Exoskeleton Hand Control by Fractional Order Models

This paper deals with the fractional order control for the complex systems, hand exoskeleton and sensors, that monitor and control the human behavior. The control laws based on physical significance variables, for fractional order models, with delays or without delays, are proposed and discussed. Lyapunov techniques and the methods that derive from Yakubovici-Kalman-Popov lemma are used and the frequency criterions that ensure asymptotic stability of the closed loop system are inferred. An observer control is proposed for the complex models, exoskeleton and sensors. The asymptotic stability of the system, exoskeleton hand-observer, is studied for sector control laws. Numerical simulations for an intelligent haptic robot-glove are presented. Several examples regarding these models, with delays or without delays, by using sector control laws or an observer control, are analyzed. The experimental platform is presented.


Introduction
The IHRG is an intelligent haptic robotic glove system for the rehabilitation of patients that have a diagnosis of a cerebrovascular accident. This system is created by a thin textile in order to have a comfortable environment for grasping exercises. An exoskeleton architecture ensures the mechanical compliance of human fingers. The driving and skin sensor system is designed to determine comfortable and stable grasping function. This paper analyzes the dynamics of an exoskeleton hand using fractional order operators and proposes control solutions.
The number of applications in the system modelling, where the fractional order calculus (FOC) is used, has increased significantly in the last few decades. Many authors proved that non-integer order integrals and derivatives are suitable for analyses of the properties of various materials. Recent achievements in the interpretation of FOC operators allowed to apply FOC for processes that are better described by fractional order models (FOM) rather than integer order models (IOM). The role of these models in soft matter physics and viscoelastic behavior, in the theory of complex materials, its quality to include effects with non-conservative forces and power-law phenomena suggest to describe the complexity of human dynamics using FOM operators [1].
The idea is supported by the evidence of s β dynamics in muscles and joint tissues throughout human musculo-skeletal system [2,3]. Interaction and dependence between biological systems and associated mechanical components was analyzed in [4][5][6]. Fractional order models for metal polymer composite was discussed in [7]. In [8,9] viscoelastic properties for a large variety of biological entities 1.
The fractional order integral of order β is the Riemann-Liouville fractional integral: The Caputo derivative of order β, 0 < β < 1 is: where β is the fractional order exponent and Γ(β) is the gamma (Euler's) function.
(a) 3D curvature sensors described by FOM Bending sensors represent a class of sensors with large applications in the control of complex systems. They convert changes in bend to an electrical parameter variation. Conventional bending sensors handle cases in which bending is produced in the 2D plane. The most common are the resistive sensors, described by IOM operators of order 0. For a special class of systems, such as the hyper-redundant robots [29] where bending is produced in a 3D space (Figure 1a), a special class of bending sensors defined by FOM operators (Figure 1b) is used.
Sensors 2019, 19, 4608 3 of 18 a) 3D curvature sensors described by FOM Bending sensors represent a class of sensors with large applications in the control of complex systems. They convert changes in bend to an electrical parameter variation. Conventional bending sensors handle cases in which bending is produced in the 2D plane. The most common are the resistive sensors, described by IOM operators of order 0. For a special class of systems, such as the hyper-redundant robots [29] where bending is produced in a 3D space (Figure 1a), a special class of bending sensors defined by FOM operators (Figure 1b) is used.  The architecture of this sensor consists of a main viscoelastic component determined by a long flexible backbone wrapped in a cylindrical elastic envelope. Three antagonist cables are implemented at the periphery of the system. In static behavior, curvature is obtained by the differential measurement of the cable lengths, Figure 1c [30]: The architecture of this sensor consists of a main viscoelastic component determined by a long flexible backbone wrapped in a cylindrical elastic envelope. Three antagonist cables are implemented at the periphery of the system. In static behavior, curvature κ is obtained by the differential measurement of the cable lengths, Figure 1c [30]: The dynamic behavior is inferred considering constant curvature along the length. Employing the same technique as that developed in [17] yields ( Figure 2): ..
where c νs , k s are distributed viscous and elastic coefficient, assumed uniform distributed along the length, b s , c s are material parameters and M is the moment that determines the bending. The transfer function is derived from Equation (2) as: The dynamic behavior is inferred considering constant curvature along the length. Employing the same technique as that developed in [17] yields ( Figure 2): where , are distributed viscous and elastic coefficient, assumed uniform distributed along the length, , are material parameters and ℳ is the moment that determines the bending. The transfer function is derived from Equation (2) as: That corresponds to an order 2 FOM operator.

