Chapter 5 Interlimb coordination in prosthetic walking: Effects of asymmetry and walking velocity
The present study focuses on interlimb coordination in walking with an above-knee prosthesis using concepts and tools of dynamical systems theory (DST). Prosthetic walkers are an interesting group to investigate from this theory because their locomotory system is inherently asymmetric, while, according to DST, coordinative stability may be expected to be reduced as a function of the asymmetry of the oscillating components. Furthermore, previous work on locomotion motivated from DST has shown that the stability of interlimb coordination increases with walking velocity, leading to the additional expectation that the anticipated destabilizing effect of the prosthesis-induced asymmetry may be diminished at higher walking velocities. To examine these expectations, an experiment was conducted aimed at comparing interlimb coordination during treadmill walking between 7 participants with an above-knee prosthesis and 7 controls across a range of walking velocities. The observed gait patterns were analyzed in terms of standard gait measures (i.e., absolute and relative swing, stance and step times) and interlimb coordination measures (i.e., relative phase and frequency locking). As expected, the asymmetry brought about by the prosthesis led to a decrease in the stability of the coordination between the legs as compared to the control group, while coordinative stability increased with increasing walking velocity in both groups in the absence of a significant interaction. In addition, the 2:1 frequency coordination between arm and leg movements that is generally observed in healthy walkers at low walking velocities was absent in the prosthetic walkers. Collectively, these results suggest that both stability and adaptability of coordination are reduced in prosthetic walkers but may be enhanced by training them to walk at higher velocities.
Stella Donker & Peter Beek Acta Psychologica, 110, 265-288, 2002
Introduction
Whereas normal unimpaired walking is more or less symmetrical, walking with a leg prosthesis is characterized by a marked asymmetry between the leg movements as well as a lower preferred walking speed (Boonstra, Fidler, & Eisma, 1993; Dingwell, Davis, & Frazier, 1996; Isakov, Burger, Krajnik, Gregorič, & Marinček, 1996, 1997; Murray, Mollinger, Sepic, Gardner, & Linder, 1983; Skinner & Effeney, 1985). On the assumption that symmetric gait is energetically most efficient (Mattes, Martin, & Royer, 2000; Waters & Mulroy, 1999), pathological gait analyses have been mainly concerned with identifying deviations of pathological gait from normal gait, both in terms of symmetry and preferred walking velocity. Partly in conjunction with this, locomotor symmetry and walking velocity have become important determinants of the quality of prosthetic walking and are often considered to be meaningful goals in rehabilitation. There are strong grounds, however, to doubt whether (re-)establishing a normal, symmetric gait pattern constitutes an appropriate goal for rehabilitation procedures aimed at improving pathological gait patterns stemming from an impairment at one side of the body. After all, in such cases, the “pathological body” is inherently asymmetric and by definition hardly equipped for producing a symmetric gait pattern. Far more important than trying to (re-)establish a normal, symmetric gait pattern, so it seems, is to promote the development of a functionally optimal walking pattern, that is, one that is both stable and adaptive, increases mobility, and minimizes the risk of falling. For example, in young healthy walkers, the preferred walking velocity corresponds with the most efficient walking velocity, whereas in prosthetic walkers (or elderly, for that matter) the comfortable walking velocity is lower than the most efficient walking velocity, that is, the energetic optimum (e.g., Jaegers, Vos, Rispens, & Hof, 1993; Martin, 1999). Possibly, an energetically less optimal walking pattern is elected which is more stable, thus providing (a feeling of) safety. In this study we focus on the stability and adaptability of walking with a leg prosthesis. Much of the pertinent literature on prosthetic walking concerns the issue of energy consumption and its determinants (Czerniecki & Gitter, 1996). The apparent deadlock in current discussions on this topic highlights the need for studying prosthetic walking in terms of its functionality. For instance, quite some research has been done on the mass distribution of the prosthetic leg but no consensus has been reached to date as to what constitutes an optimal prosthetic inertial loading. On the one hand, it has been argued that the physical properties of the prosthetic leg 98
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should match the physical properties of the normal leg so as to create an optimal symmetry between the legs. On the other hand, it has been suggested that a prosthesis should be as light as mechanically possible so as to reduce the work needed to propel it forward (for an overview see, Selles, Bussmann, Wagenaar, & Stam, 1999). A complicating factor in resolving such issues is that the amputee population is very diverse in terms of both the cause of the amputation and its anatomical characteristics. Furthermore, amputees differ widely in age, profession, daily activities, and so on. Also in view of this diversity, we deemed it useful to study the functional properties of prosthetic walking such as stability and adaptability. The conceptual approach adopted in the present study was motivated from both a clinical and a theoretical point of view. As mentioned previously, locomotory symmetry and walking velocity are often used to characterize and improve prosthetic gait in clinical settings. Yet, with the prevailing emphasis on energy consumption in the pertinent literature, the effects of these factors on coordinative stability have hardly been investigated. There are, however, at least three good (i.e., both theoretically and empirically motivated) reasons to believe that such effects are present. First, it is known from dynamical systems theory (DST) that, in general, a system of coupled oscillators with identical properties is more stable than a similarly coupled system with different, asymmetric components (e.g., Haken, Kelso, & Bunz, 1985; Rand, Cohen, & Holmes, 1988; Schmidt, Shaw, & Turvey, 1993; Sternad, Amazeen, & Turvey, 1996; Treffner & Turvey, 1996). Using