- Email: [email protected]
The synthesis of bipedal locomotion
THE SYNTHESIS
OF BIPEDAL
LOCOMOTION*
M. A. TOWNSENDt and 4. SEIREGS Xlechanical Engineering Department. University of Wisconsin. Madison. Wisconsin, U.S.A. Abstract-This paper presents a mathematical model for the synthesis and control of bipedal walkers. The model deals with the three-dimensional motion of a symmetrical rigid body with massless extensible legs. Body trajectories and controls are synthesized for optimum stability and energy expenditure. The model provides a rigid body idealization of human locomotion and a rational tool for design of prosthetic devices and walking machines. 1. INTRODUCTION
oxygen consumption. Work has been done at the University of California (Ralston, 1958; Ralston and Lukin, 1969: Saunders ef al. 1953; Lukin et al. 1967) and at Milan (Cavagna et al. I963 ; Margaria, 1968) on estimating external work and efficiency of motion. One may find values for energy input during human activities in the work of Ralston (1958: 1969) or in texts such as that of Morton and Fuller (1952). Hill (1965) has exhaustively investigated the biophysical aspects of muscle motion, energy and efficiency; is a bibliography of his work. Another area receiving attention in recent years is the development of load carrying walking machines which would be preferable to wheel chairs. Work by Seireg and Peterson (unpublished) on the actual construction of a manually controlled, load carrying tripod walker (2 legs and a pusher) indicates that such machines may be realizable. Since Elftman ( I939a.b.c), analytical (engineering) studies have been largely dormant until very recently. Frank and Vukobratovic (1969) consider the classes of a biped motion, and Frank (in preparation) has proposed a mechanical design approach to the construction of a machine similar to the model of this paper, which he has modelled
of human and animal locomotion has a long and colorful history, albeit fitful, but it has only recently begun emerging from the stages of copious (but often incomplete) data gathering and causal explanations of the data. Observations such as ‘-... the whole of this ensemble acts in unison with a single and complete rhythm, fusing the whole enormous complexity into clear and harmonious simplicity”§ may be true, but do not explain the phenomenon. Quantitative analysis of many aspects of human locomotion are now available in the clinical studies of gait characteristics by Murray et nl. (1964, 1966, 1969, 1967), Contini et al. ( 1964). and the Ducroquets ( 1968). Fischer (1898- 1904) and Braune and Fischer ( 1890) initiated the rational study of human locomotion, followed by Elftman’s studies on the dynamics of motion (1939a, b, c). These have been expanded by the studies of leg and joint forces of Bresler and Frankel (1950). Paul (1965. 1966-67), and summarized by Peizer er al. (1969). Evaluations of energy transfer, useful work done during human activity, and ‘efficiency’ have been somewhat less successful, although Ralston (1958) has determined an optimal walking velocity based on the energy equivalent of
THE STUDY
“Receiced
22 Februnrv
I97
I.
iResearch Assistant. (Now Associate Professor. Univ. of Toronto, Canada.) f Professor. SQuoted from Bernstein et al. “Investigation on Biodynamics of Locomotion”, Conrini. (ibid. p. 119) employs a similar description. 71
Vols. 1.2. MOSCOW1935. 1910 by
72
IV. A. TOWNSEND
on a hybrid computer. Kane and Scher (1970) and Smith and Kane (1968) have considered the mechanics of motion during free fall conditions. The kinematic and dynamic characteristic of other human activities standing, jumping, etc.-have been studied in terms of the support characteristics by Murray ef al. (1967). Studies of athletic events have yielded interesting results; Lascari (1970) considered the linear impulse* of the external forces as a significant measure of effort and observed that what is considered beautiful may not be most ‘efficient’. This aspect will be discussed later in the paper. 1. HYPOTHESES
The model (Fig. 1) consists of a single rigid body, which represents the body mass and inertias, and two rigid extendable,
Fig. 1. Model and nomenclature.
and A. SEIREG
massless legs. As such, the system has six degrees of freedom: the three motions of the center of mass in the direction of the space coordinates and three successive rotations of the body about a set of body-fixed coordinate axes. The system is situated (location and orientation) relative to an earth-fixed (inertial) reference frame E at OE. For a general system. each body of the system is defined by a set of body-fixed, right-hand principal coordinates axes. here 0,. In multiple-body systems (more complex models), one body of the system can be designated as the main body. Its position relative to the inertial frame is defined by the location of its center of mass (CM) and three Euler angles.* The positions of the other bodies of the system are then described relative to the main body in terms of relative coordinates (internal variables) between the bodies. In the model considered in this study, the main body is defined by axes at 0,. Each leg (‘bodies’ 2 and 3) is assumed to have a leg-fixed coordinate system with the z_-axis along the leg and the origin at the ‘hip’. The motion and the controlling forces (controls) can be evaluated if we know the location of 0, relative to OE and the three rotations necessary to orient the body in space. By computing the forces at the hip points we can directly evaluate the leg forces and support points without a priori assumptions of foot placement. An ideal dynamic leg is defined as the straight rod connecting the designated body support point (hip) to the instantaneous earth support point. The locus of the earth points for each step determines the *footprint’. The motion to be studied is level walking. For the present study it is assumed that the body is at all times supported by a single ideal dynamic leg (right or left) and that the change from one to the other is
*Linear impulse = ,z F dt. *The Euler angle system used here is successive rotation about the body fixed z, y.x axes, each rotation taking about the present location as defined by the previous rotation(s): (Shames. 1966).
Place
SYNTHESIS
OF BIPEDAL
essentially instantaneous. This assumption is a convenient device to facilitate the synthesis of body trajectories, controls and locus of support. The motion sequence and control for the acutal locomotion (including the double leg support phase) can be directly derived from the results of the synthesis. The ideal dynamic leg is the exact representation of the support between heel strike (represented by x in Fig. 5) and the onset of double leg support (or ‘toe off point, shown by 0 in the figures). During double leg support (shown by dashed lines in Fig. 6) the ideal dynamic leg represents the resultant effect of the two real legs. ‘: The assumption of a single ideal dynamic leg support at all times, therefore. considerably simplifies the problem formulation and computational procedure without affecting in any way the motion trajectories, locus of support, and computed controls. Body mass, principal moments of inertia, and salient geometry are given in Table 1. In this analysis we are mainly concerned with the body dynamics rather than jvith the kinematic relations involved in the motion (as in Paul, 1963: Bresler and Franked. 1950). The kinematic constraints of fixed-length leg segments can easily be superimposed later, with at most a constant shift in the average vertical location of the body CM. This does Table Body mass Principal
I. X40&1parameters
0.1167 lb sec’iin.
t 161 lb vveight)
moments of inertia of body lxx = 60 lb in. sec3 I,, = 60 lb in. se? I,, = 12 lb in. set’ Location of hip points in body system tx.~.:). in. right rl-’ = (0.4.0. -9.0) left r,:> = 10. -6.0. -9.0) \lean location of CM in earth system R,taverage) = (63,0”t, O., -40.0) @,(average) = rO.O.0)
LOCO~lOTION
73
not have any effect on the work-causing motions or the resulting controls (forces and moments), and the footprint size will change proportional to the new value of the average height of the CM. Actual foot location can be chosen for minimum work or arbitrarily selected, but the decision is not necessary at this point in the analysis. As shown later (Tables 2 and 3), the effect of foot placement is to change the work values. The main point is that there is no fundamental mathematical difference between an extensible leg and a leg with a knee. The proposed model can be used for the following purposes: ( I) estimating forces and moments generated by the body necessary for stable locomotion: (2) evaluating the variables and criteria fat different patterns of walking (and motion in general): (3) comparing human vs. biped-machine walk. (4) determine system (internal) work required. bvork done as a machine. and ‘efficiency’, and (5) ‘optimal’ design of a biped uaiker. Much of the above depends upon the ability of the model to generate trajectories for specified criteria of motion for the system. If the results are reasonable, such models provide insight into the real systems. For example, it can be anticipated that walking of (a man fitted with) a biped prosthesis is fundamentahy different from normal human motion since the principal mass of the human body is highly segmented and rotationally coupled. Conversely. a man fitted with a biped walker will undoubtedly be strapped-in for practical reasons: thus, this motion can be simulated by this first-generation model. Finally, a ‘good’ model may eventually be physically analogous to a real system, but this is not a requirement.
‘The control forces during the double leg support phase can be directly computed from these results without influencing the system dynamics or the synthesized trajectories. The controls of both legs at the assumed physical positions of toe off and next heel strike respectively can be evaluated. for example. as the components of the controls of the ideal dynamic leg at any instant during this phase from static equilibrium considerations.
