longwall mining automation - ieee control systems society · l ongwall mining is a method for...

Longwall mining is a method for extracting coal fromunderground mines. The mining technology involvesa longwall shearer, which is a 15-m long, 100-t

machine that has picks attached to two drums, whichrotate at 30–40 rev/min. A longwall face is the mined areafrom which material is extracted. The shearer removes coalby traversing a face at approximately 25-min intervals.Traditionally, longwall mining equipment is controlledmanually, where the face is aligned using a string line.Under manual control, the face meanders and wanders outof the coal seam, which causes rock to contaminate the coaland limits the rate of production. Coal production is maxi-mized by maintaining a straight shearer trajectory andkeeping the face within the seam. Therefore, precise esti-mates of the face locations are required so that the long-wall equipment can be repositioned after each shear.

We are automating longwall equipment to improveproduction. A Northrop Grumman LN270 inertial naviga-tion unit and an IEEE 802.11b wireless local area networkclient device are installed within a flame-proof enclosure,

which, together with an odometer, are mounted on ashearer. The inertial navigation unit and odometer mea-sure the shearer’s orientation and distance traveled acrossthe face, respectively. The inertial navigation and odome-ter measurements are stored locally and subsequently for-warded when the shearer is near an access point of thewireless local area network. Upon completion of eachshear, inertial navigation and odometer data are used toestimate the position of the face and control the roof sup-port equipment for the next shear. In particular, minimum-variance, fixed-interval smoothing is applied to the inertialand odometer measurements to calculate face positions inthree-dimensional space.

Filtering refers to the process of estimating the currentvalue of a signal from noisy measurements up to the cur-rent time. In fixed-interval smoothing, measurementsrecorded over an interval are used to estimate past valuesof a signal. Compared to filtering, smoothing can provideimproved estimation accuracy at the cost of twice the com-putational complexity. Smoothing is applicable wherevermeasurements are organized in blocks and retrospectivedata analysis is feasible. In the case of longwall mining, the

28 IEEE CONTROL SYSTEMS MAGAZINE » DECEMBER 2008

to improve process efficiency and control, not only in pre-cision positioning but also in machine reconfigurabilityand adaptability based on changes in machine and materi-als over time as shown in Figure 13.

REFERENCES[1] International Standards Organization, “Methods for the assessment of depar-ture from roundness—Measurement of variations in radius.” ISO 4291-1985(E),1985.[2] National Instruments, NI LabVIEW Control Design and Simulation Module.2008 [cited 30 May 2008]; Available from: http://sine.ni.com/nips/cds/view/p/lang/en/nid/203826.[3] National Instruments, LabVIEW Real-Time for Measurement and Control.2008 [cited 30 May 2008]; Available from: http://www.ni.com/realtime/.[4] National Instruments, NI LabVIEW FPGA—Customize your hardware with-out having to build it. 2008 [cited 30 May 2008]; Available from: http://www.ni.com/fpga/.[5] B.F. Feeny and J.W. Liang, “Decrement method for the simultaneousestimation of coulomb and viscous friction,” Journal of Sound and Vibration.vol. 195, no. 1, pp. 149–154.

AUTHOR INFORMATIONLaine Mears ([email protected]) is an assistant profes-sor at Clemson University, South Carolina, in the Depart-ment of Mechanical Engineering, and a member of theInternational Center for Automotive Research (CU-

ICAR). He previously worked in the automotive manu-facturing industry for over ten years, specializing in high-volume automated assembly. He received the B.S. inmechanical engineering from Virginia Tech and the M.S.and Ph.D. degrees in mechanical engineering from Geor-gia Institute of Technology. He can be contacted at Clem-son University, International Center for AutomotiveResearch, 343 Campbell Graduate Engineering Building, 4Research Drive, Greenville, SC 29607 USA.

Jeannie Sullivan Falcon is a senior engineer in controland simulation at National Instruments, Austin, Texas. Shehas led research in control and mechatronics at the AirForce Research Laboratory, Midé Technology Corporation,and National Instruments. She received the B.S. in physicsfrom Carnegie Mellon and the S.M. and Ph.D. in mechani-cal engineering from MIT.

Thomas Kurfess is the BMW professor of manufacturing,and director of the Campbell Graduate Engineering Cen-ter of CU-ICAR. He previously held academic positionsat Carnegie Mellon and Georgia Institute of Technology.He received the S.B. and S.M. in mechanical engineering,the S.M. in electrical engineering, and the Ph.D. inmechanical engineering from MIT.

Longwall Mining AutomationAn Application of Minimum-Variance Smoothing

GARRY A. EINICKE, JONATHON C. RALSTON, CHAD O. HARGRAVE, DAVID C. REID, and DAVID W. HAINSWORTH

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DECEMBER 2008 « IEEE CONTROL SYSTEMS MAGAZINE 29

position estimates and controls are calculated while theshearer is momentarily stationary at the ends of the face.

