Automama Vol 23 No 4 pp 437 447 1987 0005 1098 87 $:100+000 Prlnled i n Great Brtlam Pergamon Journals Lid

( 1987 Internatmnal Federation of Automatic Control

Time Optimal Control of Overhead Cranes with Hoisting of the Load*

J W AUERNIGt and H TROGER~

The time optimal control of the system which is at rest at the start pomt and has to be at rest again at the target pomt ts determined by piecing together analytically found solutions of the canonical system obtained from Pontryagm's maximum prmctple

Key Words - -M in imum t~me control, overhead crane, load homtlng, maximum principle, control constraints, state constraints, materials handling, ship unloader

Abstract- -The minimum time transfer of a load, suspended from a trolley by ropes, from an initial point at rest to a terminal point where it is required to be at rest again, IS investigated by controlling both the traversing motion of the trolley and the hoisting motmn of the load Special care is given to the modelling of the mechamcal and electrical features of the system Mathematically, a boundary value problem w~th constraints both m the control and state variables is gwen Necessary and sufficient conditions for time optimal solutions are derived using an extensmn of the Pontryagm maximum principle The nonhnear control problem is solved analytically and a detailed dlscussmn of the solutions is given Numerical results are presented which apply to ship unloaders

1 INTRODUCTION

THE REQUIREMENT of faster cargo handhng work leads to higher speeds and consequently to higher accelerations in the motions of cranes Hence swing- ing of the load hanging on ropes is an unavoidable consequence, which ts unpleasant and undesired if it continues at the target point In this paper attentmn will be focussed on the arrangement shown in Fig 1 One can expect that the crane (mamly a ship unloader or a container bridge) has a fixed position with respect to the ship and thus the whole unloading process is done by a planar motion of the load hanging at the moving trolley The basic requirement for the motion of the load now is, that starting from point 0 in Fig 1, where it is at rest, the load should arrive in point E being

*Received 25 February 1986, revised 11 August 1986, revised 5 January 1987 The original versmn of this paper was not presented at any IFAC meeting This paper was recommended for pubhcatmn in revised form by Associate Editor E Kremdler under the &rectmn of E&tor H Kwakernaak

tlnstltut fur Allgemelne Maschmenlehre und Fordertechnlk, Techmsche Umversltat Wlen, Karlsplatz 13, A-1040 Wlen, Austria

~InstltUt fur Mechanlk, Techmsche Umversltat Wlen, Karlsplatz 13, A-1040 Wlen, Austria

437

at rest again in order to be able to unload without any time delay An additional requirement ts that this transfer is done in minimum time Thus, mathematically speaking a minimum time bound- ary value problem has to be solved

Though there exist some papers concernmg such problems, only the problem without hoisting motion has been solved analytically in a satisfactory way [Manson (1977) gave the solution for a speed controlled motor and in Auernlg (1985) this was extended to the torque directed motor whose behavior is determmed by its torque-speed charac- teristic ] Alsop et al (1965), Anselmlno and Llebhng (1967) and Manson (1977) also considered the hoisting motion of the load But because they did not Include the constraint of the trolley traversmg speed, the results are only of restricted practical relevance

Besides the requirement that the load must be at rest at the end of the motion, sometimes, addition- ally, it ts required that there is to be no swingmg of the load in those sections of the transfer when the trolley traverses at constant speed (Alsop et al, 1965, Mlta and Kanal, 1979) Here, however, the authors treat the motion where only swing com- pensation at the target point is required Practically, this ts the more important case and furthermore it always results in shorter transfer times

A different type of approach has been used in Sakawa and Shmdo (1982), where, instead of minimizing the transfer time, the swing of the load is minimized As the swing of the load depends on the acceleration of the trolley, this type of optimization will always lead to considerably greater transfer times compared to the time optimal case

Moreover, the type of control of the motion of the trolley, i e posttIon control vs force control,

438 J W AUERNIG and H TROGER

III

A I J / /

, M__ S

FIG 1 Ship unloader, showing starting point 0 (rope length 1o) and end point E (IL) of the load, transfer distance S of the trolley and three different ldeahzed load transfer paths A, B and C

also plays an ~mportant role m the formulatton of the problem as will be explained below In most papers (Manson, 1977, Mlta and Kanm, 1979, Sakawa and Shmdo, 1982) the simpler case of position control, i e the usage of a speed controlled motor, has been considered or the results are vahd only for this case (Hlppe, 1970, Kuntze, 1972)

