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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 控制橋式起重機執(zhí)行一項指定的負(fù)載的軌跡預(yù)測 摘要: 操縱橋式起重機的有效載荷是具有挑戰(zhàn)性的，因為它的欠驅(qū)動系統(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。人們使用一個有效的數(shù)字編碼來解決由此產(chǎn)生的方程。一個閉環(huán)控制策略來反饋實際誤差，延伸為前饋控制法獲得的這種方法 ，以提供擾動所需的參考負(fù)載軌跡的穩(wěn)定的跟蹤 關(guān)鍵詞：起重機，動力，控制，軌跡跟蹤，微分代數(shù)方程 1.導(dǎo)言 橋式起重機屬于一個更廣泛類型的欠驅(qū)動系統(tǒng)——輸入輸出控制數(shù)小于自由度數(shù)量的受控機械系統(tǒng)?？冃繕?biāo)是一個理想的負(fù)載軌跡?？刂戚敵鰅.e.是單位時間的負(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來改變繩子的長度（見圖1）測定控制輸入的策略，迫使系統(tǒng)完成指定議案是一個具有挑戰(zhàn)性 圖1 的問題，迄今為止，反映在大量的研究的確立。目的是在這個問題上給一個新的觀點，從受限運動角度并制定數(shù)學(xué)工具，在相對高的速度和沒有擺動的情況下來控制設(shè)計執(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。必須研制一個方法來解決這種“奇異”的控制問題。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有效的代碼來解決由此產(chǎn)生的微分代數(shù)方程。一個閉環(huán)控制策略來反饋實際誤差，延伸為前饋控制法獲 得這種方法，以提供擾動所需的參考負(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的橋式起重機見圖1.它的廣義坐標(biāo)是,并且它被一個M=3的勵磁機約束.該系統(tǒng)的動力學(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是廣義動態(tài)和應(yīng)用力向量是輸出控制影響u在廣義驅(qū)動力向量下的矩陣。承擔(dān)吊裝繩索是不計質(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ΨQ的。 預(yù)期目標(biāo)是一個理想的負(fù)載軌跡i.e. m=3輸出是一個指定時間的載荷坐標(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間獲得的約束條件。 這里是m×n矩陣是一個m×n約束矩陣并且是約束誘導(dǎo)加速度。對于圖1所示的起重機，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.受限運動情況的相似軌跡控制問題可能產(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?！翱刂破娈悺钡囊粋€方法就是m×m矩陣的秩 這說明了約束和控制因子的內(nèi)積。對于手頭的情況rank（P）=1，并且這表明僅有一個控制輸入（Mn）直接促動公式2里的控制約束條件，另兩個物理量（Fx，F(xiàn)y）對控制限制并無影響。 3方程 該起重機的動力學(xué)方程（ 1 ）可投影到互補性約束和無約束子向量，定義為一個3×5的約束矩陣C和它的和它的正交補集——一個5×2的矩陣，如 則起重機的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的一個獨立條件，并且m-p=2限制著起重機，補充到原來對方程2的限制。這樣，由于混合正交正切實現(xiàn)控制的限制，匯總起來變成式子m=m-p=5=n,這樣的形式是很明確的。在部分特定條件下，這種情況對應(yīng)于平坦的欠驅(qū)動系統(tǒng)。 4.解決方案 對于手頭的資料,在10個位置的q,v和控制輸出u,方程7代表13個( 5+2+3+3 )微分代數(shù)方程.微分代數(shù)方程的指數(shù)是3,并且利用最簡單的歐拉向后差分逼近法可以解決這些問題.要解決的式子可以寫成 在時間tn時給出qn和vn,方程8代表13個非線性代數(shù)方程在時，和.通過求解方程組,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題無關(guān).截斷誤差不在時間上累積,擬訂的簡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,并且得出了保證實現(xiàn)指定方案的U(t).控制獲得這種方式可被用來作為前饋控制吊臂執(zhí)行負(fù)荷指定的方案.然后，就應(yīng)該得到加強反饋控制，以提供對負(fù)載軌跡中存在的擾動的穩(wěn)定的跟蹤.其中一個可能性是在加速度條件下引進(jìn)一個穩(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)致加速度通過加入對于約束的校正計算,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如圖2所示. 圖2 6.數(shù)值演算 該起重機使用的數(shù)據(jù)計算分別為:mb=20kg,mt=10kg,ml=100kg,r=0.1m,J=0.1kgm2.控制的任務(wù)是直線運輸一項負(fù)載任務(wù),按照安放到安放的機動操作 . 這里并且分別在時間和時，是初始的和最終的負(fù)載位置，并且。并且時，負(fù)載規(guī)格如圖3所示 圖3 逆仿真的結(jié)果是,i.e.解決這個方程7通過用8的解,得到,見圖4。控制額定這種方式可以僅僅作為一種前饋控制起重機執(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(見圖2)第一次被測試,通過在時執(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控制計劃。i.e. 并且為了整合時間步進(jìn),一個很好的選擇是p=10.結(jié)果數(shù)值模擬見圖5.可以看出，該系統(tǒng)有一個有關(guān)參考軌跡的阻尼反應(yīng) 圖6 其他實驗構(gòu)成制衡的影響建模不一致.在動態(tài)模型用于直接動態(tài)仿真,有關(guān)s1,s2和l的額外阻尼已經(jīng)計算在內(nèi),沒有考慮到在所用的模型測定混合控制。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.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（無型號不一致).該方案的負(fù)荷以及起重機的實際運動是很接近的參考運動特性.模擬被延長了,在運輸?shù)淖詈?6s)增長到了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 橋式起重機執(zhí)行一個指定負(fù)載軌跡的控制系統(tǒng)的設(shè)計的計算框架已提交.解這些方程的辦法是起重機運動特征實現(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反饋控制增強.通過使用相同的方程稍加修改得到.