Wrist tremor suppression based on repetitive control with multi-muscle electrical stimulation⁎

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Wrist tremor suppression based on Wrist Wrist tremor tremor suppression suppression based based on on Wrist tremor suppression based on repetitive control with multi-muscle repetitive controlstimulation with multi-muscle  electrical repetitive controlstimulation with multi-muscle electrical electrical stimulation  electrical stimulation ∗ ∗ ∗∗ Zan Zhang ∗∗∗ Bing Chu ∗∗ ∗∗ Yanhong Liu ∗ ∗ Haichuan Ren ∗ ∗

Zan Liu Zan Zhang Zhang Bing Bing Chu Chu ∗∗ Yanhong Yanhong Liu ∗ Haichuan Haichuan Ren Ren ∗,∗∗∗ ∗,∗∗∗ Zan Zhang ∗∗ Bing Chu Yanhong Liu ∗ Haichuan Ren ∗∗ D.H. Owens ∗,∗∗∗ D.H. ∗∗ Owens D.H. Owens ∗,∗∗∗ Zan Zhang Bing Chu Yanhong Liu Haichuan Ren D.H. Owens ∗,∗∗∗ D.H. Owens ∗ ∗ of Electrical Zhengzhou University, Zhengzhou, ∗ School of Electrical Engineering, Engineering, Zhengzhou University, Zhengzhou, School of Engineering, Zhengzhou University, ∗ School SchoolChina, of Electrical Electrical Engineering, Zhengzhou University, Zhengzhou, Zhengzhou, China, (e-mail: [email protected], [email protected], (e-mail: [email protected], [email protected], ∗ China, (e-mail: [email protected], [email protected], SchoolChina, of Electrical Engineering, Zhengzhou University, Zhengzhou, (e-mail: [email protected], [email protected], [email protected], [email protected]) [email protected], [email protected]) [email protected], [email protected]) ∗∗ China, (e-mail: [email protected], [email protected], ∗∗ [email protected], [email protected]) of Electronics and Computer ∗∗ Department Department of Electronics and Computer Science, University of Department of Electronics and Computer Science, Science, University University of of ∗∗ [email protected], [email protected]) Department of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK (e-mail: Southampton, Southampton, SO17 1BJ, UK (e-mail: ∗∗ Southampton, Southampton, SO17 1BJ, UK (e-mail: Department of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK (e-mail: [email protected]) [email protected]) [email protected]) ∗∗∗ Southampton, Southampton, SO17 1BJ, UK (e-mail: ∗∗∗ [email protected]) Department of Automatic Control and Systems Engineering, ∗∗∗ Department of Automatic Control and Systems Engineering, The Control and Systems Engineering, The The ∗∗∗ Department of Automatic [email protected]) Departmentof Automatic Control and Systems University Mappin Street, Sheffield, S1 University ofofSheffield, Sheffield, Mappin Street, Sheffield,Engineering, S1 3JD, 3JD, UK UKThe ∗∗∗ University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UKThe University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK Abstract: Abstract: Tremor is aa rhythmic, rhythmic, alternating swing motion caused by involuntary repetition of Abstract: Tremor Tremor is is a rhythmic, alternating alternating swing swing motion motion caused caused by by involuntary involuntary repetition repetition of of Abstract: Tremor is a rhythmic, alternating swing motion caused by involuntary repetition of muscle contraction and relaxation. Although it does not endanger life, it will make the work muscle contraction and relaxation. Although it does not endanger life, it will make the work muscle contraction and relaxation. Although it does not endanger life, it will make the work Abstract: Tremor is a rhythmic, alternating swing motion caused by involuntary repetition of muscle contraction and relaxation. Although it does not endanger life, it will make the work and daily life of patients difficult. Functional electrical stimulation (FES) has been shown as and life difficult. electrical stimulation (FES) has been shown as and daily daily life of of patients patients difficult. Functional Functional electrical stimulation (FES) been as aaa muscle contraction and relaxation. Although Wrist it does not endanger life, by ithas will makeshown the work and daily life of patients difficult. Functional electrical stimulation (FES) has been shown as a promising technique for tremor suppression. motion is produced a group of muscles promising technique for suppression. Wrist motion is by aa group of promising technique for tremor tremor suppression. Wrist motion is produced produced byhas group of muscles muscles and daily life of patients difficult. Functional electrical stimulation (FES) been shown asata promising technique for tremor suppression. Wrist motion is produced by a group of muscles in a collective and coordinate way. However, existing FES-based design methods mostly aim in a collective and coordinate way. However, existing FES-based design methods mostly aim at in a collective and coordinate way. However, existing FES-based design methods mostly aim at promising technique for tremor suppression. Wrist motion is produced by a group ofofmuscles in a pair collective and coordinate way. However, existing FES-based design methods mostly aim at one pair of muscles associated with the wrist motion, thus limiting the performance tremor one of muscles associated with the wrist motion, thus limiting the performance of tremor one pair of muscles associated with the wrist motion, thus limiting the performance of tremor in a pair collective and coordinate way. However, existing FES-based design methods mostly aim at one of muscles associated with the wrist motion, thus limiting the performance of tremor suppression. Furthermore, the possible high level of stimulation required for a single muscle pair suppression. Furthermore, the high level of required for aa single pair suppression. Furthermore, the possible possible high level of stimulation stimulation required for single muscle muscle pair one pair of muscles associated withofthe wrist motion, thus limiting the performance of tremor suppression. Furthermore, the possible high level of stimulation required for a single muscle pair can also accelerate muscle fatigue the patients. To address these problems, this paper uses can also accelerate muscle fatigue of the patients. To address these problems, this paper uses can muscle fatigue of the patients. address problems, this paper uses suppression. Furthermore, the possible high level of To stimulation required for a single muscle pair can also also accelerate accelerate muscle fatigue of the patients. To address these these problems, this paper uses multiple muscles FES to suppress tremor by fully considering the properties of wrist motion. multiple muscles FES to suppress tremor by fully considering the properties of wrist motion. multiple muscles FES to suppress tremor by fully considering the properties of wrist motion. can also accelerate muscle fatigue of the patients. To address these problems, this paper uses multiple muscles FES to suppress tremor by fully considering the properties of wrist motion. This paper develops a wrist musculoskeletal model with Hammerstein structure, identifies its This paper develops a wrist musculoskeletal model with structure, identifies its This paper develops wrist musculoskeletal with Hammerstein Hammerstein structure, identifies its multiple muscles FESa to suppress tremor by model fully considering the properties of wrist motion. This paper develops a wrist musculoskeletal model with Hammerstein structure, identifies