M. A. TOWNSEND
74 3. ANALYSIS
(a) Dynamic equations The conditions for the dynamic equilibrium of the model under consideration can be expressed in body coordinates as (Shames, 1966; Greenwood, 1965): Vector summation of forces
=x
4+mg+(-ma,)=O
(I)
Vector summation of moments = C, mi+ where
(-I$)
=O*
(2)
m = total mass of the body fi = the force vectors acting on the body at the hips (hip forces) mg = the weight of the body -ma, = the inertia force vector a, = the acceleration vector of the center of mass @, = the moment vectors acting on the body at the hips (hip moments) (-I&) = the inertial moment vector abopt the center of mass.
The acceleration vector a, of the center of mass, and the angular velocity o and angular acceleration C;, necessary for evaluation of & are initially determined from the displacements and rotations of the body as observed in earth coordinates. The. principal axes of inertia for the body are the body-fixed reference axes: the moments of inertia about these axes are used in calculating &. The forces fi and moments M.-i which cause the body to move are the active controls. If either all the motions or all the forces or a proper combination of these are known, equations (1) and (2) can be solved for the remaining variables. This becomes an analytic problem with only one possible solution. On the other hand, the synthesis of a system of controis fi*P& and resulting motions requires solution of a system of non-linear
and A. SEIREC
ordinary differential equations determined by equations (l), (2) and the kinematic constraints. This results in more unknowns than there are equations and consequently many possible solutions. To select the ‘best’ solutions a performance criterion for the resulting motion and controls must be defined. Even so, a solution may not be possible due to the non-linear nature of the equations of motion and each of a general existence theory for such systems of equations. Examples of the analytic and synthesis problem are given in this paper. (b) Trajectory analysis The data for normal human walking in Murray et al. (1964), is used in this analysis to provide a nominal history for the motion of the center of mass and the three body rotations. Since walking motions are periodic, each motion component can be represented by the sum of a polynomial and a Fourier series. The polynomial is usually formulated in terms of an average location and an average velocity. Hence, trajectories for walking (and other steady-state activities climbing stairs, etc.) can be represented in the form:
where
qi = different coordinates of the motion, (i coordinate) (2 steps per % = stride frequency stride) in rad./sec 4i = mean position Vi = average velocity at a ak = amplitude of fluctuation particular frequency (kw) c#+= phase shift.
In order to simplify the analysis, the following observations can be made on the experimental data. First, in the mean sagittal plane (-TE- zE) the dominant fluctuations occur at twice the stride frequency (i.e. at step frequency). This applies to the xE and zE motions
SYNTHESIS
OF BIPEDAL
LOCOMOTIOS
7s
/.Lj=Mj+rjXfi where as well as the 8? rotation about the y,-axis. Fi,Xi,Ai are the external forces acting Second, the dominant amplitudes of the other on the system and the associmotions (yE, 8, about z,-axis, and e3 about x1ated motions and velocities axis) occur at the stride frequency. A first (assumed to act at CM) approximation to the motion is to use one fjxjsj are the internal forces and term of the Fourier series in equation (3) corresponding relative angles with k = 2 for the sagittal motions (4i = XE, q3=zE. q5=Bn); and k= 1 for the other Mj,8j,aj are the internal moments and motions (qZ=yErqJ=01,46=8,). corresponding relative angles Differentiation of equation (1) gives the rj is the distance from the CM velocities and accelerations. Depending on the to the point of application of coordinate system in which the motions are the force determined, a transformation to the body Wj = function (6). coordinate system may be necessary for the Equation (5) is more convenient for the evaluation of the control forces and moments present discussion. The first integral is the at the hips and legs. work done by the external forces on the system. The second integral is the work done by CC)Motion criferia the internal forces of the system. Over one cycle (one stride) the first integral In this analysis two criteria for walking are considered. The first is the minimization of the is zero for level walking. As this is not a machine the second integralsize of the footprint. Footprint size and the conservative path of the center of support can be used as a and hence the net work done- need not be zero. It may be positive or negative; the sign measure of the system stability and the physical realizability of the motion. of each product indicates whether that control The other criterion relates to the efficiency acted as a motor or brake over the cycle. If the of performance as indicated by the useful net work of the system forces is negative, then work done and the energy expended in the the net effect is a braking action. Since equaprocess. Because of its importance to this tions (4) and (5) are for the whole system, study, a detailed discussion of work and negative work done by one part of the system energy in locomotion is given in the following cancels positive work done by another part. section. However, real systems do work at specific locations, so it is also necessary to evaluate the local net work done at each of the salient (d) Work points: hip, leg and ankle. The real processes The work done by the system under con- of doing work (motors) and braking (dissipasideration as a machine W, can be readily tors) are usually fundamentally different with evaluated from little effective energy transfer between them and minimal energy storage during braking. To estimate the work as would be performed by an equivalent physiological system which expends energy at specific locations, we use the following rationale: the human body is a contractile mechanism-the muscles do not perform work in extension and store negligible mechanical energy. The physiological work done at point of relative motion is assumed to
M. A. TOWNSEND
76
and A. SEiREG
be the absolute value of the net power output at that point integrated over time: Wp.joint
=
JI (If’
XI+ IF .ol)
dt
for each joint. The total physiological fore be estimated as
defines trajectory A (Ax,,y,,~&.&,&). The solution is obtained by substituting the trajectory motions into the dynamic equations with the proper transformations between earth and body coordinates to yield the required controls. The results are shown in Fig. 2, where the controls are the 3 moments at the hip and the leg force. Also shown are the length of the ideal dynamic leg and net moment MZF in the floor plane at the location of support. Figure 3(a) shows the footprint (focus of support points) corresponding to this trajectory. In the second phase of the study, both the trajectories and controls are synthesized concurrently to produce optimal performance for the model with respect to a particular motion criterion. The results from two exam-
(6)
work W, can there-
Thus, if a force is exerted through some relative motion, a physiological actuator does real work and stores no energy: i.e. motion resulting from the internal forces is a manifestation of positive work done by a muscle and represents a positive expenditure of energy by the system. This arbitrary definition is likely to be an underestimate for a real system: it assumes that the construction is ‘optimal’ for the motion in the sense that a instantaneously properly single actuator oriented at each joint couid provide the proper force and motion. For this model there are three components of moment at the hip and ankle acting through three angles. With a massfess leg one hip angle and one ankle angle are superfluous and therefore can be arbitrarily fixed at constant values. The leg in the model has force and extension which is mechanically equivalent to a knee. For the ideal dynamic leg, Wankle= 0. An efficiency r) can then be evaluated as the ratio of the useful machine work to the physiological energy expenditure
73.
I i
LHS T, set O-05
RHS
LHS --I~0
LHS % set 0 -0~5
RHS
LHS --I,0
(8)
P
4. SOLUTION PROCEDUREI AND RESULTS
In the first phase of this study, the data for normal walking given in Murray et al. (1964) is utilized to obtain the corresponding model controt forces and moments. The parameters a, and 4, in equation (3) with one fluctuating term were evaluated for the 6 motions. This
d
4=-H 04
P :
- 04
(0) CM Trajectory
(b) Body
Fig. 2. Trajectory A results.
Controls
SYNTHESIS LHS f, *ec 0 -oo-5-
RHS
LHS I.0
T, set
LHS 0 -@5--0
OF BIPEDAL
RHS
77
LOCOMOTION RHS
LHS
LHS RHS 7, set 0 -Cvj-l.t 2ooo Left leg ‘-Right
LHS
LHS leg
f
.g *;
(a) CM Trajectory,
E-System
o
(b)
Body
Controls
-1.0 (L) (R) I.0
, r
-I-
(o) CM Trajectory
.E-System
o
(b) Body
Controls
Fig. 3. Trajectory B results.
Fig. 4. Trajectory C results
pies are shown in Figs. 3 and 4, and the corresponding footprints are given in Figs. 5a, 5b and jc respectively. In order to insure a feasible starting condition, trajectory A is used as the initial mathematical input in the synthesis process. The solution technique proceeds to find the set of aj and $j in the assumed form of equation (3) which best satisfies the motion criterion. The motion criterion in these two examples is a weighted combination of the size of the footprint and system physiological work as defined in equations (6) and (7). The results shown in Fig. 3 and 5b are for an objective of minimum footprint size with zero emphasis on the required internal work W,. The results of Figs. 4 and jc correspond to a motion criterion which requires the minimization of a
particular combination of footprint size and internal work W,,. In all three examples the mean location of the center of mass is the same (40 in. above ground), and the average velocity is 63 in./sec in the X,-direction. Due to the nature of the synthesis problem, theoretically the number of variables is not limited as long as they influence the performance criteria. Therefore, it may not be necessary to specify some of the motion parameters, such as stride frequency, time of heel strike, foot placement, etc. if optimal values of these are also desired. Table 2 shows W, and W, values and efficiency as defined by equations (5-8) for all trajectories. Ankle work was computed by assuming the body support force reactions are along the actual leg. Literature values for
78
M. A. TOWNSEND
IO -
and A. SEIREG
t
I
I
,
I
,
I
5
IO
15
20
25
30
35
/
40
I
I
45
50
Y E -in.
