Spiking-free HGO-based Dynamic Surface Control for Flexible Joint Manipulator with Unmeasurable States

 Abstract: For the flexible joint manipulator control system (FJMCS) with unmeasurable states, a novel partial states feedback control (PSFC) is proposed. Firstly, the unmeasurable states and the uncertainties are observed by a high-gain observer (HGO) simultaneously. Then, a dynamic surface controller is proposed based on the output of the HGO. The newly proposed controller has several advantages over existing methods. First, the proposed controller not only uses the estimate states to avoid using unmeasurable states, but also uses the estimation of uncertainties to enhance the robustness of FJMCS. Second, a novel spike suppression function (SSF) is developed to avoid the estimation spike problem in the existing HGO-based controllers. The closed-loop system stability is proved by the Lyapunov theory. Simulation results demonstrate the effectiveness of the proposed controller.


Introduction
With the development of mechanical, control and sensor technology, the design of robot has made great progress and gradually replacing people in the fields of repetitive labor, emergency rescue and home service, etc. [1][2][3][4]. In robot system, the robot manipulator has a very important role as the end-effector of the robot which interacts with the external environment [5]. For the robot manipulator, many studies on control strategies for different kinds of robot manipulator components have been completed. And, among these strategies, the disturbance attenuation control takes an  Wen-Dong Wang wdwang@nwpu.edu.cn important role since the robot manipulator system is a typical nonlinear uncertain system, and the uncertainties can be caused by the unmodeled dynamics, external environment and parameter uncertainties [6][7][8][9][10]. Compared with the conventional rigid robot manipulator (RRM), the flexible joint manipulator (FJM) has become a research hotspot increasingly since it has several advantages, such as smaller actuator, light weight, lower cost and better maneuverability, etc. [11][12][13][14]. However, the flexibility of FJM brings more complex uncertain characteristics of dynamic model. The dynamic model of conventional RRM is a low-order (two-order) matched uncertain system. But the dynamic model of FJM is a typical high-order (four-order) mismatched uncertain system.
To solve the mismatched uncertain problem, the backstepping control method (BCM) is a good candidate [15]. The main advantage of BCM is that it can suppress the mismatched uncertainties through a step-by-step recursive process [16]. Some BCM-based methods for FJM have been developed, such as BCM-based adaptive control [17], BCMbased fault-tolerant control [18], BCM-based neural network control [19], BCM-based intelligent control [20] and BCMbased L2-gain control [21]. For conventional BCM, the computations needed for back-stepping is very tedious and complicated, this is so called the problem of "differential explosion". To solve this problem, the dynamic surface method (DSM) was proposed [22][23]. The DSM not only can suppress the mismatched uncertainties by the step-bystep recursive process like BCM, but also adopt a low-pass filter to avoid the explosiveness of backstepping. Then an adaptive fault-tolerant control for FJM based on DSM was proposed [24], and the complex differential calculation for 1 ·3· backstepping in [17][18][19][20][21] can be avoid by the low-pass filter. However, in order to complete the step-by-step recursive process, the BCM-based and DSM-based controllers in [17][18][19][20][21]24] not only need to measure the states of link, but also need the states of motor for feedback. Thus, the measurement systems for the controllers in [17][18][19][20][21] are much more complex than the measurement system for the conventional RRM controller which only needs the states of link.
As a control method which only needs the measurements of partial states, the output feedback control (OFC) scheme can simplify the measurement system for high-order systems [25][26][27][28], significantly. Some control schemes for FJM based on OFC have been developed to simplify the measurement system. A control scheme which only requires the feedback of position states was developed for FJM [29], in which an observer was design to estimate the unmeasurable link and the actuator velocity, but the uncertainties of FJM were not considered in [29]. The unmeasurable velocities of link and motor were estimated by an approximate differential observer, and a complex adaptive control law was proposed to eliminate the effect of the uncertainties [30]. A high-gain observer was designed to generate pseudo signals for unmeasurable states, and an adaptive law of the projection type was developed to estimate the unknown feed-forward term [31].
To guarantee the observation and control precision, the observer gains of the observers in [29][30][31] should be chosen as the high gains. Thus, the observers used in [29][30][31] are typical high-gain observer (HGO). However, the undesirable spike problem can be caused by the high observer gains when the initial values of estimate and true states are not equal [32][33]. The spike problem may cause the saturation of control input and even lead to the instability of whole system. In addition, for the schemes in [30,31], on the basis of using observer, an additional robust control scheme should be adopted to eliminate the effect of uncertainties, thus the structures of controllers in [30,31] are very complex.
In this paper, a new partial states feedback control (PSFC) scheme based on DSM and HGO is proposed for the FJM. The main contributions of this paper are summarized as follows: (1) The spike problem in the existing HGO-based schemes are solved: a novel spike suppression function (SSF) is adopted to solve the estimation spike problem in the existing HGO such as the observers in [29][30][31]. And, since the spike problem can be avoided even the observer gain is very high, a high observer gain can be adopted in the proposed scheme to guarantee the high control precision.
(2) Only partial states of FJM need to be measured: The proposed method only needs the same measurement states as conventional RRM by using the HGO to estimate the unknown states.
(3) The structure of proposed controller is simple: The unmeasurable states and uncertainties are estimated by HGO simultaneously. Then the uncertainties of FJM are compensated by using the estimation of uncertainties without requiring the additional robust control scheme such as the adaptive schemes used in [30,31]. Thus, the structure of proposed control scheme is simpler than the schemes in [30,31].
The remaining parts of this paper are as follows. In section 2, the dynamic model, the design objective, and the motivation of this paper are expounded. The main results are presented in Section 3. In section 3.1, a high-gain observer (HGO) is developed. In section 3.2, a novel spike suppression function (SSF) is developed, and then a dynamic surface controller based on HGO and SSF is proposed. In section 4, a simulation verifies the effectiveness of proposed HGO, SSF and controller. In section 5, the conclusion of the whole paper is presented. The flexible joint manipulator is shown schematically in Figure 1. The dynamic model of the flexible manipulator control system (FMCS) can be described as follows [34]: Where q and q are the rotation angle and angular velocity of link, respectively.  and  are the rotation angle and angular velocity of motor, respectively. I and J are the moment of inertia of link and motor, respectively. M and L represent the mass and the distance between the joint and the centroid of the link, respectively. g is the Jia-Fan Zhang et al.

