外文翻译--一种实用的办法--带拖车移动机器人的反馈控制

1 河北建筑工程学院 毕业设计（论文）外文资料翻译 系别： 机 械 工 程 系 专业： 机械设计制造及其自动化 班级： 姓名： 学号： 外文出处： Proceedings ofthe 1998 IEEE International Conference on Robotics & Automation 附 件： 1、外文原文； 2、外文资料翻译译文。 指导教师评语： 签字： 年 月 日 2 Proceedings ofthe 1998 IEEE International Conference on Robotics & Automation Leuven, Belgium May 1998A 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 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. 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, 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, 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 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 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. 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 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 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 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（ xr ,yr , r ） of the robot are computed by integrating the former velocities. With these notations, the control system 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 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 ( , )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 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 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 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 become 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 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). 6 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 ,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 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 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 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 virtual 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 focus 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 and (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 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. 8 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 ( , )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 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 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: 00( ( ) , ( ) )( ( ) , ( ) )c s scsd q s q sd q s q s 9 where s is the curvilinear abscissa along 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 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 controllability17. 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 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 configurations 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 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 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 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: 0rl , 125tl cm for A and 60rl cm, 90tl 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 2qcolumns 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 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 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 12 Table 2: System B: initial and final configurations,gaps between the first 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, R.S. Millman et H.H. Sussmann Eds,1983. 3C. Canudas de Wit, O.J. Sordalen. 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Bolland, Vol 18, pp 147-158, 1992. 14 一种实用的办法 -带拖车移动机器人的反馈控制 F. Lamiraux and J.P. Laumond 拉斯，法国国家科学研究中心 法国图卢兹 florent ,jpllaas.fr 摘 要 本文提出了一种有效的方法来控制带拖车移动机器人。轨迹跟踪和路径跟踪这两个问题已经得到解决。接下来的问题是解决迭代轨迹跟踪。并且把扰动考虑到路径跟踪内。移动机器人 Hilare 的实验结果说明了我们方法的有效性。 1 引言 过去的 8 年，人们对非完整系统的运动控制做了大量的工作。布洛基 2提出了关于这种系统的一项具有挑战性的任务，配置的稳定性，证明它不能由一个简单的连续状态反馈。作为替代办法随时间变化的反馈 10,4,11,13,14,15,18或间断反馈 3也随之被提出。从 5 移动机器人的运动控制的一项调查可以看到。另一方面，非完整系统的轨迹跟踪不符合布洛基的条件，从而使其这一个任务更为轻松。许多著作也已经给出了移动机器人的特殊情况的这一问题 6,7,8,12,16。 所有这些控制律都是工作在相同的假设下：系统的演变是完全已知和没有扰动使得系统偏离其轨迹。很少有文章在处理移动机器人的控制时考虑到扰动的运动学方程。但是 1提出了一种有关稳定汽车的配置，有效的矢量控制扰动领域，并且建立在迭代轨迹跟踪的基础上。 存在的障碍使得达到规定路径的任务变得更加困难，因此在执行任务的任何动作之前都需要有一个路径规划。 在本文中，我们在迭代轨迹跟踪的基础上提出了一个健全的方案，使得带拖车的 15 机器人按照规定路径行走。该轨迹计算由规划的议案所描述 17 ，从而避免已经提 交了输入的障碍物。在下面，我们将不会给出任何有关规划的发展，我们提及这个参考的细节。而且，我们认为，在某一特定轨迹的执行屈服于扰动。我们选择的这些扰动模型是非常简单，非常一般。它存在一些共同点 1。 本文安排如下：第 2 节介绍我们的实验系统 Hilare 及其拖车：两个连接系统将被视为（图 1） 。第 3节处理控制方案及分析的稳定性和鲁棒性。在第 4节，我们介绍本实验结果 。 图 1 带拖车的 Hilare 2 系统描述 Hilare是一个有两个驱动轮的移动机器人。拖车是被挂在这个机器人上的，确定了两个不同的系统取决于连接设备：在系统 A的拖车拴在机器人的车轮轴中心线上方（图 1 ，顶端），而对系统 B是栓在机器人的车轮轴中心线的后面（图 1 ，底部 )。 A对 B来说是一种特殊情况，其中 rl = 0 。这个系统不过单从控制的角度来看，需要更多的复杂的计算。出于这个原因，我们分开处理挂接系统 。两个马达能够控制机器人的线速度和角速度（ vr ， r )。除了这些速度之外，还由传感器测量，而机器人和拖 16 车之间的角度 ，由光学编码器给出。机器人的位置和方向（ xr , yr , r ）通过整合前的速度被计算。有了这些批注，控制系统 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 当考虑到现实的系统，人们就必须要考虑到在运动的执行时产生的扰动。 这可能有许多的来源，像有缺陷的电机，轮子的滑动，惯性的影响 . 这些扰动可以被设计通过增加一个周期在控 制系统（ 1） ，得到一个新的系统的形式 ( , )x f x u 在上式中可以是确定性或随机变量。 在第一种情况下，扰动仅仅是由于系统演化的不规则，而在第二种情况下，它来自于该系统一个随机行为。我们将看到后来，这第二个模型是一个更适合我们的实验系统。 为了引导机器人，从一开始就配置了目标，许多工程认为扰动最初只是机器人和目标之间的距离，但演变的系统是完全众所周知的。为了解决这个问题， 他们设计了一个可输入的时间 -状态函数，使目标达到一个渐近稳定平衡的闭环系统。现在，如果我们介绍了先前定义周期 在这个闭环系统，我们不知道将会发生什么。但是我们可以猜想，如果扰动 很小、是确定的、在平衡点（如果仍然还有一个）将接近目标，如果扰动是一个随机变数，平衡点将成为一个平衡的子集。 但是，我们不知道这些新的平衡点或子集的位置。 此外 ，在处理障碍时，随时间变化的方法不是很方便。他们只能使用在附近的目标，这附近要适当界定，以确保无碰撞轨迹的闭环系统。请注意连续状态反馈不能适用于真实情况下的机器人，因为间断的速度导致无限的加速度。 17 我们建议达成某一存在障碍特定配置的方法如下。我们首先在当前的配置和使用自由的碰撞议案所描述 17目标之间建立一个自由的碰撞路径，然后，我们以一个简单的跟踪控制率执行轨迹。在运动结束后，因为这一目标的各种扰动机器人从来没有完全达到和目标的轨迹一致，而是这一目标的左右。如果达到配置远离目标，我们计算另一个 我们之前已经执行过的一个轨迹。 现在我们将描述我们的轨迹跟踪控制率，然后给出我们的全球迭代方法的鲁棒性问题。 3.2 轨迹跟踪控制率 在这一节中，我们只处理系统 A。对系统 B 容易计算（见第 3.4 节）。 图 2 单一机器人的跟踪控制率 很多带拖车轮式移动机器人的跟踪控制律已经被提出。其中 16虽然很简单 ,但是提供了杰出的成果。 