its parameters, and proposes repetitive controllers based on frequency modified inverse algorithm parameters, and proposes repetitive controllers based on frequency modified inverse algorithm parameters, and proposes repetitive controllers based on frequency modified inverse algorithm This paper develops a wrist musculoskeletal model with Hammerstein structure, identifies its parameters, and proposes repetitive controllers based on frequency modified inverse algorithm to suppress tremor. Experimental results are presented to demonstrate its advantages over single to suppress tremor. Experimental results are presented to demonstrate its advantages over single to suppress tremor. Experimental results are presented to demonstrate its advantages over single parameters, and proposes repetitive controllers based on frequency modified inverse algorithm to suppress tremor. Experimental results are presented to demonstrate its advantages over single muscle stimulation based tremor tremor suppression. muscle stimulation based muscle stimulation tremor suppression. suppression. to suppress tremor. based Experimental results are presented to demonstrate its advantages over single muscle stimulation based tremor suppression. © 2019, stimulation IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. muscle based tremor suppression. Keywords: Tremor suppression, functional electrical stimulation (FES), repetitive control, Keywords: Tremor suppression, functional electrical Keywords: Tremor suppression, functional electrical stimulation stimulation (FES), (FES), repetitive repetitive control, control, Keywords: Tremor suppression, functional electrical stimulation (FES), repetitive control, multi-muscle electrical stimulation, experimental verification. multi-muscle electrical stimulation, experimental verification. multi-muscle electrical stimulation, experimental verification. Keywords: Tremor suppression, functional electrical stimulation (FES), repetitive control, multi-muscle electrical stimulation, experimental verification. multi-muscle electrical stimulation, experimental verification. 1. more 1. INTRODUCTION INTRODUCTION more and more tremor patients seek for the non-invasive 1. more and and more more tremor tremor patients patients seek seek for for the the non-invasive non-invasive 1. INTRODUCTION INTRODUCTION more and more tremor patients seek for the non-invasive rehabilitation treatments. Functional electrical stimularehabilitation treatments. Functional electrical stimularehabilitation treatments. Functional electrical stimula1. INTRODUCTION more and more tremor patients seek forelectrical the non-invasive rehabilitation treatments. Functional stimulation (FES) is one of the promising methods. Tremor is an involuntary, periodic motion which is a tion (FES) is one of the promising methods. Tremor is is an an involuntary, involuntary, periodic periodic motion motion which which is is aa tion (FES) is one of the promising methods. Tremor rehabilitation treatments. Functional electrical tion (FES) is one of the promising methods. Tremor is an involuntary, periodic motion which is a common movement movement disorder, disorder, occuring occuring often often in in hands, hands, arms arms FES transmits appropriate electrical stimulationstimulacommon movement disorder, occuring often in hands, arms common is one of the promising methods. FES (FES) transmits appropriate electrical stimulation signals Tremor is anthe involuntary, periodic motion whichaffects is a tion common movement disorder, occuring often in hands, arms transmits appropriate electrical stimulation signals signals and mostly in elderly (Elble (2017)). It seriously and mostly mostly in in the the elderly elderly (Elble (Elble (2017)). (2017)). It It seriously seriously affects affects FES FES transmits appropriate electricalattached stimulation signals and through surface stimulus electrodes attached to muscles through surface stimulus electrodes to muscles common movement disorder, occuring often in hands, arms and mostly in the elderly (Elble (2017)). It seriously affects through surface stimulus electrodes attached to muscles the daily life of patients, such as writing, typing and eating. transmits appropriate electrical stimulation signals the daily daily life life of of patients, patients, such such as as writing, writing, typing typing and and eating. eating. FES through surface stimulus electrodes attached to muscles the to produce muscle contraction in anti-phase with tremor muscle contraction in with tremor anddaily mostly elderlydisease (Elble (2017)). It seriously affects the lifeinofthe such as in writing, typing and65 to produce produce muscle contraction in anti-phase anti-phase with tremor The incidence ofpatients, tremor in the elderly elderly over 65eating. years to through surface stimulus electrodes attached to was muscles The incidence of tremor disease the elderly over years to produce muscle contraction in anti-phase with tremor The incidence of tremor disease in the over 65 years movement. FES-based tremor suppression system first FES-based tremor suppression system was first the daily life 10.2% ofofpatients, such as in writing, typing and65 eating. The tremor disease the elderly over years movement. movement. FES-based tremor suppression system was first old isincidence about (Louis et al. al. (1996); Louis and Ferreira to produce muscle contraction in anti-phase with tremor old is about 10.2% (Louis et al. (1996); Louis and Ferreira movement. FES-based tremor suppression system was first old is about 10.2% (Louis et (1996); Louis and Ferreira proposed by Prochazka et al. (1992), filter-based controller proposed by Prochazka et al. (1992), filter-based controller Theisincidence of tremor disease in thegetting elderly over 65 years old about 10.2% (Louis et al. (1996); Louis and Ferreira proposed by Prochazka et al. (1992), filter-based controller (2010)), and the aging problem is worse in the movement. FES-based tremor suppression systemcontroller was first (2010)), and and the the aging aging problem problem is is getting getting worse worse in in the the proposed by Prochazka et al. (1992), filter-based (2010)), was developed and three different tremor patients particiwas developed and tremor patients particiold is about (Louis etisal. (1996); Louis and Ferreira (2010)), and10.2% the aging is getting in the developed and three threeetdifferent different tremor patients particiworld (DESA (2019)), so it becoming very important to proposed by Prochazka al. (1992), filter-based controller world (DESA (DESA (2019)), soproblem it is is becoming becoming very worse important to was was developed and three different tremor patients particiworld (2019)), so it very important to pated in the study to confirm the feasibility of FES-based pated in in the the study study to to confirm confirm the the feasibility feasibility of of FES-based FES-based (2010)), the agingsoproblem is getting in the world (DESA (2019)), it is becoming very worse important to pated study theand tremor suppression methods. was developed and to three different tremor patients particistudy the tremor suppression methods. pated in the study the feasibility of FES-based study the suppression methods. In et al. aaa neural was method. In Zhang