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附錄
附錄一
英文原文
PREDICTION OF CONTROL OF OVERHEAD
CRANES EXECUTING A PRESCRIBED LOAD
TRAJECTORY
Abstract: Manipulating payloads with overhead cranes can be challenging due to the underactuated nature of the system – the number of control inputs/outputs is smaller than the number of degrees-of-freedom. The control outputs (desired load trajectory coordinates), expressed in terms of the system states, lead to control constraints on the system, and the governing equations arise as index five differential-algebraic equations, transformed then to an index three form. An effective numerical code for solving the resultant equations is used. The feedforward control law obtained this way is then extended by a closed-loop control strategy with feedback of the actual errors to provide stable tracking of the required reference load trajectories in presence of perturbations.
Key words: cranes, dynamics, control, trajectory tracking, differential-algebraic equations.
1. INTRODUCTION
Overhead cranes belong to a broader class of underactuated systems – the
controlled mechanical systems in which the number of control inputs/outputs
is smaller than the number of degrees-of-freedom. The performance goal is a
desired load trajectory, i.e. the control outputs are time-specified load coordinates
x (t ) d , y (t) d and z ( t) d . The control inputs are the forces x F and y F
actuating the trolley position and the winch torque n M changing the rope
length (see Fig. 1). The determination of control input strategy that force the
system to complete the prescribed motion is a challenging problem, reflected
in huge amount of research established hitherto.1 The purpose of this study is
to give a fresh view on the problem from the constrained motion perspectiveand to develop the mathematical tools for control design aimed at executing
prescribed load trajectories with relative high speeds and without sway.
The control outputs, expressed in terms of the system states, are treated
as control constraints on the system.2 It is noticed, however, that control
constraints differ from the classical contact constraints in several aspects.
Mainly, they are enforced by means of available control forces (control inputs),
which may have any directions with respect to the control constraint
manifold, and in the extreme may be tangent. A specific methodology must
then be developed to solve such ‘singular’ control problem. The initial governing
equations arise as index five differential-algebraic equations (DAEs).3
They are transformed then to an equivalent index three form, and an effective
code for solving the resultant DAEs is proposed. The feedforward control
law obtained this way is extended by a closed-loop control strategy with
feedback of the actual errors to provide stable tracking of the required reference
load trajectories in the presence of perturbations.
Figure 1. An overhead trolley crane.
2. MATHEMATICAL PRELIMINARIES
Consider a 5-degree-of-freedom ( n = 5 ) overhead (gantry) crane seen in
Figure 1, whose generalized coordinates are , and which is enforced by m = 3 actuators T
. The dynamic equations of the system can be written in the following generic form
where M is the generalized mass matrix, d and f are the generalized dynamic
and applied force vectors, and T B is the matrix of influence of control inputs
u on the generalized actuating force vector f B u T
a . Assumed the hoisting
rope is massless, inextensible and flexible, and neglecting for simplicity all
the forces associated with 1 s , 2 s and l motions apart from the control inputs
x F , y F and n M , the components of dynamic equations are:
where b m , t m and l m are the bridge, trolley and load masses, J and r are the
moment of inertia and radius of the winch, and g is the gravitational acceleration,
and ?? in the mass matrix M denotes a symmetric entry.
The performance goal is a desired load trajectory, i.e. the m ×3 outputs
are time-specified load coordinates
equal in number to the number of control inputs u. Expressed in terms of the system coordinates, the outputs lead to m control constraints2 in the form
whereis the m×n constraint matrix, and
is the constraint induced acceleration. For the crane shown in Figure 1 we have
While Eq. (2) is mathematically equivalent to m rheonomic holonomic
constraints c(q ) 0 , the resemblance of the trajectory control problem to
the constrained motion case may be misleading. Assumed Eq. (2) represents
contact constraints,
a in Eq. (1) must be replaced by c ,
and by assumption the contact constraint reactions are orthogonal to the
manifold of contact constrains. By contrast, the available control reactions
may have arbitrary directions with respect to the control constraint manifold,
and in the extreme some of the control reactions may be tangent. In the latter
case, not all of the desired outputs can directly be actuated by the system
inputs u. A measure of the ‘control singularity’ is the rank of m×m matrix
which represents the inner product of the constrained and controlled subspaces.
4 For the case at hand, rank( ) 1 , and this means that only one control
input ( n M ) actuates directly the control constraint conditions of Eq. (2),
and the other two actuators ( x F and y F ) have no direct influence on realization
of the control constraints.
