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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
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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.
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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.