PID CONTROL DESIGN FOR REHABILITATION BY QUASI-ISOMETRIC TRAINING IN PARAPLEGIA: A SIMULATION STUDY

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1 PID CONROL DESIGN FOR REHABILIAION BY QUASI-ISOMERIC RAINING IN PARAPLEGIA: A SIMULAION SUDY Fabio Previdi*, Maurizio Ferrarin, Sergio Savaresi #, Sergio Bittanti # * Dipartiento di Ingegneria Gestionale e dell Inforazione Università degli Studi di Bergao Via Marconi, Daline (BG), Italy previdi@unibgit Centro di Bioingegneria FDG Fondazione Don Carlo Gnocchi ONLUS IRCCS Milano, Italy # Dipartiento di Elettronica e Inforazione Politecnico di Milano Pzza Leonardo da Vinci, Milano, Italy Abstract: his paper reports siulation results about the design of a PID controller of Functional Electrical Stiulation (FES) to be used for quasi-isoetric exercises in rehabilitation of paraplegics he siulations refer to a specific experiental device developed at the Fondazione Don Gnocchi, ainly used in standing up and sitting down using FES his is a seesaw, with the patient on one side and a weight on the other side he patient is seated so that its posture can be fully known in real-tie by continuously onitoring the knee joint angle he design of feedback controllers for such a device is currently based on nonlinear strategies hese are necessary when aiing to standing and sitting control However, there is a strong deand by rehabilitation physiotherapists for easy-to-tune controllers, even though these could be used only in specific conditions So, a PID linear controller has been designed and tested by eans of siulations he controller tuning is perfored by using four different ethods specifically designed for unstable systes with delay Coparison of the perfornce obtained with each controller is provided Copyright 005 IFAC Keywords: Rehabilitation engineering, Functional Electrical Stiulation INRODUCION Functional Electrical Stiulation (FES) is a technique that uses electrical pulses to induce skeletal uscle contraction and lib oveents In a paraplegic patient who suffered a Spinal Cord Injury (SCI) at the thoracic level, partial otion activity of the lower libs can be induced by eans of FES In fact, an injury causing a coplete lesion of the spinal cord, results in an interruption of the neural pathways and thus the ipossibility for the physiological stiulus to reach the uscles innervated below the level of the lesion However, the uscles preserve their ability to contract theselves herefore, oveents can be induced by proper electrical stiulation, allowing a partial restoration of functionality of the patient legs he generation of appropriate electrical stiulation patterns to induce functional oveents can be achieved by the application of feedback control theory (Schauer and Hunt, 000; Ferrarin et al, 00; Previdi and

2 Carpanzano, 003; Previdi et al, 004a; Previdi et al, 004b) In the present work, only coplete spinal lesions are considered, where the control of uscles distal to the lesion site is copletely lost he topic of this paper is a siulation study about the design of a feedback PID linear controller of Functional Electrical Stiulation to be used in rehabilitation training of paraplegics he experiental set up, called FES - Weight Reliever (FES-WR), developed at the Fondazione Don Gnocchi Onlus and described in Ferrarin et al (00) is at the basis of the siulations carried out in the present paper he FES-WR is a seesaw, with the patient on one side and a weight on the other side he patient is seated so that its posture can be fully known in real-tie by continuously onitoring the knee joint angle By delivering a suitable electrical stiulation to the quadriceps uscles group, the patient can be raised and sit with sooth oveents he ain use of this device is sit to stand to sit exercises and it has been clearly experienced that nonlinear feedback control strategies are necessary to achieve sooth oveents of the knee joint fro the sit position to full extension (Ferrarin et al, 00; Previdi et al, 004a; Previdi et al, 004b) However, this device can be used also to perfor different exercises in rehabilitation requiring sall aplitude knee joint oveents For instance, quasi-isoetric exercises can be done, where the patient ust stand still with an assigned knee angle and only sall oveent about a given position are requested o perfor such exercises it is not necessary to use coplex control strategies, which could be