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Model-based tracking control design, implementation of embedded digital controller and testing of a biomechatronic device for robotic rehabilitation
Mechatronics 52 (2018) 70–77
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Model-based tracking control design, implementation of embedded digital controller and testing of a biomechatronic device for robotic rehabilitation☆
T
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Alessio Merola, Domenico Colacino, Carlo Cosentino , Francesco Amato School of Computer and Biomedical Engineering, Universitá degli Studi Magna Græcia di Catanzaro, Campus Universitario di Germaneto, Catanzaro 88100, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Model-based control Trajectory tracking control Pneumatic artificial muscles Rehabilitation robotics
In this paper, the tracking control problem of a biomimetic exoskeleton powered by a pair of pneumatic artificial muscles is considered. The antagonistic configuration of the pair of pneumatic muscles, which is biologically inspired, enables safe and reliable actuation in applications of orthopaedic rehabilitation. However, during the inflation-deflation process, the pneumatic muscles introduce nonlinearity and hysteresis which deteriorate the control performance. A model of the antagonistic artificial muscles is adopted to develop a computed-torque control for feedforward compensation of the nonlinear dynamics of the actuated joint. A PID control action is used in combination with the feedforward compensation to achieve fast and accurate tracking control performance. The model, which possesses a reduced set of parameters as functions of the inflation/deflation phase, enables efficient nonlinear compensation. The experimental tests on the biomechatronic device, compared with other state-of-the-art approaches for controlling pneumatic artificial muscles, show better tracking performance in terms of convergence rate and robustness, justifying the convenience of using the proposed control methodology in the design of tracking controllers for exoskeletal biomechatronic devices.
1. Introduction The emerging field of soft robotics is currently covering novel applications in rehabilitation robotics, prosthetics and surgical robotics and, more in general, several safety-critical applications involving interactions between robots and human operators. The main requirements for safe human-robot interactions [1] can be fulfilled through the inherent (and adaptable) compliance of soft actuators. Moreover, the adaptation mechanisms of the compliance of soft robotic and biomechatronic devices can mimic the behaviour of the biological musculo-skeletal system [2,3]. As an example of biomimetic and soft actuation, pneumatic artificial muscles (PAMs) have been employed in the realization of rehabilitation robots, wearable exoskeleton robots and energy-efficient walking humanoids (see [4–6]). More recently, pneumatic muscle actuation technologies are developed towards the realization of miniaturized biomechatronic devices. For instance, the work [7] focuses on the characterization of pneumatic muscles for set-point regulation of the motion of a biomechatronic finger. Together with their advantages, PAMs offer some challenges in the design and implementation of the tracking control, since the controller has to handle the strong nonlinearity of the PAMs dynamics.
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Some control techniques have been developed to solve the problems of regulation and trajectory tracking for PAMs-driven robots. Prior results focus on variable structure control [8], gain scheduling [9], adaptive backstepping [10], sliding mode [11] and PID neural network control [12]. The sliding mode control strategy recently proposed in [13] is aimed at the enhancement of the safety during collision with obstacles. Therefore, the sliding mode tracking controller is complemented by a joint compliance controller, which meets the safety requirements during collision. Model-based compensation strategies are originally proposed in [14,15], where the compensation of the hysteresis in the force characteristics of pneumatic muscles is achieved on the basis of generalized models of the hysteresis in the mechanical response of PAMs. A feedforward compensation is implemented into the feedback control schemes of linear positioning stages implementing backstepping [15] and cascade control [14] strategies. More recently, in [16,17], the authors show how the guaranteed-cost control approach can be effectively applied to the solution of a regulation problem for a PAMs-driven robot whose dynamics can be described through uncertain (bilinear and quadratic) systems. To foster the efficient implementation of model-based control strategies for PAMs-driven robots, lumped parameter models of the
This paper was recommended for publication by Associate Editor Prof Kong Kyoungchul. Corresponding author. E-mail addresses: [email protected] (A. Merola), [email protected] (D. Colacino), [email protected] (C. Cosentino), [email protected] (F. Amato).
https://doi.org/10.1016/j.mechatronics.2018.04.006 Received 12 June 2017; Received in revised form 16 February 2018; Accepted 25 April 2018 0957-4158/ © 2018 Published by Elsevier Ltd.
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nonlinear dynamics of PAMs can be exploited. The three-element phenomenological model proposed by Reynolds et al. [18] allows to accurately predict the dynamic response of PAMs; the model parameters can be easily identified through static perturbation tests at several constant values of pressure. More recently, in [19] a series of static perturbation tests have been carried out for the efficient identification of a novel model of the mechanical response of some classes of PAMs. The main contribution of this work consists in the development and the experimental evaluation of a novel model-based tracking control methodology for a biomechatronic device powered by PAMs in antagonistic configuration. The lumped parameter model by Reynolds et al. [18] is used here to develop a computed torque control for feedforward compensation of the actuated joint via nonlinear inversion. The Reynolds’s model provides an efficient description of the variability of the nonlinear dynamics of the PAMs during the pressure cycle. Therefore, the feedforward control action is a function of the model parameters which depend on the phase of inflation/deflation. The compensation loop of the tracking controller is complemented by a PID control action which enables the robust regulation to zero of the tracking error. To the best of the authors’ knowledge, this is the first time that such a combined nonlinear inversion feedforward + feedback PID control methodology has been proposed and experimentally tested for the tracking control of a PAM-based robot. The devised approach improves the tracking performance over the existing approaches in the related literature on the control of PAMs-based robots. The parsimony of the model makes this control approach very suitable for the real-time implementation on embedded microcontroller devices. Therefore, a non negligible byproduct of this work is the microcontroller-based implementation of the embedded control system, whereas the previously published results were tested by means of PC-based implementations. Some experimental tests are presented to prove the superior control performance achieved by the proposed methodology thanks to the efficient compensation of the nonlinear dynamics. Specifically, the performance of the closed loop control system has been measured in terms of convergence rate, steady-state error and robustness to load disturbance during the tracking of constant and sinusoidal trajectories. Furthermore, the micro-controller based implementation of the proposed control scheme shows the advantages related to the real-time execution, computational resources and customizable constraints on the control action. This paper is structured as follows. The main technical specifications of the robotic exoskeleton and a model of the dynamics of the actuated robot are presented in Sections 2 and 3, respectively. The model-based tracking control law is derived in Section 4, where the steps of design and implementation of the digital controller on embedded control unit are also described. The experimental tests and the discussion of the experimental results are given in Section 5. Some concluding remarks are eventually left to Section 6.
Fig. 1. Overview of CoRAnT. (a) Orthotic shoe. (b) Pneumatic muscle actuator. (c) Pressure regulator. (d) Load cell. (e) Optical encoder.
