尋線搬運(yùn)機(jī)器人模型及其控制系統(tǒng)設(shè)計(jì)含開(kāi)題及6張CAD圖,搬運(yùn),機(jī)器人,模型,及其,控制系統(tǒng),設(shè)計(jì),開(kāi)題,cad Designing approach on trajectory-tracking control of mobile robot Abstract Based on differential geometry theory, applying the dynamic extension approach of relative degree, the exact feedback linearization on the kinematic error model of mobile robot is realized. The trajectory-tracking controllers are designed by pole assignment approach. When angle speed of mobile robot is permanently nonzero, the local asymptotically stable controller is designed. When angle speed of mobile robot is not permanently nonzero, the trajectory-tracking control strategy with globally tracking bound is given. The algorithm is simple and applied easily. Simulation results show their effectiveness. Keywords: Trajectory tracking; Dynamic extension approach; Exact feedback linearization; Globally tracking bound 1. Introduction Recently, interest in the tracking control of mobile robots has increased with various theoretical and practical contributions being made. Particularly, feedback linearization has attracted a great deal of research interest in recent nonlinear control theory, and some techniques have been employed in mobile robot control Path tracking problems of several types of mobile robots have been investigated by means of linearizing the static and dynamic state feedback in [1]. The local and global tracking problems via time-varying state feedback based on the back stepping technique have been addressed in [2]. Since the wheel-driven mobile robot has nonholonomic constraints that arise from constraining the wheels of the mobile robot to roll without slipping and the linearized mobile robot with nonholonomic constraints has a controllability deficiency, it is difficult to control them. The point stabilization problem can be regarded as the generation of control inputs to drive the robot from any initial point to target point. The crucial problem in this stabilization question centers on the fact that the mobile robot model does not meet Brockett’s well-known necessary smooth feedback stabilization condition, so the mobile robot cannot be stabilized with smooth state feedback, which leads to the limitation in application. Therefore some discrete time-invariant controllers, time-varying controllers and hybrid controllers based on Lyapunov control theories have been proposed in [4]. The global trajectory-tracking problem to reference mobile robot is discussed based on the back stepping technique in [5]. The trajectory-tracking problem to reference mobile robot is discussed based on the terminal sliding-mode technique in [6], but it requires the nonzero speed of rotation. Point stabilization of mobile robot via state-space exact feedback linearization based on dynamic extension approach is proposed in [7]. The point stabilization problem in polar frame can be exactly transformed into the problem of controlling a linear time-invariant system. But its disadvantage is to require the verification of the complex involution. And the point stabilization problem is only discussed but the trajectory tracking is not solved. In the present paper, the trajectory tracking to reference mobile robot as [5] and [6] is addressed based on dynamic extension approach in [7]. The exact feedback linearization on the kinematic error model of mobile robot is realized. Its proof is simple and different from [7] since the complex process of verifying involution is avoided. By linearization, the nonlinear system is transferred to linear time-invariance system, which is equivalent to two reduced-order linear time-invariance systems that can be controlled easily. If angle speed of mobile robot is permanently nonzero, the local asymptotically stable controller is designed. If angle speed of mobile robot is not permanently nonzero, the trajectory-tracking control strategy with globally tracking bound is given. The algorithm is simple and applied easily. 