The fixed-interval smoothing techniques in use todayare innovations of the 1950s and 1960s [1]–[3]. In [1],Wiener factorizes power spectral densities and constructsestimators that minimize the error variance. The Wienersolution is called “physically unrealizable by its verynature” [1] because it requires future data. In fact, theWiener solution is a fixed-interval smoother, which can beimplemented by both forward and backward processes. Inpractice, smoother realizations involve a forward pass overa measurement record, followed by a backward pass overquantities from the forward calculations.

A state-space approach and the minimum-variance filterof [4] are applied to the fixed-interval smoother problem in[2]. The smoother given in [2] maximizes a likelihood func-tion under the assumption that the input noise processesare Gaussian. The structure of the smoother resembles theKalman filter since the state-estimate update involves asmoother gain that multiplies the error between previoussmoother and filter estimates. The fixed-interval smootherof [3] arises as a linear combination of states from a forwardKalman filter and a backward Kalman filter, weighted bythe inverses of the error covariances.

The Wiener solution [1] applies to transfer functiondescriptions of time-invariant plants. A state-space gener-alization of this solution for output estimation problems isgiven in [5] and [6], which is applicable to time-varying,nonlinear, and uncertain plants. Unlike the techniques of

[2] and [3], the smoother of [5] and [6] minimizes the vari-ance of the output estimation error.

This article reviews the development of the minimum-variance smoother and describes its use in longwallautomation. We describe both continuous- and discrete-time smoother solutions. It is shown, under suitableassumptions, that the two-norm of the smoother estimationerror is less than that for the Kalman filter. A simulationstudy is presented to compare the performance of the mini-mum-variance smoother with the methods of [2] and [3].

LONGWALL MININGLongwall coal mining is an efficient mining method thatcan extract a high percentage of coal from a seam. In alongwall coal mine, two long, horizontal and permanenttunnels known as gate-roads are cut into a coal seam toform the boundaries for a large rectangular block ofcoal, known as a longwall panel [7]. Each gate-road istypically 5–6-m wide. A typical longwall panel is200–400-m wide, and 2–5-km long, which is the lengthof the gate-roads. Figure 1 shows the ends of the gate-roads, where major pieces of mining equipment areinstalled across the longwall face.

Coal is extracted from the longwall panel at the face,which progressively moves back toward the start of thegate-roads [7]. This movement is known as retreat sincethe longwall panel is mined back toward the starting pointof the gate-roads, where the gate-roads branch off at rightangles from the main tunnel roadways. Increasingly, the

FIGURE 1 A plan view of a longwall coal mine. Coal is mined from the longwall face and conveyed out to the surface through the maingategate-road. To maximize productivity, the longwall face needs to be cut straight and perpendicular to the gate-roads. As the mining equipmentmoves forward into the void left by the shearer, the roof collapses behind the roof support equipment, resulting in the goaf being formed.

Extracted Panel (Goaf) Longwall Panels

Longwall Face

Maingate

Longwall Panel Axes

Across Panel Direction

Retreat Direction Gate-Roads Gate-Roads Gate-Roads

Tailgate

UnminedCoal

Goaf

RetreatDirection

UnminedCoal


gate-roads are developed directly from portals in a finalhighwall of an open-cut operation.

The three main items of longwall mining equipmentinstalled on the face are the shearer, which cuts the coal, thearmored face conveyor, which guides and locates the shearerand transports the coal to a conveyor system in the gate-road,and the roof support system. The armored face conveyer hastwin steel chains, which are sprocket driven by 800–1500-kWmotors situated in the gate-roads. The chains are fitted withscraper bars, which run inside the conveyor and carry thecoal along the face. Figure 2 shows the armored face convey-or and the roof support system. The shearer runs back andforth across the armored face conveyor, cutting coal from theface, while the mine roof is held up by the linked but individ-ually hydraulically actuated roof supports, which shuffle for-ward as the coal is mined, causing the face to retreat. Behindthe longwall roof support equipment, the roof collapsesunder its own weight in a region called the goaf. The goaf cancontain unstable voids, which may collapse at any time andthus is off-limits to personnel [7].

Although longwall mining is traditionally a manualprocess, CSIRO Exploration and Mining is undertaking aLongwall Automation Project, sponsored by the industry-funded Australian Coal Association Research Program.This project is automating several aspects of the longwallmining operation, including a system that aligns the faceof the retreating longwall panel perpendicularly to thegate-roads. The automation delivers significant productivi-

ty gains and also provides safety benefits since workers areremoved from the hazardous face region.

Inertial navigation system position estimates exhibit driftrates of one nautical mile per hour at best and thus cannot beused to automate a longwall shearer. Instead, the milliradian-accuracy Euler angles reported by an inertial navigation sys-tem and independent odometer measurements are used in adead-reckoning calculation to estimate the shearer positions.Optimal fixed-interval smoothing is applied to improve thedead-reckoning estimates, and then the longwall equipmentis repositioned at the completion of each shear.

