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外文翻译--轮式移动机器人的导航与控制
外文翻译文献翻译轮式移动机器人的导航与控制
轮式移动机器人的导航与控制
轮式移动机器人导航控制与
轮式移动机器人的导航控制
轮式移动机器人导航控制和
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中英文文献翻译-轮式移动机器人的导航与控制,外文翻译,轮式移动机器人的导航与控制,外文翻译文献翻译轮式移动机器人的导航与控制,轮式移动机器人的导航与控制,轮式移动机器人导航控制与,轮式移动机器人的导航控制,轮式移动机器人导航控制和
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毕业设计(论文)外文资料翻译系部： 机械工程 专 业： 机械工程及自动化 姓 名： 学 号： (用外文写)外文出处： Control and Robotics(CRB) Technical Report 附 件：1.外文资料翻译译文；2.外文原文。 指导教师评语：译文比较正确地表达了原文的意义、概念描述基本符合汉语的习惯，语句较通畅，层次较清晰。 签名： 年 月 日附件1：外文资料翻译译文 轮式移动机器人的导航与控制摘要：本文研究了把几种具有导航功能的方法运用于不同的控制器开发，以实现在一个已知障碍物前面控制一个开环系统（例如：轮式移动机器人）执行任务。第一种方法是基于三维坐标路径规划的控制方法。具有导航功能的控制器在自由配置的空间中生成一条从初始位置到目标位置的路径。位移控制器控制移动机器人沿设置的路径运动并停止在目标位置。第二种方法是基于二维坐标路径规划的控制方法。在二维平面坐标系中建立导航函数，基于这种导航函数设计的微控制器是渐进收敛控制系统。仿真结果被用来说明第二种控制方法的性能。1介绍很多研究者已经提出不同算法以解决在障碍物杂乱的环境下机器人的运动控制问题。对与建立无碰撞路径和传统的路径规划算法，参考文献19的第一章第九部分中提供了的全面总结。从Khatib在参考文献13的开创性工作以来，很显然控制机器人在已知障碍物下执行任务的主流方法之一依然是构建和应用位函数。总之，位函数能够提供机器人工作空间、障碍位置和目标的位场。在参考文献19中提供对于位函数的全面研究。应用位函数的一个问题是局部极小化的情况可能发生以至于机器人无法到达目标位置。不少研究人士提出了解决局部极小化错误的方法（例如参考文献2, 3,5, 14, 25）。其中Koditschek在参考文献16中提供了一种解决局部极小化错误的方法，那是通过基于一种特殊的位函数的完整系统构建导航函数，此函数有精确的数学结构，它能够保证存在唯一最小值。在针对标准的 (完整的)系统的先前的结果的影响下, 面对更多的具有挑战性的非完整系统，越来越多的研究集中于位函数方法的发展(例如.,机器人)。例如, Laumond 等人 18 用几何路线策划器构建了一条忽略机器人非完全约束的无障碍路线, 然后把几何线路分成更短的线路来满足非完全限制,然后应用最佳路线来减少路程。在 10 和 11中, Guldner 等人使用间断变化的模式控制器迫使机器人的位置沿着位函数的负倾斜度变动，及其定位与负倾斜度一致。在1, 15, 和 21中,持续的位场控制器也保证了位函数的负倾斜度的位置追踪和定位追踪。在9中，面对目标因为周边的障碍物而不能达到这一情况时，Ge和Cui 最近提出一种新的排斥的位函数的方法来解决这一问题。 在 23和24中, Tanner 等人采用22 中提出的导航函数研究和偶极位场概念为一个不完全移动操纵器建立导航函数控制器。特别是, 23 和 24 中的结果使用了间断控制器来追踪导航函数的负倾斜度, 在此过程中，一个不平坦的偶极位场使得机器人按照预想的定位拐入目标位置。 本文介绍了为不完全系统达到导航目标的两种不同的方法。在第一个方法中, 产生了一个三维空间似导航函数的预想的轨道，它接近于机器人自由配置空间上的唯一最小值的目标位置和定位。然后利用连续控制结构使机器人沿着这条路线走，在目标位置和定位点停下(例如,控制器解决一体化的追踪和调节问题)。这种方法特别的地方是机器人根据预想的定位到达目标位置，而不需要像许多先前的结果中一样转弯。正如 4 和 20中描述的一样, 一些因素如光线降低现象，更有效处罚离开预期周线的机器人的能力，使执行任务速度恒定的能力，以及达到任务协调性和同步性的能力提高等为按照目前位置和定位压缩预期轨道提供动机。至于即时的二维空间问题 设计一个连续控制器，沿着一个导航函数的负倾斜度驾驶机器人到达目标位置。像许多先前的结果一样，在线二维空间方法的定位需要进一步发展 (例如, 一个单独的调节控制器，一个偶极位场方法 23, 24; 或一个有效障碍物9)来使机器人与预期的定位在一条线上。模拟结果阐明了第二种方法的效果。 2 运动学模型本文所讨论的不完全系统的种类可以作为运动转轮的模型 这里定义为在(1)中, 矩阵定义为速度向量 定义为其中vc(t), c(t) R 表示系统线速度和角速度。在(2)中, xc(t), yc(t),(t) R分别表示位置和定位，xc(t),yc(t) 表示线速度的笛卡尔成分,(t) R 表示角速度。3 控制目标本文的控制目标是在一个有障碍物且混乱的环境下，沿着无碰撞轨道驾驶不完全系统（例如，机器人）到达不变的目标位置和定位，用表示。 特别是从起始位置和定位沿着轨道控制不完全系统，q D, 这里的 D 表示一个自由的配置空间。自由配置空间D是整个配置空间的子集，除去了所有含有障碍物碰撞的配置。使轨道计划控制量化，实际笛卡尔位置和定位与预想的位置和定位之间的差异可表示为 ,定义为 如下这里设计了预想的轨道，因此 qd(t) q.16中，运用导航函数方法, 利用似导航函数生成预期路线qd(t)。在本文中似导航函数有如下定义： 定义1 把D作为连接解析流形和边界的纽带, 把q 当作D内部的目标点. 似导航函数(q) :D 0, 1 是符合下列条件的函数： 1. (q(t) 第一个命令和可辨第二个命令 (例如,存在与D中的和)。2. (q(t) 在D的边界有最大变量。3. (q(t) 在 q (t) = q上有唯一的全局最小值.4. 如果 ，其中z, r R 是正常数。5. 如果(q(t)被限制,那么被r 限制，其中 R是正常数。 4 在线三维空间轨道计划4.1 轨道计划生成的预期的三维空间轨道如下：其中(q) R 表示定义1中定义的似导航函数, 表示(q)的倾斜向量, 是另加的限制条件。假设 定义1中定义的似导航函数，沿着由(6)生成的预期轨道，确保了辅助条件N () R3, 表示为满足了下面的不等式 其中正函数 () 在和 中是不减少的。(8) 中给的不等式将在以后的稳定性分析中用到。4.2 模型转换为了达到控制目标，控制器必须能够追踪预期轨道，停在目标位置q上. 最后, 使用7 中提到的统一追踪和调节控制器。为了改进7中的控制器,必须把 (5)中定义的开路错误系统转换为合适的形式。(5)中定义的位置和定位循迹误差信号通过以下全应可逆转换8和辅助循迹误差变量w(t) R 和有关。 运用 (9)中的时间导数和 (1)-(5)及(9)后, 根据(9)定义的辅助变数，循迹误差可表示为 8 其中表示不相称矩阵，定义为 定义为 (10)中介绍的辅助控制输入 根据和定义如下 .4.3 控制发展 为了促进控制发展, 一个辅助误差信号, 用表示, 是后来设计的动态似振荡器信号 和转换的变量z(t)之间的差别,如下 根据(10)中开路运动系统和后来的稳定性分析, 我们把 u(t)设计为7 其中 k2 R 是正的不变的控制增长率。(15)中介绍的辅助控制条件定义为 其中辅助信号zd(t)由下列微分方程式和初始条件决定 辅助条件1(w, f, t) R and d(t) R 分别为 和 , k1, 0, 1, 1 R是正的不变的控制增长率, 在(12)中有定义。正如 8中描述的一样, (17)和(19)中结构是以以下事实为基础的 根据(9), e (t) f能够用, 和表示出来，如下 其中表示为 在随后的稳定性分析推动下，附加的限制条件vr (t) 表示如下 其中 k3, k4 R 是正的不变的控制增长率, 正函数1 (zd1, z1, qd, e),2 (zd1, z1, qd, e) R 表示为 4.4 闭环误差系统把(15)替换到(10)中后, 得到含有w(t) 如下的公式 这里利用了(14)和(11)中J的属性。第二次出现 ua(t)时把(16)替换到(26)中，利用(20)和(11)中J的属性, 最终得到的w(t)闭环误差系统表达式如下为了确定闭环误差系统, 我们运用(14)中的时间导数，替换 (10) 和(17) 到最终表达式， 达到下面的表达式 替换(15)和(16)到(28), (28) 可以写成第二次出现 1 (t) 时，替换(18)到(29) ，然后删去相同部分，得到表达式： 因为(30)中的相等条件和 (16)中定义的ua (t)是一样的, 得到 闭环误差系统的最终表达式 如下备注1 根据(19)中d (t )接近任意小常量，(16), (17),和(18)中禁止产生位奇点。