Combining position and acceleration measurements for joint force estimation

(‘

0021Y_W91s3u)+ al I Prr~moa Ru plc

IW

COMBINING POSITION AND ACCELERATION MEASUREMENTS FOR JOINT FORCE ESTIMATION ZVI LAWN and GE Wu Biomedical Engineering Department and Neuromuscular Research Center. Boston University. Boston. MA 02215, U.S.A. Abstrscc -The calculation of joint forces in biomechanics is usually based on the measurements of the kinematics of a given body seemen& the eslimation of the inertial properties of that segment and the solution of the ‘inverse dynamics problem’. Such a process results in estimates of the joint forces and moments needed 10 sustain the monitored motion. This paper presents a new approach that combines position and acceleration measurements for the purpose of deriving high-quality joint force estimales. An enperimental system that is based on an instrumented compound pendulum was designed and tested. The joint forces necessaryto maintain a swinging motion of the pendulum were measured by an array ofstrain gauges. and were compared to the forces estimated by the integrated kinematic segment that measured the position and accelerationof the pendulum. The joint force measuremenls were also compared to rhe force estimates that were based on the calculated segmental acceleralion generated by the differentiation of the segmental position alone. The results show a high degree of correlation between the lorees estimated by rhe integrated segment and those measured by the strain gauges. The fora estimates based on the position measurements alone were less accurate and noisier. The application of Ihe integrated segment to the study of human kmctics is discussedand illusrrated by the ankle and knee forces during slow walking. The results suy~~cstthat Ihe USCof accclerometcrs is necessary for the estimation of transients and high-frequency

The evaluation of the joint forces and moments that control the motion and posture of mammals has been the focus of scicntilic research for many years. Instrumenting human joints for direct mcasuremcnls ol the dynamic vari;thlcs requires highly invasive proccdurcs (Hodgc CI ttf., 19X6) and, therefore. is limited in its applicilbility. The noninvasivc study of joint loading involves the ~~rint&~n of the joint forces and joint moments based on the externally measured kinematics of body segments.The solution processis based on the analysis of single body segments as ‘free rigid bodies’, and the formulation of Newton’s equations of motion for each segment. Using estimates of the mass and inertial properties of each segment. and estimates (or measurements) of the acceleration and angular velocity of the segment. Newton’s equations ofmotion can be formulated and solved. This process, referred to as the ‘inverse dynamics problem’, results in noninvasive estimates of the joint forces and moments that arc necessary lo obtain Ihe observed rigid body motion. Two approaches have been used over the years for the assessmentof [he kinematic derivatives necessary to calculate the joint loads. (I) Measurement of acceleration information and the USCof imcgration proccdurcs for the estimation of velocity and position. (2) Mcasurcmcnl of position information and the use of numerical dilTcrentintion for the estimation of time dcrivarivcs.

The USC of accelerometers is the more direct appreach to the mcasuremcnt of the acceleration of body segments.ahhough its wide use is limited by two factors: the sensitivity of the acceleromc(crs to the field of gravity, and the dilficulty in monitoring the segmental center of mass. The frequency content of extensively studied biomcchanical phcnomcna such as gait was reported by Antonsson and Mann (1985) IO extend from steady-state levels to values that arc under 100 Hz. Such a frequency range requires the use of accelerometers that are sensitive to the lield of gravity. The acceleration signal recorded from such devices is composed of the kinematic acceleration (i.e. the second time derivative of the position vector), and the gravitational acceleration. Since the gravitational component depends on the orientation of the accelerometer in the field of gravity, this information has to be available in order to extract the kinematic variables that are of interest. The need to measure the acceleration of the segmental center of mass, and the obvious difficulty in attaching an accelerometer to that point, presents a second limitation lo its broad applicability. Some researchers have tried to overcome the above difficulties by using multiple accelerometers to rcsolvc the full segmental kinematics, i.e. the six dcgrces of freedom that comprise rigid body motion. and then USCtime integration to calculate the spatial oricnta(ion of the rigid segments. Morris (1973) used six accelerometers, Padgaonkar et al. (1975) described a system based on nine accelerometers, and Kant et al. (1974) described a twelve accelerometer system. More rccenlly, Hayes er al. (1983) described a four accelerometer system for studying the kinematics of gait, and

1173

1174

2. LADIN and GE WV

Gilbert er al. (1984) described a system of eight uniaxial accelerometers for the study of the sagittal plane kinematics of gait. Such an approach is faced with the need to identify an initial orientation that could serve as the initial condition for the integration process. This information can be either assumed. as was done by Morris (1973) and Hayes et al. (1983), or measured, as was done by Seemann and Lustick (1981). In both cases the integration process introduces errors that increase with time, leading to significant errors in the calculated orientation of the rigid body. This was documented in the pendulum studies by Hayes et al. (1983) and in the head motion studies of Seemann and Lustick (1981). The second approach to the assessment of kinematic variables is based on the measurements of the position. and the calculation of the first and second time derivatives for the estimation of the joint forces. Since the differentiation process amplifies the highfrequency components of the noise in the position signal, it requires the use of low-pass filters that can distort some of the original signal contents. The application of this approach to the dynamics of gait was originally dcscribcd by Breslcr and Frankcl (1950) and was later followed by other rcscarchcrs including Crowninshicld ct ul. (1978). Hardt (1978). Winter and Robertson (1978). Whittlc (1982). Patriarco YI 01. (1981) and McFadycn and Winter (1988). The intcrprctation of force cstimatcs gcncratcd by this process rcquircs sonic information on the inhcrcnt accuracy or such a computational approach. Since the quality of the force cstimatcs dcpcnds directly on the quality of the accclcration data. some studies have been conducted in an attempt to assessthe accuracy of estimating second time derivatives from noisy data. These studies included experimental systems that measured both the position and acceleration of mechanical devices (Pezzack cr (I/.. 1977; Ladin er ul., 1989). theoretical algorithms that quantilied the expected noise in second time derivatives (Lanshammar, 1982a. b), and ditTerentiation techniques to minimize that noise (Usui and Amidror. 1982; Busby and Trujillo, 1985). Pezzack et ul. (1977) conducted synchronous measurements of the position and acceleration of a mechanical arm that was moved at two speeds. Ladin et al. (1989) described an electromechanical system that was designed to test the accuracy of acceleration estimates over a known frequency range (l-1 1 Hz). Both studies described the dependence of the acceleration estimates on the nature of the smoothing and dircrentiating algorithms and the cutotT frequencies sclcctcd for the low-pass filters. Thcsc studies showed further that an appropriate selection of the filtering parameters can result in high-quality estimates of the acceleration, although the quality of the estimates deteriorated by an improper selection of the parameters. By adding some assumptions on the nature of the noise in the displacement measurements and the frequency content of the original displacement signal, Lanshammar (1982a. b) calculated a lower bound on

