tl STIMATXNO DEPTH OF CUT FROM BLADE FORCES ON THE PONT COUNTER OBSTACLE VEHICLE(U) NASSACHUSETTS INST OF TECH CABRIDGE 6 3 BRECHER APR 88 CERL-TR-P-S/911

IFIED *eeaaaaalaaia! F/G /1 0

lllllllllllhl

lllllllllllhl

,/IlllIhhlll
VI

140.

% lb

wN

%

J

111111 lIIII, . 1 _A 111 1, 

'0 S

- j. . ,, a ..
USA-CERL TECHNICAL REPORT P-88/11

April 1988 k

US Army CorpsA of Engineers

Construction Engineering

Research Laboratory AD-A195 327

Estimating Depth of Cut From Blade Forces on

the Army Counter Obstacle Vehicle

The Counter Obstacle Vehicle (COY) is a heavyweight

bulldozer-type vehicle used for earthwork and general

obstacle removal in military operations. Microprocessor- . •

based controls have been considered for use on the vehicle -'EL.,-E

to automate blade control and thus enhance operator ",' ." 

performance. This study evaluates the feasibility of using MAY 3 11988

blade force measurements for controlling the depth of cut

in real-time operation of the COV. This method of depth 0

control is significantly limited because the variability of L 9

natural soil deposits causes the blade force to vary over D

time. To account for this, the study used random process

(i.e., stochastic) models of soil variation based on empirical

data to simulate natural deposits. Blade forces as the COV

plowed through the simulated soils were calculated using

soil mechanics models. Kalman filtering and optimal control techniques were used to adjust depth of cut based on

measured blade force. Study results suggest the blade force

variability may not be as great an obstacle to the use of , .

force feedback for blade control as previously thought.

e

0. V

Approved for public release; distribution is unlimited. 5 3j 01 4

~-v X.p-* - -. - ~ * ' ** %* %*~ %***
gw

The contents of this report are not to be used for advertising, publication, or

promotional purposes. Citation of trade names does not constitute an

official indorsement or approval of the use of such commercial products.

The findings of this report are not to be construed as an official Department

of the Army position, unless so designated by other authorized documents.

DESTROY THIS REPORT WHEN IT IS NO L ONGER NEEDED --

DO NOT RETURN IT To THE ORIGIN.4 TOR _ ,

N N % N 'N
Unclassified 0

SECURITY CLASSIFICATION OF THIS PAGE

Form Approved REPORT DOCUMENTATION PAGE OM8 No 0 7040188

_1 Ep Dare Jun )O 1986

la REPORT SECURITY CLASSIFICATION lb RESTRICTIVE MARKINGS

Unclassified

2a SECURITY CLASSIFICATiON AUTHORITY 3 DISTRiBuTION/AVAILABILITY OF REPORT

Approved for public release; distribution

2b DECLASSIFICATION/ DOWNGRADING SCHEDULE is unlimited.

4 PERFORMING ORGANIZATION REPORT NUMBER(S) S MONITORING ORGANIZATION REPORT NUMBER(S)

N/A USA-CERL TR P-88/11

6a NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a NAME OF MON(TORING ORGANIZATION

Massachusetts Institute of (If applicable) U.S. Army Construction Engineering

Technology Research Laboratory

6c ADDRESS (City, State. and ZIPCode) 7b ADDRESS (City, State, and ZIP Code)

Cambridge, MA 02139 P.O. Box 4005

Champaign, IL 61820-1305

Ba NAME OF FUNDING,SPONSOR'NG 8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

ORGANIZATION (If applicable)

Belvoir Research (cont'd STRBE-JNC LAO DA 2544A61L5, March 1986

8c ADDRESS (City, State. and ZIP Code) 10 SOURCE OF FUNDING NUMBERS

Fort Belvoir, VA 22060 PROGRAM PROJECT TASK WORK UNIT

ELEMENT NO NO NO ACCESSION NO

11 TITLE (Include Security Classification)

Estimating Depth of Cut From Blade Forces on the Army Counter Obstacle Vehicle (U)

12 PERSONAL AUTHOR(S)

Baecher, Dr. Gregory B., et al.

13a TYPE OF REPORT 3ub TIME COVERED 114 DATE OF REPORT (Year. Month. Day) 115 PAGE COUNT

final IFROM TO 1988, April 49

16 SUPPLEMENTARY NOTATION

Copies are nvailable from the National Technical Information Service

Springfield VA 22161

17 COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

FIELD GROUP SUB-GROUP >-.zounter Obstacle Vehicle/ control systems

19 03 minesweepers depth control "_

15 06 06 6iade's

19 ABSTRA ontinue on reverse of necessary and identify by block number)

The Counter Obstacle Vehicle (COV) is a heavyweight bulldozer-type vehicle used

for earthwork and general obstacle removal in military operations. Microprocessor-based

controls have been considered for use on the vehicle to automate blade control and thus

enhance operator performance. This study evaluates the feasibility of using blade force

measurements for controlling the depth of cut in real-time operation of the COV. This

method of depth control is significantly limited because the variability of natural soil

deposits causes the blade force to vary over time. To account for this, the study used

random process (i.e., stochastic) models of soil variation based on empirical data to

simulate natural deposits. Blade forces as the COV plowed through, the simulated soils

were calculated using soil mechanics models. Kalman filtering ahd optimal control

techniques were used to adjust depth of cut based on measured blade force. Study results

suggest the blade force variability may not be as great an obstacle to the use of force.

feedback for blade control as previously thought. k P , w"-'

20 DISTRIBUTION A4A" ^ ... ' r)F ARS'-A(T7 121 ABTRACT SECURITY CLASS ICATION

El JNCLASSIFiE,/UNL'fviITEU C3 SAME AS PPT C3 DTIC IJSERS Unclassified

22a NAME OF RESPONSIBLE INDIVIDUAL "2b TELEPHONE (include Area Code) 22. OFFCE SYMBOL

Jane Andrew (217) 352-6511, x 388I CECER-IMT

DD FORM 1473.84 MAR APR ',1 tof may he used unlI oxhausted .d ;I(-. PITy -LASSIFICATION F THIS PA(A_

Al! ,i)ther ed't ,Ov, ie ubibilvi Unclassified

% %
~Unclassified

Block 8a. (Cont'd)

Development and Engineering Center (BRDEC)

S

4

NJ4

Unclasifie
I l p -: -,42-,.4,,, . Y - l - k. - .-. .

FORE WORD

The work described in this report was performed for the Belvoir Research 0

Development and Engineering Center (BRDEC) at Fort Belvoir under Intra-Army Order

DA 2544A6IL5, dated March 1986. Under a contract administered by the U.S. Army Construction Engineering Research Laboratory, Facility Systems Division (USA-CERL-FS),

the research was done by the Department of Civil Engineering at the Massachusetts

Institute of Technology. The BRDEC Technical Monitor was Ms. Gloria Barrett (STRBEJNC), and tie USA-CERL point of contact was Mr. Robert Blackmon.

The authors would like to acknowledge the ideas and suggestions of Mr. Blackmon,

which have improved this report and the direction of the project as a whole. The technical editor was Ms. Jane Andrew, USA-CERL Information Management Office.

COL Norman C. Hintz is Commander and Director of USA-CERL, and Dr. L. R. 0

Shaffer is Technical Director. Dr. Michael J. O'Connor is Acting Chief, USA-CERL-FS.

P0

A c es1i F _or . "'

NVS.

UTIC TA [-

Urln'io i-rced

0 -,,L.. t - hty CI..I

t lit1

IN

3

N N N 
,.w . ,

o- - VFA -Wx a

CONTENTS

Page

DD FORM 1473 1

FOREWORD LIST OF FIGURES AND 3 TABLES 5

INTRODUCTION ..................................................... 7

Background

Objective

Approach

2 BLADE LOAD MODEL ............................................... 8

Simplifying Assumptions

Comparisons With Field Measurements •

3 MODELING SOIL VARIABILITY ........................................ 10

Covariance Function for Undrained Shear Strength

Stochastic Differential and Difference Equations

for Undrained Shear Strength

4 BASICS OF STOCHASTIC CONTROL THEORY ........................... 13

System with White Noise Only

System With Additive Noise

Nonlinear Observations

5 KALMAN FILTER - THEORY AND RESULTS ............................ 16

Kalman Filter

Kalman Filter Applied to a Linear, Scalar System

6 ESTIMATING DEPTH OF CUT FROM BLADE LOAD......................20

System Model

Soil Model

Linearization of Force Model

Simulation Case Studies

7 CONCLUSIONS ...................................................... 25

METRIC CONVERSIONS 25

REFERENCES 26

FIGURES 27

APPENDIX: List of Variables 47

DISTRIBUTION

4 '

NV. 

