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An adaptive dynamic controller for autonomous mobile robot rajectory tracking
abstract
This paper proposes an adaptive controller to guide an unicycle-like mobile robot during trajectory tracking. Initially, the desired values of the linear and angular velocities are generated, considering only the kinematic model of the robot. Next, such values are processed to compensate for the robot dynamics, thus generating the commands of linear and angular velocities delivered to the robot actuators. The parameters characterizing the robot dynamics are updated on-line, thus providing smaller errors and better performance in applications in which these parameters can vary, such as load transportation. The stability of the whole system is analyzed using Lyapunov theory, and the control errors are proved to be ultimately bounded.
Simulation and experimental results are also presented, which demonstrate the good performance of the proposed controller for trajectory tracking under different load conditions.
1. Introduction
Among different mobile robot structures, unicycle-like platforms are frequently adopted to accomplish different tasks, due to their good mobility and simple configuration. Nonlinear control for this type of robot has been studied for several years and such robot structure has been used in various applications,such as surveillance and floor cleaning. Other applications, like industrial load transportation using automated guided vehicles (AGVs) automatic highway maintenance and construction, and autonomous wheelchairs, also make use of the unicycle-like structure. Some authors have addressed the problem of trajectory tracking, a quite important functionality that allows a mobile robot to describe a desired trajectory when accomplishing a task.
An important issue in the nonlinear control of AGVs is that most controllers designed so far are based only on the kinematics of the mobile robot.
However, when high-speed movements and/or heavy load transportation are required, it becomes essential to consider the robot dynamics, in addition to its kinematics. Thus, some controllers that compensate for the robot dynamics have been proposed.
As an example, Fierro and Lewis (1995) proposed a combined kinematic/torque control law for nonholonomic mobile robots taking into account the modeled vehicle dynamics. The control commands they used were torques, which are hard to deal
with when regarding most commercial robots. Moreover, only simulation results were reported. Fierro and Lewis (1997) also proposed a robust-adaptive controller based on neural networks to deal with disturbances and non-modeled dynamics, although
not reporting experimental results. Das and Kar (2006) showed an adaptive fuzzy logic-based controller in which the uncertainty is estimated by a fuzzy logic system and its parameters were tuned on-line. The dynamic model included the actuator dynamics, and the commands generated by the controller were voltages for the robot motors.
The Neural Networks were used for identification and control, and the control signals were linear and angular velocities, but the realtime implementation of their solution required a high-performance computer architecture based on a multiprocessor system.
On the other hand, De La Cruz and Carelli (2006) proposed a dynamic model using linear and angular velocities as inputs, and showed the design of a trajectory tracking controller based on their model. One advantage of their controller is that its
parameters are directly related to the robot parameters.
However, if the parameters are not correctly identified or if they change with time, for example, due to load variation, the performance of their controller will be severely affected.
To reduce performance degradation, on-line parameter adaptation becomes quite important in applications in which the robot dynamic parameters may vary, such as load transportation.
It is also useful when the knowledge of the dynamic parameters is limited or does not exist at all.In this paper, an adaptive trajectory-tracking controller based on the robot dynamics is proposed, and its stability property is proved using the Lyapunov theory.
The design of the controller was divided in two parts, each part being a controller itself. The first one is a kinematic controller, which is based on the robot kinematics, and the second one is a dynamic controller, which is based on the robot dynamics. The dynamic controller is capable of updating the estimated parameters, which are directly related to physical parameters of the robot. Both controllers working together form a complete trajectory-tracking controller for the mobile robot. The controllers have been designed based on the model of a unicycle-like mobile robot proposed by De La Cruz and Carelli A s-modification term is applied to the parameter-updating law to prevent possible parameter drift.
The asymptotic stability of both the kinematic and the dynamic controllers is proven. Simulation results show that parameter drift does not arise even when the system works for a long period of time. Experimental results regarding such a controller are also presented and show that the proposed controller is capable of updating its parameters in order to reduce the tracking error. An experiment dealing with the case of load transportation is also presented, and the results show that the proposed controller is capable of guiding the robot to follow a desired trajectory with a quite small error even when its dynamic parameters change.