the expected noise in the estimated derivatives. All of the above studies point to the need to obtain additional informtion about the displacement signal and, most importantly, its frequency content before one can assessthe quality of force estimates that are based on differentiation of the displacement signal. This paper describesa new approach that combines the position information measured by an accurate optoelectronic system, a six degrees of freedom rigid body analysis and the acceleration information measured by a single triaxial accelerometer to provide an accurate and reliable estimate of joint forces.

METHODS

The noninvasive estimation of intersegmental resultant (net) joint forces in a multilink system is based on the solution of Newton’s equation of rigid body motion. Each link is isolated as a free body, and the vector sum of the forces acting on it is equated to the mass of the link times the acceleration of its center of mass. The forces acting on each link in physiological systems include the forces arising from muscles, ligaments and surface contact, the weight of the link and any other external forces applied to the link. II one considers typical multilink systems in biomcchanics. such as the lower limb during locomotion, or the upper limb in manipulation tasks, one can model the links as rigid bodies conncctcd by joints and interacting with the cnvironmcnt through forces applied at the interface. By lumping together all the internal forces acting around a joint as a single net force, the computational process of calculating the joint forces can be simplified, leaving the intcrscgmrntal resultant joint forces as the only unknowns. Joint forces in this paper will refer to intersegmental resultants, not articular contact forces. By measuring the forces at the interface of the most distal link and the environment, and measuring the acceleration of the link center of mass, the only unknown left in this equation is the force at the proximal joint. Therefore, solving this equation for the unknown force enables us to repeat the process for the more proximal link, using the recently calculated force as the known distal joint force for this link. A conceptual framework for the serial calculation of the joint forces will require therefore. two kinds of input: (I) the force components at the interface with the environment, and (2) the acceleration of the center of mass of each link. The measurement of the force components is usually carried out by a force transducer. This could be a force plate in tasks such as walking or running or an array of strain gauges in upper limb manipulation experiments. The measurement of the acceleration of the link center of mass presents a technical challenge for two reasons: the accelerometer cannot usually be

II75

Position and accclcration mcasurcmcnts for joint force cstlmalion

attached

to the center of mass in physiological

sys-

gravity from the accelerometer

output. so that equa-

tems and. as discussed earlier in the paper, its output

tion (2) becomes

signal is sensitive to the field of gravity.

a .ecri~romc,tr = *.M. + b x r + u x b x rl + Tdc.

biomechanical attached

measurement

can be

to the surface of the link (e.g. the skin over-

lying the shank)

as close to the center of mass as

possible. The following for extracting output

formation

sections describe a technique

an accurate estimate of the link’s center

of mass acceleration ometer

In a typical

the accelerometer

by correcting

The acceleration

obtained

in-

of that link.

Aa = a.,,,I~rOm~l~r- PC,M.-T8,g=Oxr+Wx(Wxr). (6) The error introduced corrected

of a point P on a rigid body at a

by the terms containing

if the angular

by differentiating

system 0. is given by the equation:

by TRACKC.

spatial orientation

If the angular

(1)

following

approximation.

B=wx(wxr); velocity

angles generated

time derivatives

0 is chosen at the center of mass of the

rigid body, and point P represents the attachment accelerometer.

describes the acceleration erometer.

Therefore,

then

a,=aC,M. and a,,

measurement

(7)

IBI=lwlZlrlsin(O,,,)sin(~,,,.,).

(8)

Since o 1 (UJx r). sin(0,.,,, I ,) = 1. Therefore.

by the accel-

a.,,.l.rom.c.r =a,,,,+ci,xr+c,,x(m~r)+g”~,

we have

IBI=lw12 IrlsW,,).

(9)

(2)

system (BCS). The error introduced

in the real accclcr-

of mass due to the o!T-center

r and the orientation

of the ;Icccleromcter

A + B. the magnitude of thevector

C

is given by

lCl=J l A I2 +

whcrc gllrs is the gravity vector in the body coordinate

location

Irlsin(0,,,).

(I) becomes

equation

at the center

x r and

site

As the vector C-

ation

A =ti

vector of the rigid lAl=lG~l

of a triaxial

Let C=Aa.

are too

using the

then

body. If the origin

of

can be obtained

noisy, one can at least assess the error a,=a,+Li)xr+Wx(Wxr),

r can be

velocity and acceleration

the link are known. This information

distance r from the origin of the body fixed coordinate

where UJ is the angular

is, there-

fore.

the raw acceler-

using the independently

about the spatial motion

The error in the center of mass acceleration

(5)

11%I2 + 21A I I BI cos(0, J.

(10)

By substituting equations (7) and (9) into (10). WC have dcrivcd an expression for the magnitude

of the accel-

cration error in terms of the angular velocity, angular acceleration

and displacement

of the accelerometer

relative to the center of mass:

in space is. therefore,

The largest error would occur if A 11 B. &I 1 r and

w 1

r.

In this case the angles become The acceleration

Equation AF-mAa.

Since the accelerometer entation

of that link should be determined

remove

the efiect of the field of gravity

with the body link, the spatial

ori-

the translation

system that

WATRACK)

from

and orientation

system

MIT-developed an

inertial

formation

BM

24:12-C

The resulting joint force error is

the

of the link

system and a

was used for this study

combined

determination

(12)

in order to

six degrees of freedom analysis of rigid body motion.

tronic

I~aIm.,~Irl(l~l+1~12).

(13)

IAFl,.,5mlrl(l~l+l~12).

output. This process can be achieved by

in space using a position measurement The

(I I), therefore, becomes

is attached rigidly to and is

together

monitoring

o”,=u&$,=90’. . .