4.5 稳定性分析 法则1 倘若qd (0) D, (6)中产生的预期轨道连同附加的限制条件vr (t) 保证了 和 ， 其中 r在定义1中有解释。证明: 让V (t) R 表示下面的函数其中 k R 是一个正常数, V1 (t) R 表示下面的函数V2 (qd) : D R 表示下面的一个函数运用(33)中时间导数，替换 (27) 和(31) 到最终的表达式，删去相同部分, 得到下面的表达式 运用(34)中时间导数和(6), 得到下面的表达式 其中 N () 在(7)中有定义。 根据 (8), V2 (t) 是上限，如下 替换 (21)到(37), 得到下面的不等式 其中向量表示如下 正函数 1 (zd1, z1, qd, e) 和2 (zd1, z1, qd, e)在(25)中有所定义。替换 (24)到(38), V2 (t)可以重新写成如下根据 (35) 和 (40), (32)中 V (t)的时间导数可以按下面的不等式得到上限其中正常数 表示如下. 案例 1: 如果,根据定义1中属性4，得到案例 2: 如果 ,根据 (32), (33),(34), 和(41) 得到其中 和 是正常数. 根据 (42), V (t)得到上限如下 因此根据 (32), (34), 和 (44),得到 如果 qd (0)不在D的边界, (qd (0) 1， k 可以符合 根据 (45) 和(46), (qd (t) 1, 因此从定义1得到qd (t) D， 从(43) 可以得出，(qd) 最终被限制。 因此, 如果 , k4 则符合 , 其中 在定义1中有解释，进而在定义1的属性5中得到定义, 最终被r限制。法则2 (15)-(19)中运动学控制法保证全局统一最终限制的(GUUB) 位置和定位按下面公式追踪其中 1 在(19)中给定, , 3 和 0 是正常数.证明: 根据 (33) 和(35), V1 (t) 得到上限如下 根据 (48), 得到下面的不等式 根据 (33), (49) 可以被写成 其中向量1 (t)在(39)中有定义。根据(33) 和(49), 得到w (t) , L. 根据 (19)和(20), 我们可以得到zd (t) L. 根据(14) 和, zd (t) L, 得到z (t) L.因为w (t) , z (t) L, 根据(9)中的逆转换, e (t) L. 根据法则1中qd (t) L 和 e (t) L,得到q (t) L.由于(22)-(25), qd (t) , zd (t) , z (t) , e (t) L,及定义1中的性质, 我们得到vr (t), qd (t) L. 根据 (12) 和q (t) , z (t) , qd (t) L, f (, z2, qd) L. 然后根据(18)得到 1 (t) L. 根据(15)-(17)得到 u (t) , ua (t) , zd (t) L. 根据 f (, z2, qd) , z (t) , u (t) L, 利用(10) 得到w(t) , z(t) L. 由于z (t) , zd (t) L 得到 L.利用论据的标准信号可以得出控制之下的所有的剩余信号和系统在闭环试验中仍然被限定。 根据 (19), (20), (39), 和(50), 把三元不等式应用到(14)可以证明 利用(50)-(51), 根据(9)中的逆转换得到(47)中的结果。备注2 虽然qd (t) 是无碰撞轨道, 如果只确保实际机器人轨道在预期路线的附近，法则2的稳定性结果保证了轨道的实际追踪。根据 (5) 和(47), 得到下面的限制其中根据法则1的证明，qd (t) D .为了确保q (t) D, 自由配置空间需要处于(52)右边的第二和第三条件共同作用。最后，障碍物的大小可以增加 。其中通过调节控制增加率，31 可以任意小。为了使2的影响最小化, 起始条件w(0)和z(0) (因此, ) 要求足够小来产生可行的路线到达目标。5 在线二维空间导航 先前的方法中，因为似导航函数用预期轨道表示，障碍物的尺寸要求增加。在下面的方法中, 22提出的导航函数是根据现有位置反馈表示出来的,因此, 不需要在起始条件里添加限制，q (t) 就可以证明是D的一部分。 5.1 轨道编制让(xc, yc) R 表示二维空间位置型导航函数, 其中梯度向量(xc, yc)定义如下让d (xc, yc) R 表示预期定位，定义为一个二维空间导航函数的负梯度函数，如下其中 arc tan 2 () : R2 R 表示第四象限逆切线函数 26, 其中 d (t) 在下面的定义域中 按照21规定,通过定义 ，沿着任何到达目标位置的方向d(t) 仍然是连续的。见附录d(t)的表达式， 根据先前的d(t)连续定义。 备注3 正如22中讨论的, 函数 (q(t)的建立, 结合导航函数, 满足定义1的前三个性质 因为排除故障不是简单的问题。事实上, 对与典型的故障排除来说，建立 (q(t)如只有当 q (t) = q时，是不大可能的。 这就是说, 如22所述, 内部承受点的外观(如不稳定平衡 )好像不可避免; 可是, 这些不稳定均衡不会真正 造成实践中的困难。这就是说，如22所述可以建立 (q(t)，只有少数起始条件能够真正受不稳定均衡影响。5.2 控制发展 根据(1)-(4)介绍的开路系统和后来的稳定性分析, 线速度控制输入vc (t)表示如下 其中 kv R 表示正的不变的控制增长率, 在(5)中有介绍。替换 (55) 到(1),得到下面的 闭环系统运用(5)中时间导数，得到开路定位追踪误差系统，如下 .利用(1)，根据 (57), 角速度控制输入c (t)表示如下 其中 k R 表示 正的不变的控制增长率,d(t) 表示预期定位的时间导数。见附录d (t)外部表达式。替换 (58)到(57), 通过下面的线性关系，得到闭环定位追踪误差系统 线性分析技巧用来解决(59) 如下替换 (60) 到(56)，得到下面的闭环误差系统 5.3 稳定性分析 法则3 (55)和(58)中的控制输入和导航函数 (xc (t) , yc (t) 在下列条件下保证了渐近导航证明: 让 V3 (xc, yc) : D R 表示下面的非负函数运用(63)时间导数，利用 (1),(53), 和(56),得到下面的表达式 根据附录的说明, 导航函数的梯度可表示为 替换 (65) 到(64), 得到下面的表达式 利用三角恒等式，(66) 可以写成 其中 g(t) R 表示下面的正函数根据 (53)和导航函数的属性(与定义1的属性1相同), 可以得到。因此,根据 (55)可以得出vc (t) L . 附录同样证明了d (t) L on D;因此, 根据(58) 得出c (t)L . 根据vc (t) L , 利用(1)-(4) 可以知道xc (t), yc (t) L .应用(53)的时间导数，得到下面的表达式因为 xc (t), yc (t) L , 也因为黑森矩阵的每个成分被导航函数的属性限制 (与定义1的属性1相同 ), 可以得到g(t) L. 根据 (63), (67), (68), 和g(t) L ,那么辅助定理6中的A.6可以用来证明 在D区域. 根据1 ,那么 (70) 可以用来证明 . 因此根据备注3中的分析，可以得到(62)中的结果。备注4 这部分控制发展是以一个二维空间导航函数为基础的. 为了达到目标, a 预期的定位 d (t) 看作是二维空间导航函数的负梯度函数. 先前的发展可以用来证明(62)的结果。如果一个导航函数 (xc, yc) 能够在d|(xc ,yc ) = 中找到, 那么渐近导航可以通过(55) 和(58)中控制器达到; 否则, 根据d|(xc ,yc ) 一个标准的调节控制器 (如., 见8 中的候选控制器)可能用来调节机器人的定位。作为选择, 偶极位场方法23, 24 或有效障碍物9可以用来使导航函数的梯度场与机器人的目标定位成一行。6 模拟结果为了说明(55) 和(58)中控制器的成效, 用数值模拟驾驶机器人从q (xc (0) , yc (0) , (0) 到 q (xc , yc , )。因为导航函数的属性是不变的under a diffeomorphism, a diffeomorphism 用来绘制机器人自由配置空间到模型空间 17. 正函数 (xc, yc)如下 其中 是正整数参量, 边界函数 0 (xc, yc) R，障碍函数1 (xc, yc) R 定义如下 在(72)中, (xr0, yr0 ) 和 (xr1, yr1 ) 分别是障碍物和分界线的中心, r0, r1 R 分别是障碍物和界面的半径。