Composite control for trajectory tracking of wheeled mobile robots with NLESO and NTSMC

This paper proposes a control strategy integrating the non-linear extended state observer (NLESO) and the non-singular terminal sliding mode control (NTSMC) for the trajectory tracking of wheeled mobile robots subject to bounded disturbances. A new transformation method of chained model in terms of Lie derivative is presented to simplify the controller design. A speciﬁc NLESO combining linear term and non-linear term is designed to estimate the disturbances with a faster convergence performance. A scheme for determining the gain range of NLESO is explicitly given to facilitate the tuning of experimental parameters. Meanwhile, the NTSMC achieves ﬁnite time convergence of the tracking error system and the chattering phenomenon in NTSMC is dramatically alleviated with the compensation from NLESO. The experimental results validate the strong robustness and good performance of the proposed control strategy.

applied for trajectory tracking with external disturbances [12]. On the other hand, WMRs are subject to non-holonomic constraints, which means that WMRs cannot move in every direction freely. To this end, simplifying the model with appropriate transformations becomes a viable solution. Actually, WMRs are non-linear affine systems. In [13], a method of transforming the non-linear affine system into a chained form was developed. Due to the simple structure of the chained model, controller design and theoretical analysis become more convenient. [14] gave a specific chained transformation and attitude control of a space robot, while it cannot meet the requirement of the global transformation. To solve this problem, we have put forward a simple and practical method of global transformation for WMRs. In general, it is interesting to develop an appropriate control strategy to guarantee accurate tracking control for WMRs with uncertainties, disturbances and non-holonomic constraints. Furthermore, most of the existing results focus on the theoretical analysis and it is hard to apply them in practice. Therefore, this paper aims to design a practical trajectory tracking control strategy.
Sliding mode control has been widely studied due to its excellent disturbances rejection performance [15][16][17]. [18] proposed a novel sliding surface for the robust control with matched uncertainties. An event-triggered sliding mode control algorithm was investigated in [19]. These results provide some theoretical support for the research of sliding mode control.
The key of sliding mode control is to design a suitable sliding mode manifold. Ordinary linear slide mode manifold can only guarantee asymptotic stability or exponential stability. In terms of non-singular terminal sliding mode control (NTSMC), [20] proposed a finite-time convergence strategy. As we all know, the chattering problem is inevitable for the aforementioned methods. This greatly hinders the application of sliding mode control in practice, although it has good robustness in theory. Fortunately, some tools have been developed to handle it. [21] utilized filtering techniques to schedule the switching control gain automatically when system entered the sliding motion. [22] switched on the derivative of control instead of the control input itself to reduce chatter phenomenon. A dynamic sliding mode control strategy was given to solve unmatched perturbations and chattering problem [23]. In [24], a robust two-dimensional observer for estimation of the state-dependent uncertainty in the sliding variable was considered for chattering reduction. It is undeniable that observer compensation is an effective technique, but a well-designed observer remains difficult. There are some advanced disturbance observers such as adaptive disturbance observer [25], finite-time disturbance observer [26] and sliding mode observer [27]. The abovementioned disturbance observers have good performance, but they bring certain difficulties for practical application due to the complex parameters and difficulties in implementation. On the other hand, ADRC has received more and more attention for its superior capability of disturbance rejection [28]. The key component named extended state observer (ESO) in ADRC [29] provides an alternative for us. ESO estimates the total disturbance including modelling deviation, external disturbances and other uncertainties. Then the total disturbance is compensated by suitable feedback control. ESO has good accuracy and real-time performance when the gains are designed properly. Many types of ESO have been reported in the literature. In [30], a high gain ESO was constructed to guarantee that the output tracks the reference signal practically. [31] proposed an adaptive ESO to reduce the estimation errors. In [32], a non-linear ESO (NLESO) constructed from piece-wise smooth functions has been given, which consists of linear and fractional power functions. The convergence of NLESO has also been discussed in [33] which provides us a method of stability analysis. Furthermore, the practical stability of ADRC for the closedloop system was considered in [34]. A higher-order ESO was designed to reconstruct the unmeasured states and disturbances online for brushless DC motor control [35]. [36] presented a radial basis function neural network ESO to estimate the unknown wind gusts. The linear ESO has been well studied and successfully applied in engineering practice in [37] for parameter tuning.
Motivated by the aforementioned discussions, our work aims to design a controller with disturbance rejection ability and accurate tracking performance without chattering for WMRs under bounded perturbations. More specific, we develop a tracking control strategy which integrates the advantages of NTSMC and NLESO. NLESO is used to estimate the total disturbance. i. A targeted NLESO is developed. Meanwhile, a scheme to determine the parameter range is explicitly given, which facilitates the practical applications. ii. A finite time control strategy in terms of NTSMC is proposed with designed NLESO to reduce the chattering phenomenon which generally appears in traditional sliding mode control.
The remainder of this paper is organized as follows. In Section 2, the kinematic model of WMRs and the chain transformation method are formulated. In addition, the explicit control strategy and some related lemmas for stability analysis are given. Section 3 concentrates on the proof of convergence about NLESO and stability of the closed-loop system. In Section 4, experimental results are given to verify the validity of the proposed strategy. Finally, Section 5 makes a summary of this paper.
Notations. ′ sign ′ denotes the sign function, ‖ ⋅ ‖ represents the Euclidean norm, L g h = ∇h ⋅ g is the Lie derivative of function h and vector field g, ad f g = ∇g ⋅ f − ∇ f ⋅ g denotes Lie bracket of vector fields f and g.