5 

-

; N.I ve J, %i .*' N~ N.5 p C-.y~~
FIGURES '

Number Page0

1 Stresses Within the Soil Mass at Failure 27

2 Typical Plowing Case for the COV 27

3 Measured and Theoretical Values of Soil Resistance for

Bulldozer Blade 28

4 Scatter Plot and Time Plot of x and R for Simulation Parameters4

of Case Iin Table 2 30

5 Scatter Plot and Time Plot of x and k for Simulation Parameters

of Case 2 in Tabl- 2 3

6 Scatter Plot and Time Plot of x and i for Simulation Parameters

of Case 3 in Table 2 32

7 Scatter Plot and Time Plot of x and i for Simulation Parameters

of Case 4 in Table 2 33

8 Scatter Plot and Time Plot of x and k for Simulation Parameters

of Case 5Sin Table 2 34

9 Scatter Plot and Time Plot of x and k for Simulation Parameters

of Case 6 in Table 2 35

10 Scatter Plot and Time Plot x and k for Simulation Parameters

of Case 7 in Table 2 36

11 True Force F vs. Linearized Force F' for d = 2 ft and c = 400 psf 37 0

12 Back-Calculated Depth (in Feet) From Linearized Force for

c = 400 psf 37

13 Time Plot of Force for Case 1 38

14 Time Plot of True Shear Strength c and Estimated Shear Strength

6 for Case 1 38

A

15 Time Plot of True Depth d and Estimated Depth d for Case 1 39

16 Estimated vs. True Shear Strength for Case 1 39

17 Estimated vs. True Depth for Case 1 40

18 Plot of Force vs. True Depth of Cut for Case 1 40

19 Time Plot of Force for Case 2a 41

5

% %C
FIG U RES (Cont'd)

Number Page

20 Time Plot of True Shear Strength c and Estimated Shear Strength

6 for Case 2a 41

A

21 Time Plot of True Depth d and Estimated Depth d for Case 2a 42

22 Etimtedvs.TrueSher Srenth fr Cse a 4

22 Estimated vs. True SearSnth for Case 2a 42

24 Time Plot of Force for Case 2b 43

25 Time Plot of True Shear Strength c and Estimated Shear Strength

6 for Case 2b 44

A

26 Time Plot of True Depth d and Estimated Depth d for Case 2b 44

27 Estimated vs. True Shear Strength for Case 2b 45

28 Estimated vs. True Depth for Case 2b 4

29 The Effect of Redundant Observation on the Accuracy of Depth

Estimation 46

0~l

TABLES

1 Contribution to Total Force 28

2 Parameters for Kalman Filter Simulation 29

3 Effect of Redundant Observation on Estimation Accuracy 46

0

... t

%'

J P..1

ad Ia.~ %1
ESTIMATING DEPTH OF CUT FROM BLADE FORCES

ON THE ARMY COUNTER OBSTACLE VEH1ICLE

1 INTRODUCTION

Background 

continuous 

Effective 

plowing 

use 

at 

of 

a 

the 

constant 

Counter 

depth. 

Obstacle 

However, 

Vehicle 

because 

(COV) 

the 

for 

COV 

minesweeping 

encounters 

requires

varia-%X

U

tions in ground topography as it moves along, the plow height must be continuously

controlled to achieve a cut of constant depth. Since it is difficult for an operator to

make such adjustments while closed inside the vehicle, automatic control is desirable.

The first requirement for such control is an accurate method for monitoring the

position of the blade. For civilian vehicles, front mounted, exterior devices can be used

to monitor blade position. As a combat vehicle, however, the COV operates in a

hazardous environment where devices mounted on the exterior of the vehicle are

vulnerable. The combat environment also precludes the use of visual inspections by the

COy operators or other personnel.

An attractive alternative for blade control is real-time force feedback, by which

the depth of cut is estimated from the force on the blade. However, the accuracy of

blade control using force feedback depends on the accuracy with which plowing depth can

be estimated from blade force. Such estimates are complicated by the dependence of

blade force on (1) spatial variations of soil properties, (2) obstacles such as roots and

boulders, and (3) dynamic effects of the blade passing through the soil.

Objective

The objective of this work was to assess the feasibility of using force feedback for

controlling depth of cut of the COY blade. Controlling depth of cut depends primarily on

the ability to accurately estimate depth at an arbitrary time from the time stream of

blade forces measured up to and including that time. Thus, this work addressed the

accuracy of depth of cut estimates made from blade forces.

Approach

To evaluate this method of estimation, a model was developed which is based on

variations in the force on the blade which are caused by variations in the properties and

topography of the soil. (The dynamic effects of the blade are not modeled.) To study the

accuracy of the estimate produced using this model, several simulations were conducted.

The development of the overall model begins with a model of the forces on the . blade (Chapter 2). These forces depend on the characteristics of the soil, and especially

on the undrained shear strength, so that property is modeled using stochastic equations

(Chapter 3). Since the ultimate goal is to use estimates of the force to control the

position of the blade, a model of position and control is developed using stochastic

control theory (Chapter 4). The final control function will require an estimate of the

blade position, which can be obtained using the Kalman filter (Chapter 5). Finally all

these elements, the force and soil models, the position equation from the control model -

(currently with zero external control), and the Kalman filter, are used to simulate the

COV plowing operation.

7
2 BLADE LOAD MODEL

The principal plowing force on the COV blade results from shearing soil in front of 0

the blade. The shearing strength of soil is usually modeled well by the Mohr-Coulomb

failure criterion, expressed as' U.

Tf = C + a ftan() [Eq 11 o?

where if = shear stress on the failure surface S

of = normal stress on the failure surface

c = undrained soil strength (cohesion) and

= soil friction angle.

The soil parameters c and s usually vary in space. '-p

In pushing the blade into the soil, horizontal stresses in the soil mass increase until

a failure surface develops. On this surface the stresses - and a satisfy the MohrCoulomb criterion (Figure 1).* The mechanics of blade action and soil failure in this

pushing case are similar to passive failure of retaining structures. Considering twodimensional equilibrium of the failed mass of soil, the force exerted on the blade can be

expressed as 2

2 F = 0.5yd N + 2cdN + qdN [Eq 2] c q

where F = force on the blade

y = unit weight of soil

d = depth of cut

c = cohesion intercept

q = surcharge on the soil surface.

N , N , and N are constants which depend on the soil friction angle and the blade's

roughness. This force model is the sum of three terms. The first term is the contribution of lifting the failed mass up the failure surface, the second term is the undrained 0

shearing of the soil, and the third term is the effect of surchartqe.

Equation 2 is simple in form, but the variability of the soil parameters y, c,

Nb, Nc, and N make the estimation of depth from force measurements impractical.

Simplifications are necessary to apply this equation for the COV application.

Simplifying Assumptions

Smooth Blade

A smooth blade means that the soil exerts no lateral traction on the blade. Soil •

will not stick to the blade as it is plowed. Assuming low friction between the blade and

soil, the parameters N , Nc, and N simplify to functions of the soil friction angle.

C q

T. W. Lambe and V. R. Whitman, Soil Mechanics (John Wiley & Sons, New York, 1968).

*All tables and figures can be found at the end of this report (p 27).

2 Lambe and Whitman.

8

%.. %' . .- - . .... .......... •.
%

N = 2(450 + 012) [Eq 31

N N = tan(450 + 012) [Eq 41 L c q 0

and the failure surface is a plane inclined at (450 + 4)/2 ) from the vertical blade surface.

Undrained Condition

"Undrained condition" means that the soil is plowed quickly enough that water is

not squeezed out of the soil during shearing. Thus, the water content of the soil is 0

constant during plowing. If it is assumed that the soil is sheared at constant water

content and that the blade is smooth, then Equation I reduces to r = c, and the blade

load (Equation 2) becomes

2 F = 0.5yd + 2cd + qd [Eq 51

where the cohesion intercept, c, is the undra*aed shear strength. The assumption of

undrained shearing of soil is reasonable for the COV operating at design plowing speeds

or in low permeability soils such as silts and clays.

To examine the importance of each term in the force model expressed in Equation

2, a typical case was chosen with a blade depth of 2 ft* aid loose soil in front of the

blade. The geometry of this case is shown in Figure 2. Choosing typical c/y ratios in %

the range 2 to 6, the contribution of each term as a percentage of total force is shown in

Table 1.

