Yangsheng Xu, Yongsheng Ou. Control of Single Wheel Robots

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A =[a

..,

] ,

B =[b

,..., b

] ,

where

∂

∂x

, ∀ i =1, 2 ,.

..,

= x

− ¯x

, ∀ i =1, 2 ,..., m.

Then,Equation (5.11)evolves to be

∂f

∂x

∂f

∂x



X =

+ AB

+ o ( ||( X −

X ||) . (5.12)

≤



( a

+ a

+ ... + a

)

( b

+ b

+ ... + b

)

≤



( a

+ a

+ ... + a

)(b

+ b

+ ... + b

)

≤



mξ

(

∂f

∂x

)

Submitting Equation (5.9) into it we have

≤|

∂f

∂x

| /η.

Thus, AB

is ahigher order smaller andthe eﬀect on thelinear relationis

small.

We assume thatthe sampling time intervalissmall enough. Then,weuse

ﬁnite-diﬀerence to approx

imate the

ﬁrst order diﬀerentials.

dY ≈ δY,dx

≈ δx

,i=1,..., m,

where δY = Y ( t +1) − Y ( t )and δx

= x

( t +1) − x

( t ).

Equation(5.10) will be

δY =

∂f

∂x



X =

δx

∂f

∂x



X =

δx

+ ... +

∂f

∂x



X =

δx

+ o ( ||δX||) . (5.13)

In fact,t

he sampledata aret

he observation forthe true values of

the

variables. From thesample data selection condition 1inEquations (5.7), we

knowthatthe sensorreading error is ahigher order smaller. By combining

the approximation errorand sensorreading error together, we have

δY =

∂f

∂x



X =

δx

∂f

∂x



X =

δx

+ ... +

∂f

∂x



X =

δx

+ 



, (5.14)

where 



partiallycomesfrom  ,and assums independentGaussian distribution

with E ( 



)=0and V ( 



)=σ

 2

Second, we use Least-squares theorytoestimate

∂f

∂X

in all k data groups.

If we let

∂f

∂X

be β ,Equation (5.14) becomes

δY = β

δx

+ β

δx

+ ... + β

δx

+ 



, (5.15)

where β =(β

,β

,..., β

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125

If we let

be thenumberofsample data in group i ,(i =1, 2 ,..., k ). For

eachgroup i ,weforma

× m data matrixXanda

× 1datavector Y.

Data

matrices

and

ctors

ritteni

oman

lettersa

riablesi

italic.

The least squares estimator

β of β is obtained by minimizing

LSE =(Y − X β )

(Y − X β ) . (5.16)

If we takethe derivativeofEquation(5.16) with respect to β ,ityields the

normale

quation

X)β =X

Y . (5.17)

As addressedabove,eventhough X aresimilar to each other, δX are

suﬃcien

tly

diﬀeren

ther.A

lthough

most

cases,

− 1

exists,

here we do notdiscuss it, howeverwewill discuss it later. Assumingthatit

exists, the linear equation(5.17) hasaunique solution. Then, pre-multiplying

Equation (5.17) by (X

− 1

gives theleast squares estimatorof β ,thatis,

β =(X

− 1

Y . (5.18)

If thea

assumptionholds, thel

east squares theory prov

ides us with

the following we

ll-known

results [38]:

The k × 1vector

β hasthe following properties:

E (

β )=β ;

that is,

β is an unbiased estimatorf

β .

β is the best linear un

biasede

stimator(BLUE) for

β ,t

hati

s, among

the class of linear un

biasede

stimators,

β hast

he smallestv

ariance. The

variance of

β is

Var (

β )=σ

 2

− 1

As the ab

eEquation sho

ws, andw

eknow

∂f

∂X



X =

= β ,therefore

β is

an ideal estimator for

∂f

∂X



X =

The sensitivitycoeﬃcient

∂Y

∂x

is an estimate of the number of unitsthat

change in thevariable Y as aresult of aunit change in theparameter x

.This

alsomeans that the sensitivityresult depends on itsphysical units of variables

andparameters, andismeaningfulonly when theunitsofthe modelare

known.Ingeneral cases, the variablesand the parameters eachhavediﬀerent

physical units, andtherefore thesensitivitycoeﬃcients cannot be compared

with eachother.Here, we assume that all data have been well calibrated. If

not, theyneed to be properly scaledaccordingtothe physical meaning.

