0021-9290/92 55.00+.00 Pergamon Press Ltd
.I. Biomahonics Vol. 25, No. 10, pp. 1209-1211, 1992. Printed in Great Britain
TECHNICAL
AN EXPLICIT
EXPRESSION
NOTE
FOR THE MOMENT SYSTEMS
IN MULTIBODY
AT L. HOF Laboratory of Medical Physics, University of Groningen, Bloemsingel lo,9712 KZ Groningen, The Netherlands Abstract-The equations of motion are formulated for a set of interacting rigid bodies. An explicit formulation could be derived for the total moment of the forces on such a system. This is applied to (1) the intersegmental moment in a multisegment rigid-body model, and (2) the interpretation of the ‘centre of pressure’ as defined in the studies of human posture.
NOMENCLATURE
ai af
e, F,i F,i F: g 9 Ii 4 Ei Mei N rs ri 5 rki *i Y, % mi
acceleration of the centre of mass of segment i,ai = ii component of a, in the direction of (vertical) z-axis unit vector along z-axis force of segment k on segment i external force acting on segment i component of F, in the direction of (vertical) z-axis acceleration due to gravity =lgl moment of inertia of segment i mass of segment i total mass of all segments moment (of muscle forces) exerted by segment k on segment i external moment acting on segment i total number of segments of the model position of the whole-body centre of mass position of the centre of mass of segment i position of point P position of joint between segments k and i velocity of the centre of mass of segment i, vi = ii position of F, in sagittal plane (planar model) angular acceleration of segment i (in planar motion) angular velocity of segment i
pressed as an explicit function of the relevant external forces and accelerations. As a second application, an expression will be derived for the ‘centre of pressure’, the point of application of the ground reaction vector, as used in studies of human posture. THEORY
A single rigid body
If a segment i from a model of interacting rigid bodies is connected with segments k, I, . . . , intersegmental forces Fki, F,i, . . . and intersegmental moments M,,, M,,, . . . will act on it. in addition, there may be, in the most genera1 case, an external force Fei and an external moment MCi. In what follows, one external force will be considered separately: the weight m,g, which acts at the segment’s centre of mass. A stationary coordinate system is de&d, in which the positions of the joints with respect to the origin 0 are given as position vectors rti and the position of the centre of mass as ri. Rigid body i is subject to two equations of motion:
i.e. the rate of change of momentum is equal to the sum of all the applied forces, and
INTRODUCTION 1 (rkix FA+(rei x Fci)+(ri x m,g)+C M,,+M,,= Biomechanical models of a moving human or animal usually consist of a set of rigid bodies (segments) connected at joints. These joints can transfer intersegmental forces. The actions of muscles and ligaments about the joints are as a rule modelled as intersegmental moments acting on each joint. For solving the inverse problem, the calculation of intersegmental moments and forces when the positions and accelerations of the segments are known, the standard procedure is to solve the equations of motion for the separate segments, proceeding from segment to segment. It will be shown here that from the basic equations of motion relations which are valid for systems of many rigid bodies, can be derived. One of these relations is the well-known law of conservation of linear momentum, the other one is a related expression about the total amount of angular momentum. It will be shown that by means of the latter relation any intersegmental moment in a multibody system can be exReceived
in final fom
29 January
1992.
i.e. the rate of change of angular momentum is equal to the sum of the moments of the applied forces and torques. The first term on the right-hand side of equation (2) can be worked out as (vi x m,v,+r, x miaJ=ri x m,a,. For a planar movement d(I,wJdt reduces to &xi, but in the general threedimensional case the evaluation of this term is more complicated (Hardt and Mann, 1980). The moments are relative to the origin of the reference frame 0, the position of which is in itself arbitrary, only it is if stationary. A set of N rigid bodies
In solving the inverse dynamics of a multi-rigid-body system equations (1) and (2) are, as a rule, applied to the separate segments (with the moments relative to the segment centre of mass), proceeding from segment to segment (Bresler and Frankel, 1950; Hardt and Mann, 1980; Winter, 1990). It
1209
1210
Technical Note
will be shown here that some useful relations can be derived for complete sets of interacting rigid bodies. The application of equation (1) to a set of N interacting rigid bodies gives the same relation as for a set of N point masses:
(W On the left-hand side of equation (3b) only the external forces are present. This is so because every intersegmental force Fki is cancelled by a reaction force F,, = -F,,. Similarly, from equation (2) follows, for the change in total angular momentum, that
Fig. 1. Multibody model of a subject walking over a force plate. Separate free-body diagrams have been given for segments 1, . . , k and I, . . . , N, respectively.