the standard deviation of relative phase as a theoretically motivated measure of coordinative stability, this general principle has been demonstrated repeatedly to apply to human rhythmic interlimb coordination as well (e.g., Amazeen, Amazeen, & Turvey, 1998; Amazeen, Sternad, & Turvey, 1996; Jeka & Kelso, 1995; Jeka, Kelso, & Kiemel, 1993; Kelso & Jeka, 1992; Peck & Turvey, 1997; Schmidt, Beek, Treffner, & Turvey, 1991; Serrien & Swinnen, 1998a, 1998b; Swinnen, Dounskaia, Verschueren, Serrien, & Daelman, 1995; Swinnen, Jardin, Meulenbroek, Dounskaia, & Hofkens-Van Den Brandt, 1997; Swinnen, Young, Walter, & Serrien ,1991; Whitall, 1989; Whitall & Caldwell, 1992). Generalizing these findings to locomotion, one would expect that the asymmetry brought about by an above-knee prosthesis is accompanied by a reduction of the stability of the coordination between the leg movements. Second, previous studies on locomotion from the perspective of DST have demonstrated that the stability of coordination between leg movements as well as between leg and arm movements, as indexed by the standard deviation of relative phase, increases with increasing walking velocity Chapter 5
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(Donker, Beek, Wagenaar, & Mulder, 2001; Wagenaar & Van Emmerik, 2000; chapter 2). This observation is at odds with one of the key findings of a large number of studies on rhythmically coordinated bimanual movements (e.g., Kelso 1981, 1984), namely that the coupling strength between two oscillating limbs is reciprocally related to movement frequency. Admittedly, it could be argued that in the cited locomotion studies walking velocity was not increased sufficiently far in that the point at which the coordinative stability of walking collapses and transitions to running may occur was not approached (cf. e.g., Diedrich & Warren, 1995). This argument, however, cannot explain the changes in the frequency coordination between arm and leg movements that have been observed at relatively low walking velocities (see the next argument). In seeking to understand the dynamical discrepancy of interest, it is important to recognize that, in contrast with bimanual studies, biomechanical factors strongly affect the dynamics of interlimb coordination during human locomotion (cf. Schöner, 1995). Not only are the muscular, intersegmental and inertial forces involved in human walking much larger than in performing rhythmic hand and finger movements, there is in walking also a clear need to maintain balance in the gravitational field, which, presumably, is especially difficult to satisfy in prosthetic walking. Thelen (1986) therefore even went as far to suggest that balance rather than walking velocity may be the control parameter that shifts the system into a new behavioral mode. In view of these considerations, it is reasonable to expect that the observed increase in coupling strength between the leg movements in healthy subjects at higher walking velocities is being preserved in prosthetic walkers. If so, this would be a potentially important result in that it could provide a means to compensate for the expected reduction in coordinative stability in prosthetic walking, namely by prompting or training prosthetic walkers to walk at higher velocities. Third, as was alluded under the previous point, walking velocity has also been shown to affect the frequency coordination between arm and leg movements during human locomotion. At low walking velocities the arms generally swing in unison at twice the frequency of the leg movements, whereas at higher walking velocities the 2:1 frequency coordination pattern is abandoned in favor of a 1:1 frequency coordination pattern in which the arms swing alternately at the same frequency as the leg movements (e.g., Craik, Herman, & Finley, 1976; Donker et al., 2001; Wagenaar & Van Emmerik, 2000; Webb & Tuttle, 1989). This phenomenon has received contrasting interpretations in the literature. Following an earlier suggestion by Webb, Tuttle, and Baksh (1994), Wagenaar and Van Emmerik (2000) 100
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argued that the 2:1 rather than the 1:1 frequency coordination is elected at low walking velocities because the arms have a tendency to move as close to their eigenfrequency as possible. Donker et al. (2001; chapter 2), however, found that the velocities at which a switch occurred to the 1:1 frequency coordination varied considerably across participants, even though the eigenfrequencies of their arms were rather similar. They therefore concluded that, at least to a considerable degree, the observed coordination patterns between arm and leg movements reflect a genuine coordination principle rather than a purely biomechanical effect. Prosthetic walking provides an interesting context for further examining the factors underlying the observed transition in frequency coordination between arm and leg movements. After all, a prosthesis brings about a marked change in the biomechanical properties of one of the legs without affecting the eigenfrequencies of the arms. Given the expected reduction of coordinative stability in prosthetic walking, and given the observation that, in general, 2:1 frequency coordination is less stable than 1:1 frequency coordination (Donker et al., 2001; Serrien & Swinnen, 1997), it is conceivable that in prosthetic walking the 1:1 frequency coordination mode is being preserved at all walking velocities. To examine these expectations and possibilities, we conducted an experiment comparing interlimb coordination during treadmill walking between 7 participants with an above-knee prosthesis and 7 controls with two intact legs across a range of walking velocities. In particular, we sought to answer the following questions: (1) To what extent does the prosthesis bring about an asymmetry in the leg movements, and to what extent is this behavioral asymmetry affected by walking velocity? (2) How is the coordinative stability between the prosthetic and normal leg affected by the asymmetry, and how is this effect modulated by walking velocity? (3) How are arm and leg movements coordinated in prosthetic walkers? In particular, is the 2:1 frequency coordination mode often observed in healthy walkers also seen in prosthetic walkers? (4) What is the effect of asymmetry and walking velocity on the relative phasing between arm and leg movements? (5) What is the effect of asymmetry and walking velocity on the stability of this relative phasing?
