Decentralized robust active disturbance rejection control of modular robot manipulators: an experimental investigation with emotional pHRI

This paper presents an active disturbance rejection control (ADRC) method for modular robot manipulators (MRMs) based on extended state observer (ESO), which solves the problem of trajectory tracking when modular robot manipulators facing the emotional physical human-robot interaction (pHRI). The dynamic model of MRMs is formulated via joint torque feedback (JTF) technique that is deployed for each joint module to design the model compensation controller. ESO is used to estimate the interconnected dynamic coupling (IDC) term and the interference term caused by emotional pHRI. An uncertainty decomposition-based robust control is developed to compensate the friction term. The terminal sliding mode control (TSMC) algorithm is introduced to the controller to provide faster convergence and higher precision control effect. Based on the Lyapunov theory, the tracking error is proved to be ultimately uniformly bounded (UUB). Finally, experiments demonstrate advantages of the proposed method.


I. INTRODUCTION
Modular robot manipulators (MRMs) have drawn widely attentions in robotics community since they possess better structural adaptability and flexibility than conventional robot manipulators. MRM consists standardized robotic modules, which consists of actuators, speed reducers, sensors and communication units. MRMs are always employed in dangerous and complex environments, such as space exploration, hazard survey and furthermore physical human-robot interaction (pHRI). Nowadays, pHRI has become a research hotspot in the field of robotics. Emotional pHRI can ensure that when humans produce different kinds of emotions, the robots can continue to complete predefined tasks according to the established requirements. Hence, appropriate control systems are required to guarantee the robustness and precision of MRMs in contact with pHRI.
Besides the properties of modularity and pHRI etc., for achieving high-precision robot control, some scholars de-velop control issues of manipulator under conditions of random delay [1], input dead zone [2], input saturation [3], visual servo [4], optimization verification [5], and micropositioning [6], etc. Biglarbegian [7] gave an interval type-2 fuzzy controller for trajectory tracking of desired motion. Kasprzak [8] proposed motion planning for multicorporate MRM based on hierarchical search algorithms. Xu [9] gave an adaptive sliding mode control to achieve interaction control. Pham [10] proposed a robust control method. All aforementioned methods used the centralized control scheme. For practicality, the centralized control strategy is not suitable for the MRMs system because of its modularity and high complexity. Decentralized control is more suitable for MRMs system compared with centralized control strategy [11]- [12]. The advantages of decentralized control are to simplify complexity of the MRM system and effectively improve the running speed [13]- [15].
In order to address the problems in enhancing interaction VOLUME 4, 2016 stability and robustness, lots of researches focused on investigating active disturbance rejection control (ADRC) method for nonlinear system [16]- [17]. ADRC algorithm includes tracking differentiator, extended state observer (ESO) and nonlinear feedback, and it has been applied in many fields. In the field of robotics, ADRC algorithm has been applied in the control system of manipulator [18], such as Stewart platform [19] and calibration-free robotic eye-hand coordination [20]. Among them, ESO is a more effective and mature algorithm in ADRC. Huang et al. [21] studied a centralized controller based on ESO compensation, and carried out control simulation for a four-jointed finger. Ren et al. [22] proposed a model predictive control method with friction compensation based on the ESO for the trajectory tracking control problem of an omnidirectional mobile robot. However, the existing ESO-based ADRC of robot control method does not considering the pHRI. Indeed, the control torques of each robotic joint may increase significantly along with the instantaneous deviations of the feedback of joint position, velocity and torque measurements when in contact with external circumstance. In order to improve the convergence speed and the noise tolerance of dynamic systems, the sliding mode control (SMC) system [23] is introduced. Linear SMC is an insightful method for robot control, and it has been widely applied for its simple algorithm, fast response, and strong robustness [24]- [27]. However, the linear SMC cannot guarantee that the system error will converge to zero within a finite time. In order to provide faster convergence speed and higher control accuracy to compensate the disturbance term caused by the uncertain external environment, the terminal sliding mode control (TSMC) is designed to reach the nonlinear sliding surface [28]- [30] instantly. At the same time, JTF technique can reduce the effect of uncertain contact force created between the end-effector and payloads. Schaffer et al. [31] gave a perspective control method for flexible manipulators. A JTF-based distributed control method for MRMs is proposed in [32]. However, the above JTF-based robot studies did not consider the pHRI tasks. Therefore, an ideal controller for MRMs based on JTF technique should guarantee the robustness of robotic systems and the ADRC when MRMs facing the external environment.
A novel decentralized robust ADRC control is proposed for MRMs in contact with emotional pHRI. First, the dynamic model of the MRM systems is formulated via JTF technique, and the robust TSMC as well as the ADRC is developed by effectively utilizing the local dynamic information in joint space. Second, based on the uncertainty decomposition scheme, a decentralized robust controller is proposed to deal with the friction model uncertainties. Then, the ESO eliminates the effect of IDC term and interference items caused by emotional pHRI. In addition, TSMC is applied for its faster convergence and higher precision control. The experimental results directly demonstrate the effectiveness of the control algorithm. The joint position errors and velocity errors are proved to be uniformly ultimately bounded (UUB). Finally, experiments demonstrate advantages of the proposed method.
The major contributions are summarized: 1.To the best of author's knowledge, it is first time to solve decentralized robust ADRC problem of MRMs with pHRI.
2.Because of different kinds of emotion, we propose a control algorithm to meet the trajectory tracking problem of people in different emotions, and the tracking performance of MRM system can be improved by experimental verification.