which can be written as
between segments k and 1.This gives after some rearranging in which the left-hand side, again, contains only moments of external forces and torques. In a similar way as in the derivation of equation (3), internal forces and moments are cancelled by their reactions. The internal reaction force pairs need not act at the same point, as is the case with contact forces, but they are always collinear, thus having the same moment with respect to 0. A restriction is that equation (4) is valid only for the moment with respect to a stationary point 0. It is possible, however, to give the moment with respect to an arbitrary moving point P, with position vector rp, in almost the same form. By taking the cross product of rp with both sides of equation (3b) we find that
(5) Subtracting equatian (5) from equation (4b) gives
It should be noted that in equation (6) the point P may be moving, but the acceleration terms mia, and d(l&/dt are still defined with respect to a stationary frame of reference.
APPLICATIONS
The intersegmental
moment
A familiar problem in human biomechanics is illustrated in Fig. 1. A moving subject has one foot on a force plate by which the reaction force F, can be measured, Positions and accelerations are determined by other means, e.g. by cinematography. It is assumed that the foot does not exert a torque on the force plate. The biomechanical model is a linked chain of N rigid bodies, the stance foot being no. 1. The question is to determine the moment M,, (due to muscular action) between the adjacent segments k and 1.To this end, we consider the subset of segments 1, . . , k, on which act the floor reaction force F,, the intersegmental force F, and the intersegmental moment M,. Thus, F, and MIk are now considered ‘external’ to this subset of segments. We then apply equation (6) with respect to rlt, the location of the joint
k
Ml,= -_(r,--Ttk)x F,- C Urt-rtt) x mill i=l
teml 3
tcml 4
The terms on the right-hand side can be recognized as follows: Term 1 =minus the moment of the reaction force(s); Term 2 = minus the moments of the weights of the segments 1, . . , k; Term 3 = the moments of the acceleration forces (ah these moments are taken with respect to the joint between segments k and I); and Term 4= the moments due to rotational acceleration. Terms 1 and 2 together form the quasistatic approximation. Obviously, it is also possible to give the solution for the same moment as a function of the accelerations of segments I,. . . , N. In biomechanical problems, however, it is a good practice to avoid the inclusion of the trunk segment, with its big mass and dubious rigidness. As a consequence, a calculation of the moments, in e.g. the swing leg, should begin with the most distal segment at the other end, the contralateral foot, where the external force is zero. In clinical gait analysis a simple technique to assess the joint moments is in common use, in which these moments are simply calculated as the cross product of the ground reaction vector and the distance from the joint centres to that vector. Wells (1981) has shown experimentally the errors of this approach (also discussed in Winter, 1990, p. 92). This technique obviously uses only the first term of equation (7). The other terms of equation (7) give directly the errors encountered in this approximation. Of course, it will depend very much on the context of the problem whether neglecting some of the terms of equation (7) is acceptable. In a clinical setting at slow walking speeds Term 1 may contain sufficient information. In many cases the rotational Term 4, which poses difficult computational problems in the three-dimensional case, may be neglected without undue error. The intersegmental force F, does not appear in solution (7). In many cases it does not need to be determined, but should this be necessary, it follows directly from equation (3b):
Technical Note
n
1211
quick horizontal arm movement might cause a sufficient displacement of the CP (i.e. outside the area between both feet) to induce a fall of the subject if there were no compensatory countermovements of other segments. The centre of pressure; three-dimensional case
In the above paragraph all segments were supposed to be flat laminae in the sagittal plane, hardly a realistic model for the study of two-legged human posture. A three-dimensional vector representation of equation (10) will be given now. Defining e, as the unit vector along the z-axis, we can form the triple-vector product of this vector with (rrxF,), for which the following relation holds (Prentis, 1979): e, x (r, x F,) = r,(e, *F,) - F,(e, *r,).
(11)
As long as the force plate is horizontal, e, is perpendicular to r,, (e;r,)=O,
e, x (r, x FJ
Y
rr=(e;F,)
Fig. 2. Multibody model of a subject standing on a force plate. CG: whole-body centre of mass. For other symbols see Nomenclature. When accelerations are present, c,, the point of application of the floor reaction force F,, is not identical to the vertical projection of CG.
