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Neural adaptive output feedback formation control of type ( m , s ) wheeled mobile robots

2016; Institution of Engineering and Technology; Volume: 11; Issue: 4 Linguagem: Inglês

10.1049/iet-cta.2016.0952

1751-8652

Khoshnam Shojaei,

Adaptive Control of Nonlinear Systems

IET Control Theory & ApplicationsVolume 11, Issue 4 p. 504-515 Research ArticleFree Access Neural adaptive output feedback formation control of type (m, s) wheeled mobile robots Khoshnam Shojaei, Corresponding Author Khoshnam Shojaei khoshnam.shojaee@gmail.com Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, IranSearch for more papers by this author Khoshnam Shojaei, Corresponding Author Khoshnam Shojaei khoshnam.shojaee@gmail.com Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, IranSearch for more papers by this author First published: 17 January 2017 https://doi.org/10.1049/iet-cta.2016.0952Citations: 21AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study introduces a general framework for the formation tracking control of all types of wheeled mobile robots (WMRs). On the basis of a coordinate transformation, a second-order input–output model is developed for the general type (m, s) WMR with the mobility m and the steerability s. Then, a saturated feedback linearising controller in conjunction with a non-linear velocity observer is proposed in order to leave out the velocity sensors in all agents. In addition, a radial basis function neural network and an adaptive robust compensator are incorporated in the formation controller design to improve the tracking performance in the presence of uncertain non-linearities and unknown parameters. Semi-global stability of the closed-loop formation system is proved using direct Lyapunov method. Moreover, a generalisation of the proposed controller is introduced for the cooperative control of mobile manipulators. Finally, simulation examples are given to illustrate the effectiveness of the proposed formation control system for two types of WMRs. 1 Introduction The formation tracking control of multiple wheeled mobile robots (WMRs) is an attractive problem in the control and robotic engineering societies. This strong motivation originates from the fact that agents teamwork obtains more robustness, redundancy and efficiency than a single-agent. Surveillance, coverage, autonomous exploration, reconnaissance and rescue operations are typical examples of the formation control of multiple mobile robotic systems. In fact, the formation control problem is related to designing stabilising and tracking controllers to force a group of mobile robots track desired positions and orientations with respect to one or more reference points [1–8]. There exist different strategies for the formation control of multiple WMRs including behavioural-based [1, 2], virtual structure [3, 4] and leader-following approaches [5, 6]. In [7], a sliding-mode formation control has been proposed for cooperative autonomous mobile robots. Dierks et al. [8] have proposed a formation controller with reduced information exchange for mobile robots using neural networks (NNs) and the optimal control method. In [9], a leader–follower formation controller has been proposed for non-holonomic mobile robots based on a bioinspired neurodynamic-based approach. An adaptive leader–follower formation controller has been proposed in [10] for non-holonomic mobile robots using active vision. Recently, attractive results on the mobile robot formation have been reported in the literatures [11–13]. However, these works are not implementable without the velocity measurements. In practise, velocity signals are not easily measurable due to communication delays and noise contamination. Moreover, velocity sensors increase the weight and the cost of the agents. Toward this end, the design of observer-based formation controllers is a desirable solution. However, since the separation principle does not hold for the non-linear systems, the design of observer–controller schemes is still a challenging and difficult task. In addition, the non-holonomic constraints of WMRs and modelling uncertainties make this problem more demanding. For this purpose, some of researchers [14–18] proposed formation controllers for a group of mobile robots without velocity measurements. Recently, some researchers have also proposed a variety of formation tracking controllers for WMRs in [19–24]. However, all of above-mentioned papers only consider a special type of WMRs such as type (2, 0) robot and their solutions are not applicable to other types with different mobility and steerability. On the basis of the presented literature review, there is no work to propose a single formation controller for different types of WMRs [1–24] to the best of author's knowledge. Therefore, main contributions of this paper are expressed as follows: Compared with previously proposed ad hoc solutions [1–24], the proposed controller is applicable to all WMRs