(4)

moving

accelerometer

0 A,s=o.

error Aa results in a joint force error

that is equal to

for

the WATSMART

position

software

measurements

package TRACK\0

of the spatial orientation reference

system.

A

(dubbed optoelecand

the

for the

of the link in

rotational

trans-

matrix T,, was used to subtract the effect of

Equation

(13) indicates that the maximum

force error

resulting from the effect of the off-center-of-mass tion of the accelerometer

of the body segment. the displacement ometer angular

relative

of the acceler-

to the link’s center of mass and the

acceleration,

square of the angular plotting

loca-

is linearly related to the mass

and

is proportional

to

the

velocity of the rigid body. By

the error in the force estimate per unit mass

and unit distance, as shown in Fig. 1. one can get a quantitative

assessment of the maximum

expected

1176

2. LADINand GE WV

Fig. I. The error in joint force estimates (per unit mass of the link per unit distance from the link center of mass) as a function of the angular velocity and angular acceleration of the link. The units of the error function are N kg- * m- I.

errors in the value of a given joint force. As an illustration of this process consider the task of estimating the hip forces of a subject who weighs 600 N walking at a pace of 0.7 ms-‘. Kinematic mcasurcments of the thigh motion provided approximate values of the angular velocity and accclcration of that link. The swing phase involed angular rotation of the thigh with approximate values of Itil= I5 rad s’~, Iwl=5 rad s-t. An accelerometer that is attached to the thigh (whose mass is estimated as 7 kg) at a distance of 10 cm from the thigh’s center of mass will, therefore, produce an error in the force estimate of 28 N. Such an error could be quite substantial considering that the joint force during swing phase is of the order of 50 N. although the absolute values of the force are quite small. A correspondng calculation for the stance phase, with values of IhIrad s-‘, Iwj = 1.5 rad s- I, would result in an error of 3 N, when the order of magnitude of the joint force is 600 N. The situation could be quite different for more dynamic applications such as running or jumping, underlying the need to use such an error analysis. By using better measurements of the angular rotation of the rigid link, coupled with estimates of the mass of the link and the displacement of the accelerometer, one can better estimate the acceleration of the center of mass, and thereby achieve a better estimate of the joint forces. The theoretical approach combining position measurement, six degrees of freedom kinematic analysis and accelerometry is illustrated on a compound two degrees of freedom pendulum instrumented with an array of strain gauges. The mechanical system was used to conduct a three-way comparison: joint forces measured by the strain gauges were compared with the estimates of the integrated segment and the estimates based on position measurements alone. As pointed out earlier, the use of accelerometers involves

overcoming two hurdles: the effect ol gravity and the oti-center-of-mass attachment of the accclcromcter. In an attempt to isolate these errors, the accclcromcter setup described in this paper was attached to the center of mass of the pendulum. enabling us to study the efTect of combining orientation information and accelerometry to dynamically calibrate the accelcromcter.

TIII: EXPERIMENTAL SYSTEM

The experimental setup shown in Fig. 2 consisted of a two degrees of freedom pendulum mounted on two vertical supports. Two sets of bearings provided the two degrees of freedom: bearings I and 2 enabled rotation around a ftxed horizontal axis (axis l-2). and bearings 3 and 4 enabled rotation around an axis that was fixed with respect to the horizontal beam (axis 34). By locking bearings 3 and 4 the pendulum could swing in a single degree of freedom rotation. The vertical support beams were instrumented with an array of strain gauges that directly measured the joint axial forces and moments. The details of the strain gauge locations, signal conditioning and processing are given in the Appendix. The integrated segment used for the synchronous kinematic measurements is described in Fig. 3. The infrared light-emitting diodes (LEDs) are used for position measurements and the triaxial accelerometer (model EGA38-f-10 manufactured by Entran Devices, Fairfield, NJ) is used for linear acceleration measurements. The LEDs and the linear accelerometer were attached to the rigid lexan segment that was bolted to the pendulum; therefore, the segment performed the

Positlon and acceleration measurements for joint fora estimation

-

w

Inlograled

I

Klnamatlc

1177

(axis l-2)

Sogmont

‘Pendulum

f:ig. 2. The expcriment;~l .sc~up:an instrumented two dcgrrxs of freedom pendulum. with an array of strain gauges attached to the stationary vertical suppurts.

TrlplrLEDMarker TrlaxlalAccelerometer

IntegratedSegment Fig. 3. The integrated kinematic segment showing the LED ’ markers and the location of the ucalcrometcr.

same rigid body motion as the pendulum. The center of mass of the pendulum was determined from the mechanical drawing. and a small hole was drilled at the center. The accelcromctcr was attached such that its location corresponded to the theoretically calculated center of mass of the pendulum. The WATSMART optoelectronic system (Northern Digital. Ontario, Canada) with two infrared-sensitive cameras was used for the three-dimensional reconstruction of

the spatial location of each LED. The analog data from the accelerometer and the strain gauges were synchronized with the position measurements of the WATSMART system using the WATSCOPE A/D unit. The kinematic data were sampled at I00 Hz and the analog data at 300 Hz. The data were saved on an IBM-AT and later transferred to a DEC.VAX I l/750 for processing. The quantitative comparison of kinematic or dynamic variables that describe the motion of a rigid body in space requires that the comparison be done in a single coordinate system. The different measurement elements described in Fig. 2 clearly illustrate the difficulties arising from such a requirement. The spatial coordinates that describe the motion of the LEDs are measured in a coordinate system attached to the cameras and, since the cameras are stationary, it can be viewed as an inertial reference system. The accelcration is measured by the accelerometer along axes that arc fixed with respect to the segment. Since the segment is rigidly attached to the pendulum, these axes can be viewed as describing a coordinate system centered on the pendulum. Such a coordina:c system is usually referred to as a body-centered coordinate system (or BCS). The strain gauges are fixed to the