b) FOM systems
A large class of systems that monitors or controls the human behavior is well described by the FOM operators. Figure 3 shows the control system of an intelligent haptic robot-glove (IHRG) for the rehabilitation of patients that have a diagnosis of a cerebrovascular accident. The IHRG is a medical device that acts in parallel to a hand in order to compensate for lost functions [16]. The exoskeleton architecture that ensures the mechanical compliance of human fingers for the driving system determines comfortable and stable grasping functions.
The dynamics of the system (EXHAND) can be accurately described by FOM operators, where is the state vector = [ , , . . . , ] that defines the motion parameters, β is the fractional order exponent, and , are ( × ), ( × 1) constant matrices. In a FOM operator of EXHAND, the vector components are defined as The nonlinear term ( ) is determined by the gravitational components and satisfies the inequality The output of the system is generated by the bending sensors. Provided that the bending of the phalange musculoskeletal system is in 2-D plane, in this project, we used an Arduino Flex Resistive Sensor network. This sensor operates as a zeroth IOM operator, That corresponds to an order 2 FOM operator.

(b) FOM systems
A large class of systems that monitors or controls the human behavior is well described by the FOM operators. Figure 3 shows the control system of an intelligent haptic robot-glove (IHRG) for the rehabilitation of patients that have a diagnosis of a cerebrovascular accident. The IHRG is a medical device that acts in parallel to a hand in order to compensate for lost functions [16]. The exoskeleton architecture that ensures the mechanical compliance of human fingers for the driving system determines comfortable and stable grasping functions. where the function is associated to initial states. For Equations (4)-(8) we used the control system from Figure 4 with a FOM operator for the EXHAND and a IOM operator for the sensor system.   The dynamics of the system (EXHAND) can be accurately described by FOM operators, where z is the state vector z = [z 1 , z 2 , . . . , z n ] T that defines the motion parameters, β is the fractional order exponent, and A 0 , b are (n × n), (n × 1) constant matrices. In a FOM operator of EXHAND, the vector components are defined as The nonlinear term f (z) is determined by the gravitational components and satisfies the inequality The output of the system is generated by the bending sensors. Provided that the bending of the phalange musculoskeletal system is in 2-D plane, in this project, we used an Arduino Flex Resistive Sensor network. This sensor operates as a zeroth IOM operator, where c is a constant (n × 1) vector.
A new model can be inferred if the delay time constant, associated with the neuro-muscular system, the driving system and the processing time, is introduced, The initial conditions are defined by where the function ϕ is associated to initial states. For Equations (4)-(8) we used the control system from Figure 4 with a FOM operator for the EXHAND and a IOM operator for the sensor system. where the function is associated to initial states. For Equations (4)-(8) we used the control system from Figure 4 with a FOM operator for the EXHAND and a IOM operator for the sensor system.

Mathematical Preliminaries
Lemma [19] For any symmetric matrix ∈ , the following inequality holds: where ( ) , ( ) denote the minimum and maximum eigenvalue, respectively, of matrix and * is the unit matrix.

Mathematical Preliminaries
Lemma [19]. For any symmetric matrix P ∈ R nxn , the following inequality holds: where λ min(P) , λ max(P) denote the minimum and maximum eigenvalue, respectively, of matrix P and I * is the unit matrix.

Control for the EXHAND Without Delays
Consider the system from Figure 4 defined by Equations (4)-(7) without a delay time. Assume a control law.
where the control gain k verifies the condition where σ is a positive constant (for simplicity, y re f (t) = 0). (10)) becomes a PD β law

Control for the EXHAND with Delay
Control System 2. The system described by Equation (8) with the control law defined by Equation (13) is asymptotically stable if: where Q = qq T , P 1 are solutions of the Lyapunov equations and Proof. Consider the following Lyapunov function: where P 1 , P 2 are (n × n) are positive definite and symmetrical matrices, P 1 > 0, P 2 > 0, P T 1 = P 1 , P T 2 = P 2 . V(z) satisfies the condition (Equation (11)) of Theorem 2.
The derivative D β V(z) is computed from: where which leads to the inequality By evaluating Equation (37) along of solutions of Equation (8) it turns out that By applying the control law Equation (28), it yields The following inequality will be used [23] Additionally, considering the YKP Lemma as in the previous Control System, yields Substituting this result into Equation (39), considering the inequalities of Equations (6) and (40), one derives that Employing Equations (30) and (31), yields Denoted by and from Equation (35) results

