What is Kinematics

Kinematics studies the motion of bodies

Joints for Robots

An Example - The PUMA 560

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

1

2 3

4

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

Concepts:- Revolute joint- DOF

Other basic joints

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

Revolute Joint1 DOF ( Variable - )

Prismatic Joint1 DOF (linear) (Variables - d)

Concepts:- Prismatic joint- Spherical joint

We are interested in two kinematics topics

Forward Kinematics (angles to position)What you are given: The length of each link

The angle of each joint

What you can find: The position of any point (i.e. it’s (x, y, z) coordinates

Inverse Kinematics (position to angles)What you are given: The length of each link

The position of some point on the robot

What you can find: The angles of each joint needed to obtain that position

Concepts:- Forward Kinematics- Inverse Kinematics

Math Review

Review of Vectors and

Matrices

Dot Product and Unit VectorDot Product: Geometric Representation:

A

Bθ

cosθBABA

Unit VectorVector in the direction of a chosen vector but whose magnitude is 1.

BBuB

y

x

aa

y

x

bb

Matrix Representation:

yyxxy

x

y

x bababb

aa

BA

B

BuConcepts:- Dot Product- Unit Vector

More on Dot Product

Review of Matrices

Quick Matrix Review

Matrix Multiplication:

An (m x n) matrix A and an (n x p) matrix B, can be multiplied since the number of columns of A is equal to the number of rows of B.

Non-Commutative MultiplicationAB is NOT equal to BA

dhcfdgcebhafbgae

hgfe

dcba

Matrix Addition:

hdgcfbea

hgfe

dcba

Concepts:- Matrix Multiplication- Matrix Addition

Inversion of Matrices

Translation

Basic TransformationsMoving Between Coordinate Frames

Translation Along the X-Axis

N

O

X

Y

VNO

VXY

Px

VN

VO

Px = distance between the XY and NO coordinate planes

Y

XXY

VV

V

O

NNO

VV

V

0P

P x

P

(VN,VO)

Notation:

Concepts:- Translation along the X

axis

NX

VNO

VXY

PVN

VO

Y O

NOO

NXXY VPV

VPV

Writing in terms of XYV NOV

X

VXY

PXY

N

VNO

VN

VO

O

Y

Translation along the X-Axis and Y-Axis

OY

NXNOXY

VPVP

VPV

Y

xXY

PP

P

Concepts:- Translation along the X axis and

Y axis

oVnV

θ)cos(90VcosθV

sinθVcosθV

VV

VNO

NO

NO

NO

NO

NO

O

NNO

NOV

on Unit vector along the N-Axis

Unit vector along the O-Axis

Magnitude of the VNO vector

Using Basis VectorsBasis vectors are unit vectors that point along a coordinate axis

N

VNO

VN

VO

O

n

o

Concepts:- Using planar trigonometry to calculate the vector from

projections

Rotation

Rotation (around the Z-Axis)X

Y

Z

X

Y

N

VN

VO

O

V

VX

VY

Y

XXY

VV

V

O

NNO

VV

V

= Angle of rotation between the XY and NO coordinate axis

Concepts:- Rotation around Z-Axis

We rotate around Z and we have two frames of reference for the same vector

X

Y

N

VN

VO

O

V

VX

VY

Unit vector along X-Axis

x

xVcosαVcosαVV NONOXYX

NOXY VV

Can be considered with respect to the XY coordinates or NO coordinates

V

x)oVn(VV ONX (Substituting for VNO using the N and O components of the vector)

)oxVnxVV ONX ()(

))

)

(sinθV(cosθV90))(cos(θV(cosθV

ON

ON

Concepts:- Rotation around Z-Axis

Similarly….

yVα)cos(90VsinαVV NONONOY

y)oVn(VV ONY

)oy(V)ny(VV ONY

))

)

(cosθV(sinθV(cosθVθ))(cos(90V

ON

ON

So….

)) (cosθV(sinθVV ONY )) (sinθV(cosθVV ONX

Y

XXY

VV

V

Written in Matrix Form

O

N

Y

XXY

VV

cosθsinθsinθcosθ

VV

V Rotation Matrix about the z-axis

Concepts:- Rotation around Z-Axis

Rotation together with

translation

X1

Y1

N

O

VXY

X0

Y0

VNO

P

O

N

y

xY

XXY

VV

cosθsinθsinθcosθ

PP

VV

V

(VN,VO)

In other words, knowing the coordinates of a point (VN,VO) in some coordinate frame (NO) you can find the position of that point relative to your original coordinate frame (X0Y0).

(Note : Px, Py are relative to the original coordinate frame. Translation followed by rotation is different than rotation followed by translation.)