The main contributions of the paper are: (1) the use of a dynamic model whose input commands are velocities, which is usual in commercial mobile obots, while most of the works in the literature deals with torque commands; (2) the design of an adaptive controller with a s-modification term, which makes it robust, with the corresponding stability study for the whole adaptive control system; and (3) the presentation of experimental results showing the good performance of the controller in a typical industrial application, namely load transportation.
2. Dynamic model
In this section, the dynamic model of the unicycle-like mobile robot proposed by De La Cruz and Carelli (2006) is reviewed. Fig. 1depicts the mobile robot, its parameters and variables of interest. u and o are the linear and angular velocities developed by the robot, respectively, G is the center of mass of the robot, C is the position of the castor wheel, E is the location of a tool onboard the robot, h is the point of interest with coordinates x and y in the XY plane, c is the robot orientation, and a is the distance between the point of interest and the central point of the virtual axis linking the traction wheels (point B). The complete mathematical model
is written as
where and are the desired values of the linear and angular velocities, respectively, and represent the input signals of the system.
A vector of identified parameters and a vector of parametric uncertainties are associated with the above model of the mobile robot, which are, respectively,
where dx and dy are functions of the slip velocities and the robot orientation, du and do are functions of physical parameters as mass, inertia, wheel and tire diameters, parameters of the motors and its servos, forces on the wheels, etc., and are considered as disturbances.
The equations describing the parameters h were firstly presented in, and are reproduced here for convenience. They are
It should be stressed that i=1,2,4,6,Parameters y3 and y5 will be null if, and only if, the center of mass G is exactly in the central point of the virtual axis linking the traction wheels。 In this paper it is assumed that b6=0.
The robot’s model presented in (1) is partitioned into inematic art and a dynamic part, as shown in Fig. 2. Therefore, two controllers are implemented, based on feedback linearization, or both the kinematic and dynamic models of the robot.
3. The kinematic controller
3.1. Design
The design of the kinematic controller is based on the
kinematic model of the robot, assuming that the disturbance
term in (1) is a zero vector. From (1), the robot’s kinematic model s given by
whose output are the coordinates of the point of interest, thus
meaning . Hence
Note 2: The stability of the whole system will be revisited in the next section, in which an adaptive dynamic controller is added to the kinematic controller in order to implement the whole control scheme of Fig. 2.
4. The adaptive dynamic controller
4.1. Design
The dynamic controller receives from the kinematic controller the references for linear and angular velocities, and generates another pair of linear and angular velocities to be delivered to the robot servos, as shown in Fig. 2.
The design of the adaptive dynamic controller is based on the parameterized dynamic model of the robot. After neglecting the disturbance terms du and do the dynamic part of Eq. (1) is
By rearranging the terms, the linear parameterization of the dynamic equation can be expressed as
which can also be rewritten as
Note 4: It is important to point out that a nonholonomic mobile robot must be oriented according to the tangent of the trajectory path to track a trajectory with small error. Otherwise, the control errors would increase. This is true because the nonholonomic platform restricts the direction of the linear velocity developed by the robot. So, if the robot orientation is not tangent to the trajectory, the distance to the desired position at each instant will increase. The fact that the control errors converge to a bounded value shows that robot orientation does not need to be explicitly controlled, and will be tangent to the trajectory path while the control errors remain small.
5. Experimental results
To show the performance of the proposed controller several experiments and simulations were executed. Some of the results are presented in this section. The proposed controller was implemented on a Pioneer 3-DX mobile robot, which admits
linear and angular velocities as input reference signals, and for which the distance b in Fig. 2 is nonzero.