Control System with Observer for the EXHAND with Delay
Consider the linearized model of Equation (8) rewritten as where the nonlinear term was approximated by Remark 2. For the EXHAND model, the pair (A L , b) is controllable and the pair (C, A L ) is observable.
Consider the system defined by Equation (46). The following observer is proposed: whereẑ ∈ R n is the observer state,ŷ ∈ R 2 is the estimated output and L 1 , L 2 are (n × 1) observability vectors. The observer error is defined by the following equation: Consider the control law The global state (ẑ, z −ẑ) = (ẑ, ∆z) is considered for the system "EXHAND-observer".
Applying the same procedures as in the previous control system, yields

EXHAND with Sensors Without Delays
Consider the IHRG system of Figure 3. The exoskeleton drive system is a decoupled one, for each finger. The following parameters of the hand and exoskeleton mechanical architecture [16] will be used: the equivalent moment of inertia is = 0.005 kg · m , the equivalent mass is = 0.015 kg, the viscous and elastic coefficients of the equivalent Kelvin-Voigt model of the joint tissues throughout phalange musculoskeletal system and exoskeleton are [6,7] = 0.22 Nm · s · rad = 2.8 Nm · rad , respectively, and the damping coefficient is = 7.8 Nm · s · rad .
where is defined by Equation (50) and Proof. Consider the Lyapunov function where P 1 , P 2 , P 3 are (n × n) are positive definite and symmetrical matrices. V(z, ∆z) satisfies the first condition (Equation (11)) of Theorem 2.
Applying the same procedures as in the previous control system, yields and using (53) yields Remark 3. The asymptotical stability conditions of Control System 2, Control System 3 are independent by the time delay τ.

EXHAND with Sensors Without Delays
Consider the IHRG system of Figure 3. The exoskeleton drive system is a decoupled one, for each finger. The following parameters of the hand and exoskeleton mechanical architecture [16] will be used: the equivalent moment of inertia is J = 0.005 kg·m 2 , the equivalent mass is m = 0.015 kg, the viscous and elastic coefficients of the equivalent Kelvin-Voigt model of the joint tissues throughout phalange musculoskeletal system and exoskeleton are [6,7] where the nonlinear component verifies the inequality (Equation (6)) for η = 0.2. The sensor is considered as an IOM operator and the output is defined as The fractional order exponent is β = 1 2 . The FOM model (Equations (8) and (9)) is defined as The pairs are (A, b), (A, c) controllable, respectively observable. The IOM sensor output without delays is given by (7). A control law (Equation (13)) (for y re f (t)) = 0) with k = 200 is applied. This control verifies the sector constraint (Equation (11)) with σ = 5 × 10 −3 . The matrix R was considered as, R = diag(3, 3, 3, 3) where A 1 = A − R is Hurwitz stable. The vector q = 0.05 × [1111] T and a matrix P were inferred with λ max(P) = 0.725. The polar plot of c T ( jωI − A 1 ) −1 b is shown in Figure 6. The closed-loop system satisfies the frequential criterion (Equation (17)), condition (18) is verified for Q = 14.5, ρ = 2.17, MATLAB/SIMULINK and techniques based on the Mittag-Leffler functions are used for the simulation [1,2]. Figure 7 shows the trajectories of fractional order variables. control verifies the sector constraint (Equation (11)) with = 5 × 10 . The matrix R was considered as, = diag(3,3,3,3) where = − is Hurwitz stable. The vector = 0.05 × [1111] and a matrix were inferred with ( ) = 0.725. The polar plot of ( − ) is shown in Figure  6. The closed-loop system satisfies the frequential criterion (Equation (17)), condition (18) is verified for ‖ ‖ = 14.5, = 2.17, MATLAB/SIMULINK and techniques based on the Mittag-Leffler functions are used for the simulation [1,2]. Figure 7 shows the trajectories of fractional order variables.   as, = diag(3,3,3,3) where = − is Hurwitz stable. The vector = 0.05 × [1111] and a matrix were inferred with ( ) = 0.725. The polar plot of ( − ) is shown in Figure  6. The closed-loop system satisfies the frequential criterion (Equation (17)), condition (18) is verified for ‖ ‖ = 14.5, = 2.17, MATLAB/SIMULINK and techniques based on the Mittag-Leffler functions are used for the simulation [1,2]. Figure 7 shows the trajectories of fractional order variables.   The control law (Equation (13)) can be rewritten as a PD control and the controller transfer function will be For the PD 0.5 control law, we have The steady error can be inferred from Equations (71)-(73) as [21,32,33] e s(PD) = e s(PD 0.5 ) = 0.0042 The behavior of the linearized model (Equation (68)) for both control laws (Equations (73) and (74)) is studied. The trajectories of angular position θ for target signal θ targ = π 6 are shown in Figure 8. The steady error can be inferred from Equations (71)-(73) as [21,32,33] ( ) = ( . ) = 0.0042 The behavior of the linearized model (Equation (68)) for both control laws (Equations (73) and (74)) is studied. The trajectories of angular position for target signal = are shown in Figure 8.  The sensor dynamics are where τ ∈ [−0.1; 0]. Substituting Equations (74) and (75) into (68) and using the control law (Equation (70)), yields J ..
The delay component of the dynamic model is defined by τ = 0.1 s. The controller parameters are selected as k 1 = 200, k 2 = 15 that satisfy Equations (29)-(31) by employing the same parameters for q, R as in the previous example. The evolution of the fractional order variables is shown in Figure 9.  Figure 10 shows the trajectories of physical significance variables, position and velocity, for the system and observer.