Fig. 5a. Loci
of support
points
Table
yE ,in
Za.
per set
(footprints).
Work of internal forces per stride. one stride at 63 in./sec average velocity. Values are given in in. lb for I set*
Table 2b. Human energy expenditure based on oxygen
consumption
(Ralston, 1958; Morton et nl. 1952)
State X$in
Fig. Sb. Enlargement of trajectory B footprint shown). c, Enlargement of trajectory C footprint
(left (left
shown). energy expenditure based on O2 consumption are given. Table 3 shows results for trajectories A and B as a function of CM average velocity. 5. DISCUSSION OF THE RESULTS The results given in Figs. 2 and 5a are calculated for trajectory A (obtained from curve-fitting human motions). In this case the mode1 gives relatively large footprints and a
Energy expenditure, (in. lb*)
BASAL Rest. alert 63-O in./sec *in. lb x 0.27
780 1240 3000 x
10e4 = kcal.
pigeon-toed gait. This can be attributed to the fact that the model does not incorporate pelvic relative rotation. In generating the trajectories, thoracic rotations were assumed to represent the total body rotations. Also, the mass and moments of inertia for the model (Table I) are the total values for the human body. Since the pelvic and thoracic motions are approximately 180” out of phase in human walking, it is expected that the footprint in the
SYNTHESIS
OF.BIPEDAL
‘79
LOCOMOTION
Table 3. Work of internal forces for one stride as a function of velocity. at I stride per sec.‘Values given in in. lb for 1 set’ CM
Total work (hip + leg i ankle)
Average
Trajectory
velocity (in./sec) 31.5
47.25 63 .O 78.75 94.5
A
W,
4102 4601 5424 6948 9184
0.301 0.230 0.142 0.053 0.018
513 498 412 2.51 5.3
2608 3 108 3884 5266 7155
0.197 0.160 0.106 0.048 -
7500 7884 8012 8228 8442
12621 13883 15600 17740 203 15
0,594 0.568 0.514 0.464 0.416
1269 2038 2989 3893 7610
5440 742 1 9878 12766 18348
0.233 0.275 0.303 0.305 0.415
31.5
“in.
lb x
0.27 x 10ei =
kcal
model could be affected by assuming a single motion for the whole body without incurring a significant effect on the’_magnitudes of the propelling hip moment MUand leg force. The effect of assuming massless legs and concentrating the entire body mass and moments 6f inertia into one rigid body could be expected to produce higher fluctuation in the synthesized trajectories and increase the magnitude of the controlling forces and moments in all cases. Although the modet.. as described, does not exactly duplicate the expected locus of support in normal human walking it could well represent highly controlled motions of a special type, such as a person with a stiff back and knee or a paraplegic in a walking machine. It is interesting to note that the footprint (Fig. 5a) is similar to that of a promenade as exemplified by a ballet dancer’s stage walk. That is, *heel strike’ and initial support occur at the ball region of the foot, eventually receding to the heel then proceeding to toe off. (Double leg support is indicated by the dotted lines between the calculated points.) This result is notable since the recorded data *This came out of conversations sin Ballet Company.
1)
for
1236 1059 770 362 -168
47.25 63.0 78.75 94.5
B
B,
Work
ideal dynam c leg (hip + leg) w, W” r)
in (1) was stated to show a demonstrable heel strike and a slight toe out. Further study of the footprint suggests that the initial strike could occur with a toe out, but that the combination of increasing floor moment MZF and decreasing leg force (Fig. 2) could allow the foot to pivot to the pigeon-toed stance. The numerical values for leg force and moments in this example bear some resemblance to the shapes shown by Bresler and Frankel ( 1950). Values in the present study are smoother and higher due to the form taken for the motion and the higher torso mass and inertias implicit in the considered model. Aside from the size of the footprint, there are several gaits similar to this case in phase relations and approximate magnitudes of motion. The previously-mentioned ballet promenade is used not only for its aesthetic value, but because it is smoother and under conscious control.* It is known that unless the heel is locked, foot strike after heel strike resembles a slap and that some other (controlled) movement during this phase is generally of a gross nature requiring substantial effort, and may be painful. Neverthless, a gait
with Prof. Tibor Zana, The University of Wisconsin, and Director of The Wiscon-
80
M. A. TOWNSEND
with a definite heel strike appears to involve less energy expenditure as in the case of trajectory C (Fig. 4 and 5c) and is physically more natural. Other motions similar to trajectory A are observed in ankle sprain gaits where the subject tends to place the whole foot down in the manner indicated, (Ducroquet et al. 1968) and in infants learning to walk where the whole foot is placed down and the body rocks slightly forward. The forces and moments for trajectory A (Fig. 3) show that the greatest effort is the hip moment, &I,, required to maintain the body upright in the lateral plane. The principal propelling moment. iV, is relatively small and the foot twisting moment MZF is also small. The length of the ideal dynamic leg increases noticeably when the support point is in the heel region and the body is relatively high. The physiological work computed is somewhat higher than oxygen consumption tests indicate (Table 3). Trajectory B (Figs. 3, Sa and 5b) demonstrates that a very small support zone (2 in. long by 0.7 in. wide) is possible as long as it provides the required frictional resistance to the ground moment. This trajectory has very high Wf and W, values. Comparing trajectory B to A, the x variation (4~~) is essentially the same, but the phase relations and amplitudes of the y, and iE motions are considerably different. The angular relations also are substantially altered, particularly &. In this trajectory there is essentially no lateral motion of the CM, (-+O.OOSin.) and the CM is at its highest point at heel strike. resulting in completely different values for leg forces and hip moments. This trajectory also shows foot-strike at a ‘mid-foot’ position, as in A, even considering the small footprint. This trajectory also shows no double-Ieg support phase. The principle stabilizing moment M, and the leg forces are very large in this case. The ground moment is negligible. However, both W, and W, are extremely high but so is the ‘efficiency’, Table 2. Trajectory C (Figs. 4, 5a and 5c) which is
and A.
SElREG
synthesized to minimize footprint size and W, shows a definite heel strike and double-leg support phase, a slight toe out, and a reasonable foot size. The sagittal plane translational motions AxE and zE have similar phasic relations as A but with reduced amplitudes. The yE motion is small but is approximately 6~ that of trajectory B. The complete reversal and relatively large amplitude of 8, (horizontal plane rotation) and large foot moments are the notable differences from what might have been expected for the normal human motion trajectory. The result is a ‘double limp’ type of stride -resembling gaits described by the Ducroquets (1968) for severe sagittal plane ankyloses (pp. 162- 174) and painful stiffness gaits (pp. 128- 129). Some of the characteristics of decreased agility observed by Murray et al. ( 1969) also can be detected in the results. In this case, the body twists about the zE axis in anticipation of the support point reiative to the position of the body. Thus the rotation is more in phase with the human pelvic rotation than the thorax if compared to human motion data. The total rotation is much larger, however. The propelling moment M, is approximately the same as for the original trajectory A, but the main stabilizing moment M, has less variability than in A. There is much less fluctuation in the length of the ideal dynamic leg and in the leg force than in any of the other trajectories. The floor moment MZF is very large, which is not desirable. The work values necessary to produce this trajectory (Table 2) are more compatible with values in the literature. Figure 6 shows the variation in footprint for trajectory A for several mean velocities of walking. At lower velocity the pigeontoeing increases and the footprint gets smaller, and vice versa. Note that if only the translational component Vi in equation (3) for the XEdirection were changed, the work done by the different forces, the footprint sizes, etc. would all remain the same as before; only the footprint spacing would change, since any constant velocity does not alter the forces.
SYSTHESIS
78.25
IO in
I
Direction X
Heel
47~75
IPS
31.5
OF BIPEDAL IPS
63.0
IPS (Design
94.5
IPS
81
LOCOMOTION Point
1
IPS
I of Movement
-
Strike
Fig. 6. Footprint
as a function of velocity, for trajectory A.