A. DYNAMIC MODEL FORMULATION
Since MRMs have many mechanical modules, we formulate the MRM subsystem dynamic model (see Fig. 1). Similarly in [33], we assume following conditions.  Based on MRMs with n modules via interconnected subsystems, the ith subsystem of MRMs is [34]: is the joint friction, z mi and z qj represent the unity vectors along axis of rotation of ith rotor and joint, τ f i /γ i is the coupling torque divided by gear ratio, where d ic (q i ) is torque sensor output disturbance. d is (q i ) indicates external disturbance which caused by emotional pHRI. Remark 1: The external disturbance which caused by pHRI d is (q i ) will change with the human's mood. When human beings are in a delighted mood, they will produce friendly actions to the robot, such as shaking hands, gently stroke, etc. When in anger or sadness emotion, humans will beat or collide with the robotic manipulator [35]- [37]. According to the reference in [38], f i (q i ,q i ) denotes: We consider f si , f τ i are approximate to actual values, and f si e (−fτiq 2 i ) can be linearized asf si ,f τ i . Hence, we have Substituting (4) into (3), we have: mi z qjqj for facilitating as follows: We obtain relationships of (7) and (8), where D i j is dot product of z mi , z qj and D i j is alignment error. We also have

B. SYSTEM STATE SPACE DESCRIPTION
Rewriting (1), we havë Next, a decentralized robust ADRC in contact with emotional pHRI is proposed to ensure tracking error UUB.

III. DECENTRALIZED ROBUST ADRC BASED ON TSMC A. DESIGN OF ROBUST SLIDING MODE CONTROLLER
According to (9), we define the actuator output torque: where u i is control input of ith joint. Define parameters: where e i1 is the position tracking error, r i is the filter error term, a i is the auxiliary error function, λ i is arbitrary positive constant, q id is the expected trajectory. Then u i is defined as: where u ic1 =I mi γ i a i +b iqi + f ci +f si e (−fτiq 2 i ) sgn(q i ) compensates for the moment of motor. u ir is a robust controller to deal with friction modeling. u in3 is active disturbance rejection controller, which compensates IDC and disturbance torque item d i (q i ).
Therefore, the decentralized robust ADRC problem has transformed obtaining u ic1 , u ir and u in3 , and in this way, to realize compensation of model uncertainty as well as emotional pHRI.
Next, we design a decentralized robust controller to compensate the model uncertainties.
First, we decomposeF i into: whereF ic is unknown constant vector,F iv is limited by:

VOLUME 4, 2016
Based on decentralized controller design, we choose: where u i u is compensate f qi (q i ,q i ). u i pc and u i pv compensatẽ F ic andF iv . The control of the ith joint u i pc , u i pv and u i u are defined as: where ζ i = Y (q 1 ) T r i , ε i , ε i pn are positive control parameters. After that, to improve the convergence rate, the TSMC algorithm is introduced in this paper, firstly, the position tracking errors of e i1 and e i2 are defined as follows: where α i denotes virtual control, and as shown follow: where η i is a positive integer. The time derivative of e i1 , e i2 is taken: Select the following Lyapunov functions for the ith subsystem: The derivation of the above formula can be obtained: For ensuring stability of SMC, the state can follow desired state within a specified finite time, and tracking error converges asymptotically to 0.
Secondly, define TSMC: where β i is arbitrary positive number, p i and q i are any positive odd numbers, and p i > q i > 0. By (22) and (23), we can obtain the derivative of s i with respect to time: The decentralized sliding mode control law is: In the end, the robust sliding mode controller is:

B. DESIGN OF ACTIVE DISTURBANCE REJECTION CONTROLLER
In this part, we introduce the ADRC based on the ESO. Based on the state space (10), the extended state is introduced to estimate IDC term and interference items which caused by contact with emotional pHRI. Define the following terms for the active disturbance rejection controller design.
where k 1 is a positive feedback gain. ESO deals with modeling uncertainty in feedback linearization control [39]. First, (10) is now described as: From (30), a linear ESO is defined as: T is the state estimation and ω 0 > 0 means the bandwidth of ESO. According to [40], we have Denotex ij = x ij −x ij , j = 1, 2, 3 as estimation error. From (31), (32), the estimation error is: Define ε ij =x ij /ω j−1 0 , j = 1, 2, 3, then, (33) can be rewritten as:ε where A i is Hurwitz inferred in (33)  So that we can obtain ESO-based control law: Combined with (11), (13), (28), and (35), the decentralized controller is: (36) Remark 2: With the combination of the sliding mode control and active disturbance rejection control, the estimated disturbance and chattering can be observed, and then adjust the switch gain to reduce chattering effect generated by the system.

