MULTIBODY MODEL OF THE HUMAN HAND FOR THE DYNAMIC ANALYSIS OF A HAND REHABILITATION DEVICE

Transcription

1 MULTIBODY MODEL OF THE HUMAN HAND FOR THE DYNAMIC ANALYSIS OF A HAND REHABILITATION DEVICE AGNIESZKA MUSIOLIK Dissertação para obtenção do Grau de Mestre em Engenharia Mecânica Júri Presidente: Prof. Nuno Manuel Mendes Maia Orientador: Prof. Miguel Pedro Tavares da Silva Vogal: Prof. Manuel Frederico Oom de Seabra Pereira JUNHO 008

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3 Resumo Esta tese descreve o desenvolvimento de um modelo matemático da mão humana utilizando uma formulação multicorpo planar com coordenadas naturais. O modelo multicorpo descreve correctamente o movimento e as características inerciais da mão, assim como a sua interacção com um dispositivo de reabilitação em desenvolvimento. A utilização de metodologias multicorpo não invasivas permite o cálculo dos momentos-de-força e das forças internas nas articulações da mão durante a eecução da tarefa de reabilitação. É também apresentada uma breve descrição da anatomia, fisiologia e das principais patologias e ferimentos da mão humana com o propósito de permitir uma melhor compreensão do modelo matemático, do dispositivo de reabilitação e dos principais conceitos que guiaram o seu projecto. O modelo matemático é cuidadosamente descrito no âmbito da formulação multicorpo com coordenadas naturais, na qual a posição e orientação dos segmentos anatómicos da mão humana são representadas utilizando as coordenadas Cartesianas de um conjunto de pontos localizados em marcos anatómicos relevantes, tais como as suas articulações e etremidades. Os dados cinemáticos que descrevem o movimento de fleão-etensão da mão foram obtidos no laboratório de análise do movimento e as coordenadas dos referidos pontos relevantes reconstruídas e filtradas utilizando filtros passa-baio Butterworth de ª ordem. Foram desenvolvidos dois códigos computacionais utilizando a formulação multicorpo em ambiente Matlab: um que realiza a análise cinemática do movimento adquirido e outro que realiza a sua análise dinâmica. Os resultados obtidos apresentam relevância biomecânica e podem ser utilizados na fase actual do projecto do dispositivo de reabilitação. i

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5 Abstract This thesis describes the development of a mathematical model of the human hand using a planar multibod formulation with natural coordinates. The multibod model correctl describes the motion and inertial characteristics of the hand as well as its interaction with a hand rehabilitation device under development. The use of non-invasive multibod dnamic tools allows the calculation of the momentsof-force and internal forces at the joints of the hand during the rehabilitation task. A brief description of the anatom, phsiolog and of the most common pathologies and injuries of the human hand is also provided for the sake of understanding the mathematical model, the associated rehabilitation device and the main concepts driving its design. The mathematical model is thoroughl described under the scope of the multibod formulation with natural coordinates, in which the position and orientation of the anatomical segments are represented using the Cartesian coordinates of points located in relevant anatomical landmarks of the hand, such as joints and etremities. The kinematic data describing the hand fleion-etension movement was acquired in the movement analsis laborator and the coordinates of the referred relevant points reconstructed and filtered using a nd order Butterworth low-pass filter. Two multibod computational codes were developed in Matlab: one that performs the kinematic analsis of the acquired motion and other that performs its dnamic analsis. The results obtained present biomechanical relevance and can be used in the actual design phase of the rehabilitation device. iii

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7 Palavras-Chave Biomecânica da Mão, Dinâmica de Sistemas Multicorpo, Modelo Biomecânico, Momentos Articulares, Dispositivo de Reabilitação. Kewords Hand Biomechanics, Multibod Dnamic Analsis, Biomechanical Model, Articular Moments, Rehabilitation device. v

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9 Contents Resumo... i Abstract... iii Palavras-Chave... v Kewords... v Contents... vii List of Tables... i List of Figures... i List of Smbols... iii Convention... iii Over script... iii Superscript... iii Latin Smbols... iii Greek Smbols... iv Chapter Introduction Motivation Literature Review Objectives Structure and organization... 3 Chapter... 5 Anatom and Phsiolog of the Hand Skeletal bones structure Muscles and tendons Neurological structure Joints Chapter Common Pathologies and Injuries of the Human Hand The congenital defects of the hand Major hand injures Diseases of the hand... 1 vii

10 Chapter Multibod Sstems with Natural Coordinates Dependent and Independent Coordinates Kinematics Analsis Initial position and finite displacement analsis Velocit and Acceleration analsis Dnamic Analsis Equations of Motion of a Multibod Sstem Chapter Multibod Model of the Hand and Rehabilitation Device Hand rehabilitation Devices Scope of Application of the Hand Rehabilitation Device Under Development The Multibod Model Natural coordinates, kinematic constraint and degrees of freedom Jacobian matri Contributions to the ν and γ vectors Kinematic Data Collection Masses, Inertias and assembl of the Global Mass Matri Kinetic Data Results of the kinematic and Dnamic Analsis of the Multibod Model of the Hand Chapter Conclusion and Future Developments Conclusions Future Developments References Anne Sizes and masses of hand links Anne Maimum values of wrist joint capacit Anne Wrist range of motion activities of dail living Anne Hand Kinematics Matlab Program Source Anne Hand dnamics Matlab Program Source viii

11 List of Tables Table 1 Jacobian Matri Table Parameters included in model Table 3. Mass matri Table 4. Sizes and masses of links in the model Table.5. Maimum values of wrist joint capacit Table 6. Wrist Range of Motion in Activities of Dail Living i

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13 List of Figures Fig. 1 Bones of the hand... 6 Fig.. Muscles of the hand... 8 Fig. 3. Muscles of the hand... 7 Fig. 4. The thenar eminences Fig. 5. Joints of the hand Fig. 6. Partial aphalangia of the right upper limb Fig. 7. Ectrodactl Fig. 8. Trisom 1- hand features Fig. 9. Poldactl Fig. 10. Sndactl Fig. 11. Finger injur Fig. 1. Finger 1 ear after the surger Fig. 13. Pre-operative absence of etension to the middle and ring fingers Fig. 14. Operative findings showing loss of the fleor tendons Fig. 15. Harvesting of the palmaris longus tendon graft Fig. 16. Grafting of the fleor tendons to the middle and ring fingers Fig. 17. Post operative results after 6 months Fig. 18. Acute fleor tendon injuries such as this require onl a good fleor tendon repair and hand therap Fig. 19. Therap and post-operative results of a 3 finger fleor tendon injuries Fig. 0. Proimal phalangeal fracture with open reduction and internal fiation done Fig. 1. Ice-skating fall... 0 Fig.. Skier's thumb... 1 Fig. 3. Amotroph of left hand... Fig. 4. Duputren s disease... Fig. 5. Partial paresis of the finger... 3 Fig. 6. An elderl woman's hand deformed b severe arthritis... 3 Fig. 7. Carpal tunnel sndrome... 4 Fig. 8. Independent coordinates sstem of particles: The sstem of particles 1 and has 4 degrees of freedom 1, 1,,. The displacements 1, 1,, are independent coordinates in the sstem of the particles Fig. 9 Rigid bod with points - dependent coordinates: the sstem uses 4 coordinates. However a planar bod has onl 3 degrees of freedom. 1, 1,, are related as the length need to be kept... 7 Fig. 30. a Unconstrained sstem of rigid bodies, b constrained sstem of rigid bodies Fig. 31. a Discs, b The ball Fig. 3. a The ball, b Meshes Fig. 33. Powerball Fig. 34. a Phsical prototpe, b Virtual prototpe i

14 Fig. 35. Multibod model of the hand: 4 rigid bodies I to IV, eight points P1 to P8. Three revolute joints R1-R3 and 6 degree-of-freedom D1 to D Fig. 36. The hand with the angular in the wrist Fig. 37. The hand with the angular θ Fig. 38. The hand with the angular θ Fig. 39. The hand with the angular θ Fig. 40. Markers on the hand Fig. 41. Force generate b the finger Fig. 43. Analsis statistics Fig. 45. Kinematics for point Fig. 46. Kinematics for point... 5 Fig. 47. Kinematics for point Fig. 48. Kinematics for point Fig. 49. Kinematics for point Fig. 50. Kinematics for point Fig. 51. Kinematics for point Fig. 5. Kinematic for point Fig. 53. Reactions of the hand Fig. 54. Moments in each joint Fig. 55. Model of the hand Fig. 56. The tpical of activities performed at the assembl line in lamp factor stud ii

15 List of Smbols Convention a, A, α Scalar a Vector A Matri Over script a& a&& First time derivative Second time derivative Superscript T T a, A Transpose of a vector or matri 1 A Inverse matri Latin Smbols c 1, c, Coordinates of segment r ip in a generic two-dimensional vector base c Vector containing previous coefficients c 1, and c C Coordinate transformation matri f c, f s f Filter s cut-off frequenc and sampling frequenc of capture device Vector with the Cartesian components of a generic force iii

16 g Vector of generalized forces i, j Generic points I,, VI Anatomical segment number I L ij, L u m m Identit matri Lengths of segment r ij and unit vector u Rigid bod mass Cartesian components of a generic moment-of-force M, M e Sstem s global mass matri and rigid bod s local mass matri n c n h q, q&, q&& r i, r j, t Number of generalized coordinates Number of holonomic kinematic constraints Vectors of generalized coordinates, velocities and accelerations Vectors with the Cartesian coordinates of generic points i, and j Time variable or current time step of analsis u, v Generic unit vectors ω, ω& Angular velocit and angular acceleration of a rigid bod, & Auiliar vectors used in the direct integration process Greek Smbols α, β Parameters used in the Baumegarte s Stabilization Φ Φ q λ ν γ Vector of kinematic constraints Jacobian matri of kinematic constraints Vector of Lagrange multipliers Right-hand-side vector of velocit equations Right-hand-side vector of acceleration equations iv

17 Chapter 1 Introduction 1.1. Motivation The hand is a ver important part of the human bod. It is used to do a lot of different things as for eample to do the precision motion of sewing with a needle or to withstand 0kg when returning home. The function of the hand is unusuall important in the human life. The hand is used as a working tool but at the same time its gestures make up the smbols of greeting, request or condemn. That is wh it is so important to quickl restore the largest efficienc of the hand function and movement of persons who, as a result of an accident or disease, have being deprived of these faculties. The fact that there are onl a few number of devices on the market to the rehabilitation of the hand creates the need to devise new solutions for a better and more efficient hand rehabilitation. It is in this perspective that the motivation for the work that develops net appears. It is well known that an medical or rehabilitation device must be put to the test before it can be applied to help people with disabilities. With the actual computers and with the right computational methods, it is possible nowadas to construct reliable and accurate computational models that are able to analze in an integrated wa the performance of the device and the effectiveness of its action on the affected biological structure. This is even more important in the earl stages of design when no prototpe is et available. This thesis describes the development of a mathematical model of the human hand using a multibod sstem formulation with natural coordinates that correctl describes the motion and behaviour of the human hand and its interaction with the rehabilitation device. From this perspective, the use multibod dnamics tools, has proved to be a successful option to describe sstems undergoing large displacements, calculating the moments of force and the internal forces developing in the joints of the human hand during the rehabilitation task. Another important characteristic provided b the use of multibod methodologies, is that the information obtained is the result of the application 1

18 of mechanical laws of phsics to living structures and, therefore, it is not obtained using invasive measuring techniques. The calculation of the reaction forces at the joints and the determination of the moments of force produced b the muscles during the prescribed task are obtained from the solution of a set of equations of motion, assembled in a sstematic was for the biomechanical sstem under analsis, instead of being obtained using specific force measuring devices. 1.. Literature Review Due to its importance in normal da living the human hand has been the focus of man scientific studies. Man of these studies focus on general information about the hand such as masses and inertias of each of its components other on its several different motions. In terms of mathematical models of the hand, the work of Birukova and Yourovskaa, 1994 is considered decisive in the development of knowledge on this subject. In that work, a model of human hand including links and joints was proposed to analze hand dnamics. This model consisted of 16 links interconnected b frictionless joints. Also in this work masses and link lengths were defined for the hand. These values were the ones used in this work and are presented in Anne 1. In the same ear, the maimum values of the joint capacit during certain performance of activities were calculated using multicriterion optimization Gielo-Perczak, 1994,see Anne. Relevant research regarding the functional anatom of finger muscles was also carried-out Mansour, In this research quantitative information about of the passive moments of the inde finger for tip pinch were presented. Other studies focused onl on the comple kinematics of the wrist bones and their relative motion Feipel, et. al., These autors defined movements of the carpal bones during wrist motion, in particular for the triquetrum, scaphoid and hamate bones. Furthermore, the defined wrist etension and fleion, wrist fleion and radial deviation. Regarding the wrist range of motion, Nelson, et. al, 1994 analzed the range of motion of this joint for several activities such as hair combing, answering a phone or eating with a knife see Anne 3. On a different perspective Wu, et. al., 1994 proposed a specific joint coordinate sstem for the bones of the hand and wrist. From the dnamic point of view, Valero-Cuevas, 005 presented a stud about static and dnamic fingertip forces. Also Kuo et. al., 006 presented a stud about inde finger joint coordination during tapping. In another stud Vigourou, et. al., 005 provided a estimation of the muscle-tendon tension and pulle forces during specific sport-climbing grip techniques. A multibod approach with dependent coordinates is applied in this work to describe hand dnamics. Using this tpe of approach, the number of coordinates is usuall higher than the number of degrees of freedom of the sstem, and therefore not minimal. This tpe of formulation serves generalpurpose applications and performs well in the simulation of general mechanical sstems. Among the