of type (m, s) with the mobility m and the steerability s, which are presented in [25]. According to the literatures [1–24], this is the first attempt to find a general formation solution for all types of WMRs, while most of previous works including [1–24] is often applied to type (2, 0) or unicycle-type WMRs. Consequently, this helps the user employ several types of WMRs in a special formation mission. In contrast to previous works [1–24], the proposed controller reduces the risk of the actuators saturation by bounding, tracking and observation errors. Consequently, the transient response of the formation control system is significantly improved. The proposed formation controller does not need the velocity measurements by designing a non-linear observer. In contrast to [14–17], the proposed observer reduces unwanted peaks in the estimated velocity signals by employing saturation functions. The proposed formation controller preserves its robustness against unknown kinematic and dynamic parameters, unmodelled dynamics, friction, external disturbances by using projection-type neural adaptive techniques. The proposed formation control system is extended to the construction of a more general formation configuration by using a chain of ranges and bearing angles with respect to a virtual leader. This extension helps us obtain a more flexible formation configuration which is useful for a simple cooperative control of mobile manipulators. The rest of this paper is organised as follows. The problem formulation including control objectives, definitions and assumptions is presented in Section 2. The main results of this paper including the formation controller design and its stability analysis are presented in Section 3. Simulation results are given in Section 4 to show the effectiveness of the proposed controller. Finally, Section 5 concludes this paper. 2 Problem formulation 2.1 Notations In this paper, () denotes the largest (smallest) eigenvalue of a matrix. is used as Euclidean norm of a vector , while the norm of a matrix A is defined as the induced norm and Frobenius norm, i.e. , where denotes the trace operator. The matrix denotes n -dimensional identity matrix. denotes a set of . represents the projection operator. To facilitate the subsequent control design and stability analysis, the following notations are also used , , where , diag[•] denotes a diagonal matrix, and are hyperbolic tangent function and its derivative, respectively. In addition, denotes positive real numbers. 2.2 General kinematic and dynamic models of WMRs Consider a group of N type (m, s) WMRs which are described by the following motion equations [25–29], where and denote their mobility and steerability, respectively: (1) (2) (3) (4)where represents the generalised coordinates where denotes the coordinates of a reference point Poi on the robot, is the heading angle according to Fig. 1, represents the steering coordinates of independent steering wheels, is the kinematic matrix, is a vector of angular velocities of the robot wheels. The vectors and are related to velocities of the mobility and steering coordinates of WMRs through the kinematic matrix , respectively. The velocity vector is transformed to pseudo-velocities vector by the transformation matrix , is a symmetric positive-definite inertia matrix, is the centripetal and Coriolis matrix, is the input transformation matrix, is the vector of actuators inputs, stands for the viscous friction and damping coefficient matrix and represents Coulomb friction matrix. The term denotes time-varying disturbances and unmodelled dynamics such as passive castor wheels, gearbox non-linear effects and actuators drivers. The kinematic model (1) and (2) and dynamic equation (3) are integrated into the following state-space representation: (5)where is the state vector, , and are smooth vector fields and the kinematic matrix is defined as follows [26]: (6)where the matrix and the vector are related to kinematics of a type (m, s) WMR which are inferred from Table 1 of [26]. The representation (6) allows applying the differential geometric control theory for the trajectory tracking problem. Fig. 1Open in figure viewerPowerPoint Planar formation of followers with steerable wheels 2.3 Formation model In this section, a second-order dynamic formation model is developed based on the virtual leader–follower approach. Since most of WMRs are under-actuated and are restricted by their non-holonomic constraints, the development of a general formation model is always a difficult task. However, motivated by the work of [26], the following smooth epimorphism transformation is introduced to develop a simple formation model: (7)where is a vector representing the coordinates of a virtual reference point in the front of the i th type (m, s) WMR in the body-fixed frame, is a rotation matrix between Earth-fixed frame and body-fixed frame , and are desired range and bearing values of the i th follower with respect to the virtual