Copyright 2002 (c) 2002 Howie Choset Introduction to ROBOTICS Images of sample robots and robot technologies

Copyright 2002 (c) 2002 Howie Choset

Introduction to

ROBOTICS

Images of samplerobots and robottechnologies







?robot: (noun) …

Insert image here

What is a robot?


ROBOTICSWith Your Host: Prof. Howie Choset

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The Categories Are…..

Control Mechanisms Mechanics

SensorsMotion

PlanningPerception

Image of Quiz Show

Computer Science

Electrical Engineering

Mechanical Engineering


Combining these fields we can create a system that can

SENSE

PLAN

ACT


Challenges

• Terrain (Locomotion and Navigation)

• Obstacles (Motion Planning)

• Low-Level Sensors (Control/Perception)

• Vision (Control/Perception)

• Manipulation (Kinematics)

• Power


Robotic Locomotion


Design Tradeoffs with Mobility Configurations

• Maneuverability• Controllability• Traction• Climbing ability• Stability• Efficiency• Maintenance• Environmental impact• Navigational considerations


Differential Drive

Where D represents the arc length of the center of the robotfrom start to finish of the movement.

Pictures from “Navigating Mobile Robots:Systems and Techniques” Borenstein, J.


Differential Drive (continued)

Advantages:• Cheap to build• Easy to implement• Simple design

Disadvantages:• Difficult straight line motion


Problem with Differential Drive: Knobbie Tires

Changing diameter makes for uncertainty in dead-reckoning error



Skid Steering

Advantages:•Simple drive system

Disadvantages:•Slippage and poor odometry results•Requires a large amount of power to turn


Synchro Drive

Advantages:•Separate motors for translation and

rotation makes control easier•Straight-line motion is guaranteed mechanically

Disadvantages:•Complex design and implementation



Distributed Actuator Arrays:Virtual Vehicle

• Modular Distributed Manipulator System• Employs use of Omni Wheels


Omni Wheels

Advantages:•Allows complicated motions

Disadvantages:•No mechanical constraints to require straight-line motion•Complicated implementation



Tricycle

Advantages:•No sliding

Disadvantages:•Non-holonomic planning required



Ackerman Steering

Advantages:Simple to implement•Simple 4 bar linkage controls

front wheels

Disadvantages:•Non-holonomic planning required



Articulated Drive:Nomad

Advantages:•Simple to implement except for turning mechanism

Disadvantages:•Non-holonomic planning is required


Mobile Robot Scale


Framewalker: Jim2

Advantages:•Separate actuation of translationand rotation•Straight-line motion is guaranteedmechanically

Disadvantages:•Complex design and implementation•Translation and rotation are excusive


SNAKE ROBOTS: Many DOF’s• Thread through tightly

packed volumes• Redundancy• Minimally invasive• Enhanced mobility• Multi-functional

Thanks to Robin Murphy


Hyper-redundant Mechanism Applications (Choset)

Thread through tightly packed volumes accessing locations that people and conventionally machinery cannot and in a non-invasive surgical manner

Bio-terrorist Sample Recovery

Urban Search and Rescue

Bridge and Bomb Inspection

Urban Reconnaissance

Robots bypass the danger and expedite the process of recovering bio-terrorist materials. Snake robots are needed to reach in and around implements commonly found in urban environments, I.e., be able to reach into a coffee cup

Reaching through the internals of tightly packed structures to locate points of failure without having to disassemble, thereby expediting the operation and making it safer

Acquire information about the enemy hiding in the many nooks and crannies the urban arena provides. This bypasses the dangerous current mode where our military blindly maneuvers until they blunder onto the enemy. Here the enemy decides when the battle begins.


Basic tasks: locomotionLinear progression/Rectilinear

Biological Snakes Robotic Snakes

•Burdick and Chirikjian, Yim

•Anchors at sites - travel backwards•Symmetric movement in axial direction•Anteroposterior flexible skin •Momentum is conserved as the snake travels at a fairly constant speed/little drag

•C. GansEnergy efficient (esp. friction surfaces)Low wiggle volume


Basic tasks: locomotionLateral undulation

Biological Snakes Robotic Snakes•Propulsion by summing the longitudinal resultants of posterolateral forces•Momentum is conserved•Efficiency* increases with lower sliding friction•Used for traversing flat clear ground with some irregularities

Gans•Hirose, Osrtowski, Miller, Haith

Gans

•*Energy Efficiency compared to tetrapods

•Jayne – comparable

•Gans/Chodrow&Taylor – more

•High endurance


Basic tasks: locomotionConcertina

Biological Snakes Robotic Snakes

•Uses static friction•Energy inefficient (7X)* due to stop and go movement•Tree climbers use some form of concertina

*Jayne

Concertina in 3DHirose

Gans


Basic tasks: locomotion

• Side winding

• Wheel

• Burrowing

• Climbing

• Swimming Osrtowski

•Energy•Terrain resolution•Step

Demonstration on new mechanism: locomotion (2d) and climbing (3d)

Wheel is good


Legged Robots

Advantages:•Can traverse any terrain a human can

Disadvantages:•Large number of degrees of freedom•Maintaining stability is complicated

Are legs better than wheels?