19 multibod formulations using this tpe of coordinates, the ones in which the mechanical sstems are described with Cartesian coordinates Haug, 1989; Nikravesh, 1988 and with full Cartesian coordinates Jalón and Bao, 1994; Nikravesh, 1994, are emphasized. The latter tpe, in particular, presents additional advantages in the simulation of biomechanical sstems since man relevant bod landmarks can be directl used, as generalized coordinates, in the description of the sstem Silva and Ambrósio, 00a. This feature is particularl important in inverse dnamics applications, in which the motion is acquired eperimentall using digitizing techniques Silva et. al., 001. A multibod formulation with full Cartesian coordinates is presented in this work and used in the description of mechanical sstems in general and biomechanical models in particular Objectives The most important objective of this thesis is to define a mathematical model of the hand and device, using a multibod methodolog which will allow to define the dnamics of the hand and the overall performance of the rehabilitation device as well as it will allow for the calculation of the internal forces at the bones, reaction forces of the joints and articular moments. Also muscle forces in a near future. The investigation was made considering the fleion-etension motion of the hand in the phsical prototpe obtained in the movement laborator Structure and organization In Chapter 1 the motivation for the work, objectives and relevant literature of the hand is presented. To better understand the essential function of the hand, it is necessar to know its anatom and the phsiolog which are described in the Chapter. In Chapter 3, knowledge about the pathologies of the human hand, is provided, and in Chapter 4 the formulation with natural coordinates is described. In Chapter 5 the formulation introduced in the previous Chapter is applied to construct a planar model of the hand with the purpose or analsing its fleion-etension movement and its infraction with the rehabilitation device. In this chapter the most important results are presented and discussed. In Chapter 6 the work is reviewed and conclusions and future developments are discussed. 3

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21 Chapter Anatom and Phsiolog of the Hand In this chapter a brief description of the anatom and phsiolog of the hand will be provided for the sense of understanding the mathematical model description that follows in a later chapter. The interested reader is referred to the works of Tortora, G., Grabowski, S. R., 001 for a more detailed and comprehensive information about the subject of this chapter. The anatom of the hand is comple and intricate. Its integrit is absolutel essential for our everda functional living. Our hands ma be affected b man disorders, most commonl from traumatic injur. The hand manus is a part of the upper limb and includes the wrist, the metacarpus and fingers. Fingers are usuall referred as the thumb, the inde, the middle, the ring and the small finger as represented in fig. 1. Few structures of the human anatom are so unique as the hand. The hand needs to present high mobilit in order to allow the proper position of the fingers and thumb. Adequate strength resulting from comple coordination and control methods forms the basis for normal hand function, as it allows the development of fine motor tasks with high-precision and different gripping strategies..1. Skeletal bone structure Bones are skeletal supporting structures that allow the transmission of the internal loads, provide stiffness to the skeletal structure and together with the joints contribute for a proper gross motion of the hand. There are 7 bones within the wrist and hand distributed b 3 major areas: the carpal, the metacarpal and the phalangeal. The wrist itself contains eight small bones, called carpals, which are showed in fig. 1: the hamate 1, the pisiform, the triquetrum 3, the capitate 4, the lunate 5, the scaphoid 6, the trapezium 7, and the trapezoid 8. All carpal bones participate in wrist function 5

22 ecept the pisiform, which is a sesamoid bone through which the fleor carpi ulnaris tendon passes. The scaphoid serves as link between each row and therefore, it is vulnerable to fractures. The distal row of carpal bones is strongl attached to the base of the second and third metacarpals, forming a fied unit. All other structures mobile units move in relation to this stable unit. The fleor retinaculum, which attaches to the pisiform and hook of hamate ulnarl, and to the scaphoid and trapezium radiall, forms the roof of the carpal tunnel. The carpals connect to the metacarpals. There are five metacarpals forming the palm of the hand. Each metacarpal is characterized as having a base, a shaft, a neck, and a head. One metacarpal connects to each finger and thumb. The first metacarpal bone thumb is the shortest and most mobile. Each finger ecluding the thumb includes the distal phalange, the middle phalange and the proimal phalange. The thumb onl has one interphalangeal joint between the two thumb phalanges, respectivel the distal phalange and the proimal phalange. The three phalanges in each finger without the thumb are separated b two joints, called interphalangeal joints IP joints, whose major phsiological characteristics will be further described in section.4. Distal phalange PHALANGES Middle phalange Distal phalange Proimal phalange Proimal phalange METACARPAL CARPAL Fig. 1. Bones of the hand 6

23 .. Muscles and tendons The muscles of the hand are divided into intrinsic and etrinsic muscle groups. The intrinsic muscles are located within the hand itself, whereas the etrinsic muscles are located proimall in the forearm and insert into the hand skeleton b long tendons...1 Etrinsic etensor muscles The etensor muscles are all etrinsic ecept for the interosseous-lumbrical comple involved in interphalangeal joint etension. All the etrinsic etensor muscles are innervated b the radial nerve. This group of muscles consists of 3 wrist etensors and a larger group of thumb and digit etensors as represente in fig. and fig. 3. Fig.. Partial view of the muscles of the hand The etensor carpi radial brevis ECRB is the main etensor of the wrist, along with the etensor carpi radialis longus ECRL and etensor carpi ulnaris ECU, which also deviate the wrist radiall and ulnarl, respectivel. The ECRB inserts at the base of the third metacarpal, while the ECRL and ECU insert at the base of the second and fifth metacarpal, respectivel. The etensor digitorum communis EDC, etensor indicis proprius EIP, and etensor digiti minimi EDM etend the digits. The insert to the base of the middle phalanges as central slips and to the base of the distal phalanges as lateral bands. The abductor pollicis longus APL, etensor pollicis brevis EPB, and etensor pollicis longus EPL etend the thumb. The insert at the base of the thumb metacarpal, proimal phalan, and distal phalan, respectivel. 7

24 The etensor retinaculum ER prevents bowstringing of tendons at the wrist level and separates the tendons into 6 compartments. The etensor digitorum communis EDC is divided in a series of tendons to each digit with a common muscle bell and with intertendinous bridges between them. The inde and small finger each have independent etension function through the etensor indicis proprius EIP and etensor digiti minimi EDM. Abductor digiti minimi brevis Adductor pollicis Opponens Digiti minimi Fleor digiti minimi brevis Fleor pollicis brevis Adductor pollicis brevis Opponens pollicis Interrosei Fig. 3. Partial view of the muscles of the hand.. Etrinsic fleors The etrinsic fleors consist of 3 wrist fleors and a larger group of thumb and digit fleors. The are innervated b the median nerve, ecept for the fleor carpi ulnaris FCU and the fleor digitorum profundus FDP to the small and ring finger, which are innervated b the ulnar nerve. The fleor carpi radialis FCR is the main fleor of the wrist, along with the fleor carpi ulnaris FCU and the palmaris longus PL, which is absent in 15% of the population. The insert at the base of the third metacarpal, the base of the fifth metacarpal, and the palmar fascia, respectivel. The FCU is primaril an ulnar deviator. The 8 digital fleors are divided in superficial and deep muscle groups. Along with the fleor pollicis longus FPL, which inserts at the thumb distal phalan, the pass through the carpal tunnel to provide fleion at the interphalangeal joints. At the palm, the fleor digitorum superficialis FDS tendon lies superficial to the palm of the hand. It then splits at the level of the proimal phalan and reunites dorsal to the profundus tendon to insert 8

25 in the middle phalan. The fleor digitorum profundus FDP perforates the superficialis tendon to insert at the distal phalan. The relationship of fleor tendons to the wrist joint, metacarpophalangeal joint and interphalangeal joint is maintained b a retinacular or pulle sstem that prevents the bowstringing effect...3 Intrinsic muscles The intrinsic muscles are situated totall within the hand. The are divided into 4 groups: thenar, hpothenar, lumbrical, and interossei muscles. The thenar group consists of the abductor pollicis brevis APB, fleor pollicis brevis FPB, opponens pollicis OP, and adductor pollicis AP muscles. All are innervated b the median nerve ecept for the adductor pollicis and deep head of the fleor pollicis brevis, which are innervated b the ulnar nerve. The originate from the fleor retinaculum and carpal bones and insert at the thumb's proimal phalan. The hpothenar group consists of the palmaris brevis PB, abductor digiti minimi ADM, fleor digiti minimi FDM, and opponens digiti minimi ODM. The are all innervated b the ulnar nerve. This group of muscles originates at the fleor retinaculum and carpal bones and inserts at the base of the proimal phalan of the small finger. The lumbrical muscles contribute to the fleion of the metacarpophalangeal joints and to the etension of the interphalangeal joints. The originate from the fleor digitorum profundus tendons at the palm and insert on the radial aspect of the etensor tendons at the digits. The inde finger lumbricals IFL and long finger lumbricals LFL are innervated b the median nerve, and the small finger lumbricals SFL and ring finger lumbricals RFL are innervated b the ulnar nerve. The interossei group consists of 3 volar and 4 dorsal muscles, which are all innervated b the ulnar nerve. The originate at the metacarpals and form the lateral bands with the lumbricals. The dorsal interossei DI abduct the fingers, whereas the volar interossei VI adduct the fingers to the hand ais..3. Neurological structure The hand is innervated b 3 nerves, the median, the ulnar, and the radial nerves respectivel. Each has both sensor and motor components. Variations from the classic nerve distribution are so common that the are the rule rather than the eception. The skin of the forearm is innervated mediall b the medial antebrachial cutaneous nerve and laterall b the lateral antebrachial cutaneous nerve. The median nerve is responsible for innervating the muscles involved in the fine precision and pinch function of the hand. It originates from the lateral and medial cords of the brachial pleus C5- T1 and etends until the fingers. In the forearm, the motor branches suppl the pronator teres, fleor 9

26 carpi radialis, palmaris longus and fleor digitorum superficialis muscles. The anterior interosseus branch innervates the fleor pollicis longus, fleor digitorum profundus inde and long finger and pronator quadratus muscles. Muscles of thenar eminences Fig. 4. The thenar eminences Proimal to the wrist, the palmar cutaneous branch provides sensation at the thenar eminence fig. 4 As the median nerve passes through the carpal tunnel, the recurrent motor branch innervates the thenar muscles abductor pollicis brevis, opponens pollicis, and superficial head of fleor pollicis brevis. It also innervates the inde and long finger lumbrical muscles. Sensor digital branches provide sensation to the thumb, inde, long, and radial side of the ring finger. The ulnar nerve is responsible for innervating the muscles involved in the power grasping function of the hand. It originates at the medial cord of the brachial pleus C8-T1 and etends until the fingers. Motor branches innervate the fleor carpi ulnaris and fleor digitorum profundus muscles to the ring and small fingers. Proimal to the wrist, the palmar cutaneous branch provides sensation at the hpothenar eminence. The dorsal branch, which branches from the main trunk at the distal forearm, provides sensation to the ulnar portion of the dorsum of the hand and small finger, and part of the ring finger. At the hand, the superficial branch forms the digital nerves, which provide sensation at the small finger and ulnar aspect of the ring finger. The deep motor branch passes through the Guon canal in compan with the ulnar arter. It innervates the hpothenar muscles abductor digiti minimi, opponens 10

27 digiti minimi, fleor digiti minimi, and palmaris brevis, all interossei, the ulnar lumbricals, the adductor pollicis, and the deep head of the fleor pollicis brevis. The radial nerve is responsible for innervating the wrist etensors, which control the position of the hand and stabilizes the fied unit. It originates from the posterior cord of the brachial pleus C6-8. At the elbow, motor branches innervate the brachioradialis and etensor carpi radialis longus muscles. At the proimal forearm, the radial nerve divides into the superficial and deep branches. The deep posterior interosseous branch innervates all the muscles in the etensor compartment, namel the supinator, the etensor the carpi radialis brevis, the etensor digitorum communis, the etensor digiti minimi, the etensor carpi ulnaris, the etensor indicis proprius, the etensor pollicis longus, the etensor pollicis brevis and the abductor pollicis longus. The superficial branch provides sensation at the radial aspect of the dorsum of the hand, the dorsum of the thumb, and the dorsum of the inde, long, and radial half of the ring finger proimal to the distal interphalangeal joints..4.joints and ligaments The hand is composed b 3 tpes of articular joints: the wrist joint comple with the intercarpal joints, the metacarpophalangeal joints and the interphalangeal joints as depicted in fig. 5. The interphalangeal joints The metacarpophalangeal jonts The wrist joint Compleintercarpal joints Fig. 5. Joints of the hand 11

28 The wrist joint is a comple multiarticulated joint that allows wide range of motion in fleion, etension, circumduction, radial deviation, and ulnar deviation. The distal radioulnar joint allows pronation and supination of the hand as the radius rotates around the ulna. The radiocarpal joint includes the proimal carpal bones and the distal radius. The proimal row of carpals articulates with the radius and ulna to provide etension, fleion, ulnar deviation, and radial deviation. This joint is supported b an etrinsic set of strong palmar ligaments that arise from the radius and ulna. Dorsall, it is supported b the dorsal intercarpal ligament between the scaphoid and triquetrum and b the dorsal radiocarpal ligament. At the intercarpal joints, motion between carpal bones is ver restricted. These joints are supported b strong intrinsic ligaments. The most important ones are the scapholunate ligament and the lunotriquetral ligament. Disruption of either one can result in wrist instabilit. The Gilula lines have been described to represent the smooth contour of a greater arc formed b the proimal carpal bones and a lesser arc formed b the distal carpal bones in normal anatom. All 4 distal carpal bones articulate with the metacarpals at the carpometacarpal CMC joints. The second and third CMC joints form a fied unit while the first CMC forms the most mobile joint. At the metacarpophalangeal joints, lateral motion is limited b the collateral ligaments, which are actuall lateral oblique in position rather than true lateral. This arrangement allows the ligaments to be tight when the joint is fleed and loose when etended. The volar plate is part of the joint capsule that attaches onl to the proimal phalan, allowing hperetension. The volar plate is the site of insertion for the intermetacarpal ligaments. These ligaments restrict the separation of the metacarpal heads. At the interphalangeal joints, etension is limited b the volar plate, which attaches to the phalanges at each side of the joint. Radial and ulnar motion is restricted b collateral ligaments, which remain tight through their whole range of motion. Knowledge of these configurations is of great importance when splinting a hand in order to avoid joint contractures. 1

29 Chapter 3 Common Pathologies and Injuries of the Human Hand In this chapter the most common pathologies of the human hand are described. This brief description is provided for the sake of understanding the problem of the rehabilitation of the hand and the driven concepts for the construction of the prototpe of the hand rehabilitation. The upper limbs have a large number of different genetic and environment derived abnormalities, some of which can be surgicall repaired, while others ma indicate other sndromes or karotpe anomalies b association. Man of the hand abnormalities are described as dactl from the Greek daktulos for finger or digit The congenital defects of the hand The congenital defects of the hand happen seldom. For eample, acheiria is a missing hand, adactl is the absence of the metacarpal or metatarsal, amelia is a complete absence of a limb and aphalangia, showed on the fig. 6, is the absent of one or more digit fingers. Fig. 6. Partial aphalangia of the right upper limb 13