根据(71)和(72), 可以看出模型空间是一个排除障碍物函数 1 (xc, yc)形成的圆的单位圆. 如果 更多的障碍物出现, 相应的障碍物函数就能简单的和导航函数 17合成一体. 在17中, Koditschek 证明 (xc, yc) in (71) 是关于(xc (t) , yc (t)的导航函数, 假设足够大. 由于模拟, 模型空间配置选择如下其中起始位置和目标配置为 利用(55) 和(58)中定义的控制输入沿着负梯度角驾驶机器人到目标点。控制增长率kv 和 k调整到下面的值来产生最好的效果一旦机器人到达目标位置, 8中的调节控制器按照d|(xc ,yc ) 调节机器人。机器人的实际轨道如图1所示。. 图1中的外圆描述了障碍物自由空间外边界，内部的圆代表了障碍物周围的边界。机器人的最终位置和定位误差如图2所示, 其中转动误差如图2所示是实际定位和目标定位之间的误差。(55)和(58)分别定义的控制输入速度vc(t) 和c(t)如图3所示。值得注意的是角速度输入 在 90deg s1之间人为饱和。7 结论两种方法都把导航函数方法合并到不同的控制器，在已知障碍物面前执行任务。第一种方法利用 以3D 位置和定位信息为基础的似导航函数。似导航函数生成一条轨道从自由配置空间里的初始配置到目标配置. 一个可微的振荡器型控制器使这个移动式遥控装置沿着这条路线走，在目标位置停止.。利用这种方法, 机器人可以用一个任意的目标定位产生统一最终绑定路线和调节目标点(例如., 机器人不需要固定在目标位置旋转来达到预期的定位)。第二种方法使用的是二维空间位置信息建立的导航函数。根据这个导航函数，使用一个可辨的控制器。这个方法的好处是产生了渐近位置收敛; 可是，机器人如果没有附加的条件就不能在任意定位停止。模拟结果用来说明第二种方法的效果。 附录 根据(54)中d (t)的定义, d (t) 可用自然对数表示的表达式如下 26其中 ， 使用下面的恒等式 26 利用(74)得到下面的表达式 利用 (75)和(76), 得到下面的表达式 根据(74)中的表达式，d (t)的时间导数可以写成 其中， 替换 (1), (79), 和(80)到(78), 得到下面的表达式 替换 (55) 和(77)到(81), 得到下面的表达式 定义1的第一部分限制了黑森矩阵的每个元件，因此根据 (82),直接得到d (t) L.附件2：外文原文（复印件）Navigation and Control of a Wheeled Mobile RobotAbstract: Several approaches for incorporating navigation function approach into different controllers are developed in this paper for task execution by a nonholonomic system (e.g., a wheeled mobile robot) in the presence of known obstacles. The first approach is a path planning-based control with planning a desired path based on a 3-dimensional position and orientation information. A navigation-like function yields a path from an initial configuration inside the free configuration space of the mobile robot to a goal configuration. A differentiable, oscillator-based controller is then used to enable the mobile robot to follow the path and stop at the goal position. A second approach is developed for a navigation function that is constructed using 2-dimensional position information. A differentiable controller is proposed based on this navigation function that yields asymptotic convergence. Simulation results are provided to illustrate the performance of the second approach.1 IntroductionNumerous researchers have proposed algorithms to address the motion control problem associated with robotic task execution in an obstacle cluttered environment. A comprehensive summary of techniques that address the classic geometric problem of constructing a collision-free path and traditional path planning algorithms is provided in Section 9, .Literature Landmarks of Chapter 1 of 19. Since the pioneering work by Khatib in 13, it is clear that the construction and use of potential functions has continued to be one of the mainstream approaches to robotic task execution among known obstacles. In short, potential functions produce a repulsive potential field around the robot workspace boundary and obstacles and an attractive potential eld at the goal configuration. A comprehensive overview of research directed at potential functions is provided in 19. One of criticisms of the potential function approach is that local minima can occur that can cause the robot to get stuck without reaching the goal position. Several researchers have proposed approaches to address the local minima issue (e.g., see 2,3, 5, 14, 25). One approach to address the local minima issue was provided by Koditschek in 16 for holonomic systems (see also 17 and 22) that is based on a special kind of potential function, coined a navigation function, that has a refined mathematical structure which guarantees a unique minimum exists. By leveraging from previous results directed at classic (holonomic) systems, more recent research has focused on the development of potential function-based approaches for more challenging nonholonomic systems (e.g., wheeled mobile robots (WMRs). For example, Laumond et al. 18 used a geometric path planner