To obtain a training performance at least comparable to the manual therapy, an essential requirement for biomechatronic rehabilitation is the safe interaction between the patient and the rehabilitation device. During the conventional clinical therapy, the manipulations performed by the therapist are adapted to the resistance force exerted by the patient. Similar adaptation mechanisms, guaranteeing the intrinsic safety of robotic rehabilitation tasks, can be realized through suitable control strategies of biomechatronic devices driven by biomimetic and soft actuators. For instance, the natural compliance of soft actuators, which utilize air as source of energy, enables to absorb potential shocks occurring during the manipulation of the patient. After the analysis of both the main issues involved in physical rehabilitation and the advantages provided by robotic and biomechatronic technologies, the main specifications are implemented in a robotic exoskeleton for rehabilitation following the design principles of soft and biomimetic robotics. Therefore, CoRAnT is actuated by soft pneumatic muscles, which guarantee safety and comfort for the patient. The mechanical structure and the main components of the robotic exoskeleton are highlighted in Fig. 1; the pneumatic muscle used for the actuation of the exoskeleton is the fluidic muscle MAS-20-200N by FESTO company. An orthotic shoe is installed on the rotational joint of the exoskeleton; the joint axis is aligned with the ankle axis of the patient (see Fig. 2). A couple of PAMs is required for the full actuation of the joint, since a single PAM actively generates motion in one direction. The PAMs are arranged in the bio-inspired configuration of Fig. 3, where the antagonistic setup can mimic the mechanisms of regulation of motion and stiffness of the joints in the human musculo-skeletal system. Therefore, in analogy with the human anatomy, the PAMs act as biceps and triceps, respectively. The conversion of the linear motion of the actuators to the joint rotation is obtained through some cables of diameter 3.5 mm tied together around a pulley of diameter 80 mm. Position and force sensors are installed on the robotic exoskeleton. An optical rotary encoder (AVAGO HEDM5500 J14) measures the angular position of the joint at 1024 counts per revolution. The encoder provides the measured variable to the tracking controller. Moreover, each PAM is connected to a load cell (Phidgets 3138 S-type). The signals of force and angle can be used for measuring the interaction between
2. CoRAnT: Compliant Robotic Ankle Trainer The traditional therapy for stroke and traumatic brain injuries requires long and intensive rehabilitation tasks performed by the physical therapist. Indeed, the quick recovery of neural and motor functions, resulting from cortical reorganization in the motor cortex of the patient, can be achieved under intensive and repetitive exercises. Unfortunately, the required burden of care may contrast with the budget and time constraints. Thanks to the accuracy and repeatability of the control performance, robotic exoskeletons performing high-intensity physical rehabilitation can offer their potential in the optimization of costs and efficiency of the rehabilitation procedures. Moreover, biomechatronic and robotic devices can collect quantitative data useful for the evaluation of both the patient’s motor performance and the progress of the motor recovery. 71
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Table 1 Model parameters.
Spring element Damping element Force element
Parameter
Value
k0 k1 b0i (b0d) b1i (b1d) f0 f1
4.82 131 1.22 (1.31) 135 (−230) 14.89 2090
[N/m] [N/m/bar] [N/m/s] [N/m/s/bar] [N] [N/bar]
sense of the contraction displacement. The nonlinearity introduced by the PAMs is described through the three-element phenomenological model by Reynolds et al. [18]. The expression of the total force ϕ exerted by a single PAM is obtained from a parallel configuration of an elastic element (spring) of stiffness k(p), a viscous element (damper) of damping coefficient b(p) and a contractile element generating the active force contribution f(p). Therefore, the PAM force reads
ϕ = f (p) − b (p) s˙ − k (p) s, where p denotes the pressure at which the PAM is inflated and s is the contraction length. The model parameters follow the constitutive laws Fig. 2. CoRAnT performing a rehabilitation task on the patient. (a) Lateral view. (b) Front view.
f (p)= f0 + f1 p
[N]
b0i + b1i p [N/m/s] inflation b (p)= b0 + b1 p = ⎧ ⎨ ⎩ b0d + b1d p [N/m/s] deflation
k (p)= k 0 + k1 p
[N/m]
The values of the parameters of the damping function b(p) vary depending on the phase of inflation/deflation. Concerning the values of the model parameters, we refer both to Table 1 and to the identification procedure described in [18]. The values listed in Table 1 are taken from [13], where the model parameters have been identified for the same FESTO fluidic muscles adopted in CoRAnT. The parameters of the elastic and damping functions of the pressure can be identified from the step response of the actuator at constant pressure and subject to a step change of the applied load. The contractile (active force) element function is determined by a step change to the inflation pressure of the actuator under constant load. The model of the actuated joint requires the moment of inertia of CoRAnT. The estimated moment of inertia amounts to Il = 0.0023 kg m2. This value can be obtained from the CAD 3D model of CoRAnT after specifying the materials of the parts of the model. The torque of the actuated joint reads
Fig. 3. Actuation scheme of CoRAnT.
τ = τb − τt = (ϕb − ϕt ) r , the robot and the patient, e.g., through the measurement of the torque opposed by the patient during the motion cycle. For instance, the patient’s participation to the rehabilitation task could be estimated through the measured data. The regulation of the pressure of the PAMs, involved in the trajectory control of the joint, is obtained through two proportional pressure regulators FESTO MPPE-3-1/8-6-010-B. The implementation of the pressure regulation into the motion control system of the exoskeleton is discussed in detail in Section 4.
(1)
where τb and τt are the torques generated by each PAM, as in
τb= (fb − bb s˙b − kb sb) r
(2a)
τt = (ft − bt s˙t − kt st ) r .
(2b)
In (2), sb and st denote the contraction length of biceps and triceps, respectively. From the antagonistic configuration of Fig. 3, a positive rotation is obtained through the active force generated by the inflation of the biceps, whereas the triceps deflates without active contribution to motion. For a negative rotation, it is required that the triceps generates the force driving the joint rotation, whereas the biceps is passive. The total torque can be expressed in terms of the joint angle θ, using the kinematic relations
3. Derivation of the equations of motion including actuator dynamics Starting from the kinematic scheme of Fig. 3, the equations of motion of CoRAnT are derived. Following the notation of Fig. 3, clockwise rotations are positive, whereas the upright vectors ϕb and ϕt denote the forces exerted by biceps and triceps, respectively. In Fig. 3, the force vectors point upward since each PAM generates its (actuation or resistance) force only in the
π sb = r ⎛θ + ⎞, 6⎠ ⎝
π st = r ⎛ − θ ⎞. ⎝6 ⎠
In a first stage of the actuation strategy, each PAM is inflated at the initial positive pressure p0. Thereafter, the joint rotation is obtained through the antagonistic control of the pressure of the pair of PAMs. A 72
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approach of the computed-torque control (see [20], Ch. 6.6). The control input can be computed in real time according to the feedforward-feedback scheme, after measuring both the kinematic variables and the model parameters of the actuated joint. The tracking control system, following a time-varying reference trajectory θd(t), requires the derivation of the tracking error dynamics as a function of the error e : =θd − θ and its successive derivative e˙: =θ˙d − θ˙ and e¨: =θ¨d − θ¨ . The nonlinear compensation is achieved through the control input
positive/negative rotation of the joint requires an increment/decrement of the pressure pb of the biceps and a decrement/increment of the pressure pt of the triceps. Therefore, the antagonistic actuation can be formulated as
pb = p0 + Δp ,
pt = p0 − Δp ,
where p0 is the initial pressure and Δp is the pressure difference from p0 used as manipulated variable in the antagonistic pressure control. It is useful to note that the antagonistic strategy simplifies the motion control, since only one control variable, the pressure difference Δp, is used for the actuation of the joint through two PAMs. Therefore, the dynamics of the biomechatronic joint is described as
Il θ¨ + ζθ˙ + τg (θ) + d = τ ,
u=
where Il denotes the link inertia, ζ is the viscous friction coefficient at the joint, τg(θ) denotes the gravity torque, τ is the control input torque and d represents the unknown external input disturbance which embeds some external perturbations, e.g., load variations or disturbance torque, other than modelling mismatches, unknown parametric uncertainties, friction nonlinearities, etc. Since the PAMs actuating the joint are identical, the spring element parameters are taken as k 0b = k 0t = k 0 and k1b = k1t = k1. The pressure difference Δp is the control input of the closed-loop system, i.e. u = Δp. After introducing in (3) the constitutive laws of the actuators, the equation of motion of the joint, including actuator dynamics, reads
v = kP e + kV e˙ + kI
⎡ ∫ edt ⎤ x: =⎢ e ⎥, ⎢ ⎦ ⎣ e˙ ⎥ then the state space representation of the closed loop system yields
x˙ = Ax + Bw, where
1 0 ⎞ ⎛ 0 0 1 A=⎜ 0 , ⎜−I −1 k −I −1 k −I −1 k ⎟⎟ I P V l l l ⎝ ⎠
with (5a)
β = r 2 [b0b + b0t + p0 (b1b + b1t )]
(5b)
π δ = 2rf1 − k1 r 2 3
(5c)
σ = r 2 (b1t − b1b).