difficult to tune for a user not specialist in control and require a big coputational effort So, we investigated the possibility of using a linear PID controller, also to fulfill the requireents of physiotherapists to have siple and easy to tune control algoriths Previous research in standing-up and sitting-down has evidenced that linear closed-loop controllers provide poor tracking perforances (Riener and Fuhr, 998), if coplete standing ust be achieved In the present work, PID controller is designed to work for sall aplitude knee joint oveents he controller paraeter tuning is perfored on the basis of a linear odel which is estiated on the basis of sall oveent I/O data he odel results to be unstable with tie delay So, tuning rules for PID controllers designed for unstable systes with delay are used Specifically, four tuning rules will be analyzed and copared on the basis of tracking perforances and difficulty in coputing the controller paraeters he paper has the following structure: in Sect the experiental setup at the basis of the siulations of this work is described In Sect 3 frequency doain identification of odel is perfored, to represent the patient and seesaw dynaics In Sect 4 the tuning rules for PID controllers with the siulation results are shown DESCRIPION OF HE SYSEM he plant of the control syste considered in this work is coposed by the patient and a Weight- Reliever (WR) device together his syste has been atheatically represented by eans of a siulation odel described in (Riener et al, 998) he odel refers to the experiental set up developed at the Fondazione Don Gnocchi Onlus (Ferrarin et al, 00), which is aied to train the paralysed quadriceps uscles of patients with spinal cord lesion, for the increase of uscle strength and endurance, in view of functional applications of FES like walking or cycling he WR device provides the following functions: to partially support the weight of the patient during the standing up oveent; to prevent patient fro falling; to reduce the degrees of freedo of the ulti-link huan usculo-skeletal syste It consists of a seesaw construction, equipped with a saddle at one extreity and with a counterweight at the other here is a four-bar linkage which iposes the saddle to aintain a horizontal position during the rotation of the ain bea he aount of counterweight is defined on the basis of patient weight and strength of quadriceps contraction elicited by electrical stiulation he huan body is described by a three-segental odel consisting of shanks, thighs, and upper body, since feet are flat on the ground Nine ono and biarticular uscle groups are odeled in the sagittal plane inducing oents about the ankle, knee, and hip joints Each uscle group has its own activation and contraction dynaics, taking into consideration the effect of spatial and teporal suation, a linear second order calciu dynaics, uscle fatigue and recovery In addition, a nonlinear relationship has been used to capture the force-frequency characteristic of an artificially stiulated uscle (see Riener and Quintern, 997 for further details) he active oent developed by a single uscle group is calculated fro its nonlinear oent ar and the uscle force, which is a function of axiu isoetric uscle force, uscle activation, nonlinear force-length, and nonlinear force-velocity relations (see Riener and Fuhr, 998 for further details) In the body-segental dynaics, total joint oent is the su of active, passive elastic and passive viscous joint oents he latter has been odeled as a linear relation between the angular joint velocity and the daping coefficient, while passive elastic joint oents have been odeled by double exponential equations which account for the influence of the adjacent joint angles Active joint oent is the su of the joint oents produced by each uscle group It is assued that the patient s hip is rigidly connected with the WR bea and that the trunk reains vertical during oveent, yielding a close chain consisting of shanks, thighs and WR bea, characterized by only one degree of freedo, since all angles can be expressed as a function of the knee joint angle his hypothesis is certainly true for sall oveents, as in the case of the present work

