外文翻譯--一種實(shí)用的辦法--帶拖車(chē)移動(dòng)機(jī)器人的反饋控制

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1、 1 河北建筑工程學(xué)院 畢業(yè)設(shè)計(jì)（論文）外文資料翻譯 系別： 機(jī) 械 工 程 系 專(zhuān)業(yè)： 機(jī)械設(shè)計(jì)制造及其自動(dòng)化 班級(jí)： 姓名： 學(xué)號(hào)： 外文出處： Proceedings ofthe 1998 IEEE International Conference on Robotics & Automation 附 件： 1、外文原文； 2、外文資料翻譯譯文。 指導(dǎo)教師評(píng)語(yǔ)： 簽字： 年 月 日 2 Proceedings ofthe 1998 IEEE International Conference on Robotics & Automation Leuven, Belgium May 1998

2、A practical approach to feedback control for a mobile robot with trailer F. Lamiraux and J.P. Laumond LAAS-CNRS Toulouse, France florent ,jpllaas.fr Abstract This paper presents a robust method to control a mobile robot towing a trailer. Both problems of trajectory tracking and steering to a given c

3、onfiguration are addressed. This second issue is solved by an iterative trajectory tracking. Perturbations are taken into account along the motions. Experimental results on the mobile robot Hilare illustrate the validity of our approach. 1 Introduction Motion control for nonholonomic systems have gi

4、ven rise to a lot of work for the past 8 years. Brocketts condition 2 made stabilization about a given configuration a challenging task for such systems, proving that it could not be performed by a simple continuous state feedback. Alternative solutions as time-varying feedback l0, 4, 11, 13, 14, 15

5、, 18 or discontinuous feedback 3 have been then proposed. See 5 for a survey in mobile robot motion control. On the other hand, tracking a trajectory for a nonholonomic system does not meet Brocketts condition and thus it is an easier task. A lot of work have also addressed this problem 6, 7, 8, 12,

6、 16 for the particular case of mobile robots.All these control laws work under the same assumption: the evolution of the system is exactly known and no perturbation makes the system deviate from its trajectory.Few papers dealing with mobile robots control take into account perturbations in the kinem

7、atics equations. l however proposed a method 3 to stabilize a car about a configuration, robust to control vector fields perturbations, and based on iterative trajectory tracking. In this paper, we propose a robust scheme based on iterative trajectory tracking, to lead a robot towing a trailer to a

8、configuration. The trajectories are computed by a motion planner described in 17 and thus avoid obstacles that are given in input. In the following.We won t give any development about this planner,we refer to this reference for details. Moreover,we assume that the execution of a given trajectory is

9、submitted to perturbations. The model we chose for these perturbations is very simple and very general.It presents some common points with l. The paper is organized as follows. Section 2 describes our experimental system Hilare and its trailer:two hooking systems will be considered (Figure 1).Sectio

10、n 3 deals with the control scheme and the analysis of stability and robustness. In Section 4, we present experimental results. The presence of obstacle makes the task of reaching a configuration even more difficult and require a path planning task before executing any motion. 2 Description of the sy

11、stem Hilare is a two driving wheel mobile robot. A trailer is hitched on this robot, defining two different systems depending on the hooking device: on system A, the trailer is hitched above the wheel axis of the robot (Figure 1, top), whereas on system B, it is hitched behind this axis (Figure l ,

12、bottom). A is the particular case of B, for which rl = 0. This system is however singular from a control point of view and requires more complex computations. For this reason, we deal separately with both hooking systems. Two motors enable to control the linear and angular velocities （ vr ， r ) of t

13、he robot. These velocities are moreover measured by odometric sensors, whereas the angle between the robot and the trailer is given by an optical encoder. The position and orientation（ xr ,yr , r ） of the robot are computed by integrating the former velocities. With these notations, the control syst

14、em of B is: 4 c o ss i ns i n ( ) c o s ( )r r rr r rrrr r rrttxvyvvlll （ 1） Figure 1: Hilare with its trailer 3 Global control scheme 3.1 Motivation When considering real systems, one has to take into account perturbations during motion execution.These may have many origins as imperfection of the m

