Disturbance-Observer-Based Fuzzy Control for a Robot Manipulator Using an EMG-Driven Neuromusculoskeletal Model

Robot manipulators have been extensively used in complex environments to complete diverse tasks. The teleoperation control based on human-like adaptivity in the robot manipulator is a growing and challenging field. This paper developed a disturbance-observer-based fuzzy control framework for a robot manipulator using an electromyography- (EMG-) driven neuromusculoskeletal (NMS) model. The motion intention (desired torque) was estimated by the EMG-driven NMS model with EMG signals and joint angles from the user. The desired torque was transmitted into the desired velocity for the robot manipulator system through an admittance filter. In the robot manipulator system, a fuzzy logic system, utilizing an integral Lyapunov function, was applied for robot manipulator systems subject to model uncertainties and external disturbances. To compensate for the external disturbances, fuzzy approximation errors, and nonlinear dynamics, a disturbance observer was integrated into the controller. The developed control algorithm was validated with a 2-DOFs robot manipulator in simulation. The results indicate the proposed control framework is effective and crucial for the applications in robot manipulator control.


Introduction
Robotic manipulators are increasingly used in welding automation, robotic surgery [1], and space, as they are able to complete diverse tasks in complex environments, such as uncertain system dynamics, time-vary delays, and unknown external disturbances. e robot manipulator may work within dangerous environments for unfriendly tasks, such as handing radioactive material and searching, and the teleoperation control of the robot manipulator has been widely utilized into the controller design. Yang et al. [2] proposed an admittance-adaptation-based methodology for robot manipulators when interacting with unknown environments and guaranteed trajectory tracking performance. Recently, there is a growing demand for the natural interface between the robotic manipulator and the user [3]. Specifically, the robotic manipulator is teleoperated by the user with human-like characteristics. A human-like learning controller was proposed to optimally adapt interaction with unknown environments [4], and the human-like adaptivity was shown well by the robot manipulator in stable and unstable tasks. e development of robot manipulator controller with human-like characteristics (the user's intention) requires accurate and robust decoding of motor function. Muscle electromyographic (EMG) signals from the central nervous system (CNS) [5,6] are widely used in the user's intention detection as EMG signals are relatively easy to acquire and process and provide essential information on human motion. EMG-based modelling methodologies have been utilized into various human-machine control algorithms for robot manipulators. Ryu et al. [7] developed a continuous position-based strategy for a robot manipulator with EMG signals and the manipulator could replicate the movements from the user well, thereby improving the control strategy for teleoperated robot manipulators. Artemiadis et al. [3] proposed a teleportation methodology for a robot manipulator based on EMG signals and position feedback and realized a good master-slave manipulator system with human-like adaptivity. Bu et al. [8] proposed a Bayesiannetwork-based model to predict occurrence probabilities of the motions with the given information of the previous motion and classify hybrid motions with EMG signals. eir results demonstrated that the EMG-based Bayesian network model could improve the robustness and stability for motion classification.
erefore, integrating an EMG-driven neuromusculoskeletal (NMS) model with human-like characteristics into the robot manipulator controller is valuable and could be crucial in the robot manipulator control field [9].
Diverse control strategies for robot manipulators have been developed in complex environments [10][11][12], as these environments could degrade the performance of the robot manipulator system [13,14]. Lin et al. [15] developed fuzzy Gaussian mixture models to approximate the objects' shape for robot manipulator grasping tasks under unknown environments, and the model has a good grasp quality. He et al. [16] proposed a disturbance-observer-based control strategy to approximate unknown parameters and disturbance for multimanipulator robots and validated on a dual-arm cooperative robot (Baxter).
eir results showed that the controller has an accurate control performance and was able to compensate for the errors due to the model uncertainties and external disturbances. Yang et al. [17] proposed a disturbance-observer-based impedance control for uncertain robot manipulators in unknown environments. Fuzzy logic system and disturbance observer are two commonly used techniques in the control system to compensate for unknown functions, parameters, and disturbance [18][19][20]. erefore, in this study, a disturbance-observer-based fuzzy algorithm was integrated into the control framework for a robot manipulator, subjected to practical problems including external disturbances and model uncertainties. e objective of this study was to design a disturbanceobserver-based fuzzy controller for a 2-DOFs robot manipulator using an EMG-driven NMS model. e main contributions of this paper are as follows: (1) a disturbanceobserver-based fuzzy control framework is developed that fully incorporates a robot manipulator system with an EMGdriven NMS model and could be applied in controlling the movement of robot manipulators with human-like characteristics; (2) the user's motion intention was able to be well-predicted by the EMG-driven model and transmitted into the desired velocity through an admittance filter, which allows the robot manipulator system behave with humanlike adaptivity; (3) in the current paper, the control framework we proposed in general is useful for obtaining the human-like characteristics, as well as simulating the control strategy in the robot manipulator system, and thus be of large benefit for teleoperation robot manipulator applications.