leader, respectively, according to Fig. 1. In addition, the steering variables , and can be extracted from Table 2 of [26] for a type (m, s) WMR. Then, a formation dynamic model is developed. For this purpose, one needs to repeatedly differentiate the transformation (7), so that it is explicitly related to torque inputs. By differentiating the output equation in (7) and substituting (1), (2) and (5), one gets (8)where , and denote the Lie derivatives of h i along the direction of the vectors f i, g i and , respectively, represents the gradient of h i with respect to . Since in (8) is not related to the actuators input, we differentiate (8) once again to obtain the following leader–follower formation model: (9)where is the decoupling matrix and and represent uncertain non-linearities which are defined as follows: (10) (11) 2.4 Radial basis function NN (RBFNN) approximator In this section, RBFNN is introduced to approximate uncertain non-linearities of the WMR system in the next section. From a review of [8], [17, 28] and references therein, this technique are widely used to estimate unknown non-linear functions. Fig. 2 shows the structure of three layers RBFNN. For a given continuous function , where is a compact set, there exists an RBFNN such that (12)where denotes the NN approximation error, and p show the number of hidden nodes and output nodes, respectively, is the j th Gaussian basis function, and are the centre vector and the standard deviation, respectively. Then, the approximated non-linearities can be expressed as follows: (13)where , is the weight matrix, and is bounded as where is a constant. Then, the estimate of the uncertain non-linearities is given by where denotes the estimated weight matrix which is updated by an appropriate update rule [28]. Fig. 2Open in figure viewerPowerPoint The structure of three layers RBFNN 2.5 Control objectives Given a set of desired ranges and bearings, i.e. , , , and given a reference trajectory which is generated by a virtual leader, the control objective discussed in this paper is to design a formation controller for a group of type (m, s) WMRs whose motion equations are given by (1)–(4) such that (i) the formation tracking errors, , , converge to a neighbourhood of the origin in the presence of modelling uncertainties in sense that where is an arbitrarily small positive constant; (ii) the controller does not need the velocity measurements; (iii) the actuators saturation should be reduced to prevent a poor tracking performance; and (iv) the controller should be applicable to all types of WMRs. Assumptions.The following assumptions are necessary for the presented control objectives: A1: Measurements of for all WMRs are available in real time. A2: The time derivatives of the desired trajectory up to second order are bounded. A3: The vector of time-varying disturbances and unmodelled dynamics, i.e. , is assumed to be bounded in the sense that where is an unknown positive constant. A4: The ideal NN weights are bounded such that where is an unknown positive constant [28]. A5: The non-linear term in (9) satisfies the following Lipchitz condition: where is an unknown constant. Lemma 1 [29].The following properties of hyperbolic functions can be proven: (i) ; (ii) and ; (iii) , ; (iv) ; (v) From item (iv), it follows that and ; and (vi) From item (i), it follows that which leads to . Lemma 2.The inequality holds for any , and for any and where is a constant which satisfies , i.e. [30, 31]. 3 Neural adaptive output feedback formation controller 3.1 Controller design In this section, a formation tracking controller is designed to force a group of type (m, s) WMRs to construct a desired formation. For this purpose, the tracking and state estimation errors are defined by and , respectively. Then, the following error variables are introduced: (14) (15)where and is a symmetric positive-definite gain matrix. By replacing (14) into (9), one obtains (16)Then, the following formation feedback linearising controller is proposed in this paper: (17) (18)where represents the best approximation of which is guessed by the designer, denotes a positive-definite symmetric gain matrix, and and are updated by the following update rules: (19) (20)where denotes the projection operator [32, 33], and denote adaptation gains. Then, the following non-linear observer is proposed: (21) (22)where is the observer gain. The initial conditions for the observer are chosen as , and . From (8) and (21), the estimated velocity signal is given by . 3.2 Closed-loop error dynamic equations By substituting (17) and (18) into (16), using (14) and (15), adding and subtracting to the right-hand side of (16), the following dynamic equation is obtained: (23)where which is bounded as follows by using Assumption A5: (24)where and are unknown positive constants and is defined as follows: (25)and is given by (26)By using RBFNN approximation properties, is approximated by where and . As a result, the closed-loop error dynamic equation is given as follows: (27)where denotes weights estimation errors matrix. By differentiating both sides of (21), it is straightforward to show that (21) and (22) are equivalent to which together with (27) yield the following observer error dynamic equation: (28)In the next section, the stability of the proposed closed-loop control system is analysed by using Lyapunov direct method. 