Image of leggedhumanoid robot


What is Motion Planning?

• Determining where to go


Overview

• The Basics– Motion Planning Statement– The World and Robot– Configuration Space– Metrics

• Path Planning Algorithms– Start-Goal Methods– Map-Based Approaches– Cellular Decompositions

• Applications– Coverage


The World consists of...

• Obstacles– Already occupied spaces of the world– In other words, robots can’t go there

• Free Space– Unoccupied space within the world– Robots “might” be able to go here– To determine where a robot can go, we need to discuss

what a Configuration Space is


Motion Planning Statement

If W denotes the robot’s workspace,

And Ci denotes the i’th obstacle,

Then the robot’s free space, FS, is defined as:

FS = W - ( Ci )

And a path c C0 is c : [0,1] FS

where c(0) is qstart

and c(1) is qgoal


Example of a World (and Robot)

Obstacles

Free Space

Robot

x,y


Start-Goal Algorithm:Lumelsky Bug Algorithms


Lumelsky Bug Algorithms

• Unknown obstacles, known start and goal.

• Simple “bump” sensors, encoders.

• Choose arbitrary direction to turn (left/right) to make all turns, called “local direction”

• Motion is like an ant walking around:

– In Bug 1 the robot goes all the way around each obstacle encountered, recording the point nearest the goal, then goes around again to leave the obstacle from that point

– In Bug 2 the robot goes around each obstacle encountered until it can continue on its previous path toward the goal


Configuration Space: Accommodate Robot Size

Obstacles

Free Space

Robot(treat as point object)x,y


The Configuration Space

• What it is– A set of “reachable” areas constructed from

knowledge of both the robot and the world

• How to create it– First abstract the robot as a point object. Then,

enlarge the obstacles to account for the robot’s footprint and degrees of freedom

– In our example, the robot was circular, so we simply enlarged our obstacles by the robot’s radius (note the curved vertices)


Configuration Space: the robot has...

• A Footprint– The amount of space a robot occupies

• Degrees of Freedom– The number of variables necessary to fully

describe a robot’s configuration in space• You’ll cover this more in depth later• fun with non-holonomic constraints, etc

x,y


Start-Goal Algorithm:Potential Functions


Potential Function Equations


Basics: Metrics

• There are many different ways to measure a path:

• Time

• Distance traveled

• Expense

• Distance from obstacles

• Etc…


Local Minimum Problem with the Charge Analogy


The Wavefront Planner

• A common algorithm used to determine the shortest paths between two points– In essence, a breadth first search of a graph

• For simplification, we’ll present the world as a two-dimensional grid

• Setup:– Label free space with 0– Label start as START– Label the destination as 2


Representations

• World Representation– You could always use a large region and distances

– However, a grid can be used for simplicity


Representations: A Grid

• Distance is reduced to discrete steps– For simplicity, we’ll assume distance is uniform

• Direction is now limited from one adjacent cell to another– Time to revisit Connectivity (Remember Vision?)


Representations: Connectivity

• 8-Point Connectivity • 4-Point Connectivity– (approximation of the L1 metric)


The Wavefront Planner: Setup


The Wavefront in Action (Part 1)

• Starting with the goal, set all adjacent cells with “0” to the current cell + 1– 4-Point Connectivity or 8-Point Connectivity?– Your Choice. We’ll use 8-Point Connectivity in our example



• Now repeat with the modified cells– This will be repeated until no 0’s are adjacent to cells

with values >= 2• 0’s will only remain when regions are unreachable



• Repeat again...



• And again...



• And again until...


The Wavefront in Action (Done)

• You’re done– Remember, 0’s should only remain if unreachable

regions exist


The Wavefront, Now What?

• To find the shortest path, according to your metric, simply always move toward a cell with a lower number

– The numbers generated by the Wavefront planner are roughly proportional to their distance from the goal

Twopossibleshortest

pathsshown


Wavefront (Overview)

• Divide the space into a grid.

• Number the squares starting at the start in either 4 or 8 point connectivity starting at the goal, increasing till you reach the start.

• Your path is defined by any uninterrupted sequence of decreasing numbers that lead to the goal.


Map-Based Approaches: Roadmap Theory

• Properties of a roadmap:– Accessibility: there exists a collision-free

path from the start to the road map

– Departability: there exists a collision-free path from the roadmap to the goal.