(5d)
∫ edt.
Let us define the state vector as
(4)
α = 2r 2 (k 0 + k1 p0 )
(6)
where v is an auxiliary control signal to be designed in order to achieve robust tracking performance. To this end, v is taken as a PID control action in the form
(3)
Il θ¨ + (ζ + β ) θ˙ + αθ + τg (θ ) + d = (δ + σθ˙ ) u,
Il θ¨d + αθ + βθ˙ + τg (θ) + v , δ + σθ˙
⎛0 ⎞ B = ⎜ 0 ⎟, ⎜ I −1⎟ ⎝l ⎠
and w defines the disturbance which embodies friction effects, model uncertainties and unknown interaction torque. The nominal value of the inertia and the measured values of θ and θ˙ are introduced in the control action (6). Moreover, the value of the parameters α, β and δ are computed offline after the regulation of the initial pressure to the desired value p0; only the parameter σ, which depends on the inflation/deflation phase, requires an online computation. The PID controller parameters, as shown in Table 2, are tuned and verified experimentally for optimal results. The tracking controller is implemented on 8-bit AVR RISC-based microcontroller ATmega 328; the control signal and the measurements are processed at 50 Hz. An I2C bus interconnects all the digital components of the embedded control system: microcontroller, optical encoder and digital-analog converter (DAC). The pressures of the PAMs are regulated through two proportional pressure regulators in the range 0–6 bar. Two complementary control signals are generated to drive the pressure regulators associated to the pair of antagonistic muscles. One signal inflates/deflates the biceps and the other one deflates/inflates the triceps in order to produce positive/ negative rotations of the joint. The control signals in the range 0–10 V, which correspond to the pressure outputs 0–6 bar, are obtained after conversion of the control signals via a dual channel 12-bit DAC TLV5618 and through amplification by a programmable
Given the initial pressure p0, all the parameters defined in (5) do not vary during the motion, except for the value of σ which depends on the phase of inflation/deflation. The parameter α can be associated to a stiffness coefficient of the elastic term αθ arising from the spring element of the three-element phenomenological model of the PAM, whereas β can be viewed as an equivalent viscous friction factor depending on both the initial pressure and damping effects of the actuator. The parameters δ and σ describe the variability of the generated torque through the damping and active force elements of the actuator model. 4. Model-based tracking control and digital implementation A satisfactory trajectory-tracking control performance can be achieved through the combined feedforward-feedback control scheme of Fig. 4, which takes into account the nonlinearity introduced by the pneumatic actuators. A control law, which attains a compensation of the nonlinear terms of the dynamics (4), is developed through the
Fig. 4. Torque inverse control scheme. 73
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feedforward-feedback PID control scheme: the control action is characterized both by smoothness in response to sudden changes in desired position and by robustness against the patient load. The experimental results show that the proposed inverse modelbased controller is capable of efficiently controlling the antagonistic mechanism of the PAMs pair, while compensating for nonlinear phenomena and external load disturbances occurring during the inflationdeflation cycle. The superior performance achieved by the proposed tracking controller can be appreciated with respect to the related approaches in the scientific literature on nonlinear control of pneumatic muscle actuators in antagonistic configuration. Our model-based tracking controller outperforms some sliding mode control strategies [11,21] and nonlinear PID control techniques [12,22]. In [11], a sliding mode approach is applied to the trajectory tracking control of a pneumatic artificial muscle manipulator; a maximum joint error of 7° is achieved for a sinusoidal reference trajectory. Chattering, lack of robustness and slow responses are the major issues involved in sliding mode control strategies. In [21], a sliding mode tracking controller of a pneumatic artificial muscle manipulator is optimized in order to attenuate the oscillations in the system response. The mitigation of the oscillations is obtained at the cost of a longer settling time since, for a step change of about 35° in the desired angle, the joint angle converges to the desired value after a minimum transient of 2.5 s. In [12], a nonlinear PID controller using a neural network is presented for controlling a planar robotic arm actuated by PAMs in antagonistic setup. After training of the neural network, a maximum tracking error of about 2.5° is obtained following a sinusoidal reference trajectory ranging from −20° to 20° at frequency 0.2 Hz. Further comparison with the nonlinear PID controller of an exoskeletal wrist in [22] shows the superior performance of our feedforward-feedback PID control scheme in terms of fast and accurate tracking. To this end, it is useful to compare the step responses of the PAM-based exoskeletal wrist, which are presented in [22] for constant angular references of −10°, −20°, −30°, 10°, 20° and 30°. The experiments of flexion-extension movements are performed on the exoskeleton equipped with a spherical joint simulating a passive wrist; the rising time in response to the references of −20° and 20° is of the order of 0.5 s. The step responses acquired on CoRAnT exhibit better tracking performance when compared to the respective responses of the exoskeletal wrist, since the absolute tracking error reduces to 0.4° after 0.28 s in presence of the patient load. The tracking control performance achieved during the flexion-extension experiments in [22] has been evaluated for sinusoidal reference trajectories at maximum frequency of 0.4 Hz. The execution frequency of the PID controller is 10 Hz, a value compatible with the time required for computing the control action. The improvement of the control performance by the feedforwardfeedback scheme of CoRAnt is confirmed also for the case of tracking of a sinusoidal reference over an extended range of frequency, up to 1 Hz. Moreover, the model-based control law can be efficiently implemented at the higher frequency of 50 Hz on embedded microcontroller, without requiring the PC-based implementations involved in [11,12,21,22]. The experimental results prove that the proposed control scheme succeeds in guaranteeing the robust and efficient control performance required for rehabilitation purposes and, also in comparison with the existing results in the related field of the control of PAMs-based exoskeletons, it yields fast and accurate tracking performance.
Table 2 Parameters of the PID controller. Parameter
Value
KP KD KI
30.1 2.05 1.2
[bar/deg] [bar s/deg] [bar/deg/s]
instrumentation amplifier INA122. Based on desirable safety attributes on the interaction between the patient and CoRAnT, some constraints on the controller output can be posed by setting into the control routine a maximum pressure limit. During the trajectory tracking with minimum joint compliance, the error from the reference trajectory should be minimized by setting a maximum pressure limit compatible with the actuator saturation, whereas a strategy of maximum compliance suggests to lower such a limit in order to safely absorb the involuntary contractions of the patient. In the following tests for the experimental validation of the tracking control performance, the maximum pressure limit is set to 5.7 bar in order to improve the tracking accuracy.