3 3 FREQUENCY DOMAIN IDENIFICAION In this section, a linear odel of the plant dynaics is estiated Black-box identification is used, ie the plant odel is derived by I/O easureents on the plant Identification will be perfored by using sall signal data, ie the odel is estiated to approxiate the plant dynaics only about a given operating point of the plant, corresponding to half of the knee joint extension (about rad) he identification procedure ust be perfored in closed loop, due to the plant instability his eans that a stabilizing PI feedback controller is applied to the plant It is worth noting that this controller is not designed to achieve any specific tracking perforance: its ai is only to provide external stability when closed in loop with the plant, so that data acquisition for identification can be done (Previdi et al, 004a) he I/O data for the odel identification has been generated as follows: the closed loop syste has been fed with a set of 7 pure-tone reference signals, w t Asin Ωi t + with frequencies ie () ( ) Ω i n 0 rad/s, where n,,3,, 9 and,0, he aplitude A has been kept sall enough (about 5 ) to reduce nonlinear distortion in the easured signals hen a direct ethod for closed loop identification has been used (see Van Den Hof and Schraa, 995; Forssell and Ljung, 999) Specifically, the control action u(t), ie the pulsewidth odulation of the electrical stiulation, and the syste output y(t), ie the knee joint angle, being pure-tone signals at the sae frequency of the reference, have been directly recorded and used for identification Using the estiated agnitude and Ω phase of both u and y, a set of points { ( e )} j i G i, K, 7 of the frequency response of the open-loop plant is available hen, the following third order paraetric odel with delay has been chosen: () G s ρe sτ ( s p )( s p )( s ) p3 () where: ρ is the Evans gain; τ is delay of the syste; p i with i, K, 3 are the poles At least one pole ust be positive, to force the syste to be unstable he choice of this structure has been done by looking at the easured frequency response (see Fig ) Fro this figure it is quite evident that the plant has a delay and a relative degree equal to 3 he estiated paraeter values are: ρ 35; p 7; p p3 86; τ 0043 In Fig the frequency response agnitude and phase values coputed fro the sinusoidal test input are shown (star plot) together with the frequency response values of the estiated odel In Fig a coparison between the plant and the odel output in tie doain is shown he reference signal is a square wave with aplitude 0 rad, period 40 s and ean value rad, ie the operating point about which the odel has been estiated his square wave reference has been filtered with a first order filter with unit gain and tie constant s Fro this figure it is possible to argue soe outcoes First of all, the plant shows nonlinear behaviour also for sall aplitude excitations In fact, the plant response to the upward step is different fro that to the downward one Secondly, the estiated odel is slightly ore daped than the plant Finally, the perforance of the PI controller used for closed loop identification are definitely poor (sall daping, long settling tie), even though the integral action is able to reduce to zero the steady state error Magnitude [db] Phase [deg] Frequency [rad/s] Frequency [rad/s] Figure Bode plots of the frequency response values estiated fro the I/O sinusoidal easureents about the operating point u 0 36; y 0 0 (*) he solid lines represent the agnitude and phase of the frequency response of the odel (Eq ) with the estiated paraeter values ie [s] ie [s] Figure Coparison between plant response (solid line) and odel siulated response (dotted line) In the upper plot the pulsewidth odulation signal of the electrical stiulation (the control action); in the lower plot the knee joint angle (the output) 4 PID CONROLLER UNING AND SIMULAION RESULS In this work, a PID controller is used he tuning of the controller has been perfored using four different tuning rules (O Dwyer, 004; Rotstein and Lewin,