CONTINUOUS-TIME MINIMUM-VARIANCE SMOOTHING

Derivation of the Minimum-Variance SmootherSmoothing problems can be solved in the continuous- anddiscrete-time domains (see “Smoothing Problem Defini-tion”). In the continuous-time case, we consider a linearplant G that operates on the input w(t) ∈ Rm to producethe output y(t) ∈ Rp and has the state-space realization

x(t) = A(t)x(t) + B(t)w(t), x(0) = 0, (1)

y(t) = C(t)x(t), (2)

where x(t) ∈ Rn is the state vector. Suppose that observations

z(t) = y(t) + v(t) (3)

are available, where v(t) ∈ Rp is measurement noise. Wedesire a linear smoother H that operates on z(t) and pro-duces an estimate of y(t) that is optimal in a minimum-variance sense.

Assume that w(t) and v(t) are uncorrelated zero-meanwhite processes with positive-definite covariances Q(t)and R(t), respectively. Define an approximate Wiener-Hopf factor � by

[x(t)δ(t)

]=

[A(t) K(t)R1/2(t)C(t) R1/2(t)

] [x(t)z(t)

], x(0) = 0, (4)

where

K(t) = P(t)CT(t)R−1(t) (5)

denotes the Kalman gain, in which P(t) ∈ Rn×n is the sym-metric solution of the Riccati differential equation

FIGURE 2 A longwall shearer, roof supports, and the armored faceconveyer. The shearer, including the picks on its drum, can beseen in the center of the figure. The armored face conveyer that iscovered in coal is under the shearer. The roof support equipmentappears on the top left. (Photo courtesy of Joy Mining.)


This article reviews the development of the minimum-variance smoother

and describes its use in longwall automation.


P(t) = A(t)P(t) + P(t)AT(t) − K(t)R(t)KT(t)

+ B(t)Q(t)BT(t). (6)

One way of initializing (6) is to use a Hamiltonian matrixsolver that employs the results of [8] to find

P(0) = maxt∈[0,T]

{P : 0 = A(t)P + PAT(t)

− PCT(t)R−1(t)C(t)P + B(t)Q(t)BT(t)}. (7)

A smoother H can be realized by substituting � into theoptimal solution (see “Smoothing Problem Definition”),which yields the system

H = I − R(t)(�H)−1�−1, (8)

where �H denotes the adjoint of � [5], [6]. Suppose that(6) is suitable initialized, and a positive-definite solutionP(t) exists for all t ∈ (0, T]. The smoother (8) requires theinverse of the approximate Wiener Hopf factor �−1 ,which is found by taking the inverse of the system (4),resulting in

[x(t)α(t)

]=

[A(t) − K(t)C(t) K(t)−R−1/2(t)C(t) R−1/2(t)

] [x(t)z(t)

], x(0) = 0.

(9)

Similarly, (�H)−1 is realized by


Consider Figure S1 in which the inputs w and v are uncorrelat-

ed zero-mean white noises with covariances Q and R, respec-

tively. The time dependence is omitted so that the notation is

applicable to both continuous-time and discrete-time domains.

Assume that the linear plant G operates on w ∈ Rm to produce out-

puts y ∈ Rp. Assuming that measurements z = y + v are avail-

able, we seek a linear system H that operates on z and generates

an estimate y of y. Consider the fictitious reference signal indicat-

ed by the dotted lines in Figure S2. It follows that the output esti-

mation error e = y − y is generated by e = Ri , where

R = [H HG − G] is the linear map from the input noises i =[

vw

]to the error e.

Let GH denote the adjoint of G. The problem is to design a sys-

tem H that minimizes ∥∥RRH

∥∥22, that is, a solution is sought that

minimizes the output estimation error covariance. Define

��H = GQGH + R, where the linear operator � : Rp → R

p is a

Wiener-Hopf factor. From the definition of R, it follows that

RRH = GQ GH − GQ GHHH − HGQ GH + H��HHH .

Complet ing the square leads to RRH = R1RH1 + R2RH

2 ,

whe re R1RH1 = GQGH − GQGH (��H )−1GQGH and R2 =

H� − GQ GH (�H)−1. Since ∥∥R1RH

1

∥∥22 excludes H, this quan-

tity defines the lower bound for the output estimation

error covariance.

We seek a solution H that minimizes ∥∥RRH

∥∥22 . Minimizing∥∥R2RH

2

∥∥22 yields the optimum solution

H = GQ GH(R + GQ GH)−1

= (��H − R)(��H)−1. (S1)

The above smoother is of order n4 complexity. An equivalent n2

order smoother can be found by simplifying (S1), which yields

H = I − R(��H)−1

= I − R(R + GQ GH)−1.