Tabie 3 shows that the work IV,, values may increase or decrease with changing walking velocity depending upon the type of trajectory. The physiological work W,, however increases with velocity in all cases. In all cases considered in this study, the largest moments are found to be associated with M,, which is mainly a stabilizing moment. Therefore, large swagger motions (0,) would tend to yield high work values at the hip joint. It appears as illustrated by trajectory B the minimum footprint size can result from allowing relatively large sagittal plane rotations and vertical motion. These tend to be costly motions on an energy basis. For less work and a ‘reasonable’ size for the footprint the tendency is toward greater ys-motion, substantial rotation in the horizontal plane and a definite heel strike, as in trajectory C. However, the contrived absence of any motion (requiring a motion to be of zero amplitude and thereby removing a degree of freedom) seems to cause either excessive variations in the other motions or yield unacceptable footprints, work, or both. Small motions are not necessarily insignificant and may be necessary for dynamic support. stability and control. In assessing the work done in locomotion
there is no reason for the values of Tables 2 and 3 to be what is expected in a human motion. These are evaluated as defined earlier and are strictly machine equivalents. Efficiencies here (and elsewhere) are questionable. Several investigators have evaluated work output as the difference between maximum and minimum system total energy levels. Efficiency is then calculated from this value divided by the total caloric value of oxygen consumed during the activity. Such a procedure implies that work is required to lift the body and that it can drop freely. It is also implied that the oxygen consumption represents the actual state of internal energy exchange. It is well known that the body does not fall freely during the decreasing energy phase. The true work of all the internal forces in a human cannot be evaluated yet, as adequate data for muscle behavior during human locomotion does not exist. Most investigations show IO-25 per cent overall ‘efficiency’ for various forms of motion. Meanwhile, Hill (1963) has shown that isolated muscles may have efficiencies up to 50 per cent based on measurements of work output and heat generated by the muscle. 6. coscLusIoss
A general theory for analysis and synthesis
82
M. A. TOWNSEND
of bipedal locomotion is presented and demonstrated with numerical examples for a model consisting of a rigid body with massless extensible legs. Data on human body locomotion has been used for guiding the form of the synthesized motions. The mode1 is adequate for design of prosthetic devices for paraplegics and bipedal walking machines. Synthesized trajectories and corresponding control forces can be evaluated for any required motion criterion. The criteria used in this study are the maximum stability (as characterized by minimizing the footprint) and combinations of stability plus minimum energy expenditure with different weighting factors. Of the examples discussed in this paper trajectory C (Figs. 4 and 5c) appears to be more suitable for a biped walker since it represents reasonable energy requirements and support locus conditions. Although the model is not completely representative of human locomotion, it can be used to clarify some necessary concepts in terms of trajectories, support, stability and energy expenditure. It can also provide insight into certain controlled motions which may be approximately represented by the model. The approach developed in this study is being extended for more complex models which can better simulate normal human motions. REFERENCES Braune, W. and Fischer, 0. (1890) Uber den Schwerpunkt des menschlichen Korpers mit Riicksicht auf auf die Ausriistung des deutchen Infanteristen. (Concerning the center of gravity of the human body with reference to the equipment of the German infantry.) Abh. d. math.-phys. cf.d.kSachs. Gesellsch. d. Wiss. 15. Bresler. B. and Frankel. J. P. (1948) The forces and moments in the leg during level walking. Trans. ASME 72 17-36 (1950). presented at 1948 Annual Meeting, New York, Nov. 28-Dec. 3, Paper no. 48-A-62. Cavagna, G. H., Saibene, F. P. and Margaria. R. (1963) External work in walking. J. appl. Physiol. 18. 1-9. Contini. R., Gage, H. and Drillis. R. (1965) Human gait characteristics. In Biomechanics and Related Bioenaineerina Topics (Ed. R. M. Kenedi). Peraamon Press, Oxford. (Proceedings of a symposium held in Glasgow, Sept. 1964.) Ducroquet, R.. Ducroquet, J. and Ducroquet. P. (1968) Walking and Limping, p. 284. Lippincott, Philadelphia and Toronto.
and A. SEIREG Elftman. H. (1939) The force exerted by the ground in walking. Human Biol. 2, 529-535. (Also Arbeirrphysiologie 10,485-49 1, 1939.) Elftman. H. I 1939) The force exerted by the ground in walking. Human Biol. 2, 529-535. (rUso Arbeitsphysiologie 10.485-49 I, 1939.) Elftman. H. (1939) The rotation of the body in walking. Arbeitsphysiologie 10,477-483. Efltman. H. (1939) Forces and energy changes in the leg during walking. Am. J. Physiol. 12%339-356. Fischer, 0. ( 1898-1904) Der Gang des Menschen. (‘Human gait’.) Abh. siichs. Ges. Wisi. Vols. 2 I-‘8. Frank. A. A. and Vukobratovic. M. (1969) On the synthesis of biped locomotion machines. 8th international Conference
on Medical
and Biological
Engineering.
Evanston, Illinois. Frank. A. A. An approach to the dynamic analysis and synthesis of biped locomotion machines. In preparation at the University of Wisconsin. Madison, Wisconsin. Greenwood, D. T. (1965) Principles of Dynamics. Prentice-Hall, Englewood Cliffs. New Jersey. Hill, A. V. (1963) The ratio of mechanical power to total power expended during muscle shortening. Proc. R. Sot. B159,3 19-334. Hill, A, V. (1965) Trails and Trials in Physiology. p. 374. Williams and Wilkins. Baltimore. Kane, T. R. and Scher. IM. P. (1970) Human self-rotation bv means of limb movements. J. Biomechanics. 3. 34-40.
Lascari, A. T. (1970) The Felge handstand- a comparative kinetic anaiysis of a gymnastics skill. PhD Thesis. Dept. of Physical Education, University of Wisconsin. Lukin. L.. Polissar. M. J. and Ralston. H. J. (1967) IMethods for studying energy costs and energy flow during human locomotion. Haman Facrors 9,603- 608. Margaria. R. ( 1968) Positive and negative work performance and their efficiencies in human locomotion. Int. 2. angew Physiol. 2% 339-35 I. Morton, D.J. and-Fuller, D. 0. (195’) Human Locomotion and Body Form, D. 185. Williams and Wilkins, Baltimore. ’ _ IMurray, M. P., Drought, A. B. and Kory. R. C. (1964) Walking patterns of normal men. J. Bone Jr. Slug. 46A. 335-360.
Murray, M. P., Kory, R. C., Clarkson. B. H. and Sepic. S. B. (1966) Comparison of free and fast speed walking patterns of normal men. Am. J. phys. Med. 4% 8-24. Murray, M. P., Seireg, A. and Scholtz. R. C. (1967) Center of gravity, center of pressure and supportive forces during human activities. J. app/. Physiol. 23. 83 1-818.
Murray, M. P., Kory. R. C. and Clarkson. B. H. (1969) walking patterns in healthy old men. J. Geront. 24, 169-178.
Murray, M. P. ( 1967) Gait as a total pattern of movement. Am. J. phys. Med. 46, 190-333. (Excellent bibliography.) Paul. J. P. (1965) Bio-engineering studies of the forces transmitted by joints: (11) Engineering analysis. In Biomechanics
and Related
Bio-Engineering
Topics
(Ed. R. M. Kenedi). Pergamon Press, Oxford. (Proceedings of a Symposium held in Glasgow, Sept. 1964.) Paul, J. P. (1967) Forces transmitted by joints in the
SYNTHESIS
OF BIPEDAL
human body. Proc. Znstn me& Engrs 181 (part 3-J). 8- 15. (Presented at the Inst. of Mech. Eng. Symposium in Lubrication and Wear in Living and Artificial Human Joints, Paper 8. London. April 7, 1967.) Peiser. E., Wright, D. W. and Mason, C. ( 1969) Human locomotion. BuK Prosthetics Res. Bpr IO - 11.18- 104. Ralston. H. J. (1958) Energy-speed relation and optimal speed during level walking. Inr. 2. angew. Physiol. 17, 277-183. (Formerly Arbeitsphysiologie.) Ralston. H. J. and Lukin. L. (1969) Energy levels of human body segments during level walking. Ergonomics 12.39-46.
LOCOMOTION
83
Saunders. J. B., Inman. V. T. and Eberhardt, H. D. (1953) The major determinant in normal and pathological gait. J. Bone Jt. Surg. 35A, 543-548. Seireg. A. and Peterson, R. 0. (Unpublished at present.) Construction of a manually controlled, self powered load-carrying tripod walker. Work being done at the University of Wisconsin, Madison, Wisconsin. Shames, I. H. (1966) Engineering Mechanics: Dynamics. Vol. 2. Prentice Hall, Englewood Cliffs, New Jersey. Smith, P. G. and Kane, R. R. (1968) On the dynamics of the human body in free fall. J. uppl. Mech. 35, l67- 168.