At average strength values (c/y = 4 to 5) the dominant term in the force Nfunction is the undrained shearing (2cd). As the depth increases, the contribution of the

2

first term (0.5yd ) increases, but the limit on blade height places a physical limit on

the magnitude of this term. For blade heights up to 4 ft, shearing is the dominant

component of force. This observation can be interpreted to mean that for optimal

control, variability in unit weight y can be ignored cokapared to variability in shear

strength: at typical operating depths (< 2 ft), random variations in unit weight result in

insignificant changes of force. For example, at c/y 4, a 10 percent change in unit

weight causes a 1 percent change in force.

Comparisons With Field Measurements

To confirm a force model similar to Equation 5 (with q = 0), O'Callaghan and

McCullen performed full-scale bulldozer tests. 3 A dynamometer was mounted on the

bulldozer blade and the force exerted on the blade was measured for various depths.

Measured and calculated values of horizontal blade force are compared in Figure 3. The -.

theoretical relationship is approximately linear, and thus satisfactory for present pur- 2

poses, because of the insignificant contribution of the lift term (0.5yd2).

*A metric conversion table is provided at the end of this report (p 25).

3 J. R. O'Callaghan and P. J. McCullen, "Soil Mechanics in Relation to Earth-Moving

Machinery," Institution of Mechanical Engineers-Proceedings (Symposium on EarthMoving Machinery), Vol 179, part 3F (1964-65), pp 23-29.

90

S\. .. ~% %9. W.~ % %I~
% P

3 MODELING SOIL VARIABILITY ,

Covariance Function for Undrained Shear Strength 0

In the force model outlined in Chapter 2, the effects of soil are represented by y,

the unit weight and c, the undrained shear strength. The latter quantity, undrained

shear strength of the soil, varies horizontally and vertically in any deposit, even those

said to be homogeneous. This variability is usually modeled as a second-order stochastic

field with statistical parameters estimated from field data.4 A second-order stochastic

field is similar to models used in other branches of science to represent fluctuating time- %

series of data (e.g., economics), only a stochastic field is defined in space rather than

time and can be defined over multiple dimensions. Stochastic field models are used to

characterize the spatial variability of soils because no amount of field testing can ever

completely characterize the point-to-point variation of natural soils.

A second-order, stationary, random process is completely specified by a mean (or

mean trend) and a covariance function. The function describes the degree of covariance

between soil properties as a function of separation in space. For a stationary process,

the covariance between any two points depends only on the (vector) separation between

points and not on their location within the domain. Thus, the covariance function is the

same everywhere. Experience with undrained shear properties of natural soils has •

demonstrated that the covariance function can be adequately modeled as an exponential

function of the form

2 -r/r 2

B(r) = o e o [Eq 6]

where B(r) = covariance function evaluated at separation r

2 = variance of the fluctuations around the mean

rO = a scale parameter called the covariance distance

r = separation between any two points.

Empirically, the covariance distance r° is usually anisotropic, shorter by a factor of

abrut i0 in the vertical compared to horizontal direction.to

De Groot used a maximum likelihood method to estimate the covariance parame- 2

ters, o and ro, for several natural soil deposits. His results suggest that horizontal

covariance distances in different soil deposits can vary from 50 to 200 ft and that

coefficients of variation (standard deviation divided by mean) range from 15 to 35

percent. These results are consistent with the findings of others . 5

'D. J. DeGroot, "Maximum Likelihood Estimation of Spatially Correlated Soil Properties," M.S. Thesis, M.I.T., May 1985.

5G. B. Baecher, et al., "Geotechnical Reliability of Offshore Gravity Platforms," M.I.T.

Sea Grant, Report MITSG 80-20, Cambridge, MA, 1980; Spikula, D., "Statistical :.

Estimation of Soil Properties: An Application," M.S. Thesis, M.I.T., May 1983; W. Tang,

"Probabilistic Evaluation of Penetration Resistance," American Society of Civil

Engineers Journal, Geotechnical Engineering Division, Vol 105, GT10 (1979) pp 1173-

1191. %

l 0 '--N

., , % .% % .. .. % , , - % % , , %" % .. - % . . % %. - ". '% . %, % .- % .% " , '- %'to . % ' . .- %n ,

[ ', -., , ."- .., - -.i -_- - . - , ,a' 6 "... ..,"." .. '.'.'../ %. '.'. -% w- '.-,e,'.,-.. .% .". %
Covariance functions and coefficients of variation reported in the literature usually

refer to point properties measured in individual tests or at specific depths in a boring.

As such, they summarize the stochastic variability among small elements of soil. A

blade, on the other hand, moves soil within a much larger volume. Variations in blade 0

load are related to variations among the average properties of such larger soil volumes.

Since variations from point to point within these larger volumes are averaged, the

coefficient of variation of volume-averaged properties is reduced compared to those of

point properties. At the same time, the covariance function is spread out, giving larger

covariance distances.

The reduction of variability with volume and the spreading of the covariance

function is counteracted by discrete changes in soil type that may occur along the path

of plowing. Soil properties across such discrete changes have little correlation with one

another, and thus both increase the volume-averaged coefficient of variation and

decrease the covariance distance.

0

In the simulation studies it was decided to strike a balance between these opposing

trends by adopting a coefficient of variation of 10 percent and a covariance distance, r,

of 50 ft. Choosing a typical average undrained shear strength of 400 psf, the stochastic

behavior of the undrained shear strength can be completely specified by its first and

second moments:

E[c] 400 psf [Eq 71

B (r) = 16 0 0e-r / 50 [Eq 8]

where E = the expectation (i.e., mean or arithmetical average).

r = the separation in feet.

Stochastic Differential and Difference Equations for Undrained Shear Strength

Optimal stochastic control (discussed in Chapter 4) requires that random processes

must be specified in stochastic differential or difference form. One of the many

stochastic processes of this form is the first-order Gauss-Markov process, which is a

normal or Gaussian process with an exponential covariance function. The covariance

function of the undrained shear strength of soil (Equation 6) is also exponential, so it is

assumed that the undrained shear strength, like the Gauss-Markov, is a normal process.

Therefore, its behavior can be described by a stochastic differential equation for a

normal process: 6

dc/dt = -(1/r 0 )c + (1/r0 )E[c] + e [Eq 9]

where c = undrained shear strength of the soil 0

ro = covariance distance of the undrained shear strength of the soil.

6A. Gelb, Applied Optimal Estimation (M.I.T. Press, Cambridge, MA, 1974).

11

~1* ~%
In Equation 9, c is Gaussian white noise with

E[E] = 0 [Eq 101

E[E2 = (2/r )2 [Eq 11] 

0 C

where E = the expectation operator

r o = the covariance distance of the undrained shear strength of the soil

2c = the variance.

In discrete time Equation 9 can be written as a stochastic difference equation,

c(t+1) = T'c(t) + v(t) [Eq 121

where v(t) is an independent Gaussian sequence, and from Gelb and Equations 7 and 8,

-l/r

= e o = 0.98 [Eq 131

E[v 2 ] G C (1 - T'2) = 62.7. [Eq 14]

% e

A~k

12
4 BASICS OF STOCHASTIC CONTROL THEORY

Stochastic control theory deals with dynamic systems whose states are described by 0

stochastic differential or difference equations. For example, simple stochastic control

involves a linear system disturbed by Gaussian white noise. Linearity refers to measurements, y(t), of the system state, x(t), which depend linearly on the values of x(t).

White noise refers to variations in the state variables, x(t), which are independent and

identically distributed for different values of t. The objective of this theory below is to

determine a function [u(t) below] that will "control" the state of the system [described 0

by x(t)] so as to minimize the deviations of that state from some ideal value or set of K

values. There are two cases to be considered: the ideal case, where measurements of

the state of the system, x(t), are exact, thus the only noise is the Gaussian white

noise, w(t); and the "noisy" case, where observations of the system, y(t) , contain

noise v(0t), in addition to white noise. The derivations below address both cases.

0

System with White Noise Only

In the first case it is presumed that the initial conditions are random, but that the

state of the system can be measured with complete certainty at any time. The system

can be represented by the equation S

x(t+1) = 'x(t) + ru(t) + w(t) [Eq 151

where x = n-dimensional state vector

u = m-dimensional control vector

w = n-dimensional vector of random perturbations •

= n*n one-step transition matrix from x(t) to x(t+l)

r = n*m one-step transfer matrix from u(t) to x(t+1).