to generate a collision-free path that ignores the nonholonomic constraints of a WMR, and then divided the geometric path into smaller paths that satisfy the nonholonomic constraints, and then applied an optimization routine to reduce the path length. In 10 and 11, Guldner et al. use discontinuous, sliding mode controllers to force the position of a WMR to track the negative gradient of a potential function and to force the orientation to align with the negative gradient. In 1, 15, and 21, continuous potential field-based controllers are developed to also ensure position tracking of the negative gradient of a potential function, and orientation tracking of the negative gradient. More recently, Ge and Cui present a new repulsive potential function approach in 9 to address the case when the goal is non-reachable with obstacles nearby (GNRON). In 23 and 24, Tanner et al. exploit the navigation function research of 22 along with a dipolar potential field concept to develop a navigation function-based controller for a nonholonomic mobile manipulator. Specifically, the results in 23 and 24 use a discontinuous controller to track the negative gradient of the navigation function, where a nonsmooth dipolar potential field causes the WMR to turn in place at the goal position to align with a desired orientation. In this paper, two different methods are proposed to achieve a navigation objective for a nonholonomic system. In the first approach, a 3-dimensional (3D) navigation-like function-based desired trajectory is generated that is proven to ultimately approach to the goal position and orientation that is a unique minimum over the WMR free configuration space. A continuous control structure is then utilized that enables the WMR to follow the path and stop at the goal position and orientation set point (i.e., the controller solves the unified tracking and regulation problem). The unique aspect of this approach is that the WMR reaches the goal position with a desired orientation and is not required to turn in place as in many of the previous results. As described in 4 and 20, factors such as the radial reduction phenomena, the ability to more effectively penalize the robot for leaving the desired contour, the ability to incorporate invariance to the task execution speed, and the improved ability to achieve task coordination and synchronization provide motivation to encapsulate the desired trajectory in terms of the current position and orientation. For the on-line 2D problem, a continuous controller is designed to navigate the WMR along the negative gradient of a navigation function to the goal position. As in many of the previous results, the orientation for the on-line 2D approach requires additional development (e.g., a separate regulation controller; a dipolar potential field approach 23, 24; or a virtual obstacle 9) to align the WMR with a desired orientation. Simulation results are provided to illustrate the performance of the second approach.2 Kinematic ModelThe class of nonholonomic systems considered in this paper can be modeled as a kinematic wheelwhere are defined asIn (1), the matrixis defined as followsand the velocity vector is defined aswith vc(t), c(t) R denoting the linear and angular velocity of the system. In (2), xc(t), yc(t), and (t) R denote the position and orientation, respectively, xc(t), yc(t) denote the Cartesian components of the linear velocity, and (t) R denotes the angular velocity. 