30 The brachdactl 7.a is when middle phalanges of both hands are ver short in length or absent. Ectrodactl, showed on the fig. 7.b, is a split or cleft hand. Fig. 7. a Branchdactl, b Ectrodactl Trisom 1 or Down Sndrome fig. 8 features short and broad hands with clinodactl curving of the fifth finger, little finger, a single fleion crease 0% and hperetensible finger joints UNSW, 008. Fig. 8. Trisom 1- hand features In the makrodaktlia one finger is bigger than the remaining and in the mikrodaktlia, is the opposite. 14

31 Poldactl also called poldactlia or poldactlism fig. 9 is when more than 5 fingers can be found in the hand. This condition is often treated surgicall in the infant. Poldactl can also be associated with a number of different sndromes including Greig cephalopolsndactl sndrome GCPS UNSW, 008. Fig. 9. Poldactl Sndactl fig. 10 is a fusion of fingers which ma be single or multiple and ma affect skin onl, skin and soft tissues or skin, soft tissues and bone. The condition is unimportant in toes but disabling in fingers and requires operative separation and is frequentl inherited as an autosomal dominant UNSW, 008. Fig. 10. Sndactl 15

32 3.. Major hand injures Hand injuries can be simple or comple. A lot of injures to the hand comes from sports, phsical works and falls. Hand injures are usuall not life threatening but without meticulous care, the can be the source of considerable disabilit Hand wounds Because hand wounds are not life threatening, the priorit for evacuation and earl surger is low. For eample the injur fig. 11 ma seem serious. But this fingertip injur can easil be attached back in place b a qualified hand surgeon. The nail bed is protected b an artificial nail splint to allow proper healing of the nail bed. On the fig. 1 is showed the hand 1 ear after the surger. Due to the site and depth of injur, the ulnar nerve is injured and results in paralsis of the intrinsic muscles to the hand. This also resulted in the loss of sensation to the ring and little fingers. Immediate treatment should be sought to ensure the nerve is repaired. A delaed repair ma result in the need for nerve graft and consequent poorer outcome. In this tpe of injures rehabilitation is often required. Fig. 11. Finger injur Fig. 1. Finger 1 ear after the surger 16

33 3... Tendon Injuries Other biological structures that often damage in a hand injur are the tendons. Tendons are collagen cables that help bending and straighten the fingers, since the muscles which move the fingers are located up in the forearm. It is therefore the tendon that connects the joints of the fingers to the muscles. The most common and difficult problem that people have after a tendon injur is the loss of the abilit to full bend or straighten the finger. Depending on the timing, surger can be done to repair or graft the tendon. Acute injuries require repair and late diagnosis ma result in the need for grafting. On figs. 13 to 19 eamples are showed of missed tendon injuries to the right middle and ring finger and the results obtained after the surger. Also in this cases, rehabilitation procedures are required in post operative phases HARMS, 005. Fig. 13. Pre-operative absence of etension to the middle and ring fingers Fig. 14. Operative findings showing loss of the fleor tendons 17

34 Fig. 15. Harvesting of the palmaris longus tendon graft Fig. 16. Grafting of the fleor tendons to the middle and ring fingers Fig. 17. Post operative results after 6 months Fig. 18. Acute fleor tendon injuries such as this require onl a good fleor tendon repair and hand therap 18

35 Fig. 19. Therap and post-operative results of a 3 finger fleor tendon injuries Fractures of the hand Fractures are also ver common injuries to the hand. This tpe of injuries usuall present pain and swelling. Fractures of the hand ma cause permanent deformities. It is usuall the intra-articular fractures the ones that etend into a joint that requires immediate or earl attention but these fractures are usuall overlooked because the present little deformities and bruising. Stable fractures can be treated conservativel with closed reduction to restore bone alignment. Unstable and intra-articular fractures usuall require open reduction and internal fiation procedures for best results fig. 0 HARMS, 005. Fig. 0. Proimal phalangeal fracture with open reduction and internal fiation done 19

36 Man hand injuries are the result of a fall. A common injur that occurs in such circumstances wrist fracture. The wrist is often fractured during a fall on an outstretched arm. In this position, the arm remains straight and the wrist takes the full force of the fall. While the two bones in the forearm, the radius and the ulna, are the most likel to fracture, it is also possible that the small bone in the wrist just behind the base of the thumb, the scaphoid bone, can fracture fig. 1. Fig. 1. Ice-skating fall Ligament Injuries Ligament are other biological structures that sustain injuries during a fall. For eample during winter sports injur to the thumb ligament has become one of the more common skiing injuries over the past 0 ears. It ranks second onl to knee sprains. Skier's thumb is a strain or tear to the thumb's major stabilizing ligament-the ulnar collateral ligament of the metacarpophalangeal joint of the thumb fig.. The ulnar collateral ligament assists us in grasping, pinching, and stabilizing items in our hands. Injur to the thumb while skiing usuall results from a fall on an outstretched hand that continues to hold the ski pole. At impact, the thumb is driven directl into the snow and is bent back or to the side, awa from the palm and inde finger, straining or tearing the ligament. When injured, the ulnar collateral ligament and other ligaments cannot support the thumb bone, making grasping or pinching with the thumb difficult HHA,

37 Fig.. Skier's thumb Thumb injuries require evaluation to establish the correct diagnosis and initiate proper treatment. Phsical signs include pain, swelling, and tenderness on the inner side of the metacarpophalangeal joint and loss of strength in the thumb. X-ras should be taken to rule out damage to the bone. Sometimes, a small piece of the bone is torn off with the ligament. Treatment for a torn ligament usuall consists of a brief period of wearing a splint or cast for immobilization. Occasionall, however, with complete tears or displacement of the ligament, surger is required to repair the injured ligament. Long-term problems can result from instabilit of the thumb. With proper treatment, however, the patient can regain full function and return to activit HHA, Diseases of the hand The hand can be affected b man diseases including muscular and neurological. Man require rehabilitation. Following is a brief description of the most common ones Hiraama Disease Hiraama Disease is a juvenile muscular atroph of the distal upper etremit is a sporadic juvenile-onset disease that presents with the gradual onset of unilateral weakness and atroph in the hand and forearm muscles. Generall, this disorder is considered a benign, non-progressive motor neuron disease. Operative reconstruction b tendon transfer improved the activities of dail living ADL of a patient with advanced Hiraama disease who showed marked intrinsic muscle atroph of the left hand fig. 3 Chiba, S., et. al.,

38 Fig. 3. Amotroph of left hand Duputren Disease Duputren's disease most often affects the pretendinous bands or fascia of the hand see the normal hand structure on the left. As the disease progresses, these areas can thicken and form a rope like cord. Eventuall, the disease can severel affect the hand's appearance and function, causing contracture fig. 4 RH, 008 Fig. 4. Duputren s disease

39 Partial paresis Partial paresis of the hand causes the atroph of muscles see fig. 5 Fig. 5. Partial paresis of the finger Arthritis Arthritis fig. 6 is a generic term for inflammator joint disease. Regardless of the cause, inflammation of the joints ma cause pain, stiffness, swelling, and some redness of the skin about the joint. Fig. 6. An elderl woman's hand deformed b severe arthritis 3

40 Carpal Tunnel Sndrome Carpal tunnel sndrome fig. 7 is a narrow bon passage in the wrist formed b the bones of the wrist carpal bones on the bottom and the transverse carpal ligament on the top. An important nerve median nerve and nine tendons that allow the wrist to fle, or move downward, pass through this small area. Carpal tunnel sndrome CTS occurs when the median nerve, the softest structure in the tunnel, becomes compressed against the transverse carpal ligament. The median nerve is responsible both for the abilit to fire the nine tendons to fle the wrist and for sensation in certain fingers in the hand. Fig. 7. Carpal tunnel sndrome 4

41 Chapter 4 Multibod Sstems with Natural Coordinates In the scope of this work, a multibod sstem is an assembl two or more planar rigid bodies also called elements that can be joined together, in such a wa that their relative movement became constrained. The joining of two rigid bodies is called a kinematic pair or simpl a joint Nikravesh, P., 1988, Haug, E., A joint permits certain degrees of freedom or relative motion and prevents or restricts others. Joint are classified in classes. In planar multibod sstems there are classes of Joints: a class I joint allows one degree of freedom, e.g. a revolute or hinge joint R allows one relative rotation, a class II joint allows two degrees of freedom e.g. a revolute-translational joint R-T allows one relative rotation and one relative translation. Multibod sstems are classified as open-chain of closed-chain sstems. If a sstem is composed of bodies without closed branches or loops, then it is called an open-chain sstem, otherwise, it is called a closed-chain multibod sstem. Multibod Sstem can be also acted up on b eternal forces. Forces can be conservative like the gravitational acceleration field or non-conservative like the friction force Dependent and Independent Coordinates In order to describe a multibod sstem, the first important point to consider is that of choosing a mathematical wa of describing its position and motion. In other words, select a set of coordinates that will allow one to unequivocall define the position, velocit, and acceleration of the multibod sstem at all times. There can be two choices in terms of coordinate selection: independent coordinates fig. 8, whose number coincides with the number of degrees of freedom of the multibod sstem and is thereb minimal, or dependent coordinates fig. 9 in a number larger than that of the degrees of freedom of the sstem and with which can the multibod sstem be much more easil described, 5

42 although interrelated through certain algebraic equations known as constraint equations. In most cases independent coordinates are not a suitable solution for a general purpose analsis, because the do not meet one important requirement: the coordinate sstem should unequivocall define the position of the multibod sstem. In fact independent coordinates directl determine the position of the input bodies of the value of the eternall driven coordinates, but not the position of the entire sstem. Therefore additional non-trivial analsis need be performed to this end. Fig. 8. Independent coordinates sstem of particles: The sstem of particles 1 and has 4 degrees of freedom 1, 1,,. The displacements 1, 1,, are independent coordinates in the sstem of the particles. Alternativel, dependent coordinates fig. 9 generate a number of constraints that is equal to the difference between the number of dependent coordinates and the number of degrees of freedom. Constraint equations are generall nonlinear, and pla a mjor role in the kinematics and dnamics of multibod sstems. These tpe of coordinates are the alternative choice to the independent set of coordinates as the uniquel determine the position of all bodies. Three major tpes of coordinates have been proposed to solve this problem: relative coordinates, reference point or Cartesian coordinates, and natural or full Cartesian coordinates. In this work, natural or full Cartesian Coordinates is the solution adopted to describe the multibod sstem under analsis. In the case of planar multibod sstems, natural coordinates can be considered as an evolution of the reference point coordinates in which the points are moved to the joints or to other important points of the elements, so that each element has at least two points. It is important to point out that since each bod has at least two points, its position and angular orientation are determined b the Cartesian coordinates of these points, and the angular variables used b reference point coordinates are no longer necessar. Thus the natural coordinates in the case of planar multibod sstems are made of Cartesian coordinates of a series of points. 6

43 1 1, 1, Fig. 9. Rigid bod with points - dependent coordinates: the sstem uses 4 coordinates. However a planar bod has onl 3 degrees of freedom. 1, 1,, are related as the length need to be kept. 4.. Kinematics Analsis Kinematics problems are of a purel geometrical nature and can be solved, regardless of the forces and inertia characteristics of elements. Kinematic analsis aims to calculate the position and orientation of all elements of the sstem, during the analsis period, giving the position of its elements. Input elements of a multibod sstem as those whose position or motion is know or specified. The position and motion of the other elements of the sstem are found in accordance with the position and motion of the input elements. There are as man input elements as there are degrees of freedom for the multibod sstem. Kinematic Analsis provides a view of the entire range of motion of a multibod sstem. The solution of the kinematics simulation encompasses the calculation of the position, velocit and acceleration of the sstem. It also permits one to detect collisions, stud the trajectories of points, analze the position of an element of the multibod sstem and calculate the rotation angles of the driving elements Initial position and finite displacement analsis The initial position or assembl problem consist in finding the position of all the elements of the multibod sstem once that of the input elements is known. This problem difficult to solve, since it leads to a sstem of nonlinear algebraic equations which has, in general, several possible solutions. Finite displacement problem is a variation of the initial position problem. Given a fied position on the multibod sstem and a known finite displacement not infinitesimal for the input bodies or elements, the problem of finite displacements consists of finding the final position of the sstem s 7

44 remaining bodies. This problem will be easier to solve than initial position, mainl because one starts from a known position of the sstem, which can be used as a starting point for the iterative process needed for the solution of the resulting nonlinear equations. As seen before, dependent coordinates generate a set of algebraic constraint equation that represent the interdependenc between these coordinates. Defining the vector of generalized coordinates q as the vector that holds all the dependent coordinates, then, the solution of the Kinematic problem resides in obtaining the solution of: Φq,t=0 4.1 where Φ is the global Kinematic constraint vector, i.e., the vector that contains all the Kinematic Constraint equations of the sstem, in the homogeneous form. As there are several constraints that are used to define the inputs of the sstem in time Φ q,t=0 can eplicitl depend on t. It is customar to resort to the well-know Newton-Raphson method, which has quadratic convergence in the neighborhood of the solution and does not usuall cause serious difficulties if one starts with a good initial approimation, to solve eq The Newton-Raphson method is based on a linearization of eq. 4.1 and consists in replacing this sstem of equations with the first two terms of its epansion in a Talor series around a certain approimation q i of the desired solution: Φq i,t+ Φ q q-q i =0 4. where matri Φ q is the Jacobian matri of the constraint equations, that is to sa, the matri of partial derivatives of these equations with respect to the dependent coordinates. Eq. 4. must be solved iterativel until a solution is found or in other words, when q i+1 -q i = q i ε where previous equation represents the iterative procedure of the Newton-Raphson method, and ε is a user specified error tolerance Velocit and Acceleration analsis Velocit and Acceleration Analsis is much easier to solve than the position problems, mainl because it is linear and has a unique solution. This means in mathematical terms that it is modelled b a sstem of linear equations. Calculation of the velocities and accelerations of the sstem considers that if there can not eist violation of the associated kinematic constraint equations 4.1, then, the same must also holds true for the velocities and accelerations associated to these constraints, i.e: 8