By the Gaussian white noise assumption, the random vectors w(O), w(1) ..... , w(t) are

independent, with mean E[w(t)] and covariance E[w(t) wT(t)] having values,

E[w(t)] = 0, E[w(t) w (t)] = Q [Eq 16]

where T is the transpose. The initial condition x(0) is taken to be unknown, but with

specified mean E[x(O)J and covariance matrix E[x(0)xT(0)] having the values,

E[x(0)] = 0, E[x(0) xT(0)] = P0 [Eq 17]

The control procedure attempts to minimize deviations of the process x(t) from

some ideal value or trajectory of values. Without loss of generality, these ideal values

are set to zero. For practical reasons, squared deviations of x(t) from the ideal values

x(t) = 0 are frequently chosen to be minimized. This leads to a performance criterion,

J, of the quadratic form,

J = Et [xT(t)Ax(t) + u T(t)Bu(t)I} [Eq 18]

where A and B are positive definite matrices which express the cost of deviations of the

components of x(t), and the cost of controls, respectively. Mathematically, the

13

.% %
matrices A and B are arbitrary. Quantitative values for A and B depend on the application in question.

Because this "ideal" situation is identical to a deterministic control problem, the O

controller u(t) which minimizes J (Equation 18) is7

u(t) = -Lx(t) [Eq 19]

where L, the feedback matrix, depends on F', ', A, and B.

System with Additive Noise

For most optimal control problems, including force feedback for the COV, the state

variables, x(t ), are observed indirectly and with measurement errors. Let y(t ) be the

vector of observations of the states x( ); then

y(t) = g(x(t)) + v(t) [Eq 201

where &(.) = a deterministic function

v(t) = a vector of independent, identically distributed (iid) measurement

errors.

In the blade control problem, the function (x( t)) is the blade load model of Equation 5,

or a linearized version of it. The state vector comprises blade depth and undrained shear

strength. In most cases a( t) is a linear function (discussed in the next section) such that

Equations 15 and 20 lead to the system of equations, S

x(t+l) = Yx(t) + ru(t) + w(t) m

y(t) = Hx(t) + v(t) [Eq 21]

where H is the deterministic matrix of coefficient terms expressing the linear function

g(.), and w(t ) and v(t ) are independent Gaussian white noise sequences with moments

E[w(t)] = O, E[w(t) wT(t1 = Q

T [Eq 221

E[v(t)] = 0, E[v(t) v (t)] = R

and x(0) is a Gaussian vector, independent of v() and w(t), with properties defined

in Equation 17.

0

'A. E. Bryson and Y. C. Ho, Applied Optimal Control (Blaisdell Publishing Co., Waltham,

MA, 1969).

14
lcvY97rgT- " -?WXA R, A

The optimal controller for the system of Equation 21, given the performance

criterion of Equation 18, is 8

u(t) : - Li(t) [Eq 231

where L = the feedback matrix

i(t) = the mean estimate of x(t)conditioned on the measurements

y(0), y(l),...,y(t)

The estimate i(t) is used as if it were exact, similar to deterministic control.

The solution of Equation 23 results from the so-called "separation theorem."' 9 This

theorem holds that stochastic control problems with a quadratic performance criterion

(e.g., Equation 18) can be separated into two independent parts: one is an optimal filter

by which k(t) is estimated from the observations y(t); the other is linear feedback

from the estimated state, i(t), obtained in the same way as in the case of no observation errors.

The process by which i(t) is estimated is based on Kalman filtering. The filter is

recursive and depends only on the system dynamics and noise characterization. The

estimate i(t) is independent of the quadratic performance criterion J. Thus, controlled

behavior for a dynamic system depends to a large extent on the performance of the

Kalman filter. As a result, a major portion of this feasibility analysis has concentrated

on the adequacy of Kalman filter estimation of depth from the time stream of the blade

load.

Nonlinear Observations 0

In many physical systems, including the blade load model of Equation 5, the

observation vector y(t) is a nonlinear function of the state variables x(t). To obtain

practical solutions, the nonlinear function g(.) can be made linear by expanding it in a

Taylor series around a known vector, for example, the expected value of x(t).10

Filtering and control operations are performed on this linearized system. This approach 0

is used in Chapter 6, where the blade load model is linearized around the desired depth of

cut and the mean undrained shear strength.

8g. J. Astrom, Introduction to Stochastic Control Theory (Academic Press, New York,

NY, 1970). O

9K. J. Astrom.

'OA. Gelb.

15
5 KALMAN FILTER - THEORY AND RESULTS

This section summarizes rudimentary Kalman filter theory as applied to the S

estimation of blade depth from force measurements. The Kalman filter is used to

estimate the conditional mean of the state variables, i(t), from which the controller is

calculated as in Equation 23. Clearly, the accuracy of the control signal u(t) and

the behavior of the controlled system x(t) depend on the accuracy with which i (t ) is

estimated.

Force on the blade is a function of two random variables, the undrained shear

strength of the soil being plowed and the depth of cut. The greater the soil strength and

the deeper the cut, the greater the force. The force feedback system simultaneously

estimates two unknowns and makes adjustments to (i.e., controls) the blade accordingly.

The extent to which the depth of cut at time t+1 can be precisely controlled depends on

the capability of the filter to adequately estimate the existing depth of cut at time t.

This estimate is hindered by random variations of the undrained shear strength and

inaccuracies in the model g(.).

Kalman Filter

Kalman filtering arises in many engineering problems where an estimation

algorithm is required to sequentially process observations of a dynamic system. Since

their development in the early 1960s,1 the properties of the Kalman filters have been

well studied.

From Equation 15 the process equation for an uncontrolled system is

x(t+l) -- x() + w(t) [Eq 24]

where x(t) = the state vector at time t

IF = a transition matrix

w(t) = a vector of lid perturbations (i.e., white noise).

The state x(t) is observed by noisy measurements (the observation includes additive

noise):

y(t) = Hx(t) + v(t) [Eq 25]

where Y() = the observation vector

H = a deterministic matrix

v() = a vector of iid observation errors.

'R. E. Kalman, "A New Approach to Linear Filtering and Prediction Problems,"

American Society of Mechanical Engineers (ASME) Journal of Basic Engineering, Vol

82 (1960), pp 34-45; R. E. Kalman and R. S. Bucy, "New Results in Linear Filtering and

Prediction Theory," ASME Journal of Basic Engineering, Vol 83 (1961), pp 95-107.

16

% % % %! % % 
The initial state x(O), perturbations w(t), and measurement errors v(t) are assumed

to be uncorrelated, with moments

E[x(0)] = 0, E[x(0)x(O) TI = P

0

T E[w(t)] = 0, E[w(t)w(t)T I = Q [Eq 261

E0(t)] = 0, E[v(t)v(t)T I = R

The Kalman filter can be obtained as the linear Bayes' least squares estimate of the

state vector x(t), given the observations y(O), y(l) ......,(t). 12 To indicate

the time at which an estimate of x(t) is made--because different amounts of information are available at different times--let the estimate conditional on observations up to

time t be denoted i(t/t), rather than i(t) as used in the previous section. In this

manner, an estimate conditioned on observations up to time (t-1) would be denoted

:i(t/t-l). The equations of the Kalman filter can be written as

i(t/t) = i(t/t-l) + K(t)[y(t) - Hi(t/t-l)] [Eq 271

where i (t/t-l) = estimate of x(t), given y(t-1)

[Eq 281 = V(t-1/t-1) .'

K(t) = Kalman gain matrix

= p(t/t-l)HT(H[P(t/t-l)]HT + R}-

I [Eq 29

where H = matrix relating the measurements y(t) to the

measured state variables x(t),

P(t/t-1) = covariance matrix of the estimate i(t/t-1) in

Equation 28

[P(t -/t l) ]' T + [Eq 30]

P(t/t) = covariance matrix of the estimate x(t/t)

in Equation 27 [Eq 31]

= (I - K(t)H)[P(t/t-l)J

where I is the identity matrix.

The Kalman filter is recursive in that the estimate of the state variable at time t,

f(t/t), is completely defined from the observation vector Z(t) and the previous step

estimate £(t-l/t-1). At time zero, the covariance matrix of x(O) is given by Equation

26. This is used to calculate the covariance matrix of the first time step estimate before

a measurement is made, P(1/0), using Equation 30. The covariance matrix after the first S

measurement is made, P(1/1), is calculated using Equation 31. Subsequent covariance '

matrices up to P(t/t) are calculated by recursive application of Equations 30 and 31.

12 A. Gelb. -- 7

17U
This recursive property reduces computation and computer memory requirements.

Also, the estimate variances and covariances of Equations 29 through 31 are independent

of the estimates in Equations 27 and 28. This means that covariance matrices and filter

gains can be calculated in advance and stored in the computer system. Thus, the 0

recursive calculations do not have to be made while the system is in operation (i.e., while

the COV is moving), thus real-time application of the Kalman filter is enhanced.