2. Preliminaries and problem formulation Consider a class of nonlinear systems described as Definition (Slotine and Li [8] and Feng and Fei[9].) Given X is an n-dimension differentiable manifold if there exists a neighborhood V of x0 and integer vector er1; r2;y; rmT such that is nonsingular 8xAV; we say that system (1)–(2) has vector relative degree er1; at point x0: Lemma (Feng and Fei [9]). The necessary and sufficient condition of exact feedback linearization at x0 for system (1) is that there exists a neighborhood V of x0 and smooth real-valued functionssuch that system (1)–(2) has vector relativeedegreeat the point, The kinematic model of wheel-driven mobile robot as follows: where (x; y) is the position of mobile robot and y is the heading angle. The control variables of mobile robot are the linear velocity v and the angular velocity o: Here, the trajectory-tracking problem is to track reference mobile robot with the known posture yr; yrT and velocities vr; as shown in Fig. 1. We have the posture error equation of mobile robot [5,6] Hence we have the posture error difference equations [5,6] From above analysis, the trajectory-tracking problem to reference mobile robot can be stated as: find the bounded inputs v and o so that for an arbitrary initial error, the state of system (5) can be held near the origin ; i.e. 3. Design of trajectory-tracking controllers It is obvious that system (5) cannot be state-space exact feedback linearization. It cannot also be partial input/output feedback linearization by choosing the outputs y1 = ye; y2 = ye: The reason is that system (5) has not the relative degree. Actually It is obvious that the decoupling matrix is singular. Proof. We differentiate the output equations y1=x2; y2=x3 then we have From (11) and (13), we have the decoupling matrix From (14) we have Hence under outputs y1=x2; y2=x3; and angle velocity oa0; system (9) has the relative degree er1; and: Using lemma in Section 2, there exists the local diffeomorphism so that system (9) can be linearized exactly. The local diffeomorphism and transformed states are defined as follows: the input transformations are defined as follows: Using (10)–(13), (16) and (17), system (9) can be transformed into a linear time-invariant system: where is the new state vector. is the new control input. The nonlinear system (9) is transformed into the linear time-invariant system (18)–(19) by the state and input transformations (16)–(17). For the linear timeinvariant system (18)–(19), we can apply the known linear control method such as pole-assignment method to implement control, hence we have the following Theorem 2. & Theorem 2. Assuming angle velocity oa; when system (9) is controlled by controller (20a), it has the local asymptotic stability. If oa0 cannot be satisfied, system (9) is controlled by controllers (20a) and (20b) alternately, it has the globally bounded tracking to reference mobile robot with the known posture ; and velocities vr; or. where e is a given arbitrary small positive number. The new control inputs where .i=1,2;are the parameters that A RTICLE IN PRESS make the matrix stable are, respectively, Proof. Under angle velocity oa0; from Theorem 1, system (9) can be transformed into linear time-invariant system (18)–(19). It is clear that system (18)–(19) is completely controllable and completely observable. It contains two reduce-order independent subsystems It is well known that linear time-invariant systems (22) can be well controlled via pole-assignment approach. If angle velocity oa0 cannot be satisfied, to guarantee the realization of control, we choose the suitable small positive number e that can be specified artificially according to the need of practice, so that when jojXe; controller (20a) is used, when jojoe; controller (20b) is used. Hence controllers (20a) and (20b) are used alternately, which can guarantee the bounded tracking to reference mobile robot with the known posture and velocities vr and or: From (20) we see that the control inputs v and o’ are bounded as long as vr and o’r are bounded. 4. Simulation research We implement the simulation research to verify the effectiveness of tracking controllers (20a) and (20b). In the simulation, the parameters such as the initial conditions, desired velocities and feedback gains are listed in Table 1. We choose the same feedback gain for two time-invariant systems (22a) and (22b) Figs. 2–5 show the responses of the trajectory tracking control of mobile robot. Figs. 2 and 3 show the simulation results that mobile robot follows the straight lines, where controllers (20a) and (20b) are used alternately in order to guarantee the bounded tracking to reference mobile robot with or=0: From Figs. 2 and 3 we see that the performance of tracking is good and bounded; though xe has a bias from zero before t =5 s, it approximates near zero after t = 5 s. Figs. 