5. Results and discussion The results of the tests, performed on CoRAnT during some rehabilitation sessions, are presented and discussed in this Section. During the phase of design of experiments, the range of motion and frequency of the sinusoidal references are selected on the basis of the biomechanical requirements for lower limb rehabilitation tasks. The trajectory tracking control performance of CoRAnT has been evaluated using the reference profiles at different frequencies. A set of tests have been carried out on the robotic exoskeleton performing a rehabilitation session on a female subject (29 y.o., weight 53 kg) sitting on a chair. The initial pressure of the PAMs is set to 3 bar; this value provides a limited stiffness of the joint in favour of the safety constraints. Fig. 5 shows the results of the sinusoidal trajectory tracking at frequency of 0.5 Hz and 1 Hz, achieving a root mean square error of 1.25° and 1.76°, respectively. The feedforward action compensating for the nonlinearity of the actuators is essential to guarantee good tracking performance, as shown from the results of Fig. 6 where the tracking error is compared for two different control configurations. For the same sinusoidal reference with amplitude of 15° and frequency of 0.5 Hz, the first configuration implements a PID without model-based compensation; in the second one, the tracking performance is improved considerably by adding to the pure feedback scheme the nonlinear compensation law of Section 4. The aforementioned experimental results show that CoRAnT can perform repetitive rehabilitation tasks through satisfactory tracking accuracy. Even in the case of increase in the frequency of the reference trajectory, the good performance of CoRAnT is guaranteed by the proposed feedforward-feedback control scheme. In this respect, the fast and accurate tracking can be evaluated from the closed loop response to step input. The tracking error obtained for the experimental step responses is represented in Fig. 7; the constant reference signals used in the setpoint tracking control cover the range of angular displacement allowed by mechanical hardware of the robot. The experimental tests, performed for the cases of flexion (Θref = +15∘) and extension (Θref = −15∘), are characterized by fast convergence to the setpoint, with smooth transient and limited control effort. In particular, after 0.28 s the absolute value of the error does not exceed 0.4°. The error at 2 s is negligible and it amounts to 0.057° (extension) and to 0.114° (flexion). Fig. 8 illustrates other important features of the combined
6. Conclusions This work sets out with the objective of designing efficient strategies of precision tracking control for a biomechatronic device, using a combined feedforward-feedback control scheme for rehabilitation tasks. 74
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Fig. 5. Tracking of sinusoidal trajectory with amplitude range of 20° (reference signal in blue line and measured angle in red dashed line). (a) Frequency of 0.5 Hz. (b) Frequency of 1 Hz. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Tracking error with (blue line) and without (red dashed line) compensation action. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The comparison of the experimental results to the respective ones obtained via other nonlinear control techniques for PAMs-based exoskeletons reveals that the improved control performance can be achieved in all the scenarios of robotic rehabilitation where fast and robust trajectory tracking is required. Future works will be directed to the control of the impedance of the robot joint using the viscoelastic properties of the PAMs. Attention will also be devoted to reference trajectory generation, e.g., using profiles taken from gait analysis data.
Both a model of the PAMs-actuated joint and a nonlinear control strategy are provided that enable reasonably accurate compensation of the nonlinear dynamics while keeping the computational burden low. The model-based control law, achieving a balance between control precision and computational requirements, allows the efficient implementation of the tracking control on embedded digital controller. The experimental tests on the rehabilitation exoskeleton, following the digital implementation and tuning of the controller on embedded control unit, have shown good tracking performance and robustness.
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Fig. 7. Setpoint tracking error of flexion (θref = +15∘ ) and extension movements (θref = −15∘ ).
Fig. 8. Commanded pressure on the PAMs pair for (a) flexion (θref = +15∘ ) and (b) extension movements (θref = −15∘ ).
References
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Domenico Colacino was born in Catanzaro, Italy, in 1984. He received the Laurea degree in Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy in 2008. He received the Ph.D. degree in Computer and Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy, in 2014. He is particularly interested in modelling and design of mechatronic devices for biomedical applications and also in analysis and control of nonlinear systems.
2004;12(3):349–59. [12] Thanh TDC, Ahn KK. Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network. Mechatronics 2006;16(9):577–87. [13] Choi T-Y, Choi B-S, Seo K-H. Position and compliance control of a pneumatic muscle actuated manipulator for enhanced safety. Control Syst Technol IEEE Trans 2011;19(4):832–42. http://dx.doi.org/10.1109/TCST.2010.2052362. [14] Vo Minh T, Tjahjowidodob T, Ramonc H, Van Brussel H. Cascade position control of a single pneumatic artificial muscle-mass system with hysteresis compensation. Mechatronics 2010;20(3):402–14. [15] Schindele D, Aschemann H. Model-based compensation of hysteresis in the force characteristic of pneumatic muscles. 2012 12th IEEE international workshop on advanced motion control (AMC). 2012. p. 1–6. [16] Amato F, Colacino D, Cosentino C, Merola A. Robust and optimal tracking control for manipulator arm driven by pneumatic muscle actuators. 2013 IEEE international conference on mechatronics (ICM). 2013. p. 827–34. [17] Amato F, Colacino D, Cosentino C, Merola A. Guaranteed cost control for uncertain nonlinear quadratic systems. European control conference (ECC). 2014. p. 1229–35. http://dx.doi.org/10.1109/ECC.2014.6862287. [18] Reynolds DB, Repperger DW, Phillips CA, Bandry G. Modeling the dynamic characteristics of pneumatic muscle. Ann Biomed Eng 2003;8(3):310–7. [19] Takosoglu JE, Laski PA, Blasiak S, Bracha G, Pietrala D. Determining the static characteristics of pneumatic muscles. Meas Control 2016;49(2):62–71. http://dx. doi.org/10.1177/0020294016629176. [20] Siciliano B, Khatib O, editors. Springer handbook of robotics. Springer; 2008. [21] Van Damme M, Vanderborght B, Verrelst B, Van Ham R, Daerden F, Lefeber D. Proxy-based sliding mode control of a planar pneumatic manipulator. Int J Rob Res 2009;28(2):266–84. http://dx.doi.org/10.1177/0278364908095842. [22] Andrikopoulos G, Nikolakopoulos G, Manesis S. Design and development of an exoskeletal wrist prototype via pneumatic artificial muscles. Meccanica 2015;50(11):2709–30. http://dx.doi.org/10.1007/s11012-015-0199-8.
Carlo Cosentino received the Laurea degree (M.Sc.) in Computer Engineering and the Ph.D. degree in Computer and Automation Engineering, both from Federico II University of Naples, Italy, in 2001 and 2005, respectively. Since 2014 he is Associate Professor of Systems and Control Theory at Magna Græcia University of Catanzaro. He has published more than 100 articles in international peer-reviewed journals, conference proceedings and edited books and is co-author of two scientific monographs. His research interests include finite-time stability of linear systems, stability of nonlinear quadratic systems and applications of systems and control theory to the fields of systems and synthetic biology.
