Gait consistency test based on the impulsemomentum theorem
R. Seliktar * M. Yekutiel * A. Bar *
Abstract
The kinematic and dynamic aspects of human locomotion have been investigated during the last eighty years. Significant contributions were made towards the understanding of the mechanics of movement and of the joints. As a consequence the field of Rehabilitation and Orthopaedic Biomechanics advanced considerably. However the complexity of the kinematic and dynamic data of locomotion prevented the various techniques from becoming clinically applicable. This paper attempts to develop a technique for clinical evaluation of gait by relatively simple means. For this purpose the six components of the ground reaction forces were chosen for an analysis. The major tool of the technique is the time integral of the forces. Since this is a quantitative parameter with a distinct physical definition, it can be very meaningful as far as investment of efforts in ambulation is concerned. As a first step towards the reinforcement of this thesis a consistency test was developed. The consistency test ensures that the use of dynamic forceplates do not impose a bias on the test procedure. The test is meant to indicate whether the results are valid for further processing. The concept of the test is based on the fact that the velocity vector is expected to be equal in two equivalent points of consecutive walk cycles. It therefore follows that the timeforce integral which equals the difference of momentum between the two points should be zero. The advantage of this test is that it does not discriminate between normal and pathological gait. The theory was tested with 28 subjects and the results have provided sufficient evidence for its verification.
Introduction
Ground reaction forces as such have been investigated extensively during the last two decades (Payne, 1975). Many attempts have been made to explore their potential in relation to the clinical examination of gait. It appears that despite all the efforts invested, the groundfoot forces have not become a clinical tool. This has been apparently due to the fact that no adequate technique of analysis has emerged from the above studies.
It is possible to classify the various techniques employed in the analysis of ground forces into five categories. The first method of analysis consisted of direct inspection of the analogue force versus time signals. Most investigators employed a comparative, nonquantitative approach, using as a reference the force data obtained from studies of normal gait. Studies of this kind revealed differences between hemiplegie and normal gait (Wortis et al., 1951 ; Marks and Hirschberg, 1958) and abnormal force patterns in cases of hip pathology (Charnley and Pusso, 1968; Hargreaves and Scales, 1975). All these studies concentrated on the vertical component of the ground force on the assumption that this is the most informative one.
The second method of analysis was based on quantifying the force records, normally by measuring values (minimummaximum) of the curve (Hirsch and Goldie, 1969; Kruse et al., 1971; Smidt and Wadsworth, 1973; Carlsoo et al, 1974). This resulted from an attempt to standardize the technique and enable the medical staff to use it. The general approach was positive but the measured parameters do not reflect the vast information concealed in the force records.
The third method of analysis was based on integration of the force or centre of gravity acceleration records (Ismail, 1968; Kruse et al., 1971 ; Plaja et al. ,1976; Nagai et al., 1976 and to a certain extent Cappozzo et al., 1976). This method seems to be the most informative, provided that the right parameters are investigated.
The fourth approach involved harmonic analysis of the force characteristics and the spectral power density (Gage, 1964; Smidt et al. 1971; Jacobs et al, 1972; Yamashita and Katoh, 1976 and Robinson et al., 1977). This approach may provide a quantitative criterion for evaluation and clinical followup. The technique can, however, be misleading in certain respects, as demonstrated by Jacobs et al.
The fifth approach is concerned with the orientation of the resultant ground force vector, the aim being what may be called vector visualization. Stick diagrams have been produced which depict the temporal evolution of the combined vector components in the sagittal plane (Boccardi et al, 1977). Alternatively, a nontemporal closed diagram can be formed by a combination of any two vectors (Matake, 1976). Both types of diagram tend to a constant form for healthy subjects walking under constant conditions and exhibit deviations in pathological cases. These are easy to appreciate visually, but do not seem amenable to quantitative clinicalkinetic assessment.
Physically the ground reaction forces are the ones which together with gravity (body weight) produce the acceleration pattern of the centre of gravity of the whole body. The forces are normally plotted against time, and the timeforce integral is the information which is considered of utmost importance.