45 Φ & q,q&,t = for the velocities, and Φ & q,q,q, & && t = for the accelerations. Epanding eq. 4.4 and 4.5 the following equations are obtained: Vector Φ qq& = Φ ν 4.6 q t Φ qq& = Φ& Φ& q& γ 4.7 q t Φ ν = Φt = is referred to as the rigid-hand-side of the velocit constraint equations. t Vector γ = ν& Φ& q& is referred to as the vector of the rigid-hand-side of the acceleration constraint q equation where q& is the vector of generalized dependent velocities, and q& & is the vector of generalized dependent accelerations. q 4.3. Dnamic Analsis Dnamic analses are much more complicated to solve than kinematic ones. The major characteristic about dnamic problems is that the involve the forces that act on the multibod sstem and its inertial characteristics namel its mass, inertia tensor and the position of its centre of gravit. There are essentiall two tpes of dnamic analsis: Inverse Dnamic Analsis and Forward Dnamic Analsis: The inverse dnamic problem aims at determining the motor of driving forces that produce a specific motion, as well as the reactions that appear at each one of the multibod sstem s joints. It is necessar to know the velocities and accelerations to be able to estimate the inertia forces which, together with the weight, the forces in the spring and dampers and all the other known eternal forces, will provide the basis to calculate the required actuating forces. The forward dnamic problem Dnamic Simulation ield the motion of a multibod sstem over a given time interval, as a consequence of the applied forces and given initial conditions. The importance of the direct dnamic problem lies in the fact that it allows one to simulate and predict the sstem s actual behaviour; as the motion is alwas the result of the forces that produce it. 9

46 Equations of Motion of a Multibod Sstem There are several was of obtaining the equations of motion of a multibod sstem. In this work one first considers the sstem of two unconstrained rigid bodies described in figure 30 a. i j i j Fig. 30. a Unconstrained sstem of rigid bodies, b constrained sstem of rigid bodies According to Newton-Euler equations, the equations of motion of such a sstem can be written as: M q& = g 4.8 where g is generalized eternal force vector and M is the mass matri containing the mass and inertial characteristics of each rigid bod. If a set of constraint bodies is considered instead, as depicted in figure 30b them the equation of motion will become: i Mq& + g = g 4.9 where an additional term g i is added representing the generalized internal force vector arising from the kinematic constraints imposed to the sstem. Using the Lagrange multiplier s method, these forces can be epressed as a function of the Jacobian matri that provides the direction of these forces and a vector λ of unknown Lagrange multipliers that describes their magnitude as: g i T = Φ λ 4.10 q Substituting eq in eq. 4.9, the resulting equation of motion becomes Mq& + Φ T q λ = g 4.11 Because λ and q& & are unknown, it is necessar to add equation 4.1 to solve this problem: 30

47 Φ q q& = γ 4.1 resulting the final epression of the sstem of the equations of motion of the model: T Mq&& + Φq λ = g Φqq&& = γ 4.13 In matri form, equation 4.13 becomes: M Φq T Φ q& q q = 0 λ γ 4.14 The equation of motion of the sstem 4.14 must be integrated in time in order to obtain the response of the model. This is done considering an Initial Value Problem IVP and the method proposed b Nikravesh, P., Mass Matri The form of mass matrices will undoubtedl depend on the tpe of coordinates chosen for the representation of multibod sstem. In this work, the use of Natural Coordinates lead to an elaborated deduction of this matri that remains outside the scope of this work. The interested reader is referred to the work of Jalón, J. and Bao, E., The mass matri of a general planar rigid bod, described with points, is given b: mg mg I i mg m 0 L ij Lij L ij Lij mg mg mg I i 0 m L ij Lij Lij Lij M = 4.15 mg I i mg I i 0 L ij Lij Lij Lij mg mg I i I i 0 L ij Lij Lij Lij where I i is polar moment of inertia with respect to the basic point i, m is mass of the element rigid bod, L ij is lengths of the bod, the point i., are the coordinates of the centres of masses with respect to G G 31

48 Eternal forces The application of eternal concentrated forces in a multibod formulation with Natural Coordinates requires a transformation of Coordinates that describes the concentrated force F, applied at point P with coordinates r p b an equivalent generalized force g e distributed b the points defining the rigid bod, given b: T g e = C F 4.16 where g e is the generalized force of the element, F is the eternal concerted force and C the transformation matri given b: 1 c1 c c1 c C = 4.17 c 1 c1 c c1 With c 1, c numerical coefficient calculate from the local coordinates of point P in the rigid bod s reference frame. The interested reader is referred to the work of Jalón, J. and Bao, E., 1994 for a more comprehensive description of the calculation of matri C. 3

49 Chapter 5 Multibod Model of the Hand and Rehabilitation Device As it was seen in Chapter 3, there are man pathologies that can affect the human hand: congenital defects, several tpes of hand injures including factures, tendon and ligament injures and hand diseases. All of these pathologies interfere directl with the hand function and movement as the produce damage in the neurological and musculo-skeletal structure of the hand. Hand rehabilitation can be accomplished using proper rehabilitation devices. These devices are able to improve hand function and movement, restoring in man cases its normal activit. There are alread several different tpes of hand rehabilitation devices as it can be seen net in this chapter. Each device serves a specific purpose on hand rehabilitation. In this chapter a computational model of a new rehabilitation device is constructed using the multibod formulation described in Chapter 4 with natural coordinates. This tpe of computational models are important in all phases of the design of a new product, but its application is particularl relevant in the earl design phase when no prototpe is et available Hand rehabilitation devices Although a search for hand rehabilitation devices reveals that there are alread available in the market several different tpes of models, with different function and purposes, it is also true that there is still a lot to develop in this field. Hand ma suffer from man pathologies, as seen briefl in a previous chapter, therefore several solutions are available. For eample, to the rehabilitation of the palm discs are being used fig. 31a or balls fig. 31b. The ball fig. 3a or the mesh fig. 3b are used for the rehabilitation of the fingers. To the rehabilitation of the wrist the powerball fig. 33 is one possible solution. 33

50 Fig. 31. a Discs, b The ball Fig. 3. a The ball, b Meshes Fig. 33. Powerball 34

51 5. Scope of Application of the Hand Rehabilitation Device Under Development The rehabilitation device that is now proposed is speciall aimed to help patients with partial paresis of the fingers acquired after a Cerebral Vascular Accident CVA, although more general applications can be considered. The device under development is presented in figure 34. There one can see the phsical prototpe made of wood and rope and its virtual counterpart made in the geometrical modeler Unigraphi 5.0. Fig. 34. a Phsical prototpe, b Virtual prototpe The device is conceived for active practices of the hand allowing the resistance relief of the rectifier muscles of the fingers. The resistive force is generated b a set of weights that are hanged in strings that attach to each finger. This weight ma var from person to person and from patholog to patholog. The attachment of the strings to the fingers is made using a simple velcro stripe. The device allows both hands to work although it can be used independentl for each hand. Hands are firml immobilized using proper hand immobilizer. The position of the vertical support can be tuned to meet patient s anthropometrics The Multibod Model Considering the biomechanical sstem to analse, represented in fig. 34a and composed b the hand, lower arm and rehabilitation device, the decision in terms of constructing a planar multibod model will natural coordinates capable of describing the phsical one, is to consider that the lower arm is irrelevant for the analsis and that it can be removed if the kinematics of the wrist were properl considered. Therefore the model includes the hand, starting at the metacarpus and finishing in the distal phalange of the indicate finger, as depicted on fig. 34a and 34b. The interaction with the rehabilitation device is onl considered in dnamic terms and is considered b introducing a constant 35

52 force with magnitude equal to the gravitational force produced b the weight and variable direction prescribed b the unit vector going from the attachment point of the hand to the attachment point in the device. P 6 = P 7 D = 8 t P 8 D IV 3 = θ4 t 4 = θ3 t 8 = 3 D 6 = θ1 t D = 1 8 t R 3 D D III 5 = θ t P 4 = P 5 R II R 1 P = P 3 I P 1 Fig. 35. Multibod model of the hand: 4 rigid bodies I to IV, eight points P1 to P8. Three revolute joints R1-R3 and 6 degree-of-freedom D1 to D Natural coordinates, kinematic constraints and degrees-of-freedom As it can be observed from fig. 35, the multibod model is described b four rigid bodies the metacarpal, the proimal phalange, the middle phalange, the distal phalange interconnected b 3 revolute joints. Considering that each rigid bod requires points to be properl described and that revolute joints are eplicitl described b independent points, then the model requires a total number of 8 points, i. e., 16 natural coordinates that are put together in vector: T {,,,,,,,,,,,,,, } q = , A closer analsis of the proposed model reveals that it possesses 6 degrees-of-freedom, respectivel the translations and 1 rotation of the wrist joint and the 3 rotations of the revolute joints. Therefore, the number of required kinematic constraints, required to epress the dependencies among the Natural Coordinates is equal to the difference between the number of coordinates and the number of the sstem s degrees of freedom, i. e., 16-6=10 kinematic constraints. 36

53 37 These kinematics constraints including the ones that drive the sstem are presented hereafter and epressed b algebraic equations with different natures: 4 kinematics constraint equations of rigid bod tpe that are used to preserve the constant length of rigid bodies I to IV; 6 kinematic constraint equations of revolute joint tpe that are used to describe the 3 revolute joints; and 6 kinematic constraint equations of kinematic drivers tpe that are used to full prescribe the motion of the sstem during the kinematics and dnamics analses preformed. In analtical terms, these equations are represented as: Rigid bod tpe constant length: Bod I: = + = = Φ L T r r 5. Bod II: = + = = Φ L T r r 5.3 Bod III: = + = = Φ L T r r 5.4 Bod IV: = + = = Φ L T r r 5.5 Where the inner product of vector was used to prescribe these relations between the coordinates. Revolute joint tpe: Joint R1: = = = Φ ,6 r r 5.6 Joint R: = = = Φ ,8 r r 5.7 Joint R3: = = = Φ ,10 r r 5.8 Kinematic drivers tpe: Drivers D1 and D translation of point 8: = = 0 0 * 8 8 * ,1 t t Φ 5.9

54 ,, Driver D3 angle of the wrist fig. 36, called θ 4 : Φ13 = r 78 X L78 sinθ 4 t = 7 8 L78 sinθ 4 t = , 7 θ 4 t 8, 8 X = 1,0 Fig. 36. The hand with the angle in the wrist Driver D4 angle between the metacarpal and the proimal phalange fig.37 called θ 3 : Φ 14 = r = 8 78 r 7 65 L 5 78 L 6 65 sinθ = L 78 L 56 sinθ t = , 6, P = * * 8 t, 8 t 8, 8 θ 3 t D = , Fig. 37. The hand with the angle θ 3 Driver D5 angle between proimal phalange and middle phalange fig. 38 called θ : Φ = r r L L sinθ = t = + L L sinθ t = 38

55 P = 6 6, 6, * * 8 t, 8 t D = 4 3 θ t 5, 5 4, 4 3 3, Fig. 38. The hand with the angle θ Driver D6 angle between the middle phalange and the distal phalange fig. 39 called θ 1 : Φ = r r L L sinθ 1 = t = + L L sinθ =, P 4 = 4, 4 D 6 = θ1 t P 3 = 3, 3 P =, Fig. 39. The hand with the angle θ 1 P 1 = 1, 1 39

56 40 The kinematic constraints presented before are grouped together in vector Φ, the vector of global kinematic constraint epressed as: 0 * sin * sin * sin * sin 0 1 * * = = t L L t L L t L L t L t t L L L L t θ θ θ θ Φq, 5.14 A closer analsis of vector Φq,t reveals that all constraints are either quadratic or linear in the coordinates, which represents an advantages from the computation point of view inherent to this tpe of formulation Jacobian matri As it was seen before, the Jacobian matri table 1 plas one of the most important roles in kinematic and dnamic analse of a multibod sstem. B definition the Jacobian matri is the partial derivative of the vector of global kinematic constraints in order to the generalise coordinates vector. In the case of Natural Coordinates, and considering that the kinematic constraints are either quadratic or linear, their contribution to this matri will be linear or constant in the coordinates, respectivel. For the case of the multibod model of the hand, the Jacobian matri assumes the following form:

57 Table 1 Jacobian Matri 41

58 Contributions to the ν and γ vectors General epressions for vector υ and γ were provided in Chapter 4. These vector present the righthand side of the velocit and acceleration constraint equations respectivel. The onl non-zero entries of the vector ν are the ones associated with the drivers constraints as these are the onl ones eplicitl depending on time. Therefore the contributions of the kinematic constraints for the computational multibod model of the hand are respectivel: = cos cos cos cos * 8 * 8 t t L L t t L L t t L L t t L t t θ θ θ θ θ θ θ θ & & & & & & ν 5.15 The calculation of the contributions for vector γ are more elaborated as the represent a nd derivative, although also because of this motive, the contributions of linear constraints are null. For the computational model of the hand, the contributions for γ vector are respectivel:

59 43 [ ] [ ] [ ] [ ] = cos sin cos sin cos sin cos sin * 8 * t t t t L L t t t t L L t t t t L L t t t t L t t & & & & & & & & && & & & & & & & & & && & & & & & & & & & && & && & && && & & & & & & & & & & & & & & & & θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ γ 5.5. Kinematic Data Collection In order to analze the movement of the hand a set of 4 fleion etension hand movements were collected using the BTS Bod Training Sstems sstem with si cameras in the Laborator of Biomechanics of the Silesian Universit of Technolog in Poland. The movement acquisition consisted in recording the coordinates of a set of 7 markers 6 in the lower arm and hand and 1 in the rehabilitation device as depicted in fig. 40. The subject was a health oung female with 3 ears old and with no hand pathologies. The measured lengths of the subject s hand were ver similar with the ones found when averaging the reconstructed values obtained from the BTS sstem and therefore these average lengths were use instead of the measured ones with the purpose of comparing the kinematics obtained with the driven multibod model with the kinematics obtained directl from the movement laborator. In table the average lengths are provided, together with the mass and inertial characteristics which are introduced in the net subsection. 5.16

