50-10t双梁桥式起重机小车副起升机构设计【全套含7张CAD图纸+PDF图】
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附录附录一英文原文PREDICTION OF CONTROL OF OVERHEADCRANES EXECUTING A PRESCRIBED LOADTRAJECTORYAbstract: 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. INTRODUCTIONOverhead cranes belong to a broader class of underactuated systems thecontrolled mechanical systems in which the number of control inputs/outputsis smaller than the number of degrees-of-freedom. The performance goal is adesired load trajectory, i.e. the control outputs are time-specified load coordinatesx (t ) d , y (t) d and z ( t) d . The control inputs are the forces x F and y Factuating the trolley position and the winch torque n M changing the ropelength (see Fig. 1). The determination of control input strategy that force thesystem to complete the prescribed motion is a challenging problem, reflectedin huge amount of research established hitherto.1 The purpose of this study isto give a fresh view on the problem from the constrained motion perspectiveand to develop the mathematical tools for control design aimed at executingprescribed load trajectories with relative high speeds and without sway.The control outputs, expressed in terms of the system states, are treatedas control constraints on the system.2 It is noticed, however, that controlconstraints 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 constraintmanifold, and in the extreme may be tangent. A specific methodology mustthen be developed to solve such singular control problem. The initial governingequations arise as index five differential-algebraic equations (DAEs).3They are transformed then to an equivalent index three form, and an effectivecode for solving the resultant DAEs is proposed. The feedforward controllaw obtained this way is extended by a closed-loop control strategy withfeedback of the actual errors to provide stable tracking of the required referenceload trajectories in the presence of perturbations.Figure 1. An overhead trolley crane.2. MATHEMATICAL PRELIMINARIESConsider a 5-degree-of-freedom ( n = 5 ) overhead (gantry) crane seen inFigure 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 formwhere M is the generalized mass matrix, d and f are the generalized dynamicand applied force vectors, and T B is the matrix of influence of control inputsu on the generalized actuating force vector f B u Ta . Assumed the hoistingrope is massless, inextensible and flexible, and neglecting for simplicity allthe forces associated with 1 s , 2 s and l motions apart from the control inputsx 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 themoment 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 outputsare time-specified load coordinatesequal 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 formwhereis the mn constraint matrix, andis the constraint induced acceleration. For the crane shown in Figure 1 we haveWhile Eq. (2) is mathematically equivalent to m rheonomic holonomicconstraints c(q ) 0 , the resemblance of the trajectory control problem tothe constrained motion case may be misleading. Assumed Eq. (2) representscontact constraints, a in Eq. (1) must be replaced by c ,and by assumption the contact constraint reactions are orthogonal to themanifold of contact constrains. By contrast, the available control reactionsmay have arbitrary directions with respect to the control constraint manifold,and in the extreme some of the control reactions may be tangent. In the lattercase, not all of the desired outputs can directly be actuated by the systeminputs u. A measure of the control singularity is the rank of mm matrixwhich represents the inner product of the constrained and controlled subspaces.4 For the case at hand, rank( ) 1 , and this means that only one controlinput ( 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 realizationof the control constraints.3. GOVERNING EQUATIONSThe crane dynamic equations (1) can be projected into complementary constrainedand unconstrained subspaces, defined by the 3 5 constraint matrixC and its orthogonal complement a 5 2 matrix D such that4and for the crane considered D can be proposed asThe projection formula isand the governing equations can be manipulated to:where Eqs. (7b) and (7c) are the projections of Eq. (1) into the unconstrainedand constrained subspaces, respectively.While Eq. (7c) stands for m =3 algebraic equations, for the case at handwe have and as such Eq. (7c) represents only oneindependent condition on u and m-p=2. restrictions on the crane motion,supplementary to original restrictions of Eq. (2). In this way, due to themixed orthogonal-tangent realization2 of control constraints, the total numberof motion specifications is thus m+m- p =5= n , and as such the motionis fully specified. The situation