2. CTGT Center, School of Astronautics, Harbin Institute of Technology, Harbin 150001, China;
3. School of Engineering, University of South Wales, Pontypridd, CF37 1DL, UK
Telerobotic systems, which are controlled through computer networks have become a popular research topic in recent years due to their potential applications in our daily life [1]. In summary, these applications contain unmanned aerial vehicles [2], [3], autonomous underwater vehicles [4], and wheeled mobile robots [5][7].
When a communication network is introduced into a control loop of a traditional control system, the design process and stability analysis of the system change, which transforms the traditional control system into a networked control system. The networkinduced delay and data packet dropouts in a networked control system will affect the performance of the system and may even make the system unstable [8]. Several researchers have made efforts to cope with the timedelay and data loss problem in control loop of networked control system [9][14].
In this paper, the remote tracking control problem of a networkbased agent is considered. Some closely related and relevant literatures are shown in what follows. In [15], [16], a joystick is used to control a remote mobile robot, and the motion of joystick is translated into desired linear and angular velocities of mobile robot. In [17], a sliding mode approach is proposed to solve the path tracking problem, where an exact discrete time model of mobile robot is developed as a timedelay system. Authors in [18] solve the problem of discrete time tracking control of an omnidirectional mobile robot by extending the continuous timedelay system to a discretetime model which is free of delay, and the feedback linearization strategy is adopted to obtain the control inputs. In [19], [20], the vehicle active suspension control problem is considered under situation of actuator input delay. In [21], a PDlike controller is applied to the delayed bilateral teleoperation of wheeled robots with force feedback in face of asymmetric and varyingtime delays. In [5], [7], [22], predictors are designed to compensate the negative effects of timedelay. Especially in [7], a predictorcontroller combination with a remote tracking controller is proposed, whose performance is demonstrated using an interconnected robotic platform located partly in Eindhoven, the Netherlands, and Tokyo, Japan.
To the best of authors' knowledge, this is the first time a remote tracking control problem is solved using networked predictive control scheme. Inspired by [7], this paper adopts a similar tracking controller in discretetime domain. Following that, a networked predictive control scheme was designed to compensate the negative effects caused by communication delays and data losses.
As the contribution of the current paper, a remote tracking controller based on networked predictive control strategy is proposed, which is capable of compensating for bilateral timedelays in a wireless network. It is worth noting that the simulation and experimental results are consistent.
The remainder of this paper is organized as follows. Section Ⅱ formulates the problem to be solved. In Section Ⅲ, the networked predictive tracking controller is proposed. Simulation and experimental results are presented in Section Ⅳ. Finally, this paper concludes in Section Ⅴ.
Ⅱ. PROBLEM FORMULATIONThe networkbased agent considered in this paper is a wheeled mobile robot, whose discretetime model can be described as
$ \begin{align}\label{system_model} \begin{bmatrix} x(k+1) \\ y(k+1) \\ \theta(k+1) \\ \end{bmatrix} = \left[ \begin{array}{c} x(k) \\ y(k) \\ \theta(k) \\ \end{array} \right]+\left[ \begin{array}{cc} T\cos\theta(k)&0 \\ T\sin\theta(k)&0 \\ 0 &T \\ \end{array} \right]\left[ \begin{array}{c} \upsilon(k) \\ \omega(k) \\ \end{array} \right] \end{align} $  (1) 
and
Supposing that the reference state to be tracked satisfy (1), which behaves as a virtual mobile robot with its state defined by
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Fig. 1 Positional relationship between reference robot and real robot. 