60 Fig. 40. Markers on the hand The markers were recorded with a sampling frequenc of 10Hz and the 3D coordinates of the points reconstructed automaticall b the software package of the BTS sstem. The four fleionetension movements of the hand had a duration of 11. sec. Sstem s drivers, respectivel the coordinates of the wrist point and the interphalangeal angles θ 1 to θ 4, were calculated from this kinematic data using an ecel worksheet. The results obtained, especiall for the angles were ver nois and therefore a filtering procedure was required. In this work a Butterworth nd order filter was used to filter the kinematic drivers Winter, D., The filter applies the double-pass technique to avoid the phase lag introduced b this tpe of filters. Residual Analsis was applied for an estimation of the cut off frequencies Winter, D., 1990 but the results suggested b this method failed to produce good results due to the still high levels of noise in the kinematic data. Therefore, cut-off frequencies of 1.5.3Hz were obtained b trial and error methods and successfull applied in the filtering of the kinematic data Masses, Inertias and assembl of the Global Mass Matri The dnamic analsis of the hand requires the prescription of the hand masses and inertias to the dnamic multibod program. The hand is a ver comple sstem of bones, soft tissues and joints and therefore it is not eas to obtain, or even estimate, those values. A search in the available literature also confirm the scarcit of such data, although the values for the masses could be obtained form Birukova, V. and Yourovskaa, 1994.These values are 44

61 presented in Table together with the values of the length, radius and polar inertias of the 4 rigid bodies of the model. It was consider that the total mass of the hand was 1 kg. Table. Parameters included in model Bod Mass [kg] Length [m] Radius [m] Inertias [kg m ] I II III IV The masses presented in previous table include the masses of the corresponding bones plus the masses of all adjoining muscles Birukova, V., and Yourovskaa, The polar inertias were calculated using the approimation to a solid clinder given the epression I = ml / 3+ mr / 4. The center of mass of each rigid bod was considered to be located in the middle of each rigid segment. The global mass matri of the multibod sstem results of the assembl of the four elementar mass matrices, provided in Chapter 4, for each one of the rigid bodies. Since there are no sharing of points, because the revolute joints are eplicitl define, the global mass matri is band diagonal and uncoupled: 45

62 Table 3. Mass matri 46

63 5.7. Kinetic Data Kinetic data refers to the values of eternal forces and moments applied upon the sstem. In this case, since the movement occurs in a plane perpendicular to the direction of the earth gravitational field, the gravitational force is not applied. Therefore according to fig. 41 the onl eternal force acting on the sstem is the one applied b the string and corresponding weight. The magnitude of this force is alwas constant and equal to F=m w g and its direction is coincident with the direction followed b the string. The origin of the string attached to the finger is moving and its insertion at the device is fied. The force is applied to the model following the procedure described in Chapter 4. Fig. 41. Force generate b the finger 5.8. Results of the Kinematic and Dnamic Analsis of the Multibod Model of the Hand The model described before was implemented in two Matlab programs, following the formulation presented in Chapter 4. The first Matlab program HAND_Kinematics.m deals onl with the Kinematic aspect of the moment under analsis and is able to calculate the position, velocit and acceleration of the sstem in 47

64 an instant of the analsis. This program was implemented in the initial part of this work and its source code can be found in Anne 4. Since it is onl a Kinematic program, it does not use masses, inertias and eternal forces. For a complete dnamic analsis a second Matlab program was implement HAND_Dnamic.m that besides the calculation of the positions, velocities and accelerations of the sstem, it also allows the calculation of the reaction forces at the joints and the articular moments of force that muscles need to develop around each joint in order for the prescribed motion to occur. The source code of this program can be found in Anne 5. In fig. 4a the initial position of the hand with respect to the rehabilitation device is presented. This figure was obtained from the Matlab program and respects the initial animation frame. Also a film sequence is presented in fig. 4b regarding one fleion-etension ccle. a b Fig. 4. a Initial position of the hand, b Kinematic sequence of animation of the fleion-etension movement 48

65 Analsis Indicators One wa to verif if the analsis is working properl is to calculate the norm of vectors Φ, Φ & and Φ & at each time step. If constraints are being respected and if the Baumgarte stabilization is working, properl, then these values should be close to zero. From the results presented in fig. 43, it is clear that from the kinematics point of view, the conditions imposed b the constraint equations are being respected as the values of the norms of Φ, Φ & and Φ & are alwas smaller than , and respectivel. Values in abscissa represent frame numbers. Fig. 43. Analsis statistics Drivers of the Multibod Model In figure 44 the values that were calculated from the kinematic data for the drivers of the sstem are presented. The filtered values of the driver are interpolated using cubic splines and their respective velocities and accelerations calculated b cubic spline differentiation. 49

66 Fig. 44. Kinematic drivers for the analsis: angles, angular velocit and angular acceleration of each joint The results clearl show the four fleion-etension ccles of the hand. It also can be seen that drivers far awa from the wrist joint present a more nois behavior, indicating that lower-cut-off frequencies could have been used. The results in terms of drivers, velocities and accelerations also show that these quantities are consistent with drivers positions and velocities respectivel. The Kinematics analsis of the drivers also show that maimum fleion-etension angular velocities oscillate between 0.5 rad/s for the wrist joint to 4 rad/s for the metacarpal-phalangeal joint. Values of 1 rad/s to rad/s can be found for the maimum angular velocities of the top and middle interphalangeal joints, respectivel. 50

67 Kinematic Analsis In this section the results of the kinematic analsis of the hand biomechanical model are presented. In figures 45-5 the positions, velocities and accelerations of each of the 8 points of the model are presented. From the analsis of the results it can be seen that in terms of positions, these are consistent with the kinematic data obtained from the laborator indicating that the drivers are working properl. Moreover, it can also be observed that the velocities are compatible with positions, and accelerations with velocities. The values for the velocities and accelerations are within the epected phsiological range for a normal hand. In absolute values velocities never eceed 0. m/s for an of the points independentl of the hand being fleing or etending. Accelerations are still a bit nois especiall for points further awa from the wrist, indicating that probabl smaller cut-off frequencies could have been used. However, in absolute values, accelerations never surpass.0 m/s and in man cases the are bellow 0.5 m/s. These results are within the epected range accelerations for a hand with no pathologies. The comparison of the kinematics of the two points of a revolute joint reveals that the are equal, meaning that the joint kinematic constraints are working properl. Fig. 45. Kinematics for point 1 51

68 Fig. 46. Kinematics for point Fig. 47. Kinematics for point 3 5

69 Fig. 48. Kinematics for point 4 Fig. 49. Kinematics for point 5 53

70 Fig. 50. Kinematics for point 6 Fig. 51. Kinematics for point 7 54

71 Fig. 5. Kinematic for point Dnamic Analsis In this section the dnamic related results are presented. These refer to the reaction forces at the joints presented in fig. 53 and the articular moments of force at the joints fig. 54. A closer observation of the first results show that reactions in revolute joint R1 are practicall null. That was to be epected since no gravit is considered in this planar movement and that there is no eternal forces applied to bod I. The small variations observer in the reaction force represent the inertial components due to the dnamics of the bod. In the revolute joints R and R3 and also at the wrist attachment point one can see that the reaction forces are following the direction and the magnitude of the eternal applied force. This results was epected since there are no gravitational component present and inertia forces are small since accelerations are also small, and proves that the model is working probabl in terms of the calculation of the joint reaction forces. 55

72 Fig. 53. Reactions of the hand In figure 54 the articular moments generated at the joints during the movement are presented. This is probabl the most relevant result of this stud as it allows the analsis of the performance of the hand rehabilitation device. The first obvious observation is that the moments at the joint R1 is almost zero, since no forces are applied to bod I. Then, observing the other graphics, it is clear that moments have similar behavior in all joints but with a tendenc to grow as one approimates the wrist joint. That is understandable since as we are moving awa from the point of application of the force the arm of the moment increases. Another result that can be observed and that is important for the analsis of the efficienc of the rehabilitation device is that all moment of force are positive higher values correspond to maimum etension, lower values to maimum fleion. This means that even when fleing, the hand is tring to sustain the weight load and therefore an eccentric contraction of the hand etensors is occurring instead of a contraction of the hand fleors. If an fleor activit is present it is in co-contraction with the hand etensor, with the purpose of stabilizing the movement. 56

73 Previous sentence moreover can onl be corroborated if a muscular analsis is preformed, in order to calculate the redundant muscle forces, which, although important, remains outside the scope of this work. Regarding the calculation of the redundant muscles forces the interested reader is referred to the work of Silva, et. al., 00a, Silva, et. al., 00b. Fig. 54. Moments in each joint 57

74

75 Chapter 6 Conclusion and Future Developments 6.1. Conclusions In this work a computational model for the kinematic and dnamic analsis of D hand motions was presented using a multibod formulation with natural coordinates. In the framework of the proposed methodolog, two problems have been analzed: kinematic problems to stud the movement of the hand, and dnamic problems to stud internal forces and articular moments. The procedures presented use reliable and accurate mathematical methodologies, that have the remarkable advantage of being able to obtain, without using intrusive techniques, qualitative and quantitative results regarding the human hand structure and motion. A comprehensive mathematical model of the human hand was assembled and thoroughl described, using a general-purpose multibod formulation in which the position and orientation of the human hand is represented using natural or full Cartesian coordinates. This formulation, was chosen among other tpes of coordinates due to the advantages it presents in the description of biomechanical sstems. It was observed that since rigid segments of the human hand are modelled as a collection of points, it becomes ver convenient, in biomechanical applications, to use the location of important anatomical landmarks, such as joints and etremities, directl in the construction of the rigid bodies describing the anatomical segments of the biomechanical model. Two model-specific gross-motion analsis programs were developed in the MATLAB environment, using the described multibod formulation with full Cartesian coordinates. These numerical tools were the computational support for all the kinematic and dnamic analses performed in this work. 59

76 It was seen that the model was able to correctl characterize the principal phsical characteristics and anthropometric dimensions of the human hand. The investigation was done on a health hand to which a weight was attached to the middle phalange to produce an eternal force during the motion eecuted in the prototpe device. Although, during the investigation the wrist was supposed to be fied, the acquired movement revealed a displacement of the wrist of about cm. This displacement was properl included in the model using a moving attachment point in the wrist joint. The kinematics were obtained in the movement laborator and properl filtered using a nd Order Butterworth low-pass filter to remove ecessive noise levels. The input kinematics for the kinematic model proved to be consistent with the eperimental data and correct phsiological values were obtained for the angles and angular velocities at the joints. It was seen that the reactions at the joints have a constant magnitude equal to the force produced the weight attached to the finger but changing direction, fact that directl interferes with the produced moment at the articulations as the moment arm changes accordingl. Regarding the articular moments, it was conclude that during the 4 fleion-etension ccles, these moments were alwas positive, meaning that, with the actual design, onl hand etensors are rehabilitated. As a final conclusion, it can be stated that the model developed in this work was able to analse the actual design of the rehabilitation device and can be easil adapted to analse other design alternatives. 6.. Future Developments The model proposed is robust but simple. This means that there are other important features that can be implement in the future. For the analsis of the rehabilitation of the hand it would be important to analse the activation patterns and forces produced b the muscles to rehabilitate. This feature was not included in the present work as it would require the implementation of optimization tools with which the redundant force sharing problem could be solved. Also it would be important to include more fingers in the model and etend it to 3D to take full advantages of the data collected b the BTS Sstem. With such a model, it would also be important to compare the results it produces with muscular Electromograph EMG patterns measured in the principal muscle groups. 60

77 References Birukova, V. and Yourovskaa, Z., A model of human hands dnamics, from Schuind F, Coorne III W.P., Garcia-Elias M: Advances in the Biomechanics of the Hand and Wrist, New York, NATO ASI Series, Chen, F., Lo, S., Meng, N., Lin, C., Chou, L.,: Effects of wrist position and contraction on wrist fleors H-refle, and its functional implications., Journal of Electromograph and Kinesiolog, 16, 006. Chiba,S., Yonekura,K., Nonaka, M., Imai, M., Matumoto, H., and Wada, T., "Advanced Hiraama Disease with Successful Improvement of Activities of Dail Living b Operative Reconstruction" vol. 5,no1, 004. Coburn, J., Upal, M., Crisco, J., Coordinate sstems for the carpal bones of the wrist, Journal of Biomechanics, 40, 007. Ettema, A., An, K., Zhao, C., O Brne, M., Amadio, P., Fleor tendor and snovial gliding during tunnel sndrome, Journal of Biomechanics, 41, 008. Feltham, M., Dieen, J., Coppieters, M., Hodges, P., Changes in joint stabilit with muscle contraction measured from transmission of mechanical vibration, Journal of Biomechanics, 39, 006. Gielo-Perczak, K., Biomechanical analsis of the wrist joint in ergonomical aspect, from Schuind F, Coorne III W.P, Garcia-Elias M: Advances in the Biomechanics of the Hand and Wrist, New York, NATO ASI Series, Halaki, M., O Dwer, N., Cathers, I.,: Sstematic nonlinear relations between displacement amplitude and joint mechanics at the human wrist, Journal of Biomechanics, 39, 006. Haug, E.,: Computer Aided Kinematics and Dnamics of Mechanical Sstems, Alln and Bacon, Boston, Jalón, J. and Bao, E., Kinematic and dnamic simulation of multibod sstems : the real-time challenge, Springer-Verlag, New York ; Hong Kong, Kuo, P., Lee, l., Jindrich, D., Dennerlein, J., Finger joint coordination during tapping, Journal of Biomechanics, 39,

78 Nelson, D., Margaret, A., Mitchell, P., Groszewski, G., Pennick, S., Manske, P., Wrist range of motion in activities of dail living, from Schuind F, Coorne III W.P, Garcia-Elias M: Advances in the Biomechanics of the Hand and Wrist, New York, NATO ASI Series, Nikravesh, P., Computer-aided analsis of mechanical sstems, Prentice-Hall, Englewood Cliffs, N.J., Seele R., Stephens T., Tate P., Anatom & Phsiolog. McGraw Hill, 006. Silva, M. and Ambrósio, J., "Kinematic Data Consistenc in the Inverse Dnamic Analsis of Biomechanical Sstems", Multibod Sstem Dnamics, 8, 19-39, 00a. Silva, M. and Ambrósio, J., "Solutions of Redundant Muscle Forces in Human Locomotion with Multibod Dnamics and Optimization Tools", Journal of Mechanics Based Design of Structures and Machines, 313, , 00b. Tortora, G., Grabowski, S. R., Introduction to the Human Bod, J. Wile, NY, 001. Valero-Cuevas, J:, An integrative approach to the biomechanical function and neuromuscular control of the fingers, Journal of Biomechanics, 38, 005 Vigourou, L., Quaine, F., Labarre-Vila, A., Moutet, F:, Estimation of finger muscle tendon tensions and pulle forces during specific sport-climbing grip techniques, Journal of Biomechanics, 39, 006. Winter, D.,: Biomechanics and motor control of human movement, Wile, New York, Wu, G., Helm, F., Veeger, D., Makhsous, M., Ro, P., Anglin, C., Nagels, J., Karduna, A., McQuade, K., Werner, W., Buchholz, B.,: ISB recommendation on definitions of joints coordinate sstems of various joints for the reporting of human joint motion Part II: shoulder, elbow, wrist and hand, Journal of Biomechanics, 38, 005. Relevant Internet References : ACPOC, June 008 CHARMS, DofOS, September, 003 HHA, UNSW Embrolog,

