外文翻譯--帶拖車移動機(jī)器人的反饋控制【中英文文獻(xiàn)譯文】

外文翻譯--帶拖車移動機(jī)器人的反饋控制【中英文文獻(xiàn)譯文】,中英文文獻(xiàn)譯文,外文,翻譯,拖車,移動,挪動,機(jī)器人,反饋,控制,節(jié)制,中英文,文獻(xiàn),譯文 南昌航空大學(xué)科技學(xué)院學(xué)士學(xué)位論文—外文譯文 1998年的IEEE 國際會議上機(jī)器人及自動化 Leuven ，比利時1998年5月 一種實用的辦法--帶拖車移動機(jī)器人的反饋控制 F. Lamiraux and J.P. Laumond 拉斯，法國國家科學(xué)研究中心 法國圖盧茲 {florent ,jpl}@laas.fr 摘 要 本文提出了一種有效的方法來控制帶拖車移動機(jī)器人。軌跡跟蹤和路徑跟蹤這兩個問題已經(jīng)得到解決。接下來的問題是解決迭代軌跡跟蹤。并且把擾動考慮到路徑跟蹤內(nèi)。移動機(jī)器人Hilare的實驗結(jié)果說明了我們方法的有效性。 1引言 過去的8年，人們對非完整系統(tǒng)的運(yùn)動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，證明它不能由一個簡單的連續(xù)狀態(tài)反饋。作為替代辦法隨時間變化的反饋[10,4,11,13,14,15,18]或間斷反饋[3]也隨之被提出。從 [5] 移動機(jī)器人的運(yùn)動控制的一項調(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，從而使其這一個任務(wù)更為輕松。許多著作也已經(jīng)給出了移動機(jī)器人的特殊情況的這一問題[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有擾動使得系統(tǒng)偏離其軌跡。很少有文章在處理移動機(jī)器人的控制時考慮到擾動的運(yùn)動學(xué)方程。但是[1]提出了一種有關(guān)穩(wěn)定汽車的配置，有效的矢量控制擾動領(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)的任何動作之前都需要有一個路徑規(guī)劃。 在本文中，我們在迭代軌跡跟蹤的基礎(chǔ)上提出了一個健全的方案，使得帶拖車的機(jī)器人按照規(guī)定路徑行走。該軌跡計算由規(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，我們將不會給出任何有關(guān)規(guī)劃的發(fā)展，我們提及這個參考的細(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。我們選擇的這些擾動模型是非常簡單，非常一般。它存在一些共同點[1]。 本文安排如下：第2節(jié)介紹我們的實驗系統(tǒng)Hilare及其拖車：兩個連接系統(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)，我們介紹本實驗結(jié)果 。 圖1帶拖車的Hilare 2 系統(tǒng)描述 Hilare是一個有兩個驅(qū)動輪的移動機(jī)器人。拖車是被掛在這個機(jī)器人上的，確定了兩個不同的系統(tǒng)取決于連接設(shè)備：在系統(tǒng)A的拖車拴在機(jī)器人的車輪軸中心線上方（圖1 ，頂端），而對系統(tǒng)B是栓在機(jī)器人的車輪軸中心線的后面（圖1 ，底部)。 A對B來說是一種特殊情況，其中 = 0 。這個系統(tǒng)不過單從控制的角度來看，需要更多的復(fù)雜的計算。出于這個原因，我們分開處理掛接系統(tǒng)。兩個馬達(dá)能夠控制機(jī)器人的線速度和角速度（，)。除了這些速度之外，還由傳感器測量，而機(jī)器人和拖車之間的角度，由光學(xué)編碼器給出。機(jī)器人的位置和方向（,,）通過整合前的速度被計算。有了這些批注，控制系統(tǒng)B是： （1） 3 全球控制方案 3.1目的 當(dāng)考慮到現(xiàn)實的系統(tǒng)，人們就必須要考慮到在運(yùn)動的執(zhí)行時產(chǎn)生的擾動。 這可能有許多的來源，像有缺陷的電機(jī)，輪子的滑動，慣性的影響... 這些擾動可以被設(shè)計通過增加一個周期在控制系統(tǒng)（1） ，得到一個新的系統(tǒng)的形式 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，它來自于該系統(tǒng)一個隨機(jī)行為。我們將看到后來，這第二個模型是一個更適合我們的實驗系統(tǒng)。 為了引導(dǎo)機(jī)器人，從一開始就配置了目標(biāo)，許多工程認(rèn)為擾動最初只是機(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。為了解決這個問題，他們設(shè)計了一個可輸入的時間-狀態(tài)函數(shù)，使目標(biāo)達(dá)到一個漸近穩(wěn)定平衡的閉環(huán)系統(tǒng)。現(xiàn)在，如果我們介紹了先前定義周期在這個閉環(huán)系統(tǒng)，我們不知道將會發(fā)生什么。但是我們可以猜想，如果擾動很小、是確定的、在平衡點（如果仍然還有一個）將接近目標(biāo)，如果擾動是一個隨機(jī)變數(shù)，平衡點將成為一個平衡的子集。 但是，我們不知道這些新的平衡點或子集的位置。 此外，在處理障礙時，隨時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，以確保無碰撞軌跡的閉環(huán)系統(tǒng)。請注意連續(xù)狀態(tài)反饋不能適用于真實情況下的機(jī)器人，因為間斷的速度導(dǎo)致無限的加速度。 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)之間建立一個自由的碰撞路徑，然后，我們以一個簡單的跟蹤控制率執(zhí)行軌跡。在運(yùn)動結(jié)束后，因為這一目標(biāo)的各種擾動機(jī)器人從來沒有完全達(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)，我們計算另一個我們之前已經(jīng)執(zhí)行過的一個軌跡。 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。對系統(tǒng)B容易計算（見第3.4節(jié)）。 圖2 單一機(jī)器人的跟蹤控制率 很多帶拖車輪式移動機(jī)器人的跟蹤控制律已經(jīng)被提出。其中[16]雖然很簡單,但是提供了杰出的成果。 如果是模擬機(jī)器人的坐標(biāo)構(gòu)成真實機(jī)器人（圖2），如果（）是輸入的參考軌跡，這種控制律表示如下： （2） 我們控制律的關(guān)鍵想法如下：當(dāng)機(jī)器人前進(jìn)，拖車不需要穩(wěn)定（見下文）。因此，我們對機(jī)器人使用公式（2）。 當(dāng)它后退時，我們定義一個虛擬的機(jī)器人（圖3）這是對稱的真實一對拖車的車輪軸： 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)在運(yùn)動學(xué)上是等同于真正的一個。因此，我們對虛擬機(jī)器人實行跟蹤控制法（2）。 圖3 虛擬機(jī)器人 現(xiàn)在的問題是：當(dāng)機(jī)器人前進(jìn)時，拖車是否真的穩(wěn)定？下一節(jié)將回答這個問題。 3.3 拖車穩(wěn)定性分析 在這里我們考慮的向前運(yùn)動情況下,虛擬機(jī)器人向后的運(yùn)動被等值轉(zhuǎn)變。讓我們把坐標(biāo)作為參考軌跡并且把坐標(biāo)作為實際運(yùn)動的系統(tǒng)。我們假設(shè)機(jī)器人完全跟隨其參考軌跡：并且我們把我們的注意力放在拖車偏差 。這一偏差的變化很容易從系統(tǒng)（1）推導(dǎo)出（系統(tǒng)A） ： 盡管是減少的 （3） 我們的系統(tǒng)而且被不等量限制了 （4） 因此和式（3）等價于 （5） 圖4顯示的范圍隨著給定的的值正在減少。