– Connectivity: there exists a collision-free path from the start to the goal (on the roadmap).

a roadmap exists a path exists Examples of Roadmaps

– Generalized Voronoi Graph (GVG)– Visibility Graph


Roadmap: GVG

• A GVG is formed by paths equidistant from the two closest objects

• Remember “spokes”, start and goal

• This generates a very safe roadmap which avoids obstacles as much as possible


Voronoi Diagram: Metrics


Voronoi Diagram (L2)

Note the curved edges


Voronoi Diagram (L1)

Note the lack of curved edges


Roadmap: Visibility Graph

• Formed by connecting all “visible” vertices, the start point and the end point, to each other

• For two points to be “visible” no obstacle can exist between them– Paths exist on the perimeter of obstacles

• In our example, this produces the shortest path with respect to the L2 metric. However, the close proximity of paths to obstacles makes it dangerous


The Visibility Graph in Action (Part 1)

• First, draw lines of sight from the start and goal to all “visible” vertices and corners of the world.

start

goal



• Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight.

start

goal




start

goal




start

goal


The Visibility Graph (Done)

• Repeat until you’re done.

start

goal


Visibility Graph Overview

• Start with a map of the world, draw lines of sight from the start and goal to every “corner” of the world and vertex of the obstacles, not cutting through any obstacles.

• Draw lines of sight from every vertex of every obstacle like above. Lines along edges of obstacles are lines of sight too, since they don’t pass through the obstacles.

• If the map was in Configuration space, each line potentially represents part of a path from the start to the goal.


Cell Decompositions: Trapezoidal Decomposition

• A way to divide the world into smaller regions• Assume a polygonal world



• Simply draw a vertical line from each vertex until you hit an obstacle. This reduces the world to a union of trapezoid-shaped cells


Applications: Coverage

• By reducing the world to cells, we’ve essentially abstracted the world to a graph.


Find a path

• By reducing the world to cells, we’ve essentially abstracted the world to a graph.


Find a path

• With an adjacency graph, a path from start to goal can be found by simple traversal

start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal


Find a path


start goal



• First, a distinction between sensor and detector must be made

• Sensor: Senses obstacles

• Detector: What actually does the coverage

• We’ll be observing the simple case of having an omniscient sensor and having the detector’s footprint equal to the robot’s footprint



• How is this useful? Well, trapezoids can easily be covered with simple back-and-forth sweeping motions. If we cover all the trapezoids, we can effectively cover the entire “reachable” world.



• Simply visit all the nodes, performing a sweeping motion in each, and you’re done.


Conclusion: Complete Overview

• The Basics– Motion Planning Statement– The World and Robot– Configuration Space– Metrics

• Path Planning Algorithms– Start-Goal Methods

• Lumelsky Bug Algorithms• Potential Charge Functions• The Wavefront Planner

– Map-Based Approaches• Generalized Voronoi Graphs• Visibility Graphs

– Cellular Decompositions => Coverage

• Done with Motion Planning!


What Is a Sensor?

• Anything that detects the state of the environment.

• Yep. You’ve already used sensors before in the Braitenburg lab.

• Vision systems and positioning devices are often treated separately from sensors.


Some types of Sensors:

• Ladar (laser distance and ranging)

• Sonar

• Radar

• Infra-red

• Light sensing

• Heat sensing

• Touch sensing


How to Choose a Sensor:There are four main factors to consider in choosing a sensor.

1) Cost: sensors can be expensive, especially in bulk.

2) Environment: there are many sensors that work well and predictably inside, but that choke and die outdoors.

3) Range: Most sensors work best over a certain range of distances. If something comes too close, they bottom out, and if something is too far, they cannot detect it. Choose a sensor that will detect obstacles in the range you need.

4) Field of View: depending upon what you are doing, you may want sensors that have a wider cone of detection. A wider “field of view” will cause more objects to be detected per sensor, but it also will give less information about where exactly an object is when one is detected.

Copyright 2002 (c) 2002 Howie ChosetSolar Cell

Digital Infrared Ranging

Compass

Touch Switch

Pressure Switch

Limit Switch

Magnetic Reed Switch

Magnetic Sensor

Miniature Polaroid Sensor

Polaroid Sensor Board

Piezo Ultrasonic Transducers

Pyroelectric Detector

Thyristor

Gas Sensor

Gieger-MullerRadiation Sensor

Piezo Bend Sensor

Resistive Bend Sensors

Mechanical Tilt Sensors

Pendulum Resistive Tilt Sensors

CDS Cell Resistive Light Sensor

Hall EffectMagnetic Field

Sensors

Compass

IRDA Transceiver

IR Amplifier Sensor

IR ModulatorReceiverLite-On IR

Remote Receiver

Radio ShackRemote Receiver

IR Sensor w/lens

GyroAccelerometer

IR Reflection Sensor

IR Pin Diode

UV Detector

Metal Detector


Intensity Based Infrared

• Easy to implement (few components)• Works very well in controlled environments• Sensitive to ambient light

time

volt

age

timevo

ltag

e

Increase in ambient light raises DC bias


Modulated Infrared

http://www.hvwtechnologies.comhttp://www.digikey.com

• Insensitive to ambient light• Built in modulation decoder (typically 38-40kHz)• Used in most IR remote control units ( good for communications)• Mounted in a metal faraday cage• Cannot detect long on-pulses• Requires modulated IR signal

limiter demodulatorbandpass filteramplifier

comparatorintegrator

600us 600us

Input

Output


Digital Infrared Ranging

position sensitive device (array of photodiodes)