79 Anne 1 Sizes and masses of hand links 63

80

81 Fig. 55. Model of the hand Table 4. Sizes and masses of links in the model Birukova, V. and Yourovskaa, Z., 1994 Link No Link length [m] Masses [kg]

82

83 Anne Maimum values of wrist joint capacit 67

84

85 Table 5. Maimum values of wrist joint capacit with the AUR profiles 1-4 during performance of activities I-X. Chen, F., Lo, S., Meng, N., Lin, C., Chou, L., 007 Value for each activit [Nm] No. of profiles I II III IV V VI VI VIII IX X Fig. 56. The tpical of activities performed at the assembl line in lamp factor stud 69

86

87 Anne 3 Wrist range of motion activities of dail living 71

88

89 Table 6. Wrist Range of Motion in Activities of Dail Living Nelson, D., Margaret, A., Mitchell, P., 1994 Radial [deg] Ulnar[deg] Fleion [deg] Etension [deg] Activit Ma Min Avg ma min avg ma min avg ma min avg Comb hair Tailbone Washcloth Button Tie shoe Use knife Use fork Spatula Drink cup Pitcher Can opener Stir bowl Open jar Turn faucet Turn page Write name Use phone Dial O Tpewriter Steer wheel Door knob Turn ke Screwdriver Throw ball

90

91 Anne 4 Hand Kinematics Matlab Program Source 75

92

93 HAND_kinematics.m % Begin % Program to analze the kinematic of the hand % This is D % Date: % Variable declaration clear all; clc; close all; nc = 16; nh = 10; nd = 6; nh_total = nh+nd; q = zerosnc,1; q_dot = zerosnc,1; q_dot = zerosnc,1; Phi = zerosnh_total,1; Phi_q = zerosnh_total,nc; niu=zerosnh_total,1; gamma=zerosnh_total,1; % Model's data L1=0.05;%m L=0.017;%m L3=0.035;%m; L4=0.085;%m; L=[L1 L L3 L4]; erroruser=1*10^-5; %Analsis data: t0 = 0.0 dt = 1/10; t_end = 11.4; load drivers.tt; n=sizedrivers,1; %Begin Kinematic Analsis t = t0; q=[ ]'; while t<=t_end % Position analsis with Newton-Raphson method error=erroruser; while error>=erroruser [Phi,Phi_q, niu,gamma]=function_evaluationt,q,q_dot,phi,phi_q,l,drivers,niu,gamma,nc,nh_total, nd; delta_q=-phi_q\phi; error=sqrtdelta_q' *delta_q; q=q+delta_q; end % Velocit analsis q_dot=phi_q\niu; 77

94 end % Acceleration analsis q_dot=phi_q\gamma; % Animate results newplot; ais [ ]; ais equal; ais manual; for j=1:nc/ j,1 = q1+*j-1,1; j,1 = q+*j-1,1; end plot,; clear,; pause0.1; hold off; % Increment time variable t=t+dt 78

95 Function_Evaluation.m function [Phi, Phi_q, niu, gamma, drv_t]=function_evaluationt,q,q_dot,phi,phi_q,l,drivers,niu,gamma,nc,nh_total,nd % Getting data from drivers - building the piecewise polinomials q15_pp = spline drivers:,,drivers:,3; dq15dt_pp = mmspderq15_pp; ddq15dt_pp = mmspderdq15dt_pp; q16_pp dq16dt_pp ddq16dt_pp = spline drivers:,,drivers:,4; = mmspderq16_pp; = mmspderdq16dt_pp; theta1_pp = spline drivers:,,drivers:,5; dtheta1dt_pp = mmspdertheta1_pp; ddtheta1dt_pp = mmspderdtheta1dt_pp; theta_pp = spline drivers:,,drivers:,6; dthetadt_pp = mmspdertheta_pp; ddthetadt_pp = mmspderdthetadt_pp; theta3_pp = spline drivers:,,drivers:,7; dtheta3dt_pp = mmspdertheta3_pp; ddtheta3dt_pp = mmspderdtheta3dt_pp; theta4_pp = spline drivers:,,drivers:,8; dtheta4dt_pp = mmspdertheta4_pp; ddtheta4dt_pp = mmspderdtheta4dt_pp; % Evaluation of the spline for time t q15_data = ppvalt,q15_pp; q16_data = ppvalt,q16_pp; theta1_data = ppvalt,theta1_pp; theta_data = ppvalt,theta_pp; theta3_data = ppvalt,theta3_pp; theta4_data = ppvalt,theta4_pp; drv_t1,1 = q15_data; drv_t,1 = q16_data; drv_t3,1 = theta1_data; drv_t4,1 = theta_data; drv_t5,1 = theta3_data; drv_t6,1 = theta4_data; % Drivers velocities for time t dq15dt_data = ppvalt,dq15dt_pp; dq16dt_data = ppvalt,dq16dt_pp; dtheta1dt_data = ppvalt,dtheta1dt_pp; dthetadt_data = ppvalt,dthetadt_pp; dtheta3dt_data = ppvalt,dtheta3dt_pp; dtheta4dt_data = ppvalt,dtheta4dt_pp; drv_t7,1 = dq15dt_data; drv_t8,1 = dq16dt_data; drv_t9,1 = dtheta1dt_data; drv_t10,1 = dthetadt_data; drv_t11,1 = dtheta3dt_data; drv_t1,1 = dtheta4dt_data; % Drivers accelerations for time t ddq15dt_data = ppvalt,ddq15dt_pp; ddq16dt_data = ppvalt,ddq16dt_pp; 79

96 ddtheta1dt_data = ppvalt,ddtheta1dt_pp; ddthetadt_data = ppvalt,ddthetadt_pp; ddtheta3dt_data = ppvalt,ddtheta3dt_pp; ddtheta4dt_data = ppvalt,ddtheta4dt_pp; drv_t13,1 = ddq15dt_data; drv_t14,1 = ddq16dt_data; drv_t15,1 = ddtheta1dt_data; drv_t16,1 = ddthetadt_data; drv_t17,1 = ddtheta3dt_data; drv_t18,1 = ddtheta4dt_data; % Kinematic constraints evaluation Phi1 = q3-q1^+q4-q^-l1^; Phi = q7-q5^+q8-q6^-l^; Phi3 = q11-q9^+q1-q10^-l3^; Phi4 = q15-q13^+q16-q14^-l4^; Phi5 = q5-q3; Phi6 = q6-q4; Phi7 = q9-q7; Phi8 = q10-q8; Phi9 = q13-q11; Phi10 = q14-q1; Phi11 = q15-q15_data; Phi1 = q16-q16_data; Phi13 = q14-q16-l4*sintheta4_data; Phi14 = -q16-q14*q9-q11+q15-q13*q10-q1- L4*L3*sintheta3_data; Phi15 = -q1-q10*q5-q7+q11-q9*q6-q8- L3*L*sintheta_data; Phi16 = -q8-q6*q1-q3+q7-q5*q-q4- L*L1*sintheta1_data; % Jacobian matri evaluation: Phi_q=zerosnh_total,nc; Phi_q1,1 = -*q3-q1; Phi_q1, = -*q4-q; Phi_q1,3 = *q3-q1; Phi_q1,4 = *q4-q; Phi_q,5 = -*q7-q5; Phi_q,6 = -*q8-q6; Phi_q,7 = *q7-q5; Phi_q,8 = *q8-q6; Phi_q3,9 = -*q11-q9; Phi_q3,10 = -*q1-q10; Phi_q3,11 = *q11-q9; Phi_q3,1 = *q1-q10; Phi_q4,13 = -*q15-q13; Phi_q4,14 = -*q16-q14; Phi_q4,15 = *q15-q13; Phi_q4,16 = *q16-q14; Phi_q5,3 = -1; Phi_q5,5 = 1; Phi_q6,4 = -1; Phi_q6,6 = 1; Phi_q7,7 = -1; Phi_q7,9 = 1; 80

97 Phi_q8,8 = -1; Phi_q8,10 = 1; Phi_q9,11 = -1; Phi_q9,13 = 1; Phi_q10,1 = -1; Phi_q10,14 = 1; Phi_q11,15 = 1; Phi_q1,16 = 1; Phi_q13,14 = 1; Phi_q13,16 = -1; Phi_q14,9 = -q16-q14; Phi_q14,10 = q15-q13; Phi_q14,11 = q16-q14; Phi_q14,1 = -q15-q13; Phi_q14,13 = -q10-q1; Phi_q14,14 = q9-q11; Phi_q14,15 = q10-q1; Phi_q14,16 = -q9-q11; Phi_q15,5 = -q1-q10; Phi_q15,6 = q11-q9; Phi_q15,7 = q1-q10; Phi_q15,8 = -q11-q9; Phi_q15,9 = -q6-q8; Phi_q15,10 = q5-q7; Phi_q15,11 = q6-q8; Phi_q15,1 = -q5-q7; Phi_q16,1 = -q8-q6; Phi_q16, = q7-q5; Phi_q16,3 = q8-q6; Phi_q16,4 = -q7-q5; Phi_q16,5 = -q-q4; Phi_q16,6 = q1-q3; Phi_q16,7 = q-q4; Phi_q16,8 = -q1-q3; % Evaluation of vector niu: niu1 = 0; niu = 0; niu3 = 0; niu4 = 0; niu5 = 0; niu6 = 0; niu7 = 0; niu8 = 0; niu9 = 0; niu10 = 0; niu11 = dq15dt_data; niu1 = dq16dt_data; niu13 = L4*costheta4_data*dtheta4dt_data; niu14 = L4*L3*costheta3_data*dtheta3dt_data; niu15 = L3*L*costheta_data*dthetadt_data; niu16 = L*L1*costheta1_data*dtheta1dt_data; % Evaluation of vector gamma: gamma1 = -*q_dot3-q_dot1^+q_dot4-q_dot^; gamma = -*q_dot7-q_dot5^+q_dot8-q_dot6^; 81

98 gamma3 = -*q_dot11-q_dot9^+q_dot1-q_dot10^; gamma4 = -*q_dot15-q_dot13^+q_dot16-q_dot14^; gamma5 = 0; gamma6 = 0; gamma7 = 0; gamma8 = 0; gamma9 = 0; gamma10 = 0; gamma11 = ddq15dt_data; gamma1 = ddq16dt_data; gamma13 = L4*- sintheta4_data*dtheta4dt_data^+costheta4_data*ddtheta4dt_data; gamma14 = L4*L3*- sintheta3_data*dtheta3dt_data^+costheta3_data*ddtheta3dt_data-*- q_dot16-q_dot14*q_dot9-q_dot11+q_dot15-q_dot13*q_dot10- q_dot1; gamma15 = L3*L*- sintheta_data*dthetadt_data^+costheta_data*ddthetadt_data-*- q_dot1-q_dot10*q_dot5-q_dot7+q_dot11-q_dot9*q_dot6- q_dot8; gamma16 = L*L1*- sintheta1_data*dtheta1dt_data^+costheta1_data*ddtheta1dt_data-*-q_dot8- q_dot6*q_dot1-q_dot3+q_dot7-q_dot5*q_dot-q_dot4; 8

99 Anne 5 Hand dnamics Matlab Program Source 83

100

101 HAND_Dnamics.m % Begin % Program to analize the dnamics of the hand % This is D % Date: April-June 008 % IST % Clears workspace clear all; close all; % % Model's description % nc = 16; % Number of natural coordinates nh = 10; % Number of holonomic kinematic constraints nd = 6; % Number of degrees of freedom nh_total = nh+nd; % Total number of constraint equations including drivers % Lengths - average values form the kinematics L1 = ;%m L = ;%m L3 = ;%m; L4 = ;%m; L_data = [L1 L L3 L4]; % Bone Radius r1 = ; %m r = ; %m r3 = 0.010; %m r4 = 0.015; %m % Mass of each bod M1 = 0.007; M = 0.014; M3 = 0.030; M4 = 0.66; M_data = [M1 M M3 M4]; %Moment of inertia clinder approach: J=m*l^/3+m*r^/4 J1_G = M1*L1^/3+M1*r1^/4; J_G = M*L^/3+M*r1^/4; J3_G = M3*L3^/3+M3*r1^/4; J4_G = M4*L4^/3+M4*r1^/4; J_G_data = [J1_G J_G J3_G J4_G]; % Local coordinates of the centre of mass of each bod rg1:,1 = [L1/ 0.0]'; rg1:, = [L/ 0.0]'; rg1:,3 = [L3/ 0.0]'; rg1:,4 = [L4/ 0.0]'; % Eternal force data r_mass = [0.*9.81 ]'; % weight of 0. kg applied over bod r_force = [0.1*L 0]'; % local coordinates of point of application of the force on bod %Fied points coordinates - for animation purposes onl 85