3.3 Stability analysis Theorem 1.Consider the general model of WMRs which is given by (1)–(4) with the coordinate transformation (7). Under Assumptions A1–A5, if the gain conditions , and are satisfied, then the proposed adaptive output feedback formation controller (17)–(22) guarantees that tracking and observation errors are semi-globally uniformly ultimately bounded (SGUUB) and converge to a small ball containing the origin. Proof.Consider the Lyapunov function for the entire formation where (29)where . By utilising item (i) of Lemma 1, one may verify that (29) can be bounded as where , , , , , and is given by (25). Consequently, the overall Lyapunov function for the entire formation system is bounded as (30)where and . From the above inequality and item (ii) of Lemma 1, it is clear that is positive-definite, radially unbounded and decrescent. By differentiating (29) along (14), (15), (19), (20), (27) and (28), one gets (31)By using the fact that and considering that , and using (31) is expressed as follows: (32)where which is bounded as where is defined by (25), and are given as follows: (33)By considering the projection properties from [32, 33] and using Lemma 2, the inequalities hold which together with help us express (32) as follows: (34)where and which is bounded as where . Then, (34) can be stated as by recalling where and . Therefore, the following inequality is obtained for the entire formation system: (35)where , and . As a result, if the conditions of Theorem 1 are satisfied, , and, consequently, the following condition is satisfied for the entire formation system: (36)where , and , is strictly negative outside the compact set . This means that is decreasing outside the set which results in the following inequality: (37)where the upper bound on V (t) in (30) has been used. From (37), is obtained. Therefore, a sufficient condition for (36) is given by (38)This means that the following region of attraction: (39)can be made arbitrarily large to include any initial condition by selecting the control gains large enough, where , , and . From the above discussion, are SGUUB and converge to a small ball containing the origin. As a result, by recalling properties of generalised saturation functions from the previous section, and also converge to a small ball containing the origin. Therefore, by considering (14) and (15), one concludes that . Finally, considering the controller (17) and (18), and assumptions in the previous section, one also concludes that . Now, the proof is complete. □ Remark 1.In practise, a full-state tracking is of interest for successful and smooth manoeuvres of WMRs in the formation construction. Wang and Xu [26] have shown that the stable full-state tracking of a single type (m, s) WMR is achieved under the asymptotic stability of its zero dynamics via the linear approximation method. Yun and Yamamoto [34] have also investigated the stability properties of the internal dynamics of a type (2, 0) WMR. Their theoretical results are consistently adopted for the stable full-state formation tracking of a group of type (m, s) WMRs in this paper. By the investigation of [26], the presented sufficient conditions for internal dynamics stability of type (m, s) WMRs is applicable to the formation tracking problem of this paper as long as . 3.4 2n Degrees-of-freedom virtual leader-following formation In this section, the proposed formation controller is extended to construct a more general formation configuration by using a chain of ranges and bearing angles with respect to the virtual leader. For this purpose, we choose 2n degrees-of-freedom reference points whose coordinates , are specified with virtual ranges and bearings , , according to Fig. 3 a. The coordinates of the point in the Earth-fixed frame is given by (7). If the control objective is chosen to force the reference points of all WMRs to track the desired trajectory , then 2n degrees-of-freedom virtual leader-following formation is obtained. Fig. 3Open in figure viewerPowerPoint 2n degrees-of-freedom formation (a) General configuration for planar formation of type (m, s) WMRs, (b) Example of cooperative type (2,0) mobile manipulators which are carrying an object on a desired trajectory It is assumed that the reference point is connected to an arbitrary point on the body of the i th mobile robot. The coordinates of in the body-fixed frame for each follower is given by (40)Then, the transformation (7) is used to construct the formation model (9) and the proposed controller is applied to meet the control objective. A direct application of this configuration is useful for the cooperative control