Hence knowledge of the plant G together with the second-order noise

statistics Q and R is sufficient to construct an optimal smoother.

FIGURE S1 The output estimation problem. The output of the lin-ear system y = Gw is contaminated with additive measurementnoise v . An estimate y = Hz is desired, where H is a linear oper-ator to be designed.

v

w

z

yy

+

+G ∑ H ˆ

FIGURE S2 Construction of the output estimation error. The errore is constructed by taking the difference between the estimate yand the fictitious reference signal y. It can be seen thate = Hv + (HG − G)w.

e

v

w z yy

+

+

+

−

G ∑ H

∑

ˆ

Smoothing Problem Definition


[−λ(t)β(t)

]=

[AT(t) − CT(t)KT(t) −CT(t)R−1/2(t)

KT(t) R−1/2(t)

]

×[

λ(t)α(t)

], λ(T) = 0. (10)

The estimate y(t|T) of y(t), given the measurements z(t) on(0, T], is calculated as

y(t|T) = z(t) − R(t)β(t). (11)

Note that y(t|T) estimates only the output of the plant,whereas the smoothers in [2] and [3] estimate theentire state.

The smoother (9)–(11) can be implemented by the fol-lowing two-step procedure. First, integrate (9) from t = 0to t = T, to obtain the output α(t). Second, integrate (10)from t = T to t = 0, to obtain β(t) for use within (11) toobtain the smoothed estimate y(t|T).

The output β(t) results from the adjoint system (�H)−1,which operates backward on α(t). In practice, (�H)−1 canbe realized by operating the forward recursion (�T)−1 onthe time-reversed α(t) and taking the time reverse of theresult. That is, β(t) can be calculated from

[λ(τ )

β(τ)

]=

[AT(τ) − CT(τ)KT(τ) −CT(τ )R−1/2(τ)

KT(τ) R−1/2(τ)

]

×[

λ(τ)

α(τ)

], λ(0) = 0, (12)

which evolves forward in the time-to-go variableτ = T − t. Thus, the smoother can be implemented by inte-grating (12) from τ = 0 to τ = T, computing β(τ), thenusing it within (11) to obtain the smoothed estimate y(t|T).

Comparison of Smoother and Filter PerformanceLet R = [H HG − G] denote the linear map from theinput noises

i(t) =[

v(t)w(t)

]

to the output estimation error e(t) = y(t|T) − y(t) . Theerror covariance can factored into two components,

namely, RRH = R1RH1 + R2RH

2 , in which R1RH1

excludes H and defines a lower bound for RRH (see“Smoothing Problem Definition”). It can be shown [6]that the smoother (9)–(11) satisfies

R2 = R(t)[(��H)−1 − (��H)−1]�

= R(t)[(��H)−1 − (��H − ε(t))−1]�, (13)

where ε(t) = C(t)G0P(t)GH0 CT(t), in which G0 is the opera-

tor whose realization is x(t) = A(t)x(t) + B(t) andy(t) = x(t). It follows that the two-norm of the output errorcovariance ‖RRH‖2 is minimized when the solution of (6)is nonincreasing. Suppose that there exists δ > 0 such thatP(t) ≥ P(t + δ) for 0 ≤ t ≤ δ, and

[B(t)Q(t)B(t) AT(t)

A(t) −CT(t)R−1(t)C(t)

]≥

[B(t + δ)Q(t + δ)B(t + δ) AT(t + δ)

A(t + δ) −CT(t + δ)R−1(t + δ)C(t + δ)

]

for all t ∈ (0, T]. Under these conditions, it follows thatP(t) ≥ P(t + δ) for all t ∈ (0, T],

limt→∞

∥∥∥R2RH2

∥∥∥2

= 0, (14)

and e(t) ∈ L2 , and thus the smoother error system isasymptotically stable [6]. That is, the smoother achievesthe best possible performance whenever the Riccati equa-tion solution reaches steady-state conditions.

The causal part H+ of the minimum-variance smootherH is given by

H+ = {I − R(�H)−1�−1}+= I − R{(�H)−1}+�−1

= I − R1/2�−1, (15)

which has the realization

x(t) = A(t)x(t) + K(t)[z(t) − C(t)x(t)] (16)

and

y(t) = C(t)x(t). (17)

It can be seen that H+ is the minimum-variance filter. Sup-pose that either the plant G is time invariant or the solutionof (6) is nonincreasing, so that the smoother achieves (14).Let R denote the map from the input noises to the filtererror e(t) = y(t) − y(t). The filter error covariance can bewritten as RRH = R1 RH

1 + R2 RH2 , in which R1 RH

1excludes H+ and


Longwall coal mining is an efficient

mining method that can extract a high

percentage of coal from a seam.