Z. LADIN and GE Wu

1178

stationary vertical supports and. therefore, their output is measured in a laboratory-fixed inertial reference system. DATA PROCESSING

is attached to the horizontal support bar (see Fig 2) and. therefore. has one axis that passesthrough bearings 1 and 2 and is fixed with respect to the inertial reference system. The coordinate transformation between the I coordinate system and the B coordinate system was obtained in two steps: rotating I around the X, axis by the angle 0t. a processthat results in the T coordinate system. A second rotation by the angle 6z transforms the T coordinate system into the final body coordinate system B. The three coordinate systems have the following relationships:

The processing of the experimental data was performed on a DEC-VAX 1 l/750 computer and is depicted in Fig. 4. First, the strain gauge outputs from five bridges were combined to obtain the joint force components in the inertial reference system (see the Appendix for details). Next, the instantaneous spatial x,=x,, position and orientation of the pendulum in the labor(14) us= Yr. atory-fixed inertial reference system was calculated MIT-developed software package using the The rotation angles 0, and 8, were directly obTRACK’P. This package (Antonsson. 1982; Antonstained from TRACK0 and substituted into the transson and Mann, 1989) calculates the six degrees of formation matrices T,, and Tr, given by the following freedom of a rigid segment moving through space, equations: expressed in an inertial reference system. The spatial orientation of the segment calculated this way was used to subtract the efTectof gravity from the accelerometer output and obtain the actual spatial kinematic acceleration of the moving segment in the inertial reference system. Since the accelerometer was mounted at the pendulum’s center of gravity, the measured acceleration was considcrcd to bc the accclcration of the ccntcr of mass. This value was then used to calculate the lumped joint forces. i.e. the forces acting T,, dcscribcs the rotation transformation rrom j to i. on the bearings. and to compare the cstimatcd WIUCS Thcreforc. the transformation from B to I is given by to the forces mcasurcd by the strain gauges. Three coordinate systems were used in the com0 sin 0, cos 0, putational process: the inertial rcfcrcncc system I which is attached to the vertical supports; the body= cos 0, -cosU,sinI), . ((7) sin U, sin 0, fixed coordinate system B moving with the ( -sin Uzcos U, sin U, cosu,cosu, i pendulum; and a transition coordinate system T that The inverse of the matrix T,, gives the transformation T,, from I to B. T,, was used to transform the vector describing the gravity acceleration g to the body coordinate system and extract the acceleration of the center of mass ac,“, from the accelerometer I signal a.,Ccl.rom.l.raccording to the following equa3-D marker LOcatiOn tion: aC.M.= a.ccclcrom.t.r-Tatg-

Segmental

Accelerstion

I Inverse

Dynamics

Equation

The acceleration of the center ol mass (measured originally in the body coordinate system) was transformed into the inertial reference system and substituted into the pendulum’s equation of motion. This resulted in equation (19) with the joint forces F as the only unknown, since m is the known mass of the pendulum. The estimates of the joint forces were compared directly to the components measured by the strain gauges as both were given in the inertial reference system. F+mg=mTn+c.~.-

I Joint

Force

Fig. 4. Flow chart for the data processing.

(18)

(19)

A second comparison was conducted by estimating the linear accelerations based on the position measurements alone as calculated by TRACK”. The kinematic variables were low-pass filtered with different

Posrtwn

cutoff frequencies noise ratio.

The

and acceleration measurements for jomt force cstlmallon

in order to improve estimated

the signal to

The spatial

were then

bined rotation

accelerations

substituted into equation (19). resulting in estimates of are based e.xclusicel_v on the

the joint

forces that

position

measurements.

compared

These

estimates

with the force measurements

were then

by the strain

trajectory

of the pendulum

Fig. 2) is shown in Fig. 5. The translation tional coordinates the laboratory

of the pendulum

coordinate

and Y coordinates bimodal

appearance

The angular The estimation

of joint forces is based on the meas-

urement or estimation of translational

of the second time derivatives

kinematic

variables.

sented in this section compare measurements

The results pre-

the estimates and the

of the accelerations

of the pendulum

rotation

system. Even though the X rotation

of the

Z

coordinates

taking

that

place. the

coordinate

clearly

two-axes rotation.

show that the bulk of the

occurs around the X and Y axes. with only a

small component amount

and rota-

are described in

seem to give the impression

identifies the motion as a combined

RESULTS

in a com-

around axes I-? and 3-4 (as defined in

there is only a single-axis

gauges.

1179

of rotation

around

the Z axis. The

of noise in the WATSMART

construction

and in the calculations

of freedom by TRACK’

marker

can be estimated

from this

center of mass, and the effects of such estimates on the

figure, as no filtering of the raw data was performed.

joint

appears that in a well-controlled

force predictions.

and their orthogonal laboratory

inertial

The resultant

components coordinate

force estimates

are described in the

system.

Dlrplacom~nt

the one where translational

[m]

environment.

the experiment noise

Aotrtlon

is

on

the

re-

of the six degrees

was performed,, order

of

[‘I

Fig. 5. Spatial trajectory of the pendulum during a combined two degrees of freedom rotation: (a-c) dcscribc the linear displaccmcnts of the intcgratcd segment; (d-f) describe the angular orientation of the pendulum. Raw unfiltcrcd data described in the laboratory coordinate system.

It

such as the

0.5 mm

1180

2. LADIN and GE Wu

[Fig. 5(c)] and the rotational noise is about I-2” [Fig. 5(f)]. Since the noise seems to be of fixed amplitude, its relative contamination of the signal depends on the amplitude of the signal: the larger the signal (e.g. the rotation around the Y axis), the less significant the noise is. The estimation ot joint forces requires the knowledge of the mass and the acceleration of the rigid body moving through space. The three components of the linear acceleration. described in the inertial reference system, are described in Fig. 6. The acceleration measurements are compared with two estimates generated by TRACK’? one with a 5 Hz low-pass filter and the other with a IS Hz low-pass filter. Filtering was obtained by applying a third-order Butterworth filter forward and backward in time to eliminate any phase shifts. A careful comparison of the two filtered trajectories shows that the estimates obtained with the 5 Hz low-pass filter arecloser to the acceleration measurements, thereby suggesting that this cutoff frequency is more appropriate to use for this particular motion. The curves shown in Fig. 6 demonstrate the known end ctfects of the Butterworth filter as the error in the acceleration cstimatcs incrcascs signilicantly at the beginning and the end of the time window of 3 s dcscribcd in the figure. The cstimatcd and mcasurcd joint forces arc shown in Fig. 7. The X. Y and % components of the joint