4 99; Lee et al, 000; McMillan, 984; Poulin and Poerleau, 996; Poulin and Poerleau, 997) and the perforances of the controllers are copared by eans of siulations he tuning rules used in the following are specifically designed for closed loop control of unstable systes with delay he controller paraeters are directly coputed on the basis of the estiated odel paraeters (See Sect 3) and, in soe cases, fine tuning of additional paraeters ust be done he controller paraeters are tuned on the basis of a plant second order odel with delay: sτ K e G() s () ( s )( + s ) As shown in Section 3, the estiated odel is a third order odel So, it ust be reduced to be put in the for of Eq () o this ai, the highest frequency pole has been neglected, taking care to preserve the value of the odel gain So, with reference to Eq (), the following odel paraeters ust be used for the controller tuning: 3 K 73 0 ; τ 050; 04406; 0 Notice that, in order to keep a good approxiation of the odel phase, the delay value has been slightly increased In the following, two different PID controller structures are considered: the so called controller with filtered derivative (Eq 3) and the classical controller (Eq 4): () sd G c s K c + + (3) s i d + s N () + s + d Gc s K c (4) s d i + s N he controller perforances will be tested using a reference square wave signal w () t with period 0 s and aplitude 05 rad, starting in equilibriu conditions with knee joint value rad his signal represents the situation of a patient during quasiisoetric exercise he perforance will be evaluated considering the quality of the tracking and the following quadratic perforance index: N J ( y() t w() t ) (5) N t he first tuning rule is due to Rotstein and Lewin (99) and it is referred to the controller structure of Eq (3) he regulator paraeters are deterined by the following equations: λ λ + + K c (6) λ K λ i λ + + (6) λ λ + d (63) λ λ + + where λ is a further paraeter to be used for fine tuning Its value ust be chosen on the basis of the ratio between the delay value τ and the largest tie constant of the poles, which is in the present case So, in the present case, λ [ 039,3] By eans of siulations, the value λ 0 45 has been easily chosen So, the resulting controller paraeters are: K c 40 ; i 48; d 0, with N 0 In Fig 3 siulation results are shown he value of the perforance index of Eq (5) is J ie [s] ie [s] Figure 3 Closed-loop siulation test using a PID controller with the structure of Eq (3) and paraeters tuned by Eqs (6) he upper plot represents the control action he lower plot siulated output (solid) he second tuning rule is due to Lee et al (000) and it is again referred to the controller structure of Eq (3) his tuning rule can be applied only under the hypothesis that the ratio between the plant delay and the highest tie constant of the poles is less than (in the present case it is about 4) he proposed regulator paraeters are coputed using the following equations: i K c (7) K λ + τ α ( ) λ + ατ 05τ i + + α (7) λ + τ α d with: α + α λ + ατ λ + τ 05τ α τ i ( 067τ 05α) λ + τ α + (73)

5 τ λ + α e and λ is the desired closed loop tie constant By choosing λ 0, which corresponds to a closed loop settling tie of about s, the following controller paraeters are obtained: K c 49 ; i 048; d 0 56, with N 0 Notice that, although this tuning procedure is based on a copletely different theoretical analysis, the obtained paraeter values are very siilar to those provided by Eqs (6) In Fig 4 siulation results are shown he value of the perforance index of Eq (5) is J ie [s] ie [s] Figure 4 Closed-loop siulation test using a PID controller with the structure of Eq (3) and paraeters tuned by Eqs (7) he upper plot represents the control action he lower plot siulated output (solid) he last tuning rule tested in the present work for the controller structure of Eq (3) is based on the work by McMillan (984): K c 065 K τ ( + ) + ( )( τ ) τ (8) 065 ( ) + i τ + ( )( ) τ τ (8) d 0 5 i (83) Notice that, in Eqs (8), there are no other paraeters but those of the odel of Eq () So, the application of these tuning rules is straightforward, but no fine tuning is allowed to the user Finally, observe that the choice of d 0 5i is done in order to obtain a controller with two real coincident zeros he controller paraeters are: K c 54 ; i 07933; d 0983, with N 0 In Fig 5 siulation results are shown he value of the perforance index of Eq (5) is J ie [s] ie [s] Figure 5 Closed-loop siulation test using a PID controller with the structure of Eq (3) and paraeters tuned by Eqs (8) he upper plot represents the control action he lower plot siulated output (solid) Finally, also a tuning rule for the classical controller (Eq 4) has been tested based on the work of Poulin and Poerleau (996, 997) It is based on a iniu IAE criterion and the regulator paraeters are obtained by the following equations: K c b K 4( τ + ) [ a 4( τ + )] ( τ + ) 4( τ + ) 4 i a + a (9) (9) d (93) where a and b are two paraeters for