R2 = H+� − GQ(t)GH(�H)−1

= R(t)[(��H)−1 − (�R1/2(t))−1]�, (18)

which implies ‖R2 RH2 ‖2 > 0. Thus, from (14),

‖RRH‖2 < ‖RRH‖2, (19)

that is, the smoother (9)–(11) outperforms the Kalman filter(16), (17).

Comparison with Alternative Fixed-Interval SmoothersWe now show that the minimum-variance smoother differsfrom the two-filter smoother [3] and the Hamiltoniansmoother [9], [10], which is equivalent to the maximum-likeli-hood smoother [2]. It is then demonstrated that the minimum-variance solution outperforms both the maximum-likelihoodsmoother [2] and the two-filter smoother [3].

The Hamiltonian smoother [9], [10] is given by

[x(t|T)

−λ(t)

]=

[A(t) B(t)Q(t)BT(t)

−CT(t)R−1(t)C(t) AT(t)

] [x(t|T)

λ(t)

]

+[

0CT(t)R−1(t)

]z(t), λ(T) = 0, (20)

and

x(t|T) = x(t) + P(t)λ(t). (21)

Substituting (21) and its derivative into the first row of(20) yields the Kalman filter (16). Substitutingλ(t) = P−1(t)(x(t|T) − x(t)) into the second row of (20)yields the maximum-likelihood smoother [2]

x(t|T) = A(t)x(t) + G(t)[x(t|T) − x(t)], (22)

where G(t) = B(t)Q(t)BT(t)P−1(t).The two-filter smoother [3] uses the Kalman filter states

from (16) within the backward recursion

−λ(t) = A(t)λ(t) + K(t)[x(t) − C(t)λ(t)], (23)

λ(T) = 0, and the linear combination

x(t|T) = [P−1(t) + P−1(t)]−1[P−1(t)x(t) + P−1(t)λ(t)], (24)

where K(t) = P(t)CT(t)R−1(t), in which

−˙P(t) = A(t)P(t) + P(t)AT(t)−K(t)R(t)KT(t) + B(t)Q(t)BT(t).

(25)

The Riccati differential equation (25) can be initialized inmany ways, such as (7). It can be seen that the optimumminimum-variance solution (9)–(11) and the smoothers

(20)–(24) have similar implementation costs since they allrely on forward and backward processes of order n2 com-plexity. A state estimate can be obtained from (10) using

x(t|T) = C†(t)[z(t) − R(t)β(t)], (26)

where C†(t) = [CT(t)C(t)]−1CT(t) is the Moore-Penrosepseudoinverse. Note that the optimal smoother (9)–(11) isapplicable to output estimation and not applicable to stateestimation when C(t) has insufficient rank.

Scalar ExampleLet A(t) = a ∈ R, B(t) = 1, and C(t) = c ∈ R denote time-invariant parameters of the plant G. Also, let p and k ∈ R

denote the solution of (6) and the Kalman gain (5), respec-tively. The transfer function of the Kalman filter (15) is

H(s) = k(s − a + kc)−1,

where s denotes the Laplace transform variable. The trans-fer function of the maximum-likelihood smoother for out-put estimation from (22) is

H(s) = cg(−s − a + g)−1H(s)

= cgk(−s − a + g)−1(s − a + kc)−1,

where g = σ 2wp−1 . Applying the two-filter formula (24)

results in


FIGURE 3 Mean-square error versus input signal-to-noise ratio(SNR) for the case of Gaussian measurement and process noises.This plot compares the performance of (i) the optimum minimum-variance smoother, (ii) the two-filter smoother [3], (iii) the maximum-likelihood smoother [2], and (iv) the Kalman filter. Note that thefixed-interval smoothers outperform the Kalman filter. This exampledemonstrates that the optimum minimum-variance smootherexhibits the lowest mean-square error. At high SNR, however, thedifference in smoother performance becomes inconsequential.

−5 0 5−32

−31

−30

−29

−28

−27

SNR, dB

MS

E, d

B

(i),(ii)

(iii)

(iv )


H(s) = 0.5kc(s − a + kc)−1 + 0.5kc(−s − a + kc)−1

= kc(−a + kc)(s − a + kc)−1(−s − a + kc)−1.

The transfer function of the optimal smoother (9)–(11) isgiven by

H(s) = 1 − σ 2v [1 − k(s − a + kc)−1]σ−2

v [1 − k(−s − a + kc)−1]

= [(−a + kc)2 − (−a + kc − k)2]

× (s − a + kc)−1(−s − a + kc)−1.

The form of these transfer functions shows that the opti-mal minimum-variance smoother differs from the fixed-interval smoothers given in [2], [3], [9], and [10].