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Method
Participants Seven volunteers (2 women and 5 men) with an above-knee prosthesis participated in the experiment. All amputations were due to trauma. One participant had suffered from a brain tumor some years later. The ages of the participants ranged from 37 to 59 years (mean age = 42 years), and the time since amputation ranged from 3 to 54 years (mean time = 22 years).1 The mean stump length, measured from the ischial tuberosity to the end of the stump, was ± 22 cm (range = 10-29 cm). All participants used their prosthesis on a daily basis. None of the participants suffered from stump problems (swelling, pain, sores etc.) on the day of the experiment. The characteristics of all prostheses except one were known. Six prostheses were equipped with a vacuum socket. Five artificial knees were driven pneumatically and one electronically (i.e., intelligent prosthesis). Two prostheses were equipped with an energy storing foot. The other four feet were equipped with a single axis for making forward and backward movements while allowing some flexibility for moving laterally. Seven healthy, unimpaired participants (3 women and 4 men) served as a control group (mean age = 34 years; range = 22-42 years). All participants were naive with regard to the purpose of the experiment. They all gave their written informed consent before the experiment proper began but after the local Medical Ethics Committee study had given their approval of the experiment. Materials The participants walked on a walking belt (Enraf Nonius, model Entred Reha) with a computer-controlled velocity (the time required to change the belt velocity by 0.5 km/h was about 1.5 s). Small lightweight triangular frames with a reflective spherical marker (diameter 20 mm) on each corner were attached to the forearms, upper and lower legs. The positions of the 6 triangular frames were recorded by means of a three-dimensional passive registration system called PRIMAS (Furnée, 1989) at a sampling rate of 100 Hz. Six video cameras were placed around the walking belt, three on each side, such that the markers were in view of at least two cameras at all times as is required for the successful three-dimensional reconstruction of their position. The reconstruction error was about 1 mm in all dimensions. Procedure and Design Prior to each experimental session the participants were acquainted with 102
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walking on the walking belt without the support of a bar. This took about 10-20 min. Subsequently, a reference measurement was taken during which the positions of the marker frames in space were recorded for future reference (Figure 1A). Participants were instructed to walk as naturally as possible. No specific instructions were given with regard to the movements of the arms. Each participant took part in a measurement session consisting of one long trial. During this trial walking velocity was increased in steps of 0.5 km/h, starting from 0.5 km/h up to 3.5 km/h. About 15 s after having set the velocity of the walking belt, the position data of the markers were recorded for 40 s at that particular walking velocity, whereafter the belt velocity was increased to the next level. An entire trial lasted about 7 min. Data Collection and Compression Initial processing. From the reference measurements a three-dimensional global coordinate system (x-, y- and z-axis) was defined that coincided with the sagittal, frontal and transverse plane of each participant (see Figure 1A and B). The rotation of the marker frames around the z-axis (i.e., the rotational movement in the sagittal xy-plane) was calculated and then filtered using a second order Butterworth lowpass filter with a cut-off frequency of 5 Hz.2 Thus, the arm and leg movements were expressed as rotations (in degrees) around the z-axis with increasing values corresponding to forward movements and decreasing values to backward movements (see Figure 1C). Temporal symmetry. The moments of toe off and heel strike were extracted by means of a peak-picking algorithm from the angular position data of the legs covering 8 representative stride cycles. The moments at which the maximal excursion of the forward movements (MFM) of the leg was reached were defined as the moments of heel strike (HS), while the moments at which the maximal excursion of the backward movements (MBM) was reached were defined as the moments of toe off (TO). From these moments, stance time (from heel strike to toe off) and swing time (from toe off to heel strike) were calculated (see Figure 1C). As a first index of the temporal symmetry between the legs we calculated the ratio between the mean swing times of both legs as well as the ratio between the mean stance times of both legs. For the prosthetic group these ratios were calculated by dividing the times of the normal leg by those of the prosthetic leg, whereas for the control group the times of the right leg were divided by those of the left leg. If the mean stance times as well as the mean swing times are the same for both legs, both ratios will be 1, indicating maximal temporal symmetry between the legs. Chapter 5
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Whereas the stance and swing time ratios index the degree of similarity between the leg movements in terms of their internal temporal organization (i.e., the division of the step cycle in step and swing times), the ratio between the step times of both legs was calculated to quantify the overall temporal correspondence between the leg movements. Specifically, for the prosthetic group the step time of the normal leg was divided by that of the prosthetic leg, while for the control group the step time of the right A. Y
Y
arm normal side - NA
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Figure 1. (A) Sagittal view of the marker configuration in the reference position and the corresponding Cartesian coordinate axes: x-axis, y-axis and z-axis. Marker frames, each holding three markers as indicated by the black circles, were placed on the forearms, upper and lower legs. (B) Sagittal view of a fictitious walker with an upper leg prosthesis. The present study was focused on the rotational movements of the limbs around the z-axis (i.e., in the sagittal xy-plane), as indicated by the arrow. The prosthetic leg, PL, and the ipsilateral arm, PA, are striped; the normal leg, NL, and the ipsilateral arm, NA, are colored white. (C) Example of an experimental trial in which the rotational movements of the normal and the prosthetic leg are shown during 3 stride cycles. The moments at which a maximal excursion of the forward movements was reached, MFM, were defined as the moments of heel strike (HS). The moments at which a maximal excursion of the backward movements was reached, MBM, were defined as the moments of toe off (TO). From these moments, stance (white bars) and swing time (black bars) were calculated. Both types of moment were further used to calculate point estimates of relative phase between the limb movements. An example is given of the calculation of the point estimate of relative phase between normal and prosthetic leg movements at the moments of maximal excursion of the forward movements (φPLNL_MFM).