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GlobalSigniﬁcance Order

Thenumberofnoisysample data pairs is limitedinalocaltime piece and it

note

nought

end

eral

ormalized

sensitivities

determine

lobal

signiﬁcance

der.

com

the

time

piece information to determinethe signiﬁcance order.

The most direct way to deﬁne the signiﬁcance order is sorting the sum-

mary of thenormalized sensitivity coeﬃcients. To avoid thenormalized sen-

sitivity

eﬃcien

canceling

ith

sitiv

neg

ative

lues

dur

ing

the

summationprocess,weuse their absolute values. Then,wedeﬁnethe Direct

Signiﬁcance Evaluation function as

I ( x



j =1

| S

( x

) | , (5.19)

where

( x

,i=1, 2 ,..., m, j

=1, 2 ,..., k.

Usually,t

his deﬁnition wo

rks well. Howe

r, in some casesw

here out-

liersare in evidence,some errordata points of aparameterare much larger

than others and as

suc

hey willgreatlya

ﬀect the av

eragev

alue. In these

conditions, we have to seek another deﬁnition for the signiﬁcance evaluation

function. According to

probabilityt

heory,the most

important

parameterh

the largest possibilitytobethe ﬁrst important parameter. We can redeﬁne

the signiﬁcance evaluationfunctioninthe following steps;

1. In eachlocal time piece, we

sort the absolute values of

normalizedsensi-

tivitycoeﬃcients.

2. Fo

achp

arameter

ollectthe times

fore

achp

ossible order. We

deﬁne n

( x

)asthe number of times that | S ( x

) | is the l

largest, where

l =1,..., m .

3. We deﬁne p

( x

he po

ssibilityt

hat

| S ( x

) | is the l

largest, where

l =1,..., m .

( x

)=n

( x

) /k × 100%,

where n

( x

) ≥ 0and



l =1

( x

)=k.

4. Foreachparameter, we deﬁnethe Statistical Importance Evaluation func-

tion as

I ( x

)=p

( x

) × 10

m − 1

+ p

( x

) × 10

m − 2

+ ...p

( x

) . (5.20)

Thus, theinput variablesselection canbeperformedafterweworkout

the signiﬁcance evaluationvalues forreal learningproblems.

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5.1.3 DependencyAnalysis

First, we will use an example to distinguishthreeterms, “relevant”, “depen-

den

t”

d“

(linear)

relativ

e”

hic

ill

usedi

this

apter.

Supp

ose

, x

and x

the

llo

wing

relations



= x

=3x

+2x

Then the relationbetween x

and x

“relev

t”,

butn

“d

den

t”

“(linear) relative”.The relation among x

, x

and x

is “dependent” and

“relevant”, but not“(linear) relative”.The relation between x

, x

and x

“(linear) relative”, “dependent” and“relevant”. The“dependent”relationis

importantinthe latertwo subtasks described in Section 5.1.3.

Let

X =[´x

, ´x

,..., ´x

¯ m

], be asubsetof X and¯m ≤ m .Let’s consider the

relation among

X .Supposedthatthe relationship of them is “dependent” and

´x

¯ m

= f



(´x

, ´x

,..., ´x

¯ m − 1

) . (5.21)

We needtouse the sample data table to verify this relation. After the data

selection andregroupingprocess,w

eknowthateachgroup of

data arev

ery

neartoeachother in the ¯m − 1dimension data space. Then,byaﬁrst-order

ylor series expansion, we

transform ournonlinearm

apping Equation (5.21)

into an almost linearone

d ´x

¯ m

∂f



∂ ´x

d ´x

∂f



∂ ´x

d ´x

+ ... +

∂f



∂ ´x

¯ m − 1

d ´x

¯ m − 1

+ o ( ||d

X ||) . (5.22)

If thereisa“dependent” relationamong

X ,then they will be almost “(linear)

relative

”a

mong

X .A

otedearlier,eventhough

X ares

imilar to

each

other,

X aresuﬃciently diﬀerentfromeachother.