problem of N type (m, s) mobile manipulators for the transportation of arbitrary objects for example. This problem is usually addressed for type (2, 0) mobile manipulators in the literatures [35, 36]. Of course, it should be noted that the cooperative control of mobile manipulators is not in the scope of this paper and a preliminary result is only indicated. It is assumed that end effectors of n -link manipulators are set to predefined desired configurations by a human operator. The desired joints angles and links lengths are represented by and , , , respectively. Let the base of the manipulator on each mobile platform be connected in the steering wheels axle. Then, the coordinates of end effectors are chosen as the reference points , . Fig. 3 b shows an example of the cooperative control of two-link type (2, 0) mobile manipulators. The desired configurations should be chosen such that maximum manipulability measures are obtained for all mobile manipulators. The manipulability measure denotes a distance measure of the manipulator configuration from singular ones at which the manipulability measure becomes zero [37]. In fact, the maximisation of the manipulability measure is of importance to keep the manipulator configuration away from singularities. As reported in [37], the manipulability measure is defined as for non-redundant manipulators where represents the Jacobian matrix of the manipulator for the i th cooperative agent in the group. A complete presentation of this problem will be devoted to future works of the author. 4 Simulation examples 4.1 Example 1: Formation of Type (1, 1) WMRs The proposed formation controller is applied to a group of type (1, 1) WMRs and simulation results are provided to illustrate the controller performance. The generalised coordinates vector is selected to be . The pseudo-velocities of the WMR are represented by where and denote the linear and steering velocities of the robot, respectively. From the review of [26, 38], the kinematic and dynamic matrices for this type of WMR are given by , , where (41)where , and are inertia parameters, denotes the wheel-base, is the distance between the points Poi and Pci according to Fig. 4 a and and , are damping and friction coefficients, respectively. Fig. 4Open in figure viewerPowerPoint WMR configuration (a) Planar configuration of a type (1,1) WMR, (b) the planar configuration of a type (2,0) WMR By following the controller design procedure which is given in the previous section, and considering for type (1, 1) WMR which is motivated by Wang and Xu [26], the following output vector is obtained: (42)Simulations have been performed by using MATLAB software. Gaussian white noise is appended to the output measurements using randn (•) function in order to simulate the uncertainty in the localisation system of WMRs. Euler approximation with a time step of 20 ms is used to simulate the proposed controller. It is assumed that inertia and damping parameters are unknown in practise. Moreover, the external disturbance is applied to all robots during the formation construction. The desired trajectory is generated by the following timing law: (43)where and . The following controller parameters achieve a satisfactory tracking performance in this simulation: , , , , , , and . The control signals are saturated within to simulate the actuator saturation. Desired ranges and bearings for a group of two WMRs are chosen as , , and . The initial posture of the WMRs is set to and . The initial conditions of the observer are selected as , . In addition, it is assumed that the WMR is initially at rest. Simulation results including robot trajectories, output tracking errors, control signals, velocity estimation errors, i.e. , Frobenius norm of estimated NN weights matrices and estimated parameters are illustrated by Fig. 5. It is obvious that the proposed controller successfully constructs the desired formation. Fig. 5Open in figure viewerPowerPoint Formation tracking results for a group of type (1,1) WMRs (a) x–y Plot, (b) Tracking errors, (c) Control signals, (d) Velocities estimation errors, (e) Frobenius norm of NN weights estimates, (f) ai Estimates 4.2 Example 2: Formation of type (2, 0) WMRs A differential drive WMR has one of the most popular locomotion systems with non-holonomic constraints among WMRs whose planar configuration has been shown by Fig. 4 b. This WMR is classified as type (2, 0) WMR according to Campion et al. [25], i.e. and . The generalised coordinates vector is chosen as . The following kinematic and dynamic matrices are defined for type (2, 0) WMR according to [27, 29]: (44)and , where and where , , , , , , , and , . The definition of these parameters is referred to [29]. The following controller parameters achieve a satisfactory tracking performance: , , , , , , and . The control signals are saturated within to simulate the actuator saturation. This time, a virtual leader with desired torque input is used to create the desired trajectory. By following