In the first experiment, the controller was initialized with the dynamic parameters of a Pioneer 2-DX mobile robot, weighing
about 10 kg (which were obtained via identification). Both robots are shown in Fig. 3, where the Pioneer 3-DX has a laser sensor weighing about 6 kg mounted on its platform, which makes its dynamics significantly different from that of the Pioneer 2-DX.
In the experiment, the robot starts at x=0.2m and y=0.0 m, and should follow a circular trajectory of reference. The center of the reference circle is at x =0.0m and y= 0.8 m. The reference trajectory starts at x=0.8m and y=0.8m and follows a circle having a radius of 0.8 m. After 50 s, the reference trajectory suddenly changes to a circle of radius 0.7 m. After that, the radius of the reference trajectory alternates between 0.7 and 0.8m each 60 s.
Fig. 4 presents the reference and the actual robot trajectories for a part of the experiment that includes a change in the trajectory radius. In this case, the parameter updating was active.
Fig. 5 shows the distance errors for experiments using the proposed controller, with and without parameter updating, to follow the described reference trajectory. The distance error is defined as the instantaneous distance between the reference and the robot position. Notice the high initial error, which is due to the fact that the reference trajectory starts at a point that is far from the initial robot position. First, the proposed controller was tested with no parameter updating. It can be seen in Fig. 5 that, in this case, the trajectory tracking error exhibits a steady-state value of about 0.17 m, which does not vary even after the change in the radius of the reference trajectory. This figure also presents the distance error for the case in which the dynamic parameters are updated. By activating the parameter-updating, and repeating the same experiment, the trajectory tracking error achieves a much smaller value, in comparison with the case in which there is no ig. 3. The robots used in the experiments.
Fig. 4. Part of the reference and real circular trajectories.
Fig. 5. Distance errors for experiments with and without parameter updating.
6. Conclusion
An adaptive trajectory-tracking controller for a unicycle-like mobile robot was designed and fully tested in this work. Such a controller is divided in two parts, which are based on the kinematic and dynamic models of the robot. The model on sidered takes the linear and angular velocities as input reference signals, which is usual when regarding commercial obile robots. It was considered a parameter-updating law for the dynamic part of the controller, improving the system performance.
A s-modification term was included in the parameter up dating law to prevent possible parameter drift. Stability analysis based on Lyapunov theory was performed for both kinematic and dynamic controllers. For the last one, stability was proved considering a parameter-updating law with and without the s-modification term. Experimental results were presented, and showed the good performance of the proposed controller for trajectory tracking when applied to an experimental mobile robot.
A long-term simulation result was also presented to demonstrate that the updated parameters converge even if the system works for a long period of time. The results proved that the proposed controller is capable of tracking a desired trajectory with a small distance error when the dynamic parameters are adapted. The importance of on-line parameter updating was illustrated for the cases where the robot parameters are not exactly known or might change from task to task. A possible application for the proposed controller is to industrial AGVs used for load transportation, because on-line parameter adaptation would maintain small tracking error even in the case of important changes in the robot load.