In this paper, the control problem of the motion of the trolley will be studied for the minimum t~me solution of the above-mentioned boundary value problem By means of the Pontryagm maximum principle, solution of the minimum time control problem is given for a trolley-load system including the hoisting motion Furthermore, the authors believe they have set up a realistic model, both in its mechanical and electrical aspects As far as possible, analytical solutions are presented, from which the influence of the parameters can easily be stud~ed Th~s, however, requires a restriction to simple load paths On the other hand, in Sakawa and Shmdo (1982) a numerical method has been used to treat a more general path of the load, but w~th strong conditions concerning the coupling between hoisting and travelling motion

2 MECHANICAL MODEL AND EQUATIONS OF MOTION

A planar motion due to hoisting of the load and traversing of the trolley for a fixed posmon of the crane wdl be treated The vahdlty and error estimates of the following slmphfymg assumptions are given m Auermg (1985) The elastic defor-

/~// / ~ / / / / " / / / / / / .~//~//, / , / / / / I / /

lmL9 FIG 2 Mechamcal model of the load-trolley system with the three degrees of freedom x r, qL I and the posmon Xs of the mass

center

mabthty of the crane will be neglected and tt wtll be assumed that all elements are of mfintte sttffness Netther dtsslpative effects hke rollmg reststance and losses tn the drive mechantsm nor wind forces are constdered The load ts a material point hangmg on a massless rope The change of the rope length due to swmgmg of the load (movement of pomt T m Ftg 2) has been neglected

The mechamcal system shown m Fig 2 has the three degrees of freedom xx, ~b and I For regular working conditions and vanishing boundary con- ditlons the amplitudes of the swinging load and also the steady-state angles ~b 0 for constant acceler- ation of the trolley remain small enough such that the equations of motion can be hnearlzed in 4~ and

Time optimal control of cranes 439

$o From the Langrangmn equatmns It follows, with ( ' ) = d/dt , that

XT+ m---~L (XT--I ~b+2 I ~b- I q~) FT m T ~1 T

I q~+2 1 ~b+g ~b=Xr

- - (1 +mE),+m_ALmH (g+Xr ~b)= Fn-mH ,1,

where m L is the mass of the load, and m T is a reduced mass includmg the mass of the trolley and the equivalent contributions of the rotating parts of the trolley travel mechamsm Furthermore, w~th the radius r of the rope drum the reduced quantities m. = ln / r 2 and F n = MH/r of the hoisting mechan- Ism have been mtroduced

There are two basically different cases for the control of the trolley traversing motion Either xr = at(t), which means that the position of the trolley is controlled (position controlled system), or Fr = FT(t), which is a force controlled system The following control variables are mtroduced

aT(t) Fr(t) U T - - o r U T - - (2)

aTm,,t fTm, ,

for the position controlled system or the force controlled system, respectwely aTma, m (2) for the position controlled system ~s given by

aTm.. = max (laT(t)l) to

440 J W AUERNIG and H TROGFR

:", 1L---.

"L J -_ -& r, M r, M -4

- I ? i " ' " I I : .... / ,

II O- n _L - -~

V,..~ 91 ....

FIG 3 Motor characteristics (1) &rect-current shunt-wound motor, (2) three-phase sqmrrel-cage motor, (3) Anselmlno and Ltebhng (1967), (4) Hippe 0970) and Kuntze (1972), (5) present

paper

optimal solutmn Addmonal ly to (8), the speed constraint

controlled and posmon controlled sectmns

{0 {O T If O" :j~ ~' T .... (12) = ' UT = If ~' = VTm..

Both real ship unloaders and most real container cranes (especmlly those eqmpped with a spreader rotating device) satisfy /~

Time optimal control of cranes 441

the first part of (13)] and for the rope length. They are

g l = VTma x - - X2 - - - - 1+~

respectwely, where

x4 > 0 and

g2=x5 - 1 >_0 (20)

Gf VTmax ~lm,. aTmal

pracUcal unloading work, and (B) and (C) are limiting cases The hmltmg case (C) (constant hoist- lng speed dunng traversing of the trolley) will now be considered, In order to determine the minimum hoisting speed The trolley traversing section of case (B) is further a llmtlng case of (C)

There is no control of the hoisting motion The hoisting speed and the rope length are determined by the fifth boundary conditions (17) and (18), for which the following are set

20 = lo/l E >_ 1, 2 E = 1 (27)