OF BIPEDAL
LOCOMOTION*
M. A. TOWNSENDt and 4. SEIREGS Xlechanical Engineering Department. University of Wisconsin. Madison. Wisconsin, U.S.A. Abstract-This paper presents a mathematical model for the synthesis and control of bipedal walkers. The model deals with the three-dimensional motion of a symmetrical rigid body with massless extensible legs. Body trajectories and controls are synthesized for optimum stability and energy expenditure. The model provides a rigid body idealization of human locomotion and a rational tool for design of prosthetic devices and walking machines. 1. INTRODUCTION
oxygen consumption. Work has been done at the University of California (Ralston, 1958; Ralston and Lukin, 1969: Saunders ef al. 1953; Lukin et al. 1967) and at Milan (Cavagna et al. I963 ; Margaria, 1968) on estimating external work and efficiency of motion. One may find values for energy input during human activities in the work of Ralston (1958: 1969) or in texts such as that of Morton and Fuller (1952). Hill (1965) has exhaustively investigated the biophysical aspects of muscle motion, energy and efficiency; is a bibliography of his work. Another area receiving attention in recent years is the development of load carrying walking machines which would be preferable to wheel chairs. Work by Seireg and Peterson (unpublished) on the actual construction of a manually controlled, load carrying tripod walker (2 legs and a pusher) indicates that such machines may be realizable. Since Elftman ( I939a.b.c), analytical (engineering) studies have been largely dormant until very recently. Frank and Vukobratovic (1969) consider the classes of a biped motion, and Frank (in preparation) has proposed a mechanical design approach to the construction of a machine similar to the model of this paper, which he has modelled
of human and animal locomotion has a long and colorful history, albeit fitful, but it has only recently begun emerging from the stages of copious (but often incomplete) data gathering and causal explanations of the data. Observations such as ‘-... the whole of this ensemble acts in unison with a single and complete rhythm, fusing the whole enormous complexity into clear and harmonious simplicity”§ may be true, but do not explain the phenomenon. Quantitative analysis of many aspects of human locomotion are now available in the clinical studies of gait characteristics by Murray et nl. (1964, 1966, 1969, 1967), Contini et al. ( 1964). and the Ducroquets ( 1968). Fischer (1898- 1904) and Braune and Fischer ( 1890) initiated the rational study of human locomotion, followed by Elftman’s studies on the dynamics of motion (1939a, b, c). These have been expanded by the studies of leg and joint forces of Bresler and Frankel (1950). Paul (1965. 1966-67), and summarized by Peizer er al. (1969). Evaluations of energy transfer, useful work done during human activity, and ‘efficiency’ have been somewhat less successful, although Ralston (1958) has determined an optimal walking velocity based on the energy equivalent of
THE STUDY
“Receiced
22 Februnrv
I97
I.
iResearch Assistant. (Now Associate Professor. Univ. of Toronto, Canada.) f Professor. SQuoted from Bernstein et al. “Investigation on Biodynamics of Locomotion”, Conrini. (ibid. p. 119) employs a similar description. 71
Vols. 1.2. MOSCOW1935. 1910 by
72
IV. A. TOWNSEND
on a hybrid computer. Kane and Scher (1970) and Smith and Kane (1968) have considered the mechanics of motion during free fall conditions. The kinematic and dynamic characteristic of other human activities standing, jumping, etc.-have been studied in terms of the support characteristics by Murray ef al. (1967). Studies of athletic events have yielded interesting results; Lascari (1970) considered the linear impulse* of the external forces as a significant measure of effort and observed that what is considered beautiful may not be most ‘efficient’. This aspect will be discussed later in the paper. 1. HYPOTHESES
The model (Fig. 1) consists of a single rigid body, which represents the body mass and inertias, and two rigid extendable,
Fig. 1. Model and nomenclature.
and A. SEIREG
massless legs. As such, the system has six degrees of freedom: the three motions of the center of mass in the direction of the space coordinates and three successive rotations of the body about a set of body-fixed coordinate axes. The system is situated (location and orientation) relative to an earth-fixed (inertial) reference frame E at OE. For a general system. each body of the system is defined by a set of body-fixed, right-hand principal coordinates axes. here 0,. In multiple-body systems (more complex models), one body of the system can be designated as the main body. Its position relative to the inertial frame is defined by the location of its center of mass (CM) and three Euler angles.* The positions of the other bodies of the system are then described relative to the main body in terms of relative coordinates (internal variables) between the bodies. In the model considered in this study, the main body is defined by axes at 0,. Each leg (‘bodies’ 2 and 3) is assumed to have a leg-fixed coordinate system with the z_-axis along the leg and the origin at the ‘hip’. The motion and the controlling forces (controls) can be evaluated if we know the location of 0, relative to OE and the three rotations necessary to orient the body in space. By computing the forces at the hip points we can directly evaluate the leg forces and support points without a priori assumptions of foot placement. An ideal dynamic leg is defined as the straight rod connecting the designated body support point (hip) to the instantaneous earth support point. The locus of the earth points for each step determines the *footprint’. The motion to be studied is level walking. For the present study it is assumed that the body is at all times supported by a single ideal dynamic leg (right or left) and that the change from one to the other is
*Linear impulse = ,z F dt. *The Euler angle system used here is successive rotation about the body fixed z, y.x axes, each rotation taking about the present location as defined by the previous rotation(s): (Shames. 1966).
Place
SYNTHESIS
OF BIPEDAL
essentially instantaneous. This assumption is a convenient device to facilitate the synthesis of body trajectories, controls and locus of support. The motion sequence and control for the acutal locomotion (including the double leg support phase) can be directly derived from the results of the synthesis. The ideal dynamic leg is the exact representation of the support between heel strike (represented by x in Fig. 5) and the onset of double leg support (or ‘toe off point, shown by 0 in the figures). During double leg support (shown by dashed lines in Fig. 6) the ideal dynamic leg represents the resultant effect of the two real legs. ‘: The assumption of a single ideal dynamic leg support at all times, therefore. considerably simplifies the problem formulation and computational procedure without affecting in any way the motion trajectories, locus of support, and computed controls. Body mass, principal moments of inertia, and salient geometry are given in Table 1. In this analysis we are mainly concerned with the body dynamics rather than jvith the kinematic relations involved in the motion (as in Paul, 1963: Bresler and Franked. 1950). The kinematic constraints of fixed-length leg segments can easily be superimposed later, with at most a constant shift in the average vertical location of the body CM. This does Table Body mass Principal
I. X40&1parameters
0.1167 lb sec’iin.
t 161 lb vveight)
moments of inertia of body lxx = 60 lb in. sec3 I,, = 60 lb in. se? I,, = 12 lb in. set’ Location of hip points in body system tx.~.:). in. right rl-’ = (0.4.0. -9.0) left r,:> = 10. -6.0. -9.0) \lean location of CM in earth system R,taverage) = (63,0”t, O., -40.0) @,(average) = rO.O.0)
LOCO~lOTION
73
not have any effect on the work-causing motions or the resulting controls (forces and moments), and the footprint size will change proportional to the new value of the average height of the CM. Actual foot location can be chosen for minimum work or arbitrarily selected, but the decision is not necessary at this point in the analysis. As shown later (Tables 2 and 3), the effect of foot placement is to change the work values. The main point is that there is no fundamental mathematical difference between an extensible leg and a leg with a knee. The proposed model can be used for the following purposes: ( I) estimating forces and moments generated by the body necessary for stable locomotion: (2) evaluating the variables and criteria fat different patterns of walking (and motion in general): (3) comparing human vs. biped-machine walk. (4) determine system (internal) work required. bvork done as a machine. and ‘efficiency’, and (5) ‘optimal’ design of a biped uaiker. Much of the above depends upon the ability of the model to generate trajectories for specified criteria of motion for the system. If the results are reasonable, such models provide insight into the real systems. For example, it can be anticipated that walking of (a man fitted with) a biped prosthesis is fundamentahy different from normal human motion since the principal mass of the human body is highly segmented and rotationally coupled. Conversely. a man fitted with a biped walker will undoubtedly be strapped-in for practical reasons: thus, this motion can be simulated by this first-generation model. Finally, a ‘good’ model may eventually be physically analogous to a real system, but this is not a requirement.
‘The control forces during the double leg support phase can be directly computed from these results without influencing the system dynamics or the synthesized trajectories. The controls of both legs at the assumed physical positions of toe off and next heel strike respectively can be evaluated. for example. as the components of the controls of the ideal dynamic leg at any instant during this phase from static equilibrium considerations.