4 and 5 show the simulation results that mobile robot follows the curves. From Figs. 4 and 5 we see that the performance of tracking is better. And all controllers are bounded, which guarantees the realization of control. ART ICLE IN PRESS 5. Conclusion In practice, to make the mobile robot obtain some postures and velocities, we can assume a reference mobile robot with these postures and velocities, and consider the trajectory-tracking problem to reference mobile robot. In this paper, the trajectory-tracking problem to reference mobile robot is addressed based on dynamic extension approach. The exact feedback linearization on the kinematic error model of mobile robot is realized. The nonlinear system is transferred to two reduced-order linear time-invariance systems that can be controlled easily. The following control is realized, i.e. if angle speed of the mobile robot is permanently nonzero, the local asymptotically stable controller is designed. If angle speed of the mobile robot is not permanently nonzero, the trajectory-trackingcontrol strategy with globally tracking bound is given. The approaches are simple and efficient. Acknowledgements The author is grateful to the associate editor and referees for the valuable comments and suggestions. 一、畢業(yè)設(shè)計(jì)（論文）的內(nèi)容 1、收集移動(dòng)機(jī)器人相關(guān)資料； 2、深入學(xué)習(xí)電動(dòng)機(jī)的單片機(jī)控制技術(shù)； 3、設(shè)計(jì)并制作尋線搬運(yùn)機(jī)器人模型及其搬運(yùn)引導(dǎo)系統(tǒng)； 4、設(shè)計(jì)移動(dòng)機(jī)器人的控制電路和程序； 5、驗(yàn)證設(shè)計(jì)結(jié)果； 6、撰寫(xiě)畢業(yè)設(shè)計(jì)說(shuō)明書(shū)、翻譯英文資料、繪制機(jī)械圖。 二、畢業(yè)設(shè)計(jì)（論文）的要求與數(shù)據(jù) 1、移動(dòng)機(jī)器人搬運(yùn)路線的彎道半徑不大于 300mm，移動(dòng)機(jī)器人的運(yùn)動(dòng)線路示意圖、移動(dòng)機(jī)器人模型的結(jié)構(gòu)圖采用 AutoCAD 或 CAXA 軟件繪制并打印； 2、移動(dòng)機(jī)器人由單片機(jī)控制，采用電池供電，能自動(dòng)前后尋線運(yùn)動(dòng)搬運(yùn)物體， 當(dāng)沒(méi)有需要搬運(yùn)的物體時(shí)自動(dòng)等待，當(dāng)運(yùn)動(dòng)線路的前方 300mm 處有障礙物時(shí)停止運(yùn)動(dòng)并報(bào)警，障礙消失后自動(dòng)繼續(xù)運(yùn)動(dòng)； 3、采用 Protel99se 軟件繪制和打印系統(tǒng)的電路原理圖，電路原理圖標(biāo)明全部零部件和元器件的型號(hào)或主要參數(shù)； 4、采用 C51 語(yǔ)言設(shè)計(jì)控制程序，控制程序同時(shí)用程序流程框圖和 C 代碼表示， 程序流程框圖盡量細(xì)化并用計(jì)算機(jī)繪制打印，C 代碼添加詳細(xì)的中文注釋； 5、畢業(yè)設(shè)計(jì)說(shuō)明書(shū) 2 萬(wàn)字左右，主要內(nèi)容包括移動(dòng)機(jī)器人概述、移動(dòng)機(jī)器人運(yùn)動(dòng)引導(dǎo)系統(tǒng)設(shè)計(jì)、移動(dòng)機(jī)器人模型結(jié)構(gòu)設(shè)計(jì)、控制系統(tǒng)電路設(shè)計(jì)、控制程序設(shè)計(jì)、以及控制效果報(bào)告，設(shè)計(jì)說(shuō)明書(shū)包括 300～500 個(gè)單詞的英文摘要； 6、翻譯與設(shè)計(jì)題目相關(guān)的英文資料，翻譯量不少于 4 萬(wàn)字符，同時(shí)附英文資料原文，翻譯資料的內(nèi)容由指導(dǎo)教師另外指定； 7、機(jī)械圖的繪圖量折合 A0 幅面 1 張，用 AutoCAD 或 CAXA 軟件繪制并打印。 三、畢業(yè)設(shè)計(jì)（論文）應(yīng)完成的工作 1、畢業(yè)設(shè)計(jì)說(shuō)明書(shū)一份； 2、英文資料翻譯一份； 3、運(yùn)動(dòng)線路示意圖、機(jī)械圖、電路圖、程序流程框圖一套； 4、移動(dòng)機(jī)器人模型一個(gè)； 5、畢業(yè)設(shè)計(jì)電子文檔一份。 四、應(yīng)收集的資料及主要參考文獻(xiàn) [1] 郭丙華, 胡躍明. 移動(dòng)機(jī)器人計(jì)算機(jī)控制[J]. 計(jì)算機(jī)測(cè)量與控制, 2003, 11(10):749～750,766. [2] 關(guān)健生. 基于單片機(jī)的聲音導(dǎo)航定位系統(tǒng)的設(shè)計(jì)[J]. 信息系統(tǒng)工程, 2009, 28(9):86～89. [3] 霍成立, 謝凡, 秦世引. 面向室內(nèi)移動(dòng)機(jī)器人的無(wú)跡濾波實(shí)時(shí)導(dǎo)航方法 [J]. 智能系統(tǒng)學(xué)報(bào), 2009, 4(4):295～302. [4] 馬兆青, 袁曾任. 基于柵格方法的移動(dòng)機(jī)器人實(shí)時(shí)導(dǎo)航和避障[J]. 機(jī)器人, 1996, 18(6):344～348. [5] 姜志兵, 趙英凱. 基于虛力柵格法的移動(dòng)機(jī)器人實(shí)時(shí)避障和導(dǎo)航[J]. 機(jī)床與液壓, 2007, 35(05):91～93. [6] 徐國(guó)華, 譚民. 移動(dòng)機(jī)器人的發(fā)展現(xiàn)狀及其趨勢(shì)[J]. 機(jī)器人技術(shù)與應(yīng)用, 2001, (3):7～14. [7] 張明路, 丁承君, 段萍. 移動(dòng)機(jī)器人的研究現(xiàn)狀與趨勢(shì)[J]. 河北工業(yè)大學(xué)學(xué)報(bào), 2004, 33(2):110～115. [8] Soonshin Han, ByoungSuk Choi, JangMyung Lee. A precise curved motion planning for a differential driving mobile robot[J]. Mechatronics, 2008, 18(9):486～494. [9] Felipe N. Martins, Wanderley C. Celeste, Ricardo Carelli, Mário Sarcinelli-Filho, Teodiano F. Bastos-Filho. An adaptive dynamic controller for autonomous mobile robot trajectory tracking[J]. Control Engineering Practice, 2008, 16(11):1354～1363. [10] Shuli Sun. Designing approach on trajectory-tracking control of mobile robot[J]. Robotics and Computer-Integrated Manufacturing, 2005, 21(1):81～85. 五、試驗(yàn)、測(cè)試、試制加工所需主要儀器設(shè)備及條件 計(jì)算機(jī) 任務(wù)下達(dá)時(shí)間： 2010 年 1 月 12 日 畢業(yè)設(shè)計(jì)開(kāi)始與完成時(shí)間： 2010 年 3 月 1 日至 2010 年 6 月 20 日 組織實(shí)施單位： 教研室主任意見(jiàn)： 簽字： 2010 年 1 月 8 日 院領(lǐng)導(dǎo)小組意見(jiàn)： 簽字： 2010 年 1 月 11 日 1．