force and its total magnitude as measured by the strain gauges are compared with the force estimates by the accelerometer and by TRACK’: estimated accelerations with 5 Hz and 15 Hz low-pass filters. The estimated forces based on the acceleration measurements are practically on top of the measured force trajectories. thus making those estimates highly accurate. The estimated forces based on the acceleration estimates show a clear dependence on the cutoff frequency of the filter used in the estimation process: the 5 Hz low-pass filter generates much better estimates of the joint forces, especially close to the beginning and the end of the trajectory. It is interesting to note that the Y and Z components of the measured force traces (by the strain gauges) show a smallamplitude (less than 1 N). high-frequency oscillation on top of the basic trace. This oscillation is captured by the force estimates based on the acceleration measurements,and does not exist in the traces generated by applying the 5 Hz low-pass filter to the acceleration rstinrotes. The traces generated by the I5 Hz low-pass filter applied to the acceleration estimcues do show the high-frequency oscillations, but their amplitude is so large that any estimates based on these values will contain a large amount of error. The magnitude of the total force calculated from the three orthogonal components shows a good corrcspondcncc bctwccn the mcasurcd force values and

Fig. 6. Linear accelerationsmeasuredby the accelcromctcr(solid tine) and TRACK” (dashedlines): a,,,,-kinematic data filteredwith a 5 Hz low-pass filter. qo,- kinematicdata filteredwith a I5 Hz lowpas.5filter.

1181

Poseion anJ acceleration measurements for jotnt lorcc csrimat~on

8-

-81

0

I 1

2

3

1

2

I 3

48 . 461 0

464

0

(a,

I 1

2 -

TIME [Sl

3

461 0 703

1

45..0

I

I

3 TIME

(b)

Fs ---Fa

I 3

2

ISI

-

Fs --+I15

46..

481 0 (cl

1

I 3

2 TIME IS1

-I3

---FtOS

Fig. 7. Measured and estimated joint lorces: the three orthogonal components F,. F, and F, and the total force F,. F,-force measured by the strain gauges; F,- force estimated by the integrated sensor: F,o,, F ,, ,-force estimated by TRACK’0 with 5 Hz and IS Hz low-pass filters, respectively.

2. LADIN and GE WV

IllI?

the values based on the acceleration The

two

traces

capturing value

practically

one

not only the basic oscillation

but also the high-frequency

examination

another,

in the force

vibrations.

The

of the force estimates based on the accel-

estimates shows

eration

measurements.

overlap

pondence is obtained

that the best overall

by applying

corres-

the 5 Hz low-pass

filter.

basic harmonic

component,

miss the high-frequency

yet it will undoubtedly

component.

tories based only on position scribed in Fig. 7 demonstrate entiated good

signal with

tracking

totally

The force trajec-

measurements

and de-

this point: the differ-

the 5 Hz low-pass

of the low-frequency

misses the high-frequency

filter shows

oscillation, oscillations.

force estimate that is based on the differentiated

The noise level is not uniform. and the examination of the 15 Hz low-pass filter trace (at points away from

frequency

oscillations,

significant

error in the values of the predicted

changes from 2 N for the X and Z components for the Y component

to 7 N

of the joint force. The resulting

noise in the total force estimates is on the order of 5 N,

signal

with the I5 Hz low-pass filter is able to track the high-

the ends of the time trajectory)

shows that its value

yet The

As an illustration

but its noisy output of the applicability

proach to the study of kinesiological

creates a force.

of this ap-

phenomena.

the

ankle and knee forces during the stance phase of slow

representing almost 50% of the measured force value.

walking

The

ments were attached to the foot and shank of the right

5 Hz

low-pass

filtered

trace

shows

maximum

are described

in Fig. 9. Two

integrated

seg-

deviations of about I N from the actual force levels (at

leg of a female walker (whose weight was 52 kgf). and

1.6 s), and much smaller values throughout

the forces during

the trajectory.

the rest of

The edge effects of the filter, i.e. estim-

ate contamination

next to the beginning

and the end

The examination

of the joint force trajectories

eratcd by the compound measured

motion of the pendulum

tion to the basic harmonic

oscillation

of 0.711 Hz. thcrc is

high-frcqucncy

that is most apparent

iI

decomposition in Fig. 8 &arty

dcscribcd

of the low-frcqucncy

force trajectory,

of

of the force shows the

component

of the

yet it also shows the distinct compon-

cnt at the frcqucncy expected

whose low-pass than

oscillation

in the X and Y components

the force. The frcqucncy

therefore

and

at a frcqucncy

mcasurcmcnts dominance

gen-

by the strain gauges rcvcals that, in addi-

also

10 Hz may

of approximatcty

that

a ditfcrentiating

12 fiz

bc able to accurately

content

ft is

algorithm

filter has a cutoff frequency

Fig. 8. Frequency

lates into a speed of about without

of the tract. extend to about 0.4s.

of less

capture

the

the stance phase of slow barefoot

walking at an average speed of 0.57 m sfiltering

’ (that trans-

1.3 mph) were calculated

the acceleration

or the orientation

data. The net joint forces show the characteristic peak’ shape, with a first maximum reprcscnting

the dcccleration

of the falling body. The

second peak has an approximate body weight. the upward

value of 115%

of

It occurs just bcforc toe-off, rctlccting accclcration

force component overall

of the body. The dominant

is in the vertical direction.

smallest component The

‘double-

after heel strike.

and the

is in the mcdial ~latcral direction.

similarity

rcllccts the dominant

in the pattcrn

of the forces

elTcct of the foot/tloor

inter-

action forces, For a lower limb in static equilibrium, the difference

bctwcen

would

be the weight of the shank,

merely

of measured and estimated joint rorces: F.-strain F.-force estimates by the inkgrated segment.

the ankle

and

gauge measured

knee

forces

atfecting

forces;

Positron and accclcration

measurements

for joint fora

Ill43

atimatlon

Ankle and Knee Joint Forces in Lab Coordinate System

-LOO

-LOO-

-600-

+

_700Ld-__

.~_-----_-_-6oQl

0

05

1

I 0

Time [S]

