KINEMATIC CHAINS

& ROBOTS

(I)

This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots.

After this lecture, the student should be able to:•Appreciate the concept of kinematic pairs (joints) between rigid bodies•Define common (lower) kinematic pairs•Distinguish between open and closed kinematic chains•Appreciate the concept of forward and inverse kinematics and dynamics analysis•Express a finite motion in terms of the transformation matrix

Kinematic Chains and Robots (I)

General Rigid-Body Motion

General Motion += Translation Rotation

The general motion is equivalent to that of the action of a screw, which can be described using 4 “screw” parameters:

The displacement component parallel to the direction of rotation auu ss ˆ

Axis of rotation a

The angle of rotation

Axis of rotation passes through the point 0A

General Rigid-Body Motion

Chasles Theorem:

A general finite rigid-body motion is equivalent to

•A pure translation (sliding) along, by an amount us, and

•A pure rotation about the axis of rotation, by an amount

The general rigid-body motion can be characterized by the screw parameters

0,,,ˆ Aua s

General Rigid-Body Motion

Given R, we can solve for the eigenvector of R corresponding to =1 to get

AP

0

APIR i.e. solve where I is the 33 identity matrix

AP

APa

n

n

ACR

aatACtACAC

ˆ]ˆ)([)( 00)( 0tAC

If given find

Get from 2

)(

)()()sin(

n

nn

AC

ACRAC

)()( 12 tCtCuc

General Rigid-Body Motion

Finally:

)]cos(1[2

)()( 210

tCtCRIRA

T

The previous deals with rotation. In general, if we are given the location of a point C at 2 time instance, i.e. given and)( 2tC

)( 1tC

cu

We can denote the motion of the point C due to both rotation and translation as

auu

uau

ss

cs

ˆ

ˆ

The displacement component of the motion parallel to isa

Kinematic Pair

A kinematic pair is the coupling or joint between 2 rigid bodies that constraints their relative motion.

The kinematic pair can be classified according to the contact between the jointed bodies:

• Lower kinematic pairs: there is surface contact between the jointed bodies

• Higher kinematic pairs: the contact is localized to lines, curves, or points

Lower Kinematic Pairs

Revolute/pin joint

Prismatic joint

s

x

y

Planar joint

Spherical joint

s

Cylindrical joint

Kinematic Chain

A kinematic chain is a system of rigid bodies which are joined together by kinematic joints to permit the bodies to move relative to one another.

Kinematic chains can be classified as:•Open kinematic chain: There are bodies in the chain with only one associated kinematic joint•Closed kinematic chain: Each body in the chain has at least two associated kinematic joints

A mechanism is a closed kinematic chain with one of the bodies fixed (designated as the base)

In a structure, there can be no motion of the bodies relative to one another

Open and Closed Kinematic Chains

Open kinematic chain

Kinematic joint

Rigid bodies

Closed kinematic chain

Base (fixed)

Structure

Robots

Robots for manipulation of objects are generally designed as open kinematic chains

These robots typically contain either revolute or prismatic joints

Link (3): Gripper

Link (0): Base

Link (1)

Link (2)

A Simple Planar Robot

This simple robot will be used throughout to illustrate simple concepts

X0

Y0

Forward Kinematics Analysis

2=90°

Y3

X3

Consider the following motion:

Given the dimensions of the linkages and the individual relative motion between links, how to find the position, velocity, acceleration of the gripper? (Forward kinematics analysis problem)

Y3

X3

?

Inverse Kinematics Analysis

X0

Y0

Y3

X3

Again consider the following motion:

Given the dimensions of the linkages and the desired motion of the gripper, how to find the individual relative motion between links? (Inverse kinematics analysis problem)

2=?°Y3

X3

Dynamics Analysis

X0

Y0

Y3

X3

Again consider the following motion:

Given the inertia and dimensions of the linkages and the desired motion of the gripper, how to find the individual relative motion between links and the actuator forces to achieve this motion? (Dynamic analysis problem)

2=?° and force requiredY3

X3

X-axis

Y-axis

Z-axis

“O”

X

A B

1e2e

3e

CX

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

Notation

Consider the following motion. We will associate basis vectors with frame {b} and the (X, Y, Z) axis with frame {a}

)ˆ,ˆ,ˆ( 321 eee

We will denote the rotation matrix “R” that brings frame {a} to frame {b} as Rab

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

Notation

Example

100

001

010

333231

232221

131211

rrr

rrr

rrr

Rab

Notation

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

The position of point “C” expressed in frame {a} is denoted by C

aP

bCaP /

For example, is the position of point “C” relative to frame {b} expressed in terms of frame {a}

Transformation matrix

Consider the 2 frames {a} and {b}. Notice that for a vector

VRV bab

a

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

V

Example:

TaB

TbB

P

P

001

010

/

/

100

001

010

Rab

aBbBab PPR //

0

0

1

0

1

0

100

001

010

)(

Transformation matrix

So far, in the example we used the origins of the two frames are at the same point. What if the origin of frame {b} is at a distance defined by the vector TOb

aOb

aOb

aOb

aaOb PPPPP 321/

In this case, the point “B” is given by: )( /// bBabaObaB PRPP

We can simplify the above equation to: )( */

*/ bB

abaB PTP

where

1000

10 3333231

2232221

1131211

/

ObaOb

aOb

a

aObaa

bab

Prrr

Prrr

Prrr

PRT

1

1 3

2

1

/*/

BbB

bB

b

bBbB

P

P

P

PP

is called the Transformation Matrix of frame {b} w.r.t. frame {a}

Tab

is called the augmented vector */321/ . bB

T

Bb

Bb

Bb

bB PPPPP

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

Example: Transformation matrix

What is the transformation matrix for the case below?

TaObP 000/

100

001

010

Rab

1000

0100

0001

0010

10/

aOb

aaba

b

PRT

X

AB

1e

2e3eC

X-axis

Y-axis

Z-axis

“O”

Example: Transformation matrix

TaB

TbB

P

P

1001

1010*/

*/

1000

0100

0001

0010

Tab

1

1

0

0

1

1

0

1

0

1000

0100

0001

0010

/*/

*/

aBaBbB

ab

PPPT

Transformation matrix and Position

)( */

*/ bB

abaB PTP

If R is the identity matrix, then there is no rotation and the transformation matrix will represent the pure displacement of the origins of the frames of reference:

1000

100

010

001

10 3

2

1

/

ObaOb

aOb

a

aObaa

bab

P

P

P

PRT

Summary

Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the the discussion on the analysis of kinematic chains with focus on robots.

The following were covered:•The concept of kinematic pairs (joints) between rigid bodies•Definition of common (lower) kinematic pairs•Open and closed kinematic chains•The concept of forward and inverse kinematics and dynamics analysis•Finite motion in terms of the transformation matrix