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Tap chi Khoa hoc va Cong nghf 102 (2014) 089-095 Shape Optimization And Fabrication of a Parametric Curved-Segment Prosthetic Foot for Amputee Toi iru hoa hinh dang va che tao khop mat ca chan gia cho nguoi khuyet tat Pham Huy Tuan'*, Nguyen Van Khien', Mai Van Trinh^ ' University of Technical Education Ho Chi Minh city No 1, Vo Van Ngan Stn, Thu Due disl Ho Chi Minh city, Viet Nam ^Orthopedic and Rehabilitation Center, Dist 3, Ho Chi Minh city Ministry of Labour - Invalids and Social Affairs, Vietnam Received August 30, 2013; accepted: August 25, 2014 Abstract A new design method and a simple manufacturing tool are developed for a novel parametric curved-segment prosthetic foot based on compliant mechanisms While current elastic energy storage and return feet have been widely commercialized, those designs still havent fully taken advantage of compliant mechanisms The design in this research would help to reduce the fabrication and assembly cost but still preserves the flexibility which is the key charactenstic to improve amputee gait A shape and size optimization scheme via genetic algorithm and finite element method is undertaken to design the device Prototypes of the devices are fabncated and tested The stiffness of the prosthetic foot predicted by theory is verified by experiments Using the proposed design methodology model stiffness and levels of strain energy stored inside the flexible segments could be easily regulated Keywords: Shape optimization, Genetic algonthm, Prosthehc foot, Elastic energy storage Tom tat Biii bdo ndyphat triin mot phwang phdp thiit ki mdi cimg vdi mot cong cu chi tao dan gian cho khdp mit ca ch§n gia dwa tren nin tang cua ca ciu dan hdi Mac du tren thwc ti rat nhiiu loai san phim mit ca chan dan hdl dang dwqfc thuang mai rong rai tren thi trwdng cac thiit ke vin chwa that sw tan dung triet di nhirng uv diem cOa nhdm ca ciu dan hdi Thiit ki bai bao khdng chi giup giam chi phi chi tao va lip rap ma van tri dwac mim deocua toan caciu Dac tinh vin la mdt nhirng tinh chit quan nhit cua san phim mit cac chan dan hii danh cho ngwdt khuyit tat Bai bao sii' dung thuat toan tii wu hda hinh dang va kich thtrdc kit htyp vdi giai thuat di truyin va mo phdng sd di thiit ki ca ciu Mot miu thir cung dwoc chi t^o di kiim nghiem vdi ly thuyit Do cirng ciia ca ciu dw doan bing ly thuyit da duac kiem tra bing thwc nghiem Sd' dung phtrang phap thiit ki dtrac di xuit cdng trinh khdng chi giiip diiu chinh mpt each iinh hoat cung cua san pham ma cdn cd kha nang diiu chinh dwac mire tich trir nang Iwq/ng dan hoi ben cac khau mim cua ca ciu I Introduction The needs in developmg prosthetic devices that I can store and return energy during gait such as elastic ) energy storage and return (ESAR) foot has proven for 'increasing numbers of researches in this field It is ^, , , „ ,, ^ supposed that their flexibihty would improve the i,.,.^ JC • I r •!_• m mobihty and functionality of transtibial amputees [1 The preference of ESAR foot by the amputees over ,u„ I / r 1, "vJU c^^TT, the conventional (solid ankle cushioned heel, SACH) J • • , , , , , , r • • idevices is evidenced by the blossom of varieties of _ , , , r, ^ ,, commercial prosthetic products Regretfiilly, lappropnate selection of prosthetic foot for amputees is still relied on patient feed-back or clinical prescription of experience physician Recent studies have tried to develop objective and quantitative selection methods based on investigating the 'Corresponding Author: Tei,: (+848)3722,0594 E-mail: phtuan@hcmute,edu,vn final prosthetic ankle stiffness and the levels of energy return [2], or based on minimizing metabolic cost and intact knee jomt loading [3] combining Vi'ith some optimization techniques Topology optimization [2] and size optimization [3] have been used to desig compliant prosthetic devices However, in order to j i u , c -a^ j i i-i modulate the level ot stinness, model thickness variation based on centerline curves while trying to maintain the overall shape IS the common method m , u -n „ • J J • j • these researches, Ihe attained design may deviate r ., , , =• J trom the aesthetic viewpoints, ^ In an effort to develop new type of ESAR feet with flexible regulation of stiffness and levels of energy, this research proposes a design methodology for a novel Bezier-curve beam-shape prosthetic foot-ankle The proposed shape optimization would not only provide a simple mean of regulating design objectives but also ensure for aesthetic contour in devices Cost-intensive material and fabrication Tap chi Khoa h^c v^ Cong ngh? 102 (2014) 089-095 technique are also suggested to enrich investigating tools in this humamty studying field, Methodology 2.1 Operational Generally, an n* order Bezier curve is determined by (n+I) control points B, as shown in Fig 3(a) A general form of ô"ã order Bezier curve can be represented as following principle A schematic