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Figure 1 (continued)
leg was divided by that of the left leg. If the step time is equal in both legs, the ratio will be 1. The step time of a particular leg was calculated from toe off of that leg to toe off of the contralateral leg. Symmetry of movement trajectories. To obtain a more detailed impression of the effect of walking velocity on the individual leg movements, we closely examined the knee angle trajectories during the stride cycle using the analytical method illustrated in Figure 2. First, knee angles were calculated by subtracting the angular position data of the lower leg from those of the upper leg. For 8 representative stride cycles, the so obtained knee angle trajectories were normalized to the moments of toe off of the leg in question, and these normalized knee angle trajectories were averaged to obtain a reference trajectory (Figure 2A). Subsequently, for each stride cycle, the difference between the normalized knee angle trajectory and the reference trajectory was calculated for each data point and squared (Figure 2C). Finally, these squared distances were averaged over all 8 stride cycles for each leg, resulting in a measure for the intralimb variability of the leg movements at each velocity level. Next, this method was extended to obtain a similar measure for the interlimb consistency or symmetry of the leg movements (or rather the lack thereof). Specifically, for the prosthetic group the kinematic differences between the legs were determined by calculating the squared distances between 8 normalized prosthetic knee angle trajectories (corresponding Chapter 5
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again to 8 representative stride cycles) and the averaged knee angle trajectory of the normal leg (which now served as reference signal; see Figure 2B and D). These squared distances were subsequently averaged over all 8 stride cycles to quantify the degree of asymmetry of the movement trajectories. For the control group essentially the same procedure was applied by using the averaged right knee angle trajectory as reference for 8 normalized left knee angle trajectories. B. Knee angle prosthetic leg
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Figure 2. Illustration of the method used to quantify the degree of asymmetry between the movement trajectories of the legs. (A) For 8 stride cycles the normalized knee angle trajectories of a normal leg are shown (dotted lines) together with the mean normalized knee angle trajectory, which served as a reference (thick line). (B) Similarly, 8 normalized knee angle trajectories of a prosthetic leg are depicted (dotted lines) together with the mean normalized knee angle trajectory of the normal leg as reference (thick line). (C) For each stride cycle and at each point in the stride cycle the squared difference between the knee angle of the normal leg and the mean normalized knee angle trajectory is plotted. (D) Similarly, the squared difference between the knee angle of the prosthetic leg and the mean normalized knee angle trajectory of the corresponding normal leg is plotted for each stride cycle and at each point of the stride cycle.
Point estimates of relative phase. For the prosthetic group, point estimates of relative phase were calculated for the following five limb pairs: prosthetic leg/normal leg (φPL/NL), arm prosthetic side/prosthetic leg (φPA/PL), 106
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arm normal side/normal leg (φNA/NL), arm normal side/prosthetic leg (φNA/PL), and arm prosthetic side/normal leg (φPA/NL; Figure 1B). For the control group, point estimates of relative phase between the limb movements were calculated in analogous fashion: right leg/left leg (φRL/LL), left arm/left leg (φLA/LL), right arm/right leg (φRA/RL), left arm/right leg (φLA/RL), and right arm/left leg (φRA/LL). All point estimates of relative phase were calculated in two ways based on two different characteristic moments in the angular position data of the limbs, namely the moments at which the maximal excursion of the backward movements was reached (φMBM) and the moments at which the maximal excursion of the forward movements was reached (φMFM; Figure 1C). All point estimates of relative phase (φMBM,MFM) were calculated using the formula: φMBM,MFM(i) =
t limby(i ) − t limbx (i ) t limbx (i +1) − t limbx (i )
⋅ 360,
where limbx and limby are the normal leg (right in the case of the control group) and the prosthetic leg (left in the case of the control group), respectively, when calculating the point estimate of relative phase between the leg movements. Similarly, when calculating the point estimate of relative phase between the arm and leg movements, limbx and limby stand for a leg and an arm movement, respectively. Thus in the latter case, the leg movement is used as a reference in that the maximal excursion of the backward or forward arm movement following that of the leg movement was used for calculating the relative phase. The stability of the coordination between the limbs was assessed by calculating the standard deviation of relative phase (SDφMBM,MFM) for all five pairs of limb movements for each participant and for each walking velocity (again using 8 stride cycles). Means and standard deviations of relative phases were calculated using directional (i.e., circular) statistics in order to avoid artifacts as a result of phase wrapping (Batchelet, 1981; BurgessLimerick, Abernethy, & Neal, 1991). Spectral analysis. To examine whether any 1:1 or 2:1 frequency-locked patterns of coordination were present between ipsilateral arm and leg movements, the collected data were analyzed in three steps. First, the dominant frequency of each limb movement was determined by applying a Fast Fourier Transform (FFT; Hamming window) algorithm to the position data. The dominant frequency of the limb movement was defined as the frequency at which the largest peak was present in the power spectrum Chapter 5
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based on the output of the FFT. To be significant, this peak had to be larger than the mean power plus twice its standard deviation. Note that the dominant frequency of the leg movements is equivalent to the stride frequency. Second, the ratio between the dominant frequencies for the arm and leg movements at each side of the body was calculated. Third, it was established whether the calculated frequency ratios were at or near the integers 1 or 2. As a demarcation criterion for frequency locking a deviation of ±0.1 from the integer values of interest (1 and 2) was used (for further details see Donker et al., 2001; chapter 2). Statistical analysis. Using SPSS software, separate analyses of variance (ANOVAs) with repeated measures were performed on the dependent variables mentioned in the preceding (i.e., swing time, stance time, swing time ratio, stance time ratio, step time ratio, squared differences, mean relative phase, Mφ, and the standard deviation of relative phase, SDφ) using various factorial designs involving the between-subjects factor group (2 levels) and the within-subjects factors limb pair (5 levels), velocity (7 levels), and moment (2 levels). To evaluate the contrast between the coordination patterns adopted at the lower walking velocities with those adopted at the higher walking velocities, additional ANOVAs were conducted in which the factor velocity was reduced to two levels, viz. low walking velocities (the data of 0.5, 1.0, and 1.5 km/h were collapsed) and high walking velocities (the data of 2.5, 3.0 and 3.5 km/h were collapsed).