M. A. TOWNSEND
74 3. ANALYSIS
(a) Dynamic equations The conditions for the dynamic equilibrium of the model under consideration can be expressed in body coordinates as (Shames, 1966; Greenwood, 1965): Vector summation of forces
=x
4+mg+(-ma,)=O
(I)
Vector summation of moments = C, mi+ where
(-I$)
=O*
(2)
m = total mass of the body fi = the force vectors acting on the body at the hips (hip forces) mg = the weight of the body -ma, = the inertia force vector a, = the acceleration vector of the center of mass @, = the moment vectors acting on the body at the hips (hip moments) (-I&) = the inertial moment vector abopt the center of mass.
The acceleration vector a, of the center of mass, and the angular velocity o and angular acceleration C;, necessary for evaluation of & are initially determined from the displacements and rotations of the body as observed in earth coordinates. The. principal axes of inertia for the body are the body-fixed reference axes: the moments of inertia about these axes are used in calculating &. The forces fi and moments M.-i which cause the body to move are the active controls. If either all the motions or all the forces or a proper combination of these are known, equations (1) and (2) can be solved for the remaining variables. This becomes an analytic problem with only one possible solution. On the other hand, the synthesis of a system of controis fi*P& and resulting motions requires solution of a system of non-linear
and A. SEIREC
ordinary differential equations determined by equations (l), (2) and the kinematic constraints. This results in more unknowns than there are equations and consequently many possible solutions. To select the ‘best’ solutions a performance criterion for the resulting motion and controls must be defined. Even so, a solution may not be possible due to the non-linear nature of the equations of motion and each of a general existence theory for such systems of equations. Examples of the analytic and synthesis problem are given in this paper. (b) Trajectory analysis The data for normal human walking in Murray et al. (1964), is used in this analysis to provide a nominal history for the motion of the center of mass and the three body rotations. Since walking motions are periodic, each motion component can be represented by the sum of a polynomial and a Fourier series. The polynomial is usually formulated in terms of an average location and an average velocity. Hence, trajectories for walking (and other steady-state activities climbing stairs, etc.) can be represented in the form:
where
qi = different coordinates of the motion, (i coordinate) (2 steps per % = stride frequency stride) in rad./sec 4i = mean position Vi = average velocity at a ak = amplitude of fluctuation particular frequency (kw) c#+= phase shift.
In order to simplify the analysis, the following observations can be made on the experimental data. First, in the mean sagittal plane (-TE- zE) the dominant fluctuations occur at twice the stride frequency (i.e. at step frequency). This applies to the xE and zE motions
SYNTHESIS
OF BIPEDAL
LOCOMOTIOS
7s
/.Lj=Mj+rjXfi where as well as the 8? rotation about the y,-axis. Fi,Xi,Ai are the external forces acting Second, the dominant amplitudes of the other on the system and the associmotions (yE, 8, about z,-axis, and e3 about x1ated motions and velocities axis) occur at the stride frequency. A first (assumed to act at CM) approximation to the motion is to use one fjxjsj are the internal forces and term of the Fourier series in equation (3) corresponding relative angles with k = 2 for the sagittal motions (4i = XE, q3=zE. q5=Bn); and k= 1 for the other Mj,8j,aj are the internal moments and motions (qZ=yErqJ=01,46=8,). corresponding relative angles Differentiation of equation (1) gives the rj is the distance from the CM velocities and accelerations. Depending on the to the point of application of coordinate system in which the motions are the force determined, a transformation to the body Wj = function (6). coordinate system may be necessary for the Equation (5) is more convenient for the evaluation of the control forces and moments present discussion. The first integral is the at the hips and legs. work done by the external forces on the system. The second integral is the work done by CC)Motion criferia the internal forces of the system. Over one cycle (one stride) the first integral In this analysis two criteria for walking are considered. The first is the minimization of the is zero for level walking. As this is not a machine the second integralsize of the footprint. Footprint size and the conservative path of the center of support can be used as a and hence the net work done- need not be zero. It may be positive or negative; the sign measure of the system stability and the physical realizability of the motion. of each product indicates whether that control The other criterion relates to the efficiency acted as a motor or brake over the cycle. If the of performance as indicated by the useful net work of the system forces is negative, then work done and the energy expended in the the net effect is a braking action. Since equaprocess. Because of its importance to this tions (4) and (5) are for the whole system, study, a detailed discussion of work and negative work done by one part of the system energy in locomotion is given in the following cancels positive work done by another part. section. However, real systems do work at specific locations, so it is also necessary to evaluate the local net work done at each of the salient (d) Work points: hip, leg and ankle. The real processes The work done by the system under con- of doing work (motors) and braking (dissipasideration as a machine W, can be readily tors) are usually fundamentally different with evaluated from little effective energy transfer between them and minimal energy storage during braking. To estimate the work as would be performed by an equivalent physiological system which expends energy at specific locations, we use the following rationale: the human body is a contractile mechanism-the muscles do not perform work in extension and store negligible mechanical energy. The physiological work done at point of relative motion is assumed to
M. A. TOWNSEND
76
and A. SEiREG
be the absolute value of the net power output at that point integrated over time: Wp.joint
=
JI (If’
XI+ IF .ol)
dt
for each joint. The total physiological fore be estimated as
defines trajectory A (Ax,,y,,~&.&,&). The solution is obtained by substituting the trajectory motions into the dynamic equations with the proper transformations between earth and body coordinates to yield the required controls. The results are shown in Fig. 2, where the controls are the 3 moments at the hip and the leg force. Also shown are the length of the ideal dynamic leg and net moment MZF in the floor plane at the location of support. Figure 3(a) shows the footprint (focus of support points) corresponding to this trajectory. In the second phase of the study, both the trajectories and controls are synthesized concurrently to produce optimal performance for the model with respect to a particular motion criterion. The results from two exam-
(6)
work W, can there-
Thus, if a force is exerted through some relative motion, a physiological actuator does real work and stores no energy: i.e. motion resulting from the internal forces is a manifestation of positive work done by a muscle and represents a positive expenditure of energy by the system. This arbitrary definition is likely to be an underestimate for a real system: it assumes that the construction is ‘optimal’ for the motion in the sense that a instantaneously properly single actuator oriented at each joint couid provide the proper force and motion. For this model there are three components of moment at the hip and ankle acting through three angles. With a massfess leg one hip angle and one ankle angle are superfluous and therefore can be arbitrarily fixed at constant values. The leg in the model has force and extension which is mechanically equivalent to a knee. For the ideal dynamic leg, Wankle= 0. An efficiency r) can then be evaluated as the ratio of the useful machine work to the physiological energy expenditure
73.
I i
LHS T, set O-05
RHS
LHS --I~0
LHS % set 0 -0~5
RHS
LHS --I,0
(8)
P
4. SOLUTION PROCEDUREI AND RESULTS
In the first phase of this study, the data for normal walking given in Murray et al. (1964) is utilized to obtain the corresponding model controt forces and moments. The parameters a, and 4, in equation (3) with one fluctuating term were evaluated for the 6 motions. This
d
4=-H 04
P :
- 04
(0) CM Trajectory
(b) Body
Fig. 2. Trajectory A results.
Controls
SYNTHESIS LHS f, *ec 0 -oo-5-
RHS
LHS I.0
T, set
LHS 0 -@5--0
OF BIPEDAL
RHS
77
LOCOMOTION RHS
LHS
LHS RHS 7, set 0 -Cvj-l.t 2ooo Left leg ‘-Right
LHS
LHS leg
f
.g *;
(a) CM Trajectory,
E-System
o
(b)
Body
Controls
-1.0 (L) (R) I.0
, r
-I-
(o) CM Trajectory
.E-System
o
(b) Body
Controls
Fig. 3. Trajectory B results.
Fig. 4. Trajectory C results
pies are shown in Figs. 3 and 4, and the corresponding footprints are given in Figs. 5a, 5b and jc respectively. In order to insure a feasible starting condition, trajectory A is used as the initial mathematical input in the synthesis process. The solution technique proceeds to find the set of aj and $j in the assumed form of equation (3) which best satisfies the motion criterion. The motion criterion in these two examples is a weighted combination of the size of the footprint and system physiological work as defined in equations (6) and (7). The results shown in Fig. 3 and 5b are for an objective of minimum footprint size with zero emphasis on the required internal work W,. The results of Figs. 4 and jc correspond to a motion criterion which requires the minimization of a
particular combination of footprint size and internal work W,,. In all three examples the mean location of the center of mass is the same (40 in. above ground), and the average velocity is 63 in./sec in the X,-direction. Due to the nature of the synthesis problem, theoretically the number of variables is not limited as long as they influence the performance criteria. Therefore, it may not be necessary to specify some of the motion parameters, such as stride frequency, time of heel strike, foot placement, etc. if optimal values of these are also desired. Table 2 shows W, and W, values and efficiency as defined by equations (5-8) for all trajectories. Ankle work was computed by assuming the body support force reactions are along the actual leg. Literature values for
78
M. A. TOWNSEND
IO -
and A. SEIREG
t
I
I
,
I
,
I
5
IO
15
20
25
30
35
/
40
I
I
45
50
Y E -in.