As a minimum time solution is required, the opti- mality condmon is given by

f? - dt ~ max (21) To solve problem (16)-(21) use is made of the Pontryagln maximum principle Necessary and sufficient conditions for the optimal solution of the problem

fto EfO(x(t),u(t),t) max (22) dt ---1. subject to

x(t) = f(x(t),u(t),t),x,f~R~,u~R ", (23)

This gives

2(~) = 20 + 2' T, 2' 1 - ;t o (28) T E - - T O

The Hamlltonlan H IS given by

H = - 1 + Pl x2 + P3 x4 - (1 + ~) P4 x3 2

+(Pf~+~+Pa) ul (29)

With the velocity constraint given by the first part of (20), this gives the Lagrangtan

- - x , (30) L= H + ql I"Tmax - - x2 1 +

X(to) = x0 and X(tE) = xE, (24) The adjoint equations for P read

u(t) e U ~ R', (25)

g(x(t), t) > 0, g e R ~ (26)

can be found In Kamlen and Schwartz (1981) on pp 220-221 and in Theorem 8 in Selerstad and Sydsaeter (1977), where in addition to the usual necessary conditions for optimality also sufficient conditions making use of the concavity of the maximized Hamlltonlan and the state constramts are formulated

5 RESULTS

Though the state equations (16) and the corres- ponding adjomt equations are nonlinear and coupled, analytic solutions can be given This is done by obtaining solutions of the unconstrained and the constrained problem and piecing them together

5 1 Constant hmsttng speed In Fig 1 three qualitatively different cases [ (A ) -

(C)] are shown (A) is the case regularly found in &OT 23 4-8

p]=O p~=( l+~)P~ 2

P2 = --Pl + ql P4 = - -P3 +T- '7 - - - ql 1W~

(31)

Since H is linear In ul, its maximum is obtained for

udO =

+1

e[ - -1 , +1] If P2(Z)

--1

+ p,(T)

>0

=0

442 J W AUERNIG and H TROGER

of the control is given If there exists a solutmn x(r) for th~s control, then the necessary condlUons for opt]mahty are satisfied Furthermore, H is hnear m x and u Thus the maximized Hamdtoman ~s concave Finally, also the first part of (20) is hnear m x Thus both suificlent condmons for opt]mahty are satisfied

Now results for two different cases are gwen

51 I Unconstrained trolley traversing speed If g~ > 0, ]e the velocity constraint is mactwe, ~t follows from the maximum prmople that q~ must be zero Thus the general solutmn of (31) reads

p] =A

P2 =B- -A r

--2 (1 + ~t) P3 -- ,~, [C Jo(z) + D Y0(z)]

pa = z [C Jr(z) + D Yt(z)] (33)

with

2 z = ~ ~/(1 + ~) ) (34)

where d0, Jr , Yo and Y1 are Bessel Functtons of the first and the second kind of order 0 and l According to (32) and (33), the opUmal control is determmed by the s~gn of the sum of a hnear and an oscillating functmn For small d~stances Y. the most general optimal solution has three switching points as shown m F~g 4 Ths bang-bang solutmn is stmllar to the well known symmetric solutmn for constant rope length (for instance see Htppe, 1970,

p2 - 1:)4 ! ,+0<

T02 T23 T~Tss

-1 ~-"

Y _%/ t~

T

[ Toe

FIG 4 Adjomt variables P2 and P4, optimal control u~ 132), speed x2 of the center of mass and a' of the trolley, swing angle q~ of the load and rope length / for constant hoisting speed and

unconstrained traversing speed

mmal values (17) and the reqmrement that the state varmbles m the three switching points z2, za and ~4 must be continuous For larger distances ~z several solutmns can occur Cons~denng first the simple case of constant rope length it can eastly be understood that also T23 + n ~ and za5 + n

values of Z, hmmng cases with one sw]tchmg point also extst

Instead of solwng the canomcal system of the first four state equations (16) and the adjomt equations (31) by determining the mmal values P(~o) or the integration constants A, B, C, D m (33) respectwely, the time intervals Zo2, T23, "t'3s and "L'56 are introduced as new unknowns They are determined by the first four end condmons (18) The reqmred values X(ZE) are gtven by the solutmn

U 1 T 2 x~ =a+b r + -

l+~ 2

x2 b + u] = "t" l+ct

boundary conditions (18) In order to get the fastest solution, the solution with the smallest values of these two time mtervals must be selected

