Human Movement Science 22 (2004) 597–610 www.elsevier.com/locate/humov Comparison of kinematic and kinetic methods for computing the vertical motion of the body center of mass during walking Steven A. Gard a,b,c,* , Steve C. Miﬀ a,b , Arthur D. Kuo d a Northwestern University Prosthetics Research Laboratory and Rehabilitation Engineering Research Program, Department of Physical Medicine and Rehabilitation, Northwestern University Medical School, 345 East Superior Street, RIC 1441, Chicago, IL 60611, USA b Department of Biomedical Engineering, Northwestern University, Evanston, IL 60208, USA c VA Chicago Health Care System, Lakeside Division, Department of Veterans Aﬀairs, Chicago, IL 60611, USA d Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA Accepted 14 November 2003 Abstract The vertical excursion of the body center of mass (BCOM) was calculated using three different techniques commonly used by motion analysis laboratories. The sacral marker method involved estimating vertical BCOM motion by tracking the position of a reﬂective marker that was placed on the sacrum of subjects as they walked. The body segmental analysis technique determined the vertical motion of the BCOM from a weighted average of the vertical positions of the centers of mass of individual body segments for each frame of kinematic data acquired during the data trial. Anthropomorphic data from standard tables were used to determine the mass fractions and the locations of the centers of mass of each body segment. The third technique involved calculating BCOM vertical motion through double integration of force platform data. Data was acquired from 10 able-bodied, adult research subjects – 5 males and 5 females – walking at speeds of 0.8, 1.2, 1.6, and 2.0 m/s. A repeated measures ANOVA indicated that at the slowest walking speed the vertical excursions calculated by all three techniques were similar, but at faster speeds the sacral marker signiﬁcantly (p < 0:001) overestimated the vertical excursion of the BCOM compared with the other two methods. * Corresponding author. Address: Northwestern University Prosthetics Research Laboratory and Rehabilitation Engineering Research Program, 345 East Superior Street, RIC 1441, Chicago, IL 60611, USA. Tel.: +1-312-238-6500; fax: +1-312-238-6510. E-mail address: [email protected] (S.A. Gard). 0167-9457/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.humov.2003.11.002 598 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 The body segmental analysis and force platform techniques were in agreement at all walking speeds. Discrepancies between the sacral marker method and the other two techniques were explained using a simple model; the reciprocal conﬁguration of the legs during double support phase signiﬁcantly raises the position of the BCOM within the trunk at longer step lengths, corresponding to faster walking speeds. The sacral marker method may provide a reasonable approximation of vertical BCOM motion at slow and freely selected speeds of able-bodied walking. However, the body segmental analysis or force platform techniques will probably yield better estimates at faster walking speeds or in persons with gait pathologies. 2003 Elsevier B.V. All rights reserved. PsycINFO classiﬁcation: 2260 Keywords: Human movement; Gait analysis; Center of mass 1. Introduction Translation of the body center of mass (BCOM) from one place to another is a fundamental objective of walking. Normal human walking is characterized by a periodic vertical displacement of the BCOM that moves through a complete cycle of vertical motion with each step, or two cycles during each stride. The peak-to-peak amplitude of the vertical BCOM displacement, referred to as the vertical excursion, is generally regarded to be about 4–5 cm for adult ambulators at their freely selected walking speed (Murray, Drought, & Kory, 1964; Inman, Ralston, & Todd, 1994; Saunders, Inman, & Eberhart, 1953). Investigators have used vertical BCOM motion during walking to estimate mechanical energy changes (Cavagna, 1975; Iida & Yamamuro, 1987; Tesio, Civaschi, & Tessari, 1985; Tesio, Lanzi, & Detrembleur, 1998a, 1998b), to gauge eﬃciency (Cavagna, Tesio, Fuchimoto, & Heglund, 1983; Saunders et al., 1953), to estimate work (Cavagna, Saibene, & Margaria, 1963; Cavagna & Margaria, 1966; Donelan, Kram, & Kuo, 2002), to describe symmetry (Cavagna et al., 1983; Crowe, Schiereck, & Keessen, 1992; Gard, Knox, & Childress, 1996), and as an indicator of the overall quality of gait (Bowker & Hall, 1975; Saunders et al., 1953). All of these applications rely on accurate determination of vertical BCOM motion. Investigators have developed a number of