Results
(1) Asymmetry between leg movements (cf. Question 1 in Introduction). Temporal symmetry. In the control group the means of both stance and swing time ratios were about 1.0, marking a temporal symmetry between the leg movements. In contrast, both the stance and swing time ratios in the prosthetic group differed significantly from those in the control group (F(1,6) = 6.17, p < .05) [stance time ratio] and F(1,6) = 13.39, p < .05 [swing time ratio]), revealing a temporal asymmetry between the leg movements in the prosthetic group. This asymmetry was due to the observation that the stance phase of the prosthetic leg was shorter than that of the normal leg as indicated by the mean stance and swing time ratios of 1.05 and 0.86, respectively. Whereas both in the prosthetic and the control group mean swing and stance time decreased significantly with increasing belt velocity (swing time: F(1,6) =9.85, p < .05 [prosthetic leg], F(1,6) = 108
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14.39 p < .01 [normal leg], F(1,6) =36.14, p < .005 [left leg], and F(1,6) = 55.83 p < .001 [right leg]; stance time: F(1,6) = 60.84, p < .001 [prosthetic leg], F(1,6) = 37.94, p < .005 [normal leg], F(1,6) = 206.29, p < .001 [left leg], and F(1,6) = 242.53, p < .001 [right leg]), velocity had no significant effect on the mean stance and swing time ratios, suggesting that the degree of similarity between the leg movements in terms of their internal temporal organization did not change systematically as a function of walking velocity. B.
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Figure 3. (A) Group-averaged step time ratios for the prosthetic group (black circles) and the control group (white circles). (B) Group-averaged squared distances between the legs for the prosthetic group (black circles) and the control group (white circles). (C) Group-averaged point estimates of relative phase between the leg movements when calculated at the moments of maximal excursion of the backward leg movements for the prosthetic group (MφPL/NL_MBM, black circles) and the control group (MφRL/LL_MBM, white circles). (D) Group-averaged standard deviations of the point estimate of relative phase between the leg movements when calculated at the moments of maximal excursion of the backward leg movements for both the prosthetic (SDφPL/NL_MBM, black circles) and the control group (SDφRL/LL_MBM, white circles)
Overall, the mean step time ratios were significantly lower in the prosthetic group than in the control group (F(1,6) = 12.62, p < .05; Figure Chapter 5
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3A), again revealing temporal asymmetry between the leg movements in the prosthetic group. In contrast to the mean stance time ratios, however, the mean step time ratios in the prosthetic group increased to values near 1 with increasing belt velocity (F(1,6) = 10.35 p < .05), indicating that the overall relative timing between the prosthetic and normal leg became more symmetric with increasing belt velocity. Since a similar effect was absent in the control group, the interaction effect between the factors group and velocity was also significant for this dependent variable (F(1,6) = 9.14, p < .05). Symmetry of movement trajectories. The mean squared distances within the prosthetic leg varied from 1.9º at 0.5 km/h to 1.4º at 3.5 km/h and from about 3.3º at 0.5 km/h to 2.9º at 3.5 km/h within the normal leg. No significant effect of velocity was observed. On average, the squared distances within the prosthetic leg were significantly smaller than in the normal leg (F(1,6) = 11.89, p < .05), reflecting a markedly smaller variability in the kinematics of the prosthetic leg. No significant differences were found when comparing the mean squared distances within the normal leg of the prosthetic group with those of the left and right legs of the control group. From Figure 3B it can be appreciated that, on average, the squared distances between the knee angular position data of the prosthetic leg and those of the intact leg were larger than between the legs of the control group. This effect was statistically significant (F(1,6) = 40.21, p < .01), implying that the discrepancy between the movement trajectories of the two legs was larger in the prosthetic group than in the control group. The absence of a significant effect of velocity in both the prosthetic and control group suggested that the degree of (a)symmetry between the movement trajectories of the legs was not systematically affected by walking velocity. Relative phasing. To compare the relative phasing of the limb movements between the prosthetic and the control group, point estimates of relative phase between the leg movements were calculated and analyzed statistically. As can be appreciated from Figure 3C, the mean relative phase between the prosthetic leg and the normal leg differed significantly from that between the legs of the control group (F(1,6) = 12.17, p < .05, Mφlegs_MBM; F(1,6) = 8.12, p < .05, Mφlegs_MFM) in that the legs did not move exactly out of phase (i.e., a relative phase of 180º).3 This implies that the temporal asymmetry, which was due to the discrepancy between the stance phase of the normal leg and that of the prosthetic leg (see Figure 4 for an example), was accompanied by an asymmetry in relative phasing. Moreover, as is illustrated in Figure 3C, the mean point estimates of relative phase between the prosthetic and normal leg increased from 153º at 0.5 110
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Figure 4. Normalized leg movements of a typical prosthetic walker (A) and of an unimpaired, normal walker (B) at different walking velocities. The thick lines represent the averaged trajectory and the thin, dotted lines the variability of the leg movements over 8 stride cycles. Note that in the prosthetic walker the moments of toe off of the prosthetic leg (PL) clearly change relative to those of the normal leg (NL) as a function of walking velocity, whereas such an effect seems absent in the left (LL) and right leg (RL) of the normal walker. Note further that the point estimate of relative phase is 180o when the moment of toe off is at 50% of the movement cycle.