3. GOVERNING EQUATIONS
The crane dynamic equations (1) can be projected into complementary constrained
and unconstrained subspaces, defined by the 3 5 constraint matrix
C and its orthogonal complement – a 5 2 matrix D such that4
and for the crane considered D can be proposed as
The projection formula is
and the governing equations can be manipulated to:
where Eqs. (7b) and (7c) are the projections of Eq. (1) into the unconstrained
and constrained subspaces, respectively.
While Eq. (7c) stands for m =3 algebraic equations, for the case at hand
we have and as such Eq. (7c) represents only one
independent condition on u and m-p=2. restrictions on the crane motion,
supplementary to original restrictions of Eq. (2). In this way, due to the
mixed orthogonal-tangent realization2 of control constraints, the total number
of motion specifications is thus m+m- p =5= n , and as such the motion
is fully specified. The situation corresponds to flatness5 of the underactuated
system in the partly specified motion
4. SOLUTION CODE
For the case at hand, Eqs. (7) represent thirteen ( 5 2 3 3 ) DAEs in ten
states q and v and three control inputs u. Index of the DAEs is three,3 and
they can be solved by using the simplest Euler backward difference approximation
method. Representing Eqs. (7b), (7c) and (7c) symbolically as
respectively, the solution code can be written as
Given n q and n v at time n t , Eqs. (8) represent thirteen nonlinear algebraic
equations in at time .By solving the equations,
the simulation is advanced from n t to n??1 t . In order to improve accuracy
of the numerical solution, the rough Euler scheme can possibly be replaced
by a higher order backward difference approximation method.3 It may
be worth noting that, due to the original control constraint equations
c(q ，t) =0 are involved in Eqs. (7), the solution is free from the constraint
violation problem, and the truncation errors do not accumulate in time. The
proposed simple code leads to reasonable and stable solutions.
5. SYNTHESIS OF CONTROL
As a solution to Eqs. (8), time-variations of state variables q( t) and v(t ) in
the prescribed motion and the control u( t) that assures the realization of the
specified motion are obtained. The control obtained this way can be used as
a feedforward control for the crane executing the load prescribed motion. It
should then be enhanced by a feedback control to provide stable tracking of
the load trajectory in the presence of perturbations. One possibility is to introduce,
instead of Eq. (2), a stabilized form of the constraint equation at the
acceleration level, where αβand χ are gain values. The modification causes that Eq. (7c) is replaced with
whose symbolic form is againIn other words, the constraint
induced accelerationsare now modified to the stabilized form
by adding the correction terms due to the constraint violations. The hybrid control can then be synthesized from such modified Eqs. (7) using the code of Eq. (8). The idea for crane control with the use of the scheme is shown in Figure 2.
6. NUMERICAL EXPERIMENTS
The crane data used in computations were: mb=20 kg b m , =10 kg t m ,=100 kg l m , r=0.1m, andJ= 0.1kgm . The control task was to move the
load along a straight line following the rest-to-rest maneuver
whereand
are the initial and final load positions at time 0 t and f t , respectively, and
for
and tf=6 s f t , the load motion specifications are illustrated in Figure 3
Figure 3. The load trajectory specifications according to Eqs. (10) and (11)..
The results of inverse simulation, i.e. the solution to the governing equations
(7) by using the code (8), obtained for △t =0.01s , are shown in Figure
4. The control rated this way can be used only as a feedforward control for
the crane executing the prescribed load trajectory.
The robustness of the hybrid control according to Eq. (9) (see Figure 2),
was first tested by applying the inconsistent rest position at 0 t – the load was
placed 0.5m below its reference position,l。=5.5m The gain values were
taken so that to assure the critical damping for the PID scheme,6 i.e.
and a good choice for the integration time step △t= 0.01s was β=10 . The
results of numerical simulations are shown in Figure 5. It can be seen that
the system has a damped response about the reference trajectory.