15、otors, slippage of the wheels, inertia effects . These perturbations can be modeled by adding a term in the control system (l),leading to a new system of the form ( , )x f x u where may be either deterministic or a random variable.In the first case, the perturbation is only due to a bad knowledge of

16、 the system evolution, whereas in the second case, it comes from a random behavior of the system. We will see later that 5 this second model is a better fit for our experimental system. To steer a robot from a start configuration to a goal, many works consider that the perturbation is only the initi

17、al distance between the robot and the goal, but that the evolution of the system is perfectly known. To solve the problem, they design an input as a function of the state and time that makes the goal an asymptotically stable equilibrium of the closed loop system. Now, if we introduce the previously

18、defined term in this closed loop system, we dont know what will happen. We can however conjecture that if the perturbation is small and deterministic, the equilibrium point (if there is still one) will be close to the goal, and if the perturbation is a random variable, the equilibrium point will bec

19、ome an equilibrium subset.But we dont know anything about the position of these new equilibrium point or subset. Moreover, time varying methods are not convenient when dealing with obstacles. They can only be used in the neighborhood of the goal and this neighborhood has to be properly defined to en

20、sure collision-free trajectories of the closed loop system. Let us notice that discontinuous state feedback cannot be applied in the case of real robots, because discontinuity in the velocity leads to infinite accelerations. The method we propose to reach a given configuration tn the presence of obs

21、tacles is the following. We first build a collision free path between the current configuration and the goal using a collision-freemotion planner described in 17, then we execute the trajectory with a simple tracking control law. At the end of the motion, the robot does never reach exactly the goal

22、because of the various perturbations, but a neighborhood of this goal. If the reached configuration is too far from the goal, we compute another trajectory that we execute as we have done for the former one. We will now describe our trajectory tracking control law and then give robustness issues abo

23、ut our global iterative scheme. 3.2 The trajectory tracking control law In this section, we deal only with system A. Computations are easier for system B (see Section 3.4). 6 Figure 2: Tracking control law for a single robot A lot of tracking control laws have been proposed for wheeled mobile robots

24、 without trailer. One of them 16,a lthough very simple, give excellent results.If ,xy are the coordinates of the reference robot in the frame of the real robot (Figure 2), and if 00,rrvare the inputs of the reference trajectory, this control law has the following expression: 01032c o ss i nr rrrv k

25、xk k yv （ 2） The key idea of our control law is the following: when the robot goes forward, the trailer need not be stabilized (see below). So we apply (2) to the robot.When it goes backward, we define a virtual robot ( , , )r r rxy (Figure 3) which is symmetrical to the real one with respect to the

26、 wheel axis of the trailer: 2 c o s2 s i n2r r t tr r t tr t rx x ly y l Then, when the real robot goes backward, the virtual robot goes forward and the virtual system ( , , , )r r rxyis kinematically equivalent to the real one. Thus we apply the tracking control law (2) to the virtual robot. Figure

27、 3: Virtual robot A question arises now: is the trailer really always stable when the robot goes forward ? The following section will answer this question. 3.3 Stability analysis of the trailer 7 We consider here the case of a forward motion ( 0)rv , the backward motion being equivalent by the virtu

28、al robot transformation. Let us denote by 0 0 0 0 0( , , , , )r r r r rx y va reference trajectory and by( , , , , , )vyx r rr r r the real motion of the system. We assume that the robot follows exactly its reference trajectory: 0 0 0 0 0( , , , , , ) ( , , , , )r r r r rv x y vyx r r r r r and we f

29、ocus our attention on the trailer deviation 0 .The evolution of this deviation is easily deduced from system (1) with 0rl (System A): 00 ( s i n s i n )2c o s ( ) s i n ( )22rtrtvlvl is thus decreasing iff 0 2 2 2 2 （ 3） Our system is moreover constrained by the inequalities 0,22 （ 4） so that 0 2 an

30、d (3) is equivalent to 00000220 22 and orand（ 5） Figure 4 shows the domain on which is decreasing for a given value of 0 . We can see that this domain contains all positions of the trailer defined by the bounds (4). Moreover, the previous computations permit easily to show that 0 is an asymptoticall