EXHAND with Delay and Observer
An observer (Equations (49)-(51)) with L 1 = L 2 = [1.5 1.5 1.5 0] T is associated to the linearized dynamic model. The matrix R = diag(1 1 1 1) verifies the condition as (A L − R) to be Hurwitz matrix. For q = [0.5 0.5 0.5 0.5] T , solution of P 1 is obtained with λ max(P 1 ) = 0.0085. A control law (55) with k 1 = 20, k 2 = 8.5, σ = 0.05 were selected. Equations (57)-(61) are easily verified. Figure 10 shows the trajectories of physical significance variables, position and velocity, for the system and observer.  Figure 10 shows the trajectories of physical significance variables, position and velocity, for the system and observer.

IHRG Experimental Platform
The IHRG is an exoskeleton that supports the human hand and hand activities by using a control architecture for dexterous grasping and manipulation. IHRG is a medical device that acts in parallel to a hand in order to compensate for lost function. It is easy to use and can be a helpful tool in the home [16,34].
The mechanical architecture consists of articulated serial elements of which design covers functional and anatomic finger phalanges. The glove is created by a thin textile that represents an infrastructure suitable for actuation wires and sensors. A distributed actuation system is used for implementing the operations of the hand. An Arduino Flex Sensor network (with zeroth order sensors) is used to control the motions. An Arduino Mega 2560 hardware platform determines the movement of the glove's actuators for exercises like opening or closing of the fingers (Figures 11 and 12).

IHRG Experimental Platform
The IHRG is an exoskeleton that supports the human hand and hand activities by using a control architecture for dexterous grasping and manipulation. IHRG is a medical device that acts in parallel to a hand in order to compensate for lost function. It is easy to use and can be a helpful tool in the home [16,34].
The mechanical architecture consists of articulated serial elements of which design covers functional and anatomic finger phalanges. The glove is created by a thin textile that represents an infrastructure suitable for actuation wires and sensors. A distributed actuation system is used for implementing the operations of the hand. An Arduino Flex Sensor network (with zeroth order sensors) is used to control the motions. An Arduino Mega 2560 hardware platform determines the movement of the glove's actuators for exercises like opening or closing of the fingers (Figures 11 and 12). All the movements of the hand are controlled by the software of the hybrid IHRG system, which was developed in MATLAB and Simulink. The performance of each patient following the exercises program can be recorded by the same software. The control system of Control System 1 is implemented. In Figure 13   All the movements of the hand are controlled by the software of the hybrid IHRG system, which was developed in MATLAB and Simulink. The performance of each patient following the exercises program can be recorded by the same software. The control system of Control System 1 is implemented. In Figure 13    All the movements of the hand are controlled by the software of the hybrid IHRG system, which was developed in MATLAB and Simulink. The performance of each patient following the exercises program can be recorded by the same software. The control system of Control System 1 is implemented. In Figure 13