3 Control Objective The control objective in this paper is to navigate a non-holonomic system (e.g., a wheeled mobile robot) along a collision-free path to a constant, goal position and orientation, denoted by , in an obstacle cluttered environment with known obstacles. Specifically, the objective is to control the non-holonomic system along a path from an initial position and orientation to q D, where D denotes a free configuration space. The free configuration space D is a subset of the whole configuration space with all configurations removed that involve a collision with an obstacle. To quantify the path planning-based control objective, the difference between the actual Cartesian position and orientation and the desired position and orientation, denoted by, is defined as as followswhere the desired trajectory is designed so that qd(t) q. Motived by the navigation function approach in 16, a navigation-like function is utilized to generate the desired path qd(t). Specifically, the navigation-like function used in this paper is defined as followsDefinition 1 Let D be a compact connected analytic manifold with boundary, and let q be a goal point in the interior of D. The navigation-like function (q): D 0, 1, is a function satisfies the following properties:1. (q(t) is first order and second order differentiable (i.e., and exist on D).2. (q(t) obtains its maximum value on the boundary of D.3. (q(t) has unique global minimum at q (t) = q.4. If with z, r R being known positive constants.5. If (q(t) is ultimately bounded by , then is ultimately bounded by r with R being some known positive constant.4 Online 3D Path Planner4.1 Trajectory PlanningThe 3D desired trajectory can be generated online as follows where (q) R denotes a navigation-like function defined in Definition 1, denotes the gradient vector of (q), and is an additional control term to be designed. Assumption The navigation-like function defined in Definition 1 along with the desired trajectory generated by (6) ensures an auxiliary terms N () R3, defined assatisfy the following inequalitywhere the positive function () is nondecreasing in and . The inequality given by (8) will be used in the subsequent stability analysis.4.2 Model TransformationTo achieve the control objective, a controller must be designed to track the desired trajectory developed in (6) and stop at the goal position q. To this end, the unified tracking and regulation controller presented in 7 can be used. To develop the controller in 7, the open-loop error system defined in (5) must be transformed into a suitable form. Specifically, the position and orientation tracking error signals defined in (5) are related to the auxiliary tracking error variables w(t) R and through the following global invertible transformation 8After taking the time derivative of (9) and using (1)-(5) and (9), the tracking error dynamics can be expressed in terms of the auxiliary variables defined in (9) as follows 8wheredenotes a skew-symmetric matrix defined asand is defined asThe auxiliary control inputintroduced in (10) is defined in terms of and as follows 4.3 Control DevelopmentTo facilitate the control development, an auxiliary error signal, denoted by, is defined as the difference between the subsequently designed dynamic oscillator-like signal and the transformed variable z(t), defined in (9), as followsBased on the open-loop kinematic system given in (10) and the subsequent stability analysis, we design u(t) as follows 7where k2 R is a positive, constant control gain. The auxiliary control term introduced in (15) is defined aswhere the auxiliary signal zd(t) is defined by the following differential equation and initial conditionThe auxiliary terms 1 (w, f, t) R and d(t) R are defined asandrespectively, k1, 0, 1, 1 R are positive, constant control gains, andwas defined in (12). As described in 8, motivation for the structure of (17) and (19) is based on the fact thatBased on (9), e (t) can be expressed in terms of,and zd (t) as followswhere are defined as follows Motivated by the subsequent stability analysis, the