31、y stable value for the variable . Thus if the real or virtual robot follows its reference forward trajectory, the trailer is stable and will converge toward its own reference trajectory. 8 Figure 4: Stability domain for 3.4 Virtual robot for system B When the trailer is hitched behind the robot, the

32、 former construction is even more simple: we can replace the virtual robot by the trailer. In this case indeed, the velocities of the robot ( , )rrv and of the trailer ( , )ttv are connected by a one-to-one mapping.The configuration of the virtual robot is then given by the following system: c o s c

33、 o s ( )s i n s i n ( )r r r r r rr r r r t rx x l ly y l lrr and the previous stability analysis can be applied as well, by considering the motion of the hitching point. The following section addresses the robustness of our iterative scheme. 3.5 Robustness of the iterative scheme We are now going t

34、o show the robustness of the iterative scheme we have described above. For this,we need to have a model of the perturbations arising when the robot moves. l model the perturbations by a bad knowledge of constants of the system, leading to deterministic variations on the vector fields. In our experim

35、ent we observed random perturbations due for instance to some play in the hitching system. These perturbations are very difficult to model. For this reason,we make only two simple hypotheses about them: 00( ( ) , ( ) )( ( ) , ( ) )c s scsd q s q sd q s q s 9 where s is the curvilinear abscissa along

36、 the planned path, q and 0q are respectively the real and reference configurations, csdis a distance over the configuration space of the system and , are positive constants.The first inequality means that the distance between the real and the reference configurations is proportional to the distance

37、covered on the planned path. The second inequality is ensured by the trajectory tracking control law that prevents the system to go too far away from its reference trajectory. Let us point out that these hypotheses are very realistic and fit a lot of perturbation models. We need now to know the leng

38、th of the paths generated at each iteration. The steering method we use to compute these paths verifies a topological property accounting for small-time controllability17. This means that if the goal is sufficiently close to the starting configuration, the computed trajectory remains in a neighborho

39、od of the starting configuration. In 9we give an estimate in terms of distance: if 1qand 2qare two sufficiently close configurations, the length of the planned path between them verifies 141 2 1 2( ( , ) ) ( , )csl P a t h q q d q qwhere is a positive constant. Thus, if is the sequence of configurat

40、ions reached after i motions, we have the following inequalities: 11,( , )( ) ( , )c s g o a lc s i g o a l c s i g o a ld q qd q q d q qThese inequalities ensure that distCS( , )i goalqqis upper bounded by a sequence 1,2.()iid of positive numbers defined by 1141iidddand converging toward 43 after e

41、nough iterations. Thus, we do not obtain asymptotical stability of the goal configuration, but this result ensures the existence of a stable domain around this configuration.This result essentially comes from the very general model of perturbations we have chosen. Let us repeat that including such a

42、 perturbation model in a time varying control law would undoubtedly make it lose its asymptotical stability.The experimental 10 results of the following section show however, that the converging domain of our control scheme is very small. 4 Experimental results We present now experimental results ob

43、tained with our robot Hilare towing a trailer, for both systems A and B. Figures 5 and 6 show examples of first paths computed by the motion planner between the initial Figure 5: System A: the initial and goal configurations and the first path to be tracked Figure 6: System B: the initial and goal c

44、onfigurations, the first path to be tracked and the final maneuver configurations (in black) and the goal configurations (in grey), including the last computed maneuver in the second case. The lengths of both hooking system is the following: 0rl , 125tl cm for A and 60rl cm, 90tl cm for B. Tables 1

45、and 2 give the position of initial and final configurations and the gaps between the goal and the reached configurations after one motion and two motions, for 3 different experiments. In both cases, the first experiment corresponds to the figure.Empty 2qcolumns mean that the precision reached after

46、the first motion was sufficient and that no more motion was performed. Comments and Remarks: The results reported in the tables 1 and 2 lead to two 11 main comments. First,the precision reached by the system is very satisfying and secondly the number of iterations is very small (between 1 and 2). In

47、 fact, the precision depends a lot on the velocity of the different motions. Here the maximal linear velocity of the robot was 50 cm/s. 5 Conclusion We have presented in this paper a method to steer a robot with one trailer from its initial configuration to a goal given in input of the problem. This