the controller design procedure in the previous sections, and taking into account for type (2, 0) WMR which is motivated by Wang and Xu [26], the following output vector is achieved: (45)A group of five type (2, 0) WMRs are considered to construct a desired formation with the following desired ranges and bearings: , , , , , , , , and . The initial posture of the WMRs are set to , , , and . Simulation results for the formation tracking are illustrated by Fig. 6. As shown by this figure, all of WMRs track the desired formation successfully without velocity measurements and without the actuator saturation. Fig. 6Open in figure viewerPowerPoint Formation tracking results for a group of type (2,0) WMRs (a) x–y Plot, (b) Tracking errors, (c) Control signals, (d) Velocities estimation errors, (e) Frobenius norm of NN weights estimates,(f) ai Estimates 4.3 Cooperative control of mobile manipulators In this section, another simulation has been carried out to demonstrate the effectiveness of the proposed controller for the cooperative control of three type (2, 0) three-link mobile manipulators. Control parameters are chosen same as the previous simulation, except that , and . The initial posture of mobile manipulators are set to , and . The coordinates of the reference point for each follower are given by (46)where and denote the coordinates of manipulator base on the i th mobile platform which are set to zero. Then, , represents the coordinates of the reference points in the Earth-fixed frame according to Fig. 3. The desired configurations of the manipulators are chosen as , , , , , and for this simulation. To generate the desired trajectory, a virtual leader with the desired torque input is used. Fig. 7 illustrates simulation results including x–y plot, tracking errors and control signals. Other signals are similar to the previous simulation and are intentionally omitted here. Fig. 7 shows that three mobile manipulators successfully perform the cooperative tracking along the desired trajectory. Fig. 7Open in figure viewerPowerPoint Cooperative control of three mobile manipulators (a) x–y Plot, (b) Enlarged portion of the x–y plot, (c) Tracking errors, (d) Control signals 4.4 Comparative results In this section, the proposed NN output feedback formation controller will be compared with a feedforward approximator-based formation controller, which is designed based on the control approach in [39], to assess its tracking performance. This controller has been designed by employing backstepping approach, RBFNN and high-gain observer as follows: (47)where , contains the approximation parameters, represents radial basis functions and . Moreover, and show high-gain observer state variables and parameters. The readers may see [39] for a detailed discussion of the controller design and its parameters. For an impartial comparison, both of controllers have been adjusted carefully to obtain their best tracking performances. In this simulation, larger initial posture are selected as , , , and . This time, is chosen to generate a linear trajectory. Other simulation parameters are similar to Section 4.2. Fig. 8 illustrates the formation tracking performance of both controllers. The proposed controller of this paper shows smoother tracking performance and flatter control signals while high-gain observer-based controller in [39] illustrates undershoots and overshoots in the tracking error signals and suffers the actuator saturation in control signals. Fig. 8Open in figure viewerPowerPoint Comparative simulation results for the proposed controller (left column) and high-gain observer-based NN controller in [39] (right column) (a) x–y Plot, (b) Tracking errors, (c) Control signals 5 Conclusion In this paper, the output feedback formation control problem of general type (m, s) WMRs was addressed by using the feedback linearisation, RBFNN, and adaptive robust control techniques. For this purpose, a general input–output model was developed via choosing a virtual reference point in the front of type (m, s) WMRs. Then, a neural adaptive observer-based controller was designed to obtain a desired formation by a simple virtual leader-following approach. A Lyapuonv-based stability analysis has proved that the tracking and state estimation errors are SGUUB. A useful generalisation of the formation control system was introduced which is effective for the cooperative control of type (m, s) mobile manipulators. Simulation results were provided to illustrate the effectiveness of the proposed controller for the formation of groups of type (1,1) and type (2,0) WMRs and the cooperative control of mobile manipulators. In the future work, more theoretical results will be provided for the cooperative control of mobile manipulators based on the presented idea in this paper. 6 Acknowledgments This research work has been supported by the research and technology programme funded by the Najafabad branch, Islamic Azad University. 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