Optical lenses

Modulated IR beam

• Optics to covert horizontal distance to vertical distance• Insensitive to ambient light and surface type • Minimum range ~ 10cm • Beam width ~ 5deg• Designed to run on 3v -> need to protect input• Uses Shift register to exchange data (clk in = data out)• Moderately reliable for ranging

+5voutputinput

gnd

1k 1k


Polaroid Ultrasonic Sensor

•

Mobile Robot Electric Measuring Tape

Focus for Camerahttp://www.robotprojects.com/sonar/scd.htm


Theory of Operation

• Digital Init• Chirp

– 16 high to low– -200 to 200 V

• Internal Blanking• Chirp reaches object

– 343.2 m/s– Temp, pressure

• Echoes– Shape– Material

• Returns to Xducer• Measure the time


Beam Pattern

Not Gaussian!!


(Naïve) Sensor Model


Problem with Naïve Model


Certainty Grid Approach

Combine info withBayes Rule (Morevac and Elfes)


Arc Transversal Method

• Uniform Distribution on Arc

• Consider Transversal Intersections

• Take the Median


Mapping Example


Creative Uses:

• Sharp IR sensors are very accurate and operate well over a large range of distances proportional to the size of a lego robot. However, they have almost no spread. This can cause a robot to miss an obstacle because of a narrow gap. One solution is to make the sensor pan.

• One could also use a light sensor to detect obstacles indoors. Inside, there tend to be lights at many angles and locations. Thus, around the edges of most obstacles, a slight shadow will be cast. A light sensor could detect this shadow and thus the associated object. Warning: this could be a very fickle design.

• Touch sensors can have their spread increased with large bumpers, and can be used for wall following to implement bug2. They are also dirt cheap.


Computer Vision


Optics

• Focal length– Length f of projection through lens on image

plane

• Inversion– Projection on image plane is inverted

f

Image planeObject


Projection on the image plane

• Size of an image on the image plane is inversely proportional to the distance from the focal point

h’h

f

d

f

h

d

h

Focal point

h’

h fd

f

h

d

h

Focal point

By conceptually moving the image plane, we can eliminate the negative sign

Image plane

Image plane


Move to three dimensions

• Similarity holds when three dimensions are considered

x

y

z

Image plane

f

(x,y,z)

(x’,y’,z’)

z

z

y

y

f

x

x

x


Perspective

• 1 Point Perspective– Using similar triangles, it is possible to determine the

relative sizes of objects in an image

– Given a calibrated camera (predetermine a mathematical relationship between size on the image plane and the actual object)

f

Image planeObject


Preliminary Math: Trigonometry

• Pythagorean Theorem– a2 + b2 = c2

• Law of sines

• Law of cosinescba

sinsinsin

cos2222 abbac


Images

• Discrete representation of a continuous function– Pixel: Picture Element – cell of constant color in a digital image

– An image is a two dimensional array of pixels– Pixel: numeric value representing a uniform portion of an image

• Grayscale– All pixels represent the intensity of light in an image, be it red, green, blue,

or another color • Like holding a piece of transparent colored plastic over your eyes

– Intensity of light in a pixel is stored as a number, generally 0..255 inclusive

• Color– Three grayscale images layered on top of eachother with each layer

indicating the intensity of a specific color light, generally red, green, and blue (RGB)

– Third dimension in a digital image


Images

• Resolution– Number of pixels across in horizontal– Number of pixels in the vertical– Number of layers used for color

• Often measured in bits per pixel (bpp) where each color uses 8 bits of data

– Ex: 640x480x24bpp

• Binary images: Two color image– Pixel is only one byte of information– Indicates if the intensity of color is above or below

some nominal value– Thresholding


Grayscale vs. Binary image

Grayscale Binary threshold


Thresholding

• Purpose– Trying to find areas of high color intensity– Highlights locations of different features of the image (notice

Mona’s eyes)– Image compression, use fewer bits to encode a pixel

• How done– Decide on a value – Scan every pixel in the image

• If it is greater than , make it 255• If it is less than , make it 0

– Picking a good • Often 128 is a good value to start with• Use a histogram to determine values based on color frequency

features


Histogram

• Measure the number of pixels of different values in an image.