The other experiment consisted in checking the influence of modeling inconsistency. In the dynamic model used for the direct dynamic simulation, additional damping forces related to 1 s , 2 s and l motions have been involved, not considered in the model used fort the determination of hybrid control. The additional forces were and added respectively to the first, second and third entry of f described in Eq. (1), and the damping coefficient used were k1=k2=35[Nsm -1] and k3=75[Nsm -1] The motion disturbed this way was then stabilized along the reference motion by using the hybrid control. Some results of numerical simulations are shown in Figure 6. While the control characteristics are now decidedly different from the reference control (with no model inconsistencies), the motion of the load as well as the actual motion of the crane are very close to the reference motion characteristics. The simulation was extended over the end of the transfer maneuver (6s) up to 8 seconds, to show that the residual oscillations of the load are damped to the rest position as well.
7. CONCLUSION
A computational framework for control design of overhead cranes executing
a prescribed load trajectory has been presented. The solution to the governing
equations are the crane motion characteristics in the reference motion
and the control required for its realization. The feedforward control scheme
obtained this way is then enhanced by a feedback control, obtained by using
the same governing equations in a slightly modified form.
外文文獻(xiàn)翻譯
控制橋式起重機(jī)執(zhí)行一項(xiàng)指定的負(fù)載的軌跡預(yù)測(cè)
摘要: 操縱橋式起重機(jī)的有效載荷是具有挑戰(zhàn)性的，因?yàn)樗那夫?qū)動(dòng)系統(tǒng)——輸入輸出的控制數(shù)量要小于自由度的數(shù)量。輸出控制（理想的負(fù)載坐標(biāo)），體現(xiàn)在該系統(tǒng)的形式，導(dǎo)致系統(tǒng)的制約因素，并且該方程出現(xiàn)指數(shù)為5的微分代數(shù)方程，然后轉(zhuǎn)化成指數(shù)為3的形式。人們使用一個(gè)有效的數(shù)字編碼來(lái)解決由此產(chǎn)生的方程。一個(gè)閉環(huán)控制策略來(lái)反饋實(shí)際誤差，延伸為前饋控制法獲得的這種方法 ，以提供擾動(dòng)所需的參考負(fù)載軌跡的穩(wěn)定的跟蹤
關(guān)鍵詞：起重機(jī)，動(dòng)力，控制，軌跡跟蹤，微分代數(shù)方程
1.導(dǎo)言
橋式起重機(jī)屬于一個(gè)更廣泛類型的欠驅(qū)動(dòng)系統(tǒng)——輸入輸出控制數(shù)小于自由度數(shù)量的受控機(jī)械系統(tǒng)。績(jī)效目標(biāo)是一個(gè)理想的負(fù)載軌跡。控制輸出i.e.是單位時(shí)間的負(fù)載坐標(biāo)x(t)y(t)和z(t).控制輸出為力Fx和Fy作用于手推車的位置，并且結(jié)合絞車力矩Mn來(lái)改變繩子的長(zhǎng)度（見(jiàn)圖1）測(cè)定控制輸入的策略，迫使系統(tǒng)完成指定議案是一個(gè)具有挑戰(zhàn)性
圖1