Consider an ideal scenario in which the system parame-ters are known precisely and the noise processes areGaussian. Simulations are conducted for the case a = −1,c = 1, σ 2

w = 1, T = 100 s, and δt = 1 ms, using 1000 realiza-tions of zero-mean noise processes. Figure 3 shows theresulting mean-square error versus input signal-to-noiseratio (SNR). The figure illustrates (19), namely, that theminimum-variance smoother outperforms the causal com-ponent. Although the optimal smoother exhibits improvedmean-square error, at high SNR the difference in smootherperformance becomes inconsequential. For this scalarexample, the two-filter formula (24) results in poles com-mon to the optimal solution and outperforms the maxi-mum-likelihood smoother.

The minimum-variance smoother (9)–(11) does notassume that the underlying noise processes are Gaussian.In contrast, the development of the maximum-likelihoodsmoother in [3] assumes that the noises are Gaussian. Sup-pose now that the process noise is the unity-variance deter-ministic signal w(t) = (sin(t)/σsin(t)), where σ 2

sin(t) denotesthe sample variance of sin(t). In this case, Figure 4 showsthat the optimal smoother can improve on the performanceof maximum-likelihood smoother by several decibels.

THE DISCRETE-TIME MINIMUM-VARIANCE SMOOTHERIn the discrete-time case, assume that G has the state-spacerealization

xk+1 = Akxk + Bkwk, (27)

yk = Ckxk, (28)

where xk ∈ Rn, Ak ∈ Rn×n, Bk ∈ Rn×m and Ck ∈ Rp×n withE{wkwT

k } = Qk ∈ Rm×m . Suppose that the observations

zk = yk + vk (29)

are available, where vk is measurement noise andE{vkvT

k } = Rk ∈ Rp×p. The development of the discrete-time,minimum-variance smoother mirrors the continuous-time


FIGURE 5 A block diagram of the discrete-time minimum-variancefixed-interval smoother. The smoother can be implemented by oper-ating �−1 on the measurements zk, operating (�H )−1 on αk, andconstructing yk/N = zk − Rkβk .

Rk

+yk|N

_

Zk

ˆ(ΔH)−1

ˆΔ−1

ßk

αk

∑ ˆ

–

FIGURE 4 Mean-square error versus input signal-to-noise ratio forsinusoidal process noises for (i) the optimum minimum-variancesolution, (ii) the two-filter smoother [3], (iii) the maximum-likelihoodsmoother [2], and (iv) the Kalman filter. The optimal minimum-vari-ance smoother, which does not assume that the input noises areGaussian, outperforms the maximum-likelihood smoother.

−5 0 5

6

8

10

12

14

16

SNR, dB

MS

E, d

B

(i)

(ii),(iii)

(iv )

By automating manual processes, face

workers can be removed from these

hazardous areas.


case. The same smoother structure arises, but the Kalmangain is calculated differently. The optimum output estima-tion solution (see “Smoothing Problem Definition”) sug-gests a smoother of the form

H = I − Rk(�H)−1�−1, (30)

in which � is an approximate Wiener-Hopf factor. A state-space realization of (30) is given by

[xk+1αk

]=

⎡⎣ Ak − KkCk Kk

−�−1/2k Ck �

−1/2k

⎤⎦[

xkzk

], x0 = 0, (31)

[λk−1βk

]=

⎡⎣ AT

k − CTk KT

k CTk �

−1/2k

−KTk �

−1/2k

⎤⎦[

λkαk

], λN = 0,

(32)

yk/N = zk − Rkβk, (33)

where Kk = AkPkCTk �−1

k , �k = CkPk−1CTk + Rk , and

Pk = PTk > 0 is the solution of the Riccati difference

equation

Pk+1 = AkPkATk − Kk�kKT

k + BkQkBTk . (34)

The Riccati equation (34) can be initialized analogouslyto (7). A block diagram of the smoother is shown inFigure 5.

LONGWALL MINE POSITION ESTIMATIONThe nominal drift rate of a high-quality military inertialnavigation system is typically one nautical mile per hour.This high drift error precludes the use of inertial positionestimates alone to control an underground longwallshearer. Instead, the Euler angles reported by an inertialsystem in real time can be used together with odometermeasurements within a dead-reckoning calculation to esti-mate the position of the face at the completion of eachshear. Dead reckoning is the process of estimating the cur-rent position based on previous estimates of position anddirection of travel. Figure 6 shows the shearer’s roll, pitch,and yaw Euler angles, which represent the direction oftravel in three-dimensional space. Dead-reckoning posi-tion estimates can be calculated as the forward differenceof the odometer measurements multiplied by the sine ofthe roll, pitch, and yaw angles.

In the absence of underground truth position refer-ences, offline simulations demonstrate the error perfor-mance of smoother-based position estimation. The Eulerangles θk, ϕk, and φk are generated by

⎡⎣ θk+1

ϕk+1φk+1

⎤⎦ = A

⎡⎣ θk

ϕkφk

⎤⎦ + wk,

where

A =⎡⎣ 0.95 0 0

0 0.95 00 0 0.95

⎤⎦ ,

and wk ∈ R3 with

Q =⎡⎣ 0.01 0 0

0 0.01 00 0 0.01

⎤⎦ .