畢業(yè)設(shè)計(jì)的主要內(nèi)容、重點(diǎn)和難點(diǎn)等 一、畢業(yè)設(shè)計(jì)（論文）的內(nèi)容： 收集移動(dòng)機(jī)器人相關(guān)資料；深入學(xué)習(xí)電動(dòng)機(jī)的單片機(jī)控制技術(shù)；設(shè)計(jì)并制作尋線搬運(yùn)機(jī)器人模型及其搬運(yùn)引導(dǎo)系統(tǒng)；設(shè)計(jì)移動(dòng)機(jī)器人的控制電路和程序；驗(yàn)證設(shè)計(jì)結(jié)果；撰寫(xiě)畢業(yè)設(shè)計(jì)說(shuō)明書(shū)、翻譯英文資料、繪制機(jī)械圖。 二、重點(diǎn)是：電動(dòng)機(jī)的單片機(jī)控制技術(shù)，尋線搬運(yùn)機(jī)器人模型的設(shè)計(jì)和制作，移動(dòng)機(jī)器人運(yùn)動(dòng)引導(dǎo)系統(tǒng)設(shè)計(jì)、移動(dòng)機(jī)器人模型結(jié)構(gòu)設(shè)計(jì)、控制系統(tǒng)電路設(shè)計(jì)、控制程序設(shè)計(jì)。 三、難點(diǎn)是：巡線機(jī)器人控制電路及傳感器信號(hào)處理電路的設(shè)計(jì)與實(shí)現(xiàn)，控制系統(tǒng)電路設(shè)計(jì)、控制程序設(shè)計(jì)。 2．準(zhǔn)備情況（查閱過(guò)的文獻(xiàn)資料及調(diào)研情況、現(xiàn)有設(shè)備、實(shí)驗(yàn)條件等） 一、文獻(xiàn)資料： [1] 郭丙華, 胡躍明. 移動(dòng)機(jī)器人計(jì)算機(jī)控制[J]. 計(jì)算機(jī)測(cè)量與控制, 2003. [2] 關(guān)健生. 基于單片機(jī)的聲音導(dǎo)航定位系統(tǒng)的設(shè)計(jì)[J]. 信息系統(tǒng)工程, 2009. [3] 霍成立, 謝凡, 秦世引. 面向室內(nèi)移動(dòng)機(jī)器人的無(wú)跡濾波實(shí)時(shí)導(dǎo)航方法[J]. 智能系統(tǒng)學(xué)報(bào), 2009. [4] 馬兆青. 基于柵格方法的移動(dòng)機(jī)器人實(shí)時(shí)導(dǎo)航和避障[J]. 機(jī)器人, 1996. [5] 姜志兵. 基于虛力柵格法的移動(dòng)機(jī)器人實(shí)時(shí)避障和導(dǎo)航[J]. 機(jī)床與液壓. [6] 徐國(guó)華, 譚民. 移動(dòng)機(jī)器人的發(fā)展現(xiàn)狀及其趨勢(shì).機(jī)器人技術(shù)與應(yīng)用, 2001. [7] [美]David Cook 機(jī)器人制作入門篇，北京航天航空大學(xué)出版社，2005. [8] 黃賢武，鄭莜霞，傳感器原理及應(yīng)用，2004.3. [9] 張毅，羅元，移動(dòng)機(jī)器人技術(shù)及其應(yīng)用，2007.9. [10][美] Gordon McComb，Myke Predko，機(jī)器人設(shè)計(jì)與實(shí)現(xiàn)，2007. 二、實(shí)驗(yàn)設(shè)備及條件： 實(shí)驗(yàn)室提供的通用儀器及自己購(gòu)買的一些電子元器件。 3．實(shí)施方案、進(jìn)度實(shí)施計(jì)劃及預(yù)期提交的畢業(yè)設(shè)計(jì)資料 一、實(shí)施方案: 收集移動(dòng)機(jī)器人相關(guān)資料，深入學(xué)習(xí)電動(dòng)機(jī)的單片機(jī)控制技術(shù)，制作尋線搬運(yùn)機(jī)器人模型及其搬運(yùn)引導(dǎo)系統(tǒng)，設(shè)計(jì)移動(dòng)機(jī)器人的控制電路和程序，驗(yàn)證設(shè)計(jì)結(jié)果。 二、進(jìn)度實(shí)施計(jì)劃： 3月至4月初搜集文獻(xiàn)及資料，深入對(duì)其控制原理和控制方法的理解，掌握各種電子元器件的原理及應(yīng)用，對(duì)電路設(shè)計(jì)和控制部分重點(diǎn)學(xué)習(xí)。4月初至5月初設(shè)計(jì)和制作的尋線搬運(yùn)機(jī)器人模型，設(shè)計(jì)運(yùn)動(dòng)引導(dǎo)系統(tǒng)、設(shè)計(jì)模型結(jié)構(gòu)、設(shè)計(jì)控制系統(tǒng)電路、設(shè)計(jì)控制程序，制作模型及其搬運(yùn)引導(dǎo)系統(tǒng)。5月調(diào)試系統(tǒng)；撰寫(xiě)說(shuō)明書(shū)、繪制機(jī)械圖。 三、應(yīng)提交的畢業(yè)設(shè)計(jì)資料： 畢業(yè)設(shè)計(jì)說(shuō)明書(shū)；英文資料翻譯；運(yùn)動(dòng)線路示意圖、機(jī)械圖、電路圖、程序流程框圖；移動(dòng)機(jī)器人模型；畢業(yè)設(shè)計(jì)電子文檔一份。 指導(dǎo)教師意見(jiàn) 該生對(duì)設(shè)計(jì)任務(wù)和設(shè)計(jì)要求已基本明確，進(jìn)度計(jì)劃安排基本可行，準(zhǔn)備工作已基本就緒，同意按計(jì)劃展開(kāi)設(shè)計(jì)工作。還應(yīng)進(jìn)一步分析資料，把握設(shè)計(jì)重點(diǎn)，抓緊制作機(jī)器人模型，以便后續(xù)電路和程序的設(shè)計(jì)與調(diào)試。要特別注意控制程序的設(shè)計(jì)方法。 指導(dǎo)教師（簽字）： 2010年3月5日 開(kāi)題小組意見(jiàn) 開(kāi)題小組組長(zhǎng)（簽字）： 2010年3月　　日 院（系、部）意見(jiàn) 主管院長(zhǎng)（系、部主任）簽字： 2010年3月　　日 - 3 - A precise curved motion planning for a differential driving mobile robot abstract A single curvature trajectory with a fixed rotation radius is proposed for an optimal trajectory plan for a differential driving mobile robot to capture a moving object. Generally, when the differential driving mobile robot moves along a path whose rotation radius is not constant, the tracking error of the mobile robot increases with the travel distance. Also, tracking errors increase greatly when the mobile robot follows a trajectory, with a small rotation radius. Based on these two observations, a single curvature trajectory, which has a constant and large rotation radius, is proposed as an optimal trajectory, in order to minimize the tracking error of the differential driving mobile robot. This paper first reviews the characteristics of a single curvature trajectory. Next, an algorithm to capture a moving object precisely is proposed using the single curvature trajectory. With the pre-determined initial states (i.e., position and orientation of the mobile robot and the final states), the mobile robot is made to capture a moving object. Through simulations and real experiments using a two DOF wheel-based mobile robot, the effectiveness of the proposed algorithm is verified. 