1

I

.

t

05

. 1

Tlme [S]

Fig.9. The forces in the ankle and knee joints during barefoot slow walking. The joint force components applied to the distal body segment arc dcscribcd in the laboratory coordinate system. The ankle forccs dcscribc the forces transmitted to the foot. and the kncv forces dcscribc the foras transmitted to the shank. The axes arc lab&d according to the following convention: X is the medial-lateral direction, with the medial direction b&g positive; Y is the anterior-posterior direction. with the posterior direction as posilivc and % is the vertical direction. with the upward direction being positive.

only the % component of the force. Since this weight represents only a small fraction of the Z foot/floor interaction force. it has a small effect on the overall magnitude of the net joint force. The most notable difference between the two patterns is the transient right after heel strike. The transient in the total knee force estimate includes a short peak that is due to the transients in the anterior/posterior and vertical directions and does not exist in the ankle joint force. Even though the magnitude of this transient may be small for such a slow activity, it represents a piece of information that would be eliminated by applying the standard low-pass filters that are currently used in kinesiological analysis. Furthermore, since this transient originates from the inertial acceleration of the shank, one would expect its value to increase in faster activities such as running or jumping. making the use of accelerometers imperative in such studies. DlSCU.SSlON

The integrated kinematic segment described in this study was daigncd to improve the quality of joint fora estimates that arc based on the solution of the ‘inverse dynamics problem’. The comparison of the joint force measurements by the strain gauges with the estimates based on the integrated segment and with those based on the position measurement alone

clcurly shows the bcncfit arising from the integration of position and acceleration measurements. The close correspondence between the estimated and the mcasurcd forces suggests that by integrating the three elements that comprise such an estimation-namely, accurate position measurements. rigid body kinematic analysis that extracts the six coordinates of spatial motion of a rigid body, and the use of this information to dynamically calibrate the acceleration measurements of the linear acalcrometer-one can obtain highquality ‘noninvasive’ estimates of joint forces. The long-term goal of using the integrated sensor centers on the improvement in the estimates of scgmental acceleration offered by this technique. Such an improvement is based on the elimination of the smoothing and/or filtering that is required for the derivation of segmental acaleration estimates from position data. The immediate result of such an improvement would be the increase in the frequency range of useful joint load estimates. The current approach for estimating human joint forces in kinesiological applications is based on measurcmcnts of the three-dimensional position of body-fixed markers and the calculation of the second time derivatives of the position data. This procedure is prone to errors and requires the use of low-pass filters or smoothing algorithms to reduce the noise generated by this process. Clearly, any information whose frequency con-

Z LADIN and GE Wu

118-l

tent exceeds the filter’s cutoff frequency will be elimin-

estimate the acceleration

ated

frequencies

in this process. Such a smoothing

might be appropriate frequencies

procedure

for the description

of slow activities

of the basic

such as slow walking;

however, it fails to capture transients that exist during such activities content.

and

have a much

Antonsson

and Mann

higher

quency range of I5 Hz in measurements plate while 4&60

Light

et al. (1980)

a fre-

using a force

described

a range of

bone up to

provided

that

the

is properly preloaded to the skin. These

estimates could be even further improved

by introdu-

cing models of the soft tissue, and computationally correcting

the measurements

celerometer. tially

of the skin-mounted

ac-

The end result will enable us to substan-

increase the frequency

estimates of joint

range of the dynamic

forces.

Any physical activity that is faster than

slow walking derirahes

ofposirion

to accurately

by cofcularing

one has to use accelerometers

accelerometers

for introducing

to the skin presents the

skin motion

ing to new and possibly significant

artifacts.

this problem

c’tul. (1980) described a study that used two one attached to the tihicl and the other

plate. They

the skin-mounted similar

the tibia using a poly-

reported

that the tracings from

accelerometer

’ . . . broadly

were

to those from the tibia’ although

some distortions

in the form

frcqucncy components swing’, rcndcring the estimation

of the loss of high-

(higher than 50 J Jz) and ‘ovcr-

this measurement

the skeletal

inappropriate

IJ~ hudi~tg.

of the r(iIr

to use the skin-mounted evaluate

they noted

They proceeded

acceleromctcr acceleration

for

as the input to

transients

during

(2) in a series of studies dealing with the propagation of vibrations examined.

in bone, the effects olsoft

of a skin-mounted

tissues on

accelerometer

Saha and Lakes (1977)

reported

were

that the

soft tissue artifacts could be reduced by spring-loading the accelerometer, determinded ometer,

whereas

that by properly

i.e. pushing

Nokes

et ul. (1984)

preloading

the acceler-

the accelerometer

skin, one can overcome

the damping

against

the

effects of the

interposed soft tissue. One needs to keep in mind that the frequency

range

requirements

for

these appli-

cations extend to IOOOHz, substantially range of interest in kinesiological (3) Trujillo

and

tissue covering

Busby (1990)

programming

bone acceleration

simulated

the soft

approach

system. based on

that was able to estimate

the

Even though

their model is one-

and is based on a linear characterization

could be incorporated

that

into any scheme using skin-

that arises from the above publi-

cations is that skin-mounted noninvasive

triaxial

and a software package

the six degrees of freedom

segment (TRACK’?)

sys-

measureof a rigid

was used to calculate the spatial

of a compound

accelerometer

pendulum

with a single

that was attached

to the seg-

mental center of mass. The accelerometer

output was

corrected

to account

for the effects of the JieJd of

gravity, and then used to solve the ‘inverse dynamics problem’ and calculate the joint forces. The estimated joint forces were compared the joint

to direct measurements

forces that were produced

strain gauges attached the pendulum.

by an array

to the stationary

The application

of of

supports of

of this approach

to the

study of human joint loading rcvcalcd high-frcqucncy transients

that exist even in such slow activities

as

slow walking. multiple

accelerometer

ted in the literature of joint ration

measurement

force estimation. provided

fully subtract

The dynamic

by TRACK”

enabled

the elTect of gravity

the accelerometer,

to a single triaxial

proach provided

the computational of required

accelerometer.

by the integrated

the need to perform

spatial calibus to success-

from the output ol

thus simplifying

process and reducing the amount mentation

schemes sugges-

are not necessary for the purpose

instruThe ap-

segment eliminates

time integration

of the acceler-

ometer signal for the purpose of estimating

the posi-

tion of the body segment. The excellent correlation and the force measurements culations i&y

of the segmental

orientation

on position measurements

to perform omctcr

dynamic

output.

combining

estimation

based exclus-

are accurate enough

‘recalibration’

of the acceier-

It appears that such an approach

accelerometry

is more accurate the multiple

segment

suggests that spatial cal-

and computationally

accelerometer

of

and position measurements simpler

than

system and the parameter

error model of the accelerometer

described

in the literature.

accelerometers.