如果 ,xy 是模拟机器人的坐标构成真实机器人（图 2），如果（ 00,rrv）是输入的参考轨迹，这种控制律表示如下： 01032c o ss i nr rrrv k xk k yv （ 2） 我们控制律的关键想法如下：当机器人前进，拖车不需要稳定（见下文）。因此，我们对机器人使用公式（ 2）。 当它后退时，我们定义一个虚拟的机器人 ( , , )r r rxy （图3）这是对称的真实一对拖车的车轮轴： 2 c o s2 s i n2r r t tr r t tr t rx x ly y l 然后，当真正的机器人退后，虚拟机器人前进和虚拟系统 ( , , , )r r rxy 在运动学上是 18 等同于真正的一个。因此，我们对虚拟机器人实行跟踪控制 法（ 2）。 图 3 虚拟机器人 现在的问题是：当机器人前进时，拖车是否真的稳定？下一节将回答这个问题。 3.3 拖车稳定性分析 在这里我们考虑的向前运动情况下 ( 0)rv ,虚拟机器人向后的运动被等值转变。让我们把坐标 0 0 0 0 0( , , , , )r r r r rx y v作为参考轨迹并且把坐标 ( , , , , , )vyx r r r r r 作为实际运动的系统。我们假设机器人完全跟随其参考轨迹：0 0 0 0 0( , , , , , ) ( , , , , )r r r r rv x y vyx r r r r r 并且我们把我们的注意力放在拖车偏差0 。这一偏差的变化很容易从系统（ 1）推导出 0rl （系统 A） ： 00 ( s i n s i n )2c o s ( ) s i n ( )22rtrtvlvl 尽管 是减少的 0 2 2 2 2 （ 3） 我们的系统而且被不等量限制了 0,22 （ 4） 因此 0 2 和式（ 3）等价于 0000022022 且或且（ 5） 19 图 4 显示 的范围随着给定的 0 的值正在减少。我们可以看到，这个范围包含了拖车的所有的位置，包括式（ 4）所界定的范围。此外，以前的计算许可轻松地表明对于变量 0 ， 0 是一个渐近稳定值的变量。 因此，如果实际或虚拟的机器人按照它的参考轨迹前进，拖车是稳定的，并且将趋于自己的参考轨迹。 图 4 的稳定范围 3.4 虚拟机器人系统 B 当拖车挂在机器人的后面，之前的结构甚至更简单：我们可以用拖车取代虚拟的机器人。在这种实际情况下，机器人的速度 ( , )rrv 和拖车 ( , )ttv 一对一映射的连接。然后虚拟的机器人系统表示为如下： 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 和以前的稳定性分析可以被很好的使用通过考虑悬挂点的运动。 下面一节讨论了我们迭代计划的鲁棒性。 3.5 迭代计划的鲁棒性 我们现在正在显示上文所提到的迭代计划的鲁棒性。为此，我们需要有一个当机器人的运动时产生扰动的模型。 1扰动的模型系统是一个不规则，从而导致矢量场确定性的变化。在我们的实验中，我们要看到由于随机扰动导致的例如在一些悬挂系统 20 中发挥作用。这些扰动对模型是非常困难的。出于这个原因， 我们只有两个简单的假说有： 00( ( ) , ( ) )( ( ) , ( ) )c s scsd q s q sd q s q s其中 s 是沿曲线横坐标设计路径， q 和 0q 分别是真正的和参考的结构，csd是结构空间系统的距离并且 ， 是正数。 第一个不等量意味着实际和参考结构之间的距离成正比的距离覆盖计划路径。第二个不等量是确保轨迹跟踪控制率，防止系统走得太远远离其参考轨迹。让我们指出，这些假设是非常现实的和适合大量的扰动模型。 我们现在需要知道在每个迭代路径的长度。我们使用指导的方法计算这些路径验证拓扑短时间的可控性 17。这个也就是说，如果我们的目标是充分接近起初的结构，轨迹的计算依然是起初的结构的附近。在 9 我们给出的估算方面的距离：如果 1q和2q是两种不够紧密的结构，规划路径的长度验证它们之间的关系 141 2 1 2( ( , ) ) ( , )csl P a t h q q d q q这里 是一个正数。 因此，如果1,2.()iiq 是配置依次获得的，我们有以下不等式： 11,( , )( ) ( , )c s g o a lc s i g o a l c s i g o a ld q qd q q d q q这些不等式确保 distCS( , )i goalqq是上界序列1,2.()iid 的正数 1141iiddd和趋近于足够反复后的。 因此，我们没有获得渐近稳定性配置的目标，但这一结果确保存在一个稳定的范围处理这个配置。 这一结果基本上是来自我们选择非常 传统扰动的模型。让我们重复这包括诸如扰动模型的时间不同的控制律无疑将使其失去其渐近稳定。 实验结果如下节显示，收敛域的控制计划是非常小的。 21 4 实验结果 现在，我们目前获得的带拖车机器人 Hilare 系统 A 和 B 的实验结果。图 5 和图 6显示第一路径计算的例子所规划初始配置（黑色）和目标配置（灰色）之间的运动。在第二种情况下包括上一次计算结果。 连接系统的长度如下：系统 A 中 0rl ， 125tl 厘米，系统 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标和期望配置在第一次动作和第二次动作之间的不足， 3 个不同的实验。在这两种情况下，第一次试验相当于图表。2q意味着，在第一动作后精度十分充足，没有更多可进行的动作。 评论和意 见：表 1 和表 2 的报告结果显示了两个主要的见解。首先， 系统达成非常令人满意的精密程度，其次迭代次数是非常小的（介于 1 和 2 之间）。事实上，精密程度取决于很多的速度和不同的动作。在这里，机器人的最大线速度是 50 厘米 /秒 。 5 结论 我们已经提出了一种方法来控制机器人与拖车从初始结构到一个已知输入问题的目标。这种方法是以迭代于开环和闭环控制相结合为前提的办法。它对大范围的扰动模型已经显示出健全的一面。这个鲁棒性主要来自拓扑性能指导方法介绍 17 。即使该方法不完全趋于机器人的最终目标，但是在真正实验期间达到 的精度程度是非常令人满意的。 图 5：系统 A：初始、目标配置跟踪第一路径 图 6：系统 B：初始、目标配置跟踪第一路径和最终结果 表 1：系统 A： 目标和期望配置在第一次动 表 2：系统 B：目标和期望配置在第一次动 作和第二次动作之间的差距 作和第二次动作之间的差距 参考文献 1M. 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