1. Introduction Research on mobile robots can be globally classified into three categories: path planning, position estimation, and driving control. The trajectory planning of a mobile robot aims at providing an optimal path from an initial position to a target position. Optimal trajectory planning for a mobile robot provides a path, which has minimal tracking error and the shortest driving time and distance. Tracking errors of mobile robots cause collisions with obstacles due to deviations from the planned path and also cause the robot to fail to accomplish the mission successfully. Tracking error can be reduced through feedback control. However, this requires excessive control efforts due to high control gains. Tracking errors also cause an increase of traveling time, as well as travel distance, due to the additional adjustments needed to satisfy the driving states. Therefore, trajectory planning to decrease tracking error is very important and needs to be handled carefully. One of the major reasons for tracking error is the discontinuity of the rotation radius on the path of the differential driving mobile robot. The rotation radius changes at the connecting point of the straight line route and curved route, or at a point of inflection. At these points, it can be easy for the differential driving mobile robot to secede from its determined orbit due to the rapid change of direction. Therefore, in order to decrease tracking error, the trajectory of the mobile robot must be planned so that the rotation radius is maintained at a constant , if possible. The increase of tracking error due to the small rotation radius interferes with the accurate driving of the mobile robot. The path of the mobile robot can be divided into curved and straight-line segments, globally. While tracking error is not generated in the straight-line segment, significant error is produced in the curved segment due to centrifugal and centripetal forces, which cause the robot to slide over the surface. Also, tracking error increases when the rotation radius is small. In fact, the straight line segment can be considered as a curved segment whose rotation radius is infinity. As the tracking error becomes larger at the curved segment, the possibility of a tracking error increases with the decrease of the rotation radius of the curved path. Note that a relatively small error occurs at the straight-line path. Therefore, it is important to maintain simultaneously a large and constant rotation radius in order to decrease the tracking error of the differential driving mobile robot. This paper proposes a single-curvature trajectory, which has a constant and large rotation radius. In view of the size and discontinuity of the rotation radius,a single-curvature trajectory and a double-curvature trajectory are compared to show that the tracking error increases with the decrease of the rotation radius. Through the tracking experiments along each trajectory, it is proved that a single-curvature trajectory has the least tracking error. With precise trajectory planning, an algorithm for the mobile robot to capture a moving object is proposed. With the pre-specified initial position and orientation of a mobile robot and the final states, and assuming that the velocity of the moving object is pre-estimated, an optimal capturing path of a mobile robot is produced as a result of this research. In Section 2, the driving characteristics of a mobile robot are analyzed using a kinematics analysis while the mobile robot is making curved motions. The single-curvature trajectory and double-curvature trajectory are described in detail in Section3. In Section4, a curved trajectory formation algorithm, according to a single-curvature trajectory, is introduced with theoretical formulas. Section 5 shows a comparison between a single-curvature and a double-curvature trajectory, with respect to tracking error,in real capturing experiments. Section 6 concludes this paper and provides an agenda for future work. 2. Moving characteristics of a differential driving mobile robot To form a trajectory for a differential driving mobile robot, a kinematics analysis first needs to be conducted. Based on the kinematics analysis, the driving characteristics of a mobile robot with a differential driving mechanism are modeled, which provides the theoretical basis for trajectory planning. 2.1. Kinematics analysis of the mobile robot As shown in Fig. 1a, a mobile robot with a differential driving mechanism has two wheels on the same axis, and each wheel is controlled by an independent motor. Let us define vL as the velocity of the left wheel, vR as that of the right wheel, and l as the distance between the two wheels. The robot’s motion can be determined by the two wheel velocities, vL and vR, and the linear and angular velocities, vl and , of the mobile robot can be described in terms of vL and vR as follows : A kinematics model of a mobile robot with a differential driving mechanism can be described as shown in Fig. 1b. In two-dimensional X _ Y Cartesian coordinates, the position of the