自主移動機器人跟蹤的自適應(yīng)動態(tài)控制器
摘要
?? 本文提出了一種自適應(yīng)控制器像在移動機器人軌跡跟蹤指導(dǎo)的獨輪車。最初，線性和角速度的期望值產(chǎn)生，只考慮機器人的運動學(xué)模型。其次，這種價值觀念被處理以補償機器人動力學(xué)，從而產(chǎn)生交付給機器人執(zhí)行器線性和角速度的命令。參數(shù)機器人動力學(xué)特征的更新上線，從而提供更小的錯誤，更好地應(yīng)用這些參數(shù)變化性能，如交通負荷。整個系統(tǒng)的穩(wěn)定性進行了分析利用Lyapunov理論和控制錯誤被證明是最終有界。仿真和實驗結(jié)果還提出，這表明了對建議的軌跡跟蹤控制在不同的負載條件下的表現(xiàn)良好。
1 導(dǎo)言
?? 在不同的移動機器人的結(jié)構(gòu)，像平臺獨輪車一些國家往往通過完成不同的任務(wù)，由于其良好的流動性和簡單的配置。非線性這種類型的機器人使用已經(jīng)好幾年，這種機器人控制結(jié)構(gòu)的研究已被用于多種應(yīng)用，如監(jiān)測和地面清洗。其它應(yīng)用，如工業(yè)負荷運輸，使用自動引導(dǎo)車輛（AGV）自動公路維修和建設(shè)，自主輪椅，還利用了獨輪車狀結(jié)構(gòu)。有些作者討論了軌跡跟蹤的問題，一個相當(dāng)重要的功能，使移動機器人來跟蹤理想的軌跡時，完成任務(wù)。
? ?在自動導(dǎo)引車系統(tǒng)的非線性控制的重要問題是，迄今為止，控制器的設(shè)計是基于移動機器人運動學(xué)。
但是，當(dāng)高速運動和重負荷交通運輸需要，就必須在考慮機器人動力學(xué)，除了其運動學(xué)。因此，一些控制器補償機器人動力學(xué)已被提出。作為一個例子，菲耶羅和Lewis（1995）提出了結(jié)合運動學(xué)/力矩控制法的非完整移動機器人考慮到車輛動力學(xué)模型。那個控制命令，他們用的扭矩，這是難以應(yīng)付當(dāng)大多數(shù)與商業(yè)有關(guān)的機器人。此外，只有仿真結(jié)果的報告。菲耶羅和劉易斯（1997年）也提出了魯棒自適應(yīng)控制器神經(jīng)網(wǎng)絡(luò)的處理干擾和非動力學(xué)模型，雖然沒有報告實驗結(jié)果。Das（2006年）顯示，自適應(yīng)模糊邏輯為基礎(chǔ)的控制器，其中的不確定性估計一模糊邏輯系統(tǒng)及其參數(shù)調(diào)整在網(wǎng)上。動態(tài)模型，包括執(zhí)行器動態(tài)，由控制器生成的命令是為機器人的電機電壓。
在神經(jīng)網(wǎng)絡(luò)被用于識別和控制，控制信號，線性和角速度，但他們的解決方案實時實現(xiàn)，需要一個高性能計算機體系結(jié)構(gòu)，多處理器系統(tǒng)為基礎(chǔ)。
?? ?另一方面，de la Cruz和Carelli（2006）提出了一個動態(tài)模型作為投入使用線性和角速度，并表現(xiàn)了軌跡跟蹤控制器設(shè)計的模型。他們控制的一個優(yōu)勢是，它的參數(shù)有直接關(guān)系的機器人參數(shù)。
但是，如果參數(shù)不正確認識，或者他們與時間的變化，例如，由于負荷變化，其控制器的性能將受到嚴(yán)重影響。為了減少性能下降，在線參數(shù)調(diào)整，就變得很重要的應(yīng)用中，機器人的動態(tài)參數(shù)可能會有所不同，如負載運輸。??這也是有用的動態(tài)參數(shù)知識是有限的，或者不存在。本文的自適應(yīng)軌跡跟蹤的機器人動力學(xué)為基礎(chǔ)的控制器提出，它的穩(wěn)定性證明利用Lyapunov理論。