• Yields information such as the brightness of an image, important color features, possibilities of color elimination for compression

• Thresholding– Make pixels above a value one color and values below

that value a different color– Binary threshold often used to transform a grayscale

image into black and white– Also usable for compression and feature extraction


Mona’s Histogram

0 255


Connectivity• Two conventions on considering two pixels next

to eachother

• To eliminate the ambiguity, we could define the shape of a pixel to be a hexagon

8 point connectivityAll pixels sharing a side or corner are considered adjacent

4 point connectivityOnly pixels sharing a side are considered adjacent


Object location - Segmentation

• One method of locating an object is through the use of a wave front

• Wavefront– Assume a binary image with values of 0 or 1

– 1. Choose 1st pixel with value 1, make it a 2

– 2. For each neighbor, if it is also a 1, make it a 2 as well

– 3. Repeat step two for each neighbor until there are no neighbors with value 1

– 4. All pixels with a value 2 are are a continuous object


Edge detection• Scanline: one row of pixels in an image• Take the first derivative of a scanline

• The derivative becomes nonzero when an edge (pixels change values) is encountered


Implementing 1st derivative edge detection digitally

• Derivative is defined as• With a scan line, the run (x – c) is 1, and the rise

(f(x) – f(c)) is B[m+1] – B[m]• This becomes

where I is the resulting image of edges

• This is really just a dot product of the vector [-1 1] repeated each pixel in the resulting image

cx

cfxf

cx

)()(lim

][1]1[1][ mBmBmI

11]1[][][ mBmBmI


More Math: Convolution

• This operation of moving a mask across an image has a name, called convolution

• In order to mathematically apply a filter to a signal, we must use convolution– If you know laplace transforms, this is a

multiplication in the laplace domain


Convolution: Analog

dττthτxty

)()()(

Given a symetric h (common in image processing), simplifies to

dττhτxty

)()()(

h(t) = [-1 1]

Move across the signal x (possibly a scanline in an image)


Convolution: Digital

k

knhkxny ][][][

More useful in image processing on a digital computerx[n] is a pixel in an image, y[n] is the resulting pixel

0 2 2 0 1 1 3 0 1 1

40 2 2 0 1 1 3

0 1 1

0 2 2 0 1 1 3

0 1 14 2

1.

2.


Convolution example, cont

4 2 10 2 2 0 1 1 3

0 1 13.

4 2 1 20 2 2 0 1 1 3

0 1 14.

4 2 1 2 40 2 2 0 1 1 3

0 1 15.


Convolution: Two-dimensional

• Rotate your mask 180 degrees about the origin• Do the same dot product operation, this time using

matrices instead of vectors• Repeat the dot product for every pixel in the resulting

image• In the boundary case around the edges of the image there

are two options– extend the original image out using the pixel values at the edge– Make the resulting image y smaller than the original and don’t

compute pixels where the mask would extend beyond the edge of the original

0 0

),(),(),( 0000m n

nnmmhnmxnmy


Convolution: Summary

• Analog

• Digital

• Two dimensional, digital

0 0

),(),(),( 0000m n

nnmmhnmxnmy

dττthτxty

)()()(

k

knhkxny ][][][


Filters, Masks, Transforms

• Edge detection– Wide masks

• Smoothing

• Object detection


An Introduction to Robot Kinematics

Renata Melamud


Kinematics studies the motion of bodies


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

23

4

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


Other basic joints

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

Revolute Joint1 DOF ( Variable - )

Prismatic Joint1 DOF (linear) (Variables - d)


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


Quick Math ReviewDot Product: Geometric Representation:

A

Bθ

cosθBABA

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

B

BuB

y

x

a

a

y

x

b

b

Matrix Representation:

yyxxy

x

y

xbaba

b

b

a

aBA

B

Bu


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

dhcfdgce

bhafbgae

hg

fe

dc

ba

Matrix Addition:

hdgc

fbea

hg

fe

dc

ba


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

V

VV

O

NNO

V

VV

0

PP x

P

(VN,VO)

Notation:


NX

VNO

VXY

PVN

VO

Y O

NO

O

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

O

Y

NXNOXY

VP

VPVPV

Y

xXY

P

PP


oV

nV

θ)cos(90V

cosθV

sinθV

cosθV

V

VV

NO

NO

NO

NO

NO

NO

O

NNO

NOV

o

n Unit vector along the N-Axis

Unit vector along the N-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


Rotation (around the Z-Axis)X

Y

Z

X

Y

N

VN

VO

O

V

VX

VY

Y

XXY

V

VV

O

NNO

V

VV

= Angle of rotation between the XY and NO coordinate axis


X

Y

N

VN

VO

O

V

VX

VY

Unit vector along X-Axisx

xVcosαVcosαVV NONOXYX

NOXY VV

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

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

)oxVnxVV ONX ()(

))

)

(sinθV(cosθV

90))(cos(θV(cosθVON

ON


Similarly….

yVα)cos(90VsinαVV NONONOY

y)oVn(VV ONY

)oy(V)ny(VV ONY

))

)

(cosθV(sinθV

(cosθVθ))(cos(90VON

ON

So….