The centre of pressure: two-dimensional case
The second application concerns the study of balance and posture. A subject is standing on a force plate (Fig. 2), which measures the floor reaction force vector F, and its point of application r, (which, of course, is always in the plane of the force plate’s surface). The position of rr on the force plate is known as the centre of pressure (CP). Often, it is incorrectly interpreted as the vertical projection of the whole-body centre of gravity (CG), see a discussion in Winter (1990). First, we will treat the two-dimensional case, i.e. consider only forces and movements in the sagittal yz-plane. The CP is then determined as Y, =
Irr x F,If’:l, .
with yr the distance of F, to the origin of a coordinate system fixed to the force platform. F, and rr x F, can be obtained from equations (3b) and (4b), respectively, to give
y= I
-(rcxMg)+C(rixmiai)+Cliai MC7+Cmiaiz
(10)
The summation is over all segments. It is seen that only in the completely static case the CP would be identical-to the nroiection of the CG. Winter (1990) and Rilev et al. 11990) have shown experimentally that, even in quiet stance, the acceleration terms in equation (10) are of a significant amplitude. It is easy to estimate from equation (10) that e.g. a
=
e,s [-(r,
’ x Mg)+C(ri x~miai)+~d(~~41dtl (12) Mg+Cw%
CONCLUSIONS
Equations (l)-(3) can be found in any elementary mechanics textbook (e.g. Prentis, 1979) and the derivation of equations (4)-(6) is straightforward. The only remarkable thing is that they seem not to have been published before. The two applications, which were qualitatively discussed in Winter (1990), may show that the equations given here may have some value for didactical purposes. Acknowledgements-We
thank Henk Grootenboer, Lubbers and Bert de Pater for helpful discussions.
Jaap
REFERENCES
Bresler, B. and Frankel, J. P. (1950) The forces and moments in the leg during level walking. Trans. ASME 72,27-36. Hardt, D. E. and Mann, R. W. (1980) A five body/threedimensional dynamic analysis of walking. J. Biomechanics 13,455-457.
Prentis, J. M. (1979) Engineering Mechanics. Oxford University Press, Oxford. Riley, P. O., Mann, R. W. and Hodge, A. W. (1990) Modelling of the biomechanics of posture and balance. J. Biomechanics 23, 503-506.
Wells, R. P. (1981) The projection of the ground reaction force as a predictor of internal joint moments. Bull. Prosthet. Res. BPR HJ-35. 15-19. Winter, D. A. (1990) Biomhchanics and Motor Control of Human Movement (2nd Edn). Wiley, New York.
.I. Biomahonics Vol. 25, No. 10, pp. 1209-1211, 1992. Printed in Great Britain
TECHNICAL
AN EXPLICIT
EXPRESSION
NOTE
FOR THE MOMENT SYSTEMS
IN MULTIBODY
AT L. HOF Laboratory of Medical Physics, University of Groningen, Bloemsingel lo,9712 KZ Groningen, The Netherlands Abstract-The equations of motion are formulated for a set of interacting rigid bodies. An explicit formulation could be derived for the total moment of the forces on such a system. This is applied to (1) the intersegmental moment in a multisegment rigid-body model, and (2) the interpretation of the ‘centre of pressure’ as defined in the studies of human posture.
NOMENCLATURE
ai af
e, F,i F,i F: g 9 Ii 4 Ei Mei N rs ri 5 rki *i Y, % mi
acceleration of the centre of mass of segment i,ai = ii component of a, in the direction of (vertical) z-axis unit vector along z-axis force of segment k on segment i external force acting on segment i component of F, in the direction of (vertical) z-axis acceleration due to gravity =lgl moment of inertia of segment i mass of segment i total mass of all segments moment (of muscle forces) exerted by segment k on segment i external moment acting on segment i total number of segments of the model position of the whole-body centre of mass position of the centre of mass of segment i position of point P position of joint between segments k and i velocity of the centre of mass of segment i, vi = ii position of F, in sagittal plane (planar model) angular acceleration of segment i (in planar motion) angular velocity of segment i
pressed as an explicit function of the relevant external forces and accelerations. As a second application, an expression will be derived for the ‘centre of pressure’, the point of application of the ground reaction vector, as used in studies of human posture. THEORY
A single rigid body
If a segment i from a model of interacting rigid bodies is connected with segments k, I, . . . , intersegmental forces Fki, F,i, . . . and intersegmental moments M,,, M,,, . . . will act on it. in addition, there may be, in the most genera1 case, an external force Fei and an external moment MCi. In what follows, one external force will be considered separately: the weight m,g, which acts at the segment’s centre of mass. A stationary coordinate system is de&d, in which the positions of the joints with respect to the origin 0 are given as position vectors rti and the position of the centre of mass as ri. Rigid body i is subject to two equations of motion:
i.e. the rate of change of momentum is equal to the sum of all the applied forces, and
INTRODUCTION 1 (rkix FA+(rei x Fci)+(ri x m,g)+C M,,+M,,= Biomechanical models of a moving human or animal usually consist of a set of rigid bodies (segments) connected at joints. These joints can transfer intersegmental forces. The actions of muscles and ligaments about the joints are as a rule modelled as intersegmental moments acting on each joint. For solving the inverse problem, the calculation of intersegmental moments and forces when the positions and accelerations of the segments are known, the standard procedure is to solve the equations of motion for the separate segments, proceeding from segment to segment. It will be shown here that from the basic equations of motion relations which are valid for systems of many rigid bodies, can be derived. One of these relations is the well-known law of conservation of linear momentum, the other one is a related expression about the total amount of angular momentum. It will be shown that by means of the latter relation any intersegmental moment in a multibody system can be exReceived
in final fom
29 January
1992.