PROBLEM FORMULATION
The kinematic model with non-holonomic constraints of WMRs is plotted in Figure 1.
Its kinematics is described as where (X , Y ) is the centre position of WMRs in Cartesian coordinates, is the heading angle measuring the wheel orientation with respect to the positive direction of axis X . The angular velocity around the vertical axis and the linear velocity v are two control variables for trajectory tracking.
We can see that (1) is a two-input three-output non-linear affine system asẋ T are smooth and linearly independent vector fields defined on an open set U . We aim to find appropriate control variables u 1 = , u 2 = v to steer (2) from the initial state x 0 ∈ U to the final state x t ∈ U . In order to effectively use NLESO, the system needs to be transformed into a chained form. To this end, we define the following three distributions associated with system (2) as In [13], sufficient conditions were proposed to determine whether a non-linear affine system can be converted to the chained form or not. The detailed information is as follows: Obviously, (1) satisfies conditions (3) and (4). Choosing h 1 = x 1 , h 2 = x 2 sin x 1 − x 3 cos x 1 , it can be easily seen that h 1 and h 2 meet conditions (5)- (8). Using the transformation we can convert (2) into the following chained forṁ where z 1 , z 2 , z 3 are new state variables and v 1 , v 2 are new control variables. Note that transformation (9)-(13) is globally differential homoeomorphic. Then we give a feasible and smooth desired reference trajectory of the WMRs centre position in Cartesian coordinates, that is X r , Y r . From the kinematic model (1), global transformation (9)-(13) and reference trajectory X r , Y r , it is straight forward to obtain the reference chained systeṁ Then we define tracking errors as According to (17), the kinematics of tracking errors becomė Analyzing the tracking error system (18)- (20) we can find that e 1 is controlled by v 1 independently. e 3 converges to zero leading e 2 to approach an invariant set. Furthermore, considering the external disturbances, we change (18)-(20) into a new forṁ where e 4 is a new error state which is equal to the derivative of e 3 and v * , f 1 is the unknown total disturbance contains system uncertainties and external disturbances and f 2 = (z r2 v r1 ) (1) + f * 2 includes reference trajectory, system uncertainties and external disturbances, which is statedependent too. Both of these two disturbances are continuously differentiable. Figure 2 is the architecture of the closed-loop system from reference trajectory to WMRs.
Controller 1, ESO 1 and Controller 2, ESO 2 for subsystem (21) and subsystem (23) and (24) will be designed respectively. NLESO reconstructs the total disturbance in terms of error and control signal. Feedback control signal is calculated through the sliding mode controller. Then combining the control signal with disturbance estimation, we get v * 1 and v * 2 . The chattering phenomenon caused by sign function in sliding mode controller can be reduced via decreasing the gain of sign function. Accurate estimation of disturbances ensures its rationality. Finally, through (9)-(13) we can obtain the actual control inputs to achieve trajectory tracking of WMRs.
For subsystem (21), NLESO 1 and Controller 1 are designed to regulate the angular velocity of WMRs. Their specific forms are as follows: NLESO 1:̂e whereê 1 ,f 1 are estimated states and 01 , 02 , 01 are constant gains and pertinent constants of NLESO 1, respectively. (e) is a segmented continuous function described as where R is a positive constant.