Zhang et confirm al. (2011), (2011), neural oscillator oscillator was world (DESA (2019)), so it is becoming study the tremor tremor suppression methods. very important to method. method. In Zhang et al. (2011), neural oscillator was pated in the study to confirm the feasibility of FES-based method. In Zhang et al. (2011), a neural oscillator was designed for the tremor suppression via FES and feedback A number of tremor suppression methods have been prodesigned for the tremor suppression via FES and feedback study the tremor suppression methods. A number of tremor tremor suppression methods have have been been propro- designed for the tremor suppression via FES and feedback A number of suppression methods method. In Zhang et al. (2011), a neural oscillator was designed for the tremor suppression via FES and feedback A number of tremor suppression methods have been proPID controller was developed to refine the intensity of posed, such such as as pharmacological pharmacological treatment, treatment, invasive invasive sursur- PID was developed to the intensity of posed, such as pharmacological treatment, invasive surPID controller controller was developed to refine refine theand intensity of posed, designed for In theCopur tremor suppression via FES feedback A number of as tremor suppressionrehabilitation methods have been proPID controller was developed to refine the intensity of posed, such pharmacological treatment, invasive surstimulation. et al. (2019), repetitive control (RC) gical methods and non-invasive treatments. stimulation. In Copur et al. (2019), repetitive control (RC) gical methods and non-invasive rehabilitation treatments. stimulation. In Copur et al. (2019), repetitive control (RC) gical methods and non-invasive rehabilitation treatments. PID controller was developed to refine the intensity of posed, such as pharmacological treatment, invasive surstimulation. In Copur et al. (2019), repetitive control (RC) gical methods and non-invasive rehabilitation treatments. based on FES was used to reduce tremor and allow a Most medication methods are unsatisfactory, and patients on FES was used to reduce tremor and allow a Most medication medication methods methods are are unsatisfactory, unsatisfactory, and and patients patients based based on FES was used to reduce tremor and allow a Most stimulation. In Copur et al. (2019), repetitive control (RC) gical methods and non-invasive rehabilitation treatments. based on FES was used to reduce tremor and allow a Most medication methods are unsatisfactory, and patients smoother pattern of movement. often stop taking drugs on their own due to the side effects smoother pattern of movement. often stop stop taking taking drugs drugs on on their their own own due due to to the the side side effects effects smoother pattern of movement. often based on FES was used to reduce tremor and allow a Most medication methods are unsatisfactory, and patients smoother pattern of movement. often stop taking drugs on their own due to the side effects of drugs drugs (Hedera (Hedera et et al. al. (2013)). (2013)). Patients Patients with with severe severe tremor tremor Although the above design has shown the promising results of drugs (Hedera et al. (2013)). Patients with severe tremor of pattern ofdesign movement. Although the above has shown the promising results often stop taking drugs onsuch their own duewith to the sidetremor effects smoother of drugs (Hedera et al. (2013)). Patients severe Although the above design has shown the promising results can be treated surgically, as stereotactic thalamotomy can be be treated treated surgically, surgically, such such as stereotactic stereotactic thalamotomy thalamotomy Although the above design has shown the promising results can in suppressing suppressing tremor via FES, the electrical stimulation in tremor via FES, the stimulation of drugs (Hedera et al.stimulation (2013)). Patients with severe tremor can treated such as as stereotactic thalamotomy suppressing tremor via has FES, the electrical electrical stimulation (ST) and deep brain (DBS). However due to Although the above design shown the promising results (ST)beand and deepsurgically, brain stimulation stimulation (DBS). However However due to to in in suppressing tremor via FES, the electrical stimulation (ST) deep brain (DBS). due input pattern is only based on one pair of muscles (one pattern only on pair (one can high beand treated surgically, such as stereotactic thalamotomy (ST) deep (DBS). However due to input input pattern is istremor only based based on one one pair of of muscles muscles (one the risk and costs of surgery (Michmizos et al. (2017)), in suppressing viaInFES, the electrical stimulation the high high risk andbrain costsstimulation of surgery surgery (Michmizos (Michmizos et al. al. (2017)), (2017)), input pattern is only based on one pair of muscles (one the risk and costs of et extensor and one flexor). fact, even for single degree of extensor and and one one flexor). flexor). In In fact, fact, even even for for single single degree degree of (ST)high andrisk deep brain stimulation (DBS). However due to extensor the and costs of surgery (Michmizos et al. (2017)), of input pattern is only based on one pair of muscles (one  extensor and one flexor). In fact, even for single degree of freedom (DOF) wrist movement, such as wrist flexion and This work is supported by the National Natural Science Founda  freedom (DOF) wrist movement, such as wrist flexion and the high riskis and costs of surgery (Michmizos et al. (2017)), This work supported by the National Natural Science Foundafreedom (DOF) wrist movement, such as wrist flexion and This work is supported by the National Natural Science Foundaextensor and onewrist flexor). In fact, even forwrist single degree of  This freedom (DOF) movement, such as flexion and tion of China (No. 61803344, and the Innovation extension (WFE), four muscles are involved at least (Ramwork is supported by 61473265) the National Natural Science Research Foundation of China (No. 61803344, 61473265) and the Innovation Research extension (WFE), four muscles are involved at least (Ramtion of China (No. 61803344, 61473265) and the Innovation Research extension (WFE), fourmovement, muscles aresuch involved at least (Ram freedom (DOF) wrist as wrist flexion and This work is supported by the National Natural Science FoundaTeam of Science & Technology of Henan Province (No. 17IRTSTHtion of China (No. 61803344, 61473265) and the Innovation Research extension (WFE), four muscles are involved at least (Ramsay et al. (2009)). Stimulating only single pair of wrist Team of of Science Science & & Technology Technology of of Henan Henan Province Province (No. (No. 17IRTSTH17IRTSTHsay Stimulating only single of wrist Team say et et al. al. (2009)). (2009)).four Stimulating only singleatpair pair wrist tion ofof China (No.&61803344, 61473265) the Innovation Research extension muscles are involved leastof N013). Team Science Technology of Henanand Province (No. 17IRTSTHsay et al. (WFE), (2009)). Stimulating only single pair of(Ramwrist N013). N013). Team N013).of Science & Technology of Henan Province (No. 17IRTSTHsay et al. (2009)). Stimulating only single pair of wrist N013). 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.617