Controller Framework.
A disturbance-observer-based fuzzy control framework ( Figure 1) was developed for a robot manipulator system in complex environments. In the robot manipulator controller design, to complete tasks with human-like characteristics, an EMG-driven NMS model through the EMG signals and joint angles from the user was used to estimate the desired torque. e desired torque was transmitted into the desired velocity through an admittance filter. A disturbance-observer-based adaptive controller with fuzzy compensation was developed for the robot manipulator with unknown model dynamics.

EMG-Driven Neuromusculoskeletal Model.
A previously developed EMG-driven NMS model [21] was used in this study. It reproduces the transformations from EMG signal generation and joint angles to musculotendon forces and joint torques. e NMS model consists of four components: musculotendon kinematics, muscle contraction dynamics, muscle activation dynamics, and joint dynamics [22]. e musculotendon kinematics component used the 3D joint angles to calculate musculotendon lengths and moment arms of individual musculotendon units (MTUs) through a musculoskeletal model. e muscle activation dynamics component calculated muscle activation based on filtered EMG signals. e relation between neural activation u(t) and filtered EMG signal e(t) was represented by a recursive filter [23] as shown in the following equation: where e i (t) is the linear envelope of the EMG signal of i th muscle; u i (t) is the neural activation of i th muscle; d is the electromechanical delay; μ is the muscle gain coefficient; and β 1 and β 2 are the recursive coefficients and are subject to the following constraints to obtain a stable solution [21,23,24]: e relationship from neural activation to muscle activation is nonlinear and was formulated to describe muscle activation dynamics as follows: where a i (t) is the i th muscle activation and A i is the nonlinear shape factor of i th muscle and subjected to the interval (− 3, 0) with zero representing a linear relationship and negative values introduce a nonlinear relationship [24,25]. Musculotendon forces were computed through muscle contraction dynamics, based on a 3-element Hill-type muscle model; a series elastic element (SE), a contractile element (CE), and a parallel elastic element (PE). Each MTU's force (F mt ) could be represented as a function of 2 Complexity muscle activation and muscle kinematics as shown in the following equation: where F m 0 is the maximum isometric muscle force; f a (l m ) is the active force-length relationship that describes the ability of muscle fibres to generate forces at different lengths; l m is the fibre length normalized with the optimal fibre length; f v (v m ) is the force-velocity relationship that represents the muscle fibre force contribution of the fibres' contraction velocity (v m ), and the velocity was normalized with maximum contraction velocity and optimal fibre length; f p (l m ) is the passive force-length relationship that expresses the force response to the fibres to strain; d m is the muscle damping coefficient which represents the muscle damping characteristics; and ϕ is the pennation angle of the fibres.
Joint torques were then estimated by the product of musculotendon forces F mt and moment arms r mt through the joint dynamics component as shown in the following equation:

Admittance Filter.
To control the robot manipulator based on the user's adaptivity (desired torque estimated from EMG signals), an admittance filter was used to transform the joint torque into the desired angular velocity [26,27].