Francesco Amato was born in Naples on February 2, 1965. He received the Laurea and the Ph.D. Degree both in Electronic Engineering from the University of Naples in 1990 and 1994 respectively. From 2001 to 2003 he has been Full Professor of Automatic Control at the University of Reggio Calabria. In 2003 he moved to the University of Catanzaro, where, since 2010, he is Professor of Bioengineering. He is currently the Dean of the School of Computer and Biomedical Engineering, the Coordinator of the Doctorate School in Biomedical and Computer Engineering, the Director of the Biomechatronics Laboratory, and member and vice-president of the Concilium of the School of Medicine and Surgery. The scientific activity of Francesco Amato has developed in the fields of systems and control theory with applications to the contexts of the computational biology and of the modeling and control of biomedical systems. He has published about 250 papers in international Journals and conference proceedings and two monographies with Springer Verlag entitled “Robust Control of Linear Systems subject to Uncertain Time-Varying Parameters” and “Finite-Time Stability and Control”.
Alessio Merola was born in Catanzaro, Italy, in 1979. He received the Laurea degree (summa cum laude) in Mechanical Engineering from the University of Calabria, Italy, and the Ph.D. degree in Computer and Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy, in 2003 and 2008, respectively. He is currently Assistant Professor of Systems and Control Engineering at the Department of Experimental and Clinical Medicine of the University of Catanzaro Magna Græcia. He is coauthor of about 40 scientific papers, published on international journals and proceedings of international conferences. His current research interests include analysis and control of nonlinear systems and control of biomechatronic devices.
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Model-based tracking control design, implementation of embedded digital controller and testing of a biomechatronic device for robotic rehabilitation☆
T
⁎
Alessio Merola, Domenico Colacino, Carlo Cosentino , Francesco Amato School of Computer and Biomedical Engineering, Universitá degli Studi Magna Græcia di Catanzaro, Campus Universitario di Germaneto, Catanzaro 88100, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Model-based control Trajectory tracking control Pneumatic artificial muscles Rehabilitation robotics
In this paper, the tracking control problem of a biomimetic exoskeleton powered by a pair of pneumatic artificial muscles is considered. The antagonistic configuration of the pair of pneumatic muscles, which is biologically inspired, enables safe and reliable actuation in applications of orthopaedic rehabilitation. However, during the inflation-deflation process, the pneumatic muscles introduce nonlinearity and hysteresis which deteriorate the control performance. A model of the antagonistic artificial muscles is adopted to develop a computed-torque control for feedforward compensation of the nonlinear dynamics of the actuated joint. A PID control action is used in combination with the feedforward compensation to achieve fast and accurate tracking control performance. The model, which possesses a reduced set of parameters as functions of the inflation/deflation phase, enables efficient nonlinear compensation. The experimental tests on the biomechatronic device, compared with other state-of-the-art approaches for controlling pneumatic artificial muscles, show better tracking performance in terms of convergence rate and robustness, justifying the convenience of using the proposed control methodology in the design of tracking controllers for exoskeletal biomechatronic devices.
1. Introduction The emerging field of soft robotics is currently covering novel applications in rehabilitation robotics, prosthetics and surgical robotics and, more in general, several safety-critical applications involving interactions between robots and human operators. The main requirements for safe human-robot interactions [1] can be fulfilled through the inherent (and adaptable) compliance of soft actuators. Moreover, the adaptation mechanisms of the compliance of soft robotic and biomechatronic devices can mimic the behaviour of the biological musculo-skeletal system [2,3]. As an example of biomimetic and soft actuation, pneumatic artificial muscles (PAMs) have been employed in the realization of rehabilitation robots, wearable exoskeleton robots and energy-efficient walking humanoids (see [4–6]). More recently, pneumatic muscle actuation technologies are developed towards the realization of miniaturized biomechatronic devices. For instance, the work [7] focuses on the characterization of pneumatic muscles for set-point regulation of the motion of a biomechatronic finger. Together with their advantages, PAMs offer some challenges in the design and implementation of the tracking control, since the controller has to handle the strong nonlinearity of the PAMs dynamics.
☆ ⁎
Some control techniques have been developed to solve the problems of regulation and trajectory tracking for PAMs-driven robots. Prior results focus on variable structure control [8], gain scheduling [9], adaptive backstepping [10], sliding mode [11] and PID neural network control [12]. The sliding mode control strategy recently proposed in [13] is aimed at the enhancement of the safety during collision with obstacles. Therefore, the sliding mode tracking controller is complemented by a joint compliance controller, which meets the safety requirements during collision. Model-based compensation strategies are originally proposed in [14,15], where the compensation of the hysteresis in the force characteristics of pneumatic muscles is achieved on the basis of generalized models of the hysteresis in the mechanical response of PAMs. A feedforward compensation is implemented into the feedback control schemes of linear positioning stages implementing backstepping [15] and cascade control [14] strategies. More recently, in [16,17], the authors show how the guaranteed-cost control approach can be effectively applied to the solution of a regulation problem for a PAMs-driven robot whose dynamics can be described through uncertain (bilinear and quadratic) systems. To foster the efficient implementation of model-based control strategies for PAMs-driven robots, lumped parameter models of the
This paper was recommended for publication by Associate Editor Prof Kong Kyoungchul. Corresponding author. E-mail addresses: [email protected] (A. Merola), [email protected] (D. Colacino), [email protected] (C. Cosentino), [email protected] (F. Amato).
https://doi.org/10.1016/j.mechatronics.2018.04.006 Received 12 June 2017; Received in revised form 16 February 2018; Accepted 25 April 2018 0957-4158/ © 2018 Published by Elsevier Ltd.
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nonlinear dynamics of PAMs can be exploited. The three-element phenomenological model proposed by Reynolds et al. [18] allows to accurately predict the dynamic response of PAMs; the model parameters can be easily identified through static perturbation tests at several constant values of pressure. More recently, in [19] a series of static perturbation tests have been carried out for the efficient identification of a novel model of the mechanical response of some classes of PAMs. The main contribution of this work consists in the development and the experimental evaluation of a novel model-based tracking control methodology for a biomechatronic device powered by PAMs in antagonistic configuration. The lumped parameter model by Reynolds et al. [18] is used here to develop a computed torque control for feedforward compensation of the actuated joint via nonlinear inversion. The Reynolds’s model provides an efficient description of the variability of the nonlinear dynamics of the PAMs during the pressure cycle. Therefore, the feedforward control action is a function of the model parameters which depend on the phase of inflation/deflation. The compensation loop of the tracking controller is complemented by a PID control action which enables the robust regulation to zero of the tracking error. To the best of the authors’ knowledge, this is the first time that such a combined nonlinear inversion feedforward + feedback PID control methodology has been proposed and experimentally tested for the tracking control of a PAM-based robot. The devised approach improves the tracking performance over the existing approaches in the related literature on the control of PAMs-based robots. The parsimony of the model makes this control approach very suitable for the real-time implementation on embedded microcontroller devices. Therefore, a non negligible byproduct of this work is the microcontroller-based implementation of the embedded control system, whereas the previously published results were tested by means of PC-based implementations. Some experimental tests are presented to prove the superior control performance achieved by the proposed methodology thanks to the efficient compensation of the nonlinear dynamics. Specifically, the performance of the closed loop control system has been measured in terms of convergence rate, steady-state error and robustness to load disturbance during the tracking of constant and sinusoidal trajectories. Furthermore, the micro-controller based implementation of the proposed control scheme shows the advantages related to the real-time execution, computational resources and customizable constraints on the control action. This paper is structured as follows. The main technical specifications of the robotic exoskeleton and a model of the dynamics of the actuated robot are presented in Sections 2 and 3, respectively. The model-based tracking control law is derived in Section 4, where the steps of design and implementation of the digital controller on embedded control unit are also described. The experimental tests and the discussion of the experimental results are given in Section 5. Some concluding remarks are eventually left to Section 6.
