Identification of muscle forces in human lower limbs during sagittal plane movements Part I: Human body modelling

Transcription

1 Acta of ioenineerin and iomechanics Vol. No. 00 Identification of muscle forces in human lower limbs durin saittal plane movements Part I: uman bod modellin WOJCIEC LAJER KRZYSZOF DZIEWIECKI ZENON MAZUR Institute of Applied Mechanics echnical Universit of Radom he paper presents a human biomechanical model aimed at determinin muscle forces durin planar movements such as standin lon jumps vertical jumps and jumps down from a heiht. Onl the hip knee and ankle joints are modelled as directl enforced b the muscle forces applied to their tendon attachment points and the actuation of the other joints is simplified to torques representin the muscle action. A sstematic construction of the related dnamic equations in independent coordinates is provided followed b some uidelines for their use in analsin motion durin the flin and support phases.. Introduction he understandin of how human beins are maneuvered can be interestin from man viewpoints. he determination of muscle forces durin movement contributes to the elucidation of the underlin neural control and is essential for a comple analsis of internal loads actin on bones and joints; both of considerable importance to clinicians []. he was the humans are moved/controlled are then also widel imitated in enineerin desins like robots manipulators and walkin machines. Since the measurin of muscle forces directl within the livin beins ma be difficult and even injurious the other possibilit is to estimate them computationall based on a biomechanical model and some input data obtained from noninvasive measurements. he need for reliable results obtained this wa has stimulated an increased interest in thorouh human bod modellin and advanced computational alorithms [] []. his paper is another contribution in this field. he inherent compleit of human bod alwas calls for some modellin simplifications and the mathematical models should in eneral be as comple as necessar and as simple as possible which is alwas a compromise between the analsis thorouhness and modellin feasibilities. he biomechanical model developed in this paper is aimed at

2 W. LAJER et al. analzin human movements such as standin lon jumps vertical jumps and jumps down from a heiht in which the lower and upper etremities are movin parallel to each other in the saittal plane. For these reasons we decided for a planar model of human bod in which the modelled muscle forces are in fact their projections onto the saittal plane. hen since the attention is focused on lower limb control and loadins onl the hip knee and ankle joints are modelled as enforced directl b the muscle forces applied to their tendon attachment points and the actuation of the other joints is simplified to torques representin the muscle action. In this first part of the paper we present the foundations for the desin of the biomechanical model. A sstematic construction of the related dnamic equations in independent coordinates is then provided followed b some uidelines for their use in the analsis of motion durin the flin and support phases.. Modellin preliminaries he human bod is modelled as a planar kinematic structure consistin of N = 9 riid bodies branchin from the pelvis in the open chain linkaes (see fiure ). he n = eneralized coordinates that describe the position of the flin model with respect to the inertial reference frame are q = [ ϕ... 9 ] ϕ where and are the coordinates of the hip point and the anular coordinates ϕ i ( i =... N ) are all measured from the vertical direction. he n-deree-of-freedom sstem is actuated b k = torques u = [ τ... τ] representin the muscle action in the joints the sstem is thus underactuated in the flin phase k < n. 9 r = K A Fi.. he human biomechanical model 9 * Fi.. he muscles in lower limbs 0

3 Identification of muscle forces in human lower limbs. Part I he attention of this stud is focused on the lower limb control and loadins durin saittal movements. he actuation in the hip knee and ankle joints needs thus a more detailed modellin. We identified m F = muscles that produce the control torques in the three joints (see fiure ) and the respective muscle forces F... F applied to their tendon attachment points are treated as control inputs instead of τ τ and τ. he total vector of m = mf + mτ = 0 control inputs is then u = [F... F τ where m = are the control torques in the other joints (note that... τ ] τ included in u is slihtl different from τ represented in u ). Since the m F muscle forces actuate the motion in the three joints K and A a local control redundanc is faced which will be discussed in more detail in part II of this paper. * τ C A l C K A F F C A Fi.. he tendon attachment points of the -th muscle Usin N = absolute coordinates θi [ C C θ... C 9 C9 θ9] p = where Ci Ci and are the coordinates of the mass center C i and the orientation anle (here θ i = ϕ i ) of the i-th bod sement with respect to the inertial frame i =... N the dnamic equations of the flin model epressed in p are the constrained Newton Euler equations in the form M p = f + u C () where M = dia( m m JC... m9 m9 JC9) is the eneralized mass matri related to p m i and J Ci are the mass and mass moment of inertia with respect to C i of the i-th sement f = 0 m m 0 ] contains the ravitational forces f = u [ 9 u