As the similar analysis before we have

δ ´x

¯ m

∂f



∂x



X =

δ ´x

∂f



∂x



X =

δ ´x

+ ... +

∂f



∂x

¯ m − 1



X =

δ ´x

¯ m − 1

+¯, (5.23)

where¯ is similar to 



whichisahigher order smaller with respect to δ ´x

¯ m

.If

we let

∂f



∂X



X =

β ,Equation (5.23) becomes

δ ´x

¯ m

δ ´x

+ ... +

¯ m − 1

δ ´x

¯ m − 1

+¯, (5.24)

where

β =(

,...,

¯ m − 1

We use Least-squares theorytoestimate

β in all k data groups in the

following Equation.

β =(

− 1

Y . (5.25)

To eliminate the possibilitythatthe terms in Equation (5.24), will cancel

eachother,let

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||δ ´x

¯ m

|| = |

δ ´x

| + |

δ ´x

| + ... + |

¯ m − 1

δ ´x

¯ m − 1

| . (5.26)

We deﬁne the “(linear) relative” coeﬃcientas

ρ =

¯

||δ ´x

¯ m

. (5.27)

Foreachdata pointineachdata group, we will calculate the “(linear)

relativ

e”

eﬃcien

roup,

the

roup

has

data points, we

deﬁne the average“(linear) relative” coeﬃcientas,

¯ρ =









i =1

, (5.28)

where¯ρ oughttobeasmall number around 0 . 1(≤ 0 . 2).

Accordingtothe information derivedfromthesecoeﬃcients, we can de-

terminet

he “(linear)relative”

relationa

mong the ﬁrst-order derivativeof

and, further, the nonlinear“dependent” relationship amongthem.

rthermore, if the relation among

X is not dependent,

then we

can

use

the aboveanalysistoverifythe self-contained relation between Y and

X .If

they have

elf-contained relation, they will have

ependentrelationand

their ﬁrst-order deriva

tives

will

exhibit anearly “(linear) relative”

relation. It

is possible that some other featureswill aﬀect Y .However, since the “(linear)

relative”

eﬃcien

tissmall, the relevance of features should be

ak, which

means they will be removed fromthe keyfeature set.

Thus, we

can

summarize our input selectiona

pproachb

he following

steps.

1. Perform thesigniﬁcance analysis between the fullfeatures andthe learning

output. Reorder the featuresa

ccordingt

heirsigniﬁcance.

2. Selectthe ﬁrst importantfeature with learning outputtocarry outthe

self-contained analysis. If the learning outputisdependentonthe ﬁrst

signiﬁcantfeature, stop and report theresult.

3. If not, combine the ﬁrst signiﬁcantfeature with the second signiﬁcant

feature to perform adependentanalysisbetween them. If the second sig-

niﬁcantfeature is dependentonthe ﬁrst signiﬁcantfeature, remove the

feature andgotothe nexts

tep. If the relations are notdependent,

execute

the self-con

tained analysis betwe

en these two

featureswith the learning

output. If thelearningoutput is dependentonthesetwo features, stop

andreport the result.

4. Combine thefeatures in the former step with the next signiﬁcantfeature to

perform adependentanalysisamong them. If the next featureisdependent

on theformerfeatures, remove the feature andgobacktothe beginning

of this step,until thereisnofeature left, stop andreportthe result.Ifthe

relations arenot dependent, execute theself-contained analysis between

this new setoffeatures with the learning output. If the learningoutput

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is dependentonthesenew set of features,gotothe nextstep. If not, go

back to the beginning of this step,until thereisnofeature left, stop and

reportthe result.

Remo

the

atures

form

elf-con

tained

analysis

eliminate

edundan

eatures

still

remaining,

til

eatures

are

critical.

res

ult.

Sinces

ome

atures

end

ome

later

atures

om-

bination of someformerfeatures and later features, thus,the last step is

necessary andimportant.