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

Y

XXY

V

VV

Written in Matrix Form

O

N

Y

XXY

V

V

cosθsinθ

sinθcosθ

V

VV

Rotation Matrix about the z-axis


X1

Y1

N

O

VXY

X0

Y0

VNO

P

O

N

y

x

Y

XXY

V

V

cosθsinθ

sinθcosθ

P

P

V

VV

(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


O

N

y

x

Y

XXY

V

V

cosθsinθ

sinθcosθ

P

P

V

VV

HOMOGENEOUS REPRESENTATIONPutting it all into a Matrix

1

V

V

100

0cosθsinθ

0sinθcosθ

1

P

P

1

V

VO

N

y

xY

X

1

V

V

100

Pcosθsinθ

Psinθcosθ

1

V

VO

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

100

Pcosθsinθ

Psinθcosθ

H y

x

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


Rotation Matrices in 3D – OK,lets return from homogenous repn

100

0cosθsinθ

0sinθcosθ

R z

cosθ0sinθ

010

sinθ0cosθ

Ry

cosθsinθ0

sinθcosθ0

001

R z

Rotation around the Z-Axis

Rotation around the Y-Axis

Rotation around the X-Axis


1000

0aon

0aon

0aon

Hzzz

yyy

xxx

Homogeneous Matrices in 3D

H is a 4x4 matrix that can describe a translation, rotation, or both in one matrix

Translation without rotation

1000

P100

P010

P001

Hz

y

x

P

Y

X

Z

Y

X

Z

O

N

A

O

N

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

V

V

V

HV

1

A

O

N

zzzz

yyyy

xxxx

XY

V

V

V

1000

Paon

Paon

Paon

V

Homogeneous Continued….

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 MatrixEX.

Y

X

Z

J

I

K

N

OA

T

P

A

O

N

W

W

W

A

O

N

W

W

W

K

J

I

W

W

W

Z

Y

X

W

W

W Point relative to theN-O-A frame

Point 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

W

W

W

aon

aon

aon

P

P

P

W

W

W

1

W

W

W

1000

Paon

Paon

Paon

1

W

W

W

A

O

N

kkkk

jjjj

iiii

K

J

I


Y

X

Z

J

I

K

N

OA

TP

A

O

N

W

W

W

k

J

I

zzz

yyy

xxx

z

y

x

Z

Y

X

W

W

W

kji

kji

kji

T

T

T

W

W

W

1

W

W

W

1000

Tkji

Tkji

Tkji

1

W

W

W

K

J

I

zzzz

yyyy

xxxx

Z

Y

X

Substituting for

K

J

I

W

W

W

1

W

W

W

1000

Paon

Paon

Paon

1000

Tkji

Tkji

Tkji

1

W

W

W

A

O

N

kkkk

jjjj

iiii

zzzz

yyyy

xxxx

Z

Y

X


1

W

W

W

H

1

W

W

W

A

O

N

Z

Y

X

1000

Paon

Paon

Paon

1000

Tkji

Tkji

Tkji

Hkkkk

jjjj

iiii

zzzz

yyyy

xxxx

Product of the two matrices

Notice that H can also be written as:

1000

0aon

0aon

0aon

1000

P100

P010

P001

1000

0kji

0kji

0kji

1000

T100

T010

T001

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

J

I

K

N

OA

TP

A

O

N

W

W

W

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)

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


What is an INS?

• Position (dead reckoning)

• Orientation (roll, pitch, yaw)

• Velocities

• Accelerations


Sampling of INS Applications

Image of dynamic stabilized personal vehicle

Image of autonomous vehicle


Accelerometers


Accelerometers

Proof Mass

d1d2

Fixed fingers

Moving finger

Suspension Springs

• F = ma

(Newton’s 2nd Law)

• F = kx

(Hooke’s Law)


Accelerometers

Proof Mass

d1d2

Fixed fingers

Moving finger

Suspension Springs

• C = ε0A/d

(parallel-plate capacitor)– ε0 = permitivity constant

Voltage Capacitance Surface Area and distance Spring displacement Force Acceleration

Integrate to get velocity and displacement

• Q = CV


Gyroscopes


Gyroscopes How does it maintain angular orientation?

Disk on an axis Disk stationary Disk rotating

Red pen indicates applied force


Gyroscopes – Precession

As green force is applied to axis of rotation, red points will attempt to move in blue directions

These points rotate and continue to want to move in the same direction causing precession

Rotating around red axis, apply a moment around axis coming out of paper on red axis


Gyroscopes – Gimbaled • Rotor Axle wants to keep pointing in the same

direction

• Mounting in a set of gimbals allows us to measure the rotation of the body


Gyroscopes – MEMS

• Coriolis effect – “fictitious force” that acts upon a freely moving object as observed from a rotating frame of reference


Gyroscopes – MEMS • Comb drive fingers can be actuated by applying

voltage

• Coriolis effect induces motion based on rotation

• Capacitive sensors (similar to accelerometers) detect the magnitude of this effect and therefore the rotation

Tuning Fork Gyroscope Vibrating Ring Gyroscope


Fiber Optic Gyroscope (FOG)

= attitude rate, 1 = laser light source, 2 = beamsplitter, 3 = wound optical fiber, 4 = photosensor.