Δτ(s)
where Δτ and Δq are the torque and position regulation and M, B and K are the mass, damping, and stiffness parameters of the admittance filter, respectively.

Robot Manipulator Dynamic
Modelling. e dynamics of a robot manipulator was generally modeled as follows [28]: where M(q) is inertial matrix; q is the joint angle; C(q, _ q) is centripetal and Coriolis force; G(q) is the gravitational force; τ is the joint torque; and f dis is the unknown disturbance.
We rewrite this robot manipulator dynamics as follows: where We aim to design an adaptive controller that could ensure the robot manipulator system has satisfactory tracking performance under input nonlinearity and uncertain model dynamics.
e B(x) could be divided as follows: where  (A1) the model's musculotendon kinematics were used to compute musculotendon lengths and moment arms; (A2) muscle activation dynamics were employed to calculate the level of muscle activation involved in the processed EMG signals; (A3) muscle contraction dynamics according to a Hill-type muscle model were applied to predict musculotendon force, using calculated musculotendon length and muscle activation as inputs; (A4) joint dynamics was used to compute joint torques, using the calculated musculotendon forces and moment arms as inputs. e EMG signals and joint angles from the user were used to estimate the motion intention (desired torque) through the EMG-driven NMS model. en, the desired torque was transmitted to the desired velocity through an admittance filter. A disturbance-observer-based adaptive fuzzy controller was developed for the robot manipulator system with model uncertainties.

Integral Lyapunov
Analysis. An integral Lyapunov function is modeled as where According to (14), then (13) is remodelled as a highdimensional Lyapunov function Based on the definition of B α , B α has minimum and maximum eigenvalues λ min (B α ) and λ max (B α ), Considering ϑ component in B α function, which is independent of s, x, and v, we can obtain us, V 1 ≥ 0 is proved. As B α and B ϑ are symmetric, the derivative of V 1 is modeled as with Considering the following equations, we have As ϑ is a scalar, and σ � ϑs i , we have Combining (21) and (22), (18) can be rewritten as Considering (12), we rewritten (23) as Since B d , α, and B d α are symmetric, we have and rewrite (24) as Considering (14) and B α � B d α, where 4 Complexity 2.6. Disturbance-Observer-Based Fuzzy Controller. As the accurate and complete dynamics of a robot manipulator is difficult to obtain, especially in complex environments with unknown external disturbance, a fuzzy logic system is adopted for the estimation of unknown functions in the robot manipulator. Generally, the fuzzy rules are given as where h i represents i th fuzzy rule.
where Z � [Z 1 , Z 2 , . . . , Z n ] ∈ R n is the input variable; S i � (( n j�1 φ(Z j ))/( m l�1 ( n j�1 φ(Z j )))) are the fuzzy basis functions; m is the number of fuzzy rules; and W denotes the adaptable weight parameters.
In this study, the uncertain term F(x) is estimated as where Z � [x T 1 , x T 2 ] T and ς represents the approximate error and ‖ς‖ ≤ ς * (ς * > 0). en, we rewritten (9) as To eliminate the effect of the external disturbance and unknown terms in the system, a disturbance observer is utilized in the controller design. We define D � R + p − ς, and (32) can be represented as As in the robotic system, the actuator and input saturation exist [29]; the motor torque is normally assumed to be constrained with saturation in practical. erefore, p � − Δ B B − 1 (x)τ is bounded. In this study, the r in this robot manipulator system is assumed as bounded with ‖r‖ ≤ d c , where d c is a positive constant. As the derivative of a differential continuous function with bounded constraints is also bounded [30], we can obtain ‖ _ D‖ ≤ ϱ, where ϱ > 0. For the disturbance observer design, an auxiliary variable Ω is defined as Ω � D − Ψx 2 with Ψ � Ψ T > 0. With (33), we can obtain en, the estimate of _ Ω is calculated as where D � Ω + Ψx 2 and is the estimate of D. en, we define D � D − D as the disturbance estimate error, and we have en, the derivative of D is computed as with Θ � Θ − Θ * . Based on the disturbance observer [31] and fuzzy logic rules, the control law of this robot manipulator is developed as e updated law of the fuzzy controller is developed as where Γ i and ε are positive constants.

