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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.
移動(dòng)機(jī)器人軌跡跟蹤的控制設(shè)計(jì)方法
摘要
基于微分幾何理論，運(yùn)用相對(duì)階動(dòng)態(tài)擴(kuò)展的方法，對(duì)移動(dòng)機(jī)器人的運(yùn)動(dòng)誤差模型的精確反饋線性化的實(shí)現(xiàn)。軌跡跟蹤控制器被用來(lái)設(shè)計(jì)極點(diǎn)配置法。當(dāng)角的移動(dòng)機(jī)器人的速度永遠(yuǎn)為零，當(dāng)?shù)貪u近穩(wěn)定控制器的設(shè)計(jì)。當(dāng)角的移動(dòng)機(jī)器人的速度并非永久為零，軌跡跟蹤全程跟蹤與控制策略的約束給出。該算法簡(jiǎn)單，應(yīng)用方便。仿真結(jié)果表明其有效性。
關(guān)鍵詞：軌跡跟蹤;動(dòng)態(tài)擴(kuò)展的方法;精確反饋線性化;在全程范圍內(nèi)跟蹤約束
1 導(dǎo)言
最近，在移動(dòng)機(jī)器人跟蹤控制利益與不同的理論和實(shí)踐作出的貢獻(xiàn)正在增加。特別是，反饋線性化，吸引近非線性控制理論的研究的極大興趣，一些技術(shù)已在路徑跟蹤控制的移動(dòng)機(jī)器人有幾種類型由線性化的靜態(tài)和動(dòng)態(tài)方法研究了移動(dòng)機(jī)器人的就業(yè)問(wèn)題反饋。當(dāng)?shù)睾腿騿?wèn)題，通過(guò)跟蹤時(shí)變狀態(tài)反饋基于反演技術(shù)都已經(jīng)解決了。由于四輪驅(qū)動(dòng)移動(dòng)機(jī)器人非完整約束，從制約了移動(dòng)機(jī)器人的滾動(dòng)車輪出現(xiàn)打滑的情況下與線性非完整約束移動(dòng)機(jī)器人具有可控性不足，很難控制他們。穩(wěn)定問(wèn)題，這一點(diǎn)可以被視為新一代的控制輸入驅(qū)動(dòng)從任何初始點(diǎn)機(jī)器人的目標(biāo)點(diǎn)。在這一個(gè)事實(shí)，即移動(dòng)機(jī)器人模型不符合售書(shū)的著名必要的順利反饋鎮(zhèn)定的條件，因此移動(dòng)機(jī)器人不能穩(wěn)定與穩(wěn)定問(wèn)題的中心，關(guān)鍵的問(wèn)題平穩(wěn)狀態(tài)反饋，從而導(dǎo)致在應(yīng)用的限制。因此，一些離散時(shí)不變的控制器，時(shí)變控制器和基于Lyapunov混合控制器控制理論基礎(chǔ)已提出。
全面軌跡跟蹤問(wèn)題，參考討論移動(dòng)機(jī)器人基于反演技術(shù)在[5]。軌跡跟蹤問(wèn)題的參考移動(dòng)機(jī)器人的基礎(chǔ)上，討論終端滑模在[6技巧]，但它需要非零的旋轉(zhuǎn)速度。點(diǎn)穩(wěn)定的移動(dòng)機(jī)器人通過(guò)狀態(tài)空間的精確反饋線性化動(dòng)態(tài)擴(kuò)展的方法，提出了[7]。在極地幀點(diǎn)鎮(zhèn)定問(wèn)題可以較準(zhǔn)確地轉(zhuǎn)化為控制線性時(shí)不變系統(tǒng)的問(wèn)題。但其缺點(diǎn)是要求復(fù)雜合核查。而點(diǎn)鎮(zhèn)定問(wèn)題只討論軌跡跟蹤，但沒(méi)有得到解決。
在本文件中，軌跡跟蹤的提法，[5]和[6]是針對(duì)移動(dòng)機(jī)器人的基礎(chǔ)上在[7動(dòng)態(tài)擴(kuò)展的方法]。關(guān)于移動(dòng)機(jī)器人的運(yùn)動(dòng)誤差模型的精確反饋線性化的實(shí)現(xiàn)。其證明是簡(jiǎn)單的，從[7個(gè)不同]自核查合復(fù)雜的過(guò)程，是可以避免的。通過(guò)線性，非線性系統(tǒng)轉(zhuǎn)移到線性時(shí)不變系統(tǒng)，相當(dāng)于兩個(gè)降階線性時(shí)不變的，可以很容易控制系統(tǒng)。如果角度的移動(dòng)機(jī)器人的速度永遠(yuǎn)為零，當(dāng)?shù)貪u近穩(wěn)定控制器的設(shè)計(jì)。如果角
移動(dòng)機(jī)器人的速度并非永久為零，軌跡跟蹤全球跟蹤與控制策略的約束給出。該算法簡(jiǎn)單，應(yīng)用方便。
2.預(yù)備知識(shí)及問(wèn)題描述
考慮一個(gè)描述為一類非線性系統(tǒng)
定義（Slotine和李和鋒和費(fèi)）。鑒于X是一個(gè)n維微流形，如果存在的x0附近V和整數(shù)向量使得
?