48、 method is based on an iterative approach combining open loop and close loop controls. It has been shown robust with respect to a large range of perturbation models. This robustness mainly comes from the topological property of the steering method introduced in 17. Even if the method does not make t

49、he robot converge exactly to the goal, the precision reached during real experiments is very satisfying. Table 1: System A: initial and final configurations,gaps between the first and second reached configurations and the goal 12 Table 2: System B: initial and final configurations,gaps between the f

50、irst and second reached configurations and the goal References 1M. K. Bennani et P. Rouchon. Robust stabilization of flat and chained systems. in European Control Conference,1995. 2R.W. Brockett. Asymptotic stability and feedback stabilization. in Differential Geometric Control Theory,R.W. Brockett,

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59、Conference on Robotics and Automation, Sacramento, California, pp 1136-1141,1991. 17 S. Sekhavat, F. Lamiraux, J.P. Laumond, G. Bauzil and A. Ferrand. Motion planning and control for Hilare pulling a trader: experzmental issues. IEEE Int. Conf. on Rob. and Autom., pp 3306-3311, 1997. 18 O.J. Splrdal

60、en et 0. Egeland. Exponential stabzlzsation of nonholonomic chained systems. in IEEE Transactions on Automatic Control, Vol 40, No 1, 1995. Bolland, Vol 18, pp 147-158, 1992. 14 一種實(shí)用的辦法 -帶拖車(chē)移動(dòng)機(jī)器人的反饋控制 F. Lamiraux and J.P. Laumond 拉斯，法國(guó)國(guó)家科學(xué)研究中心 法國(guó)圖盧茲 florent ,jpllaas.fr 摘 要 本文提出了一種有效的方法來(lái)控制帶拖車(chē)移動(dòng)機(jī)器人。軌跡

61、跟蹤和路徑跟蹤這兩個(gè)問(wèn)題已經(jīng)得到解決。接下來(lái)的問(wèn)題是解決迭代軌跡跟蹤。并且把擾動(dòng)考慮到路徑跟蹤內(nèi)。移動(dòng)機(jī)器人 Hilare 的實(shí)驗(yàn)結(jié)果說(shuō)明了我們方法的有效性。 1 引言 過(guò)去的 8 年，人們對(duì)非完整系統(tǒng)的運(yùn)動(dòng)控制做了大量的工作。布洛基 2提出了關(guān)于這種系統(tǒng)的一項(xiàng)具有挑戰(zhàn)性的任務(wù)，配置的穩(wěn)定性，證明它不能由一個(gè)簡(jiǎn)單的連續(xù)狀態(tài)反饋。作為替代辦法隨時(shí)間變化的反饋 10,4,11,13,14,15,18或間斷反饋 3也隨之被提出。從 5 移動(dòng)機(jī)器人的運(yùn)動(dòng)控制的一項(xiàng)調(diào)查可以看到。另一方面，非完整系統(tǒng)的軌跡跟蹤不符合布洛基的條件，從而使其這一個(gè)任務(wù)更為輕松。許多著作也已經(jīng)給出了移動(dòng)機(jī)器人的特殊情況的這一問(wèn)

62、題 6,7,8,12,16。 所有這些控制律都是工作在相同的假設(shè)下：系統(tǒng)的演變是完全已知和沒(méi)有擾動(dòng)使得系統(tǒng)偏離其軌跡。很少有文章在處理移動(dòng)機(jī)器人的控制時(shí)考慮到擾動(dòng)的運(yùn)動(dòng)學(xué)方程。但是 1提出了一種有關(guān)穩(wěn)定汽車(chē)的配置，有效的矢量控制擾動(dòng)領(lǐng)域，并且建立在迭代軌跡跟蹤的基礎(chǔ)上。 存在的障礙使得達(dá)到規(guī)定路徑的任務(wù)變得更加困難，因此在執(zhí)行任務(wù)的任何動(dòng)作之前都需要有一個(gè)路徑規(guī)劃。 在本文中，我們?cè)诘壽E跟蹤的基礎(chǔ)上提出了一個(gè)健全的方案，使得帶拖車(chē)的 15 機(jī)器人按照規(guī)定路徑行走。該軌跡計(jì)算由規(guī)劃的議案所描述 17 ，從而避免已經(jīng)提 交了輸入的障礙物。在下面，我們將不會(huì)給出任何有關(guān)規(guī)劃的發(fā)展，我們提及這個(gè)參