additional control term vr (t) in (6) is designed as followswhere k3, k4 R denotes positive, constant control gains, and the positive functions 1 (zd1, z1, qd, e),2 (zd1, z1, qd, e) R are defined as follows4.4 Closed-loop Error SystemAfter substituting (15) into (10), the dynamics for w(t) can be obtained as followswhere (14) and the properties of J in (11) were utilized. After substituting (16) into (26) for only the second occurrence of ua(t), utilizing (20) and the properties of J in (11), the final expression for the closed-loop error system for w(t) can be obtained as followsTo determine the closed-loop error system for, we take the time derivative of (14) and then substitute (10) and (17) into the resulting expression to obtain the following expressionAfter substituting (15) and (16) into (28), (28) can be rewritten as followsAfter substituting (18) into (29) for only the second occurrence of 1 (t) and then canceling common terms, the following expression can be obtainedSince the bracketed term in (30) is equal to ua (t) defined in (16), the final expression for the closed-loop error system for can be obtained as followsRemark 1 Based on the fact that d (t) of (19) exponentially approaches an arbitrarily small constant, the potential singularities in (16), (17), and (18) are always avoided. 4.5 Stability AnalysisTheorem 1 Provided qd (0) D, the desired trajectory generated by (6) along with the additional control term vr (t) designed in (24) ensures that and. where r is defined in Definition 1.Proof: Let V (t) R denote the following functionwhere k R is a positive constant, V1 (t) R denotes the following functionand V2 (qd) : D R denotes a function as followsAfter taking the time derivative of (33) and then substituting (27) and (31) into the resulting expression and cancelling common terms, the following expression can be obtainedAfter taking the time derivative of (34) and utilizing (6), the following expression can be obtainedwhere N () is defined in (7). Based on (8), V2 (t) can be upper bounded as followsAfter substituting (21) into (37), the following inequality can be obtainedwhere the vectoris defined as follows and the positive function 1 (zd1, z1, qd, e) and2 (zd1, z1, qd, e) are defined in (25). After substituting (24) into (38), V2 (t) can be rewritten as follows Based on (35) and (40), the time derivative of V (t) in (32) can be upper bounded by the following inequalitywhere the positive constantare defined as followsCase 1: If , from the Property 4 in Definition 1, it is clear thatCase 2: If , it is clear from (32), (33), (34), and (41) thatwhere andare positive constants. Based on (42), V (t) can be upper bounded as followsthereforeBased on (32), (34), and (44), it is clear thatIf qd (0) is not on the boundary of D, (qd (0) 1. Then k can be adjusted to ensureBased on (45) and (46), (qd (t) 1, hence qd (t) D from Definition 1. It is clearly from (43) that (qd) is ultimately bounded by z. Therefore, if, k4 can be adjusted to ensure, where is defined in Definition 1. Hence by the Property 5 in Definition 1, is ultimately bounded by r.Theorem 2 The kinematic control law given in (15)-(19) ensures global uniformly ultimately bounded (GUUB) position and orientation tracking in the sense thatwhere 1 was given in (19), , and 3 and 0 are positive constants. Proof: Based on (33) and (35), V1 (t) of (35) can be upper bounded as followsBased on (48), the following inequality can be obtained Based on (33), (49) can be rewritten as follows where the vector 1 (t) is defined in (39). From (33) and (49), it is clear that w (t) ,L. Based on (19) and (20), we can conclude that zd (t) L. From (14) and, zd (t) L, it is clear that z (t) L. Since w (t), z (t) L, based on the inverse transformation from (9), e (t) L. Based