C. STABILITY PROOF
Theorem 1: Consider n modules MRMs system with (1), and the model uncertainty is defined by (7) and (8), as well as the disturbance term caused by the emotional pHRI in (2). The tracking error can be guaranteed UUB under control law proposed by (36). Proof: Select Lyapunov candidate function: Then, the derivative of (37): Therefore, the derivative of Lyapunov function is shown: SinceV i ≤ 0 is negative semidefinite, which implies s i ) is integrated from 0 to t, which is: Since V (0) is bounded, V (t) is non-increasing with time, so that : According to Barbalat's lemma, V p (t) → 0 is known when t → ∞ is used. It means e i1 → 0, s i (t) → 0. and, if e i1 = 0, s i (t) = 0. The tracking error of the MRMs system will asymptotically be 0, in the end the theorem is completed.

A. EXPERIMENTAL SETUP
The experimental platform of this experiment is 2-DOF MRM (See Fig. 2) which produced by Quanser Company. The platform is composed of QPIDe data acquisition board, a linear power amplifier (LPA), two sets of joint modules and  each of them contains a DC motor, a joint torque sensor, a speed reducer and an absolute encoder. The joint torque sensor measures the joint torque, the absolute encoder measures the position of the connecting rod end, the incremental encoder measures the motor end displacement in the joint and sends all the measured data back to the host computer through the QPIDe data acquisition board. The speed reducers are coupled with the DC motor. We consider two kinds of emotional pHRI which include delighting and sadness (See Fig. 3). For the first situation, the player feels happy thus shaking hands with the endeffector of the manipulator. In the second scene, the player is dissatisfied and tap the manipulator at 30s and 90s. For the third situation, people beat the manipulator to demonstrate the motion, and after a while at 60s, people shake hands with manipulator initiatively to demonstrate the apology. The robotic joints are followed with the trajectories of q 1d = 3 10 sin( π 20 )t + π 18 sin( π 10 )t, for the first situation and for the second scene and the third situation is )t.
The model parameters and control parameters of different emotion are given in Table 1. The physics parameters of various mood are shown in Table 2.

B. EXPERIMENTAL RESULTS
For comparing the advantages between existing [41]- [43] and proposed method, two different control scheme are taken into account.
(1) Position tracking performance Figs. 4-7 are trajectory tracking curves with delighted emotion, respectively. From these figures, both methods are effective. Figs. 8-11 are trajectory tracking and error curves with sadness and fusion emotion, respectively. We can obtain the negative emotion appear, the position trajectory can also follow the desired trajectory.
(2) Velocity tracking performance Figs. 12 and 13 are velocity error tracking curves under existing and proposed control methods with delighted emotion. The existing and proposed method of up-bound velocity error are less than 3e-3rad/s, 1e-3rad/s. That is because the   existing method has not considered ADRC to compensate the IDC effects and emotional pHRI. Fig. 14 is velocity error tracking curves under proposed control method with sadness emotion. When emotional pHRI happens, the error values decrease to normal ranges within short time, which may attribute to active disturbance rejection control. Fig.  15 is velocity error tracking curves under proposed control method with fusion emotion. We can get the similar results with delighted and sadness emotion, that is attributed to the effective decentralized robust ADRC method. increase when emotional pHRI occurred and lead to the robotic system out of control. The proposed active disturbance rejection control realizes the optimization of tracking errors. For the experimental results one can get the proposed ADRC method can guarantee stability as well as accuracy.

V. CONCLUSION
A decentralized ADRC scheme for MRMs in contact with emotional disturbance is proposed. Based on JTF technique, we obtain the dynamic model of MRM. When MRM systems facing external environment, we transform the control problem with emotional pHRI into active disturbance rejection. Based on the strong estimation ability of ESO, we design ADRC, and it is used to estimate IDC term and the interference term caused by emotional pHRI. TSMC is used to guarantee high precision control as well as fast convergence. According to Lyapunov method, trajectory tracking error is proved to be UUB. Experiments are performed to confirm effectiveness. In our future work, there will be a more detailed and completed analysis of what kind of interaction behavior human will do for different emotions.