我們可以看到，這個范圍包含了拖車的所有的位置，包括式（4）所界定的范圍。此外，以前的計算許可輕松地表明對于變量，0是一個漸近穩(wěn)定值的變量。 因此，如果實際或虛擬的機(jī)器人按照它的參考軌跡前進(jìn)，拖車是穩(wěn)定的，并且將趨于自己的參考軌跡。 圖4 的穩(wěn)定范圍 3.4虛擬機(jī)器人系統(tǒng)B 當(dāng)拖車掛在機(jī)器人的后面，之前的結(jié)構(gòu)甚至更簡單：我們可以用拖車取代虛擬的機(jī)器人。在這種實際情況下，機(jī)器人的速度和拖車一對一映射的連接。然后虛擬的機(jī)器人系統(tǒng)表示為如下： 和以前的穩(wěn)定性分析可以被很好的使用通過考慮懸掛點的運(yùn)動。 下面一節(jié)討論了我們迭代計劃的魯棒性。 3.5迭代計劃的魯棒性 我們現(xiàn)在正在顯示上文所提到的迭代計劃的魯棒性。為此，我們需要有一個當(dāng)機(jī)器人的運(yùn)動時產(chǎn)生擾動的模型。 [1]擾動的模型系統(tǒng)是一個不規(guī)則，從而導(dǎo)致矢量場確定性的變化。在我們的實驗中，我們要看到由于隨機(jī)擾動導(dǎo)致的例如在一些懸掛系統(tǒng)中發(fā)揮作用。這些擾動對模型是非常困難的。出于這個原因， 我們只有兩個簡單的假說有： 其中s是沿曲線橫坐標(biāo)設(shè)計路徑，和分別是真正的和參考的結(jié)構(gòu)，是結(jié)構(gòu)空間系統(tǒng)的距離并且，是正數(shù)。 第一個不等量意味著實際和參考結(jié)構(gòu)之間的距離成正比的距離覆蓋計劃路徑。第二個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)是非常現(xiàn)實的和適合大量的擾動模型。 我們現(xiàn)在需要知道在每個迭代路徑的長度。我們使用指導(dǎo)的方法計算這些路徑驗證拓?fù)涠虝r間的可控性[17]。這個也就是說，如果我們的目標(biāo)是充分接近起初的結(jié)構(gòu)，軌跡的計算依然是起初的結(jié)構(gòu)的附近。在[9] 我們給出的估算方面的距離：如果 和是兩種不夠緊密的結(jié)構(gòu)，規(guī)劃路徑的長度驗證它們之間的關(guān)系 這里是一個正數(shù)。 因此，如果 是配置依次獲得的，我們有以下不等式： 這些不等式確保distCS是上界序列的正數(shù) 和趨近于足夠反復(fù)后的。 因此，我們沒有獲得漸近穩(wěn)定性配置的目標(biāo)，但這一結(jié)果確保存在一個穩(wěn)定的范圍處理這個配置。 這一結(jié)果基本上是來自我們選擇非常傳統(tǒng)擾動的模型。讓我們重復(fù)這包括諸如擾動模型的時間不同的控制律無疑將使其失去其漸近穩(wěn)定。 實驗結(jié)果如下節(jié)顯示，收斂域的控制計劃是非常小的。 4實驗結(jié)果 現(xiàn)在，我們目前獲得的帶拖車機(jī)器人Hilare系統(tǒng)A和B的實驗結(jié)果。圖5和圖6顯示第一路徑計算的例子所規(guī)劃初始配置（黑色）和目標(biāo)配置（灰色）之間的運(yùn)動。在第二種情況下包括上一次計算結(jié)果。連接系統(tǒng)的長度如下：系統(tǒng)A中，厘米，系統(tǒng)B厘米，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作和第二次動作之間的不足，3個不同的實驗。在這兩種情況下，第一次試驗相當(dāng)于圖表。意味著，在第一動作后精度十分充足，沒有更多可進(jìn)行的動作。 評論和意見：表1和表2的報告結(jié)果顯示了兩個主要的見解。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）。事實上，精密程度取決于很多的速度和不同的動作。在這里，機(jī)器人的最大線速度是50厘米/秒 。 5結(jié)論 我們已經(jīng)提出了一種方法來控制機(jī)器人與拖車從初始結(jié)構(gòu)到一個已知輸入問題的目標(biāo)。這種方法是以迭代于開環(huán)和閉環(huán)控制相結(jié)合為前提的辦法。它對大范圍的擾動模型已經(jīng)顯示出健全的一面。這個魯棒性主要來自拓?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)，但是在真正實驗期間達(dá)到的精度程度是非常令人滿意的。 13 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 表2：系統(tǒng)B：目標(biāo)和期望配置在第一次動 作和第二次動作之間的差距 作和第二次動作之間的差距 參考文獻(xiàn) [1].M. K. Bennani et P. Rouchon. Robust stabilization of flat and chained systems. in European Control Conference,1995. [2].R.W. Brockett. Asymptotic stability and feedback stabilization. in Differential Geometric Control Theory,R.W. Brockett, R.S. Millman et H.H. Sussmann Eds,1983. [3].C. Canudas de Wit, O.J. Sordalen. Exponential stabilization of mobile robots with non holonomic constraints.IEEE Transactions on Automatic Control,Vol. 37, No. 11, 1992. [4].J. M. Coron. Global asymptotic stabilization for controllable systems without drift. in Mathematics of Control, Signals and Systems, Vol 5, 1992. [5].A. De Luca, G. Oriolo et C. Samson. Feedback control of a nonholonomic car-like robot, "Robot motion planning and control". J.P. Laumond Ed., Lecture Notes in Control and Information Sciences, Springer 'Verlag, to appear. [6].R. M. DeSantis. Path-tracking for a tractor-trailerlike robot. in International Journal of Robotics Research,Vol 13, No 6, 1994. [7].A. Hemami, M. G. Mehrabi et R. M. H. Cheng. Syntheszs of an optimal control law path trackang an mobile robots. in Automatica, Vol 28, No 2, pp 383-387, 1992. [8].Y. Kanayama, Y. Kimura, F. Miyazaki et T.Nogushi.A stable tracking control method for an autonomous mobile robot. in IEEE International Conference on Robotics and Automation, Cincinnati, Ohio, 1990. [9].F. Lamiraux.Robots mobiles ci remorque : de la planification de chemins d: l ' e x h t i o n de mouuements,PhD Thesis N7, LAAS-CNRS, Toulouse, September 1997. [l0].P. Morin et C. Samson. Application of backstepping