DSP 4000

turret, antenna, and optical stabilization systems


GPS – Global Positioning System

• Constellation 27 satellites in orbit

• Originally developed by U.S. military

• Accuracy ~ 10 m

• 3D Trilateration


GPS – 2D Trilateration

AB

CYou are here

50 mi75 mi

30 mi


GPS – 3D Trilateration• Location of at least three satellites

(typically 4 or more)

• Distance between receiver and each of those satellites

– Psudo-random code is sent via radio waves from satellite and receiver

– Since speed of radio signal is known, the lag time determines distance


GPS – Improvements • Some sources of error

– Earth’s atmosphere slows down signal

– Radio signal can bounce off large objects

– Misreporting of satellite location

• Differential GPS (DGPS)– Station with known location calculates receiver’s inaccuracy

– Broadcasts signal correction information

– Accuracy ~ 10 m


GPS – Improvements

• WAAS (Wide Area Augmentation System)– Similar to DGPS

– Geosynchronous Earth Orbiting satellites instead of land based stations

– Accuracy ~ 3 m

Image of GPS navigation device


Encoders


Encoders – Incremental

LED Photoemitter

Photodetector

Encoder disk


Encoders - Incremental


Encoders - Incremental

• Quadrature (resolution enhancing)


Encoders - Absolute

4 Bit Example

More expensive

Resolution = 360° / 2N

where N is number of tracks


Pros and ConsPros Cons

Accelerometer Inexpensive, small

Integration drift error

Gyroscope Large selection Integration drift error

GPS No drift Data at 1 Hz

Encoders Inexpensive Slip


F o r w a r d K i n e m a t i c s


The Situation:You have a robotic arm that

starts out aligned with the xo-axis.You tell the first link to move by 1 and the second link to move by 2.

The Quest:What is the position of the end

of the robotic arm?

Solution:1. Geometric Approach

This might be the easiest solution for the simple situation. However, notice that the angles are measured relative to the direction of the previous link. (The first link is the exception. The angle is measured relative to it’s initial position.) For robots with more links and whose arm extends into 3 dimensions the geometry gets much more tedious.

2. Algebraic Approach Involves coordinate transformations.


X2

X3Y2

Y3

1

2

3

1

2 3

Example Problem: You are have a three link arm that starts out aligned in the x-axis. Each

link has lengths l1, l2, l3, respectively. You tell the first one to move by 1 , and so on as the diagram suggests. Find the Homogeneous matrix to get the position of the yellow dot in the X0Y0 frame.

H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 )

i.e. Rotating by 1 will put you in the X1Y1 frame. Translate in the along the X1 axis by l1. Rotating by 2 will put you in the X2Y2 frame. and so on until you are in the X3Y3 frame.

The position of the yellow dot relative to the X3Y3 frame is(l1, 0). Multiplying H by that position vector will give you the coordinates of the yellow point relative the the X0Y0 frame.

X1

Y1

X0

Y0


Slight variation on the last solution:Make the yellow dot the origin of a new coordinate X4Y4 frame

X2

X3Y2

Y3

1

2

3

1

2 3

X1

Y1

X0

Y0

X4

Y4

H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) * Tx3(l3)

This takes you from the X0Y0 frame to the X4Y4 frame.

The position of the yellow dot relative to the X4Y4 frame is (0,0).

1

0

0

0

H

1

Z

Y

X

Notice that multiplying by the (0,0,0,1) vector will equal the last column of the H matrix.


More on Forward Kinematics…

Denavit - Hartenberg Parameters


Denavit-Hartenberg Notation

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

IDEA: Each joint is assigned a coordinate frame. Using the Denavit-Hartenberg notation, you need 4 parameters to describe how a frame (i) relates to a previous frame ( i -1 ).

THE PARAMETERS/VARIABLES: , a , d,


The Parameters

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

You can align the two axis just using the 4 parameters

1) a(i-1)

Technical Definition: a(i-1) is the length of the perpendicular between the joint axes. The joint axes is the axes around which revolution takes place which are the Z(i-1) and Z(i) axes. These two axes can be viewed as lines in space. The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines.


a(i-1) cont...

Visual Approach - “A way to visualize the link parameter a(i-1) is to imagine an expanding cylinder whose axis is the Z(i-1) axis - when the cylinder just touches the joint axis i the radius of the cylinder is equal to a(i-1).” (Manipulator Kinematics)

It’s Usually on the Diagram Approach - If the diagram already specifies the various coordinate frames, then the common perpendicular is usually the X(i-1) axis. So a(i-1) is just the displacement along the X(i-1) to move from the (i-1) frame to the i frame.