The constants A, B, C and D are now obtained from the swltchmg condition (32) at the switching points z2, z3 and ~4 and the condition that the Hamdtontan becomes zero at the free end pomt rE (Kamlen and Schwartz, 1981, pp 143-150) The solutton ts ttme optimal, if no further swltchmg points exist This is always the case

Figure 5 shows, for increasing traversmg dstance E, the vanatmn of the locatmn of the switching points L(E) and of the corresponding time intervals L~(Y.), together wtth some typical relatlonshtps of the velocity x2 of the center of mass

U 1 " (3 - - l+ct

- - 2 + z [c J l (z) + d Yl(z)]

Ul 2 ,+2 (1 +or) x , - 1 + ~ 2' [c Jo(Z) + d Yo(z)]

(35)

of the first four state equations (16), the first four

5 1 2 Constramed trolley traversing speed For short trolley traversing distances (Z small), case 5 1 1 always holds because the trolley does not reach the maximum traversmg speed VT=.. HOW- ever, if 5". is big enough such that VT=.. tS reached, the constramt gwen by the first part of (20) is actwated, resulting m gl = 0 Differentiating the

Time optimal control of cranes

2s X l-

~-0 50 - : - / / , / - - - -~ = -t X , -2 O0 / / " , , '

/ ~ / ,/ /

-- 1 B i I t t l t

0 ~ , "1 ' I ~ - I ~ I ' I , I , I ' I ' l ' I '

O 1~ 20 30 40 50 50 70 80 90 ~O0 l l0 120

---.-Z

F~G 5 Dependance of the traversing time Zo6, the switching points L and the corresponding ttme intervals % on the traversing distance Z (see also Fig 4)

443

first part of (20) with respect to z and introducing x2 and x, from (16) gives

In the case of an actwe constraint g~, the general solutmn of (16) reads

X3 = X 3 ul = g - - (36) xl a '{- VTmax ~" 1 + /.

If ul saUsfies the first part of (19), whmh always holds according to our chome of ~, there is a singular solution to the control problem, ~e the switching funcUon gwen by the second part of (32) is zero Dlfferentmtmg the second part of (32) and introducing (31) gwes the multlpher function qt = Pt/( 1 + ~) + P3, which allows one to find the general solution of (31) as

p l=,4

P2 = - - (1 + ~X) P4

- -2(1 +~x) [C Jo(z)+O Yo(z)] P3-- 2'

cx

+ 1- -~ Pl

P4 = 2 [(~ J l ( z ) +/ ) YI(Z)] (37)

with

2 z = i-~?1 ,,//~" (38)

~x = - - - - x 4 X2 VTm'x 1 +

x 3 = z [6 Jl(z) + i [ YI(z)]

2 x4=-~ [6 Jo(z) + 7a yo(z)] (39)

To obtain the optimal solution of the motion of the trolley one must combine the two solutions (33) and (37) for the adjomt variables and the solutions (35) and (39) for the state variables under the condmon that the state varmbles must be continu- ous The adjomt variables Pl and P3 are continuous, P2 and P4 could have jumps However, it turned out that for the problem at hand P2 and P4 are found to be continuous too

Starting from the system w~thout velocity con- stramt (Fig 4), the system behavior with velocity constraint as presented in Fig 6 will be explained As long as a' at zl or z4 is smaller than VTmax , the constraint gl ~s inactive and the control shown m F~g 4 exists However, ff in "c 1 or m z4, a' = VTmax ~s reached, an additional sectmn of constant velocity motion must be included Its t~me period can be calculated from the condmon that a'(z) = vr... for z = zt or r = z4, respectively Depending on the system parameters this period can occur before z2 or z5 or z2 and zs

444 J W AUERNIG and H TROGER

p2 - ~.4

llx ~ 2 3 _ __ - ~ -

u ~

X2~ (7'

l+~ 'T'

I

FIG 6 As m Fig 4 but for constrained trolley traversing speed

TABLE 1 OPERATING (YCLES OF SEVERAL SHIP UNLOADERS, EQUIPPED WITH SPEED CONTROLLED MOTORS (0~ = 0)

Ship unloader VK.., ZO Z E* A 4 18 278 3597 3602 B 4 40 2 06 34 31 34 63 C 321 1 94 1776 2437

2 29 22 44 25 62 D 461 1 13 2998 3013 E 3 04 1 76 23 08 22 39

Section 5 1 2 does not result m a time optimal solution and as m&cated at the end of th~s section a new strategy w~th more control intervals ~s necessary