4 W. LAJER et al. is the N-vector of eneralized control forces with bein the N m matri of distribution of control inputs u in the directions of p and fc = C is the N-vector of eneralized reaction forces due to the l = k = kinematical joint constraints with C bein the l N constraint matri and = [ λ... λ containin the reaction forces in the joints. he formulation of M and f in equaqtion () is evident and the same concerns the last mτ columns of related to τ... τ their entries are either 0 or (see [] for more details). For eample the 9-th column of related to τ has all the entries equal to zero ecept the -th and -th entries (related to θ and θ ) which are equal to and respectivel. he determination of the first m F columns of related to F... F is a little more challenin. Let us demonstrate this for the case of the -th column of related to F (fiure ). Firstl we must state that the -th muscle tendons are attached to sements and and onl these two bodies are directl affected b F. his also means that onl those entries of the -th column of that correspond to these two bodies must be found and the other entries are in principle equal to zero. he indices of nonzero entries are from to (bod ) and from to (bod ). he second step is determination of the inertial frame coordinates r A = [ A A] and r = [ ] of tendons attachment points A and (see fiure ) which are: l ] A A = = = = ξ + ξ A A + ξ l ξ A A () where A = [ ξ A ηa] and = [ ξ η] are the coordinates of A and in the local coordinate sstems of sements and respectivel and l is the lenth of sement. he nonzero entries of the -th column of are then: = ( = ( C C A = sinα = cosα )sinα ( )sinα + ( C = sinα = cosα C A )cosα )cosα ()

5 Identification of muscle forces in human lower limbs. Part I where C = [ C C] r and r ] of mass centers C and C can be determined from equation () usin and and of A sin C = [ C C C = [ ξc ηc] and C = [ ξc ηc ] A A α = cos = ( A ) + ( A ) ( A) + ( A ) instead α. () Followin the above procedure for all forces F j j = mf the first m F columns of can be determined. hen aumentin them b the last mτ columns of defined above the whole N m matri can be formulated. z z K z z K A z z A Fi.. he open-constraint coordinates and the reaction forces in lower etremit joints he l N constraint matri C in equation () follows from the joint constraint equations (p) = 0 = C( p) p = 0 ()() i.e. C = p. Since all the k joints in the sstem are rotar at each joint two constraint equations epressin the prohibited relative and translations are introduced denoted open-constraint coordinates z = [ z... z l ] where l = k []. he constraint equations are then z = (p) = 0 and a particular constraint equation Φ i ( p) = 0 ( i =... l ) depends onl on the absolute coordinates of the adjacent bodies in the respective joint. he open-constraint coordinates in the lower etremit joints K and A and the related reaction forces are illustrated in fiure. As an eample the two constraint equations related to K joint are: Φ = Φ = C C ξ + ξ C C C C [ [ C C + ( l ξ ( l ξ C C ) ) C C ] = 0 ] = 0. ()