basic methodologies for calculating BCOM motion during walking. Some of the methods utilize kinematic data acquired from markers that are placed on the body, and others utilize kinetic data acquired from force platforms. One of the simplest kinematic methods, the sacral marker method, uses a single marker placed on the sacrum to approximate BCOM motion. A more sophisticated approach, the segmental analysis method, uses multiple markers to measure body segment positions, and incorporates an anthropometric model to calculate segmental center of mass positions. These segmental center of mass positions are then used to calculate the BCOM. In contrast, the force platform method uses measured ground reaction forces to calculate BCOM motion based upon Newton’s Second Law, which states that the net external force acting upon a body is S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 599 equal to its mass multiplied by its acceleration. The displacement of the BCOM can be calculated through double integration of the acceleration term with proper consideration for the integration constants. Comparisons of BCOM motion between the kinematic and kinetic methods, utilizing data that have been simultaneously acquired from walking individuals, have been shown to yield results with signiﬁcantly diﬀerent excursion magnitudes, especially in the vertical direction (Saini, Kerrigan, Thirunarayan, & Duﬀ-Raﬀaele, 1998; Thirunarayan, Kerrigan, Rabuﬀetti, Croce, & Saini, 1996; Whittle, 1997). Investigators using the force platform method have generally reported much smaller BCOM excursions than those who have used the sacral marker method (Gard & Childress, 1997; Lee & Farley, 1998; Tesio et al., 1998a). In theory, the BCOM motion calculated from kinematic and kinetic data should match, but the reason for this inconsistency has not yet been adequately explained. The discrepancies reported in the literature may be due to the assumptions associated with each method. The sacral marker method assumes that the BCOM can be closely approximated by the motion of a single marker. The segmental analysis method assumes that segmental masses and center of mass locations can be determined accurately. The force platform method makes none of these assumptions (Elftman, 1939), and does not rely on accurate placement of markers on the body as required for a kinematic analysis. Instead, it assumes that integration constants can be determined accurately. In principle, the segmental analysis and force platform methods should agree well, and a likely explanation for the diﬀerence with the sacral marker method is the motion of the limbs (Whittle, 1997). The conﬁguration of the limbs varies between double support and mid-stance, so that the BCOM’s location may vary relative to any single point on the pelvis. The purpose of the current investigation was to account for diﬀerences in vertical BCOM excursion calculated from kinematic data and kinetic data. The vertical BCOM excursions were measured during walking in able-bodied adults using the sacral marker, segmental analysis, and force platform methods. We propose a simple kinematic model that explains the diﬀerences observed between the measurement techniques. Comparisons are made between the excursions predicted by the model and those calculated from empirical data. 2. Theoretical model A simple, three-link model (Fig. 1) illustrates why motion of the limbs may aﬀect estimates of BCOM excursion. Analysis of the model is used to show how deviation between BCOM calculations and the sacral marker method may occur as walking speed varies. The three rigid links of the model represent the trunk and two rigid legs having rocker feet. Motion of two legs is suﬃcient to cause the BCOM position to change relative to a sacral marker, because each leg has its own COM that moves with the leg, and both leg COMs contribute to the overall BCOM position. The rocker feet are added because an inverted pendulum model having legs without rocker feet, referred to as a Ôcompass gait’ model (Saunders et al., 1953), is known to 600 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 Fig. 1. A simple model can be used to explain the diﬀerences observed when calculating the vertical excursion of the BCOM using the sacral marker method, the segmental analysis method, and the force platform method. In the model, the trunk COM and the BCOM are at their highest elevations in midstance when the legs are vertical (ﬁgure at left), and they are at their lowest elevations in the middle of double support (ﬁgure at right). As step length increases, the trunk COM and BCOM both undergo greater vertical excursions during the