Simulation
e EMG-driven NMS model was implemented in OpenSim 3.3 through the calibrated EMG-informed NMS modelling toolbox (CEINMS) [21]. OpenSim was also used to calculate musculotendon lengths and moment arms using joint angles through the scaled musculoskeletal model [32]. CEINMS was then employed to calibrate a subject-specific EMGdriven NMS model to predict joint torques. A generic reaching task was simulated with a musculoskeletal arm model (Figure 2), and the data were acquired from a publicly available database (https://simtk.org/frs/index.php? group_id�657) [33]. e input data include EMG signals and joint kinematics of right arm shoulder and elbow. e EMG signals of 6 muscles of the right arm were simulated through static optimization tool (resolving the net joint moments into individual muscle forces by minimizing the sum of squared muscle activations) in OpenSim, including triceps long, triceps lateral, triceps medial, biceps long, biceps short, and brachialis.
Before using the EMG-driven NMS model to predict joint torque, a calibration process was applied to obtain a subject-specific EMG-driven NMS model with individual's muscle-tendon properties. To perform the calibration, inverse dynamics was used to calculate the joint moment as measured joint moments. Inverse dynamics calculated the forces and torques of joints by solving dynamic equation of motion of the user as shown in the following equation: where q, _ q, € q are the position, velocity, and acceleration of the generalised coordinates; M(q) is the mass matrix; C(q, _ q) is the centripetal and Coriolis forces matrix; G(q) is the gravitational forces matrix; and τ is the vector of unknown generalised forces.

Complexity
During the calibration of the subject-specific EMGdriven NMS model, optimal fibre length l m 0 and tendon slack length l t s of each MTU were bounded within ±15% from their initial values, and muscle activation dynamics parameters A, C 1 , C 2 were calibrated globally. e shape factor A was bounded between − 3 and 0 and coefficients C 1 , C 2 were bounded between − 1 and 1. A strength coefficient constrained between 0.5 and 2.5 was assigned to each MTU and was used to calibrate maximum isometric force. During the calibration, these subject-specific parameters were refined by an optimization algorithm to minimize the error between estimated and measured/actual ankle joint torques [21].
Correlation analysis of joint torque estimated via inverse dynamics and EMG-driven NMS model was carried out across all trials using MATLAB (MatlabR2018a, MathWorks Inc., Natick, MA, USA). A significance level of 0.05 was set for all statistical tests.
We transform (41) into the model we used in (7), which is , r � − f dis , and τ is the joint torque.
Let θ i (0) � [0, . . . , 0] T be the initial values of the adaptive law (39). e design parameters' values were set to α � I 2×2 , e joint torque (normalized by the arm mass) estimated via inverse dynamics and EMG-driven NMS model is shown in Figures 3 and 4. e results of the correlation analysis showed that there was a significant correlation of joint torque estimated via inverse dynamics and EMG-driven NMS model (p < 0.01).
e Pearson coefficients r 1 � 0.991 and r 2 � 0.507 were observed at the shoulder and elbow flexion/extension DOFs in the arm reaching movement, respectively. e root mean square error (RMSE) of shoulder flexion/extension torque between inverse dynamics and EMG-driven NMS model was 0.048 Nm/kg. Similar RMSE 0.046 Nm/kg was observed at the elbow flexion/extension DOFs in the arm reaching movement. Figures 5 and 6 demonstrated the desired and actual joint angular velocity of the robot manipulator in the reaching task, and the results validated the effectiveness of the proposed controller which could complete the task with users-like adaptivity and had a good performance.