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gravity acceleration. K represents the elastic stiffness of the flexible joint.  is the motor torque.
MgL  represents the unmodeled dynamics of the FMCS. 1 d and 2 d denote the disturbance from the environment.
Define the lumped uncertainties 1 d and 2 d as Then, Equation (1) can be rewritten as The states are defined as The control objective is to design the control input u such that the tracking error 1 x approaches to zero, while ensuring that all closed loop signals are bounded.

Problem Description and Purposes of this paper
For the traditional control scheme for FJM in [17][18][19][20][21]24], the link states q , q and the motor states  ,  are both assumed to be measurable, but the RRM controllers only need the link states q and q . Thus, the measurement system for the controller in [17][18][19][20][21]24] is much more complex than the measurement system for RRM. This motivates the first research topic of this paper, that is, designing a control scheme which only needs the same measurement states as RRM to simplify the measurement system of FJM. Thus, we have the following assumption: Assumption 1: For the FJM system (1), q and q are measurable,  and  are unmeasurable.
To solve the unmeasurable problem, some schemes in [26][27][28] have adopted an observer to estimate unmeasurable states. And a high observer gain should be chosen to guarantee the observation precision. However, an undesirable spike problem may be caused by the high observer gain (see [29][30]). To explain the spike problem, the high-gain observer in [28] can be give as following: where 1 y denotes the angle state, 1 y and 2 y are the estimate angle state and the estimate angular velocity state, respectively. 0 k is the observer gain. Let estimate errors 1 The solutions for the differential equation (10) can be easily obtained as  (11) It can be known that cannot be known in advance for the most of cases. Thus, the initial estimate state error ( ) 1 0 y maybe not equal to 0 and even a large value. If the initial value of true states 1 (0) y is unknown, to avoid the potential spike problem, the HGObased controllers in [29][30][31] cannot adopt the enough high observer gain to guarantee the control precision. This motivates the second research topic of this paper, that is, design a new scheme not only can guarantee high precision, but also eliminate the undesirable spike problem in any cases.
On the other hand, from Equation (4), it is clear that FJMCS is a typical mismatched uncertain system. To suppress the matched and mismatched uncertainties, the conventional control schemes in [30,31] for FJMCS need an additional robust scheme. Thus, the structures of the controllers in [30,31] are very complex. This motivates the third research topic of this paper, that is, avoiding using additional robust scheme. The motivation of this paper is summarized as follows: Motivation of This Paper: (1) To avoid using the unmeasurable states, the unknown states will be estimated by designing a HGO in following section 3.1. (2) To use the high observer gain, while avoiding the spike problem, a novel SSF will be proposed in the following section 3.2. (3) To suppress the uncertainties without using additional complex method, the proposed HGO are used to estimate not only the unmeasurable state but also the uncertainties.