Kalman Filter Applied to a Linear, Scalar System

To illustrate the performance of Kalman filters for simple system dynamics, a

series of simulations were performed. The system model was taken as linear and scalar:

x(t+t) = x(t) + w(t)

[Eq 321

y(t) = x(t) + v(t)

where the notation follows that of Equations 15 and 20.

Let the covariance function of x be

B Bx (r) (r= = a2 e -r/r03o [Eq 33]

xx x

where Bxx(r) = the covariance of x(i), x(i+r)

072 = the variance of x across its domain x

r = the separation distance

ro = the covariance distance (a soil parameter).

For the covariance function of Equation 33, the system components become

= er/ro [Eq 34]

E[w 2(t)] = Q = (1 _ v ) [Eq 351 x

E[x(0)] = 0, E[x 2(0)] = P = [Eq 361

0 X

2 E[v (t)j = R = constant. [Eq 37]

Table 2 is a summary of seven simulated cases. Case 3 is designated the base case

because its covariance distance is that used for the depth-of-cut model in later

sections. The other cases show sensitivity of filter estimates to covariance distance to

2 observation variance R, and variance of the state a x . The last column (designated P)

is the estimation variance, P(t/t), for large time. P is a measure of the limiting

precision of the estimate. Figures 4 through 10 show estimated vs. true values of x and

time plots of those values.

18
Figures 4 through 6 show sensitivity with respect to the covariance distance, and as

a result the perturbation variance Q; i.e., as the covariance distance (rd) increases, the%

estimates icesinaccuracy (P., decreases). This is caused by the higher signal-tonoise ratio ( P 0 /Q ) as Q decreases. In the limiting case r 0 and the process becomes

a random constant with P CO 0 (after a large number of observations the constant is

exactly estimated).

As expected, cases 3, 4, and 5 show that an increase in observation variance, R,

increases the uncertainty in the estimate of x. Cases 6 2and 7 demonstrate the analogous

result that an increase in the fluctuation of X (PO = a ) decreases the accuracy of the

estiate Theimpicatonsof tesemathematical properties for blade control are

consistent with the intuitive notions that (1) the less precisely blade force are measured,

the less precisely depth of cut can be estimated, and (2) the more variable soil properties

are, the less precisely depth of cut can be estimated. The behavior of the asymptotic

estimator variance P '0indicates the performance properties of the estimator when the

COV has plowed a long way (i.e., when initial effects have been attenuated).

190

p~ %9
6 ESTIMATING DEPTH OF CUT FROM BLADE LOAD

This chapter presents results of simulation studies to assess the feasibility of S

estimating the depth of cut from measured blade force. The problem is modeled as a

stochastic control (estimation) problem, where the state variables are depth of cut and

undrained shear strength of the soil, and force is a nonlinear response to the state

variables.

System Model

The dynamics of depth of cut depend on mechanical characteristics of the blade,

the dynamics of control, and the magnitude of noise in the system. The difference

equation governing depth of cut is taken as

d(t+l) = T d d(t) + ru(t) + w d(t) [Eq 381

where d(t) = the (scalar) depth of cut at time t

u(t) = the (scalar) control input

Wd (t) = (scalar) random noise.

Note that Equation 38 implies that the depth of cut, at zero control [wd(t) = 01,

follows a one-step Gauss-Markov process.

In reality the dynamics of the blade may differ from the system of Equation 38.

However, as long as the state equation can be presented in a linear stochastic difference

equation, the details of the estimation process are only slightly changed.

For the present simulation studies, the following parameters were taken as

representative. 2@

E[d(t)] = 2 ft, E[(d(t) - 2) 2 1 = P = 0.50 ft2

-l/r -1/10

Id = e o = e = 0.905 [Eq 39]

2 ( ) (1 2

E[wd(t)] = 0, E[w d2 (t)] = ( - d ) = P = 0.091

Note that the chosen parameters indicate that the depth-of-cut process has a mean of

2 ft, covariance distance of 10 ft, and standard deviation of 0.5 ft. The mean (2 ft) is

approximately the design specification (target value) for mine plowing, the standard

deviation (0.5 ft) is a conservative guess at the uncertainty in initial depth of cut, and

the covariance distance (10 ft) is a conservative estimate presuming on erratic topography.

N

20

% r - W .1C I r -
In the simulation study it was assumed that the control signal, u(t) is zero,

indicating a fixed setting for the blade. For this case, random variations of blade depth

are due to noise and changes in topography. The addition of control terms u(t) is to be

part of a second phase of the work reported here. •

Soil Model

Several assumptions were made to model the soil. A balance was struck between

the reduction of variability with volume and discrete changes in soil type (see Chapter 3)

by assuming a coefficient of variation of 10 percent, a covariance distance, ro, of 50 ft,

and an average undrained shear strength of 400 psf. ro50t

Linearization of Force Model

The force function which was developed in Chapter 2,

F = 0.5yd 2 + 2cd + qd [Eq 401

in a nonlinear function of the state variables d = depth of cut and c = undrained soil

strength (cohesion). This is the function used to relate observations y(t) to the state 0

variables x(t). To make the force function linear, it was expanded in a Taylor series

about the mean values of d and c and truncated to the first two terms. Denoting the

mean values of depth and undrained shear strength by md and mc ,

F md,m + - m)-- + (c_ m [Eq 41]

d (d 6d c 6c

where the partial derivatives are evaluated at m and m . Choosing y = 120 psf,

q = 0 (no surcharge), and the mean values md = 2 ft and m = 400 psf, the specific

blade force (lb/unit width) is,

F = -1840 + 1040d + 4c [Eq 42]

To assess the magnitude of error in the linearized form, several comparisons were

made. Figure 11 shows the comparison for c = 400 psf, the average soil strength. The

approximation is good for depths in the range 1 to 3 ft. Figure 12 is a comparison of -

estimated vs. true depth in which depth is back-calculated from the linearized force,

d = (F + 240)/1040. Knowing the true depth, a force is calculated from Equation 5, then

the linear approximation is used to back-calculate an approximate depth; then real and

approximated depths are compared. The back-calculated depth in Figure 12 is close to

the true depth in the range 0.8 to 3 ft.

These comparisons indicate that the error in the linearized force function is

small. However, more robust linearization techniques are available if needed. 13

'A. Gelb.

21

.'.~~~~~ vVm V N-0a "Ve 'ra . ~j ~ V
Simulation Case Studies

Based on the above discussion, the following system and observation equations were

used to characterize the estimation problems. 0

d(t+l) = T dd(t) + Wd(t) [Eq 431

c(t+l) = C Ct + w (t) [Eq 441

F(t) = -1840 + 1040d(t) + 4c(t) + vF(t) [Eq 451

Parameter values are taken from Equations 39 and 42. Equation 45 is the observation of

the state parameters through the force model, where a random noise term v (t) ex- F

presses errors in force measurement (e.g., derived from data such as those in Figure 3).

Three simulation studies were performed on a MicroVAX minicomputer using

specially written software. The simulation used (pseudo-) random number generators to

obtain values of the noise terms. A sequence of "actual" depths of cut and a sequence of

"actual" soil strengths along a presumed plow path were first generated by Equations 43

and 44, respectively. Then, those sequences were used to calculate a sequence of blade

forces using Equation 45. The parameters of Equations 43, 44, and 45 were kept the

same in each simulation. The sequence of blade forces was "observed" with measurement

error (noise) and the "observed" blade forces put through a Kalman filter to simultaneously estimate depth of cut and soil strength at discrete points along the plow path.

Case 1: Perfect Observations

In Case 1 the observations of depth of cut (force) were assumed to be made without ,-

errors. The parameters of the noise term in Equation 45 are L%

E[vF(t)] = 0, E[vF(t)] = 0 [Eq 46]

Figure 13 shows the time stream of blade force for the simulated shear strength

and blade depth profiles of Figures 14 and 15, respectively. Figures 14 and 15 also show .

the estimated shear strength and blade depths obtained by applying the Kalman filter of

Equation 27 to the time stream of blade force in Figure 13. Scatter plots of estimated

and true state variables are shown in Figures 16 and 17. ,€.

The accuracy of the estimated depth (Figure 17) was greater than expected. To 0

investigate this finding, force was plotted against depth as shown in Figure 18. The -

relationship, except for large deviations at around 1.8 to 2.4 ft, is approximately linear,

which explains the accuracy of the estimated depth. The reason for this linear behavior

is the low coefficient of variation (10 percent) assumed for undrained shear strength

("Soil Model," this chapter). Although c is variable, its variation is small compared to the

mean shear strength (400 psf).