km/h to 172º at 3.5 km/h when calculated at the maximal excursion of the backward movements (MφPL/NL_MBM) and from 146º at 0.5 km/h to 172º at 3.5 km/h when calculated at the maximal excursion of the forward movements (MφPL/NL_MFM). However, this tendency toward a more antiphase Chapter 5
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movement between the prosthetic and normal leg with increasing walking velocity was only significant for MφPL/NL_MBM (F(1,6) = 9.04, p < .05) as was the interaction effect of group and velocity (F(1,6) = 9.52, p < .05). In the control group the mean relative phase between the leg movements increased from 175º at 0.5 km/h to 177º at 3.5 km/h and from 168º at 0.5 km/h to 178º at 3.5 km/h, for MφRL/LL_MBM and MφRL/LL_MFM, respectively, but these effects were not significant. (2) Variability of relative phasing between leg movements (cf. Question 2 in Introduction). To determine whether the stability of coordination between the movements of the prosthetic and the normal leg was less than that between the leg movements of the control group, the SDφ’s for the leg movements were compared statistically for the two groups (i.e., SDφPL/NL_MBM, SDφPL/NL_MFM, SDφRL/LL_MBM, and SDφRL/LL_MFM). In both groups the mean SDφ’s decreased monotonically with increasing velocity (see Figure 3D): SDφPL/NL_MFM decreased from 18.4º at 0.5 km/h to 3.9º at 3.5 km/h (F(1,6) = 16.18, p < .01) and SDφPL/NL_MBM from 19.2º at 0.5 km/h to 5.0º at 3.5 km/h (F(1,6) = 17.17, p < .01), while SDφRL/LL_MFM decreased from 17.0º at 0.5 km/h to 2.2º at 2.5 km/h (F(1,6) = 30.18, p < .01) and SDφRL/LL_MBM from 11.0º at 0.5 km/h to 3.1º at 5.0 km/h (F(1,6) = 67.45, p < .01). No significant differences were found for the two moments (cycle points) at which the standard deviations of the relative phase between the leg movements were calculated (i.e., SDφMFM versus SDφMBM). As can be seen in Figure 3D, the mean SDφ’s for the leg movements were larger in the prosthetic group than in the control group at all walking velocities, resulting in significant effects of group (F(1,6) = 6.04, p < .05, SDφlegs_MBM; F(1,6) = 6.47, p < .05, SDφlegs_MFM). This implies that the observed asymmetry in relative phasing in the prosthetic group was accompanied by a lower coordinative stability. No significant interaction effects of group and velocity were found. (3) Frequency coordination between arm and leg movements (cf. Question 3 in Introduction). The prosthetic walkers had a significantly higher mean stride frequency (i.e., dominant frequency of the leg movements) than the control group (F(1,6) = 8.49, p < .05). In Figure 5, the dominant frequencies of the leg movements (i.e., the stride frequencies) are plotted against the dominant frequencies of the arm movements for all participants of the prosthetic (Figure 5A) and the control group (Figure 5B), respectively. As can be seen 112
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in Figure 5, the prosthetic walkers generally showed a 1:1 frequency relation between arm and leg movements at all walking velocities. The control group, on the other hand, generally exhibited a 2:1 frequency relation between arm and leg movements at velocities lower than 3.0 km/h. Although at 0.5 km/h, 4 out of 8 prosthetic walkers showed a frequency relation of 2:1 between ipsilateral arm and leg movements, this frequency relation was not a genuine “double swinging” mode as it occurred at one side of the body only. Prosthetic Group Dominant frequency of arm movement (Hz)
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Figure 5. Frequency coordination between arm and leg movements for all walking velocities for all participants in the prosthetic (A) and the control group (B). The dominant frequencies of the leg movements (i.e., stride frequency) are plotted against the dominant frequencies of the ipsilateral arm movements. Prosthetic or left body side are marked by a circle, normal or right body side by a cross.