Fig. 5a. Loci
of support
points
Table
yE ,in
Za.
per set
(footprints).
Work of internal forces per stride. one stride at 63 in./sec average velocity. Values are given in in. lb for I set*
Table 2b. Human energy expenditure based on oxygen
consumption
(Ralston, 1958; Morton et nl. 1952)
State X$in
Fig. Sb. Enlargement of trajectory B footprint shown). c, Enlargement of trajectory C footprint
(left (left
shown). energy expenditure based on O2 consumption are given. Table 3 shows results for trajectories A and B as a function of CM average velocity. 5. DISCUSSION OF THE RESULTS The results given in Figs. 2 and 5a are calculated for trajectory A (obtained from curve-fitting human motions). In this case the mode1 gives relatively large footprints and a
Energy expenditure, (in. lb*)
BASAL Rest. alert 63-O in./sec *in. lb x 0.27
780 1240 3000 x
10e4 = kcal.
pigeon-toed gait. This can be attributed to the fact that the model does not incorporate pelvic relative rotation. In generating the trajectories, thoracic rotations were assumed to represent the total body rotations. Also, the mass and moments of inertia for the model (Table I) are the total values for the human body. Since the pelvic and thoracic motions are approximately 180” out of phase in human walking, it is expected that the footprint in the
SYNTHESIS
OF.BIPEDAL
‘79
LOCOMOTION
Table 3. Work of internal forces for one stride as a function of velocity. at I stride per sec.‘Values given in in. lb for 1 set’ CM
Total work (hip + leg i ankle)
Average
Trajectory
velocity (in./sec) 31.5
47.25 63 .O 78.75 94.5
A
W,
4102 4601 5424 6948 9184
0.301 0.230 0.142 0.053 0.018
513 498 412 2.51 5.3
2608 3 108 3884 5266 7155
0.197 0.160 0.106 0.048 -
7500 7884 8012 8228 8442
12621 13883 15600 17740 203 15
0,594 0.568 0.514 0.464 0.416
1269 2038 2989 3893 7610
5440 742 1 9878 12766 18348
0.233 0.275 0.303 0.305 0.415
31.5
“in.
lb x
0.27 x 10ei =
kcal
model could be affected by assuming a single motion for the whole body without incurring a significant effect on the’_magnitudes of the propelling hip moment MUand leg force. The effect of assuming massless legs and concentrating the entire body mass and moments 6f inertia into one rigid body could be expected to produce higher fluctuation in the synthesized trajectories and increase the magnitude of the controlling forces and moments in all cases. Although the modet.. as described, does not exactly duplicate the expected locus of support in normal human walking it could well represent highly controlled motions of a special type, such as a person with a stiff back and knee or a paraplegic in a walking machine. It is interesting to note that the footprint (Fig. 5a) is similar to that of a promenade as exemplified by a ballet dancer’s stage walk. That is, *heel strike’ and initial support occur at the ball region of the foot, eventually receding to the heel then proceeding to toe off. (Double leg support is indicated by the dotted lines between the calculated points.) This result is notable since the recorded data *This came out of conversations sin Ballet Company.
1)
for
1236 1059 770 362 -168
47.25 63.0 78.75 94.5
B
B,
Work
ideal dynam c leg (hip + leg) w, W” r)
in (1) was stated to show a demonstrable heel strike and a slight toe out. Further study of the footprint suggests that the initial strike could occur with a toe out, but that the combination of increasing floor moment MZF and decreasing leg force (Fig. 2) could allow the foot to pivot to the pigeon-toed stance. The numerical values for leg force and moments in this example bear some resemblance to the shapes shown by Bresler and Frankel ( 1950). Values in the present study are smoother and higher due to the form taken for the motion and the higher torso mass and inertias implicit in the considered model. Aside from the size of the footprint, there are several gaits similar to this case in phase relations and approximate magnitudes of motion. The previously-mentioned ballet promenade is used not only for its aesthetic value, but because it is smoother and under conscious control.* It is known that unless the heel is locked, foot strike after heel strike resembles a slap and that some other (controlled) movement during this phase is generally of a gross nature requiring substantial effort, and may be painful. Neverthless, a gait
with Prof. Tibor Zana, The University of Wisconsin, and Director of The Wiscon-
80
M. A. TOWNSEND
with a definite heel strike appears to involve less energy expenditure as in the case of trajectory C (Fig. 4 and 5c) and is physically more natural. Other motions similar to trajectory A are observed in ankle sprain gaits where the subject tends to place the whole foot down in the manner indicated, (Ducroquet et al. 1968) and in infants learning to walk where the whole foot is placed down and the body rocks slightly forward. The forces and moments for trajectory A (Fig. 3) show that the greatest effort is the hip moment, &I,, required to maintain the body upright in the lateral plane. The principal propelling moment. iV, is relatively small and the foot twisting moment MZF is also small. The length of the ideal dynamic leg increases noticeably when the support point is in the heel region and the body is relatively high. The physiological work computed is somewhat higher than oxygen consumption tests indicate (Table 3). Trajectory B (Figs. 3, Sa and 5b) demonstrates that a very small support zone (2 in. long by 0.7 in. wide) is possible as long as it provides the required frictional resistance to the ground moment. This trajectory has very high Wf and W, values. Comparing trajectory B to A, the x variation (4~~) is essentially the same, but the phase relations and amplitudes of the y, and iE motions are considerably different. The angular relations also are substantially altered, particularly &. In this trajectory there is essentially no lateral motion of the CM, (-+O.OOSin.) and the CM is at its highest point at heel strike. resulting in completely different values for leg forces and hip moments. This trajectory also shows foot-strike at a ‘mid-foot’ position, as in A, even considering the small footprint. This trajectory also shows no double-Ieg support phase. The principle stabilizing moment M, and the leg forces are very large in this case. The ground moment is negligible. However, both W, and W, are extremely high but so is the ‘efficiency’, Table 2. Trajectory C (Figs. 4, 5a and 5c) which is
and A.
SElREG
synthesized to minimize footprint size and W, shows a definite heel strike and double-leg support phase, a slight toe out, and a reasonable foot size. The sagittal plane translational motions AxE and zE have similar phasic relations as A but with reduced amplitudes. The yE motion is small but is approximately 6~ that of trajectory B. The complete reversal and relatively large amplitude of 8, (horizontal plane rotation) and large foot moments are the notable differences from what might have been expected for the normal human motion trajectory. The result is a ‘double limp’ type of stride -resembling gaits described by the Ducroquets (1968) for severe sagittal plane ankyloses (pp. 162- 174) and painful stiffness gaits (pp. 128- 129). Some of the characteristics of decreased agility observed by Murray et al. ( 1969) also can be detected in the results. In this case, the body twists about the zE axis in anticipation of the support point reiative to the position of the body. Thus the rotation is more in phase with the human pelvic rotation than the thorax if compared to human motion data. The total rotation is much larger, however. The propelling moment M, is approximately the same as for the original trajectory A, but the main stabilizing moment M, has less variability than in A. There is much less fluctuation in the length of the ideal dynamic leg and in the leg force than in any of the other trajectories. The floor moment MZF is very large, which is not desirable. The work values necessary to produce this trajectory (Table 2) are more compatible with values in the literature. Figure 6 shows the variation in footprint for trajectory A for several mean velocities of walking. At lower velocity the pigeontoeing increases and the footprint gets smaller, and vice versa. Note that if only the translational component Vi in equation (3) for the XEdirection were changed, the work done by the different forces, the footprint sizes, etc. would all remain the same as before; only the footprint spacing would change, since any constant velocity does not alter the forces.
SYSTHESIS
78.25
IO in
I
Direction X
Heel
47~75
IPS
31.5
OF BIPEDAL IPS
63.0
IPS (Design
94.5
IPS
81
LOCOMOTION Point
1
IPS
I of Movement
-
Strike
Fig. 6. Footprint
as a function of velocity, for trajectory A.