6 W. LAJER et al. he third and fourth rows of C can then be constructed not reported here for shortness whose nonzero entries will relate onl the absolute coordinates of sements and.. Dnamic equations in independent coordinates he projective formulation of joint coordinate method described in [] is applied to derive the dnamic equations of the human biomechanical model in coordinates q. he scheme is based on the relationships between the absolute (dependent) coordinates p and the independent coordinates q which state the joint constraint equations iven eplicitl. At the position velocit and acceleration levels the eplicit constraint equations are: p = ( q) p = D( q) q p = D( q) q + ( q q ) ()(9)(0) where D = q is the N n matri and = D q is the N vector. As an eample the absolute coordinates of sement can be epressed in q as follows: C C = = l l + ξ ξ θ = ϕ C C C C which make the seventh eihth and ninth entries of equation (). After introducin the other components of p = (q) D and defined in equations (9) and (0) can be obtained. Since ( ( q)) 0 after insertin equations () and (9) into equation () it can be deduced that C( ( q)) D( q) q 0 and thus C D = 0 D C = 0 the N n matri D is an orthoonal complement to the l N constraint matri C in the Ndimensional confiuration space related to p []. he dnamic equations () defined in the N-dimensional linear space related to p can then be projected into the n- dimensional tanential (velocit allowed) and l-dimensional constrained (null velocit) subspaces defined b the vectors represented as columns of D and rows of C. he projection formula is (see [] for more details) D CM ( M( Dq + ) f u + C ) = 0 () () where equation (0) was substituted for p. Considerin that D C = 0 the first n equations lead to the requested dnamic equations in q whose matri eneric form is

7 Identification of muscle forces in human lower limbs. Part I 9 M q d = f + u () + where M( q) = D M D is the n n eneralized mass matri related to q; d( qq ) = D M and f = D f are the n-vectors of eneralized dnamic forces related to q due to the centrifual accelerations and ravitational forces respectivel and = D is the n m matri of the distribution of the control inputs u in the directions of q. a) b) Fi.. he foot in contact with the round he dnamic equation () describes the human bod dnamics in the flin phase. Durin the support phase when the feet are in contact with the round r = additional constraints are imposed on the sstem written smbolicall as ( q) = 0. For the two contact modes seen in fiure these constraint equations are: Φ = Φ = l l 0 0 = 0 = 0 ( ) Φ a ( ) = ϕ 0 or Φ b = ϕ 0 = = () where 0 and 0 are the inertial frame coordinates of the point to which the reactions from the round = [ λ λ ] are reduced. For both modes of the foot round λ contact seen in fiure we reduce the round reaction to the same point the end of instep (sement ). he round-contact constraint equations at the velocit and acceleration levels are C q = 0 and C q = 0 respectivel where C( q) = q is the r n constraint matri and ( q q ) = Cq is the n-vector of constraint-induced accelerations []. he dnamic equations () modif then to M q + d = f n k ( k = n r = ) matri D such that C D = 0 D C = 0 + u C. Introducin an and usin the

8 0 W. LAJER et al. projection formula similar to that used in equation () the tanential and orthoonal projections ive respectivel: D M q D ( f d + u) () = ( q q u) = ( C M C ) ( C M ( f d + u) ). () Eqation () constitutes the round reaction-free dnamic equations of the modeled human durin the support phase. Note that the number k of these equations is equal to the number of control torques u at the k joints distributed then in joints K and A into m F muscle forces. Equation () supplies one with a formula for determination of the round reaction values.. Conclusions In this first part of the paper we presented foundations and sstematic construction of a human bod biomechanical model aimed at analzin planar movements such as vertical or standin lon jumps. A hbrid model of control was used modelled b means of muscle forces in the lower etremit joints and simplified to the torques representin the muscle action at the other joints. wo different sets of motion equations were derived separatel for the flin phase and the support phase. he round reactions durin the support phase can also be determined. In the second part of the paper [] applications of the developed mathematical model to the inverse dnamics analsis of human movements will be discussed. References [] WISMANS J.S..M. JANSSEN E.G. EUSENERG M. KOPPENS W.P. LUPKER.A. Injur biomechanics Course notes Facult of Mechanical Universit Eindhoven Universit of echnolo Eindhoven 99. [] ÖZEREN A. uman bod dnamics: classical mechanics and human movement Spriner New York 000. [] AMAGUCI G.. Dnamic modelin of musculoskeletal motion Kluwer Norwell 00. [] LAJER W. CZAPLICKI A. Contact modelin and identification of planar somersaults on the trampoline Multibod Sstem Dnamics [] LAJER W. Methods of multibod dnamics (in Polish) Monoraphs No. Press of the echnical Universit of Radom Radom 99. [] LAJER W. DZIEWIECKI K. MAZUR Z. Identification of muscle forces in lower limbs durin saittal plane movements. Part II: Computational alorithms (ibidem).