mobile robot is described by (t) and (t), whereas the orientation is represented as (t). Then, and represent the linear velocities, whereas represents the angular velocity. The velocity vector of the mobile robot is defined as Now, the kinematics model of the mobile robot can be represented as 2.2. Driving principle of the mobile robot Through the kinematics analysis, it can be recognized that the motion states of a mobile robot with a differential driving mechanism change according to the velocities of the two wheels. When a robot with multiple wheels rotates about a rotation center instantaneously, this rotation center is defined as the ICC (Instantaneous Center for Curvature). As shown in Fig. 2a, the ICC is located at the cross-section point of the extension-lines of the wheel centers. For a mobile robot with a differential driving mechanism, as shown in Fig. 2b, the ICC can be located at any point on the wheel axis, since the two wheel axes are on the same line. In this case, the ICC will be determined by the velocity ratio between the two wheels. Fig. 3 illustrates the ICC along with the robot’s motion and position. There is a proportional relationship between the wheel velocities and the distance from the wheel to the ICC, which is represented as In addition, Eq. (5) can be simply represented as： Note that the rotation radius of the mobile robot is determined by the values of the left and right wheel velocities. When the robot is following a straight line, R =1 and vR = vL. When vR != vL, the robot follows a curved trajectory of a certain rotation radius. Therefore, the velocity and acceleration of the robot is changed if the rotation radius is varied. When the mobile robot is moving from A, where the robot is locatedon at time = t, to B, where the position is at time = t + d t, the coordinates of the ICC can be dynamically determined as Also, the mobile robot’s position，at time = t + dt, can be expressed according to the position of the ICC and the angular velocity, vx, as follows: ， Now, the total distance, d, and the rotation angle, u, of the mobile robot’s movement from location A to B can be obtained as follows: Using these equations, when the rotation radius, the distance of movement, and the rotation angle of the mobile robot are determined beforehand, the desired linear and angular velocities, vL, vR, and vx can be dynamically obtained while the robot is moving in a curved path [16]. 3. Curvature trajectory 3.1. Curved motion characteristics The curvature, k, is defined as the ratio of Dh to Ds when a mobile robot is rotating from a point P to Q, as illustrated in Fig. 4. That is, the curvature is defined as [17] The rotation radius can be defined as the inverse of the curvature, p=1/k, where k > 0. From Eq. (11), the total traveling distance along the curve, Ds, is the length of the arc and is proportional to the curveradius. Since the curvature, k, is inversely proportional to Ds, the curvature k is also inversely proportional to the curve-radius. If k = 0, then the curve-radius, that is, the rotation radius, becomes infinite. Note that k = 0 implies a straight line, which is a circle of infinite radius. When the mobile robot is moving along a curved trajectory, the rotation radius has a severe effect on tracking error. Generally, a mobile robot has a smaller chance of error along a straight line (k = 0) than on a curved path (k – 0). Theoretically, a curved trajectory can be calculated with Eqs. (6)–(8), assuming pure-rolling and non-slipping conditions. However, practical driving might result in some divergences from the theoretical values. When a mobile robot is following a curved path, there are combined centrifugal and centripetal forces. The friction force between the surface and the wheels acts as a centripetal force on the ICC and maintains the curved motion of the mobile robot. Under ideal conditions without slippage, the tracking error becomes zero. However, in practical circumstances, there is always a tracking error caused by slippage. The centrifugal force can be formulated as a function of rotation radius, R, and velocity, v, where m is the mass of the robot and c is a proportional constant. Fig. 5 illustrates a driving situation of a mobile robot, starting from A and following a curved path. Under ideal conditions, the estimated position of the robot becomes B1. However, in practical circumstances, the robot arrives at B2 via a tracking