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muscles will partly considerate the physiological property of the wrist muscles, possibly increase the intensity of electrical stimulation signals, accelerate muscle fatigue and get poor effect of tremor suppression. To address the above limitations, this paper proposes RC method with multi-stimulus electrode input to eliminate tremor signal. RC is suitable for tremor suppression because tremor can be seen as periodic signal. The output of RC system can reject repeating disturbance acting on the system completely. To achieve this, the wrist dynamics is firstly analyzed and a nonlinear Hammerstein structure wrist model is obtained. A pre-feedback controller is utilised to linearize the wirst model. Frequency modified inverse RC (FMI-RC) algorithm is proposed to design the compensator. The experimental results demonstrate the advantages over single muscle stimulation in tremor suppression. 2. PROBLEM SET UP Tremor is produced by abnormal activation signals causing unnecessary wrist movement and preventing the patient from voluntary motion. FES method can stimulate and contract muscles to produce force and make joint move, and then achieve the purpose of restoring and rebuilding motor function. The block diagram of the system for suppressing wrist tremor using FES is shown in Fig. 1, which contains two major parts: the wrist musculoskeletal model and the repetitive controller.

Fig. 1. The block diagram of the system for suppressing wrist tremor using FES The arm and wrist joints are considered to be the controlled plant. The musculoskeletal model consists of muscle model and wrist skeletal dynamic model. Tremor can be regarded as disturbance, which derived from the tremor torque acting on the skeletal dynamic system. Angle sensor collects real-time angle changing data of wrist joint and feeds it back to controller input. Based on the angle error, the repetitive controller outputs the corresponding electric stimulation control signal to the stimulator. According to the input signals, the stimulator outputs the current of the corresponding amplitude to stimulate the extensors and flexors of the arm respectively, and to generate the wrist joint motion contrary to tremor. 3. ELECTRICALLY STIMULATED WRIST MUSCULOSKELETAL MODELLING In this section we propose a Hammerstein model for the electrically stimulated wrist musculoskeletal system. Noting that the WFE motion occurs more frequently in daily life, we just consider the dynamic modelling of the electrically stimulated wrist musculoskeletal system associated with WFE motion. For WFE motion, the mainly involved