5.1.4Experimental Study

Thea

this

exp

erimen

llustrate

the

pro

sed

criter

ion

to realize theinput variablesselection andvalidate the approach.

The controlproblemconsistsoftracking Gyroverinacircle within ade-

ﬁned radius.Inthe experimental system,for eachsampling process,wecan

collect 20 sensorreadings. Their

deﬁnitions andphysical meanings areshown

in Table 5.1. Among them, thereare 11 sensor readings, ‘ADC0’∼ ‘ADC10’,

correspondingtothe system statesvariables. Thereare alsotwo majorsen-

sorr

eadings, ‘PIR0’ (PIC-IN-READ 0) and ‘PIR1’(PIC-IN-READ 1), corre-

spondingtothe two control inputs: U

controlling the rolling speed of the

single wheel ˙

γ ,and U

controlling

the

angular position of

theﬂywheel

.For

the manual-model (i.e. controlled by ahuman), U

and U

areinput by joy-

sticks, and in the auto-model, they arederived fromthe software controller.

During all experiments,w

nly fo

cusonthe value of

andﬁ

he va

lue

of U

to some suitable value. Since the capabilityofspinning motor forthe

ﬂywheelislimited, after it drives the ﬂywheel to the working spinning speed

˙γ

,the motorwill try to maintain this speed. Thus,‘PIR3’ (PIC-IN-READ 3)

can notbeacontrol input anditisﬁxed duringthe data producing process,

too.

Ahumanexpert controlled Gyrovertotrack aﬁxed 3-meter radius circular

path andproducedaround N =30744 trainingsampleswith identicaltime

intervals ( δt =0. 025s ). Table 5.2displays some rawsensor data fromthe

human expert controlprocess.Inthe learningcontrol trainingprocess,the

system states variablesare the learningmodel inputs andtilt command U

themodel output. If we put all of the11sensor readings, ‘ADC0’ ∼ ‘ADC10’,

into thelearningprocess,the input dimensionistoo large to process areliable

model duetothe ‘

curse of dimensionality’. Thereforeweneed to perform input

variablesselection. Here, we use

signiﬁcance analysis

approachand dependence

analysis approachfor it.

SigniﬁcanceAnalysis Study

It is the aimofthis step to rank the 11 system variables(model inputs) in

signiﬁcance order with respect to the tilt command (modeloutput).

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Column Sensor

name

nction

descriptions

1 Counter read Spinning angle of ﬂywheel

2 ADC0 Tilt servoposition,i.e., tilt angle of ﬂywheel β

3 ADC1 Tilt servocurrent

4 ADC2 Drivemotor current

5 ADC3 Roll rateofGyro,i.e., roll rate of single wheel

6 ADC4 Pitchrate of Gyro

7 ADC5 Yawrate of Gyro, i.e., precession rate of single wheel ˙α

8 ADC6 Xaccelerometer

9 ADC7 Yaccelerometer

10 ADC8 Zaccelerometer

11 ADC9 Xtilt sensor

12 ADC10 Ytilt sensor, i.e., lean angle of single wheel β

13 PIC-IN-READ 0 Drive command U

14 PIC-IN-READ 1 Tilt command U

15 PIC-IN-READ 3 Command for spinning speedofﬂywheel

16 PIC-OUT-READ 0 Drive command

17 PIC-OUT-READ

Tilt

command

18 PIC-OUT-READ 3 Command for spinning speedofﬂywheel

19 instantaneous speed instantaneous driving speedofsingle wheel ˙γ

20 average speed instantaneous driving speedofsingle wheel

˙γ in 5sampling

intervals

Table 5.1. Gyrover’s sensor data string

Input Output

ADC0 ADC1 ADC2 ADC3 ADC4 ADC5 ADC6 ADC7 ADC8 ADC9 ADC10 PIR1

2.614 0.029 -0.004 0.422 0.617 1.696 2.440 2.589 3.624 5.821 5.874 180

2.646 1.970 1.989 0.410 0.500 2.104 2.885 2.528 3.178 6.060 6.168 179

2.585 6.245 1.935 -0.300 0.659 2.717 2.724 2.393 3.221 6.021 6.407 178

2.419 3.308 3.475 0.505 0.478 0.310 2.172 2.489 3.214 6.075 5.400 178

2.382 3.352 3.636 -0.678 0.468 1.616 2.601 2.739 3.769 6.075 5.997 175

Table 5.2. Sample of human control data

First, we obtainthe sensorreading error information as follows. If we ﬁx

Gyroverinsome position andstart it, all sensor readings should keep in some

lues. In

fact, they are changinga

round their true values forthe noisesake.