非奇異8xAV，我們說(shuō)系統(tǒng)(1)–(2)公式的相對(duì)程度er1，R2的：y在點(diǎn)x0：引理（豐和費(fèi)）。
系統(tǒng)必要和精確反饋線性化的充要條件為x0（1）是存在一個(gè)x0附近V和光滑實(shí)值函數(shù)，使得系統(tǒng)(1)–(2)的相對(duì)是矢量車輪運(yùn)動(dòng)的模型驅(qū)動(dòng)的移動(dòng)機(jī)器人如下：
其中（x，y）是移動(dòng)機(jī)器人的位置和y是航向。移動(dòng)機(jī)器人的控制變量的線性速度v和角速度?：在這里，軌跡跟蹤的問(wèn)題是要跟蹤已知的姿態(tài)不怕參考移動(dòng)機(jī)器人; yrT和速度vr;，如圖1所示。我們擁有的移動(dòng)機(jī)器人姿態(tài)誤差方程[5,6]
因此，我們的姿態(tài)誤差差分方程
從上述的分析，軌跡跟蹤問(wèn)題的參考移動(dòng)機(jī)器人可以表述為：發(fā)現(xiàn)有界輸入v和?使一個(gè)任意初始誤差，系統(tǒng)狀態(tài)可由附近,也就是說(shuō)
3 設(shè)計(jì)的軌跡跟蹤控制器
顯然，系統(tǒng)（5）不能精確反饋空間狀態(tài)線性化。它不能輸入/輸出反饋線性化選擇出y1 = ye; y2 = ye：這是因?yàn)橄到y(tǒng)（5）沒(méi)有相對(duì)程度。其實(shí)很明顯，去耦矩陣是奇異的。
證明,區(qū)分輸出方程y1 = x2，y2 = X3的話，我們有
從（11）和（13），我們有解耦矩陣
從(14)我們有
因此，根據(jù)y1 = x2，y2 = X3,角速度oa0;系統(tǒng)（9）有相對(duì)程度和;使用第2引理，存在局部的微分，使系統(tǒng)（9）線性準(zhǔn)確。局部的微分和改變狀態(tài)的定義如下：
輸入變換的定義如下：使用式 (10)–(13), (16)和 (17), 系統(tǒng) (9)可以轉(zhuǎn)化為線性時(shí)不變系統(tǒng)：
其中是新的矢量；是新的控制輸入。
非線性系統(tǒng)（9）轉(zhuǎn)化為線性時(shí)不變系統(tǒng)的狀態(tài)和輸入轉(zhuǎn)換（16） - （17）對(duì)于線性時(shí)不變系統(tǒng)（18） - （19），我們可以應(yīng)用已知,極點(diǎn)配置方法的線性控制方法實(shí)施控制，因此，我們有以下定理和定理2。假設(shè)角速度Oa，當(dāng)系統(tǒng)（9）是由控制器（20a）控制，它具有局部漸近穩(wěn)定。如果oa0不能滿足，系統(tǒng)（9）是由控制器（20A）和（20B）交替控制，它已在全局范圍內(nèi)引用跟蹤與已知的移動(dòng)機(jī)器人坐標(biāo)和速度Vr。其中e是一個(gè)給定的任意小的正數(shù)。證明，根據(jù)角速度oa0，從定理1，系統(tǒng)（9）可轉(zhuǎn)化為線性時(shí)不變系統(tǒng)（18） - （19）。顯然，式（18） - （19）是完全可控和完全觀察。它包含兩個(gè)減少，為了獨(dú)立的子系統(tǒng)
其中.i=1,2;其參數(shù)為：
使矩陣 ， ，它們分別是：