Discussion
We developed a disturbance-observer-based fuzzy control framework that fully incorporates a robot manipulator system with an EMG-driven NMS model and could be applied in the controlling of the movement of robot manipulators with human-like characteristics. In the developed framework, a reaching task for a robot manipulator was studied to test performance of the disturbance-observerbased fuzzy controller. We found that the user's intention was well-predicted by the EMG-driven NMS model and then transmitted to the desired velocities of the robot manipulator system, which allows the robot manipulator system behave with human-like adaptivity. In the robot manipulator system with external disturbances and model uncertainties, the proposed disturbance-observer-based fuzzy control was also able to provide a good performance of motion tracking. In the current paper, the control framework we proposed in general is useful for obtaining the human-like characteristics, as well as simulating the control strategy in the robot manipulator system, and thus be of large benefit for teleoperation robot manipulator applications.
EMG-driven NMS model is a popular method in users' motion intention (joint torque) estimation [34][35][36]. e EMG-driven NMS model uses mathematical equations to reproduce the transformations from EMG signal generation and joint angles to musculotendon forces and joint torques. To better predict joint torque, subject-specified EMG-driven model was calibrated with personalized musculoskeletal geometry such as moment arms and muscle characteristics. During the calibration, these subject-specific parameters were refined by an optimization algorithm to minimize the error between estimated and measured/actual ankle joint torques. With the mathematical equations approximately reproducing the transformations and optimization algorithm included in the calibration process, inevitable error existed in the EMG-driven NMS model when mapping EMG signals and angles into the joint torque. Despite this, the EMG-driven NMS model was able to estimate joint torque in close agreement to the reference data (with RMSE lower than 0.048 Nm/kg).
It is important to notice that unknown dynamics and external disturbance commonly exist in robotic manipulator systems. Fuzzy logic system [37], neural network [38], and disturbance observer [39] are commonly applied to solve these problems and maintain the system stability. Su et al. [40] integrated a fuzzy compensator into a teleoperation controller for a robot manipulator, and this controller was able to guarantee a safety-enhanced behavior in the null space. Li et al. [41] utilized a disturbance observer to compensate for the external disturbance in an admittance control for an upper robotic exoskeleton and guaranteed the robustness of the robotic arm. In the current study, a disturbance-observer-based controller with a high-dimensional integral-type Lyapunov function was developed for a robotic      (Figures 5 and 6) demonstrate the effectiveness of the proposed control framework and the semiglobally uniformly ultimate boundness of the closed-loop control system is also established.
One limitation of this study was that only reaching task at one speed was studied as an example in the proposed framework. Reaching task at different speeds as well as different situations and other daily activities such as grasp could be also tested in the proposed framework. Another limitation was that we only tested the control framework in a simulation environment with one subject data. Large sample size of subjects could be enrolled in the future study to see whether the proposed control framework would be able to generalize across subjects.

Conclusions
We developed an adaptive control framework that fully incorporates a robot manipulator system with an EMGdriven NMS model and use it to control the motion of the robot manipulator with human-like characteristics. In the developed framework, an example was studied to test performance of the disturbance-observer-based fuzzy controller, which was applied to a robot manipulator during a reaching task. We found that the EMG-driven NMS model was able to predict the user's intention (desired torque) and transmitted these human-like characteristics to the desired trajectories of the robot manipulator system. Moreover, external disturbances as well as model uncertainties were simulated in the robot manipulator system, and the proposed adaptive fuzzy controller integrated with a disturbance observer was able to provide a good performance of motion tracking. In the current paper, the control framework we proposed in general is useful for obtaining the human-like characteristics, as well as simulating the control strategy in the robot manipulator system that is subjected to realistic conditions, such as model uncertainties and external disturbances, and thus be of large benefit for teleoperation robot manipulator applications. where κ � min (A.7) By multiplying e κt and integrating the both sides of inequality (A.6), we have Based on (A.8), V 2 is ultimately bounded as t ⟶ ∞; D, s, and Θ are also bounded.

Complexity
Data Availability e data supporting this study is from a previously reported study, which has been cited at relevant place within the text as reference [33]. e dataset are available at (https://simtk. org/frs/index.php?group_id�657).