的問(wèn)題，迄今為止，反映在大量的研究的確立。目的是在這個(gè)問(wèn)題上給一個(gè)新的觀點(diǎn)，從受限運(yùn)動(dòng)角度并制定數(shù)學(xué)工具，在相對(duì)高的速度和沒(méi)有擺動(dòng)的情況下來(lái)控制設(shè)計(jì)執(zhí)行指定的負(fù)載軌跡的目的?？刂戚敵?，體現(xiàn)在該系統(tǒng)的體系，被當(dāng)成系統(tǒng)的控制限制。然而，控制的限制在某些方面不同于傳統(tǒng)的接觸約束，已經(jīng)引起關(guān)注。
主要是他們被現(xiàn)有的可控制力約束著，這對(duì)遵照約束流形有一定的指導(dǎo)意義，并且在極端位置與可能正切。必須研制一個(gè)方法來(lái)解決這種“奇異”的控制問(wèn)題。初始方程出現(xiàn)指數(shù)為5的微分代數(shù)方程（DAEs）。然后他們被轉(zhuǎn)化成指數(shù)為3的形式，并且人們建議用一個(gè)有效的代碼來(lái)解決由此產(chǎn)生的微分代數(shù)方程。一個(gè)閉環(huán)控制策略來(lái)反饋實(shí)際誤差，延伸為前饋控制法獲
得這種方法，以提供擾動(dòng)所需的參考負(fù)載軌跡的穩(wěn)定的跟蹤。
2 數(shù)學(xué)預(yù)算
假設(shè)一架自由度為5的橋式起重機(jī)見(jiàn)圖1.它的廣義坐標(biāo)是,并且它被一個(gè)M=3的勵(lì)磁機(jī)約束.該系統(tǒng)的動(dòng)力學(xué)方程可以寫成下面這種形式
這里M是廣義質(zhì)量矩陣，d和f是廣義動(dòng)態(tài)和應(yīng)用力向量是輸出控制影響u在廣義驅(qū)動(dòng)力向量下的矩陣。承擔(dān)吊裝繩索是不計(jì)質(zhì)量的，靈活的并且忽略除了由控制輸出的所有與s1,s2和l有關(guān)的微小因素。Fx,Fy和Mn的動(dòng)力學(xué)方程是
mb，mt和ml是導(dǎo)軌，小車和負(fù)載質(zhì)量。J和r是此刻轉(zhuǎn)動(dòng)慣量和絞車半徑。g是重力加速度，并且x在質(zhì)量矩陣?yán)锩媸菍?duì)稱的。
預(yù)期目標(biāo)是一個(gè)理想的負(fù)載軌跡i.e. m=3輸出是一個(gè)指定時(shí)間的載荷坐標(biāo)。和輸出控制U相等。體現(xiàn)在該系統(tǒng)的坐標(biāo)。輸出導(dǎo)致m控制限制于這樣的形式
在加速條件下，初始方程的控制約束兩次不同于遵照時(shí)間獲得的約束條件。
這里是m×n矩陣是一個(gè)m×n約束矩陣并且是約束誘導(dǎo)加速度。對(duì)于圖1所示的起重機(jī)，我們有：
然而公式2在數(shù)學(xué)上相當(dāng)于m完全約束C(q，t）=0.受限運(yùn)動(dòng)情況的相似軌跡控制問(wèn)題可能產(chǎn)生誤導(dǎo)。假定公式2表示接觸約束，公式里的可以被 替代，并且由假設(shè)接觸約束的反應(yīng)是正交的多方面的聯(lián)系，制約了。相比之下，現(xiàn)有的控制反應(yīng)可能有任意方向方面的控制約束形式，并且在極端是一些控制反應(yīng)可正切。在后一種情況下，并不是所有理想的輸出可以被輸入系統(tǒng)u直接驅(qū)動(dòng)?！翱刂破娈悺钡囊粋€(gè)方法就是m×m矩陣的秩
這說(shuō)明了約束和控制因子的內(nèi)積。對(duì)于手頭的情況rank（P）=1，并且這表明僅有一個(gè)控制輸入（Mn）直接促動(dòng)公式2里的控制約束條件，另兩個(gè)物理量（Fx，F(xiàn)y）對(duì)控制限制并無(wú)影響。
3方程
該起重機(jī)的動(dòng)力學(xué)方程（ 1 ）可投影到互補(bǔ)性約束和無(wú)約束子向量，定義為一個(gè)3×5的約束矩陣C和它的和它的正交補(bǔ)集——一個(gè)5×2的矩陣，如
則起重機(jī)的D可以轉(zhuǎn)化為
推算公式為
并且方程可以被轉(zhuǎn)化為
方程7c 7d分別是方程1在無(wú)約束和受限情況下的投影
然而方程7c是m=3的代數(shù)方程，我們使用.，由于方程7c是u的一個(gè)獨(dú)立條件，并且m-p=2限制著起重機(jī)，補(bǔ)充到原來(lái)對(duì)方程2的限制。這樣，由于混合正交正切實(shí)現(xiàn)控制的限制，匯總起來(lái)變成式子m=m-p=5=n,這樣的形式是很明確的。在部分特定條件下，這種情況對(duì)應(yīng)于平坦的欠驅(qū)動(dòng)系統(tǒng)。
4.解決方案
對(duì)于手頭的資料,在10個(gè)位置的q,v和控制輸出u,方程7代表13個(gè)( 5+2+3+3 )微分代數(shù)方程.微分代數(shù)方程的指數(shù)是3,并且利用最簡(jiǎn)單的歐拉向后差分逼近法可以解決這些問(wèn)題.要解決的式子可以寫成
在時(shí)間tn時(shí)給出qn和vn,方程8代表13個(gè)非線性代數(shù)方程在時(shí)，和.通過(guò)求解方程組,模擬是從到的.為了提高數(shù)值解的精確性.歐拉公式可以被高階落后差逼近法取代.也許它應(yīng)該引起注意.由于原始的控制約束方程c(q,t)=0,參與方程7,解決方法與違反約束的問(wèn)題無(wú)關(guān).截?cái)嗾`差不在時(shí)間上累積,擬訂的簡(jiǎn)單的式子引出合理穩(wěn)定的解決方案.