It is assumed that measurements of the Euler angles areavailable from gyro sensors. The rectangular position coor-dinates xk, yk, and zk are propagated by


FIGURE 6 The Euler angles measured by the inertial navigation sys-tem mounted on a longwall shearer. Yaw refers to the horizontalheading of the shearer across the face. Pitch and roll are rotationsaround the shearer’s transverse and longitudinal axes, respectively.

+ Pitch

Pitch Axis

Roll AxisYaw Axis

+ Yaw

+ Roll

FIGURE 7 Mean-square error of the position estimate versus inputsignal-to-noise ratio. The plot shows the performance of (i) deadreckoning, (ii) the Kalman filter, and (iii) the minimum-variancesmoother. Dead reckoning provides the worst performance becausethe Euler angle measurements are not filtered. The optimal fixed-interval smoother exhibits the best mean-square error since itexploits all of the data available during the measurement interval.

−10 0 10 20−30

−25

−20

−15

−10

−5

0

SNR [dB]

MS

E [m

]

(i)

(ii)

(iii)


⎡⎣ xk+1

yk+1zk+1

⎤⎦ =

⎡⎣ xk

ykzk

⎤⎦ + (dk − dk−1)

⎡⎣ sin(θk)

sin(ϕk)

sin(φk)

⎤⎦ , (35)

where it is assumed that distance measurements dk areavailable from an odometer sensor. Figure 7 shows themean-square error versus input SNR resulting from 1000realizations of zero-mean Gaussian noise processes, underthe simplifying assumption that dk − dk−1 = 1. In Figure 7the dead-reckoning estimates are calculated from (35). Fig-ure 7 demonstrates that Kalman filtering [11] of the dead-reckoning estimates can yield several decibels improvementin mean-square error. The mean-square error exhibited bythe minimum-variance smoother (31)–(33) shows that theoptimal smoother outperforms the optimal filter.

Figure 8 shows some shearer trajectories calculatedfrom underground-mine inertial and odometer measure-

ments. The blue and black curves indicate the estimatedroof and floor positions, respectively, for 45 shears of a250-m coalface. The vertical changes in the trajectories,which can be seen in the figure, reflect the horizon adjust-ments made by the operators to steer the shearer withinthe naturally undulating coal seam.

Inertial navigation-system position estimates exhibit driftrates of one nautical mile per hour at best. Consequently, ifused without additional sensors, the error range of an iner-tial navigation system increases by one nautical mile pereach hour of operation. Therefore, these estimates cannot beused to automate a longwall shearer without additional sen-sor inputs. We instead combine the yaw, pitch, and rollfrom the inertial system, which are accurate to better than 1mrad, together with odometer measurements. The odome-ter has a resolution of 65,536 counts per revolution that hasa gear and ratchet coupling to the shearer rail, resulting in


FIGURE 8 Position estimates calculated from underground longwall mine inertial and odometer measurements. The blue and black tracesindicate the estimated roof and floor positions, respectively, collected over several shifts. The face is about 250-m wide, and its height variesbetween 4.5 m and 5 m. The vertical changes in the trajectories reflect the horizon adjustments made by the operators to steer the shearerwithin the naturally undulating coal seam.

8.0

0 20 40 60 80 100 120 140

Face160 180 200 220 240 260 280 300

10.012.014.016.0

18.020.0

22.024.026.0

28.030.0

32.036.038.0

40.042.0

44.0

Panel

23456789101112131415161718192021222324252627282930313233343536373839

404142434445

Longwall mine equipment is traditionally operated manually, which results in the

face wandering out of the seam and coal product being contaminated by rock.


approximately ±50-cm error over 250 m. From anecdotalobservations, obtained when the shearer reaches surveyedpoints at the gate-roads, the combination of dead reckoningand smoothing yields an average three-dimensional posi-tioning error of about ±20 cm. The estimated shearer posi-tions are used to keep the face straight and in the seam.

Compared to manual control of the mine equipment,the automated system yields improved production rates of140 tons per hour. In addition to productivity gains,automating longwall equipment leads to safety benefits.The coalface is a hazardous area because methane and car-bon monoxide are present, while the area is hot and humidsince water is sprayed over the face to minimize the likeli-hood of sparks occurring when the shearer picks strikerock. By automating manual processes, face workers canbe removed from these hazardous areas.

CONCLUSIONSLongwall mine equipment is traditionally operated manu-ally, which results in the face wandering out of the seamand coal product being contaminated by rock. Since drifterror rates from an inertial navigation unit are greaterthan one nautical mile per hour, they cannot be usedalone to control underground mine equipment. We thususe Euler angles from an inertial system along withodometer measurements within fixed-interval smoothingto estimate the face positions and control longwall equip-ment at the completion of each shear. The automated sys-tem keeps the face straight and thus within the seam,while minimizing creep into the gate-roads.