There has been much doubt in relation to the reliability of the gait test employing force plate dynamometers. The reservations centred on the biases caused to the walker by the necessity to target himself on the force plate. However, most investigators performed their studies without testing reliability. Cappozzo et al. (1976) made a suggestion to test reliability by calculating the mean value of the AP shear force on one leg, assuming that it has to be zero. Their approach was not wholly correct, especially with regard to pathological gait. The intention of the present work is to suggest a reliability test and to support it with experimental evidence.
The timeforce integral
The timeforce integral represents the impulse function. As far as the whole body translatory dynamics is concerned, a single leg forceintegral does not supply the complete information. Only the forces on both legs can define the dynamic state of the body:
W + F_{L} + F_{R} = mr_{G}
W stands for weight of the whole body, the suffixes L and R stand for left and right. F is the ground force vector and r_{G} is the acceleration of the centre of gravity. Integration of this equation gives the impulse momentum theorem.
integral(Wdt) + integral(F_{L}dt) + integral(F_{R}dt) =integral(mr_{G}dt) = m(r_{G2}  r_{G1})
Where r_{G}, and r_{G2} are the velocities of the body at the limits of the integration. In other words, the time integration of the force gives the total increase of linear momentum of the body. It is therefore considered a very important parameter, since ambulation of the human body consists of alternation of propelling and restraining actions. The quantities of the "propulsion" and "restraint" are represented by the net increase in positive or negative momentum correspondingly. It is therefore most obvious that locomotor malfunctions should be quantitatively expressed by this parameter.
The above integral and the body momentum are described graphically in Fig. 1 . The area under the forcetime characteristic represents the change in momentum in the integration interval t_{1}t_{2}.
The impulse momentum theorem can be written and described for all three translatory degrees of freedom and, using the conventional notation of coordinates (vertical; anteroposterior and mediolateral), three impulsemomentum equations can be written:
integral(F_{ZL} + F_{ZR} + W)dt = m(F_{Z2}  r_{Z1})
integral(F_{YL} + F_{YR})dt = m(r_{Y2} r_{Y1})
integral(F_{XL} + F_{XR})dt = m(r_{X2}r_{X1})
The coordinate system is described in Fig. 2 . Graphically the forces on both legs can be superimposed as illustrated in Fig. 3 for the vertical force.
The result of the superimposition of the vertical forces and the weight (Fig. 3b) also represents the vertical acceleration of the centre of gravity.
The same procedure is applicable to the anteroposterior (AP) forces and the mediolateral (ML) forces except for the fact that the gravitation has no direct contribution to these components.
The concept of repeatability
The term walk cycle has been in use in locomotion studies ever since this field has been investigated. Its introduction was based on the assumption that gait is composed of a sequence of periodical events and each cycle is a representative one. Since the walk cycle has been defined by the action of the legs only, for example, one double stride, the term needs further expansion.
During one double stride there are subevents occurring at different time sequences (for instance, lateral oscillations of the body, movement of the head, movements of the arms, etc). These subevents could be out of phase with the defined walk cycle. This means that if it is assumed that the basic frequency or first harmonic is the double stride, then some of the named subevents would not be higher harmonics of this phenomenon. In other words, not all the events of gait would be repeated every walk cycle.
The basic assumption made is that all events are repeatable within one walk cycle and they are either of the same frequency of occurrence or higher harmonics of the basic one. This implies that dynamically and kinematically every two corresponding points in two subsequent walk cycles should be identical. This implication is of great value since its practical meaning is that the total change of momentum between every two equivalent points is zero.
Since the momentum equation is in a vector form, then the zero momentum concept will apply to every degree of freedom separately. The meaning of the zeroing of the momentum between every two equivalent points is that the absolute velocities in these points are equal, which conforms with the concept of repeatability.
It is therefore possible to use this concept assuming that if the momentum between, say, heel strike to heel strike of the same leg is not equal to zero, then the walk cycle is not a representative one. In other words, if this happens, then the walker must have altered his walking pattern during the cycle under consideration.
Force plate reliability test
Force plate transducers have been introduced extensively into biomechanics departments and gait clinics. A significant contribution to the advancement of the force plate technology was recently made by Kistler Instruments AG, of Switzerland which produced a sophisticated and reliable piezoelectric force platform. This has been acquired by many institutions and increased the popularity of the use of force plates.