step. However, reciprocal action of the legs eﬀectively raises the BCOM position within the trunk, thereby causing the vertical excursion of the BCOM over the gait cycle to be less than that of the trunk. produce excessive vertical excursion. Rocker feet eﬀectively lengthen the leg and result in a ﬂattened trajectory that more closely matches human walking (Gard & Childress, 2001). We used this model to predict diﬀerences between the BCOM calculation methods. The model’s trunk mass was set to the combined masses of the head, arms, and thorax, totaling approximately 64% of body mass (Table 1). Each leg comprised approximately 18% of body mass. Body segment dimensions were scaled according to an assumed individual’s height of 178 cm. Leg length was assumed to be 96 cm (54% of height), and the foot rockers were deﬁned as circular arcs with radii equal to 35 cm. The vertical positions of the trunk COM and the BCOM were calculated for two geometric conﬁgurations of the model corresponding to the times during the gait cycle when the body is at its highest and lowest elevations. The body reaches its highest elevation at midstance when the legs are vertical, and it reaches its lowest elevation during double support when the legs are outstretched. For the calculations, the trunk is assumed to remain vertical in both conﬁgurations. The highest elevations of the trunk COM and BCOM are independent of step length, but their lowest elevations decrease with longer step lengths. The lowest elevations of the trunk COM and BCOM were calculated for step lengths ranging from 50 to 90 cm, corresponding to the range of step lengths observed in our research subjects’ data at speeds from 0.8 to 2.0 m/s. The vertical excursions of the trunk COM and the BCOM were calculated as the diﬀerence between their respective maximum and minimum elevations S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 601 Table 1 Deﬁnitions, lengths, and center of mass locations for body segments (Drillis, Contini, & Bluestein, 1964) Segment Deﬁnition (proximal/distal) % Body weight Segment COM location from proximal endpoint Foot Shank Thigh Forearm, hand Upper arm Head, neck, trunk Heel marker/virtual toe marker Knee center/ankle center Hip center/knee center Elbow marker/wrist marker Shoulder marker/elbow marker Shoulder markers midpoint/hip joints midpoint 1.8 4.4 11.2 2.6 3.3 53.4 0.506 0.433 0.433 0.682 0.430 0.540 8.0 Vertical Excursion (cm) 7.0 6.0 Trunk 5.0 4.0 BCOM 3.0 2.0 1.0 0.0 50 60 70 80 90 Step Length (cm) Fig. 2. The vertical excursions of the trunk and BCOM, from the simple model depicted in Fig. 1, plotted as functions of step length. As step length increases, the reciprocal action of the legs raise the position of the BCOM relative to the trunk, causing its vertical excursion to be less. at these two body conﬁgurations. The diﬀerence between these two excursions is small for very short step lengths, but increases substantially for longer step lengths (Fig. 2). This analysis therefore predicts that the sacral marker method, which tracks trunk motion, will tend to over-estimate vertical trunk excursion compared to the segmental analysis and force platform methods. 3. Methods Kinematic and force plate data were recorded from 10 subjects – 5 males and 5 females – using non-invasive procedures routinely employed in clinical gait analysis facilities. All subjects were considered to be non-pathologic ambulators in good health. The subjects were presented with a simple description of the experimental protocol and were asked to sign consent forms approved by Northwestern University’s Institutional Review Board. 602 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 Fig. 3. The measurement volume of the kinematic measurement system occupies an area on the walkway of 4.9 m · 1.2 m, with a height of approximately 1.8 m. The six platforms, numbered 1–6, are arranged along the walkway such that the array measures 2.5 m end-to-end, which enables a minimum of three foot strikes (one stride) to be captured at the fastest walking speeds of able-bodied subjects. The total length of the walkway is approximately 11 m. Data collection was conducted at the VA Chicago Motion Analysis Research Laboratory (VACMARL) of the VA Chicago Health Care System. The laboratory is equipped with an eight-camera Eagle Digital Real-Time motion measurement system from Motion Analysis Corporation. 