Translation along P followed by rotation by

Concepts:- Rotation around Z-Axis

HOMOGENEOUS REPRESENTATION for robot kinematics

1. One of important representations in robotics2. Putting it all into a Matrix

O

N

y

xY

XXY

VV

cosθsinθsinθcosθ

PP

VV

V

HOMOGENEOUS REPRESENTATIONPutting it all into a Matrix

1

VV

1000cosθsinθ0sinθcosθ

1PP

1VV

O

N

y

xY

X

1

VV

100PcosθsinθPsinθcosθ

1VV

O

N

y

xY

X

What we found by doing a translation and a rotation

Padding with 0’s and 1’s

Simplifying into a matrix form

100PcosθsinθPsinθcosθ

H y

x

Homogenous Matrix for a Translation in XY plane, followed by a Rotation around the z-axis

Concepts:- All transformations can be put into one Matrix Notation.

Rotation Matrices

in 3D

How to represent a sequence of translations and rotations, a case common in robot arm design?

10000aon0aon0aon

Hzzz

yyy

xxx

Homogeneous Matrices in 3DH is a 4x4 matrix that can describe a translation, rotation, or both in one matrix

Translation without rotation

1000P100P010P001

Hz

y

x

P

Y

X

Z

Y

X

Z

O

N

A

ON

ARotation without translation

Rotation part: Could be rotation around z-axis, x-axis, y-axis or a combination of the three.

1

A

O

N

XY

VVV

HV

1

A

O

N

zzzz

yyyy

xxxx

XY

VVV

1000PaonPaonPaon

V

Homogeneous Continued…. combining rotation and translation to one matrix

The (n,o,a) position of a point relative to the current coordinate frame you are in.

The rotation and translation part can be combined into a single homogeneous matrix IF and ONLY IF both are relative to the same coordinate frame.

xA

xO

xN

xX PVaVoVnV

Finding the Homogeneous MatrixEXAMPLE.

Y

X

Z

J

I

K

N

OA

TP

A

O

N

WWW

A

O

N

WWW

K

J

I

WWW

Z

Y

X

WWW Point relative to the

N-O-A framePoint relative to theX-Y-Z frame

Point relative to theI-J-K frame

A

O

N

kkk

jjj

iii

k

j

i

K

J

I

WWW

aonaonaon

PPP

WWW

1WWW

1000PaonPaonPaon

1WWW

A

O

N

kkkk

jjjj

iiii

K

J

I

Different notation for the same thing

Y

X

Z

J

I

K

N

OA

TP

A

O

N

WWW

k

J

I

zzz

yyy

xxx

z

y

x

Z

Y

X

WWW

kjikjikji

TTT

WWW

1WWW

1000TkjiTkjiTkji

1WWW

K

J

I

zzzz

yyyy

xxxx

Z

Y

X

Substituting for

K

J

I

WWW

1WWW

1000PaonPaonPaon

1000TkjiTkjiTkji

1WWW

A

O

N

kkkk

jjjj

iiii

zzzz

yyyy

xxxx

Z

Y

X

1WWW

H

1WWW

A

O

N

Z

Y

X

1000PaonPaonPaon

1000TkjiTkjiTkji

Hkkkk

jjjj

iiii

zzzz

yyyy

xxxx

Product of the two matrices

Notice that H can also be written as:

10000aon0aon0aon

1000P100P010P001

10000kji0kji0kji

1000T100T010T001

Hkkk

jjj

iii

k

j

i

zzz

yyy

xxx

z

y

x

H = (Translation relative to the XYZ frame) * (Rotation relative to the XYZ frame) * (Translation relative to the IJK frame) * (Rotation relative to the IJK frame)

The Homogeneous Matrix is a concatenation of numerous translations and rotations

Y

X

Z

JI

K

N

OA

TP

A

O

N

WWW

One more variation on finding H:

H = (Rotate so that the X-axis is aligned with T)

* ( Translate along the new t-axis by || T || (magnitude of T))

* ( Rotate so that the t-axis is aligned with P)

* ( Translate along the p-axis by || P || )

* ( Rotate so that the p-axis is aligned with the O-axis)

1. This method might seem a bit confusing, but it’s actually an easier way to solve our problem given the information we have.

2. Here is an example…

Frames of Reference and transformations:

EULER ANGLES

Rotation Matrices in 3D - towards homogenous representation

1000cosθsinθ0sinθcosθ

R z

cosθ0sinθ010

sinθ0cosθRy

cosθsinθ0sinθcosθ0001

R z

Rotation around the Z-Axis

Rotation around the Y-Axis

Rotation around the X-Axis

The Rotation Matrix (called also Direction Cosine

Matrix)