The state deviation can be derived as
$ \begin{align} \begin{bmatrix} e_x(k) \\ e_y(k) \\ e_\theta(k) \\ \end{bmatrix} = R(k) \begin{bmatrix} x_r(k)x(k) \\ y_r(k)y(k) \\ \theta_r(k)\theta(k) \\ \end{bmatrix} \end{align} $  (2) 
which is mapped into local coordinate frame of real mobile robot, with
$ \begin{align*} R(k) &= \begin{bmatrix} \cos\theta(k)&\sin\theta(k)&0 \\ \sin\theta(k)&\cos\theta(k)&0 \\ 0&0&1 \\ \end{bmatrix} \end{align*} $ 
and
$ \begin{align}\label{state_error_sys_0} \nonumber \begin{bmatrix} e_x(k+1) \\ e_y(k+1) \\ e_\theta(k+1) \\ \end{bmatrix} = & \begin{bmatrix} 1&T\omega(k)&0 \\ T\omega(k)&1& 0 \\ 0&0& 1 \\ \end{bmatrix} \begin{bmatrix} e_x(k) \\ e_y(k) \\ e_\theta(k) \\ \end{bmatrix} \\ &+ T \begin{bmatrix} \upsilon_r(k)\cos e_\theta(k)\upsilon(k) \\ \upsilon_r(k)\sin e_\theta(k) \\ \omega_r(k)\omega(k) \\ \end{bmatrix} \end{align} $  (3) 
with
$ \begin{align}\label{basic_controller} \begin{bmatrix} \upsilon(k) \\ \omega(k) \\ \end{bmatrix} = \begin{bmatrix} \upsilon_r(k) \\ \omega_r(k) \\ \end{bmatrix} + \begin{bmatrix} k_x&k_y\omega_r(k)& 0 \\ 0 & 0 &k_\theta \\ \end{bmatrix} \begin{bmatrix} e_x(k) \\ e_y(k) \\ e_\theta(k) \\ \end{bmatrix} \end{align} $  (4) 
where
$ E(k+1) = f(k)E(k) $  (5) 
with
$ \begin{align*} f(k) &= \begin{bmatrix} A(k)&g(k, E(k)) \\ 0 &C \\ \end{bmatrix} \end{align*} $ 
and
$ \begin{align*} A(k) &= \begin{bmatrix} 1Tk_x&T\omega_r(k)(1+k_y) \\ T\omega_r(k) &1 \\ \end{bmatrix} \\ g(k, E(k)) &= \begin{bmatrix} k_\theta e_y(k)+\upsilon_r(k)\dfrac{\cos e_\theta(k)1}{e_\theta(k)} \\ k_\theta e_x(k)+\upsilon_r(k)\dfrac{\sin e_\theta(k)}{e_\theta(k)} \end{bmatrix}T \\ C &= 1Tk_\theta. \end{align*} $ 
With tracking controller (4) and methods proposed in [23], the closedloop state error system (5) can be proved to be uniformly globally asymptotically stable (UGAS). When the communication network is introduced into the control loop, especially when networkinduced delay exists in the communication channel, the design process of the controller and stability analysis of the system change, which transform the traditional control system into a networked control system.
In the current problem setting, the mobile robot subjects to networkinduced bilateral timedelays consisting of a forward and a backward timedelay (see Fig. 2).
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Fig. 2 Schematic of remote tracking control. 
In Fig. 2, constant timedelays
To cope with this problem, which also is the control goal of this paper, the state
As mentioned in Section Ⅱ, there exists constant timedelays
It should be noted that the future reference states
If there are no internal and external uncertainties in (1), the following result is derived on the stability of the closedloop predictive control system.
Theorem 1: Consider the discretetime kinematics model of mobile robot (1) with constant timedelays
Proof: At stepping time
$ E_{k\tau_b} = R_{k\tau_b}(q_{r, k\tau_b}q_{k\tau_b}) $  (6) 
with
$ \begin{align*} R_{k\tau_b} &= \begin{bmatrix} \cos\theta_{k\tau_b}&\sin\theta_{k\tau_b}&0 \\ \sin\theta_{k\tau_b}&\cos\theta_{k\tau_b}&0 \\ 0&0&1 \\ \end{bmatrix}. \end{align*} $ 
Substitute state deviation (6) into controller (4) gives
$ \begin{align*} \hat{E}_{k\tau_b+1k\tau_b} &= f_{k\tau_b}E_{k\tau_b} \\ &= E_{k\tau_b+1} \end{align*} $ 
where
$ \begin{align*} f_{k\tau_b} &= \begin{bmatrix} A_{k\tau_b}&g_{k\tau_b, E_{k\tau_b}} \\ 0 &C \\ \end{bmatrix} \end{align*} $ 
then the control inputs are calculated as
$ \begin{align*} \hat{u}_{k\tau_b+1k\tau_b} = u_{k\tau_b+1}. \end{align*} $ 
Similarly, it gives