Fig. 1. Overview of CoRAnT. (a) Orthotic shoe. (b) Pneumatic muscle actuator. (c) Pressure regulator. (d) Load cell. (e) Optical encoder.
To obtain a training performance at least comparable to the manual therapy, an essential requirement for biomechatronic rehabilitation is the safe interaction between the patient and the rehabilitation device. During the conventional clinical therapy, the manipulations performed by the therapist are adapted to the resistance force exerted by the patient. Similar adaptation mechanisms, guaranteeing the intrinsic safety of robotic rehabilitation tasks, can be realized through suitable control strategies of biomechatronic devices driven by biomimetic and soft actuators. For instance, the natural compliance of soft actuators, which utilize air as source of energy, enables to absorb potential shocks occurring during the manipulation of the patient. After the analysis of both the main issues involved in physical rehabilitation and the advantages provided by robotic and biomechatronic technologies, the main specifications are implemented in a robotic exoskeleton for rehabilitation following the design principles of soft and biomimetic robotics. Therefore, CoRAnT is actuated by soft pneumatic muscles, which guarantee safety and comfort for the patient. The mechanical structure and the main components of the robotic exoskeleton are highlighted in Fig. 1; the pneumatic muscle used for the actuation of the exoskeleton is the fluidic muscle MAS-20-200N by FESTO company. An orthotic shoe is installed on the rotational joint of the exoskeleton; the joint axis is aligned with the ankle axis of the patient (see Fig. 2). A couple of PAMs is required for the full actuation of the joint, since a single PAM actively generates motion in one direction. The PAMs are arranged in the bio-inspired configuration of Fig. 3, where the antagonistic setup can mimic the mechanisms of regulation of motion and stiffness of the joints in the human musculo-skeletal system. Therefore, in analogy with the human anatomy, the PAMs act as biceps and triceps, respectively. The conversion of the linear motion of the actuators to the joint rotation is obtained through some cables of diameter 3.5 mm tied together around a pulley of diameter 80 mm. Position and force sensors are installed on the robotic exoskeleton. An optical rotary encoder (AVAGO HEDM5500 J14) measures the angular position of the joint at 1024 counts per revolution. The encoder provides the measured variable to the tracking controller. Moreover, each PAM is connected to a load cell (Phidgets 3138 S-type). The signals of force and angle can be used for measuring the interaction between
2. CoRAnT: Compliant Robotic Ankle Trainer The traditional therapy for stroke and traumatic brain injuries requires long and intensive rehabilitation tasks performed by the physical therapist. Indeed, the quick recovery of neural and motor functions, resulting from cortical reorganization in the motor cortex of the patient, can be achieved under intensive and repetitive exercises. Unfortunately, the required burden of care may contrast with the budget and time constraints. Thanks to the accuracy and repeatability of the control performance, robotic exoskeletons performing high-intensity physical rehabilitation can offer their potential in the optimization of costs and efficiency of the rehabilitation procedures. Moreover, biomechatronic and robotic devices can collect quantitative data useful for the evaluation of both the patient’s motor performance and the progress of the motor recovery. 71
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Table 1 Model parameters.
Spring element Damping element Force element
Parameter
Value
k0 k1 b0i (b0d) b1i (b1d) f0 f1
4.82 131 1.22 (1.31) 135 (−230) 14.89 2090
[N/m] [N/m/bar] [N/m/s] [N/m/s/bar] [N] [N/bar]
sense of the contraction displacement. The nonlinearity introduced by the PAMs is described through the three-element phenomenological model by Reynolds et al. [18]. The expression of the total force ϕ exerted by a single PAM is obtained from a parallel configuration of an elastic element (spring) of stiffness k(p), a viscous element (damper) of damping coefficient b(p) and a contractile element generating the active force contribution f(p). Therefore, the PAM force reads
ϕ = f (p) − b (p) s˙ − k (p) s, where p denotes the pressure at which the PAM is inflated and s is the contraction length. The model parameters follow the constitutive laws Fig. 2. CoRAnT performing a rehabilitation task on the patient. (a) Lateral view. (b) Front view.
f (p)= f0 + f1 p
[N]
b0i + b1i p [N/m/s] inflation b (p)= b0 + b1 p = ⎧ ⎨ ⎩ b0d + b1d p [N/m/s] deflation
k (p)= k 0 + k1 p
[N/m]
The values of the parameters of the damping function b(p) vary depending on the phase of inflation/deflation. Concerning the values of the model parameters, we refer both to Table 1 and to the identification procedure described in [18]. The values listed in Table 1 are taken from [13], where the model parameters have been identified for the same FESTO fluidic muscles adopted in CoRAnT. The parameters of the elastic and damping functions of the pressure can be identified from the step response of the actuator at constant pressure and subject to a step change of the applied load. The contractile (active force) element function is determined by a step change to the inflation pressure of the actuator under constant load. The model of the actuated joint requires the moment of inertia of CoRAnT. The estimated moment of inertia amounts to Il = 0.0023 kg m2. This value can be obtained from the CAD 3D model of CoRAnT after specifying the materials of the parts of the model. The torque of the actuated joint reads
Fig. 3. Actuation scheme of CoRAnT.
τ = τb − τt = (ϕb − ϕt ) r , the robot and the patient, e.g., through the measurement of the torque opposed by the patient during the motion cycle. For instance, the patient’s participation to the rehabilitation task could be estimated through the measured data. The regulation of the pressure of the PAMs, involved in the trajectory control of the joint, is obtained through two proportional pressure regulators FESTO MPPE-3-1/8-6-010-B. The implementation of the pressure regulation into the motion control system of the exoskeleton is discussed in detail in Section 4.
(1)
where τb and τt are the torques generated by each PAM, as in
τb= (fb − bb s˙b − kb sb) r
(2a)
τt = (ft − bt s˙t − kt st ) r .