techniques to the time-varying exponential stabitisation of chained form systems. European Journal of Control, Vol 3, No 1, 1997. [11].J. B. Pomet. Explicit design of time-varying stabilizang control laws for a class of controllable systems without drift. in Systems and Control Letters, North [12].M. Sampei, T. Tamura, T. Itoh et M. Nakamichi.Path tracking control of trailer-like mobile robot. in IEEE International Workshop on Intelligent Robots and Systems IROS, Osaka, Japan, pp 193-198, 1991. [13].C. Samson. Velocity and torque feedback control of a nonholonomic cart. International Workshop in Adaptative and Nonlinear Control: Issues in Robotics, Grenoble, France, 1990. [14].C. Samson. Time-varying feedback stabilization of carlike wheeled mobile robots. in International Journal of Robotics Research, 12(1), 1993. [15].C. Samson. Control of chained systems. Application to path following and time-varying poznt-stabilization. in IEEE Transactions on Automatic Control, Vo l 40,No 1, 1995. [16].C. Samson et K. Ait-Abderrahim. Feedback control of a nonholonomic wheeled cart zncartesaan space.in IEEE International 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. Splrdalen 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. 外文資料翻譯 題目： 帶拖車移動機(jī)器人的反饋控制 系 別 航空與機(jī)械工程系 專業(yè)名稱 機(jī)械設(shè)計制造及自動化 班級學(xué)號 088105406 學(xué)生姓名 鄧文文 指導(dǎo)教師 許瑛 二O一二 年 六 月 南昌航空大學(xué)科技學(xué)院學(xué)士學(xué)位論文-外文 Proceedings ofthe 1998 IEEE International Conference on Robotics & Automation Leuven, Belgium May 1998 13 A practical approach to feedback control for a mobile robot with trailer F. Lamiraux and J.P. Laumond LAAS-CNRS Toulouse, France {florent ,jpl}@laas.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 configuration 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 given rise to a lot of work for the past 8 years. Brockett’s 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, 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 Brockett’s condition and thus it is an easier task. A lot of work have also addressed this problem [6, 7, 8, 12, 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 kinematics equations. [l] however proposed a method to stabilize a car about a configuration, robust to control vector fields perturbations, and based on iterative trajectory tracking. The presence of obstacle makes the task of reaching a configuration even more difficult and require a path planning task before executing any motion. In this paper, we propose a robust scheme based on iterative trajectory tracking, to lead a robot towing a trailer to a 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 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).Section 3 deals with the control scheme and the analysis of stability and robustness. In Section 4, we present experimental results. 2 Description of the system 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 , bottom). A is the particular case of B, for which = 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 （，) of the 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（,,）of the robot are computed by integrating the former velocities. With these notations, the control system of B is: （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 motors, 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 where may be either deterministic or a random variable.In the first case, the perturbation is only due to a bad knowledge of the system evolution, whereas in the second case, it comes from a random behavior of the system. We will see later that 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 initial 