$ \begin{align*} \hat{E}_{k\tau_b+2k\tau_b} &= \hat{f}_{k\tau_b+1k\tau_b}\hat{E}_{k\tau_b+1k\tau_b} \\ &= f_{k\tau_b+1}E_{k\tau_b+1} \\ &= E_{k\tau_b+2} \end{align*} $ 
with
$ \begin{align*} \hat{f}_{k\tau_b+1k\tau_b} &= \begin{bmatrix} A_{k\tau_b+1}&\hat{g}_{k\tau_b+1, \hat{E}_{k\tau_b+1k\tau_b}} \\ 0 &C \\ \end{bmatrix} \\ &= \begin{bmatrix} A_{k\tau_b+1}&g_{k\tau_b+1, E_{k\tau_b+1}} \\ 0 &C \\ \end{bmatrix} \\ &= f_{k\tau_b+1}. \end{align*} $ 
Furthermore, we have
$ \begin{align*} \hat{E}_{k\tau_b+s+1k\tau_b} &= \hat{f}_{k\tau_b+sk\tau_b}\hat{E}_{k\tau_b+sk\tau_b} \\ &= f_{k\tau_b+s}E_{k\tau_b+s} \\ &= E_{k\tau_b+s+1} \end{align*} $ 
with
$ \begin{align*} \hat{f}_{k\tau_b+sk\tau_b} &= \begin{bmatrix} A_{k\tau_b+s}&\hat{g}_{k\tau_b+s, \hat{E}_{k\tau_b+sk\tau_b}} \\ 0 &C \\ \end{bmatrix} \\ &= \begin{bmatrix} A_{k\tau_b+s}&g_{k\tau_b+s, E_{k\tau_b+s}} \\ 0 &C \\ \end{bmatrix} \\ &= f_{k\tau_b+s} \end{align*} $ 
for
$ \hat{u}_{k\tau_b+s+1k\tau_b} = u_{k\tau_b+s+1}. $ 
Let
$ \begin{align*} \hat{E}_{k+\tau_fk\tau_b} &= \hat{f}_{k+\tau_f1k\tau_b}\hat{E}_{k+\tau_f1k\tau_b} \\ &= f_{k+\tau_f1}E_{k+\tau_f1} \\ &= E_{k+\tau_f} \end{align*} $ 
where
$ \begin{align*} \hat{f}_{k+\tau_f1k\tau_b} &= \begin{bmatrix} A_{k+\tau_f1}&\hat{g}_{k+\tau_f1, \hat{E}_{k+\tau_f1k\tau_b}} \\ 0 &C \\ \end{bmatrix} \\ &= \begin{bmatrix} A_{k+\tau_f1}&g_{k+\tau_f1, E_{k\tau_b+s}} \\ 0 &C \\ \end{bmatrix} \\ &= f_{k+\tau_f1}. \end{align*} $ 
After iterations for
$ \hat{u}_{k+\tau_fk\tau_b}=u_{k+\tau_f}. $ 
Replacing
$ \begin{align}\label{state_error_sys_cascaded} \hat{E}_{k+1k\tau_b} &= \hat{f}_{kk\tau_b}\hat{E}_{kk\tau_b} = f_{k}E_{k}= E_{k+1} \end{align} $  (7) 
with
$ \begin{align*} \hat{f}_{kk\tau_b} &= \begin{bmatrix} A_{k}&\hat{g}_{k, \hat{E}_{kk\tau_b}} \\ 0 &C \\ \end{bmatrix} = \begin{bmatrix} A_{k}&g_{k, E_{k}} \\ 0 &C \\ \end{bmatrix} = f_{k} \end{align*} $ 
and the predictive tracking controller at mobile robot side could be described as
$ \begin{align}\label{networked_predictive_tracking_controller} u_k = \hat{u}_{kk\tau_f\tau_b}. \end{align} $  (8) 
We see that the closedloop state error system in (7) is exactly the same as (5), which implies that the stability performance of (7) is equivalent to that of (5).
Remark 1: In above results, it is assumed that the kinematic model of mobile robot is consistent with its dynamic model, thus it allows us to use networked predictive control scheme to compensate the bilateral timedelays. In real applications, the kinematics models cannot fully describe the dynamic characteristics of the system. In other words, there are errors between the kinematics and dynamics model of mobile robots. As previously mentioned, the value of timedelay determines the iteration times. When using an inaccurate model (kinematics model) to predict future states of mobile robot, the cumulative error will increase rapidly as the iteration times increase. Consequently, the maximum value of timedelay
In this section, the experimental platform is first introduced, then some simulation and experimental results which are obtained using the same parameters are provided to demonstrate the control performance of the proposed networked predictive tracking controller.
The reference trajectory to be tracked is a circular line which satisfies the kinematics model of the mobile robot (1) and has the following form:
$ \begin{equation*} \begin{cases} x_k = x_{rc}+r\sin \theta_k \\ y_k = y_{rc}r\cos \theta_k \end{cases} \end{equation*} $ 
the reference circular line is centered at
In general, the experimental platform designed in this paper, as shown in Fig. 3, consists of four parts: networked controller, wireless router, vicon system, and mobile robot.