error. There have been many studies conducted to reduce the tracking error caused by a small rotation radius and high traveling velocity. Fig. 6 shows the error characteristics of a real mobile robot, according to the traveling radius and speed [19]. The speed of the right wheel is maintained at a constant, whereas that of the left wheel is changed in order to move the mobile robot in a curved motion. With increases of the left wheel velocity, the tracking error increases as well. Also note that the tracking error increases with a small rotation radius even though the speed is kept constant. From an analysis of Fig. 6, it can be concluded that the tracking error of a mobile robot increases with a smaller rotation radius and with a higher speed while the mobile robot is traveling along a curved path. 3.2. Single-curvature trajectory Fig. 7a and b represent single-curvature and double-curvature trajectories, respectively. The double-curvature trajectory has a symmetrical shape with respect to the point of inflection. The single- curvature trajectory keeps the same curvature, while the double- curvature trajectory changes its traveling direction and curvature at the point of inflection. Fig. 7c represents a trajectory that changes its curvature randomly; that is, the points of inflection exist at several locations. While a mobile robot is moving along the trajectory of Fig. 7c, the wheel velocities need to be changed whenever the rotation radius and traveling direction are changed, according to Eq. (7). For the same travel distance, Fig. 7a shows the biggest rotation radius, whereas the others have various smaller radii. Therefore, it can be predicted that when a mobile robot is traveling along a single- curvature trajectory, it has the least tracking error. Fig. 8 shows the simulation shapes of single-curvature and double-curvature trajectories. Using Eqs. (6)–(11), the rotation radius, traveling distance, rotation angle, and location of the ICC are obtained and are presented in Table 1. The total traveling distances are the same for the single-curvature and double-curvature trajectories. However, the rotation radius of the single-curvature trajectory is two times larger than that of the double-curvature trajectory. Specifically, the rotation radius of the single-curvature trajectory is 5.0 m, whereas that of the double-curvature is 2.5 m, which has the point of inflection at (2.5, 2.5). Close observations of Figs. 6 and 7, led to the expectation that the single-curvature trajectory will have a smaller tracking error than the double-curvature trajectory. Through real experiments, the tracking errors of the single-curvature trajectory and double-curvature trajectory can be compared to each other quantitatively. The orientation of the mobile robot at the final position is not restricted for the double-curvature trajectory since another point of inflection is needed in order to match the orientation of the mobile robot to the single-curvature trajectory. 4. Algorithm for curved trajectory planning As a typical example for the precise curved motion, the capturing of a moving object by the mobile robot has been demonstrated. This section has proposed an algorithm to produce an optimal trajectory path for capturing a moving object. The proposed algorithm is decomposed into four processes. 4.1. Pre-assumptions Depending on the states of a moving object and a mobile robot, the optimal trajectory for the mobile robot may be different. Therefore, in this paper, an optimal trajectory for a mobile robot has been planned, which considers the restrictions of the motion of the moving object and the mobile robot. First, the linear velocity of moving object, , and angular velocity, ; From Eqs. (14) and (15), the motion of the moving object is limited to a straight-line motion in constant velocity from the initial position. It is assumed that the mobile robot is kept static at the beginning, and it should have the same velocity and orientation as the moving object at the capturing moment. For the optimal path planning of a mobile robot, there are two major factors to be considered: the traveling time and the tracking error. If the mobile robot moves rapidly for the minimum driving time, the tracking error will increase in a curved path. Therefore, the conditions for minimum driving time and smallest tracking error can be obtained within the range of the mobile robot’s linear velocity and acceleration, which are represented as follows: where is the maximum allowable linear velocity and is the maximum allowable acceleration. For the minimum driving time with a given path, the acceleration of a mobile robot is selected at the maximum speed within the range of allowable acceleration. Also, the maximum velocity of the mobile robot is limited to minimize the tracking error within the maximum allowable range. 