102 r_link = [ ]'; % coordinates of a point located in the lower arm r_anchor = [ ]'; % coordinates of the anchor point on the device %Analsis data: t0 = 0.0; dt = 1/30; t_end = 11.19; Baum_data = [ ]; % These are the values of alpha and beta of the Baumegarte's stabilization procedure load drivers.tt; n=sizedrivers,1; %Begin Dnamic Analsis - Defines IVP initial step q = [ ]'; q_dot = [ ]'; Y_0 = [q; q_dot]; % Solves the IVP using a Direct Intergration Procedure t_stat = cputime; [t, Y] = ode45@solve_eofm_ode45, [t0 t_end], Y_0, [], L_data, M_data, J_G_data, rg, Baum_data, drivers, r_mass, r_force,r_anchor, nc, nh_total, nd; % Get values from variables i_rep = 1; w_rep = 6; t_report = t0; for i = 1:sizet if ti >= t_report [q_dot_au, lambda_au, data_au] = Solve_EofMti, Yi,1:nc', Yi,nc+1:*nc', L_data, M_data, J_G_data, rg, Baum_data, drivers, r_mass, r_force,r_anchor, nc, nh_total, nd; t_repi_rep = ti; q_repi_rep,1:nc = Yi,1:nc; q_dot_repi_rep,1:nc = Yi,nc+1:*nc; q_dot_repi_rep,1:nc = q_dot_au'; % Etracts reaction and internal forces lambda_repi_rep,1:nh_total-6 = lambda_au1:nh_total-6,1'; lambda_repi_rep,nh_total-5 = -lambda_aunh_total-5,1; lambda_repi_rep,nh_total-4 = -lambda_aunh_total-4,1; % Calculates articular moments lambda_repi_rep,nh_total-3 = -lambda_aunh_total- 3,1*L_data4*cosdata_au10,1; lambda_repi_rep,nh_total- = -lambda_aunh_total-,1*l_data4*l_data3*cosdata_au9,1; lambda_repi_rep,nh_total-1 = -lambda_aunh_total- 1,1*L_data3*L_data*cosdata_au8,1; lambda_repi_rep,nh_total = -lambda_aunh_total- 0,1*L_data*L_data1*cosdata_au7,1; data_repi_rep,1:w_rep = data_au'; i_rep = i_rep + 1; t_report = t_report + dt; end 86

103 end % Get the values for the LAB kinematics load LabKinematics.tt; for i = 1:i_rep-1 j=1; while LabKinematicsj,<=t_repi j=j+1; end for k = 1:nc if LabKinematicsj, == t_repi q_kini,k = LabKinematicsj-1,+k; else q_kini,k = LabKinematicsj-1,+k + LabKinematicsj,+k- LabKinematicsj-1,+k*t_repi-LabKinematicsj-1,/LabKinematicsj,- LabKinematicsj-1,; end end end t_stat = cputime - t_stat/60; disp'analsis done!'; disp'see STATISTICS bellow:'; Iterations = sizet,1 Elapsed_CPU_time = t_stat disp'press ENTER to begin animation of results'; pause % Animate results fig = figure; setfig,'doublebuffer','on'; setgca,'lim',[-80 80],'lim',[-80 80],'NetPlot','replace','Visible','off'; mov = avifile'dnamicanalsisarticular moments.avi'; for i = 111: 180; 1:sizet_rep' newplot; title[t_repi,i]; ais [ ]; ais equal; ais manual; % Plot the LAB Kinematics for j=1:nc/ j,1 = q_kini,1+*j-1; j,1 = q_kini,+*j-1; end plot,,'-o','color',[ ],'LineWidth',1,'MarkerEdgeColor',[ ],'MarkerFaceColor','w'; clear ; % % Plot the string and force vector 1,1 = r_anchor1,1; 1,1 = r_anchor,1;,1 = data_repi,5;,1 = data_repi,6; 3,1 = data_repi, *data_repi,3; 3,1 = data_repi, *data_repi,4; 87

104 plot1:,1,1:,1,'- ','Color','k','LineWidth',1,'MarkerEdgeColor','k','MarkerFaceColor','k'; plot:3,1,:3,1,'- <','Color','c','LineWidth',,'MarkerEdgeColor','c','MarkerFaceColor','c'; % Plot the hand and arm for j = 1:nc/ j+3,1 = q_repi,1+*j-1; j+3,1 = q_repi,+*j-1; end 1,1 = r_link1,1; 1,1 = r_link,1; plot4:11,1,4:11,1,'- o','color','b','linewidth',3,'markeredgecolor','r','markerfacecolor','r'; plot11:1,1,11:1,1,':s','color','b','linewidth',3,'markeredgecolor','r','ma rkerfacecolor','r'; % % Plot velocities % for j = 1:nc/ % j+1,1 = q_repi,1+*j-1 + q_dot_repi,1+*j-1; % j+1,1 = q_repi,+*j-1 + q_dot_repi,+*j-1; % _au = [3+j,1 1+j,1]'; % _au = [3+j,1 1+j,1]'; % plot_au,_au,'-- <','Color','','LineWidth',1,'MarkerEdgeColor','','MarkerFaceColor',''; % end % % % Plot accelerations % for j = 1:nc/ % j+0,1 = j+1,1 + q_dot_repi,1+*j-1; % j+0,1 = j+1,1 + q_dot_repi,+*j-1; % _au = [1+j,1 0+j,1]'; % _au = [1+j,1 0+j,1]'; % plot_au,_au,'-- <','Color','g','LineWidth',1,'MarkerEdgeColor','g','MarkerFaceColor','g'; % end % % Plot joint reaction forces % clear ; % for j = 1:4 % _au = [q_repi,1+**j-1 q_repi,1+**j *lambda_repi,5+*j-1]'; % _au = [q_repi,+**j-1 q_repi,+**j *lambda_repi,6+*j-1]'; % plot_au,_au,'-- >','Color','c','LineWidth',1,'MarkerEdgeColor','c','MarkerFaceColor','c'; % end % % Plot articular moments of force clear ; for j = 1:4 _au = [q_repi,1+**j-1 q_repi,1+**j-1]'; _au = [q_repi,+**j-1 q_repi,+**j-1]'; M_rep = round50.0*lambda_repi,17-j; if M_rep >= 0 M_rep = M_rep + 1; 88

105 plot_au,_au,'- o','color','g','linewidth',1,'markeredgecolor','g','markerfacecolor','g','markersiz e',m_rep; else M_rep = M_rep - 1; plot_au,_au,'- *','Color','r','LineWidth',1,'MarkerEdgeColor','r','MarkerFaceColor','r','MarkerSiz e',-m_rep; end end clear ; F = getframegca; mov =addframemov,f; pause0.3; hold off; end mov = closemov; %Plot statistic figure; H = SUBPLOT3,1,3; SUBPLOT3,1,1, plotdata_rep10:336,1, 'g' ; title'constraint violation' SUBPLOT 3,1,, plotdata_rep10:336,, 'r '; title'velocit of constraint violation' SUBPLOT3,1, 3, plotdata_rep10:336,3, 'b'; title'acceleartion of constraint violation' %Plot drivers figure; H = SUBPLOT3,6,18; SUBPLOT3,6,1, plotdata_rep10:336,5, 'g'; AXIS[ ] title'xwrist [m]' SUBPLOT 3,6,, plotdata_rep10:336,6, 'r '; title'ywrist [m]' SUBPLOT3,6, 3, plot-data_rep10:336,7*180/pi, 'b'; title'theta1 [deg]' SUBPLOT3,6,4, plotdata_rep10:336,8*180/pi, 'g'; title'theta [deg]' SUBPLOT 3,6, 5, plotdata_rep10:336,9*180/pi, 'r '; title'theta3 [deg]' SUBPLOT3,6, 6, plotdata_rep10:336,10*180/pi,'b'; 89

106 title'thera4 [deg]' SUBPLOT3,6,7, plotdata_rep10:336,11, 'g'; title'dxwrist m/s]' SUBPLOT 3,6, 8, plotdata_rep10:336,1,'r '; title'dywrist [m/s]' SUBPLOT3,6, 9, plotdata_rep10:336,13*180/pi, 'b'; title'dtheta1 deg/s]' SUBPLOT3,6,10, plotdata_rep10:336,14*180/pi,'g'; title'dtheta deg/s]' SUBPLOT 3,6,11, plotdata_rep10:336,15*180/pi,'r '; title'dtheta3 [deg/s]' SUBPLOT3,6, 1, plotdata_rep10:336,16*180/pi, 'b '; title'dtheta4 [deg/s]' SUBPLOT3,6, 13, plotdata_rep10:336,17*180/pi, 'g'; title'ddxwrist [m/s^]' SUBPLOT3,6,14, plotdata_rep10:336,18, 'g'; title'ddywrist [m/s^]' SUBPLOT 3,6, 15, plotdata_rep10:336,19*180/pi, 'r'; title'ddtheta1 [deg/s^]' SUBPLOT3,6, 16, plotdata_rep10:336,0*180/pi, 'b '; title'ddtheta [deg/s^]' SUBPLOT3,6,17, plotdata_rep10:336,1*180/pi, 'g'; title'ddtheta3 deg/s^]' SUBPLOT 3,6, 18, plotdata_rep10:336,*180/pi, 'r '; title'ddtheta4 [deg/s^]' %Plot reactions figure; H = SUBPLOT3,4,1; SUBPLOT3,4,1, plotlambda_rep:,5, 'g'; AXIS[ ] title'r1 [N]' SUBPLOT 3,4,, plotlambda_rep:,7, 'r '; title'r [N]' SUBPLOT3,4, 3, plotlambda_rep:,9, 'b'; AXIS[ ] title'r3 [N]' SUBPLOT3,4,4, plotlambda_rep:,11, 'g'; AXIS[ ] title'r4 [N]' SUBPLOT 3,4, 5, plotlambda_rep:,6, 'r '; 90

107 title'r1 [N]' SUBPLOT3,4, 6, plotlambda_rep:,8, 'b'; title'r [N]' SUBPLOT 3,4, 7, plotlambda_rep:,10, 'r '; title'r3 [N]' SUBPLOT3,4, 8, plotlambda_rep:,1, 'b '; AXIS[ ] title'r4 [N]' A=lambda_rep:,6; B=lambda_rep:,5; SUBPLOT 3,4, 9, plota,b, 'r '; title'r1r1' C= lambda_rep:,8; D=lambda_rep:,7; SUBPLOT3,4, 10, plotc,d, 'b '; title'rr' E= lambda_rep:,10; F=lambda_rep:,9; SUBPLOT 3,4, 11, plote,f, 'r '; title'r3r3' G=lambda_rep:,1; H=lambda_rep:,11; SUBPLOT3,4, 1, plotg, H, 'b'; AXIS[ ] title'r4r4' %Plot moments figure; H = SUBPLOT4,1,4; SUBPLOT4,1,1, plotlambda_rep:,16,'r'; title'm1 [Nm]' SUBPLOT 4,1,, plotlambda_rep:,15, 'g'; title'm [Nm]' SUBPLOT4,1, 3, plotlambda_rep:,14, 'r'; title'm3 [Nm]' SUBPLOT4,1, 4, title'm4 [Nm]' plotlambda_rep:,13, 'b '; 91

108 % Plot kinematics % X1, Y1 figure; H = SUBPLOT,3,6; SUBPLOT,3,1, plot q_rep:,1,'r'; AXIS[ ] title'x1' SUBPLOT,3,, plot q_dot_rep:,1, 'g'; title'dx1' SUBPLOT,3, 3, plot q_dot_rep:,1, 'b '; AXIS[ ] title'ddx1' SUBPLOT,3,4, plot q_rep:,,'r'; title'y1' SUBPLOT,3, 5, plot q_dot_rep:,, 'g'; title'dy1' SUBPLOT,3, 6, plot q_dot_rep:,, 'b '; AXIS[ ] title'ddy1' % X, Y figure; H = SUBPLOT,3,6; SUBPLOT,3,1, plot q_rep:,3,'r'; AXIS[ ] title'x' SUBPLOT,3,, plot q_dot_rep:,3, 'g'; AXIS[ ] title'dx' AXIS[ ] SUBPLOT,3, 3, plot q_dot_rep:,3, 'b '; AXIS[ ] title'ddx' SUBPLOT,3,4, plot q_rep:,4, 'r'; AXIS[ ] title'y' SUBPLOT,3, 5, plot q_dot_rep:,4, 'g'; AXIS[ ] title'dy' SUBPLOT,3, 6, plot q_dot_rep:,4, 'b '; AXIS[ ] title'ddy' 9

109 % X3, Y3 figure; H = SUBPLOT,3,6; SUBPLOT,3,1, plot q_rep:,5, 'r'; AXIS[ ] title'x3' SUBPLOT,3,, plot q_dot_rep:,5, 'g'; title'dx3' SUBPLOT,3, 3, plot q_dot_rep:,5, 'b '; AXIS[ ] title'ddx3' SUBPLOT,3,4, plot q_rep:,6, 'r'; AXIS[ ] title'y3' SUBPLOT,3, 5, plot q_dot_rep:,6, 'g'; AXIS[ ] title'dy3' SUBPLOT,3, 6, plot q_dot_rep:,6, 'b '; AXIS[ ] title'ddy3' % X4, Y4 figure; H = SUBPLOT,3,6; SUBPLOT3,3,1, plot q_rep:,7, 'r'; title'x4' SUBPLOT,3,, plot q_dot_rep:,7, 'g'; AXIS[ ] title'dx4' SUBPLOT,3, 3, plot q_dot_rep:,7, 'b '; AXIS[ ] title'ddx4' SUBPLOT,3,4, plot q_rep:,8, 'r'; title'y4' SUBPLOT,3, 5, plot q_dot_rep:,8, 'g'; title'dy4' SUBPLOT,3, 6, plot q_dot_rep:,8, 'b '; AXIS[ ] title'ddy4' % X5, Y5 figure; H = SUBPLOT,3,6; 93

110 SUBPLOT,3,1, title'x5' SUBPLOT,3,, title'dx5' SUBPLOT,3, 3, title'ddx5' plot q_rep:,9, 'r'; plot q_dot_rep:,9, 'g'; plot q_dot_rep:,9, 'b '; SUBPLOT,3,4, plot q_rep:,10, 'r'; title'y5' SUBPLOT,3, 5, plot q_dot_rep:,10, 'g'; title'dy5' SUBPLOT,3, 6, plot q_dot_rep:,10, 'b '; AXIS[ ] title'ddy5' % X6, Y6 figure; H = SUBPLOT,3,6; SUBPLOT3,3,1, plot q_rep:,11, 'r'; AXIS[ ] title'x6' SUBPLOT,3,, plot q_dot_rep:,11, 'g'; AXIS[ ] title'dx6' SUBPLOT,3, 3, plot q_dot_rep:,11, 'b '; AXIS[ ] title'ddx6' SUBPLOT,3,4, title'y6' SUBPLOT,3, 5, plot q_rep:,1, 'r'; plot q_dot_rep:,1, 'g'; title'dy6' SUBPLOT,3, 6, plot q_dot_rep:,1, 'b '; AXIS[ ] title'ddy6' % X7, Y7 figure; H = SUBPLOT,3,6; SUBPLOT,3,1, plot q_rep:,13, 'r'; AXIS[ ] title'x7' 94