The conclusion

ment system (WATSMART) that calculates

position

of

of calcu-

forces is presented. An experimental used an optoelectronic

between the force estimates by the integrated

of the soft tissue, it does suggest an approach mounted

the

using the output from a skin-moun-

ted accelcromcter. dimensional

beyond

applications.

the bone as a second-order

They described a computational dynamic

tem that

measurements

for the purpose

The results of this study suggest that the elaborate

heel strike for diJTercnt footwear.

the reading

that combines

and acceleration

orientation

on the skin overlying

cthylcne

to suggest that

can be avoided,

accelerometers: mounted

lead-

errors. However,

there is some evidence in the literature

A new approach position

lating joint

measure the segmental acceleration.

Attaching

Light

the time

If’ there is a need to

infirmution.

study such phenomena

potential

CONCLCSIOK

would, therefore, involve force compon-

ents that cannot be e&mated

viable,

accelerometer

of Hz,

Hz in a study that used a bone-pin-mounted

accelerometer.

(I)

frequency

(1985) reported

of the underlying

ol hundreds

accelerometers

approach

that could

provide a accurately

Acknowlrdypmenr-This work was supported by the National Science Foundation Grant No. EET-8809060 and by a grant from the Whitakcr Foundation.

Position and acceleration

measurements

REFERENCES

for jomt fora

wave-propagation

1185

estimation

and vibration

tests for determining

the

in oiso properties of bone. J. Eiomeckanics IO. 393401. Seemann. M. R. and Lustick. L. S. (1981) Combination of Antonsson. E. K. (1982) A threedimensional kinematic acquisition and interscgmcntai dynamic analysis system for human motton. Ph.D. thesis. Department of Mechanical Engineering Massachusetts Institute of Technology. Antonsson. E. K. and Mann. R. W. (1989) Automatic 6-d.o.f. kinematic trajectory acquisition and analysis. 1. Dynam. S.rsr. Measuremum and Control 111, 31-39. Anlonsson. E. K. and Mann. R. W. (1985) The frequency content of gait. J. Biomcchanics 1% 3947. Bresler. 8. and Frankel. J. P. (1950) The forces and moments in the leg during level walking. Trans. ASME 72, 27-36. Busby. H. R. and Trujillo. D. M. (1985) Numerical experiments with a new differentiation Jilter. J. biomrch. Enyny 107. 293-299. Crowninshield. R. D.. Johnson. R. C.. Andrcws. J. C. and Brand. R. A. (1978) A biomechanical investigation of the human hip. J. Biomuchanics II, 75-85. Gilbert. J. A.. Maxwell Maret, G.. McElhaney, J. H. and Clippinger. F. W. (1984) A system to measure the forces and moments at the knee and hip during level walking. 1. Orthop. Rcs. 2. 28 I-288. Hardt, D. E. (1978) Determining muscle forces in the leg during normal human walking-an application and evaluation of optimization methods. J. hiomcch. Engny 100. 72-78. Hayes. W. C.. Gran. J. D.. Nagurka. M. L.. Feldman. J. M. and Oatis. C. (19X3) Lc8 motiun analysis during gait by multiaxial accclcromctry: thcoreticai foundations and preliminary validations. J. hiomoch. Engny 105, 283-289. Ilodge. W. A.. Fijan. R. S.. Carlson K. L.. Burgess. R. G., Jiarris. W. Il. and Mann. R. W. (1986) Contact pressure in the human hip joint measured in viro. I’roc. narn. Acud. Sri. U.S.A. 83, 2873-2883. Kane T. R.. liayes. W. C. and Priest, J. D. (1974) Experimental determination of fiwccs cxertcd in lcnnis play. In Birmtrchunics, Vol. IV. pp. 284-290. Univcnity Park Press. Baltimore. Ladin. 2.. Flowers. C. W. and Mcssner. W. (1989) A quantitative comparison of a position mcasurcmcnl system and accelcromctry. J. Iliomrchunics 22, 295-308. Lanshnmmar. H. (1912r) On practical evaluation of dilferentiation techniques for human gait analysis. J. liomrckunits 15.99-105 Lanshammar. H. (19XZb) On precision limils for derivative numrrio;llly calculated from noisy data. J. Biomrchunics 15.459-t70. Light, L. H.. McLellan. G. E. and Klenerman, L. (1980) Skeletal transients on heel slrikc in normal walking with different footwear. J. Biomrchunics 13. 477480. McFudyen. B. J. and Winter. D. A. (1988) An integrated biomechanical analysis of normal stair ascent and descent. J. Biomrchunics 21. 733-744. Morris. J. R. W. (1973) Acccleromctry-a technique for the mcasurcmcnt of human body movements. 1. Biomrchanics 6. 729-736. blokes. L.. Fairclough. J. A., Mintowt-Czyr.. W. J. and Wilhams. J. (1984) Vibration analysis of human tibia: the effect of soft tissue on the output from skin-mounted accelerometrs. 1. hiomrd. Engng 6. 223-226. Padaaonkar. A. J.. Kriener. K. W. and Kinn. A. I. (1975) M\asurement of angular acceleration of a rigid body‘using linear accclerometurs. J. appl. Afech. 42. 552-556. Patriarco. A. G.. Mann. R. W.. Simon, S. R. and Mansour. J. M. (1981) An evaluation of the approaches of optimization models in the prediction of muscle forces during gait. J Biomrchunics 14, 513-525. Pczzack. J. C., Norman. R. W. and Winter. D. A. (1977) An assessment of derivative determining techniques used for motion analysis. J. Eiomechanirs 10. 377-382. Saha, S. and Lakes, R. S. (1977) The effect of soft tissue on

accelerometer and photographically variables defining thncdimcnsional

derived kinematic rigid body motion.