Examination of the frequency ratios of the dominant frequencies of the ipsilateral arm and leg movements of the prosthetic group revealed that, on the adopted criterion, only 6.5% of the frequency ratios between ipsilateral arm and leg movements differed from the 1:1 frequency coordination. The fact that the observed ratios, except those obtained for two prosthetic walkers at 0.5 km/h, never deviated more than one decimal place from their corresponding integer values suggests that the arm and leg movements were frequency locked at all walking velocities. A similar result was found for the control group in that the observed ratios, except those obtained for two Chapter 5
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healthy walkers at 0.5 km/h, always satisfied the ±0.1 criterion for frequency locking. It should be noted, however, that a discrepancy was present between the 2:1 and 1:1 frequency coordination in the control group. Whereas only 3.6% of the near 1:1 frequency ratios deviated from an integer, 30% of the near 2:1 frequency ratios deviated from an integer, suggesting that, at least on the criterion adopted, the 2:1 frequency locking was less stable than the 1:1 frequency locking. (4) Relative phasing between arm and leg movements (cf. Question 4 in Introduction). If calculated at the moments at which the maximal excursion of the forward movements was reached, the Mφ between contralateral arm and leg movements decreased from about 73º at 0.5 km/h to about 13º at 3.5 km/h (F(1,6) = 22.27, p < .01). This indicates that with increasing walking velocity contralateral arm and leg movements increasingly tended to move in-phase (i.e., 0º). In a similar fashion, the Mφ between ipsilateral arm and leg movements increased from about 110º at 0.5 km/h to about 173º at 3.5 km/h (F(1,6) = 27.01, p < .01), indicating an increased tendency to antiphase (180º) coordination at higher walking velocities. In contrast, when calculating the relative phase at the moments at which the maximal excursion of the backward movement was reached neither in-phase nor antiphase relationships were found: The Mφ between contralateral arm and leg movements decreased from about 105º at 0.5 km/h to about 84º at 3.5 km/h (F(1,6) = 7.2, p < .05), and the Mφ between ipsilateral limb movements increased from about 76º at 0.5 km/h to about 95º at 3.5 km/h (F(1,6) = 5.08, p = .065). No significant differences between control and prosthetic group were found for the coordination between arm and leg movements. (5) Variability of relative phasing between arm and leg movements (cf. Question 5 in Introduction). The stability of relative phase between ipsilateral arm and leg movements did not differ significantly from that between contralateral arm and leg movements. This was the case for both the control and the prosthetic group. In contrast, as can be seen in Figure 6A, coordinative stability was lower for limb pairs in which the arm at the prosthetic side was involved (i.e., SDφPA/PL and SDφPA/NL) than for other limb pairs. This difference in stability was significant at low walking velocities, that is, 0.5-1.5 km/h (F(1,6) = 14.73, p < .01). On average, all SDφ’s that were calculated, except SDφNA/NL_MBM, 114
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decreased with increasing walking velocity (Figure 6A and B), indicating an increase of coordinative stability (SDφNA/NL_MFM: F(1,6) = 14.45, p < .01; SDφPA/NL_MFM: F(1,6) = 13.99, p < .05; SDφNA/PL_MFM: F(1,6) = 12.96, p < .05; SDφPA/PL_MFM: F(1,6) = 8.89, p < .05; SDφPA/NL_MBM: F(1,6) = 12.46, p < .05; SDφNA/PL_MBM: F(1,6) = 0.07, p < .05; SDφPA/PL_MBM: F(1,6) = 19.74, p < .01), which is consistent with the results obtained by Donker et al. (2001) and Wagenaar and Van Emmerik (2000) for healthy walking. Furthermore, and also in line with those previous studies, the stability of coordination between the arm and leg movements was significantly lower than that of the coordination between the leg movements in both the prosthetic (SDφMFM: F(6,6) = 10.04, p < .05; SDφMBM: F(6,6) = 11.74, p < .05) and the control group (SDφMFM: F(6,6) = 6.26, p < .05; SDφMBM: F(6,6) = 12.10, p < .05). Again, no significant effect of moment was found. Maximal excursion of Backward Movement
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Figure 6. Averaged standard deviations of the point estimate of relative phase between ipsilateral (black markers) and contralateral (white markers) arm and leg movements calculated at the moments of maximal excursion of both backward (A) and forward (B) limb movements.
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Discussion
The goal of the present study was to examine the effects of prosthesisinduced asymmetry and walking velocity on the coordinative stability of prosthetic walking patterns. We first focused on the nature of the asymmetry brought about by the prosthesis in the leg movements. In confirmity with the literature on prosthetic gait (e.g., Baker, & Hewison, 1990; Isakov et al., 1996; Mattes et al., 2000), it was found that the stance time of the prosthetic leg was significantly shorter than that of the normal leg, which was also reflected in the finding that the step time ratio of the prosthetic group (NL/PL) was significantly lower than in the control group. In all likelihood, the stance time of the prosthetic leg is shortened in order to avoid pain and to minimize the need for controlling the mechanical interaction between the prosthetic leg and the environment. The observed temporal asymmetry was accompanied by a marked deviation of the mean relative phase between the leg movements away from the antiphase (180º) relation. Conceivably, the frequency-detuning term (∆ωlegs ≠ 0) introduced by the different physical properties of the prosthetic leg relative to those of the normal leg caused a displacement in the attractor point (i.e., stable coordinative state), and as such may have been responsible for the observed drift in relative phase away from 180º (Amazeen et al., 1996). From DST it is known that oscillators with different properties are less stably coupled than a similarly coupled system with identical oscillators (e.g., Haken et al., 1985). DST further dictates that the antiphase and inphase relations are more stable than other phase relations are (Carson, Goodman, Kelso, & Elliott, 1995; Yamanishi, Kawato, & Suzuki, 1980; Zanone & Kelso, 1992). Hence, in view of the physical asymmetry brought about by the prosthesis and the resulting asymmetry in the relative phasing of the leg movements, we expected an inverse relation between the degree of deviation from a genuine antiphase relation and the stability of the resulting coordination. The observed asymmetry in the prosthetic group was indeed accompanied by an overall lower coupling (larger standard deviation of relative phase) between the prosthetic and the normal leg movements than observed in the control group. The significant interactions between the factors group and velocity that were found for the step time ratio and the relative phasing between the backward leg movements (MφMBM) suggested that the difference between the two groups decreased as function of walking velocity (at least for the independent variables of interest). These results confirmed our expectation that the difference in coordinative stability between the two groups would 116