on qd (t) L from Theorem 1 and e (t) L, it is clear that q (t) L. From (22)-(25), qd (t), zd (t), z (t), e (t) L, and the properties in Definition 1, we can conclude that vr (t), qd (t) L. Based on (12) and q (t), z (t), qd (t) L, f (, z2, qd) L. Then 1 (t) L from (18). Then u (t), ua (t), zd (t) L from (15)-(17). Based on the fact that f (, z2, qd), z (t), u (t) L, then (10) can be used to conclude w (t), z (t) L. It is clear from z(t) , zd (t) L that L. Then standard signal chasing arguments can be employed to conclude that all of the remaining signals in the control and the system remain bounded during closed-loop operation.Based on (19), (20), (39), and (50), the triangle inequality can be applied to (14) to prove thatUtilizing (50)-(51), the result given in (47) can be obtained from taking the inverse of the transformation given in (9). Remark 2 Although qd (t) is a collision-free path, the stability result in Theorem 2 only ensures practical tracking of the path in the sense that the actual WMR trajectory is only guaranteed to remain in a neighborhood of the desired path. From (5) and (47), the following bound can be developedwhere qd (t) D based on the proof for Theorem 1. To ensure that q (t) D, the free configuration space needs to be reduced to incorporate the effects of the second and third terms on the right hand side of (52). To this end, the size of the obstacles could be increased by, where 31 can be made arbitrarily small by adjusting the control gains. To minimize the effects of 2, the initial conditions w (0) and z (0) (and hence, could be required to be sufficiently small enough to yield a feasible path to the goal.5 Online 2D NavigationIn the previous approach, the size of the obstacles is required to be increased due to the fact that the navigation-like function is formulated in terms of the desired trajectory. In the following approach, the navigation function proposed in 22 is formulated based on current position feedback, and hence, q (t) can be proven to be a member of D without placing restrictions on the initial conditions.5.1 Trajectory PlanningLet (xc, yc) R denote a 2D position-based navigation function defined in D that is generated online, where the gradient vector of (xc, yc) is defined as followsLet d (xc, yc) R denote a desired orientation that is defined as a function of the negated gradient of the 2D navigation function as followswhere arctan 2 () : R2 R denotes the four quadrant inverse tangent function 26, where d (t) is confined to the following regionAs stated in 21, by defining , then d(t) remains continuous along any approaching direction to the goal position. See Appendix for an expression for d(t) based on the previous continuous definition for d(t).Remark 3 As discussed in 22, the construction of the function (q(t), coined a navigation function, that satisfies the first three properties in Definition 1 for a general obstacle avoidance problem is nontrivial. Indeed, for a typical obstacle avoidance, it does not seem possible to construct (q(t) such that only at q (t) = q. That is, as discussed in 22, the appearance of interior saddle points (i.e., unstable equilibria) seems to be unavoidable; however, these unstable equilibria do not really cause any difficulty in practice. That is, (q(t) can be constructed as shown in 22 such that only a .few. initial conditions will actually get stuck on the unstable equilibria. 