63、考的細(xì)節(jié)。而且，我們認(rèn)為，在某一特定軌跡的執(zhí)行屈服于擾動(dòng)。我們選擇的這些擾動(dòng)模型是非常簡(jiǎn)單，非常一般。它存在一些共同點(diǎn) 1。 本文安排如下：第 2 節(jié)介紹我們的實(shí)驗(yàn)系統(tǒng) Hilare 及其拖車(chē)：兩個(gè)連接系統(tǒng)將被視為（圖 1） 。第 3節(jié)處理控制方案及分析的穩(wěn)定性和魯棒性。在第 4節(jié)，我們介紹本實(shí)驗(yàn)結(jié)果 。 圖 1 帶拖車(chē)的 Hilare 2 系統(tǒng)描述 Hilare是一個(gè)有兩個(gè)驅(qū)動(dòng)輪的移動(dòng)機(jī)器人。拖車(chē)是被掛在這個(gè)機(jī)器人上的，確定了兩個(gè)不同的系統(tǒng)取決于連接設(shè)備：在系統(tǒng) A的拖車(chē)拴在機(jī)器人的車(chē)輪軸中心線上方（圖 1 ，頂端），而對(duì)系統(tǒng) B是栓在機(jī)器人的車(chē)輪軸中心線的后面（圖 1 ，底部 )。 A對(duì)

64、B來(lái)說(shuō)是一種特殊情況，其中 rl = 0 。這個(gè)系統(tǒng)不過(guò)單從控制的角度來(lái)看，需要更多的復(fù)雜的計(jì)算。出于這個(gè)原因，我們分開(kāi)處理掛接系統(tǒng) 。兩個(gè)馬達(dá)能夠控制機(jī)器人的線速度和角速度（ vr ， r )。除了這些速度之外，還由傳感器測(cè)量，而機(jī)器人和拖 16 車(chē)之間的角度 ，由光學(xué)編碼器給出。機(jī)器人的位置和方向（ xr , yr , r ）通過(guò)整合前的速度被計(jì)算。有了這些批注，控制系統(tǒng) B是： c o ss i ns i n ( ) c o s ( )r r rr r rrrr r rrttxvyvvlll （ 1） 3 全球控制方案 3.1 目的 當(dāng)考慮到現(xiàn)實(shí)的系統(tǒng)，人們就必須要考慮到在運(yùn)動(dòng)的執(zhí)行時(shí)產(chǎn)

65、生的擾動(dòng)。 這可能有許多的來(lái)源，像有缺陷的電機(jī)，輪子的滑動(dòng)，慣性的影響 . 這些擾動(dòng)可以被設(shè)計(jì)通過(guò)增加一個(gè)周期在控 制系統(tǒng)（ 1） ，得到一個(gè)新的系統(tǒng)的形式 ( , )x f x u 在上式中可以是確定性或隨機(jī)變量。 在第一種情況下，擾動(dòng)僅僅是由于系統(tǒng)演化的不規(guī)則，而在第二種情況下，它來(lái)自于該系統(tǒng)一個(gè)隨機(jī)行為。我們將看到后來(lái)，這第二個(gè)模型是一個(gè)更適合我們的實(shí)驗(yàn)系統(tǒng)。 為了引導(dǎo)機(jī)器人，從一開(kāi)始就配置了目標(biāo)，許多工程認(rèn)為擾動(dòng)最初只是機(jī)器人和目標(biāo)之間的距離，但演變的系統(tǒng)是完全眾所周知的。為了解決這個(gè)問(wèn)題， 他們?cè)O(shè)計(jì)了一個(gè)可輸入的時(shí)間 -狀態(tài)函數(shù)，使目標(biāo)達(dá)到一個(gè)漸近穩(wěn)定平衡的閉環(huán)系統(tǒng)。現(xiàn)在，如果我們介