控制器的設(shè)計分為兩部分，每一部分是一個控制器本身。第一個是運動控制器，它是在機器人運動學(xué)為基礎(chǔ)的，第二個是一個動態(tài)的控制，這是對機器人動力學(xué)為基礎(chǔ)。動態(tài)控制器能夠更新估計參數(shù)，它直接關(guān)系到機器人的物理參數(shù)。兩個控制器一起形成一個完整的軌跡跟蹤的移動機器人控制器。該控制器的設(shè)計基礎(chǔ)上的獨輪車模型，如移動機器人，de la Cruz和Carelli擬議的第S -修改長期應(yīng)用于參數(shù)更新的法則，以防止可能的參數(shù)漂移。
兩者的運動學(xué)和動力學(xué)控制漸近穩(wěn)定性證明。仿真結(jié)果表明，參數(shù)漂移，甚至不會出現(xiàn)在系統(tǒng)的長期工程。對于這樣的控制器實驗結(jié)果還介紹表明，該控制器是能夠更新其參數(shù)，以減少跟蹤誤差。實驗與交通負荷的情況，并給出了處理，結(jié)果表明，該控制器是引導(dǎo)機器人遵循一個非常小的錯誤期望的軌跡甚至可當(dāng)其變化動態(tài)參數(shù)。
該文件的主要貢獻是：（1）一個用動態(tài)模型的輸入命令的速度，這是通常在商用移動機器人，而在涉及扭矩命令的文學(xué)作品中；（2）與一個S -修改來說，這使得它的自適應(yīng)魯棒控制器設(shè)計，與整個相應(yīng)的穩(wěn)定性研究自適應(yīng)控制系統(tǒng)，以及（3）實驗顯示在一個典型的工業(yè)應(yīng)用的控制器具有良好的表現(xiàn)，即裝載運輸介紹。
2 動態(tài)模型
?? 在本節(jié)中，該獨輪車動態(tài)模型，如移動由克魯斯和Carelli（2006）提出的機器人進行審查。圖1描述移動機器人，它的參數(shù)和感興趣的變量。 U和O的線性和角度的機器人，分別對應(yīng)的速度，G是機器人的重心，C是小輪的位置，E是一種工具，機上的機器人的位置，h是點感興趣的坐標(biāo)x和在xy Y軸，C是機器人的前進方向和之間有利益點和連接虛擬軸中心點的距離牽引輪（B點）。完整的數(shù)學(xué)模型被寫為
uref和oref是線性和角速度，分別為所需的值，并代表該系統(tǒng)的輸入信號。??一個確定的參數(shù)向量和向量參數(shù)，不確定性是與上述型號的機器人，它們分別是
圖1 雙輪樣移動機器人
這里dx和dy是滑移速度職能和機器人定位，Do和Du是慣性參數(shù)，車輪和輪胎直徑的電機和伺服系統(tǒng)，車輪上，等等力參數(shù)被視為干擾。
? 該方程描述的參數(shù)h的首先提出，并在這里為方便起見轉(zhuǎn)載。他們是
應(yīng)當(dāng)強調(diào)指出，參數(shù)Y3和Y5，將是無效，當(dāng)且僅當(dāng)中心G與接牽引車輪的虛擬軸中心點是完全相同的。本文假定B6= 0。
個機器人的模型，介紹了如1劃分靜態(tài)和動態(tài)部分，如圖2所示。因此，實施兩個控制器，反饋線性化的基礎(chǔ)上，或兩者兼而有之的機器人運動學(xué)和動力學(xué)模型。
3運動控制器
3.1設(shè)計
該運動控制器的設(shè)計是基于運動學(xué)模型的機器人，假設(shè)干擾是一個零向量。從機器人運動學(xué)模給出
它的輸出是感興趣的點的坐標(biāo)，這里，所以：
注2：整個系統(tǒng)的穩(wěn)定性將再次在下一節(jié)，其中自適應(yīng)動態(tài)控制器添加到運動控制器，以執(zhí)行圖整個管制計劃。
4自適應(yīng)動態(tài)控制器
4.1設(shè)計
動態(tài)控制器接收來自運動控制器對線性和角速度參數(shù)，生成另一種線性和角速度付給機器人伺服系統(tǒng)，如圖2所示。