Tabie 3 shows that the work IV,, values may increase or decrease with changing walking velocity depending upon the type of trajectory. The physiological work W,, however increases with velocity in all cases. In all cases considered in this study, the largest moments are found to be associated with M,, which is mainly a stabilizing moment. Therefore, large swagger motions (0,) would tend to yield high work values at the hip joint. It appears as illustrated by trajectory B the minimum footprint size can result from allowing relatively large sagittal plane rotations and vertical motion. These tend to be costly motions on an energy basis. For less work and a ‘reasonable’ size for the footprint the tendency is toward greater ys-motion, substantial rotation in the horizontal plane and a definite heel strike, as in trajectory C. However, the contrived absence of any motion (requiring a motion to be of zero amplitude and thereby removing a degree of freedom) seems to cause either excessive variations in the other motions or yield unacceptable footprints, work, or both. Small motions are not necessarily insignificant and may be necessary for dynamic support. stability and control. In assessing the work done in locomotion
there is no reason for the values of Tables 2 and 3 to be what is expected in a human motion. These are evaluated as defined earlier and are strictly machine equivalents. Efficiencies here (and elsewhere) are questionable. Several investigators have evaluated work output as the difference between maximum and minimum system total energy levels. Efficiency is then calculated from this value divided by the total caloric value of oxygen consumed during the activity. Such a procedure implies that work is required to lift the body and that it can drop freely. It is also implied that the oxygen consumption represents the actual state of internal energy exchange. It is well known that the body does not fall freely during the decreasing energy phase. The true work of all the internal forces in a human cannot be evaluated yet, as adequate data for muscle behavior during human locomotion does not exist. Most investigations show IO-25 per cent overall ‘efficiency’ for various forms of motion. Meanwhile, Hill (1963) has shown that isolated muscles may have efficiencies up to 50 per cent based on measurements of work output and heat generated by the muscle. 6. coscLusIoss
A general theory for analysis and synthesis
82
M. A. TOWNSEND
of bipedal locomotion is presented and demonstrated with numerical examples for a model consisting of a rigid body with massless extensible legs. Data on human body locomotion has been used for guiding the form of the synthesized motions. The mode1 is adequate for design of prosthetic devices for paraplegics and bipedal walking machines. Synthesized trajectories and corresponding control forces can be evaluated for any required motion criterion. The criteria used in this study are the maximum stability (as characterized by minimizing the footprint) and combinations of stability plus minimum energy expenditure with different weighting factors. Of the examples discussed in this paper trajectory C (Figs. 4 and 5c) appears to be more suitable for a biped walker since it represents reasonable energy requirements and support locus conditions. Although the model is not completely representative of human locomotion, it can be used to clarify some necessary concepts in terms of trajectories, support, stability and energy expenditure. It can also provide insight into certain controlled motions which may be approximately represented by the model. The approach developed in this study is being extended for more complex models which can better simulate normal human motions. REFERENCES Braune, W. and Fischer, 0. (1890) Uber den Schwerpunkt des menschlichen Korpers mit Riicksicht auf auf die Ausriistung des deutchen Infanteristen. (Concerning the center of gravity of the human body with reference to the equipment of the German infantry.) Abh. d. math.-phys. cf.d.kSachs. Gesellsch. d. Wiss. 15. Bresler. B. and Frankel. J. P. (1948) The forces and moments in the leg during level walking. Trans. ASME 72 17-36 (1950). presented at 1948 Annual Meeting, New York, Nov. 28-Dec. 3, Paper no. 48-A-62. Cavagna, G. H., Saibene, F. P. and Margaria. R. (1963) External work in walking. J. appl. Physiol. 18. 1-9. Contini. R., Gage, H. and Drillis. R. (1965) Human gait characteristics. In Biomechanics and Related Bioenaineerina Topics (Ed. R. M. Kenedi). Peraamon Press, Oxford. (Proceedings of a symposium held in Glasgow, Sept. 1964.) Ducroquet, R.. Ducroquet, J. and Ducroquet. P. (1968) Walking and Limping, p. 284. Lippincott, Philadelphia and Toronto.
and A. SEIREG Elftman. H. (1939) The force exerted by the ground in walking. Human Biol. 2, 529-535. (Also Arbeirrphysiologie 10,485-49 1, 1939.) Elftman. H. I 1939) The force exerted by the ground in walking. Human Biol. 2, 529-535. (rUso Arbeitsphysiologie 10.485-49 I, 1939.) Elftman. H. (1939) The rotation of the body in walking. Arbeitsphysiologie 10,477-483. Efltman. H. (1939) Forces and energy changes in the leg during walking. Am. J. Physiol. 12%339-356. Fischer, 0. ( 1898-1904) Der Gang des Menschen. (‘Human gait’.) Abh. siichs. Ges. Wisi. Vols. 2 I-‘8. Frank. A. A. and Vukobratovic. M. (1969) On the synthesis of biped locomotion machines. 8th international Conference
on Medical
and Biological
Engineering.
Evanston, Illinois. Frank. A. A. An approach to the dynamic analysis and synthesis of biped locomotion machines. In preparation at the University of Wisconsin. Madison, Wisconsin. Greenwood, D. T. (1965) Principles of Dynamics. Prentice-Hall, Englewood Cliffs. New Jersey. Hill, A. V. (1963) The ratio of mechanical power to total power expended during muscle shortening. Proc. R. Sot. B159,3 19-334. Hill, A, V. (1965) Trails and Trials in Physiology. p. 374. Williams and Wilkins. Baltimore. Kane, T. R. and Scher. IM. P. (1970) Human self-rotation bv means of limb movements. J. Biomechanics. 3. 34-40.
Lascari, A. T. (1970) The Felge handstand- a comparative kinetic anaiysis of a gymnastics skill. PhD Thesis. Dept. of Physical Education, University of Wisconsin. Lukin. L.. Polissar. M. J. and Ralston. H. J. (1967) IMethods for studying energy costs and energy flow during human locomotion. Haman Facrors 9,603- 608. Margaria. R. ( 1968) Positive and negative work performance and their efficiencies in human locomotion. Int. 2. angew Physiol. 2% 339-35 I. Morton, D.J. and-Fuller, D. 0. (195’) Human Locomotion and Body Form, D. 185. Williams and Wilkins, Baltimore. ’ _ IMurray, M. P., Drought, A. B. and Kory. R. C. (1964) Walking patterns of normal men. J. Bone Jr. Slug. 46A. 335-360.
Murray, M. P., Kory, R. C., Clarkson. B. H. and Sepic. S. B. (1966) Comparison of free and fast speed walking patterns of normal men. Am. J. phys. Med. 4% 8-24. Murray, M. P., Seireg, A. and Scholtz. R. C. (1967) Center of gravity, center of pressure and supportive forces during human activities. J. app/. Physiol. 23. 83 1-818.
Murray, M. P., Kory. R. C. and Clarkson. B. H. (1969) walking patterns in healthy old men. J. Geront. 24, 169-178.
Murray, M. P. ( 1967) Gait as a total pattern of movement. Am. J. phys. Med. 46, 190-333. (Excellent bibliography.) Paul. J. P. (1965) Bio-engineering studies of the forces transmitted by joints: (11) Engineering analysis. In Biomechanics
and Related
Bio-Engineering
Topics
(Ed. R. M. Kenedi). Pergamon Press, Oxford. (Proceedings of a Symposium held in Glasgow, Sept. 1964.) Paul, J. P. (1967) Forces transmitted by joints in the
SYNTHESIS
OF BIPEDAL
human body. Proc. Znstn me& Engrs 181 (part 3-J). 8- 15. (Presented at the Inst. of Mech. Eng. Symposium in Lubrication and Wear in Living and Artificial Human Joints, Paper 8. London. April 7, 1967.) Peiser. E., Wright, D. W. and Mason, C. ( 1969) Human locomotion. BuK Prosthetics Res. Bpr IO - 11.18- 104. Ralston. H. J. (1958) Energy-speed relation and optimal speed during level walking. Inr. 2. angew. Physiol. 17, 277-183. (Formerly Arbeitsphysiologie.) Ralston. H. J. and Lukin. L. (1969) Energy levels of human body segments during level walking. Ergonomics 12.39-46.
LOCOMOTION
83
Saunders. J. B., Inman. V. T. and Eberhardt, H. D. (1953) The major determinant in normal and pathological gait. J. Bone Jt. Surg. 35A, 543-548. Seireg. A. and Peterson, R. 0. (Unpublished at present.) Construction of a manually controlled, self powered load-carrying tripod walker. Work being done at the University of Wisconsin, Madison, Wisconsin. Shames, I. H. (1966) Engineering Mechanics: Dynamics. Vol. 2. Prentice Hall, Englewood Cliffs, New Jersey. Smith, P. G. and Kane, R. R. (1968) On the dynamics of the human body in free fall. J. uppl. Mech. 35, l67- 168.