5.合成控制
作為方程8的解決方案,時(shí)間變化的狀態(tài)變量Q(t)和V(t)的指定方案,并且得出了保證實(shí)現(xiàn)指定方案的U(t).控制獲得這種方式可被用來(lái)作為前饋控制吊臂執(zhí)行負(fù)荷指定的方案.然后，就應(yīng)該得到加強(qiáng)反饋控制，以提供對(duì)負(fù)載軌跡中存在的擾動(dòng)的穩(wěn)定的跟蹤.其中一個(gè)可能性是在加速度條件下引進(jìn)一個(gè)穩(wěn)定的形式約束方程，而非方程2, 這里的和都是增益值.修改導(dǎo)致方程7c被下面式子取代
它的基本形式又是0=(b,v,u,t).換言之,約束導(dǎo)致加速度通過(guò)加入對(duì)于約束的校正計(jì)算,現(xiàn)在修改成穩(wěn)定的形式.該混合控制可以再被合成.用這種方法控制起重機(jī)如圖2所示.
圖2
6.數(shù)值演算
該起重機(jī)使用的數(shù)據(jù)計(jì)算分別為:mb=20kg,mt=10kg,ml=100kg,r=0.1m,J=0.1kgm2.控制的任務(wù)是直線運(yùn)輸一項(xiàng)負(fù)載任務(wù),按照安放到安放的機(jī)動(dòng)操作
.
這里并且分別在時(shí)間和時(shí)，是初始的和最終的負(fù)載位置，并且。并且時(shí)，負(fù)載規(guī)格如圖3所示
圖3
逆仿真的結(jié)果是,i.e.解決這個(gè)方程7通過(guò)用8的解,得到,見(jiàn)圖4?？刂祁~定這種方式可以僅僅作為一種前饋控制起重機(jī)執(zhí)行指定負(fù)載軌跡.
圖4
圖5
根據(jù)方程9,這穩(wěn)定的混合控制(見(jiàn)圖2)第一次被測(cè)試,通過(guò)在時(shí)執(zhí)行不一致的安放位置,負(fù)載應(yīng)該放在參考位置下0.5m, .增益值分別被,采取以保證臨界阻尼為PID控制計(jì)劃。i.e.
并且為了整合時(shí)間步進(jìn),一個(gè)很好的選擇是p=10.結(jié)果數(shù)值模擬見(jiàn)圖5.可以看出，該系統(tǒng)有一個(gè)有關(guān)參考軌跡的阻尼反應(yīng)
圖6
其他實(shí)驗(yàn)構(gòu)成制衡的影響建模不一致.在動(dòng)態(tài)模型用于直接動(dòng)態(tài)仿真,有關(guān)s1,s2和l的額外阻尼已經(jīng)計(jì)算在內(nèi),沒(méi)有考慮到在所用的模型測(cè)定混合控制。新增外力為和在方程1里面分別以第一，第二和第三次進(jìn)入的f描述了,并且阻尼系數(shù)分別為和。通過(guò)混合控制,擾亂這種方法的事物順著參考方案穩(wěn)定了下來(lái).一些結(jié)果的數(shù)值模擬如圖6所示.而控制特性現(xiàn)在斷然不同于基準(zhǔn)控制（無(wú)型號(hào)不一致).該方案的負(fù)荷以及起重機(jī)的實(shí)際運(yùn)動(dòng)是很接近的參考運(yùn)動(dòng)特性.模擬被延長(zhǎng)了,在運(yùn)輸?shù)淖詈?6s)增長(zhǎng)到了8秒.這表明了負(fù)荷的震蕩也延伸到了其它位置。
7.結(jié)論
橋式起重機(jī)執(zhí)行一個(gè)指定負(fù)載軌跡的控制系統(tǒng)的設(shè)計(jì)的計(jì)算框架已提交.解這些方程的辦法是起重機(jī)運(yùn)動(dòng)特征實(shí)現(xiàn)參考方案和控制的需求.前饋控制方案獲得了這種方式被一個(gè)反饋控制增強(qiáng).通過(guò)使用相同的方程稍加修改得到.