SPIE-Biomechonics Cinematography 291. 133-140. Trujillo, D. M. and Busby. H. R. (1990) A mathematical method for the measurement of bone motion with skinmounted accelerometer. J. biomech. Ettgng 112, 229-231. Usui. S. and Amidror, 1. (1982) Digital low-pass differentiation for biological signal process&g. fEEE.Frans. biomcd. Engnq BME-29.686693. Whittles M. W. (1982) Calibration and prformana of a 3dimensional television system for kinematic analysis. 1. Biomechanics 15, 1S-t- 196. Winter. D. A. and Robertson. D. G. E. (1978) Joint torque and energy patterns in normal gait. Biol. Cghern. 29. 137-142.

APPENDIX STRAIN GAUGE SYSTEM FOR JOINT MEASUREMENTS

FORCE

The dynamic forces due IO the motion of the pendulum cause deformations of the two vertical supports that suspend it. The strains resulting from the deformations are linearly related to the local stresses according to Hooke’s law. This appendix describes the installation, calibration and signal conditioning used to measure lhc local strains, and the calculations used IO derive the joint loads tha: gave rise to the measured strains. Forcr and .~lrtw unulysis The free-body diagram of the pendulum-supporting tcm is shown in Fig. Al. The lumped joint loads are F= {F,. M = {M,.

sys-

F”. F,}. M,. M,}.

(A))

The supporting forces and moments at points 0, and 0, arc given by the following termr F,=IFz,. F, =

F,,, Fz,I.

W

i F,,, F,,. F,, I.

M,=IM,,. M,,. M,,ls M,=tM,,. M,,. Mm}. The equilibrium

equations

for the forces are:

F, = -(F,,

+ F,,).

F,=

-(F,,+F&

F,=

-(F,,+F,,)+mg.

(A3)

Since there are no torsional loads transferred from the pendulum to the supporting structure. the moments &f,, and M,, arc zero. The moment equations for a cross-section located at a distance of I, from 0, on the left support are, therefore, given by the following equations: M L,.=

-(M,,-F,,:,).

Mz,,=

-(M,,+F.,:,).

(A4)

M 2,s -0. Four separate sites on the surfaa of the support (lahelcd pl. ~2, p3. p4) were chosen for stressanalysis. where pl and p3

were in the Y-Z plane along the - Y and + Y directions, respectively, p2 and p4 were in the X-Z plane along the + X and

-X

directions.

respectively

(see Fig

Al).

The tensile

1186

2. LADIN and GE WV

crons

Y

P3

P2 x

X

P4

view of crosn srction I

Ilz!%PI 2d u

Fig. Al. Free-body diagram of the pendulum supporting system.

stressesat these four points arc

used to calculate the unknown forces at the support 0, according to the following equations:

F,, (M,,-F,,l2,I)‘!

%(z*)‘~+ %h)=~+

I

F,, W,,+F,,I~*I)J 1

F,, (MS,-F,,Ir,IV uIJ(fd=~I u,.w~-

(~,1(~,)-Q,*(z‘)-~,*(~~)+~,.(~~))f

’ *

Fd’

(AS)

F,,=

’

’ bw

W4-k,l) L

F,, Of,, + F,, 121I Id I

Wlz,l-Id) (~,l(~l)-~,,(~,)-~,,(~*)+~,,(~,))~

’

where A is the cross-sectionalarea, 1 is the moment of inertia of the cross-sectionwith respect to the X or Y axis, d is half of the side length of the cross-section.The set of four redundant equations described above rcquircd the instrumentation of a

-,

The analysis for the right vertical support was done in a similar manner by instrumenting two cross-sections on the right vertical support. The equations for the forces at the supports 0, and 0, were then substituted into equation (A3), with the following set of cquations’for the three orthogonal components of the joint force:

.

WI-*I-l-,1)

F,_-~~lz~~,)+~,*~--,)+u.,(~,)+u,,(.-,))A

(A71

+my.

2 second site. so that the foras at 0, could be uniquely determined. The second set of measurements was obtained by instrumenting a cross-section at a distance of :s from 0,. The local stressesobtained from the strain gauges were then

Srrain gauge iastallarion, condirioning and calibration The strain is linearly related to the stress at the same location; therefore. equation (A7) is still valid when the

Position and acceleration measurements for joint force estimation

appropriately scakd stressese are substituted for the strains .s. The strains used for computing the joint loads were measured by strain gauges FAE-0635-S6EL from BLH Electronics. Inc. (Waltham. MA) with a gauge resistance of 350 n The locations of the corresponding cross-sectionson both supports were chosen in such a way that they had equal distanas to 0, and 0,. The strain gauges were glued carefully to the desired positions on both vertical supports at four cross-sections, with the sensitive axis parallel to the longitudinal axes of the vertical supports, and left to dry for 24 h at room temperature. The leads of appropriate gauges were then connected in Wheatstone bridge configurations to provide an output that is proportional to the resistance changes of each arm. Each Wheatstone bridge was conditioned using one strain gauge amplifier. It included a high-performance strain gauge conditioner 2B3lJ from Analog Devias (Norwood. MA) and a second-stage amplifier to increase the total gain. The

1187

strain gauge conditioner consists of an adjustable bridge excitation. a highquality instrumentation amplifier, and a three-pole low-pass filter. The excitation voltage was adjusted to 4 V in order to increase the molutioa and compensate for the heat drifting. With a total gain of lO.ooO the maximum drift was less than IO0 mV h- ’ after IS min of warm up. The calibration was done on each channel separately. First, the position of the sensitive axis of the channel was adjusted to coincide with the direction of the gravity vector, while the amplifier’s output was adjusted to zero. Then. a series of standard weights of 5 lb were hung off the support, and a calibration curve was plotted for each channel. The linearity of each channel depended on the installation of the strain gauges. The maximum nonlinearity was less than 1% for a full operation range of f IS V. The error due to the strain gauge’s transverse sensitivity and installation was less than 0.8%.