muscles are extensor carpi radialis (ECR), extensor carpi ulnaris (ECU), flexor carpi radialis (FCR) and flexor carpi ulnaris (FCU). FCR, FCU contribute to flexion motion, ECR, ECU contribute to extension motion (Ramsay et al. (2009)). The sketch of the four muscles is shown in Fig. 2. In this paper, all four muscles are considered, in contrast, existing methods only consider two of them.

Fig. 2. Sketch of FCR, ECR, FCU and ECU To suppress the flexion and extension tremor, we need to apply proper electrical stimulation signals to the corresponding muscles. Therefore FES-based WFE tremor suppression system is a multi-input single-output (MISO) system. The inputs are the electrical stimulation signals and the output is the angle of wrist joint motion.

Fig. 3. The MISO Hammerstein electrically stimulated musculoskeletal model The structure of MISO Hammerstein model is shown in Fig. 3, where uf cr (k), uecr (k), uf cu (k) and uecu (k) are the stimulation signals acting on the FCR, ECR, FCU and ECU muscles respectively; wf cr (k), wecr (k), wf cu (k) and wecu (k) are the steady-state isometric muscle torques; ff cr (uf cr ), fecr (uecr ), ff cu (uf cu ) and fecu (uecu ) are static nonlinear functions representing the relationship between the stimulation signals on the muscles and the corresponding output muscle torques, which are generally called as the isometric recruitment curves (IRCs) of the muscles; df cr (k), decr (k), df cu (k) and decu (k) are tremor signals. Gf cr (z), Gecr (z), Gf cu (z) and Gecu (z) represented the linear muscle contraction dynamics and τ (k) = τf cr (k) + τf cu (k) − τecr (k) − τecu (k) is the overall torque. GRBD(z) is the skeletal dynamics of the wrist; y(k) is the overall angle of the WFE motion. Impulse response method (Durfee and MACLean (1989)) is used to get IRCs, because it causes little muscle fatigue. In order to simplify the model structure and the related identification problem, the four-input four-output IRC model is transformed into two-input two-output model by combining the two inputs uf cr (k) and uecr (k) into a single input u1 (k), uf cu (k) and uecu (k) into a single input u2 (k).



Zan Zhang et al. / IFAC PapersOnLine 52-29 (2019) 31–36

Since df cr (k), decr (k), df cu (k) and decu (k) are Np -periodic, and can be assumed to contain a maximum Nf harmonic, they can be written as Nf 

2πjk 2πjk di (k) = Ai0 + {Aij cos( ) + Bji sin( )}, Np Np j=1

(1)

where i = f cr, ecr, f cu and ecu and hence their sum is given by the Np -periodic signal d1 (k) = A10 +

Nf  j=1

{A1j cos(

2πjk 2πjk ) + B1j sin( )}, Np Np

Nf 

(2)

2πjk 2πjk {A2j cos( ) + B2j sin( )}, d2 (k) = A20 + N Np p j=1

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4. REPETITIVE CONTROL FOR WRIST TREMOR SUPPRESSION 4.1 Linearing controller To suppress tremor by RC, the internal model (IM) should be determined according to the tremor signal D(k), which equals to the sum of d1 (k), d2 (k) and comprises a combination of Np -periodic signals. In order to design repetitive controller C(z), the recruitment nonlinearity f1 (u1 ) and f2 (u2 ) are cancelled by the inverse function ¯1 ) and f2−1 (w ¯2 ). The control problem requires C(z) f1−1 (w generate muscle torque w¯1 (k) and w¯2 (k) to remove the influence of disturbance d1 (k) and d2 (k). The schematic is shown in Fig. 4, where R(k) is the reference signal (voluntary wrist motion), Y (k) is the output of the plant model G(z) , E(k) is the error between R(k) and Y (k).

f cu with A10 = Af0 cr + Aecr + Aecu 0 , A20 = A0 0 , A1j = f cr f cr ecr ecr and Aj + Aj , B1j = Bj + Bj , A2j = Afj cu + Aecu j f cu ecu B2j = Bj + Bj .