66、紹了先前定義周期 在這個(gè)閉環(huán)系統(tǒng)，我們不知道將會(huì)發(fā)生什么。但是我們可以猜想，如果擾動(dòng) 很小、是確定的、在平衡點(diǎn)（如果仍然還有一個(gè)）將接近目標(biāo)，如果擾動(dòng)是一個(gè)隨機(jī)變數(shù)，平衡點(diǎn)將成為一個(gè)平衡的子集。 但是，我們不知道這些新的平衡點(diǎn)或子集的位置。 此外 ，在處理障礙時(shí)，隨時(shí)間變化的方法不是很方便。他們只能使用在附近的目標(biāo)，這附近要適當(dāng)界定，以確保無(wú)碰撞軌跡的閉環(huán)系統(tǒng)。請(qǐng)注意連續(xù)狀態(tài)反饋不能適用于真實(shí)情況下的機(jī)器人，因?yàn)殚g斷的速度導(dǎo)致無(wú)限的加速度。 17 我們建議達(dá)成某一存在障礙特定配置的方法如下。我們首先在當(dāng)前的配置和使用自由的碰撞議案所描述 17目標(biāo)之間建立一個(gè)自由的碰撞路徑，然后，我們以一個(gè)簡(jiǎn)單

67、的跟蹤控制率執(zhí)行軌跡。在運(yùn)動(dòng)結(jié)束后，因?yàn)檫@一目標(biāo)的各種擾動(dòng)機(jī)器人從來(lái)沒(méi)有完全達(dá)到和目標(biāo)的軌跡一致，而是這一目標(biāo)的左右。如果達(dá)到配置遠(yuǎn)離目標(biāo)，我們計(jì)算另一個(gè) 我們之前已經(jīng)執(zhí)行過(guò)的一個(gè)軌跡。 現(xiàn)在我們將描述我們的軌跡跟蹤控制率，然后給出我們的全球迭代方法的魯棒性問(wèn)題。 3.2 軌跡跟蹤控制率 在這一節(jié)中，我們只處理系統(tǒng) A。對(duì)系統(tǒng) B 容易計(jì)算（見(jiàn)第 3.4 節(jié)）。 圖 2 單一機(jī)器人的跟蹤控制率 很多帶拖車(chē)輪式移動(dòng)機(jī)器人的跟蹤控制律已經(jīng)被提出。其中 16雖然很簡(jiǎn)單 ,但是提供了杰出的成果。 如果 ,xy 是模擬機(jī)器人的坐標(biāo)構(gòu)成真實(shí)機(jī)器人（圖 2），如果（ 00,rrv）是輸入的參考軌跡，這種控制

68、律表示如下： 01032c o ss i nr rrrv k xk k yv （ 2） 我們控制律的關(guān)鍵想法如下：當(dāng)機(jī)器人前進(jìn)，拖車(chē)不需要穩(wěn)定（見(jiàn)下文）。因此，我們對(duì)機(jī)器人使用公式（ 2）。 當(dāng)它后退時(shí)，我們定義一個(gè)虛擬的機(jī)器人 ( , , )r r rxy （圖3）這是對(duì)稱(chēng)的真實(shí)一對(duì)拖車(chē)的車(chē)輪軸： 2 c o s2 s i n2r r t tr r t tr t rx x ly y l 然后，當(dāng)真正的機(jī)器人退后，虛擬機(jī)器人前進(jìn)和虛擬系統(tǒng) ( , , , )r r rxy 在運(yùn)動(dòng)學(xué)上是 18 等同于真正的一個(gè)。因此，我們對(duì)虛擬機(jī)器人實(shí)行跟蹤控制 法（ 2）。 圖 3 虛擬機(jī)器人 現(xiàn)在的問(wèn)題是：

69、當(dāng)機(jī)器人前進(jìn)時(shí)，拖車(chē)是否真的穩(wěn)定？下一節(jié)將回答這個(gè)問(wèn)題。 3.3 拖車(chē)穩(wěn)定性分析 在這里我們考慮的向前運(yùn)動(dòng)情況下 ( 0)rv ,虛擬機(jī)器人向后的運(yùn)動(dòng)被等值轉(zhuǎn)變。讓我們把坐標(biāo) 0 0 0 0 0( , , , , )r r r r rx y v作為參考軌跡并且把坐標(biāo) ( , , , , , )vyx r r r r r 作為實(shí)際運(yùn)動(dòng)的系統(tǒng)。我們假設(shè)機(jī)器人完全跟隨其參考軌跡：0 0 0 0 0( , , , , , ) ( , , , , )r r r r rv x y vyx r r r r r 并且我們把我們的注意力放在拖車(chē)偏差0 。這一偏差的變化很容易從系統(tǒng)（ 1）推導(dǎo)出 0rl （系統(tǒng)