There has always been some doubt in relation to the reliability of this tool from the psychological point of view of the walker. It is evident in some examinations of gait that the walking subject makes certain adjustments when targeting himself on the force plate. This becomes especially evident with certain hemiplegic patients. In addition, other biases may exist which alter the behaviour of the walker in the vicinity of the force plate. Our observations also indicate that there is a certain correlation between the maintenance of consistency during gait on the instrumented walkway and the intelligence of the walker. Certain patients are totally unable to conform with the test requirements even after several runs of training.
In some of the examinations it is obvious from observation that the recorded gait cycle was not a "typical" one and it can be eliminated from the data. In certain cases, however, inconsistency is very difficult to detect. One is aware of the fact that the force plate is limited to the population which can perform the test from both functional and psychological aspects. It is desirable, however, to exclude the false results from the data analysis so as to draw conclusions only from the reliable tests. This is the aim of the reliability test.
It was stated earlier that the change in total momentum between two equivalent points should be zero. Since the momentum is a vector and all three components zero simultaneously, testing has been confined to one component only which is the AP shear force. This is a test for detecting inconsistency but it is not sufficient for proving consistency. It is however very improbable that one of the momentum components will zero without the other two zeroing. This test is therefore regarded for the time being as adequate for clinical examinations.
Since most laboratories have at the most two force plates installed in the walkway, the force data for one full cycle are not complete. During one full cycle there are two double support phases (Fig. 4 ). The two force plates provide information only on one double support phase. The interrupted lines in the force traces in Fig. 4 describe the missing information. On the other hand the shaded area would be excessive if the missing information would have been available.
In a typical cycle the shaded area would be identical to the missing area enclosed by the interrupted line. When the cycle deviates from the typical one, the two areas will not be identical and, at the worst, one of them will be reduced to zero. It is most improbable that change of sign will occur since "restraint" is almost impossible at the "push off stage and vice versa.
If therefore one assumes identity between the two areas in order to provide the missing information, a certain error can be expected of which the boundary values can be estimated. The lower boundary will be zero (typical cycle) and the upper boundary will be A1 which is the shaded area. It is however emphasized that the inconsistency of the cycle is caused by distortion of the whole curve during the period T+D.S. and therefore the introduction of this error is not severe. The following gives an estimate of the extent of this error.
If the total absolute momentum area of a walk cycle is defined as 100% and the dashed area as X% of the total, then the following is obtained:
A=integral((F/W)(dt/T))=1
and
X= — A_{1}/A x 100
Where A_{1} is the dashed area then the relative error by definition will be e=X and this, as stated before, will be the maximum error obtained due to the above assumption. The minimum will naturally be zero. Normally this error will be closer to its lower boundary (i.e. zero) than to its upper boundary. To obtain such extreme inequality, bearing in mind that the two areas assumed equal are due to the action of the same leg during the same phase in two consecutive cycles, it takes an extreme distortion of the walk cycle which can easily be detected visually by observing the actual gait. The upper boundary was estimated from experimental results and was found to be about 8 per cent.
Test procedure
Nine normal subjects and 19 patients suffering from various locomotor disabilities were examined. The subject's height and weight were taken prior to each test. The subject was then asked to walk at his convenient pace along the walkpath. The starting point was marked on the ground and was altered until the subject hit both force plates. It was difficult however to eliminate the subject's awareness of the necessity "to hit the target". In certain cases when it was impossible to obtain the desired performance the patient was instructed to target himself consciously on the force plate. In a few cases even this was found impossible and those tests were excluded from the record.
The force measuring system included two Kistler Z3482 piezoelectric force platforms arranged as illustrated in Fig. 5 . The forcetransducers were connected via 16 charge amplifiers and 4 summing amplifiers to a Bell and Howell multichannel ultra violet recorder.
Simultaneously the walker was filmed by two TV cameras from the side and the front and this was recorded via a screen splitting special effect generator by an IVC 801 P video tape recorder.
The video information was meant for maintaining a visual record of the gait only and not for further processing. Altogether, six force channels were recorded for both legs. In order to test the repeatability of the technique, one of the subjects was recorded four times.
At the end of each experiment the subject was asked to stand on each of the forceplates and static records were taken for body weight reference.