Because the biophysical properties of the radial muscles (FCR and ECR) are similar, we assume that the FCR and ECR have similar linear activation dynamics (LAD), that is, Gf cr (z) ≈ Gecr (z). The same with ulnar muscles (FCU and ECU), so Gf cu (z) ≈ Gecu (z). Experimental results has shown the rationality of above assumption (Colacino et al. (2012)). We denote the LAD model as GLAD1 (z) and GLAD2 (z). Further noticing that the skeletal dynamics of human limbs can be modeled as rigid body dynamics (RBD), in which the damping and elastic functions are linear. We can combine GLAD1 (z), GLAD2 (z) and rigid body dynamics GRBD (z) to get an equivalent linear musculoskeletal model G(z) = [GLAD1 (z) GLAD2 (z)]GRBD (z) = [G1 (z) G2 (z)]. Noting that each stimulus electrode attaches to corresponding muscle, so the recruitment characteristics of the four wrist muscles are not coupled, the IRCs model of each muscle can be formulated as f1 (u1 (k)) = ff cr (u1 (k)) − fecr (−u1 (k)) = r01 + r11 u1 + r21 u21 + · · · + rs1 us1 ,

(3)

f2 (u2 (k)) = ff cu (u2 (k)) − fecu (−u2 (k)) = r02 + r12 u2 + r22 u22 + · · · + rs2 us2 ,

(4)

with the recruitment curve monotonicity conditions dfl (ul (k)) = r1l + 2r2l ul + · · · + srsl us−1 ≥ 0, l dul

(5)

where r0l , r1l , · · · , rsl are the nonlinear parameters to be identified and l = 1, 2. The transfer function matrix of G(z) takes the form of G(z) =

B(z −1 ) B1 z −1 + ... + Bnb z −nb = , A(z −1 ) 1 + a1 z −1 + ... + ana z −na

(6)

where Bi = [b1i b2i ], i = 1, 2, ..., nb , and a1 , · · · , ana (na > nb ) are parameters to be identified. In this paper, least square (LS) identification method is used to identify the parameters of the proposed model. See Section 5 for details.

Fig. 4. The closed feedback RC scheme 4.2 Design of repetitive controller Consider the RC action ¯ (k) = [w¯1 (k) w (7) W ¯2 (k)]T = C(z)E(k). The repetitive controller C(z) takes the form of z −Np (8) C(z) = [K1 H1 (z) K2 H2 (z)]T · , 1 − z −Np z −Np where 1−z −Np is the IM, K1 > 0, K2 > 0 are the gains of the RC controller, and H(z)=[H1 (z) H2 (z)]T are the compensators that can improve the stability and dynamic performance of the system. Theorem 1: The RC system depicted in Fig. 4 is asymptotically stable and can completely reject the Np periodic disturbance, if the control gains K1 > 0, K2 > 0 and the compensators H1 (z), H2 (z) satisfy the following inequalities |1 − (K1 G1 (z)H1 (z) + K2 G2 (z)H2 (z))| < 1. (9) Especially, when H1 (z) = G1 (z)−1 and H2 (z) = G2 (z)−1 , the condition (9) becomes 0 < K1 + K2 < 2. (10) Proof : The relationship among error E(k), reference R(k) and disturbance D(k) can be written as G(z) 1 R(k) − D(k). (11) E(k) = 1 + G(z)C(z) 1 + G(z)C(z) Inserting (8) into (11), we get the characteristic equation z −Np (12) . 1 + G(z)[K1 H1 (z) K2 H2 (z)]T 1 − z −Np

The stability condition of the system following a similar manner (Longman (2010)) as the proof of (9), (13) |1 − G(z)[K1 H1 (z) K2 H2 (z)]T | < 1,

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or equivalently.



|1 − (K1 + K2 )| < 1.

(14)

Remark 1: The magnitude of all frequency components of the error decays from one period to the next, which suggests asymptotic stability, with a rate of |1−(K1 +K2 )|. The smaller |1−(K1 +K2 )| is, the faster the error decays. If K1 + K2 = 1, the speed of error attenuation is the fastest. Due to the model parameter uncertainties and the presence of zeros outside the unit circle, the exact inverse of G1 (z), G2 (z) are often unrealizable. In this paper, we use the frequency modified inverse RC (FMI-RC)((Longman (2010)) to determine the parameters of H(z).

Fig. 5. Experimental platform for tremor suppression of wrist musculoskeletal model and to verify the proposed control algorithms.