70、A） ： 00 ( s i n s i n )2c o s ( ) s i n ( )22rtrtvlvl 盡管 是減少的 0 2 2 2 2 （ 3） 我們的系統(tǒng)而且被不等量限制了 0,22 （ 4） 因此 0 2 和式（ 3）等價(jià)于 0000022022 且或且（ 5） 19 圖 4 顯示 的范圍隨著給定的 0 的值正在減少。我們可以看到，這個(gè)范圍包含了拖車(chē)的所有的位置，包括式（ 4）所界定的范圍。此外，以前的計(jì)算許可輕松地表明對(duì)于變量 0 ， 0 是一個(gè)漸近穩(wěn)定值的變量。 因此，如果實(shí)際或虛擬的機(jī)器人按照它的參考軌跡前進(jìn)，拖車(chē)是穩(wěn)定的，并且將趨于自己的參考軌跡。 圖 4 的穩(wěn)定范圍 3.

71、4 虛擬機(jī)器人系統(tǒng) B 當(dāng)拖車(chē)掛在機(jī)器人的后面，之前的結(jié)構(gòu)甚至更簡(jiǎn)單：我們可以用拖車(chē)取代虛擬的機(jī)器人。在這種實(shí)際情況下，機(jī)器人的速度 ( , )rrv 和拖車(chē) ( , )ttv 一對(duì)一映射的連接。然后虛擬的機(jī)器人系統(tǒng)表示為如下： c o s c o s ( )s i n s i n ( )r r r r r rr r r r t rx x l ly y l lrr 和以前的穩(wěn)定性分析可以被很好的使用通過(guò)考慮懸掛點(diǎn)的運(yùn)動(dòng)。 下面一節(jié)討論了我們迭代計(jì)劃的魯棒性。 3.5 迭代計(jì)劃的魯棒性 我們現(xiàn)在正在顯示上文所提到的迭代計(jì)劃的魯棒性。為此，我們需要有一個(gè)當(dāng)機(jī)器人的運(yùn)動(dòng)時(shí)產(chǎn)生擾動(dòng)的模型。 1擾動(dòng)的模

72、型系統(tǒng)是一個(gè)不規(guī)則，從而導(dǎo)致矢量場(chǎng)確定性的變化。在我們的實(shí)驗(yàn)中，我們要看到由于隨機(jī)擾動(dòng)導(dǎo)致的例如在一些懸掛系統(tǒng) 20 中發(fā)揮作用。這些擾動(dòng)對(duì)模型是非常困難的。出于這個(gè)原因， 我們只有兩個(gè)簡(jiǎn)單的假說(shuō)有： 00( ( ) , ( ) )( ( ) , ( ) )c s scsd q s q sd q s q s其中 s 是沿曲線橫坐標(biāo)設(shè)計(jì)路徑， q 和 0q 分別是真正的和參考的結(jié)構(gòu)，csd是結(jié)構(gòu)空間系統(tǒng)的距離并且 ， 是正數(shù)。 第一個(gè)不等量意味著實(shí)際和參考結(jié)構(gòu)之間的距離成正比的距離覆蓋計(jì)劃路徑。第二個(gè)不等量是確保軌跡跟蹤控制率，防止系統(tǒng)走得太遠(yuǎn)遠(yuǎn)離其參考軌跡。讓我們指出，這些假設(shè)是非?，F(xiàn)實(shí)的和適

