Puma 560

The Tool Transform

A robot will be frequently picking up objects or tools.

Standard practice is to to add an extra homogeneous transformation that relates the frame of the object or tool to a fixed frame in the end-effector.

Kinematic Calibration

How one knows the DH parameters? Certainly when robots are built, there are design specifications. Yet due to manufacturing tolerances, these nominal parameters will not be exact. The process of kinematic calibration determines these nominal parameters experimentally. Kinematic calibration is typically accomplished with an external metrology system, although alternatives that do not require a metrology system exist.

Exercise

To be posted on CEAUT site:exercise1.pdf

Due: 83/7/27

Next Course

Inverse Kinematics