1 Kinematic data for this study were acquired at 120 Hz. Bilateral gait analysis was performed on all subjects using a modiﬁed Helen Hayes marker set. A total of 21 markers were placed on feet, legs, pelvis, and upper extremities. Locations of the markers were on the dorsum of the feet at the MTP joints, the heels of the feet, the lateral malleolli, the lateral femoral epicondyles, on wands extending laterally from the shank and thigh segments, the anterior superior iliac spines (ASIS), the sacrum at the midpoint between the posterior superior iliac spinae (PSIS), the tips of the acromion process, the lateral epicondyles of each humerus, and at the midpoint of the wrists between the styloid processes. This marker set allowed the creation of a 12-segment rigid-link model of the body, consisting of a seven-segment model of the locomotor system and a ﬁve-segment model of the trunk and arms. VACMARL has six AMTI 2 force platforms embedded in the walkway for measuring ground reaction forces as subjects walked across them. Simultaneous with the acquisition of the marker position data, ground reaction forces were acquired at 960 Hz as the subjects walked along the walkway and stepped on the force platforms embedded in the ﬂoor. The measurement volume and the number and layout of force platforms (shown in Fig. 3) are designed so that subjects can walk with a wide range of step lengths without needing to pay attention to stepping cleanly from one force platform to another, even for an able-bodied subject who is walking at his or her fastest comfortable speed. The cross-over of a foot strike from one plate to another 1 Motion Analysis Corporation 3617 Westwind Boulevard Santa Rosa, CA 95403-1067, Tel: +707-5796500. 2 AMTI, 176 Waltham Street Watertown, MA 02172, Tel.: +617-926-6700. S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 603 during a trial was not a problem since the summed vertical ground reaction forces between all plates were utilized for the experiment. However, only those foot strikes that occurred within the perimeter of the force platform array were analyzed. Subjects were instructed to walk at four pre-determined speeds between 0.8 and 2.0 m/s. The walkway for the experiment was marked with two lines 3.66 m apart. The time needed to traverse the distance between the lines at a given steady-state velocity was monitored by the investigator with a stop watch. The subjects were instructed to walk faster or slower from trial-to-trial in order to achieve the desired walking speed. Three good trials were recorded at each walking speed; a trial was considered acceptable if the subject’s speed was within 5% of the target speed. Data processing was initially performed using Motion AnalysisTM Software (EvA and Orthotrak). Raw marker position data were ﬁltered using a fourth order bidirectional low pass ﬁlter with an eﬀective cutoﬀ frequency of 6 Hz. The force platform data were unﬁltered. The kinematic and kinetic data were further processed and analyzed using custom macros and templates for Matlab 5.3 and Microsoft Excel 2000. Vertical BCOM excursion was calculated using three diﬀerent methods: (1) the sacral marker method, (2) the segmental analysis method, and (3) the force platform method. Vertical excursion data derived from these methods were analyzed using intra-class correlation and a repeated measures ANOVA at each of the four speeds to determine if there were signiﬁcant diﬀerences between the three techniques. Post hoc analyses were performed using paired t-tests with Bonferroni corrections when signiﬁcant diﬀerences were observed in the ANOVA. The sacral marker method simply involved tracking the vertical position of the marker that was placed between the PSIS as the subject walked through the measurement volume (Miﬀ, Childress, & Gard, 2001). For an able-bodied adult in quiet standing, the BCOM has been reported to lie in the midline of the body at a distance from the ground corresponding to about 55% of the person’s height, at a position just anterior to the second sacral vertebra (Saunders et al., 1953). The sacral marker technique is probably accurate if one assumes that (1) pelvic tilt is negligible, as is the case in able-bodied walking (Murray et al., 1964), and (2) the BCOM does not move signiﬁcantly relative to the pelvis. For the segmental analysis method, the locations of the segmental centers of mass were determined from the positions of markers that were placed on the body and the calculated joint centers. Body mass segment fractions were based upon data from Harless (Drillis et al., 1964), and the segmental lengths were from data by Dempster (Drillis et al., 1964). Body segment deﬁnitions, mass fractions, and center of mass locations are provided in Table 1. For each subjects’ data, the BCOM vertical position was determined for each frame of kinematic data by calculating the vertical positions of the centers of mass of each body segment, and then using a