Discussion
I. We designed, built, and tested an intelligent haptic robotic glove for the rehabilitation of the patients that have a diagnosis of a cerebrovascular accident. The glove is created by a thin textile in order to have a comfortable environment for the grasping exercises. This thin textile creates an infrastructure suitable for wire actuation and sensors. This exoskeleton architecture ensures the mechanical compliance of human fingers. The driving and skin sensor system is designed to determine comfortable and stable grasping function. The dynamics of the exoskeleton hand are modeled by fractional order operators. To our knowledge, this paper is the first paper in which the interaction between biological systems (human hand) and mechanical associated components (exoskeleton) is analyzed by fractional order models. These new models are used to develop a class of algorithms for the control of the stable grasping function. The control systems are based on the physical significance variable control that are generated by sensor classes implemented in the system. These sensors are also modeled as operators with delays. The paper proposes control solutions and determines the criterions for controller parameter tuning for several classes of models. The observer techniques are also discussed and implemented. The quality and the stability of motion, are analyzed

Discussion
I. We designed, built, and tested an intelligent haptic robotic glove for the rehabilitation of the patients that have a diagnosis of a cerebrovascular accident. The glove is created by a thin textile in order to have a comfortable environment for the grasping exercises. This thin textile creates an infrastructure suitable for wire actuation and sensors. This exoskeleton architecture ensures the mechanical compliance of human fingers. The driving and skin sensor system is designed to determine comfortable and stable grasping function. The dynamics of the exoskeleton hand are modeled by fractional order operators. To our knowledge, this paper is the first paper in which the interaction between biological systems (human hand) and mechanical associated components (exoskeleton) is analyzed by fractional order models. These new models are used to develop a class of algorithms for the control of the stable grasping function. The control systems are based on the physical significance variable control that are generated by sensor classes implemented in the system. These sensors are also modeled as operators with delays. The paper proposes control solutions and determines the criterions for controller parameter tuning for several classes of models. The observer techniques are also discussed and implemented. The quality and the stability of motion, are analyzed by Lyapunov methods and techniques that derive from Yakubovici-Kalman-Popov Lemma.
Despite of the model complexity, the control systems are very simple, and the controllers are easily implemented in an Arduino Mega 2560 hardware platform. There were many advantages for using this platform since this hardware board has ports for PWM signals that are useful to be sent to the actuators and ports for reading the signals coming from the bending sensors.
In order to help patients to follow an after-stoke recovery program, the system uses a set of predefined rehabilitation exercises like open the hand, close it, try to grab an object or simple wave. The system is very easy to use at home, with minimal training. The predefined rehabilitation set of exercises was created to be used.
II. The control systems discussed in the previous sections are focused on the control problems of the IHRG system, where the EXHAND model is described by FOM operators and the sensor system is based on zeroth order sensors. These control solutions can be also used for a larger class of complex systems as hyper-redundant systems, that use complex FOM sensors ( Figure 14). III. In addition, we consider that control systems discussed in the previous sections can be applied to a class of control problems associated to the persons with disabilities. Figure 15 presents a wheelchair control system for this class of persons. In this case, the human operator is represented by the persons with hemiparesis/hemiplegia, with motor restriction (arm or leg-emphasized hemiparesis) and with serious brain damage [35], The transfer function of this human operator has a model that corresponds to a time delay fractional order model with time constant and fractional order . These parameters are determined by the characteristics of the damaged brain, viscoelastic properties of the atrophied muscles, and propagation time along the nervous terminals. We consider that these models can be studied by using the techniques developed in this paper.  III. In addition, we consider that control systems discussed in the previous sections can be applied to a class of control problems associated to the persons with disabilities. Figure 15 presents a wheelchair control system for this class of persons. In this case, the human operator is represented by the persons with hemiparesis/hemiplegia, with motor restriction (arm or leg-emphasized hemiparesis) and with serious brain damage [35], III. In addition, we consider that control systems discussed in the previous sections can be applied to a class of control problems associated to the persons with disabilities. Figure 15 presents a wheelchair control system for this class of persons. In this case, the human operator is represented by the persons with hemiparesis/hemiplegia, with motor restriction (arm or leg-emphasized hemiparesis) and with serious brain damage [35],

(s) =
The transfer function of this human operator has a model that corresponds to a time delay fractional order model with time constant and fractional order . These parameters are determined by the characteristics of the damaged brain, viscoelastic properties of the atrophied muscles, and propagation time along the nervous terminals. We consider that these models can be studied by using the techniques developed in this paper.  The transfer function of this human operator has a model that corresponds to a time delay fractional order model with time constant τ and fractional order β. These parameters are determined by the characteristics of the damaged brain, viscoelastic properties of the atrophied muscles, and propagation time along the nervous terminals.
We consider that these models can be studied by using the techniques developed in this paper.