73、合大量的擾動(dòng)模型。 我們現(xiàn)在需要知道在每個(gè)迭代路徑的長(zhǎng)度。我們使用指導(dǎo)的方法計(jì)算這些路徑驗(yàn)證拓?fù)涠虝r(shí)間的可控性 17。這個(gè)也就是說(shuō)，如果我們的目標(biāo)是充分接近起初的結(jié)構(gòu)，軌跡的計(jì)算依然是起初的結(jié)構(gòu)的附近。在 9 我們給出的估算方面的距離：如果 1q和2q是兩種不夠緊密的結(jié)構(gòu)，規(guī)劃路徑的長(zhǎng)度驗(yàn)證它們之間的關(guān)系 141 2 1 2( ( , ) ) ( , )csl P a t h q q d q q這里 是一個(gè)正數(shù)。 因此，如果1,2.()iiq 是配置依次獲得的，我們有以下不等式： 11,( , )( ) ( , )c s g o a lc s i g o a l c s i g o a ld

74、q qd q q d q q這些不等式確保 distCS( , )i goalqq是上界序列1,2.()iid 的正數(shù) 1141iiddd和趨近于足夠反復(fù)后的。 因此，我們沒(méi)有獲得漸近穩(wěn)定性配置的目標(biāo)，但這一結(jié)果確保存在一個(gè)穩(wěn)定的范圍處理這個(gè)配置。 這一結(jié)果基本上是來(lái)自我們選擇非常 傳統(tǒng)擾動(dòng)的模型。讓我們重復(fù)這包括諸如擾動(dòng)模型的時(shí)間不同的控制律無(wú)疑將使其失去其漸近穩(wěn)定。 實(shí)驗(yàn)結(jié)果如下節(jié)顯示，收斂域的控制計(jì)劃是非常小的。 21 4 實(shí)驗(yàn)結(jié)果 現(xiàn)在，我們目前獲得的帶拖車(chē)機(jī)器人 Hilare 系統(tǒng) A 和 B 的實(shí)驗(yàn)結(jié)果。圖 5 和圖 6顯示第一路徑計(jì)算的例子所規(guī)劃初始配置（黑色）和目標(biāo)配置（灰色）

75、之間的運(yùn)動(dòng)。在第二種情況下包括上一次計(jì)算結(jié)果。 連接系統(tǒng)的長(zhǎng)度如下：系統(tǒng) A 中 0rl ， 125tl 厘米，系統(tǒng) B 60rl 厘米， 90tl 厘米。表 1 和表 2 提供的初始和最后配置位置以及目標(biāo)和期望配置在第一次動(dòng)作和第二次動(dòng)作之間的不足， 3 個(gè)不同的實(shí)驗(yàn)。在這兩種情況下，第一次試驗(yàn)相當(dāng)于圖表。2q意味著，在第一動(dòng)作后精度十分充足，沒(méi)有更多可進(jìn)行的動(dòng)作。 評(píng)論和意 見(jiàn)：表 1 和表 2 的報(bào)告結(jié)果顯示了兩個(gè)主要的見(jiàn)解。首先， 系統(tǒng)達(dá)成非常令人滿意的精密程度，其次迭代次數(shù)是非常小的（介于 1 和 2 之間）。事實(shí)上，精密程度取決于很多的速度和不同的動(dòng)作。在這里，機(jī)器人的最大線速度是

76、 50 厘米 /秒 。 5 結(jié)論 我們已經(jīng)提出了一種方法來(lái)控制機(jī)器人與拖車(chē)從初始結(jié)構(gòu)到一個(gè)已知輸入問(wèn)題的目標(biāo)。這種方法是以迭代于開(kāi)環(huán)和閉環(huán)控制相結(jié)合為前提的辦法。它對(duì)大范圍的擾動(dòng)模型已經(jīng)顯示出健全的一面。這個(gè)魯棒性主要來(lái)自拓?fù)湫阅苤笇?dǎo)方法介紹 17 。即使該方法不完全趨于機(jī)器人的最終目標(biāo)，但是在真正實(shí)驗(yàn)期間達(dá)到 的精度程度是非常令人滿意的。 圖 5：系統(tǒng) A：初始、目標(biāo)配置跟蹤第一路徑 圖 6：系統(tǒng) B：初始、目標(biāo)配置跟蹤第一路徑和最終結(jié)果 表 1：系統(tǒng) A： 目標(biāo)和期望配置在第一次動(dòng) 表 2：系統(tǒng) B：目標(biāo)和期望配置在第一次動(dòng) 作和第二次動(dòng)作之間的差距 作和第二次動(dòng)作之間的差距 參考文獻(xiàn) 1

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