(2b)
In (2), sb and st denote the contraction length of biceps and triceps, respectively. From the antagonistic configuration of Fig. 3, a positive rotation is obtained through the active force generated by the inflation of the biceps, whereas the triceps deflates without active contribution to motion. For a negative rotation, it is required that the triceps generates the force driving the joint rotation, whereas the biceps is passive. The total torque can be expressed in terms of the joint angle θ, using the kinematic relations
3. Derivation of the equations of motion including actuator dynamics Starting from the kinematic scheme of Fig. 3, the equations of motion of CoRAnT are derived. Following the notation of Fig. 3, clockwise rotations are positive, whereas the upright vectors ϕb and ϕt denote the forces exerted by biceps and triceps, respectively. In Fig. 3, the force vectors point upward since each PAM generates its (actuation or resistance) force only in the
π sb = r ⎛θ + ⎞, 6⎠ ⎝
π st = r ⎛ − θ ⎞. ⎝6 ⎠
In a first stage of the actuation strategy, each PAM is inflated at the initial positive pressure p0. Thereafter, the joint rotation is obtained through the antagonistic control of the pressure of the pair of PAMs. A 72
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approach of the computed-torque control (see [20], Ch. 6.6). The control input can be computed in real time according to the feedforward-feedback scheme, after measuring both the kinematic variables and the model parameters of the actuated joint. The tracking control system, following a time-varying reference trajectory θd(t), requires the derivation of the tracking error dynamics as a function of the error e : =θd − θ and its successive derivative e˙: =θ˙d − θ˙ and e¨: =θ¨d − θ¨ . The nonlinear compensation is achieved through the control input
positive/negative rotation of the joint requires an increment/decrement of the pressure pb of the biceps and a decrement/increment of the pressure pt of the triceps. Therefore, the antagonistic actuation can be formulated as
pb = p0 + Δp ,
pt = p0 − Δp ,
where p0 is the initial pressure and Δp is the pressure difference from p0 used as manipulated variable in the antagonistic pressure control. It is useful to note that the antagonistic strategy simplifies the motion control, since only one control variable, the pressure difference Δp, is used for the actuation of the joint through two PAMs. Therefore, the dynamics of the biomechatronic joint is described as
Il θ¨ + ζθ˙ + τg (θ) + d = τ ,
u=
where Il denotes the link inertia, ζ is the viscous friction coefficient at the joint, τg(θ) denotes the gravity torque, τ is the control input torque and d represents the unknown external input disturbance which embeds some external perturbations, e.g., load variations or disturbance torque, other than modelling mismatches, unknown parametric uncertainties, friction nonlinearities, etc. Since the PAMs actuating the joint are identical, the spring element parameters are taken as k 0b = k 0t = k 0 and k1b = k1t = k1. The pressure difference Δp is the control input of the closed-loop system, i.e. u = Δp. After introducing in (3) the constitutive laws of the actuators, the equation of motion of the joint, including actuator dynamics, reads
v = kP e + kV e˙ + kI
⎡ ∫ edt ⎤ x: =⎢ e ⎥, ⎢ ⎦ ⎣ e˙ ⎥ then the state space representation of the closed loop system yields
x˙ = Ax + Bw, where
1 0 ⎞ ⎛ 0 0 1 A=⎜ 0 , ⎜−I −1 k −I −1 k −I −1 k ⎟⎟ I P V l l l ⎝ ⎠
with (5a)
β = r 2 [b0b + b0t + p0 (b1b + b1t )]
(5b)
π δ = 2rf1 − k1 r 2 3
(5c)
σ = r 2 (b1t − b1b).
(5d)
∫ edt.
Let us define the state vector as
(4)
α = 2r 2 (k 0 + k1 p0 )
(6)
where v is an auxiliary control signal to be designed in order to achieve robust tracking performance. To this end, v is taken as a PID control action in the form
(3)
Il θ¨ + (ζ + β ) θ˙ + αθ + τg (θ ) + d = (δ + σθ˙ ) u,
Il θ¨d + αθ + βθ˙ + τg (θ) + v , δ + σθ˙
⎛0 ⎞ B = ⎜ 0 ⎟, ⎜ I −1⎟ ⎝l ⎠
and w defines the disturbance which embodies friction effects, model uncertainties and unknown interaction torque. The nominal value of the inertia and the measured values of θ and θ˙ are introduced in the control action (6). Moreover, the value of the parameters α, β and δ are computed offline after the regulation of the initial pressure to the desired value p0; only the parameter σ, which depends on the inflation/deflation phase, requires an online computation. The PID controller parameters, as shown in Table 2, are tuned and verified experimentally for optimal results. The tracking controller is implemented on 8-bit AVR RISC-based microcontroller ATmega 328; the control signal and the measurements are processed at 50 Hz. An I2C bus interconnects all the digital components of the embedded control system: microcontroller, optical encoder and digital-analog converter (DAC). The pressures of the PAMs are regulated through two proportional pressure regulators in the range 0–6 bar. Two complementary control signals are generated to drive the pressure regulators associated to the pair of antagonistic muscles. One signal inflates/deflates the biceps and the other one deflates/inflates the triceps in order to produce positive/ negative rotations of the joint. The control signals in the range 0–10 V, which correspond to the pressure outputs 0–6 bar, are obtained after conversion of the control signals via a dual channel 12-bit DAC TLV5618 and through amplification by a programmable
Given the initial pressure p0, all the parameters defined in (5) do not vary during the motion, except for the value of σ which depends on the phase of inflation/deflation. The parameter α can be associated to a stiffness coefficient of the elastic term αθ arising from the spring element of the three-element phenomenological model of the PAM, whereas β can be viewed as an equivalent viscous friction factor depending on both the initial pressure and damping effects of the actuator. The parameters δ and σ describe the variability of the generated torque through the damping and active force elements of the actuator model. 4. Model-based tracking control and digital implementation A satisfactory trajectory-tracking control performance can be achieved through the combined feedforward-feedback control scheme of Fig. 4, which takes into account the nonlinearity introduced by the pneumatic actuators. A control law, which attains a compensation of the nonlinear terms of the dynamics (4), is developed through the
Fig. 4. Torque inverse control scheme. 73
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feedforward-feedback PID control scheme: the control action is characterized both by smoothness in response to sudden changes in desired position and by robustness against the patient load. The experimental results show that the proposed inverse modelbased controller is capable of efficiently controlling the antagonistic mechanism of the PAMs pair, while compensating for nonlinear phenomena and external load disturbances occurring during the inflationdeflation cycle. The superior performance achieved by the proposed tracking controller can be appreciated with respect to the related approaches in the scientific literature on nonlinear control of pneumatic muscle actuators in antagonistic configuration. Our model-based tracking controller outperforms some sliding mode control strategies [11,21] and nonlinear PID control techniques [12,22]. In [11], a sliding mode approach is applied to the trajectory tracking control of a pneumatic artificial muscle manipulator; a maximum joint error of 7° is achieved for a sinusoidal reference trajectory. Chattering, lack of robustness and slow responses are the major issues involved in sliding mode control strategies. In [21], a sliding mode tracking controller of a pneumatic artificial muscle manipulator is optimized in order to attenuate the oscillations in the system response. The mitigation of the oscillations is obtained at the cost of a longer settling time since, for a step change of about 35° in the desired angle, the joint angle converges to the desired value after a minimum transient of 2.5 s. In [12], a nonlinear PID controller using a neural network is presented for controlling a planar robotic arm actuated by PAMs in antagonistic setup. After training of the neural network, a maximum tracking error of about 2.5° is obtained following a sinusoidal reference trajectory ranging from −20° to 20° at frequency 0.2 Hz. Further comparison with the nonlinear PID controller of an exoskeletal wrist in [22] shows the superior performance of our feedforward-feedback PID control scheme in terms of fast and accurate tracking. To this end, it is useful to compare the step responses of the PAM-based exoskeletal wrist, which are presented in [22] for constant angular references of −10°, −20°, −30°, 10°, 20° and 30°. The experiments of flexion-extension movements are performed on the exoskeleton equipped with a spherical joint simulating a passive wrist; the rising time in response to the references of −20° and 20° is of the order of 0.5 s. The step responses acquired on CoRAnT exhibit better tracking performance when compared to the respective responses of the exoskeletal wrist, since the absolute tracking error reduces to 0.4° after 0.28 s in presence of the patient load. The tracking control performance achieved during the flexion-extension experiments in [22] has been evaluated for sinusoidal reference trajectories at maximum frequency of 0.4 Hz. The execution frequency of the PID controller is 10 Hz, a value compatible with the time required for computing the control action. The improvement of the control performance by the feedforwardfeedback scheme of CoRAnt is confirmed also for the case of tracking of a sinusoidal reference over an extended range of frequency, up to 1 Hz. Moreover, the model-based control law can be efficiently implemented at the higher frequency of 50 Hz on embedded microcontroller, without requiring the PC-based implementations involved in [11,12,21,22]. The experimental results prove that the proposed control scheme succeeds in guaranteeing the robust and efficient control performance required for rehabilitation purposes and, also in comparison with the existing results in the related field of the control of PAMs-based exoskeletons, it yields fast and accurate tracking performance.