Smoothers can provide performance improvement com-pared to filtering at the cost of twice the implementationcomplexity. The minimum-variance smoother involves acascade of Kalman and adjoint Kalman predictors withinthe Wiener solution. For time-varying plants, this smootherminimizes the error variance. In the time-invariant case, thesmoother is equivalent to the unrealizable Wiener solution,and minimizes the mean-square error. It is demonstratedby an example that minimum-variance solutions can out-perform other fixed-interval smoothers. Under ideal condi-tions, where the system parameters are known preciselyand the noises are white Gaussian processes, the perfor-mance benefit provided by the optimal smoother can beinconsequential. It is demonstrated by example that, whenthe noise processes are not Gaussian, the reduction inmean-square error can be significant. The minimum-vari-ance smoother described herein is an output estimatorrather than a state estimator, and may not be applicable tostate estimation problems in which the output mappingsare of insufficient rank.

REFERENCES[1] N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary TimeSeries, John Wiley & Sons Inc., New York, 1949.[2] H.E. Rauch, F. Tung, and C.T. Striebel, “Maximum likelihood estimatesof linear dynamic systems,” AIAA J., vol. 3, no. 8, pp. 1445–1450, 1965.

[3] D.C. Fraser and J.E. Potter, “The optimum linear smoother as a combi-nation of two optimum linear filters,” IEEE Trans. Automat. Contr., vol.AC-14, no. 4, pp. 387–390, 1969. [4] R.E. Kalman, “A new approach to linear filtering and prediction prob-lems,” ASME Journal of Basic Engineering, ser. D, vol. 82, pp. 34–45, 1960.[5] G.A. Einicke, “Optimal and robust noncausal filter formulations,” IEEETrans. Signal Process., vol. 54, no. 3, pp. 1069–1077, 2006. [6] G.A. Einicke, “Asymptotic optimality of the minimum-variance fixed-interval smoother,” IEEE Trans. Signal Process., vol. 55, no. 4, pp. 1543–1547,Apr. 2007.[7] R.D. Singh, Principles and Practices of Modern Coal Mining. New Age Pub-lishers, New Delhi, 2004.[8] A.J. Laub, “A Schur method for solving algebraic Riccati equations,” IEEETrans. Automat. Control, vol. 24, no. 6, pp. 913–921, 1979.[9] T. Kailath and P. Frost, “An innovations approach to least-squares esti-mation Part II: Linear smoothing in additive white noise,” IEEE Trans.Automat. Control, vol. 13, no. 6, pp. 655–660, 1968.[10] T. Kailath and L. Ljung, “Two filter smoothing formulae by diagonaliza-tion of the Hamiltonian equations,” Int. J. Control, vol. 36, no. 4, pp. 663–673,1982. [11] B.D.O. Anderson and J.B. Moore, Optimal Filtering. Prentice-Hall, Engle-wood Cliffs, New Jersey, 1979 (reprinted by Dover).

AUTHOR INFORMATIONGarry A. Einicke ([email protected]) received thebachelor’s, master’s, and doctoral degrees, all in electricaland electronic engineering, from the University of Adelaide,in 1979, 1991, and 1996, respectively. He is an adjunct asso-ciate professor at the University of Queensland and a prin-cipal research scientist in CSIRO Exploration and Mining,where he is the leader of the signal processing team. Hechairs the signal processing and communications chapter inthe Queensland section of the IEEE. He can be contacted atCSIRO, Technology Court, Pullenvale 4069, Australia.

Jonathon C. Ralston received the B.Eng. degree in elec-tronics and computing and the Ph.D. degree in signal pro-cessing from the Queensland University of Technology,Brisbane, in 1991 and 1996, respectively. He is a principalresearch engineer in CSIRO Exploration and Mining,where he is the leader of the mine automation group.

Chad O. Hargrave received the B.Eng. degree in com-munications and electronics and the B.A. degree fromthe University of Queensland, Brisbane, in 1994 and1999, respectively. He is a Ph.D. candidate at the Univer-sity of Queensland and a senior research engineer inCSIRO Exploration and Mining, where he leads mineautomation projects.

David C. Reid received the B.Eng. degree in electronicsand computing and the Ph.D. in signal processing from theQueensland University of Technology, Brisbane, in 1991and 1996, respectively. He is a principal research engineerin CSIRO Exploration and Mining, where he is the leaderof the longwall automation team.

David W. Hainsworth received the B.Eng. degree incommunications and electronics and the Ph.D. in signalprocessing from the University of Queensland, Brisbane, in1971 and 1976, respectively. He is a senior principalresearch engineer and leader of the science and engineer-ing program within CSIRO Exploration and Mining.



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