The limiting standard deviation of depth-of-cut estimate (--"-) in this simulation

is 0.075 ft (0.9 in). This suggests that depth of cut might be controlled to within an inch

(rms). This standard deviation, however, should be used with caution since this value is

calculated for the linearized system.

22 P

V-- V N.. % .
Case 2: Noisy Observations of Blade Depth

Case 2 was designed to assess the sensitivity of estimates to noise in the force

measurement. Two simulations were performed, with the noise in the force model

characterized as

E[v ()] = 0, .[v = 40000, [Eq 471

and

E[vF(t)] = 0, E[v (t) = 160000. [Eq 481

Case 2a. In this simulation the variance of the white noise in Equation 45 corresponds, approximately, to a 10 percent coefficient of variation in the force at mean

values of the state variables (d = 2 ft and c = 400 psf).

Figure 19 shows the time stream of blade force for the simulated shear strength

and depth processes in Figures 20 and 21, which are the same as for case 1.

Figures 22 and 23 compare the Kalman filter estimates of state variables to true

values. As expected, the estimates in case 2 are less accurate than case 1, since errors

in the force measurement are translated into estimation errors of the state variables. S

The limiting standard deviation of depth-of-cut estimate in this simulation is 0.16

ft, about twice as large as in case 1.

Case 2b. The coefficient of variation in the force model in this simulation ,-

(Equation 48) is approximately 20 percent at mean values of the state parameters. This _

value is the same as that calculated from experimental results in Figure 3.

Figure 24 shows the time stream of blade force for the simulated shear strength

and depth processes shown in Figures 25 and 26. Note that the true states in this N'

simulation [c(t) and d(t)] are identical to those in case 2a and for case 1.

Figures 27 and 28 compare the Kalman filter estimates of state variables to true

values. The estimates are less accurate than simulation 2a, with the shear strength

estimate particularly biased. The bias is a combined result of the linearization and the

large noise variance.

Case 3: Redundant Observations 0

The characteristics of the force measurement in case 3 were assumed identical to

those in simulation 2b; however, it was assumed that a second, independent, but noisy

estimate of depth could be made. The second measurement was modeled as

y(t) = d(t) + v () [Eq491

Y

with noise parameters

2

E[v (t)] 0, E[v (t)] = v [Eq 50]

y y

This provides a two-dimensional vector of measurements. 0

3'.,

23

%"

%~.~'..~
To assess the improvements in the accuracy of depth estimate as a result of the

redundant observation, six simulations were performed for different values of the error

variance, v 2 , in Equation 50. Table 3 summarizes the results of the simulation. In the

table, (-\/P-)- is the limiting standard deviation in case 2b, and (-/P- )+ is the t CO

limiting standard deviation with redundant observations. The last column in the table,

designated (-\/ )+ is an estimate of (-v/- ) obtained as

(-V ) + l e /v 2+ (1/V-) [Eq 511 .

The implicit assumption in Equation 51 is that depth of cut is an independent process

(r o = 0). The estimate, however, is an upper limit for the correlated process that depth

of cut is in reality. 0

Figure 29 compares ('/- + to (-VP) for values of v in Equation 50. The t e

comparison is close, with a maximum 6 percent deviation. The implication of these

results is that redundant measurements of blade depth, even if noisy, substantially

improve the precision with which depth of cut can be estimated.

0

2-4

V .:r 

% %
7 CONCLUSIONS

The principal conclusion of this study is that the depth of cut for the COV blade 0

can be estimated with reasonable accuracy from measured blade forces. This means that

blade force measurement may provide a satisfactory approach to automating blade

control.

In the accepted equation for blade force, if undrained conditions (i.e., rapid

shearing) and a smooth blade are assumed, the term for undrained shear strength of the

soil being plowed dominates the calculation of blade force. Therefore, the effect of

variations in soil unit weight can be and generally are neglected in calculating variations

in blade force. The spatial variation of undrained shear strength has been found

empirically to be well-modeled by stationary, Gaussian processes with an exponential

convariance function. Thus, soil strengths and resulting blade loads are modeled as a

Gauss-Markov process. Optimal control theory was used to develop a control equation 0

incorporating random variations in topography and measurement errors, which could be

used to control blade depth on the COV. This control equation requires estimates of

depth of cut, obtained using a Kalman filter. Simulation studies indicate that as

covariance distances in soil properties and topographic variations increase, the precision

of estimates of blade depth increases. Similarly, as the variance of observation errors

increases, the precision of blade depth estimates decreases.

The simulation studies were based on a linearized form of the blade force equation

which was adopted to simplify the control equations. The error introduced by the

linearization was found to be small. Parameter values for the simulations were selected

to conform to operation specifications for the COV and to empirically typical soil

properties. When measurements were assumed to be error free, the standard error of the -

estimate of depth of cut was found to be as low as 0.9 in., much smaller than expected.

When measurement noise was increased to a 10 percent coefficient of variation (i.e., 10

percent of the force), the standard error in the estimate of depth of cut rose to 1.9 in.,

still considered modest. Further increases in measurement noise led to further increases

in the standard error of the depth of cut estimate.

Simulation experiments were also undertaken presuming that a second, independent

measurement of blade cut could be made in conjunction with blade force. Significant

increases in estimate precision (i.e., decreases in standard error) can be realized using

such a redundant measurement, even if the noise in that second measurement is large.

While direct force measurement appears a fruitful approach to controlling blade

depth, increased precision can be gained by using one or more redundant sensor systems.

These redundant systems need not be highly accurate to be effective. ..

..

METRIC CONVERSIONS

1 in. = 2.54 cm .

1 ft = 0.3048 m

1 psf = 4.882 x 10- kg/m 2

6.

AN

25

' ' . , . " ." . " ." . , % ' .% ." A . . , * " .e ol " ." %% . . %. " " o .% " ." . .. . •l .. - .
II

REFERENCES

Astrom, K. J., Introduction to Stochastic Control Theory (Academic Press, New York, 0

NY, 1970).

Baecher, G. B., M. Chan, T. S. Ingra, T. Lee, and L. R. Nucci, "Geotechnical Reliability

of Offshore Gravity Platforms," M.I.T. Sea Grant, Report MITSG 80-20, Cambridge,

MA, 1980.

Bryson, A. E., and Y. C. Ho, Applied Optimal Control (Blaisdell Publishing Co., Waltham,

MA, 1969).

De Groot, D. J., "Maximum Likelihood Estimation of Spatially Correlated Soil Properties," M.S. Thesis, M.I.T., May 1985. 0

Gelb, A., Applied Optimal Estimation (M.I.T. Press, Cambridge, MA, 1974).

Kalman, R. E., "A New Approach to Linear Filtering and Prediction Problems," American

Society of Mechanical Engineers (ASME) Journal of Basic Engineering, Vol 82

(1960), pp 34-45.

Kalman, R. E., and R. S. Bucy, "New Results in Linear Filtering and Prediction Theory,"

ASME Journal of Basic Engineering, Vol 83 (1961), pp 95-107.

Lambe, T. W., and V. R. Whitman, Soil Mechanics (John Wiley & Sons, New York, 1968).

O'Callaghan, J. R., and P. J. McCullen, "Soil Mechanics in Relation to Earth-Moving

Machinery," Institution of Mechanical Engineers-Proceedings (Symposium on

Earth-Moving Machinery), Vol 179, part 3F (1964-65), pp 23-29.

Spikula, D., "Statistical Estimation of Soil Properties: An Application," M.S. Thesis,

M.I.T., May 1983.

Tang, W., "Probabilistic Evaluation of Penetration Resistance," American Society of

Civil Engineers Journal, Geotechnical Engineering Division, Vol 105, GT10 (1979),

pp 1173-1191.

• ~26 .,

• N'
BLADE

SOIL

BLADE

SURCHARGE -

FS

--

SOIOL

Figure 2. Typical plowing ease for the COV.

27-

..LADE- ,.

10- LP . y A~~~~~ PPP6 1?. e SURCARG

, A I"F

;6a PR
Table 1

Contribution to Total Force

c/y F = 0.5yd 2cd qd

Lift Shear Surcharge

2 100% 16.5% 67% 16.5%

3 100 12.5 75 12.5

4 100 10.0 80 10.0

5 100 8.3 83 8.3

6 100 7.0 86 7.0

3000 i

2500

2000 0 0

0 .#.0

T1500 OA

00

500

EXPERIMENTAL VALUE 0

0 2__ 3__ 4__ 5

DEP TH OF BLADE-Inches

Figure 3. Measured and theoretical values of soil resistance

for bulldozer blade. c = 3.8 lb/in./in., T = 0.0.