4.3 Frequency modified inverse RC

The platform embeds an artificial tremor producing mechanism. The torque of the DC motor is transmitted to the In this paper a high-order FIR filter is used to obtain an purple U-shaped splint through the shaft. The hand of participant is placed in the middle of the splint. When approximation inverse of G(z). The cost function is the motor applies torque, wrist and hand will vibrate at N  iωj T iωj T iωj T iωj T ∗ Jl = [1 − Gl (e )Hl (e )][1 − Gl (e )Hl (e )] , the desired frequency. That means the experiment can be done on the participants without wrist tremor disease. j=0 (15) The encoder is mounted on the shaft which can obtain the WFE motion angle of the participant according to the where rotation of the shaft. Encoder data is acquired by a reall m−2 l 0 l m−1 Hl (z) = c1 z + c2 z + · · · + cm z time hardware dSpace 1202, which can interface directly (16) + · · · + cln−1 z −(n−m−1) + cln z −(n−m) , with Matlab/Simulink to obtain the wrist angle data and l = 1, 2, m and n are positive integers and ωj is a discrete to generate the FES signals to a four-channel electrical set of frequencies selected from zero to Nyquist frequency. stimulator. The parameters cl = [cl1 cl2 · · · cln ]T appearing in definition (16) are chosen by minimizing (15). 5.2 Identification of wrist model Let Gl (e(iωj T ) ) = MGl (ωj )eiϕGl (ωj ) , where MGl (ωj ) and ϕGl (ωj ) represent the magnitude and phase of Gl (z), respectively. The solution is given by, Dl cl = pl , where N  2 Dl = MG (ωj ) × Θ, l

Before starting an experiment, four surface electrodes are placed on FCR, ECR, FCU and ECU muscles of participant respectively. Parameters identification of Hammerstein model consists of two steps.

IRC Nonlinearity Estimation: Peak impulse response method estimates IRC by mapping the peak value of muscos((m − 1)ωj T + ϕGl (ωj )) N  cle impulse response to a single stimulus pulse against the  cos((m − 2)ωj T + ϕGl (ωj ))  MGl (ωj ) ×  pl = , width of the pulse. A group of test consisted of 30 pulses. ... j=0 It contains six different pulse widths (50µs, 100µs, 150µs, cos((m − n)ωj T + ϕGl (ωj )) (17) 200µs, 250µs and 300µs), each of which repeats five times. 300µs is the maximum pulse width that humans endure Θ=   1 cos(ωj T ) · · · cos((n − 1)ωj T ) when the pulse current is 20mA. The whole identification 1 · · · cos((n − 2)ωj T )  process takes less than two minutes. The five responses for  cos(ωj T ) pulse width are averaged to get the last results.  each  .. .. .. .. .  . . . . The formulas (3), (4) and (5) can be written as follows 1 cos((n − 1)ωj T ) cos((n − 1)ωj T ) · · · f1 (u1 (k)) = [us1 · · · u1 1][rs1 · · · r11 r01 ]T (18) (19) ¯ 1 r1 , =u Choosing appropriate m and n can produce various patterns for the optimized zero locations of the compensator f2 (u2 (k)) = [us2 · · · u2 1][rs2 · · · r21 r02 ]T (20) H(z). The larger m and n, the more accurately H(z) ¯ 2 r2 , =u approximates G−1 (z). T  dfl (ul (k))  s−1 = sul · · · 1 0 rsl · · · r1l r0l 5. EXPERIMENTAL VERIFICATION dul ¯ l rl . =q 5.1 Experimental platform (21) Then we use quadratic programming approach to solve An FES-based tremor suppression experimental platform, the following optimal problem and get the parameters and as shown in Fig. 5, is set up to identify the parameters order of IRC, j=0





Zan Zhang et al. / IFAC PapersOnLine 52-29 (2019) 31–36

¯ l rl 2 arg min ¯ y−u xl

(22)

¯ l rl < 0, s.t. q ¯ are ¯ are the measurement outputs, u ¯l, q where l = 1, 2, y the input pulse signals and rl are the parameters to be identification. Linear Dynamics Identification: In order to generate test data for estimating the linear part of the muscle model, the approximated recruitment nonlinearity is cancelled via the inverse function, shown in Fig. 4. All the parameters of formula (6) and tremor signal D(k) are written in a explicit form and denote it as θ. The D(k) is a NP -periodic signal and comprises Nf harmonics frequency components, which can be written in the matrix form, 2πk 2πNf k 2πk ) sin( ) · · · cos( ) D(k) = [1 cos( Np Np Np 2πNf k T )][λ0 λA λB · · · λA λB sin( 1 1 Nf Nf ] Np = hv θv . (23) Let θ = [a1 , a2 , ..., ana , B1 , B2 , ..., Bnb , θv ]T , h(k) = [−Y (k − 1), −Y (k − 2), ..., −Y (k − na ), U (k − 1), U (k − 2), ..., U (k − nb ), hv ], the linear dynamics can be written as Y (k) = h(k)θ. (24) The consistency and unbiased estimation of θ can be obtained by the LS identification algorithm. The input signals are sine waves with different amplitudes and different frequencies at 0.5Hz, 1Hz, 2Hz and 4Hz respectively. Y (k) is the angle data of WFE motion measured by encoder. 5.3 Experimental Validation The FMI-RC method proposed above is evaluated by experiment. The tremor suppression experiment consists of 3 steps. Test 1 (T1): Tracking task without induced tremor and FES. Test 2 (T2): Tracking task with induced tremor but without FES. (Tremor frequency generated by motor is set to 2Hz) Test 3 (T3): Tracking task with induced tremor and FES. The sample period Ts is 0.005s. The delay period of the repetitive controller is Np = 100. Gains of the RC controller are K1 = K2 = 0.5. Then we use the FMIRC algorithm mentioned above to get the order and parameters of compensator H(z). The experiment is carried on to compare the tremor suppression performance between FMI-RC algorithm and PID-HF algorithm with multi-muscle electrical stimulation. For FMI-RC, the parameters of m and n are selected by minimizing cost function (15). PID regulator parameter optimization is determined by trial and error method. The parameters of filter order and cut-off frequency of