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Fig. 3 Experimental platform of mobile robot. 
The networked controller is designed and developed by authors of this paper. It is capable of executing files which are derived from simulink blocks in MATLAB/Simulink, and it builds a bridge between theoretical research and engineering practice. The hardware resources of the networked controller include twochannel analog to digital converter, twochannel digital to analog converter, twochannel digital input, twochannel digital output and fourchannel pulse width modulation. Moreover, the networked controller could communicate with PCs and/or other networked controllers using network sendandreceive modules using UDP protocol.
The mobile robot used in this paper is equipped with two driven wheels, and the wheel is individually actuated by stepper motor. An omnidirectional wheel is placed at the back to keep balance. There is an additional networked controller, which plays the role of receiving control inputs, embedded within the mobile robot.
There are four Vicon markers placed on the mobile robot which can be captured by Vicon cameras at each sampling period. After that, the positional information of mobile robot is obtained in the Vicon system, and this positional information will be sent to the networked controller through wireless router. Based on the positional information of mobile robot and the reference states, control inputs of the mobile robot are calculated in the networked controller and transmitted to mobile robot using wireless network. Thus, the closedloop tracking control system of mobile robot is achieved.
B. Small TimeDelay in Bilateral Communication ChannelsIn the laboratory environment, the transmition of data packets was achieved by a wireless router, which used UDP protocol. The timedelay was usually small (less than 10 ms), and was smaller than one sampling period. When studying the case of time delay being larger than the sampling period, an artificial delay function should be implemented. In this paper, the delay function block in Simulink Library Browse is adopted, which can be used in simulation research and can also be downloaded into the networked controller.
Further more, during the simulation study, the execution time of each individual block was synchronized since PC clock was used, whereas for the experimental part, it is assumed that the execution time of each individual parts were synchronized (due to the timedelay existing in the communication channel is smaller than one sampling period). In other words, timedelays in the bilateral communication channels were introduced on the controller side artificially, thus allowing us to realize a networkinduced delay both in simulation and experiment.
In this subsection, small timedelays, e.g.
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Fig. 4 Simulation and experimental results with small timedelay. 
When the timedelays existing in the communication channels are large enough, e.g.,
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Fig. 5 Linear speed of mobile robot under large timedelay. 
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Fig. 6 Simulation and experimental results with large timedelay. 
Obviously, the moving trajectory of the mobile robot finally converges to the reference states, with tracking errors bounded within
When the wireless network is introduced into the control loop of mobile robot, the data packets losses are inevitable due to signal strength and network congestion in wireless router.
Generally speaking, in a real application of mobile robot, data packet drops will happen in two cases.
Case 1: The timedelay in the communication channel exceeds the upper bound
Case 2: If the sensor fails for a finite time period, the state of the mobile robot will never be transmitted to the controller, in which case it is also considered that the data packet is lost.
In both the above cases, the state of the mobile robot is regarded as lost on the controller side. To cope with this problem, the estimated state
To simulate data loss and to make the experiment more interesting, the Vicon markers are covered and perturbations are introduced in the experimental process. When the Vicon markers are covered (sensor fails), Vicon system cannot acquire any information about the mobile robot, then a NAN response is sent to networked controller. Hence, the data packets in feedback channel are lost.
The experimental results are shown in Fig. 7. It is obvious that the linear and angular speed of the mobile robot converges to the reference speed (see Fig. 7(a), Fig. 7(b)), and the moving trajectory converges to the reference states within
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Fig. 7 Experimental results with consecutive data loss in the feedback channel. 
Based on above results, it can be concluded that the networked predictive tracking controller (8) is capable of compensating consecutive data losses in the feedback channel. Whereas, if a perturbation occurs when the data packets are lost, controller (8) could not eliminate the perturbation until networked controller received the positional information of mobile robot.
In addition, the data loss problem in forward communication channel can be solved by sending a control sequence
In this paper, the remote tracking control problem of a networkbased Agent subject to a networkinduced bilateral timedelays was considered. The overshoot of system increased as the network delay went up, but the moving trajectory finally converged to the reference states when the timedelays were sufficiently small. Whereas, when timedelays were big enough, the moving trajectory could not converge to the reference states. To solve these problems, a networked predictive tracking control scheme was proposed, which was capable of compensating the bilateral timedelays actively. Simulation results were clearly verified by experiments, which demonstrated the effectiveness of the proposed scheme. Moreover, consecutive data losses in the feedback channel could be compensated actively based on above approach, and it could be obviously seen when lateral movement perturbations were introduced.
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