If the link is prismatic, then a(i-1) is a variable, not a parameter. 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


2) (i-1)

Technical Definition: Amount of rotation around the common perpendicular so that the joint axes are parallel.

i.e. How much you have to rotate around the X(i-1) axis so that the Z(i-1) is pointing in

the same direction as the Zi axis. Positive rotation follows the right hand rule.

3) d(i-1)

Technical Definition: The displacement along the Zi axis needed to align the a(i-1) common perpendicular to the ai common perpendicular.

In other words, displacement along the

Zi to align the X(i-1) and Xi axes.

4) i Amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi

axis.

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


The Denavit-Hartenberg Matrix

1000

cosαcosαsinαcosθsinαsinθ

sinαsinαcosαcosθcosαsinθ

0sinθcosθ

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

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

1)(iii

d

d

a

Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix 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


3 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 Parameter Table

Notice that the table has two uses:

1) To describe the robot with its variables and parameters.

2) To describe some state of the robot by having a numerical values for the variables.


Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2

d2

a0 a1

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

i

0 0 0 0 0

1 0 a0 0 1

2 -90 a1 d2 2

1

V

V

V

TV2

2

2

000

Z

Y

X

ZYX T)T)(T)((T 12

010

Note: T is the D-H matrix with (i-1) = 0 and i = 1.


1000

0100

00cosθsinθ

00sinθcosθ

T 00

00

0

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

i

0 0 0 0 0

1 0 a0 0 1

2 -90 a1 d2 2

This is just a rotation around the Z0 axis

1000

0000

00cosθsinθ

a0sinθcosθ

T 11

011

01

1000

00cosθsinθ

d100

a0sinθcosθ

T22

2

122

12

This is a translation by a0 followed by a rotation around the Z1 axis

This is a translation by a1 and then d2 followed by a rotation around the X2 and Z2 axis

T)T)(T)((T 12

010


I n v e r s e K i n e m a t i c s

From Position to Angles


A Simple Example

1

X

Y

S

Revolute and Prismatic Joints Combined

(x , y)

Finding :

)x

yarctan(θ

More Specifically:

)x

y(2arctanθ arctan2() specifies that it’s in the

first quadrant

Finding S:

)y(xS 22


2

1

(x , y)

l2

l1

Inverse Kinematics of a Two Link Manipulator

(x , y)

l 2

l1

l 2l

1


The Geometric Solution

l1

l2

2

1

(x , y) Using the Law of Cosines:

21

22

21

22

21

22

21

22

212

22

122

222

2arccosθ

2)cos(θ

)cos(θ)θ180cos(

)θ180cos(2)(

cos2

ll

llyx

ll

llyx

llllyx

Cabbac

2

2

22

2

Using the Law of Cosines:

x

y2arctanα

αθθ

yx

)sin(θ

yx

)θsin(180θsin

sinsin

11

22

2

22

2

2

1

l

c

C

b

B

x

y2arctan

yx

)sin(θarcsinθ

22

221

l

Redundant since 2 could be in the first or fourth quadrant.

Redundancy caused since 2 has two possible values


21

22

21

22

2

2212

22

1

211211212

22

1

211212

212

22

12

1211212

212

22

12

1

2222

2

yxarccosθ

c2

)(sins)(cc2

)(sins2)(sins)(cc2)(cc

yx)2((1)

ll

ll

llll

llll

llllllll

The Algebraic Solution

l1

l2

2

1

(x , y)

21

21211

21211

1221

11

θθθ(3)

sinsy(2)

ccx(1)

)θcos(θc

cosθc

ll

ll

Only Unknown

))(sin(cos))(sin(cos)sin(

))(sin(sin))(cos(cos)cos(

:

abbaba

bababa

Note


))(sin(cos))(sin(cos)sin(

))(sin(sin))(cos(cos)cos(

:

abbaba

bababa

Note

)c(s)s(c

cscss

sinsy

)()c(c

ccc

ccx

2211221

12221211

21211

2212211

21221211

21211

lll

lll

ll

slsll

sslll

ll

We know what 2 is from the previous slide. We need to solve for 1 . Now we have two equations and two unknowns (sin 1 and cos 1 )

2222221

1

2212

22

1122221

221122221

221

221

2211

yx

x)c(ys

)c2(sx)c(

1

)c(s)s()c(

)(xy

)c(

)(xc

slll

llllslll

lllll

sls

ll

sls

Substituting for c1 and simplifying many times

Notice this is the law of cosines and can be replaced by x2+ y2

22

222211

yx

x)c(yarcsinθ

slll

Documents

Copyright 2002 (c) 2002 Howie Choset Introduction to ROBOTICS Images of sample robots and robot technologies