Results and analysis
The analogue data obtained from the six force plate channels recorded on ultraviolet paper were digitized and prepared for computer processing. The consistency test was performed with the AP force data only and the results were promising. The forces were integrated in their normalized form and the total impulse was calculated for one complete walk cycle. Four out of the 19 patients were unable to take a long enough stride in order to place one foot on each force plate. Those patients were instructed to walk across the walkway at right angles, thus using the short side of the force plate. The repeatability tests showed fluctuations of the results within a narrow range and the percentage deviation from zero momentum per walk cycle amounted from 0 up to 8 per cent. An example of four tests on one normal subject are given in Table 1 . Table 2 summarizes the percentage deviation for the 9 normal subjects and Table 3 gives the same for the pathological cases.
It is evident that the deviations in both normal and pathological cases are in the same range. This verifies the assumption that the test is independent of the pattern of gait as long as sequential consistency is maintained. The degree of consistency is determined by the percentage deviation. A deviation up to 8 per cent can be considered normal and is believed to be caused by errors in the technique and limited deviation between two consecutive cycles. The 8 per cent was calculated by excluding the five extreme deviations from the 28 results and taking the mean values of the other 23 and this gave a mean value of 4.5 per cent and a standard deviation ± 2.8 per cent. Larger deviations will be considered an indication of inconsistency and the degree of inconsistency will be graded by the percentage deviation.
Discussion
The aim of the present study was to examine the feasibility of using force plate dynamometers as clinical tools. Two facts are evident to the experienced user of the force plates. 1. The force versus time records are extremely informative in relation to deviations from normal gait. 2. There is a group of patients and even some normal subjects who are totally unable to walk on the force plates without severely distorting their gait pattern. The intention was to ensure that the measured gait cycle under consideration represents the walking performance of the patient and not an atypical event. Once this is established the impulsemomentum can be accepted as an accurate measure of the subject's capacity to perform the various phases of ambulation.
In Table 1, Table 2 and Table 3 the absolute momentum describes the total measured area which is
H _{abs} = integral^{1} (F/W) (dt/T)
This gives a measure of the impulsiveness of gait. The range of the nondimensional absolute momentum, however, seems to be quite narrow with a mean value of 0.0757. In other words it seems that the mean value of the AP forces applied by a subject is proportional to his body weight, and therefore the above ratio which represents this is maintained in the vicinity of one quarter of the product WxT.
The concept of the zeroing impulse function has been demonstrated by integrating the experimental results by time. The results were close to zero except in some odd cases where inconsistency is suspected. As previously stated, the data were recorded on a uv paper and were digitized and transferred to punched cards. The scaling factors for the two force plates may have differed slightly. The accumulated error therefore may account for the deviations from zero. It is possible, however, that part of the percentage deviation is caused by limited inconsistency between the gait cycles and this can be seen from the results of the repeatability tests (Table 1 ). A more precise calibration and use of magnetic tape for data recording will definitely lead to a more reliable test.
The advantage of the present technique is that it employs physical variables to represent ambulation by quantity of movement. The zero impulse test is a correct procedure for testing repeatability of kinematic cycles. In order to find out whether gait cycles are perfectly repeatable, further sophistication is required. Still, it is possible to conclude from the present study that repeatability exists within a reasonable degree of accuracy.
It is also possible to conclude from the limited number of patients examined that the technique is as reliable with patients as it is with normal subjects.
The momentum calculations are simple ones and provide a good measure both of ambulation and of reliability. The test can be processed online on a computer and the conclusions can be obtained on the spot.
At present, a further investigation is underway to establish quantitative scale for representation of the activity during the various phases of gait. This is essential for three purposes:
Establishment of a quantitative procedure for prosthetic alignment.
Establishment of a quantitative followup technique in gait training.
Diagnostic aid for detection of the nature and degree of locomotor disability and determination of the muscles responsible.
Acknowledgements
The experimental work was carried out at the Celine Hodiamont Biomechanics Laboratory of the Loewenstein Rehabilitation Hospital. The data processing and the analysis were performed at the Julius Silver Institute of Biomedical Engineering. The research was sponsored by internal funds from both institutions. The authors wish to acknowledge the contribution to the research programme made by Professor T. Najenson, Medical Director of the Hospital and Dr. Z. Susak, Director of the Department of Amputees.
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