自適應(yīng)的動態(tài)控制器的設(shè)計是基于參數(shù)化動態(tài)模型的機器人。忽略了干擾條件和做為方程動態(tài)的一部分。式子1是：
重新安排的規(guī)定，線性參數(shù)化動力學(xué)方程可以表示為
這也可以寫成
注4：重要的是，一非完整移動機器人必須面向根據(jù)切線路徑軌跡跟蹤與小錯誤軌跡。否則，控制失誤會增加。這是因為非完整平臺限制的機器人開發(fā)的線速度方向。所以，如果機器人的方向不是相切軌跡，對每一個瞬間所需位置的距離將增加。事實上，控制誤差收斂到一個有界值表明，機器人化并不需要明確控制，將相切軌跡路徑，對照錯誤仍然很小。
5 驗結(jié)果
為了顯示控制器性能的若干建議實驗和模擬被解決。一些結(jié)果載于本節(jié)。擬議控制器實施一先鋒3DX移動機器人，它將性和角速度作為輸入?yún)⒖夹盘?，并在圖2中距離b是非零。
一個先鋒2DX機器人的動態(tài)參數(shù)，稱重約10公斤（其中獲得通過識別）。這兩個機器人如圖3，其中的先鋒3DX 光傳感器約6千克在自己的平臺，這使得它的動力顯著的不同于先鋒2 –DX。
在實驗中，機器人在x =0.2米和y = 0.0米開始，并應(yīng)遵守通告的參考軌跡。該參考圓心在x =0.0米和y = 0.8米參考軌跡開始在x = 0.8和y = 0.8米，并遵循一個圓圈具有0.8米的半徑經(jīng)過50秒，參考軌跡突然更改為半徑0.7米的圓之后，參考軌跡之間交替的半徑0.7 0.8/秒。圖4列出的參考和完整的實驗，包括在部分軌道半徑變化的實際機器人軌跡。在這種情況下，參數(shù)更新活躍。
圖5顯示使用實驗的距離誤差建議控制器，無參數(shù)的更新，以按照描述的參考軌跡。距離誤差定義為參考和機器人之間的的位置瞬時距離。注意高初始錯誤，這是由于事實，即參考軌跡在一個點，就是遠離最初的機器人位置開始。首先，建議的控制器進行了測試，沒有更新參數(shù)。圖5可以看出，在這種情況下，軌跡跟蹤誤差約為0.17米不變，即使在以后的參考軌跡的半徑變化穩(wěn)態(tài)值。這個數(shù)字還顯示出了在其中的動態(tài)參數(shù)更新的情況距離誤差。在實驗中使用的機器人，通過激活參數(shù)更新和重復(fù)同樣的實驗中，軌跡跟蹤誤差可達到較小的值。
圖3 用于實驗中的機器人
圖4 部分參考和真正的圓形軌道
圖5 距離誤差與實驗
6結(jié)論
?? 自適應(yīng)軌跡跟蹤控制的獨輪車，如移動機器人的設(shè)計和充分參與這一工作的測試。這種控制器分為兩部分，分別是關(guān)于機器人的運動學(xué)和動力學(xué)模型為基礎(chǔ)。該模型將機器人的線性和角速度作為輸入?yún)⒖夹盘?，這是通常的商業(yè)機器人。這被認為是一個參數(shù)更新為控制器的動態(tài)組成部分的法則，提高了系統(tǒng)的性能。
一長期被列入限制法則，以防止可能的參數(shù)漂移。穩(wěn)定性基于Lyapunov理論的分析，為進行運動學(xué)和動力學(xué)控制器。在過去的穩(wěn)定性證明考慮參數(shù)更新的法則。實驗結(jié)果顯示了對應(yīng)用到移動機器人實驗提出的軌跡跟蹤控制性能良好。
一項長期的仿真結(jié)果也表明，提出更新的參數(shù)收斂即使系統(tǒng)的長期工程。結(jié)果證明，這種控制器是跟蹤一個小的距離時，動態(tài)參數(shù)錯誤適應(yīng)期望的軌跡的重要性行參數(shù)更新的情況作了說明。任務(wù)中，機器人的參數(shù)不完全知道發(fā)生或可能發(fā)生變化。一個建議的控制器可能的用途是用于裝載運輸所用的工業(yè)自動導(dǎo)引車系統(tǒng)，因為參數(shù)將保持適應(yīng)性，即使在機器人中的重大變化的情況下也會減小跟蹤誤差。