Table 2 Parameters of the PID controller. Parameter
Value
KP KD KI
30.1 2.05 1.2
[bar/deg] [bar s/deg] [bar/deg/s]
instrumentation amplifier INA122. Based on desirable safety attributes on the interaction between the patient and CoRAnT, some constraints on the controller output can be posed by setting into the control routine a maximum pressure limit. During the trajectory tracking with minimum joint compliance, the error from the reference trajectory should be minimized by setting a maximum pressure limit compatible with the actuator saturation, whereas a strategy of maximum compliance suggests to lower such a limit in order to safely absorb the involuntary contractions of the patient. In the following tests for the experimental validation of the tracking control performance, the maximum pressure limit is set to 5.7 bar in order to improve the tracking accuracy.
5. Results and discussion The results of the tests, performed on CoRAnT during some rehabilitation sessions, are presented and discussed in this Section. During the phase of design of experiments, the range of motion and frequency of the sinusoidal references are selected on the basis of the biomechanical requirements for lower limb rehabilitation tasks. The trajectory tracking control performance of CoRAnT has been evaluated using the reference profiles at different frequencies. A set of tests have been carried out on the robotic exoskeleton performing a rehabilitation session on a female subject (29 y.o., weight 53 kg) sitting on a chair. The initial pressure of the PAMs is set to 3 bar; this value provides a limited stiffness of the joint in favour of the safety constraints. Fig. 5 shows the results of the sinusoidal trajectory tracking at frequency of 0.5 Hz and 1 Hz, achieving a root mean square error of 1.25° and 1.76°, respectively. The feedforward action compensating for the nonlinearity of the actuators is essential to guarantee good tracking performance, as shown from the results of Fig. 6 where the tracking error is compared for two different control configurations. For the same sinusoidal reference with amplitude of 15° and frequency of 0.5 Hz, the first configuration implements a PID without model-based compensation; in the second one, the tracking performance is improved considerably by adding to the pure feedback scheme the nonlinear compensation law of Section 4. The aforementioned experimental results show that CoRAnT can perform repetitive rehabilitation tasks through satisfactory tracking accuracy. Even in the case of increase in the frequency of the reference trajectory, the good performance of CoRAnT is guaranteed by the proposed feedforward-feedback control scheme. In this respect, the fast and accurate tracking can be evaluated from the closed loop response to step input. The tracking error obtained for the experimental step responses is represented in Fig. 7; the constant reference signals used in the setpoint tracking control cover the range of angular displacement allowed by mechanical hardware of the robot. The experimental tests, performed for the cases of flexion (Θref = +15∘) and extension (Θref = −15∘), are characterized by fast convergence to the setpoint, with smooth transient and limited control effort. In particular, after 0.28 s the absolute value of the error does not exceed 0.4°. The error at 2 s is negligible and it amounts to 0.057° (extension) and to 0.114° (flexion). Fig. 8 illustrates other important features of the combined
6. Conclusions This work sets out with the objective of designing efficient strategies of precision tracking control for a biomechatronic device, using a combined feedforward-feedback control scheme for rehabilitation tasks. 74
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Fig. 5. Tracking of sinusoidal trajectory with amplitude range of 20° (reference signal in blue line and measured angle in red dashed line). (a) Frequency of 0.5 Hz. (b) Frequency of 1 Hz. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Tracking error with (blue line) and without (red dashed line) compensation action. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The comparison of the experimental results to the respective ones obtained via other nonlinear control techniques for PAMs-based exoskeletons reveals that the improved control performance can be achieved in all the scenarios of robotic rehabilitation where fast and robust trajectory tracking is required. Future works will be directed to the control of the impedance of the robot joint using the viscoelastic properties of the PAMs. Attention will also be devoted to reference trajectory generation, e.g., using profiles taken from gait analysis data.
Both a model of the PAMs-actuated joint and a nonlinear control strategy are provided that enable reasonably accurate compensation of the nonlinear dynamics while keeping the computational burden low. The model-based control law, achieving a balance between control precision and computational requirements, allows the efficient implementation of the tracking control on embedded digital controller. The experimental tests on the rehabilitation exoskeleton, following the digital implementation and tuning of the controller on embedded control unit, have shown good tracking performance and robustness.
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Fig. 7. Setpoint tracking error of flexion (θref = +15∘ ) and extension movements (θref = −15∘ ).
Fig. 8. Commanded pressure on the PAMs pair for (a) flexion (θref = +15∘ ) and (b) extension movements (θref = −15∘ ).
References
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Domenico Colacino was born in Catanzaro, Italy, in 1984. He received the Laurea degree in Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy in 2008. He received the Ph.D. degree in Computer and Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy, in 2014. He is particularly interested in modelling and design of mechatronic devices for biomedical applications and also in analysis and control of nonlinear systems.
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Carlo Cosentino received the Laurea degree (M.Sc.) in Computer Engineering and the Ph.D. degree in Computer and Automation Engineering, both from Federico II University of Naples, Italy, in 2001 and 2005, respectively. Since 2014 he is Associate Professor of Systems and Control Theory at Magna Græcia University of Catanzaro. He has published more than 100 articles in international peer-reviewed journals, conference proceedings and edited books and is co-author of two scientific monographs. His research interests include finite-time stability of linear systems, stability of nonlinear quadratic systems and applications of systems and control theory to the fields of systems and synthetic biology.
Francesco Amato was born in Naples on February 2, 1965. He received the Laurea and the Ph.D. Degree both in Electronic Engineering from the University of Naples in 1990 and 1994 respectively. From 2001 to 2003 he has been Full Professor of Automatic Control at the University of Reggio Calabria. In 2003 he moved to the University of Catanzaro, where, since 2010, he is Professor of Bioengineering. He is currently the Dean of the School of Computer and Biomedical Engineering, the Coordinator of the Doctorate School in Biomedical and Computer Engineering, the Director of the Biomechatronics Laboratory, and member and vice-president of the Concilium of the School of Medicine and Surgery. The scientific activity of Francesco Amato has developed in the fields of systems and control theory with applications to the contexts of the computational biology and of the modeling and control of biomedical systems. He has published about 250 papers in international Journals and conference proceedings and two monographies with Springer Verlag entitled “Robust Control of Linear Systems subject to Uncertain Time-Varying Parameters” and “Finite-Time Stability and Control”.
Alessio Merola was born in Catanzaro, Italy, in 1979. He received the Laurea degree (summa cum laude) in Mechanical Engineering from the University of Calabria, Italy, and the Ph.D. degree in Computer and Biomedical Engineering from the University of Catanzaro Magna Græcia, Italy, in 2003 and 2008, respectively. He is currently Assistant Professor of Systems and Control Engineering at the Department of Experimental and Clinical Medicine of the University of Catanzaro Magna Græcia. He is coauthor of about 40 scientific papers, published on international journals and proceedings of international conferences. His current research interests include analysis and control of nonlinear systems and control of biomechatronic devices.
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