weighted average based upon the segment mass fractions to calculate overall BCOM position (Winter, 1990). The peak-to-peak amplitudes of the resulting vertical displacement waveforms were averaged for each trial in order to calculate the vertical excursion. For the force platform method, BCOM kinematics were estimated from ground reaction force data. A step was deﬁned as the interval from initial contact of one foot 604 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 to the initial contact of the opposite foot. The BCOM velocity and position were determined through integration over a complete step of the vector sum of vertical ground reaction forces acting under both limbs (Cavagna, 1975). The vertical acceleration of the BCOM, az ðtÞ, was calculated from the summed vertical ground reaction forces, Fz ðtÞ, less the body weight, m g, and divided by body mass, m: az ðtÞ ¼ Fz ðtÞ m g : m The vertical velocity of the BCOM was calculated by integrating the BCOM acceleration over a single step: Z Z t 1 t vz ðtÞ ¼ v0 þ az ðsÞ ds ¼ v0 þ ðFz ðsÞ m gÞ ds; m 0 0 where v0 is the integration constant representing the vertical velocity of the BCOM at the beginning of the step cycle. The integration constant was determined by requiring the average vertical BCOM velocity to be zero (Donelan et al., 2002). The vertical position, zBCOM ðtÞ, was found by integrating the vertical velocity over a single step: Z t vz ðsÞ ds; zBCOM ðtÞ ¼ z0 þ 0 where z0 is the integration constant representing the vertical position of the BCOM at the beginning of the step cycle. The integration constant z0 was set to zero since only the vertical displacement, rather than absolute position, of the BCOM was desired. The vertical excursion of BCOM was calculated as the peak-to-peak amplitude of the vertical displacement waveform. 4. Results The vertical displacement waveforms that were generated from the empirical data using the three techniques that were investigated were fundamentally similar in appearance at all walking speeds (Fig. 4). Vertical excursion of the BCOM was found to increase with walking speed in all three methods (Fig. 5), with the vertical excursions calculated at 2.0 m/s being about two to three times those at 0.8 m/s. The intraclass correlation between all three methods was 0.78. Among methods, the two kinematic methods appeared to be in relatively good agreement with the force platform technique at speeds less than about 1.4 m/s. However, at faster speeds, the sacral marker method produced estimates of BCOM excursion that were considerably greater than those of the other two methods. The force platform and the body segmental analysis methods showed good agreement in the vertical excursion calculations at all speeds (Table 2). Statistical analyses revealed no signiﬁcant diﬀerences between the three methods at the walking speed of 0.8 m/s. The diﬀerences between the calculated BCOM vertical excursions using the sacral marker method and the S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 605 Fig. 4. The vertical displacement of the BCOM was calculated at each of four walking speeds with the segmental analysis, force platform, and sacral marker methods. The displacement trajectories are similar in appearance between the three methods, but their amplitudes begin to diﬀer at higher speeds. The data shown here are the average BCOM trajectories from the trials of one representative subject. The shaded region indicates the entire range of trials from the same subject, computed with the segmental analysis method. Because the force platform and sacral marker methods do not compute absolute position, their trajectories were aligned with the highest elevation of the segmental analysis method. According to the model, the segmental analysis and sacral marker methods correspond at that point. other two methods were found to be statistically signiﬁcant (p < 0:001) at walking speeds of 1.2, 1.6 and 2.0 m/s. The discrepancies between the techniques increased with walking speed in the empirical data in a manner comparable to the predictions from the simple walking model (Fig. 2). Generally, in able-bodied walking step length increases with walking speed. In the model, vertical excursions of the trunk and BCOM are small at the shorter step lengths, diﬀering by only about 5 mm. This diﬀerence increases with step length, reaching approximately 18 mm at the longest step length, which corresponds with the fastest walking speed. The observed diﬀerences between the techniques in empirical data were lower than that predicted by the model at the slower walking speeds (0.8 and 1.2 m/s), but at higher speeds the vertical excursions were nearly identical to those predicted by the model. 