High-gain Observer
For the system (13), since  and  are unmeasurable, the states 3 x and 4 x are unknown. Combining unknown states with uncertainties to construct the new states. The new states are defined as Since 11 px = , the control objective can be equivalent to If the states 3 p and 4 p are known, the system (12) is a matched uncertain system. Let the matched uncertainty 5 f dp = , where 5 p is the extended state. The system (12) can be extended as (13) where 12 12 , (14) ( )  (15) The following assumption is needed: To satisfy the Assumption 1, we can know that the lumped uncertainties Then a five-order HGO is proposed in the following theorem to estimate unknown states 3 p , 4 p and uncertainty f d , simultaneously. Theorem 1: For the system (12), a HESO is constructed as ( )  (18) If assumption 1 and 2 are valid and the observer gain satisfies , then the estimate error − p = p p will converge to following small region: (19) Jia-Fan Zhang et al.

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Proof (Proof of Theorem 1): Define the estimate error vector as − p = p p (20)   Differentiating p gives − p = p p (22) Substituting (13) and (13) into (22) yields Construct Lyapunov function 1 V as Then calculating the time derivative of 1 V along the trajectories of Equation (23), we have ( ) T T  T T  T  T  T  T  p  p   T  T  T  T  T  T  p (29) Let 0 0 ck −，then p is always bounded as is a unknown positive constant. From Equation (29), it can be known that 1 0  p ，thus p will converge to following region: (31), it can be known that the observer gain 0 c should be chosen large enough to ensure the estimation error is small enough. Thus, the proposed observer (20) also is a typical HGO. The undesirable spike problem may be caused by the high observer gain as mentioned above in section 1.2. To solve this potential problem, a novel spike suppression function (SSF) will be proposed in the following control design.

Arm Dynamic Surface Controller Based on HGO
and SSF Based on the estimate states 3 p and 4 p , and the estimation of uncertainty 5 p , a DSM control scheme for the FJMCS (12) will be proposed by following four backstepping steps. And a novel SSF is adopted in the proposed controller to deal with the aforementioned spike problem of HGO.
Step 1：Define a tracking error as 11 sp = , the virtual control of 2 p is designed as Define a low-pass filter as if t →, the stability and control precision of the proposed controller will not be affected by the SSF. The detail stability and control precision analysis of the proposed controller with SSF will be given in the Theorem 2.
Before giving the Theorem 2, the following assumption which is commonly used in DSM (such as in the section 3.2 in [22], the author assumed that the derivative of virtual control input is bounded as a positive constant) should be assumed to be valid throughout this paper: Then we have where min{ } • is the minimum function.   Remark 3.5: From the above research in section 3.1 and 3.2, the three purposes of this paper mentioned in section 2.2 are achieved: 1) From (38), it is clear that the proposed controller only needs the same measurement states as RRM; 2) By designing a novel SSF, the spike problem can be solve, and the control effect also will be discussed in the following simulation section; 3) Since HGO (16) estimates not only the unmeasurable state ( ) 1,2,3,4 i pi = but also the uncertainty f d , the roust of controller can be guaranteed without using additional complex method.

Simulation Results
In this section, to illustrate the effectiveness of the proposed control scheme, the mathematical simulation is presented. Consider the parameters for FJMCS system (1) The desired rotation angle of link is chosen as To ensure the estimation error is small enough, a high observer gain of HGO is selected as The initial value of HGO is set as         (1) Proposed partial state feedback controller (PSFC)： Like case 1, the expression of controller is given in (38). The parameters of PSFC and observer are given in (80) And () sigmf • is used to avoid the chattering problem and suppress the uncertainties. The explanation of () sigmf • can be found in previous work 35 . The unmodeled dynamics and the disturbance from the environment are chosen as in (79).
The above PSFC and FSFC are used for tracking two kinds of desired angles 1 d q = rad and cos( ) d qt = rad, respectively. The simulation results for case 2 are shown in the Figures 10~13. From the simulation results, it is clear that both the conventional FSFC and the proposed PSFC can guarantee the fast and accurate convergence of tracking error for the two kinds of desired angles. Thus, compared with the conventional full states feedback scheme, the proposed method of this paper can achieve a similar excellent control performance even with partial states feedback.

Conclusions
In this paper, a novel PSFC is proposed for the FJM with unmeasurable system states based on DSM and HGO. The main contributions here are that: 1) The proposed method only needs the same measurement states as conventional RRM; 2) The estimation spike problem in the existing HGObased control is solved by developing a novel SSF; 3) Without using additional complex robust method which is wildly used in the conventional PSFC schemes, the proposed control scheme still can suppress the uncertainties by using HGO to estimate the unmeasurable states and uncertainties simultaneously. Finally, mathematical simulation result demonstrated that the performances of the proposed controller is excellent.

Declaration
Funding This work is supported by Natural Science Foundation of Shaanxi Province (Grant No. 2020JM-131 and 2020KW-058). This work is supported by the Key Research and