Then we keep recording the sensor readings for aperiodoftime and collect

hundreds of

sensor

readings foreachsensor. From thesedata, we

obtain the

noise variance σ forall variables, ‘ADC0’∼ ‘ADC10’and ‘PIR1’asshown in

Table5.3.

ADC0 ADC1 ADC2 ADC3 ADC4 ADC5 ADC6 ADC7 ADC8 ADC9 ADC10 PIR1

0.007 0.003 0.006 0.006 0.004 0.004 0.006 0.006 0.007 0.002 0.003 0.019

Table 5.3. Thenoise variance

σ informationfor variables

Second, we select and regroupdata points T

=[X

](j =1, 2 ,..., 30744)

fromthe whole data table as in thefollowing steps.

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1. AccordingtoEquation (5.6) we calculate the ﬁrst derivatives forall data

points.

2. AccordingtoEquation (5.7) we select and place thedata points into a

new

table

with

14,634d

ata

remov

ing

the

that

their

ﬁrs

eriv

ativ

are

less

than

the

noise

(

η =1

3. According to Equations(5.8) and(5.9),weworkout

N =22and r =0. 5.

4. Accordingto r =0. 2, we cluster the 14634data points into 6340 groups.

5. Accordingto N =22, we select 45 data groupsfromthe 6340 groupby

removing the groups in which thedata numbers areless than 22.

Third, we perform thelocal sensitivity analysis(Local SA). Accordingto

Equation (5.18), we work outthe sensitivity coeﬃcients β =

∂f

∂X

forall 45

data groups. Thelocal sensitivitycoeﬃcients of the ﬁrstfourmost signiﬁcant

variablesare shown in Figure 5.2.

0 5 10 15 20 25 30 35 40 45

−15

−10

−5

Local sensitivity coefficents

groups

ADC0

ADC9

ADC10

ADC4

Fig. 5.2. Thelocal sensitivitycoeﬃcients of the ﬁrst four signiﬁcance variables.

Last, we rank allthe system variablesinthe global signiﬁcance order.

According to theDirect Signiﬁcance Evaluation function in (5.19), we work

outthe global signiﬁcance value I ( x

)for eachsystems variable andorder

them in Table 5.3.

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ADC0 ADC1 ADC2 ADC3 ADC4 ADC5 ADC6 ADC7 ADC8 ADC9 ADC10

I ( x

) 331.53 3.07 5.23 25.48 73.99 51.71 15.29 24.91 11.26 173.14 97.89

rank 1 11 10 6 4 5 8 7 9 2 3

ble

5.4.

Thes

igniﬁcance

ordert

able

rformedt

signiﬁcance

alysise

erimen

sev

eral

times

with

diﬀerentdata setcombinations. Theresults are quite similar fromtime to

time.

DependenceAnalysis Study

The aimofthis step is to obtainthe smallestset of system variablesand the

controlinput is dependentonthis setoffeatures.

First,

execute

the

self

-con

ined

alysis

twe

all

m =1

system variableswith the tilt command.

1. We

select and regroupdata po

ints

=[X

](j =1, 2 ,..., 30744) from

the whole data table.

2. According to

Equation (5.6), we

calculate

theﬁ

rst derivatives for alldata

points.

3. AccordingtoEquation (5.7) we select and place thedata points into a

new data table

with 1,085 data po

ints by

removing the data points

that

their ﬁrstderivatives are less than ten times the noise variance σ ( η =10).

4. Accordingt

r, N ,w

luster and select

the 1,085 data po

ints into 22

groups.