(Source: J. R. O'Callaghan and P. J. McCullen, -,N- " 

"Soil Mechanics in Relation to Earth-Moving

Machinery," Institution of Mechanical EngineersProceedings (Symposium on Earth-Moving Machi- 01

nery), Vol 179, part 3F (1964-65), pp 23-29.

28 •

%%Z%- SV i.; -. '-i.'l - .-. ,.'i . -" -. ' . .% -. - -, ", > L- ,-. -' _. % _ ". % ". . " ". ". . ". ._ . -w • . . " * *- * .%. ,'

j d,' ..- , .,,',. .,. . -,..". ., ...- ,- " ---. .' -. , " ' . .". .,..-.. .-. . ... ....' " .. '.., .,
Table 2

Parameters for Kalman Filter Simulation

CASE ro PO R Q SENSITIVITY P

1 0.1 0.5 0.5 0.000 0.50 ro 0.69

2 1.0 0.5 0.5 0.368 0.43 ro 0.50

3 10 0.5 0.5 0.905 0.09 BASE 0.39

4 10 0.5 0.1 0.905 0.09 R 0.20

5 10 0.5 1.0 0.905 0.09 R 0.50

6 10 1.0 0.5 0.905 0.18 PO 0.42

7 10 0.1 0.5 0.905 0.02 Po 0.33

S

29

% % %W, 'r 29"K, r W %.r 
2.0.

1.0- 0

000 0

o 0 0

0 5 0 0 0 0 C

0 0130 0

0- 012 0 0

a 0 0C

-0.5- C

0C

-10- C)

-I5

-2.0-

-20 -10G 0 1.0 2.0

x

(a)