i.e. the rate of change of angular momentum is equal to the sum of the moments of the applied forces and torques. The first term on the right-hand side of equation (2) can be worked out as (vi x m,v,+r, x miaJ=ri x m,a,. For a planar movement d(I,wJdt reduces to &xi, but in the general threedimensional case the evaluation of this term is more complicated (Hardt and Mann, 1980). The moments are relative to the origin of the reference frame 0, the position of which is in itself arbitrary, only it is if stationary. A set of N rigid bodies
In solving the inverse dynamics of a multi-rigid-body system equations (1) and (2) are, as a rule, applied to the separate segments (with the moments relative to the segment centre of mass), proceeding from segment to segment (Bresler and Frankel, 1950; Hardt and Mann, 1980; Winter, 1990). It
1209
1210
Technical Note
will be shown here that some useful relations can be derived for complete sets of interacting rigid bodies. The application of equation (1) to a set of N interacting rigid bodies gives the same relation as for a set of N point masses:
(W On the left-hand side of equation (3b) only the external forces are present. This is so because every intersegmental force Fki is cancelled by a reaction force F,, = -F,,. Similarly, from equation (2) follows, for the change in total angular momentum, that
Fig. 1. Multibody model of a subject walking over a force plate. Separate free-body diagrams have been given for segments 1, . . , k and I, . . . , N, respectively.
which can be written as
between segments k and 1.This gives after some rearranging in which the left-hand side, again, contains only moments of external forces and torques. In a similar way as in the derivation of equation (3), internal forces and moments are cancelled by their reactions. The internal reaction force pairs need not act at the same point, as is the case with contact forces, but they are always collinear, thus having the same moment with respect to 0. A restriction is that equation (4) is valid only for the moment with respect to a stationary point 0. It is possible, however, to give the moment with respect to an arbitrary moving point P, with position vector rp, in almost the same form. By taking the cross product of rp with both sides of equation (3b) we find that
(5) Subtracting equatian (5) from equation (4b) gives
It should be noted that in equation (6) the point P may be moving, but the acceleration terms mia, and d(l&/dt are still defined with respect to a stationary frame of reference.
APPLICATIONS
The intersegmental
moment
A familiar problem in human biomechanics is illustrated in Fig. 1. A moving subject has one foot on a force plate by which the reaction force F, can be measured, Positions and accelerations are determined by other means, e.g. by cinematography. It is assumed that the foot does not exert a torque on the force plate. The biomechanical model is a linked chain of N rigid bodies, the stance foot being no. 1. The question is to determine the moment M,, (due to muscular action) between the adjacent segments k and 1.To this end, we consider the subset of segments 1, . . , k, on which act the floor reaction force F,, the intersegmental force F, and the intersegmental moment M,. Thus, F, and MIk are now considered ‘external’ to this subset of segments. We then apply equation (6) with respect to rlt, the location of the joint
k
Ml,= -_(r,--Ttk)x F,- C Urt-rtt) x mill i=l
teml 3
tcml 4
The terms on the right-hand side can be recognized as follows: Term 1 =minus the moment of the reaction force(s); Term 2 = minus the moments of the weights of the segments 1, . . , k; Term 3 = the moments of the acceleration forces (ah these moments are taken with respect to the joint between segments k and I); and Term 4= the moments due to rotational acceleration. Terms 1 and 2 together form the quasistatic approximation. Obviously, it is also possible to give the solution for the same moment as a function of the accelerations of segments I,. . . , N. In biomechanical problems, however, it is a good practice to avoid the inclusion of the trunk segment, with its big mass and dubious rigidness. As a consequence, a calculation of the moments, in e.g. the swing leg, should begin with the most distal segment at the other end, the contralateral foot, where the external force is zero. In clinical gait analysis a simple technique to assess the joint moments is in common use, in which these moments are simply calculated as the cross product of the ground reaction