5 2 Ttme opttmal traversmg and hmstmg motton Th~s problem has not yet been treated analyti-

cally m the l iterature, obviously, because it poses great mathematical difficulties In order to under- stand the behavior of the system quahtatwely the authors conmder the special case where

/o = /E, (41)

l e the load ~s at the same height at the beginning and at the end point of the mot ion Similarly as m Section 5 1, one can write down H, L and the system of the adjoint equations which is of fifth order now In a straightforward calculation H becomes a maximum if one selects ut and U 2 as

u l (O =

+1

~[+1, -1 ] ff P2 l+ct

- l

- -+p+

>0

=0

Ttme optimal control of cranes 445

X=~ 0"

'l UH.o= T

FIG 7 Opt imal control ut of the traversing mot ion and u 2 of the hoist ing motion for the same load height at the start ing and end point and the parameter values ct = 0 5, ,t o = 2, Z = 15, vtt.., = 0 2 The numbers (1)-(4)

correspond to the transfer times toe given m Table 2

TABLE 2 TRANSFER TIME ZOE CORRESPONDING TO THE IDEALIZED

LOAD PATHS IN FIG 7

Case vH.., toe

1 02 967 2 01 997 3 0 10 26 4 -01 10 54

and

2 u! Ps = E + [C Jo(z) + D YI(Z)]

U2 4 (1 + ~t) 2

+ u3 {[C Jo(z) + O Yo(z)]

[c Jo(z) + d Yo(z)] + [C Jl(z) + D Yl(z)] [c J,(z) + d Y,(z)]} (45)

From the control law (42) and the solution for the adlomt variables it can be seen that we have the same situation as in part 5 1 Hence the optimal solutton for the traversing motton of the trolley ~s slmdar to that shown m Fig 4

Startmg from the ltmltmg case VH=.x = 0, the time opUmal control of the traversing and the hoisting motion can be obtained Three typtcal examples are gtven m Figs 7-9 In the dmgrams for xs also other ldeahzed subopttmal rope length relatlonshtps are shown The corresponding traversmg ttmes for Figs 7-9 are given m Tables 2 -4

In Ftg 7 and Table 2 the solution for short distances Z is presented That the load is hfted m the first half of the motton and lowered m the second is physically plausible because for decreasing rope length the period of osctllatlon of the load also decreases It is obvious that with increasing hoisting velocity the traversing time is reduced Increasing the d~stance ~ results m a solution as shown m Fig 8 By means of a comparison with subopUmal control strategtes indicated by the different rope length relationships m Table 3 tt is

1

u:% T -~"-'- t 7"

ko

x,,, V . . . . . ~ . . . . . . ~2 FIG 8 As Fig 7, but for the parameter values c = 0 5, z o = 2,

= 50, v . . . = 0 1 The numbers (1)-(4) correspond to Table 3

TABLE 3 TRANSFER TIME "t'OE CORRESPONDING TO THE IDEALIZED LOAD PATHS IN FIG 8

Case roe

1 17 39 2 1748 3 17 57 4 17 82

shown that the found strategy Is optimal Increasing the hoisting speed m comparison to F~g 8 results m Fig 9, from which it can be seen that two additional switching intervals occur In Table 4 suboptimal strategies are compared w~th the opti- mal again

Because the Hamlltontan IS nonlinear m x, the proof of concawty of the max~mtzed Hamdtoman could not be gwen If several solutions exist satlsfy- mg the necessary condmons the fastest one of them must be picked

6 CONCLUSIONS

Whereas the mechamcal modelling of the system with three degrees of freedom ~s qmte straightfor- ward, special care Is given to the modelhng of the electrical drive of the trolley Here it is possible to

446 J W AUERNIG and H TROGER

1"

' 2 4

Xs ~ x o . . . . . . ~'22-- . . . . . . . . FJG 9 As F~g 7, but for the parameter values :t = 0 5, z o = 2, I: = 50, VH.,, = 0 15 The numbers (1)-(4) correspond to Table 4

TABLE 4 TRANSFER TIME TOE CORRESPONDING TO THE IDEALIZED LOAD PATHS IN FIG 9

Case TOE

1 17 329 2 17 331 3 17 498 4 17 568

include m one mathematical descnptton both cases of motors used m operat ing systems, l e the speed control led and the torque directed motor, where m the model used the former is a specml case of the latter