606 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 8.0 Sacral marker Vertical Excursion (cm) 7.0 Segmental analysis 6.0 Force platform 5.0 4.0 3.0 2.0 1.0 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Walking speed (m/sec) Fig. 5. The vertical excursion of the BCOM for the sacral marker method, the segmental analysis method, and the force platform method, plotted as a function of walking speed. All three methods yield comparable results at the slowest walking speed of 0.8 m/s, but the sacral marker method appears to overestimate the BCOM vertical excursion at faster speeds. The vertical bars represent one standard deviation from the mean; the thicker, gray error bars correspond with the segmental analysis data. Table 2 Mean and standard deviations for walking speed and the vertical excursions calculated from the three methods Walking speed (m/s) 0.83 1.22 1.62 2.02 (±0.04) (±0.05) (±0.08) (±0.08) Vertical excursion (cm) Sacral marker method Body segment analysis method Force platform method 2.7 4.0 5.8 6.7 2.7 3.5 4.7 4.9 2.4 3.4 4.6 4.8 (±0.4) (±0.8)a (±0.8)a (±1.1)a (±0.4) (±0.7)b (±0.8)b (±0.9)b (±0.4) (±0.6)c (±0.8)c (±1.0)c Paired t-test comparisons were made between methods within each velocity. a Statistically signiﬁcant diﬀerences at p < 0:001. b Diﬀerences between sacral marker and segment analysis methods. c Diﬀerences between sacral marker and force platform methods. 5. Discussion Previous investigations that have analyzed vertical motion of the BCOM during walking have shown imperfect agreement between kinematic methods based upon body segment markers and kinetic methods that utilize force platform data. Our empirical data show that the segmental analysis and force platform methods for calculating BCOM vertical excursion during walking yield similar results (Fig. 5). This agreement between these two techniques is believed to be associated with the assumption that we utilized for calculating the integration constant, v0 , generated with the force platform method. The sacral marker method diﬀered from the other two methods by an amount that increased with walking speed, leading us to believe S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 607 that segmental analysis and force platform methods are more accurate for estimating vertical BCOM motion than the sacral marker method. Statistically signiﬁcant differences at p < 0:001 were found between the sacral marker method and the other two techniques at speeds of 1.2, 1.6 and 2.0 m/s. No statistically signiﬁcant diﬀerences were found between the vertical BCOM excursions calculated with the force platform and the body segmental methods at any of the walking speeds, suggesting that both are reasonably consistent measures of BCOM vertical motion during ablebodied walking. The discrepancies observed in the vertical excursion calculations using the sacral marker method and the other two techniques can be explained with a simple, threelink model (Fig. 1). As step length increases, the trunk COM and BCOM both undergo greater vertical excursions during each step. The reciprocal conﬁguration of the legs during the double support phase of walking eﬀectively raises the BCOM position within the trunk, thereby causing the vertical excursion of the BCOM to be less than that of the trunk COM. While the vertical excursions of the trunk COM and the BCOM are of comparable magnitude at slower walking speeds when step lengths are relatively short, the diﬀerences become appreciable at the longer step lengths that occur with faster walking speeds (Figs. 2 and 5). We believe that this simple model accounts for the diﬀerences observed between the sacral marker estimation and the other two BCOM calculation techniques, especially at the higher walking speeds. A more complex model may have matched our empirical data better, had we included knee joints to account for the actual conﬁguration of the legs during mid-stance, or had we added reciprocal action of the arms during gait. We could have also developed a custom model for each research subject and based its dimensions on the actual heights of our research subjects. However, we felt that this added complexity was not necessary, and would possibly even detract from the primary reason for the observed diﬀerence in the vertical excursion measures: the reciprocal conﬁguration of the legs during the double support phase of gait. The simple model remains useful as a means to clearly illustrate the reason why the sacral marker method will diﬀer from the other methods. Another reason why other investigators may not have shown good agreement between the body segmental