4.2. Acquisition of the states of a moving object and mobile robot Since the optimal path of a mobile robot can be created according to the states of a moving object and mobile robot, the states of the mobile robot and the moving object need to be defined precisely for the successful trajectory planning of the mobile robot. As shown in Fig. 9, the position of a moving object and the mobile robot is defined by two-dimensional X _ Y Cartesian variables, x and y, and an orientation variable, h. Then, the orientation of the moving object, is represented as The initial position of moving object,, can be represented as Now, the linear velocity of the moving object, , can be calculated as follows: The initial position of the mobile robot is represented as The mission of the mobile robot requires that it start to capture a moving object with a constant velocity from the stationary initial position. If the mobile robot has the same velocity and orientation with the moving object at the final position, it is natural to assume that the moving object will be captured by the mobile robot. 4.3. Determination of path According to the initial states of the mobile robot and moving object, either a single-curvature or a double-curvature trajectory is going to be selected. The existence condition for the single curvature trajectory is obtained as, which is an interesting observation that arose from this research. As shown in Fig. 10, when the mobile robot moves along the single-curvature trajectory, the tracking error from A to D can be minimized, since the slippage of the curved motion can be minimized. As a result, the mobile robot can capture the moving object precisely when it has the same velocity and orientation as the moving object. If the trajectory and location of the moving object can be estimated accurately, the expected capturing position, , is obtained as In that case, the coordinates of C (xC,yC), can be obtained from the initial states of a moving object and the mobile robot as follows: where (xI,yI) represents the initial position of the moving object. The ICC coordinates of the single-curvature trajectory, , and the rotation radius, R, can be represented as follows: There are some cases where a single-curvature trajectory cannot be produced, since the trajectory depends on the states of the mobile robot and the moving object. As shown in Fig. 11, if the difference between the orientation angles of a moving object and a mobile robot is non-positive, a single-curvature trajectory cannot satisfy the capturing condition at the final position. In this case, a double-curvature trajectory is selected as the optimal path, instead of a single-curvature one. In such a case, the trajectory of a mobile robot is decomposed into path-1 and path-2. The rotation radius and the coordinates of the ICC are also determined according to the states of the moving object and the mobile robot (see Fig. 12). From the initial states, the coordinates of ICC1, ,and ICC2, , can be represented as follows: The rotation radius of Path-1 and Path-2 are selected to be identical to decrease tracking error, and the rotation radius can be represented as Note that the direction of the mobile robot is changed at the point of inflection in a double-curvature trajectory. 4.4. Design of velocity profile The velocity of a mobile robot is determined according to the velocity of the moving object and the driving distance to capture the object. In the single-curvature trajectory, the linear velocity of the mobile robot is divided into three sections: acceleration, uniform velocity, and deceleration. The linear velocity of the mobile robot can be represented as follows: where T is the total driving time of the mobile robot. From the kinematics of a mobile robot, the wheel velocities, and , and the angular velocity of the mobile robot, , are determined as follows: For the double-curvature trajectory, two velocity profiles are necessary. That is, at the point of inflection, the mobile robot