STABILITY ANALYSIS
In this section, our main results are formulated. Firstly, we prove the convergence of NLESO. Then the closed-loop system stability theorem is given. Let where 2 , 3 are positive constants.

then NLESO (28)-(30) is effective. The observation error satisfies
Proof. In actual experiments, when the coordinate origin is determined, the system states X , Y and are bounded, which means z 1 , z 2 and z 3 are bounded. At the same time, since the reference signal z r1 , z r2 and z r3 are bounded, the errors e i are bounded eventually. For the external disturbances in the experiment, its amplitude and derivative are bounded such that f 1  , y 2 (t ) = e 4 ( 02 t )−ê 4 ( 02 t ) 02 , y 3 (t ) = f 2 ( 02 t ) −f 2 ( 02 t ). From (28)- (30) and (23) and (24), the estimation error of NLESO 2 can be expressed aṡ y 1 = y 2 − 02 l 1 (y 1 ) where l 1 (y 1 ) = 01 y 1 + (y 1 ), l 2 (y 1 ) = 02 y 1 + (y 1 ), l 3 (y 1 ) = 03 y 1 + (y 1 ). Let y = [y 1 , y 2 , y 3 ] T and there exist positive definite function V (y) = y T Py + ∫ y 1 0 (s)ds and W (y) such that It is easy to see that we can always find a set of positive constants a, b, c satisfying (32)-(34) which makes sure that W (y) is positive definite. Therefore, condition (1) in H2 is satisfied and (1), (1) of H2 are clearly established. Finally, all the conditions of Lemma 1 hold so that the convergence of NLESO (28)-(30) is guaranteed. And the range of NLESO (28)- (30) gain is only determined by a, b and c. □ where 2 is a positive constant.
Proof. Introducing (27) into the tracking error equation (21) yields the closed-loop systeṁ Let the Lyapunov function be The derivative of V 1 along (21) iṡ where k 11 and k 12 are two parts of k 1 . It should be ensured that ) which means that e 1 converges to a certain invariant set. Then we haveV ∈ (0, 1), k * 1 = k 11 2 * > 0. Following Lemma 2, e 1 converges to an invariant set within time V 1 (e 1 (t 0 )) 1− * . Theorem 1 guarantees that M 1 is sufficiently small so that the error e 1 can converge to an invariant set near the zero.
For the closed-loop system (23) and (24) and (31), we choose the Lyapunov function as Take the derivative of V 2 along (23) and (24) to geṫ Integrating (35), we have The finite time is thus T 2 = t r + t s . From (18)- (20) we havė | as e 3 → 0. We know thaṫe 1 , z r2 are bounded, so e 2 is bounded. This completes the proof of Theorem 2. □ Remark 1. ( [39]) It should be noted that the finite time convergence here is only for the tracking errors e 1 , e 2 and e 3 rather than the whole closed-loop system due to the existence of NLESO. However, the asymptotic convergence characteristic of NLESO does not hinder the finite time convergence result of the tracking error. Even without NLESO compensation, the tracking error can still converge in a finite time, which is guaranteed by NTSMC. It is undeniable that tracking errors are precisely the most important performance indicators. Figure 3 is the experimental platform of WMRs. The whole WMRs experimental platform is made by Quanser company for For different linear velocities, the inverse model will calculate what the forward velocity would be for each wheel and generate a velocity command. In terms of sensors, the wheels of WMRs are equipped with two encoders with 2578 counts/rotation to measure linear velocity and transmit it to the host computer in real time via WiFi. The measurement noise can be attenuated by a digital filter to reduce the impact on the controller. In order to obtain accurate pose information, there are multiple reflective balls attached to the WMRs. Then optical cameras called optitrack camera measure the pose information of WMRs by tracking the reflective balls. The information measured by the optitrack camera is transmitted to the host computer via USB interface. The high-precision encoder and the optitrack camera constitute the sensor module. In the host computer, the reference trajectory can be parameterized. Combining with the feedback information from the sensor module, the controller is designed according to the proposed strategy and the control signals are sent to WMRs via WiFi to execute the trajectory tracking task. According to Theorem 1, a positive definite matrix P that satisfying (32)-(34) is calculated, the corresponding NLESO gain range can be determined accordingly. Figure 4 depicts the gain range of NLESO with a = 30, b = c = 1. In fact, there are many choices for matrix P, and the gain range will be large or small. The larger the range, the more the combination, as long as it does not exceed the boundary, the convergence of the NLESO can be guaranteed. The parameters of controllers and NLE-SOs (25)-(31) are listed in Table 1. And the trial and error procedure of parameter tuning is given as follows:

EXPERIMENTAL RESULTS AND DISCUSSION
Step 1) Initialize the parameters properly in terms of Theorem 1 and Theorem 2.
Step 3) Keep , , , D, p and q unchanged and increase k 1 and k 2 .
Step 4) If the control performance is not significantly improved, keep the controller and observer gains  unchanged, then fine-tune R 1 , R 2 and , , , D, p and q.
Experimental results for tracking a circular trajectory X r = 0.8 cos(0.2t + 3 ∕2), Y r = 0.8 sin(0.2t + 3 ∕2) with bounded disturbances at 20 s are depicted in Figure 5. WMR starts from the point (0, −0.8) and tracks the circular trajectory counterclockwise with a constant linear velocity and angular velocity. Here, we compare the proposed method with classical NTSMC in [20] and the ESO-based method in [40]. It can be seen from Figure 5 that all these methods can track the reference trajectory stably before the bounded disturbance appears. However, our proposed controller can tend to the reference trajectory more quickly when there is external disturbance,

FIGURE 6
Tracking errors indicating its stronger robustness. Although the ESO-based method in [40] can modify the control signal, there is still a large tracking error because the convergence speed of classical ESO estimation is not as fast as NLESO. While the NTSMC method has the largest error since there is no disturbance compensation. The tracking error results are plotted in Figure 6. The real-time peak error of the existing methods exceeds 37% of the tracking radius, while the tracking error for our proposed method is significantly smaller than this value, which just only about 13%. Furthermore, our method can re-track the reference trajectory more quickly, while the other methods require longer transient time. Figure 7 shows the estimated disturbances. We emphasize that [40] is the traditional NLESO based on fal (⋅) function. While this paper incorporates both linear and non-linear terms for NLESO design, which can provide faster convergence speed. Both disturbance observers can realize disturbances estimation. However, due to the lack of linear term compensation, the convergence speed of [40] is slower and the disturbances estimation effect is not good enough. From the control signals in Figure 8, it is found that the proposed controller can adjust the signal amplitude in real time to deal with bounded disturbances, so as to achieve good tracking effect. The performance indexes are listed in Tables 2 and 3, where RMSE=

CONCLUSION
A NLESO combining linear term and non-linear term has been constructed to estimate external disturbances. The convergence of NLESO has been proved and a scheme of determining the NLESO gains range is suggested. Compared with traditional ESO, it has a faster convergence rate. A controller involving