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 defined term in this closed loop system, we don't 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 become an equilibrium subset.But we don't 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 ensure 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 obstacles 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 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 about 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). Figure 2: Tracking control law for a single robot A lot of tracking control laws have been proposed for wheeled mobile robots without trailer. One of them [16],a lthough very simple, give excellent results.If are the coordinates of the reference robot in the frame of the real robot (Figure 2), and if are the inputs of the reference trajectory, this control law has the following expression: （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 (Figure 3) which is symmetrical to the real one with respect to the wheel axis of the trailer: Then, when the real robot goes backward, the virtual robot goes forward and the virtual system is kinematically equivalent to the real one. Thus we apply the tracking control law (2) to the virtual robot. Figure 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 We consider here the case of a forward motion , the backward motion being equivalent by the virtual robot transformation. Let us denote by a reference trajectory and bythe real motion of the system. We assume that the robot follows exactly its reference trajectory: and we focus our attention on the trailer deviation.The evolution of this deviation is easily deduced from system (1) with (System A): is thus decreasing iff （3） Our system is moreover constrained by the inequalities （4） so that and (3) is equivalent to （5） Figure 4 shows the domain on which is decreasing for a given value of . 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 asymptotically 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. Figure 4: Stability domain for 3.4 Virtual robot for system B When the trailer is hitched behind the robot, the former construction is even more simple: we can replace the virtual robot by the trailer. In this case indeed, the velocities of the robot and of the trailer are connected by a one-to-one mapping.The configuration of the virtual robot is then given by the following system: 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 to 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 experiment 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: where s is the curvilinear abscissa along the planned path, and are respectively the real and reference configurations, is 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 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 length of the paths generated at each iteration. The steering method we use to compute these paths verifies a topological property accounting for small-time controllability[17]. This means that if the goal is sufficiently close to the starting configuration, the computed trajectory remains in a neighborhood of the starting configuration. In [9]we give an estimate in terms of distance: if and are two sufficiently close configurations, the length of the planned path between them verifies where is a positive constant. Thus, if is the sequence of configurations reached after i motions, we have the following inequalities: These inequalities ensure that distCS is upper bounded by a sequence of positive numbers defined by and converging toward after enough 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 perturbation model