fine tuning In the present application case, we experienced that the choice of these paraeters is not an easy task As a guideline for their use, the following criteria ust be considered First, the paraeter a could not take any value, in order not to have infinite values for the integral tie and the controller gain Siilarly, it has a lower bound value to avoid negative integral tie values he paraeter b (proportionally) influences only the controller gain Finally, notice that the present controller exploits the filtered derivative action to cancel the stable pole of the plant oving it at higher frequency By choosing a 3, b 5 the following controller paraeters are obtained: K c 97 ; i 466; d 0, with N 0 In Fig 6 siulation results are shown he value of the perforance index of Eq (5) is J 3 0

6 ie [s] ie [s] Figure 6 Closed-loop siulation test using a PID controller with the structure of Eq (4) and paraeters tuned by Eqs (9) he upper plot represents the control action he lower plot siulated output (solid) 5 CONCLUSIONS In this work, the proble of control of Functional Electrical Stiulation in paraplegic patients has been considered Specifically, control of sall aplitude oveents on a weight reliever device is considered o this ai, a linear PID controller has been used and tested by eans of siulations based on a coplete siulation odel Four different tuning rules for the PID controller have been considered hese are specifically designed for unstable plants with delay All the proposed tuning ethods show good perforances while doing a typical rehabilitation exercises, but they need different efforts in tuning he ain liitations of the present study are about soe specific clinical aspects Specifically, the presence of spass, typical of paraplegic subjects, has not been considered he otivation is that these issues are very specific to each subject and difficult to be odelled So, their evaluation will be deanded to the experiental work 6 REFERENCES Ferrarin M, Pavan E, Spadone R, Cardini R, Frigo C (00) Standing Up Exerciser based on Functional Electrical Stiulation and Body Weight Relief, Medical and Biological Engineering and Coputing, 40, 3, pp8-89 Forssell U, L Ljung (999) Closed-loop identification revisited Autoatica, 35, 7, pp 5-4 Lee Y, J Lee, S Park (000) PID Controller tuning for integrating and unstable processes with tie delay Che Eng Science, 55, 7, pp McMillan GK (984) Control loop perforance, Proc of the ISA/84 Int Conf and Exhibit, Houston, exas (USA), 39, pp O Dwyer A (003) Handbook of PI and PID controller tuning rules, Iperial College Press, London (UK) Poulin E, A Poerleau (996) PID tuning for integrating and unstable processes IEE Proceedings - Control theory and applications, 43, 5, pp Poulin E, A Poerleau (997) Unified PID design ethod based on a axiu peak resonance specification IEE Proceedings - Control theory and applications, 44, 6, pp Previdi F, E Carpanzano (003) Design of a gain scheduling controller for knee-joint angle control by using Functional Electrical Stiulation, IEEE Contr Syst ech,, 3, pp Previdi F, M Ferrarin, S M Savaresi, S Bittanti (004a) Closed-loop control of FES supported standing up and sitting down using Virtual Reference Feedback uning Contr Eng Practice, in press Previdi F, M Ferrarin, S M Savaresi, S Bittanti (004b) Gain scheduling control of Functional Electrical Stiulation for assisted standing up and sitting down in paraplegia: a siulation study, Int Jour Adapt Contr Signal Proc, in press Riener R, Fuhr, (998) Patient-driven control of FES-supported standing up: a siulation study, IEEE rans Rehab Eng, 6,, pp 3-4 Riener R, Fuhr, Ferrarin M, Frigo C (998) Patient-driven control of FES-induced standing up and sitting-down supported by a echanical syste: a siulation study, Proc of the 6 th Vienna Int Workshop on Functional Electrostiulation, Vienna, (AU), pp 7-0 Riener R, Quintern J (997) Physiologically based odel of uscle activation evaluated by electrical stiulation, Journal of Bioelectrocheistry and Bioenergetics, 43, pp Rotstein GE, DR Lewin (99) Siple PI and PID tuning for open-loop unstable syste, Ind Eng Che Res, 30, pp Schauer, KJ Hunt (000) Linear controller design for the single lib oveent of paraplegics Proc of the 4 th IFAC Syposiu on Modeling and Control in Bioedical Syste, Greifswald (GER) Van Den Hof PMJ, RJP Schraa (995) Identification and Control Closed loop issues Autoatica, 3,, pp