5. According to Equations(5.24)-(5.28),for eachgroup, we calculate the

average ”(linear) relative” coeﬃcients that areinTable 5.5.

group 1 2 3 4 5 6 7 8 9 10 11

¯ρ 0.1011 0.1041 0.1103 0.1252 0.1280 0.1103 0.1232 0.1022 0.1235 0.0903 0.1250

group 12 13 14 15 16 17 18 19 20 21 22

¯ρ 0.1257 0.0983 0.1011 0.0865 0.1250 0.1078 0.0990 0.1030 0.1003 0.0316 0.099

Table 5.5. Theaverage “(linear) relative”coeﬃcients ¯ρ in full system variables

Table 5.5exhibits that the tilt command is dependentonthesesystem vari-

ables. However,

m =11istoo hightotrain alearningcontroller.

Second, according to thesigniﬁcance order in Table 5.4, we addthe vari-

able onebyone andexecute the dependenceanalysisbetween the former fea-

tures with thenew features andself-contained analysis between the selected

set of features withthe tilt command. After thesimilar process with the

full variable self-contained analysis, we stopwith aset variablesof(‘ADC0’,

‘ADC9’, ‘ADC10’, ‘ADC4’, ‘ADC5’, ‘ADC3’). Table 5.6 shows part of theav-

erage“(linear) relative” coeﬃcients of theself-contained analysis between the

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set variablesof(‘ADC0’, ‘ADC9’, ‘ADC10’, ‘ADC4’, ‘ADC5’) with tilt com-

mand.Table 5.7shows theaverage “(linear)relative” coeﬃcients of theself-

contained analysisbetween the set variablesof(‘ADC0’, ‘ADC9’, ‘ADC10’,

‘ADC4’,

‘ADC5’,‘ADC3

’)

with

the

tilt

command.F

romT

able

5.6,

thes

fea-

tures

(‘ADC0’,

‘ADC9’,

‘ADC10’,

‘ADC4’,

‘ADC5’)

can

notd

eterminet

tilt command and fromTable 5.7, thetilt command is almost dependenton

the variablesof(‘ADC0’, ‘ADC9’, ‘ADC10’, ‘ADC4’, ‘ADC5’,‘ADC3’).

group 1 2 3 4 5 6 7 8 9 10 11

¯ρ 0.5851 0.7654 0.5944 0.6203 0.5336 0.5574 0.5112 0.5799 0.5863 0.6530 0.5334

Table 5.6. Part of ¯ρ in the5variables

group 1 2 3 4

¯ρ 0.1862 0.1654 0.1862 0.1247

Table 5.7.

¯ρ in the6v

ariables

Last, we

remove

theset features of

(‘ADC0’,

‘ADC9’,

‘ADC10’, ‘ADC4’,

‘ADC5’,‘ADC3’), one by onetoobtain the smallestset of features.The ﬁnal

result is (‘ADC0’, ‘ADC10’, ‘ADC5’, ‘ADC3’).The self-cont

ained analysis

between the set variablesof(‘ADC0’, ‘ADC10’, ‘ADC5’, ‘ADC3’) with the

tilt command is shown in Table 5.8. Thetable showsthattilt command is

almost dependentonthe variablesof(‘ADC0’, ‘ADC10’, ‘ADC5’,

‘ADC3’).

group 1 2 3 4 5 6 7 8 9 10 11

¯ρ 0.1250 0.1198 0.1224 0.1219 0.1168 0.1219 0.1198 0.1250 0.1198 0.1168 0.1219

Table 5.8. Part of ¯ρ in the4variables

Learning ControlResult

Based on theseinput variables, we builtupanartiﬁcial neuralnetwork(ANN)

learning controller, basedonlearningimparted fromexpert human demon-

strations. Here, the ANN is asupportvector machine learningarchitecture

(SVM), in [81], whichcan realize avery high learning accuracy with limited

data points. We trainedanSVM learning controller to control Gyrover. The

learningcontroller works well and the model is very reliable.Figure 5.3 shows

the SVMlearningcontrol results.