analysis and the force platform technique could be diﬀerences in the conﬁguration of laboratory force platforms. Many gait analysis laboratories have two force platforms that are mounted rigidly in the ﬂoor and cannot easily be moved to accommodate diﬀerent step lengths of individuals. Even if force platform separation is satisfactory for a particular subject walking at their slow or freely selected speeds, as they walk faster and their step length increases they will have diﬃculty striking both platforms unless they alter their gait and target the plates. Step length ultimately aﬀects vertical BCOM motion (Miﬀ et al., 2001); the vertical excursion of the BCOM during the gait cycle increases directly with step length. In this situation where the two force platforms are immovable, kinematic data may in fact be collected over multiple gait cycles, while force platform data is only collected over one step (from initial contact of one leg through toe-oﬀ of the opposite leg). If the subject alters their step length to strike both plates, then their vertical BCOM motion will be changed for that one step. In contrast, the vertical 608 S.A. Gard et al. / Human Movement Science 22 (2004) 597–610 BCOM excursions from the kinematic data may be averaged across several steps, so that the altered vertical excursion of a single step has a relatively smaller eﬀect on the ﬁnal kinematic result. Having a constrained step length on the force platforms may also explain why some of the previous studies show much less variability between subjects in the vertical BCOM excursions calculated from force platform data than in the kinematic data. If the step lengths of the research subjects were dictated by the force platform separation, then the subjects’ vertical excursions would probably be approximately the same. Diﬀerent methods for performing the force platform method could be another reason why previous studies have shown poor agreement between kinetic and kinematic techniques. Integration tends to ﬁlter out noise, but it is particularly sensitive to drift and to the choice of integration constants. The integration constant not only determines the initial value of a quantity, but also can contribute to drift if that quantity is then integrated, as is the case with the BCOM velocity at the beginning of a step. Working with steady-state walking data actually simpliﬁes the removal of these errors, since no net change in the vertical velocity or displacement of the BCOM can occur over a gait cycle during level walking of able-bodied subjects (Donelan et al., 2002). Under these conditions, any linear trends in the integrals can be removed, and the force plate method agrees well with the segmental analysis method. However, poor selection of integration constants, or a lack of symmetry or periodicity in gait, can lead to signiﬁcant errors. Our results suggest that the sacral marker method for estimating vertical BCOM motion should generally be avoided at faster walking speeds of able-bodied ambulators, but it may provide a reasonably good estimate of vertical BCOM motion at relatively short step lengths that occur at or below the freely selected walking speed of about 1.4 m/s. As walking speed increases above the freely selected speed, the reciprocal action of the legs extending forwards and backwards raises the BCOM position relative to the trunk and pelvis, so that the sacral marker overestimates vertical excursion of the BCOM. In addition, the sacral marker may potentially yield poor estimates of vertical BCOM motion in some pathological gaits because of signiﬁcantly increased pelvic rotations, most notably pelvic tilt. Large pelvic tilt rotations can appreciably change the magnitude of the vertical motion of a body surface marker on the sacrum with respect to the actual BCOM position due to apparent translation that occurs with out-of-plane rotations (Gard et al., 1996). Comparing all three methods, each has complementary advantages and disadvantages. The sacral marker method, despite its inaccuracies at large step lengths, remains attractive because of its simplicity. The segmental marker method is potentially more accurate, but it is also dependent on marker measurements that can be noisy due to relative motion between marker and bone. It also assumes that the body segments behave as rigid bodies, and are well characterized by anthropometric parameters that can be estimated with limited accuracy. The force platform method makes no such assumptions, but requires accurate measurements of ground reaction forces and of the BCOM position and velocity at the beginning of a step. 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