in a time varying control law would undoubtedly make it lose its asymptotical stability.The experimental 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 obtained 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 configurations, 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: ,cm for A and cm,cm for B. Tables 1 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 columns mean that the precision reached after 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 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 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 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 the 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 Table 2: System B: initial and final configurations,gaps between the first and second reached configurations and the goal References [1].M. K. Bennani et P. Rouchon. Robust stabilization of flat and chained systems. in European Control Conference,1995. [2].R.W. Brockett. Asymptotic stability and feedback stabilization. in Differential Geometric Control Theory,R.W. Brockett, R.S. Millman et H.H. Sussmann Eds,1983. [3].C. Canudas de Wit, O.J. Sordalen. Exponential stabilization of mobile robots with non holonomic constraints.IEEE Transactions on Automatic Control,Vol. 37, No. 11, 1992. [4].J. M. Coron. Global asymptotic stabilization for controllable systems without drift. in Mathematics of Control, Signals and Systems, Vol 5, 1992. [5].A. De Luca, G. Oriolo et C. Samson. Feedback control of a nonholonomic car-like robot, "Robot motion planning and control". J.P. Laumond Ed., Lecture Notes in Control and Information Sciences, Springer 'Verlag, to appear. [6].R. M. DeSantis. Path-tracking for a tractor-trailerlike robot. in International Journal of Robotics Research,Vol 13, No 6, 1994. [7].A. Hemami, M. G. Mehrabi et R. M. H. Cheng. Syntheszs of an optimal control law path trackang an mobile robots. in Automatica, Vol 28, No 2, pp 383-387, 1992. [8] .Y. Kanayama, Y. Kimura, F. Miyazaki et T.Nogushi.A stable tracking control method for an autonomous mobile robot. in IEEE International Conference on Robotics and Automation, Cincinnati, Ohio, 1990. [9]. F. Lamiraux.Robots mobiles ci remorque : de la planification de chemins d: l ' e x h t i o n de mouuements,PhD Thesis N7, LAAS-CNRS, Toulouse, September 1997. [l0].P. Morin et C. Samson. Application of backstepping techniques to the time-varying exponential stabitisation of chained form systems. European Journal of Control, Vol 3, No 1, 1997. [11].J. B. Pomet. Explicit design of time-varying stabilizang control laws for a class of controllable systems without drift. in Systems and Control Letters, North [12].M. Sampei, T. Tamura, T. Itoh et M. Nakamichi.Path tracking control of trailer-like mobile robot. in IEEE International Workshop on Intelligent Robots and Systems IROS, Osaka, Japan, pp 193-198, 1991. [13].C. Samson. Velocity and torque feedback control of a nonholonomic cart. International Workshop in Adaptative and Nonlinear Control: Issues in Robotics, Grenoble, France, 1990. [14].C. Samson. Time-varying feedback stabilization of carlike wheeled mobile robots. in International Journal of Robotics Research, 12(1), 1993. [15].C. Samson. Control of chained systems. Application to path following and time-varying poznt-stabilization. in IEEE Transactions on Automatic Control, Vol 40,No 1, 1995. [16].C. Samson et K. Ait-Abderrahim. Feedback control of a nonholonomic wheeled cart zncartesaan space.in IEEE International 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. Splrdalen 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.