35

highpass Butterworth filter are selected as 6th and 1.2Hz respectively. The results of T1, T2 and T3 of the two control methods with multi-muscle electrical stimulation are shown in Fig. 6. 60

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Fig. 6. Wrist angular positions of T1, T2 and T3 with FMI-RC and PID-HF methods It can be seen from Fig. 6 that the tremor suppression performance of FMI-RC algorithm is much better than that of PID-HF algorithm with multi-muscle electrical stimulation. We use the following performance indicators to quantify the performance of each algorithm on tremor suppression, RM SE = Y − R2 , (25) where RM SE is the root mean square error, Y is the output signal of T3, R is the output signal of T1 and 2 denotes the 2-norm. Steady-state error performance (maximum steady-state error, MSSE) is used to judge convergence of error near steady state, (Y − R)∞ M SSE = , (26) V ∞ where Y is the output signal of T3, R is the output signal of T1, V is the tremor signal (T2) and ∞ denotes the ∞-norm. The steady state time is chosen as 15s to 20s, 35s to 40s and 55s to 60s. In order to better quantify the performance of tremor suppression, the output (Yv ) of T2 and T3 were first subtracted from the output (R) of T1, which is denoted by ∆Y = R − Yv 2 . (27) The value of ∆Y from T2 indicates tremor movements. The value of ∆Y from T3 is the tremor suppression output by FMI-RC and PID-HF methods. Tremor suppression rate (TSR) is the ratio of ∆Y . The performance of tremor suppression by different control methods with two pairs of stimulus electrodes (TPSE) inputs and one pair of stimulus electrode (OPSE) inputs (FCR and ECR) are shown in Table 1. Table 1. Experimental results of different controllers Controller Type FMI-RC PID-HF

Input Style TPSE OPSE TPSE OPSE

RMSE 1.3780 1.9061 3.3598 4.0302

MSSE 0.1250 0.1816 0.4708 0.5090

TSR 87.23% 78.94% 65.68% 54.54%

As shown in Table 1, either one pair of stimulus electrodes inputs or two pairs of stimulus electrodes inputs, the RMSE of FMI-RC is the lowest. During the steady state time, we also get the similar result. The MSSE of FMI-RC is the smallest, which means that the tremor attenuation

Zan Zhang et al. / IFAC PapersOnLine 52-29 (2019) 31–36

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rate is the highest at steady state. Moreover, the tremor suppression rates of FMI-RC method with two input patterns are up to 87.23% and 78.94%(reducing ∆y from 0.0117 to 0.0015 and to 0.0025). The tremor suppression rate of PID-HF method are 65.68% and 54.54%(reducing ∆y to 0.0040 and to 0.0053). The results show that the performance of tremor suppression using multi-muscle electrical stimulation is substantially improved. Meanwhile, the level of input stimulation signal with multi-muscle electrical stimulation is reduced too. With two input patterns, the input signals level of electrical stimulation are shown in Fig. 7 and Fig. 8.

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Fig. 7. Control input applied by FMI-RC with multimuscle electrical stimulation

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While the above results are promising, there are some issues to be solved. For tremor patients, the frequency of tremor varies with time. The design of adaptive repetitive controller based on frequency variation can obtain better effect of tremor suppression and improve the practical application value of the system. Further more the dynamics of wrist musculoskeletal model is time vary. Improving the robustness of repetitive controller is also a problem to be considered in future research. REFERENCES

300

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Further more, the FES stimulation level required is significant reduced and therefore delaying the muscle fatigue.

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Fig. 8. Control input applied by FMI-RC with single muscle (FCR and ECR) electrical stimulation It can be seen from Fig. 7 and Fig. 8, the level of stimulation input signals with multi-muscle is much lower than that of single muscle electrical stimulation, which means it can make participant more comfort and delay the muscle fatigue. In summary, all the major muscles involved in WFE motion need to be considered comprehensively in order to achieve better tremor suppression effect and deduce muscle fatigue. 6. CONCLUSION This paper proposed a repetitive controller to suppress wrist tremor by means of multiple muscles electrical stimulation input. To achieve this, nonlinear and linear parameters are obtained for individual participant by system identification of wrist musculoskeletal model with Hammerstein structure. Then FMI-RC algorithm and PIDHF algorithm have been designed based on MISO system through linearising action and tested experimentally using the self-designed experimental platform. The experimental results show that using multi-muscle electrical stimulation leads to substantial improvement in tremor suppression.

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