corresponds to flatness5 of the underactuatedsystem in the partly specified motion4. SOLUTION CODEFor the case at hand, Eqs. (7) represent thirteen ( 5 2 3 3 ) DAEs in tenstates q and v and three control inputs u. Index of the DAEs is three,3 andthey can be solved by using the simplest Euler backward difference approximationmethod. Representing Eqs. (7b), (7c) and (7c) symbolically asrespectively, the solution code can be written asGiven n q and n v at time n t , Eqs. (8) represent thirteen nonlinear algebraicequations in at time .By solving the equations,the simulation is advanced from n t to n1 t . In order to improve accuracyof the numerical solution, the rough Euler scheme can possibly be replacedby a higher order backward difference approximation method.3 It maybe worth noting that, due to the original control constraint equationsc(q ，t) =0 are involved in Eqs. (7), the solution is free from the constraintviolation problem, and the truncation errors do not accumulate in time. Theproposed simple code leads to reasonable and stable solutions.5. SYNTHESIS OF CONTROLAs a solution to Eqs. (8), time-variations of state variables q( t) and v(t ) inthe prescribed motion and the control u( t) that assures the realization of thespecified motion are obtained. The control obtained this way can be used asa feedforward control for the crane executing the load prescribed motion. Itshould then be enhanced by a feedback control to provide stable tracking ofthe load trajectory in the presence of perturbations. One possibility is to introduce,instead of Eq. (2), a stabilized form of the constraint equation at theacceleration level, where and are gain values. The modification causes that Eq. (7c) is replaced withwhose symbolic form is againIn other words, the constraintinduced accelerationsare now modified to the stabilized formby 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 EXPERIMENTSThe 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 theload along a straight line following the rest-to-rest maneuverwhereand are the initial and final load positions at time 0 t and f t , respectively, andforand tf=6 s f t , the load motion specifications are illustrated in Figure 3Figure 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 Figure4. The control rated this way can be used only as a feedforward control forthe 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 wasplaced 0.5m below its reference position,l。=5.5m The gain values weretaken 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 =10 . Theresults of numerical simulations are shown in Figure 5. It can be seen thatthe 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=35Nsm -1 and k3=75Nsm -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. CONCLUSIONA computational framework for control design of overhead cranes executinga prescribed load trajectory has been presented. The solution to the governingequations are the crane motion characteristics in the reference motionand the control required for its realization. The feedforward control schemeobtained this way is then enhanced by a feedback control, obtained by usingthe same governing equations in a slightly modified form.外文文献翻译控制桥式起重机执行一项指定的负载的轨迹预测摘要: 操纵桥式起重机的有效载荷是具有挑战性的，因为它的欠驱动系统输入输出的控制数量要小于自由度的数量。输出控制（理想的负载坐标），体现在该系统的形式，导致系统的制约因素，并且该方程出现指数为5的微分代数方程，然后转化成指数为3的形式。人们使用一个有效的数字编码来解决由此产生的方程。一个闭环控制策略来反馈实际误差，延伸为前馈控制法获得的这种方法 ，以提供扰动所需的参考负载轨迹的稳定的跟踪关键词：起重机，动力，控制，轨迹跟踪，微分代数方程1.导言桥式起重机属于一个更广泛类型的欠驱动系统输入输出控制数小于自由度数量的受控机械系统。绩效目标是一个理想的负载轨迹。控制输出i.e.是单位时间的负载坐标x(t)y(t)和z(t).控制输出为力Fx和Fy作用于手推车的位置，并且结合绞车力矩Mn来改变绳子的长度（见图1）测定控制输入的策略，迫使系统完成指定议案是一个具有挑战性图1的问题，迄今为止，反映在大量的研究的确立。目的是在这个问题上给一个新的观点，从受限运动角度并制定数学工具，在相对高的速度和没有摆动的情况下来控制设计执行指定的负载轨迹的目的。控制输出，体现在该系统的体系，被当成系统的控制限制。然而，控制的限制在某些方面不同于传统的接触约束，已经引起关注。主要是他们被现有的可控制力约束着，这对遵照约束流形有一定的指导意义，并且在极端位置与可能正切。必须研制一个方法来解决这种“奇异”的控制问题。初始方程出现指数为5的微分代数方程（DAEs）。然后他们被转化成指数为3的形式，并且人们建议用一个有效的代码来解决由此产生的微分代数方程。一个闭环控制策略来反馈实际误差，延伸为前馈控制法获得这种方法，以提供扰动所需的参考负载轨迹的稳定的跟踪。2 数学预算 假设一架自由度为5的桥式起重机见图1.它的广义坐标是,并且它被一个M=3的励磁机约束.该系统的动力学方程可以写成下面这种形式这里M是广义质量矩阵，d和f是广义动态和应用力向量是输出控制影响u在广义驱动力向量下的矩阵。承担吊装绳索是不计质量的，灵活的并且忽略除了由控制输出的所有与s1,s2和l有关的微小因素。Fx,Fy和Mn的动力学方程是mb，mt和ml是导轨，小车和负载质量。J和r是此刻转动惯量和绞车半径。g是重力加速度，并且x在质量矩阵里面是对称的。预期目标是一个理想的负载轨迹i.e. m=3输出是一个指定时间的载荷坐标。和输出控制U相等。体现在该系统的坐标。输出导致m控制限制于这样的形式在加速条件下，初始方程的控制约束两次不同于遵照时间获得的约束条件。这里是mn矩阵是一个mn约束矩阵并且是约束诱导加速度。对于图1所示的起重机，我们有： 然而公式2在数学上相当于m完全约束C(q，t）=0.受限运动情况的相似轨迹控制问题可能产生误导。假定公式2表示接触约束，公式里的可以被 替代，并且由假设接触约束的反应是正交的多方面的联系，制约了。相比之下，现有的控制反应可能有任意方向方面的控制约束形式，并且在极端是一些控制反应可正切。在后一种情况下，并不是所有理想的输出可以被输入系统u直接驱动。“控制奇异”的一个方法就是mm矩阵的秩这说明了约束和控制因子的内积。对于手头的情况rank（P）=1，并且这表明仅有一个控制输入（Mn）直接促动公式2里的控制约束条件，另两个物理量（Fx，Fy）对控制限制并无影响。3方程该起重机的动力学方程（ 1 ）可投影到互补性约束和无约束子向量，定义为一个35的约束矩阵C和它的和它的正交补集一个52的矩阵，如则起重机的D可以转化为推算公式为并且方程可以被转化为方程7c 7d分别是方程1在无约束和受限情况下的投影然
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