vector and the distance from the joint centres to that vector. Wells (1981) has shown experimentally the errors of this approach (also discussed in Winter, 1990, p. 92). This technique obviously uses only the first term of equation (7). The other terms of equation (7) give directly the errors encountered in this approximation. Of course, it will depend very much on the context of the problem whether neglecting some of the terms of equation (7) is acceptable. In a clinical setting at slow walking speeds Term 1 may contain sufficient information. In many cases the rotational Term 4, which poses difficult computational problems in the three-dimensional case, may be neglected without undue error. The intersegmental force F, does not appear in solution (7). In many cases it does not need to be determined, but should this be necessary, it follows directly from equation (3b):
Technical Note
n
1211
quick horizontal arm movement might cause a sufficient displacement of the CP (i.e. outside the area between both feet) to induce a fall of the subject if there were no compensatory countermovements of other segments. The centre of pressure; three-dimensional case
In the above paragraph all segments were supposed to be flat laminae in the sagittal plane, hardly a realistic model for the study of two-legged human posture. A three-dimensional vector representation of equation (10) will be given now. Defining e, as the unit vector along the z-axis, we can form the triple-vector product of this vector with (rrxF,), for which the following relation holds (Prentis, 1979): e, x (r, x F,) = r,(e, *F,) - F,(e, *r,).
(11)
As long as the force plate is horizontal, e, is perpendicular to r,, (e;r,)=O,
e, x (r, x FJ
Y
rr=(e;F,)
Fig. 2. Multibody model of a subject standing on a force plate. CG: whole-body centre of mass. For other symbols see Nomenclature. When accelerations are present, c,, the point of application of the floor reaction force F,, is not identical to the vertical projection of CG.
The centre of pressure: two-dimensional case
The second application concerns the study of balance and posture. A subject is standing on a force plate (Fig. 2), which measures the floor reaction force vector F, and its point of application r, (which, of course, is always in the plane of the force plate’s surface). The position of rr on the force plate is known as the centre of pressure (CP). Often, it is incorrectly interpreted as the vertical projection of the whole-body centre of gravity (CG), see a discussion in Winter (1990). First, we will treat the two-dimensional case, i.e. consider only forces and movements in the sagittal yz-plane. The CP is then determined as Y, =
Irr x F,If’:l, .
with yr the distance of F, to the origin of a coordinate system fixed to the force platform. F, and rr x F, can be obtained from equations (3b) and (4b), respectively, to give
y= I
-(rcxMg)+C(rixmiai)+Cliai MC7+Cmiaiz
(10)
The summation is over all segments. It is seen that only in the completely static case the CP would be identical-to the nroiection of the CG. Winter (1990) and Rilev et al. 11990) have shown experimentally that, even in quiet stance, the acceleration terms in equation (10) are of a significant amplitude. It is easy to estimate from equation (10) that e.g. a
=
e,s [-(r,
’ x Mg)+C(ri x~miai)+~d(~~41dtl (12) Mg+Cw%
CONCLUSIONS
Equations (l)-(3) can be found in any elementary mechanics textbook (e.g. Prentis, 1979) and the derivation of equations (4)-(6) is straightforward. The only remarkable thing is that they seem not to have been published before. The two applications, which were qualitatively discussed in Winter (1990), may show that the equations given here may have some value for didactical purposes. Acknowledgements-We
thank Henk Grootenboer, Lubbers and Bert de Pater for helpful discussions.
Jaap
REFERENCES
Bresler, B. and Frankel, J. P. (1950) The forces and moments in the leg during level walking. Trans. ASME 72,27-36. Hardt, D. E. and Mann, R. W. (1980) A five body/threedimensional dynamic analysis of walking. J. Biomechanics 13,455-457.
Prentis, J. M. (1979) Engineering Mechanics. Oxford University Press, Oxford. Riley, P. O., Mann, R. W. and Hodge, A. W. (1990) Modelling of the biomechanics of posture and balance. J. Biomechanics 23, 503-506.
Wells, R. P. (1981) The projection of the ground reaction force as a predictor of internal joint moments. Bull. Prosthet. Res. BPR HJ-35. 15-19. Winter, D. A. (1990) Biomhchanics and Motor Control of Human Movement (2nd Edn). Wiley, New York.