of a monolithic compliant prosthetic ankle-foot is illustrated in Fig Parametric curve-shaped beams are integrated to imitate flexible movements of the human ankle The possibility of these beams to deflect and store energy in order lo release later to perform a certain task when necessary is the main fimction of these elastic prosthetic devices While there might be a long road for the ESAR feet to fiilly recover the biomechanical functions of the intact ankle complex which is a complicated musculoskeletal system, recent researches in this field have made considerable contributions for the rehabilitation of transtibial amputees [1] A typical gait cycle with ESAR foot for level-ground walking is illustrated in Fig, It includes two basic components: stance phase (ST) (60% gait cycle) which can be accounted from the first heel-strike of one foot when it touches the ground and finish with the toe-off action when it completely lifts off the ground, and swing phase (SW) (40% gait cycle) The SW is the duration when the foot is off the ground and prepares for the next gait cycle 2.2 Design The design of the flexural-based prosthetic ankle-foot is based on an optimization procedure where the shape of the heel, the base, and the keel is optimized via the parameters of a fourth order and two fifth order Bezier curves, respecdvely Fig 3(a) is a schematic of the compliant ankle-foot ; '[" yA = 1,- i+I (1) PW-Zfi,*,, ,!') '^[0.1] \.(')=("]''(i-'r where b,„[t) is Berstein polynomial, t is independent parameter, P[t) and 5, denotes position vectors of points on the curve and control points, respectively The shape of each Bezier curve is opdmized by allowing its control points to move in a specified space The positions of the four points A, C, D, and E are fixed, and the Cartesian coordinates (x,y) of the remaining control points HAx,y\, B, {x,y), and K^ (^^ J') ^^e design variables for the heel, base, and keel, respectively While most control points are freely moved by the opfimization codes in their designated space, the two points Hf[x,y) and A", {x,y) are constramed to sfick to the base curve to ensure for the continuity of the structure Fig 3(b} indicates the dimension bound for the design space, In this design, the in-plane thickness of the heel, the base, and the keel are also design variables Hence, the total number of the design variables is 29, The out-of-plane thickness of the three beams is taken as 50 mm, Compliant beams Foot shell Fig Human gait cycle with ankle biomechanics for level-ground walking Fig (a) A schematic of the compliant foot-ankle, (b) Bound dimensions for design space T a p chi K h o a hpc va C o n g ngh? 102 (2014) 089-095 Given expected stiffness, ratios among three stifFness, design constraints, number of population (N), number of generations (M) Loop for creation of initial population n^ Genetic optimization U^' Genetic operators on j ""generation to create offsprings X Input each oftepnng into ABAQUS to calculate objective functions Non dominated sorting genetic algorithm Choose the best population size members for (j+l)"" generation •"""""T""""I Combine population | I Final design candidates | Fig Flowchart of the optimization procedure Adjusfing the stiffness of prosthetic foot-ankle compliant devices is supposed to have a considerable improvement on the walking mechanics [4] as well as on the metabolic consumption [3], Following Saunders, et al [5], three loading configurations are considered to explore component stiffness through-out the gait cycle The heel-only (HO) loading simulates the heel-strike action, the foot-flat (FF) configuration is at the middle of stance phase, and toe-only (TO) mimics the toe-off action The three simulated stiffness found durmg these condition are called /T^p, K^.y, and Kj.^,, respectively An optimum shape of a prosthetic foot-ankle device with preferred values of these stiffness and their ratios is expected in this research An effective optimization procedure for the design of compliant mechanisms was developed [6] and is used for the synthesis of the current compliant prosthetic foot, Fig,4 is the flowchart of this opfimization procedure The nondominated sorting genetic algorithm [7] is applied to the optimization of the shape of curved beams The algorithm is suitable for solving constrained multiobjecfive problems The efficiency of the nondominated sorting genetic Tap chi Khoa hpc va Cong nghe 102 (2014) 089-095 algonthm lies in a way muftiple objectives are reduced to a dummy fitness fiinction using nondominated sorting procedure [8] Let F{X') , defined on the design space X' , be an objective space If a vector F{X^) is partially less than another vector F{X^) , we say that the solution f ( X ' ) dominates F{X') , where no value of F{X'^) is less than F ( X ' ) and at least one value of F{X^) IS stnctly greater than F{X^) The optimal solutions to a multiobjective minimization problem can be taken as the nondominated