2.0

'.5 I

100

-Cx

0-0

~~~ v l l

A

x Ko

-2.0

0 20 40 60 80 100

TIME

(b)

Figure 4. (a) Scatter plot and (b) time plot of x and i for simulation

parameters of case 1 in Table 2.

30

-45.
2.0-

1.5 0 0

o 0

1.0 0 0

0.5 - 0 0 0 0 0

05 00 0 00 0-0 0

o3 C13 0 o3 0 o3D

-0.5 o a W

-0 . 0 0

C)0 00

00

0 0

-2.0

-2.0 -1.0 0 1.0 2.0

x

(a)

2.0-

'.5-

0I0 0.5 I

-1.0 I ' 

-15 A

x

-20-

0 20 40 60 80 110

TIME

(b)

Figure 5. (a) Scatter plot and (b) time plot of x and i for simulation

parameters of case 2 In Table 2.

31
200

20-

0 0

050 00 0 3 0

00 o 1 o0 o

-15 03 0 0

-1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1. 2 1.6

x

(a)

20-

x 1 5 __ _ A

IlI\ x

05/

oo S

-05 ' A I It

-1) 0

0° o 40 0 0016

-TM E

(b)

2O32

V V

gU.

TIME

parameters ofcae3 n ale2

-I0v

"1'd
1.5

1.0 - 0

0.5

-0.5-

003 0

-2.0 V

0-2.02.

x

(a)

2-

1.5 -I '_ _ _ _ A

x

1 .0 - 1

0.5-

0-A

x -0.5

1.0

-2.0

TIME

(b)

Figure 7. (a) Scatter plot and (b) time plot of x and x^ for simulation

parameters of case 4 In Table 2.

33

ie ,, '-q- *p*.*I
20

15

0 o

0 % 0

0.5 o 0 o o3 o

o~ Q

0K 0

0 rr 1

-05 0 o

° oqb 0 00 0 -05 o 0o 03% o 3 191

0 0000 

00 0 0

0

-1.5 - 0

-2.0 , , , , ,

-16 -1.2 -0.8 -0.4 0 0.4 0.8 12 1.6

x

(a)

18-

1

4 x

1.2 i A

10I x

08I

04

04-

00

02- I I

-0.4 /A -

- 0.6

-08-

0 20 40 60 80 100

TIME

(b)

Figure 8. (a) Scatter plot and (b) time plot of x and i for simulation

parameters of case 5 in Table 2.

34 CM.
3.0 -

2.0-

1.0-

-10a

0

0

-2.0 a

-30-1.0 1.0 3.0

x

250- A

0

)z 1 4A.

-0.5 A '

-20- \-

-2.5

0 20 40 0810 tooP

TIME

(b) 1

Figure 9. (a) Scatter plot anid (b) time plot of x and i for simulation

parameters of case 6 in Table 2.

35

7V.J

% N

: ~ *'.~.,~ S~*. . *%.N**..,%.U
08

07

06

05 0 0

04 0 0

02 o 0 0 0 go o 6 0

0 a

-04 00 0 0 ~ 

C 0 0 1

-0.6 0

-02 - o _ _ _ _ _-_

-03 - 0 0 o,9

-04 -06 04 -0 0 - .5 - 0

-0.6 o o 0

-08

-08 -06 -04 -02 0 0.2 04 06 08•

(a) X@

xV

a.>

08 •09

0.7

06 Il .\,

05 I A

04 - "

0.3 I r 02 (A 1 ' IV lI \ ,, ::'

<X 0 1 I e"-\ 

0.4 i1 V I ,-. -0 I /

-0. 1 l

-07 -04 -

0 20 40 60 80 100

TIME

(b)

Figure 10. (a) Scatter plot and (b) time plot of x and i for simulation

parameters of case 7 in Table 2.

36
3.0

28 .

2.6 - - - - - - F

2.4 F' 0

2.2

20

1.8 I- .

0c 1.4

1.2

0.6

.0

00

-0.4

-0.2

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

DEPTH (FEET)

Figure 11. True force F vs. linearized force F' for d = 2 ft and

c = 400 psf.

30 '-V

28"

26 TRUE DEPTH %.

24 -ESTIMATED DEPTH

22 "It" 210

I 16

14 %

2

10

08-

0.6I

0.2 , %,"

0 " ! . I l% T . ' 0 .4 8 12 16 i0 2 4 2.8

TRUE DEPTH ,

Figure 12. Back-calculated depth (in feet) from linearized force for e 400 psf. ,"

b 37
3.2 -

3.0

2.8 .

2.6 ,

2.4

2.2

20 - -jo

w d)

()0 1.8 -

LL 1.6

1.4-

1.2

10 Pf

0.8-

0.6- - .-;,0-

0 20 40 60 80 100 0

TIME

Figure 13. Time plot of force for case 1.

440

430 \

420 - / I- I

410 A

400

390 1' " 

370 -

0 350 z\

3 40

320 C 1k '

3.. C D A JA

30 CA I,%

0 20 40 60 80 100 ,;

TIME

Figure 14. Time plot of true shear strength c and estimated shear

strength C for case 1.

38

,"--'

%5 ,,

I i l ' i t lliIt' m

r m[-I !i ! f"I ! I [ I I" I 

I I

-il % I f 
32"'

2.6 -d

2.41

2 -0 INIi J ..- 1.00

2.4 '

I- o- Iv

0.8

0 20 40 60 80 1O0/

TIME

A Figure 15. Time plot of true depth d and estimated depth d for case 1.

0 0/

420-

410 204 80 1

400-.

4390- _

3020 

400

39 30 - DO

I-I

3601 0

S350-9

340 -B

310 0

300-

290-

280 300 320 -340 360 380 400 420 4-40 0%

TRUE STRENGTH0

Figure 16. Estimated vs. true shear strength for case 1.

39,".

V~~~~~. w. .
3.6

3.2 0 •

3.0 -l ,-Y

2.8 -

2.6

I-U 0 0 2.2 C

Q 99 Li 2.01 a

1.44

Li 1.24 0

i.0

0.8-

0.6-

0.4-- I I I I I I I

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

TRUE DEPTH

Figure 17. Estimated vs. true depth for case 1. ?.

3.2 -- - 1

3.0____ _ _

2.8 ___ __

2.6 l i ' ".

2.4 0 fir 

2.2 -- E] 202 __ _ 01___ _

O~.1.6- - ___-. __

1.0'" u o _______

0.8 _ _ '-_,__ 

0.6 . C ,, ,,__

0.8 1.2 1.6 3 0 2.4 2.8 3.2

DEPTH (P'T) S, .

Figure 18. Plot of force vs. true depth of cut for case 1. '

40

0,8 %
3.4-

3.2 -

3.0

2.8

2.6-

2.4

t 2.2 -

g 2.0

00w" 3' 1.8

LA 1.4

1.2-0

1.0

0.8

0.6-

0.4 I

0 20 40 60 80 100 0

TIME

Figure 19. Time plot of force for case 2a.

440 -

430

420 -, ,I

410 V

400 \ LA-

- 390 LA- 380 I'\

370 \ -r \

S360 = 350 \

*-340 \

_ 330 "

320 . . . . . C

310 - A

C

* 290 -

0 20 40 60 80 100 0,

TIME

Figure 20. Time plot of true shear strength c and estimated shear

strength 8 for case 2a.

41
3.4

32 -

30 A

28

2.6-

j 24 -

22

a. 20 I ,

1 6 \j /

1' 4

1.2 I lI'

1.0 -

08-

0 20 40 60 80 100

TIME

A

Figure 21. Time plot of true depth d and estimated depth d for

case 2a.

440 - a ,_"___"

430-

420

10

380

t I3

I 370- 9

z 360 0 0, -- 350 -

340- n

00 330 0 , p:: 320- 0 0 L -P 0..f 0~ [] rb ] ]....

LL 310-

300 -,

290-

280 300 320 340 360 380 400 420 440

TRUE STRENGTH(LB/FT,/FT)

Figure 22. Estimated vs. true shear strength for case 2a.

42

,-' ,_"... W:*S4
3.6-

3.2-n

3.0 D

2.8 -

2.6

2.4- 

1 a @ O

2.2 a q 6 o 

2.0-

1.8 %a

1.6 o o ao

1.4 0o

1.2 -

1.0-

0.8

0.6

0.4- , , .

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

TRUE DEPTH(FT) C

Figure 23. Estimated vs. true depth for case 2a.

3.5

3.0 -

2.5-

2.0

090*0

-0-

0_ .5-

00

05. .:' q

0 20 40 60 80 100

TIME

Figure 24. Time plot of force for case 2b.

43

L =.Cw C MV9 M1 " ,'I ,"
440 ,

430- -

420 - J %V' A

410 - C

400 f

N390

" 370 ,

z 360 1

350

C' 340Q

1- 320 \\",

300 -

290-

280 - 1 1 1 1 1 1 1 1

0 20 40 60 80 100

TIME

Figure 25. Time plot o true shear strength c and estimated shear

strength 8 for case 2b.

32-

30 - d

28 - A

26 

AIId

24 -

2.2 AAIle

20 A

-V I I

-1 8-

1 6 I , 

I

12 - L

1 0

08 

0 20 40 60 

I

80 00

TIME

A Figure 26. Time plot of true depth d and estimated depth d for

case 2b.

44 1

... *.JI
430

IL- 420 2

--- 410 m400-

390- C

Z 370- 03

360-aaW

S350- DP a 3

~340- 13 C

S330-

w 320 0

3.-

- 310 0

C

280 300 320 340 360 380 400 420 440

TRUE SHEAR STRENGTH

Figure 27. Estimated vs. true shear strength for case 25.

3.6-o01

3.4-

3.2 0,

3.0-

2.8

tZ 2.4 E

a. 2.2-

o2.0

1.8 -Q

1.6- . 1.4 - C3

0.8

0.6

0.4- , I 

0.8 1.2 1.6 2.0 2.4 2.8 3.2

TRUE DEPTH (FT)'"

Figure 28. Estimated vs. true depth for case 2b.

45

-Iqj
Table 3

Effect of Redundant Observation on Estimation Accuracy

v (1F) (--/) t )+ % Error

0.071 0.243 0.066 0.068 2.25

0.010 0.243 0.089 0.092 3.70

0.243 0.243 0.162 0.172 6.17

0.316 0.243 0.182 0.192 5.50

0.447 0.243 0.205 0.213 3.90

0.707 0.243 0.224 0.229 2.23

0.22

0.21 -

0.20 -

O. 19~ 

0 0.15

2 0.15

0.14 - + eu~n bevto

0.13 ' - . 2\v -( p = limiting standard deviation with /-o x,t redundant observation

o 0.12

o 0.1V1 +-eestimate of above (Eq 51)

z : < 0.10

ul o.og . ,

0.08 - 0.07

C.,.06 - ----- , . ,'

0 0.2 0.4 0.6 0.8

STANDARD DEVIATION OF ERROR (FT)

Figure 29. The effect of redundant observation on the accuracy of

depth estimation.

46

W1, ,1,11 W, If% ,r% I III .N , - .)..., % %, , " • I , %
APPENDIX:

LIST OF VARIABLES

T f shear stress on the failure surface

c cohesion intercept; equal to undrained shear strength under assumptions

of this report

of normal stress on the failure surface

*soil friction angle

F force on the blade

Y unit weight of soil

d depth of cut

q surcharge on the soil

No, Nc , N constants depending on the soil properties, blade roughness, I,.

N ~ q and geometry

B(r) covariance function at separation distance

2

2 variance of the fluctuations around the mean

r distance between any two points

r 0scale parameter called the covariance distance

E expectation operator 0

B cc(r) covariance function of undrained shear strength of soil at separation

distance r

Gaussian white noise

2 variance of undrained soil strength

v(t) independent Gaussian sequence

v() vector of independent identically distributed (iid) measurement errors " -- r

e-1/ro ; Eq. 12

n*n one-step transition matrix; Eq 15

x(t) n-dimensional state vector

U(t) r-dimensional control vector

47
w(t) n-dimensional vector of random perturbation

independent Gaussian white noise sequence

r n*m one-step transfer matrix

A,B arbitrary positive definite matrices

j performance criterion

L feedback matrix, depending on A, B, T, and r"

identity matrix

T matrix transpose

Q covariance matrix of the depth-of-cut perturbations w(t)

P0 initial covariance matrix of the state vector x(o)

H deterministic matrix

Y(t) vector of observations of the states x(t)

g(. ) deterministic function

R covariance matrix of the measurement errors v(t)

x(t) mean estimate of x(t) conditioned on the measurements 0

P asymptotic variance of the estimated state vector for long

lengths of plowing

_/_asymptotic standard error of the estimated state vector for long

lengths of plowing .*.

48

%®
USA-CERL DISTRIBUTION

Chief of Engineers FORT BELVOIR, VA 22060 (11) 0 41

ATTN: Tech Monitor (10) ATTN: Canadian Liaison Officer

ATTN: CEIM-SL (2) ATTN: British Liaison Officer

ATTN: CECC-P ATTN: Australian Liaison Officer

ATTN: CECW ATTN: French Liaison Officer

ATTN: CECW-O ATTN: German Liaison Officer 0

ATTN: CECW-P ATTN: Water Resources Support Ctr

ATTN: CEEC ATTN: Engr Studies Center

ATTN: CEEC-C ATTN: Engr Topographic Lab.

ATTN: CEEC-E ATTN: ATZA-TE-SU

ATTN: CERD ATTN: ATZA-TE-EM

ATTN: CERD-C ATTN: STRBE-BLURE

ATTN: CERD-M

ATTN: CERM CRREL, ATTN: Library 03755

ATTN: DAEN-ZCE

ATTN: DAEN-ZCF WES, ATTN: Library 39180

ATTN: DAEN-ZCI

ATTN: DAEN-ZCM AFESC, Tyndall AFB, FL 32403 •

ATTN: DAEN-ZCZ

NAVFAC

EHSC, ATTN: Library 22060 ATTN: Engineering Command (9)

ATTN: DET III 79906 ATTN: Division Offices (11)

ATTN: Naval Public Works Center (9)

US Military Academy 10966 ATTN: Naval Civil Engr Lab. (2)

ATTN: Facilities Engineer ATTN: Naval Constr Battalion Ctr

ATTN: Dept of Geography &

Computer Science NCEL

ATTN: Library, Code L08A 93043 - AMC - Dir., Inst., & Servc 1

ATTN: DEH (23) Defense Technical Info. Center 22314

ATTN: DDA (2) FORSCOM

FORSCOM Engr, ATTN: Spt. Det. SETAF Engineer Design Office 09019

ATTN: DEH (28) ... V\7 Engr Societies Library, NY 10017

TRADOC

HQ, TRADOC, ATTN: ATEN-DEH Natl Guard Bureau Instl. Div 20310

ATTN: DEH (19)

US Govt Print Office 22304

SHAPE 09055 Receiving Sect/Depository Copies (2)

ATTN: Surv. Section, CCB-OPS

Infrastructure Branch, LANDA US Army Env. Hygiene Agency

ATTN: HSHB-E 21010

National Bureau of Standards 20899

158

+2

04/88
p A. -- ~ ~ ..-.- 

- -- -

0SSSS

Ia

S

m~q0

4,,.

4,'r

bS

* a'.

I. ~a

.5-