證明，根據(jù)角速度oa0，從定理1，系統(tǒng)（9）可轉(zhuǎn)化為線性時(shí)不變系統(tǒng)(18)–(19)。顯然，系統(tǒng)(18)–(19)是完全可控和完全觀察。它包含兩個(gè)減少為獨(dú)立的子系統(tǒng)。
眾所周知，線性時(shí)不變系統(tǒng)（22）可以很好地通過(guò)極點(diǎn)配置的方法控制。
? 角速度oa0如果不能滿足，以保證控制的實(shí)現(xiàn)，我們選擇合適的小的正數(shù)e，可以指定人為根據(jù)實(shí)踐的要求進(jìn)行，這樣，jojXe用來(lái)控制（20A），jojoe用來(lái)控制器（20B）。因此，控制器（20A）和（20）被交替使用，可以保證職權(quán)范圍內(nèi)的跟蹤與移動(dòng)機(jī)器人
已知的動(dòng)態(tài)yrT和速度VR，從（20）我們可以看出，控制輸入v和o的范圍內(nèi)，只要在O'R的范圍內(nèi)。
4 仿真研究
我們實(shí)施仿真研究，驗(yàn)證了跟蹤控制器（20A）和（20B）的有效性。在模擬，如初始條件的參數(shù)，所需的速度和反饋增益列于表1。我們選擇了兩個(gè)時(shí)間不變系統(tǒng)（22 ）和（22B），同圖2-5顯示了移動(dòng)機(jī)器人反饋增益軌跡跟蹤控制的反應(yīng)。圖2和3顯示了模擬結(jié)果如下，移動(dòng)機(jī)器人的直線，其中控制器（20A）和（20B）被交替使用，以保證有界跟蹤引用或移動(dòng)機(jī)器人or=0。從無(wú)花果。 2和3，我們看到，跟蹤性能好，雖然T=0之前5秒，它在接近零。從圖 4和5的仿真結(jié)果表明，移動(dòng)機(jī)器人如下的曲線。從圖4和5，我們看到，跟蹤性能更好。和所有有界控制器，保證了控制的實(shí)現(xiàn)。
圖3 引用具有時(shí)變速度的移動(dòng)機(jī)器人
圖4 引用具有恒定速度的移動(dòng)機(jī)器人
圖5 引用具有時(shí)變速度的移動(dòng)機(jī)器人
5 結(jié)論
?? 在實(shí)踐中，使移動(dòng)機(jī)器人得到一些狀態(tài)和速度，我們可以假設(shè)參考這些姿態(tài)和速度移動(dòng)機(jī)器人，并考慮軌跡跟蹤問(wèn)題的參考移動(dòng)機(jī)器人。
在這個(gè)文件中，軌跡跟蹤問(wèn)題，引用是針對(duì)移動(dòng)機(jī)器人基于動(dòng)態(tài)擴(kuò)展的方法。確切的反饋對(duì)移動(dòng)機(jī)器人運(yùn)動(dòng)誤差模型線性化的實(shí)現(xiàn)。非線性系統(tǒng)轉(zhuǎn)移到二降階線性時(shí)不變的，可以很容易控制系統(tǒng)。下面的控制得以實(shí)現(xiàn)，也就是說(shuō)，如果在移動(dòng)機(jī)器人角速度永久為零，當(dāng)漸近穩(wěn)定控制器的設(shè)計(jì)。如果角度的移動(dòng)機(jī)器人的速度并非永久為零，軌跡跟蹤的全局策略的約束給出跟蹤軌跡。該方法是簡(jiǎn)單而有效的。
鳴謝
作者是感謝副主編并提出了寶貴的意見(jiàn)和建議裁判