In the mathematical formulatson of the opt imum control problem an extension of the Pontryagm maximum principle is used which allows one to obtain necessary and sufficient condmons for optl- mahty Though the state equations and the adjomt equations are coupled the authors were able to give analytic solutzons of the canomcal equations m the considered case of ptecewlse constant hoisting speed The opttmal solution basically is a bang- bang control which ~s determined by a switching function containing Bessel functions

By means of the solution of the specml case of constant hoisting speed during trolley traversmg [(C) m Fig 1], the authors were able to evaluate practical ly used other control strategies especmlly for ship unloader systems The solution gwen by Manson (1977) is expanded to systems wtth constrained traversing speed and systems driven by torque directed motors The Hamt l toman is concave and hence all the necessary and sutficlent condtt~ons for opttmahty are sat]stied The most ~mportant parameters m all the authors' cons~der- attons are the transfer d~stance Z and the maximum speed vr.,.x of the trolley, which together wtth the osctl latlon behavtor of the load strongly influence the control strategy General ly speaking, one can say that the shorter the period of oscdlatton of the load the faster the transer can be made Thus the

torque dtrected motor is superior to the speed control led motor The longer the transfer distance E of the trolley, the more complicated the control strategy will be However, the authors have shown for cranes operat ing m European ports, that the strategies whtch they have designed almost always apply

Further, they also treated the time optimal con- trol of the hoisting motion, however, only for the simple case where the two end pomts of the transfer are at the same height In this case only the necessary condit ions of the opt lmahty theorem are satisfied and the authors had to pick out the fastest solution from the set of admtsstble solutions

Fmally, the authors beheve it to be necessary to comment shortly on a second important aspect of the opttmlzatton of such an unloading system, J e the energy consumption The authors did not consider this aspect at all, however, they want to remark that the time opt imal control of the homtmg mot ion results m a higher energy consumption As for practical ly operat ing systems, the rating of hoisting motors is about three to five times that for traversing motors, m the next step of opt imizat ion the energy consumption of the hoisting motion should be included

Ackno~ledgements--Th~s work was partly supported by the Hochschul.lubdaumsstlftung der Stadt Wren and by a grant of the Fonds zur Forderung der Wlssenschafthchen Forschung m Austria under project P5519

REFERENCES Abramowltz, M and I A Stegun (1984) Pocketbook of Math-

ematlcal Functions Abridged edition of Handbook of Math- ematwal Functions Harrl Deutsch, Thun- Frankfurt am Main, FRG

Alsop, C F, G A Forster and F R Holmes (1965) Ore unloader automatlon--a feasibility study Proc 1965 IFAC Syrup, 295-305

Anselmlno, E and T M Llebhng(1967) ZeltOptlmale Regelung der Bewegung elner hangenden Last zwmchen zwel beheblgen Randpunkten Proc 1967 Int Analogue Computation Mee- tings, i, 482-492

Auermg, J W (1985) Steuerstrateglen fur Laufkatzkrane zur Vermeldung des Lastpendelns im Zlelpunkt Dissertation, Techn Umv Wlen, Austria

Franke, R (1973) Ober die Unterdruckung des Lastpendelns be1 Laufkatzkranen, insbesondere von Contamerkranen Dis- sertation, Techn Hochschule Munchen, F R G

Hlppe, P (1970) Zeltopumale Steuerung emes Erzentladers Regelungstechmk und Prozefl-Datenverarbettung, 18, 346- 350

Kamlen, M I and N L Schwartz(1981) DynamwOptlmlzatlon Elsevier North Holland, Amsterdam, New York

Kuntze, H -B (1972) Bletrag zur zeltopumalen Steuerung und Regelung spezleller schwmgungsf'ahlger Systeme (zum Betsplel yon Laufkranen) Dissertation, Hochschule fur Verkehrs- wesen, Dresden, G D R

Manson, G A (1977) Time optimal control methods arising from the study of overhead cranes Ph D Thesis, Umv of Strathclyde, Glasgow, U K

Mlta, T and T Kanal (1979) Optimal control of the crane system using the maximum speed of the trolley (m Japanese with Enghsh abstract) Trans Soc Instrum Control Eng (Japan), 15, 833-838

Sakawa, Y and Y Shmdo (1982) Optimal control of container

T ime opt imal contro l of cranes 447

cranes Automatlca, 18, 257-266 Schlemmmger, K (1970) Angabe des Gutdurchsatzes von

Schlffsentladern mR Grelferbetneb Hebezeuge F6rderm~ttel, 10, 321-325

Selerstad, A and K Sydsaeter (1977) Sufficient conditions m optimal control theory lnt Econ Rev, 18, 367-391