solutions In the optimization process, initially, an expected total stiffness K^ of the structure, the ratios r^, r^ among three stiffness, constraints on the design variables, number of generations, and size of populafions are specified The opfimization problem m this research is formulated into three objective functions: H„(\K,, -'^,.1 '"•"[km _K, '"'Ikro _K^ (4) (5) (6) where r^, r, are the relative ratios between Kp^ and A'jjQ, Kj^ respectively The purpose of Eq (4) is to find a foot-ankle design with the foot-flat stiffness (K^p) equal to a preferred sfiffness ( K ^ ) , normally stated by pafient feed-back or clinical experience of the physician or prosthetist In this investigafion, K^, is taken as 175 (N/mm) and r^, /•j are set as and 3, respectively The objective functions in Eq (5) and Eq (6) are used to seek for a certain relation between K^^ and Ku^ and Kj-^ Due to the geometry complexity, large motions and flexible beam behaviors of the prosthetic ankle-foot, the reaction force versus displacement [f~d) curve of the device may not be calculated analyfically Finite element method can be used to analyze geometncally nonlinear behaviors of the mechanism Finite element analysis by a commercial software ABAQUS is utilized to obtain the {f-S) curve In order to save resources and enhance simulation time, a simple static analyze of the model with beam element is employed in order to obtain the (f-d) curve in the design stage In this invesfigation, Ihe beams are assumed to be linear elastic materials A polyoxymethylene (POM) material is used for the ankle-foot The mechanical properties of the simulated POM are shown in Table 1[6] Table Matenal properties for POM Materials Density POM Rilsan™ D80 142 1.04 2.3 Yield stress, MPA 71 35 Tensile modulus (MPa) 2600 868 Poisson's ratio 0.25 0.39 Optimization In the optimization process, the number of generafions is set to be 40, and the population of each generation is taken as 20, A static finite elemenl analysis of each population is performed m order to find their ( / —(5) curves Fig shows the distribution of the population of the P' and * generations in the optimization process The x , y and z -coordinates represent the values of the objective functions in eq (4), eq, (5), and eq, (6), respecdvely As shown in the figure, 4c values of the objective functions are decretBed drastically from the P ' generation to the lasl generation After the optimum beam-element design is found A more accurate FEM model is used to veri^ for the stiffness and structural integrity of this design A two-dimensional sagittal plane geometry of the prosthetic foot with plane strain element (CPE8R)is built to analyze for the behavior of practical devi« Fig shows the f-S curve of the optimum design of the device Fig and Fig illustrate the deformed shapes and the Von-Mises stress distribution and the slored strain energy as functions of displacement of tte optimum design in three loading configurahons, respectively Fabrication and testing In order to prove the design of the prosth foot-ankle for flexibility, prototypes of the device «e fabricated The prototypes are carved by a computet numerical control milling machine (MIKRON UCP 600, GF AgieCharmilles, Switzerland) from POM material Fig 9(a) is a photo of a fabricated device Fig 9(b) is a photo of the experimental setup for measurement of the stiffness of the device Thf prosthetic foot-ankle is mounted between the W compression fixtures of a computer display hydra# universal testing machine (CHT-4106, SANS Testing Machine Co., Ltd., China) Tap chi Khoa hoc v^ Cdng ngh^ 102 (2014) 089-095 X a.ô/. m"", / ,ã K.lD!r./ằn, Displacement (mm) 'pulati Fig Distribution of the population of several generations in the optimizafion process Fig Simulated stiffiiess data for the compliant prosthetic ankle-foot in the heel only, foot flat and toe only conditions Displacemem (mm) Fig Deflected shape for Heel Only (a); Foot Flat (b) and Toe Only (c) Fig Stored strain energy in the heel only, f toe only conditions Tap chi Khoa hoe va Cong ngh? 102 (2014) 089-095 \\m\ li '^ (ol ilar# Fig Photos of a fabricated device (a) and the experimental setup (b) Fig 10 Snapshots for the initial loading (a) and final deformed shape (b) -S,^{b) ^c) Displacement (mm) Fig 11 f-S curves of the fabricated device Results and discussions Using the experimental setup, the foot-flat stiffness of the device is demonstrated and verify with the simulation results Fig 10(a) shows the as-fabricated of the prosthetic foot-ankle device and Fig 10(b) is its deformed shape in foot-flat loading configuration The experimental f-S curve of the opfimum design of the device is also obtained (Fig II) The experimental result is slighfiy lower than that based on the finite element analyses This discrepancy can be attributed to the uncertamfies in geometry and loading conditions of the experiments The fabricated device has slighfiy smaller beam widths than the