111 SUBPLOT,3,, plot q_dot_rep:,13, 'g'; AXIS[ ] title'dx7' SUBPLOT,3, 3, plot q_dot_rep:,13, 'b '; AXIS[ ] title'ddx7' SUBPLOT,3,4, plot q_rep:,14, 'r'; title'y7' SUBPLOT,3, 5, plot q_dot_rep:,14, 'g'; AXIS[ ] title'dy7' SUBPLOT,3, 6, plot q_dot_rep:,14, 'b '; AXIS[ ] title'ddy7' % X8, Y8 figure; H = SUBPLOT,3,6; SUBPLOT,3,1, plot q_rep:,15, 'r'; AXIS[ ] title'x8' SUBPLOT,3,, plot q_dot_rep:,15, 'g'; AXIS[ ] title'dx8' SUBPLOT,3, 3, plot q_dot_rep:,15, 'b '; AXIS[ ] title'ddx8' SUBPLOT,3,4, plot q_rep:,16, 'r'; title'y8' SUBPLOT,3, 5, plot q_dot_rep:,16, 'g'; title'dy8' SUBPLOT,3, 6, plot q_dot_rep:,16, 'b '; AXIS[ ] title'ddy8' 95

112 Function_evaluation.m function [Phi, Phi_q, niu, gamma, drv_t] = Function_evaluationt,q,q_dot,Phi,Phi_q,L,drivers,niu,gamma,nc,nh_total,nd % Getting data from drivers - building the piecewise polinomials q15_pp = spline drivers:,,drivers:,3; dq15dt_pp = mmspderq15_pp; ddq15dt_pp = mmspderdq15dt_pp; q16_pp dq16dt_pp ddq16dt_pp = spline drivers:,,drivers:,4; = mmspderq16_pp; = mmspderdq16dt_pp; theta1_pp = spline drivers:,,drivers:,5; dtheta1dt_pp = mmspdertheta1_pp; ddtheta1dt_pp = mmspderdtheta1dt_pp; theta_pp = spline drivers:,,drivers:,6; dthetadt_pp = mmspdertheta_pp; ddthetadt_pp = mmspderdthetadt_pp; theta3_pp = spline drivers:,,drivers:,7; dtheta3dt_pp = mmspdertheta3_pp; ddtheta3dt_pp = mmspderdtheta3dt_pp; theta4_pp = spline drivers:,,drivers:,8; dtheta4dt_pp = mmspdertheta4_pp; ddtheta4dt_pp = mmspderdtheta4dt_pp; % Evaluation of the spline for time t q15_data = ppvalt,q15_pp; q16_data = ppvalt,q16_pp; theta1_data = ppvalt,theta1_pp; theta_data = ppvalt,theta_pp; theta3_data = ppvalt,theta3_pp; theta4_data = ppvalt,theta4_pp; drv_t1,1 = q15_data; drv_t,1 = q16_data; drv_t3,1 = theta1_data; drv_t4,1 = theta_data; drv_t5,1 = theta3_data; drv_t6,1 = theta4_data; % Drivers velocities for time t dq15dt_data = ppvalt,dq15dt_pp; dq16dt_data = ppvalt,dq16dt_pp; dtheta1dt_data = ppvalt,dtheta1dt_pp; dthetadt_data = ppvalt,dthetadt_pp; dtheta3dt_data = ppvalt,dtheta3dt_pp; dtheta4dt_data = ppvalt,dtheta4dt_pp; drv_t7,1 = dq15dt_data; drv_t8,1 = dq16dt_data; drv_t9,1 = dtheta1dt_data; drv_t10,1 = dthetadt_data; 96

113 drv_t11,1 drv_t1,1 = dtheta3dt_data; = dtheta4dt_data; % Drivers accelerations for time t ddq15dt_data = ppvalt,ddq15dt_pp; ddq16dt_data = ppvalt,ddq16dt_pp; ddtheta1dt_data = ppvalt,ddtheta1dt_pp; ddthetadt_data = ppvalt,ddthetadt_pp; ddtheta3dt_data = ppvalt,ddtheta3dt_pp; ddtheta4dt_data = ppvalt,ddtheta4dt_pp; drv_t13,1 = ddq15dt_data; drv_t14,1 = ddq16dt_data; drv_t15,1 = ddtheta1dt_data; drv_t16,1 = ddthetadt_data; drv_t17,1 = ddtheta3dt_data; drv_t18,1 = ddtheta4dt_data; % Kinematic constraints evaluation Phi1 = q3-q1^+q4-q^-l1^; Phi = q7-q5^+q8-q6^-l^; Phi3 = q11-q9^+q1-q10^-l3^; Phi4 = q15-q13^+q16-q14^-l4^; Phi5 = q5-q3; Phi6 = q6-q4; Phi7 = q9-q7; Phi8 = q10-q8; Phi9 = q13-q11; Phi10 = q14-q1; Phi11 = q15-q15_data; Phi1 = q16-q16_data; Phi13 = q14-q16-l4*sintheta4_data; Phi14 = -q16-q14*q9-q11+q15-q13*q10-q1- L4*L3*sintheta3_data; Phi15 = -q1-q10*q5-q7+q11-q9*q6-q8- L3*L*sintheta_data; Phi16 = -q8-q6*q1-q3+q7-q5*q-q4- L*L1*sintheta1_data; % Jacobian matri evaluation: Phi_q=zerosnh_total,nc; Phi_q1,1 = -*q3-q1; Phi_q1, = -*q4-q; Phi_q1,3 = *q3-q1; Phi_q1,4 = *q4-q; Phi_q,5 = -*q7-q5; Phi_q,6 = -*q8-q6; Phi_q,7 = *q7-q5; Phi_q,8 = *q8-q6; Phi_q3,9 = -*q11-q9; Phi_q3,10 = -*q1-q10; Phi_q3,11 = *q11-q9; Phi_q3,1 = *q1-q10; Phi_q4,13 = -*q15-q13; Phi_q4,14 = -*q16-q14; Phi_q4,15 = *q15-q13; 97

114 Phi_q4,16 = *q16-q14; Phi_q5,3 = -1; Phi_q5,5 = 1; Phi_q6,4 = -1; Phi_q6,6 = 1; Phi_q7,7 = -1; Phi_q7,9 = 1; Phi_q8,8 = -1; Phi_q8,10 = 1; Phi_q9,11 = -1; Phi_q9,13 = 1; Phi_q10,1 = -1; Phi_q10,14 = 1; Phi_q11,15 = 1; Phi_q1,16 = 1; Phi_q13,14 = 1; Phi_q13,16 = -1; Phi_q14,9 = -q16-q14; Phi_q14,10 = q15-q13; Phi_q14,11 = q16-q14; Phi_q14,1 = -q15-q13; Phi_q14,13 = -q10-q1; Phi_q14,14 = q9-q11; Phi_q14,15 = q10-q1; Phi_q14,16 = -q9-q11; Phi_q15,5 = -q1-q10; Phi_q15,6 = q11-q9; Phi_q15,7 = q1-q10; Phi_q15,8 = -q11-q9; Phi_q15,9 = -q6-q8; Phi_q15,10 = q5-q7; Phi_q15,11 = q6-q8; Phi_q15,1 = -q5-q7; Phi_q16,1 = -q8-q6; Phi_q16, = q7-q5; Phi_q16,3 = q8-q6; Phi_q16,4 = -q7-q5; Phi_q16,5 = -q-q4; Phi_q16,6 = q1-q3; Phi_q16,7 = q-q4; Phi_q16,8 = -q1-q3; % Evaluation of vector niu: niu1 = 0; niu = 0; niu3 = 0; niu4 = 0; niu5 = 0; niu6 = 0; niu7 = 0; niu8 = 0; niu9 = 0; niu10 = 0; niu11 = dq15dt_data; niu1 = dq16dt_data; niu13 = L4*costheta4_data*dtheta4dt_data; niu14 = L4*L3*costheta3_data*dtheta3dt_data; 98

115 niu15 = L3*L*costheta_data*dthetadt_data; niu16 = L*L1*costheta1_data*dtheta1dt_data; % Evaluation of vector gamma: gamma1 = -*q_dot3-q_dot1^+q_dot4-q_dot^; gamma = -*q_dot7-q_dot5^+q_dot8-q_dot6^; gamma3 = -*q_dot11-q_dot9^+q_dot1-q_dot10^; gamma4 = -*q_dot15-q_dot13^+q_dot16-q_dot14^; gamma5 = 0; gamma6 = 0; gamma7 = 0; gamma8 = 0; gamma9 = 0; gamma10 = 0; gamma11 = ddq15dt_data; gamma1 = ddq16dt_data; gamma13 = L4*- sintheta4_data*dtheta4dt_data^+costheta4_data*ddtheta4dt_data; gamma14 = L4*L3*- sintheta3_data*dtheta3dt_data^+costheta3_data*ddtheta3dt_data-*- q_dot16-q_dot14*q_dot9-q_dot11+q_dot15-q_dot13*q_dot10- q_dot1; gamma15 = L3*L*- sintheta_data*dthetadt_data^+costheta_data*ddthetadt_data-*- q_dot1-q_dot10*q_dot5-q_dot7+q_dot11-q_dot9*q_dot6- q_dot8; gamma16 = L*L1*- sintheta1_data*dtheta1dt_data^+costheta1_data*ddtheta1dt_data-*-q_dot8- q_dot6*q_dot1-q_dot3+q_dot7-q_dot5*q_dot-q_dot4; Get_C.m function [C] = get_cl, r_p % Calculate the c coeficients c = 1/L.*ee*r_p; % Evaluation of the mass matri of the rigid element C = [ 1-c1,1 c,1 c1,1 -c,1; -c,1 1-c1,1 c,1 c1,1]; 99

116 ElementMass.m function [M_el] = ElementMassm, J_G, L, rg % Calculation of the Polar Moment of Inertia with respect to point i X_G = rg1,1; Y_G = rg,1; J_i = J_G + m*x_g^ + Y_G^; % Evaluation of the mass matri of the rigid element M_el = zeros4,4; M_el1,1 = m - *m*x_g/l + J_i/L^; M_el1,3 = m*x_g/l - J_i/L^; M_el1,4 = -m*y_g/l; M_el, = m - *m*x_g/l + J_i/L^; M_el,3 = m*y_g/l; M_el,4 = m*x_g/l - J_i/L^; M_el3,1 = M_el1,3; M_el3, = M_el,3; M_el3,3 = J_i/L^; M_el4,1 = M_el1,4; M_el4, = M_el,4; M_el4,4 = J_i/L^; 100

117 Mmspder.m function z = mmspder,,i % MMSPDER Cubic Spline Derivative Interpolation %YI = MMSPDERX,Y,XI finds spline through X and Y and evaluates %at points XI. %PPD = MMSPDERPP finds piecewise polnomial vector PPD describing %the derivative of the spline stored in PP. %YI = MMSPDERPP,XI evaluates the derivative of the spline stored %in PP at positions XI. if nargin == 3 pp = spline,; else pp = ; end [br,co,np,nco,dim] = unmkpppp;% take apart pp % if pp1 ~= 10 % error'spline Data Does Not Have PP Form.' % end sf = nco-1:-1:1;% scale factors for differentiation dco = sfonesnp,1,:.*co:,1:nco-1;% derivative coefficients ppd = mkppbr,dco;% build pp form for derivative if nargin == 1 z = ppd; elseif nargin == z = ppvalppd,; else z = ppvalppd,i; end return Solve_EofM_ODE 45.m function [ddt] = Solve_EofM_ODE45t,,L_data, M_data, J_G_data, rg, Baum_data, drivers, r_mass, r_force,r_anchor, nc, nh_total, nd % Copies Y to q and q_dot q1:nc,1 q_dot1:nc,1 = 1:nc,1; = nc+1:*nc,1; % Solve Equations of Motion [q_dot, lambda, rep_data] = Solve_EofMt, q, q_dot, L_data, M_data, J_G_data, rg, Baum_data, drivers, r_mass, r_force,r_anchor, nc, nh_total, nd; % Construct vector Ydot ddt1:nc,1 ddtnc+1:*nc,1 = q_dot; = q_dot 101

118 Solve_EofM.m function [q_dot, lambda, data_rep] = Solve_EofMt, q, q_dot, L_data, M_data, J_G_data, rg, Baum_data, drivers, r_mass, r_force, r_anchor, nc, nh_total, nd % Define kinematic variables Phi = zerosnh_total,1; Phi_q = zerosnh_total,nc; niu=zerosnh_total,1; gamma=zerosnh_total,1; % Define dnamics variables q_dot = zerosnc,1; lambda = zerosnh_total,1; M = zerosnc,nc; g = zerosnc,1; % Calculate mass matri for i = 1:sizeL_data' % Size L gives the number of rigid bodies ii = 4*i-1; Mii+1:ii+4,ii+1:ii+4 = ElementMassM_datai, J_G_datai, L_datai, rg:,i; end % Calculate the g vector bn = r_mass,1; C = get_cl_databn,r_force; rf = r_anchor - C*q4*bn-1+1:4*bn,1; g4*bn-1+1:4*bn,1 = r_mass1,1/normrf.*c'*rf; f_t = [r_mass1,1/normrf.*rf; C*q4*bn-1+1:4*bn,1]; and application point for report % Holds force % Do the Function Evaluation [Phi,Phi_q, niu, gamma, drv_t]=function_evaluationt,q,q_dot,phi,phi_q,l_data,drivers,niu,gamma,nc,nh_total,nd; % Assemble the leading matri and leading vector gamma_baum = gamma - *Baum_data1*Phi_q*q_dot - niu - Baum_data^*Phi; A = [M Phi_q'; Phi_q zerosnh_total,nh_total]; b = [g; gamma_baum]; = A\b; % Pass the result to the right coordinates q_dot = 1:nc,1; lambda = nc+1:nc+nh_total,1; % Builds Report Data variable data_rep = [normphi; normphi_q*q_dot - niu; normphi_q*q_dot - gamma; normphi_q*q_dot - gamma_baum; drv_t; f_t]; 10