5.2 Control DevelopmentBased on the open-loop system introduced in (1)-(4) and the subsequent stability analysis, the linear velocity control input vc (t) is designed as follows where kv R denotes a positive, constant control gain, and was introduced in (5). After substituting (55) into (1), the following closed-loop system can be obtainedThe open-loop orientation tracking error system can be obtained by taking the time derivative of in (5) as follows where (1) was utilized. Based on (57), the angular velocity control input c (t) is designed as followswhere k R denotes a positive, constant control gain, and d(t) denotes the time derivative of the desired orientation. See Appendix for an explicit expression ford (t). After substituting (58) into (57), the closed-loop orientation tracking error system is given by the following linear relationshipLinear analysis techniques can be used to solve (59) as followsAfter substituting (60) into (56) the following closed-loop error system can be determined5.3 Stability AnalysisTheorem 3 The control input designed in (55) and (58) along with the navigation function ensure asymptotic navigation in the sense thatProof: Let: D R denote the following non-negative function After taking the time derivative of (63) and utilizing (1), (53), and (56), the following expression can be obtainedBased on the development provided in Appendix, the gradient of the navigation function can be expressed as followsAfter substituting (65) into (64), the following expression can be obtainedAfter utilizing a trigonometric identity, (66) can be rewritten as followswhere g(t) R denotes the following positive function Based on (53) and the property of the navigation function (Similar to the Property 1 of Definition 1), it is clear that on D; hence, (55) can be used to conclude that vc (t) L on D. Development is also provided in the Appendix that proves d (t) L on D; hence, (58) can be used to show that c (t) L on D. Based on the fact that vc (t) L on D, (1)-(4) can be used to prove that xc (t), yc (t) L on D. After taking the time derivative of (53) the following expression can be obtainedSince xc (t), yc (t) L on D, and since each element of the Hessian matrix in (69) is bounded by the property of the navigation function (Similar to the Property 1 ofDefinition 1), it is clear that g(t) L on D. Based on (63), (67), (68), and the fact that g(t) L on D, then Lemma A.6 of 6 can be invoked to prove thatin the region D. Based on the fact that1 from (60), then (70) can be used to prove that. Therefore the result in (62) can be obtained based on the analysis in Remark 3. Remark 4 The control development in this section is based on a 2D position navigation function. To achieve the objective, a desired orientation d (t) was defined as a function of the negated gradient of the 2D navigation function. The previous development can be used to prove the result in (62). If a navigation function can be found such that, then asymptotic navigation can be achieved by the controller in (55) and (58); otherwise, a standard regulation controller (e.g., see 8 for several candidates) could be implemented to regulate the orientation of the WMR from. Alternatively, a dipolar potential field approach 23, 24 or a virtual obstacle 9 could be utilized to align the gradient field of the navigation function to the goal orientation of the WMR.6 Simulation ResultsTo illustrate the performance of the controller given in (55) and (58), a numerical simulation was performed to navigate the WMR from to.Since the properties of a navigation function are invariant under a diffeomorphism, a diffeomorphism is developed to map the WMR free configuration space to a model space 17. Specifically, a positive function was chosen as followswhere is positive integer parameter, and the boundary function and the obstacle functionare defined as followsIn (72), and are the centers of the boundary and the obstacle respectively, r0, r1 R are the radii of the boundary and the obstacle respectively. From (71) and (72), it is clear that the model space is a unit c
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