designed value due to the manufacturing error in the milling process The boundary condition is also a little different between the experiment and the simulation The two ends of the base are completely fixed in FEM while in practical two more rigid segments are coupled The contact of their bottom surfaces and the lower fixture Fig 12 Foot shell and stainless pylon attached to the prosthetic foot of the testing machine is not fixed where sliding m ^ occur Also a two dimensional FEM model has been used in device design and analysis Researches have shown that three-dimensional effect may considerably influence the behaviors of the device where off-axis and eccentric loads could be notable issues Apart from the above negligible difference, thc proposed design methodology for the new configurafion of ESAR foot ankle has shown its effectiveness Besides that a cheap material and a simple manufacturing tool is also recommended in this invesfigafion Compared with other common materials used for ESAR devices such as Rilsan™ D80 [2], POM has a larger density (Table 1) However the tensile modulus and yield stress of POM are larger than tbe later material which is complementary advantage for the final mass of products The fabricated curved-segments prosthetic foot would be used for a medical experiment The foot shell is also made and a stainless pylon is T^p chi Khoa hoc va Cong ngh$ 102 (2014) 089-095 attached lo the foot ankle as in Fig 12 Conclusions In this research, we have proposed a simple and efficient method for the design of a compliant prosthetic ankle-foot Shape optimization coupled with genetic algorithm has been used in the design step Taking full advantage of compliant mechanisms, without any movable joints, the use of the designed device would result in reduced wear, reduced need for lubrication, and an increased performance by increasing precision The presented device has a capability of storing elastic energy in early to mid-stance and releasing it to assist in forward propulsion for the transtibial amputee in late stance In order to confirm the effectiveness of the device, prototypes of the device are fabricated using a simple milling process and are successfully validated by experiments The ease of fabrication possibilities for this monolithic design is another benefit over some current commercial products which require for troublesome assembly procedures The proposed optunization methodology could be incorporated with forward dynamics simulations [9, 10] of amputee walking and replace fixed setting component stiffiiess by a proper metabolic cost and join loading objecfive functions [3] in order to account for a practical feeling of amputees Acknowledgements The authors are thankful for the financial support from the Research Management and Intemafional Relations Office of UTE, under Grant NO.T20I3-35TD References [IJ.Hafiier, B, J., etal., "Transtibial Energy - Storage - and - Return Prosthetic Devices: A Review of Energy Concepts and a Proposed Nomenclature," J Rehabilitation Research and Development, 39(1), pp 1-11,(2002) [2],South, B J,, et al., "Manufacture of Energy Storage and Return Prosthetic Feet Using Selective Laser Sintenng," J Biomechanical Engineering, 132 (2010) 015001 [3].Fey, N P., et al, "Optimization of Prosthetic Foot Stiffness to Reduce Metabolic Cost and Intact Knee Loading During Below-Knee Amputee Walking: A Theoretical Study" J Biomechanical Engineering, 134 (2012)111005 ' [4], Fey, N, P., et al, "The Influence of Energy Storage and Return Foot Stiffiiess on Walking Mechanics and Muscle Activity in Below-Knee Amputees," Clinical Biomechanics, 26 (2011) 1025-1032 '[5].Saunders, M.M., et al., "Finite Element Analysis as a Tool for Parametric Prosthetic Foot Design and Evaluation Technique Development in the Solid Ankle Cushioned Heel (SACH) Foot," Comput Methods Bio-mech Biomed Eng, (2003) 75-87 [6] Pham, H, T, and Wang, D, A,, "A Constant- Force Bistable Mechanism for Force Regulation and Overload Protection," Mechanism and Machine Theory, 46 (2011) 899-909, [7], Deb, K , Pratap, A, Agarwal, S., and Meyarivan, T., "A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II," IEEE Trans on Evolutionary Computation, 6(2002)182-197, [8].N, Srinivas, K Deb, "Multiobjective Optimizanon Using Nondominated Sorting m Genehc Algonthms", J Evolutionary Computation, (1995) 221-248, [9],Neptune, R R., et al, "Contributions of the Individual Ankle Plantar Flexors to Support, Forward Progression and Swing Initiation During Walking," J Biomechanics, 34(2001)1387-1398[10], Neptune, R R, et al., "Muscle Force Redistributes Segmental Power for Body Progression During Walking," Gait and Posture, 19 (2004) 194-205 ... each oftepnng into ABAQUS to calculate objective functions Non dominated sorting genetic algorithm Choose the best population size members for (j+l)"" generation •"""""T""""I Combine population |
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