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decrease with increasing walking velocity. However, in spite of the finding that the coupling strength between the leg movements increased with walking velocity in both groups, no significant interactions were found between group and velocity. In this context it is interesting to note that the behavior of the individual movement components was also unaffected by belt velocity (Figure 3B). Hence, it seems that the coordinative properties of prosthetic walking patterns cannot be accounted for by (changes in) the temporal organization of the legs (as indexed by the stance and swing time ratios) or by (changes in) the shape of the trajectories of the individual limb movements. In view of the preceding findings and considerations it appears that the increase in stability between the leg movements with increasing walking velocity was not specifically due to an increase in symmetry but mainly to velocity-related factors as such. That is, walking velocity is the product of stride frequency and step length, implying that a certain walking velocity can be achieved by different combinations of these two parameters. A low walking velocity, for example, can be realized by various modes of locomotion ranging from walking with small steps performed at high frequencies to walking with very large steps performed at low frequencies. However, with increasing walking velocity the number of possible combinations decreases due to biomechanical limitations of the musculoskeletal apparatus to increase step length and stride frequency (cf. Nilsson & Thorstensson, 1987). The finding that the temporal asymmetry between the prosthetic and normal legs, as indexed by the step time ratio and the relative timing between the legs at toe off (Figure 3A and C), decreased with increasing walking velocity is likely to be a result of this limitation. It is quite possible that this reduction in freedom of variation is accompanied by greater stability. After having studied the asymmetry and coordinative stability of prosthetic walking patterns, we examined the effect of the prosthesisinduced asymmetry on the coordination between arm and leg movements, focusing first on their frequencies. Given the marked change in the biomechanical properties brought about by the prosthesis, our finding that prosthetic walkers showed a clear preference for a 1:1 frequency coordination between arm and leg movements at all walking velocities has some interesting theoretical implications. The occurrence of a 2:1 frequency coordination between arm and leg movements at low walking velocities, as observed in unimpaired walkers, has been explained by the tendency of the arms to move as closely as possible to their eigenfrequencies (cf. Craik et al., 1976; Van Emmerik, Wagenaar, & Van Wegen, 1998; Wagenaar & Van Chapter 5
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Emmerik, 2000; Webb & Tuttle, 1989). Following this reasoning, it may be conjectured that the disappearance of the 2:1 frequency coordination in the prosthetic group was due to the increased stride frequency in this group. However, the mean stride frequencies at which switches from 2:1 to 1:1 frequency coordination were found in the control group ranged from 0.37 to 0.64 Hz (mean 0.51 Hz; Fig. 5), values well within the range of stride frequencies observed in the prosthetic group. Another study that seems to contradict the eigenfrequency hypothesis is that of Mattes et al. (2000), who examined the hypothesis that increasing the mass and moment of inertia of the prosthetic limb to match the mass and moments of inertia of the normal limb improves walking symmetry without extra energy cost. Contrary to this hypothesis, however, they found that such a loading configuration resulted in greater gait asymmetry and higher energy cost. Hence, it seems that the biomechanical properties of the arm and legs alone are insufficient for explaining the coordinative properties of prosthetic walking patterns. The insight obtained here is that the coordination patterns observed in prosthetic walkers are to be understood primarily in terms of stability and instability. For instance, people with an above-knee prosthesis commonly flex their trunk forwards early in the stance phase in order to keep the prosthetic knee fully extended. As a consequence, relatively large trunk movements are made, resulting in an unstable system (Whittle, 1998). Given the strong tendency of the human movement system to move its limbs in synchrony (Beek, Peper & Stegeman, 1995; Byblow & Goodman, 1994; Kelso & DeGuzman, 1992; Serrien & Swinnen, 1997, 1998b; Sternad, Turvey, & Saltzman 1999a, 1999b; Swinnen et al., 1991; Von Holst, 1939/1973), it is conceivable that the destabilizing effect of the prosthesis resulted in a preference for the (most stable) 1:1 frequency coordination between arm and leg movements at all walking velocities. Further examination of the arm and leg movements suggested that the coordination of both unimpaired (cf. Muzzi, Warburg, & Gentille, 1984) and prosthetic walking is strongly determined by the footfalls. A clear indication of this interpretation was that the relative phase between arm and leg movements at the maximal excursion of the forward movements exhibited a tendency toward in-phase and antiphase movements of contraand ipsilateral limb pairs, respectively, whereas other phase relations were found at the maximal excursion of the backward movements. Another finding supporting this interpretation was that there was no significant difference between antiphase and in-phase movements (which stands in contrast with the results of Baldissera, Cavallari, & Civaschi, 1982; Baldissera, Cavallari, Marini, & Tassone, 1991). The observation that the 118
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coordination between arm and leg movements during walking varies within a stride cycle (cf. Donker et al., 2001; chapter 2) possibly suggests that, in order to maintain in-phase and antiphase relations at the moments of heel strike, adaptations occur during other parts of the stride cycle. This seems to be consistent with the finding that the relative timing between prosthetic and normal legs when calculated at the maximal excursion of backward movement (φPL/NL_MBM) was influenced significantly by belt velocity, whereas this was not the case when the relative timing was calculated at the maximal excursion of forward movement (i.e., heel strike). On the basis of this finding it may be argued that especially when arm movements at the prosthetic side are involved, adaptations are necessary in order to maintain the in-phase and antiphase relation at the moments of footfall (PA/PL and PA/NL; see Figure 6A). Probably, in-phase and antiphase relations at the moments of footfall provide stability and as such dominate the coordination during locomotion. The presented results suggest that prosthetic walkers may have an increased risk of falling because, in comparison to non-prosthetic walkers, their walking pattern is less stable across a wide range of velocities and less adaptive to variations in walking velocity in that they do not exhibit the 2:1 frequency coordination between arm and leg movements at low velocities. It is questionable, however, whether explicitly promoting the 2:1 frequency coordination between arm and leg movements at low velocities or establishing a more symmetric gait pattern in prosthetic walkers are appropriate goals for rehabilitation as the asymmetry introduced by the prosthesis is an inherent property of the locomotor system of a prosthetic walker. Instead, practicing walking at different velocities may be a more appropriate treatment for regaining stability (i.e., dynamic balance) and adaptability. At first blush, one would perhaps be inclined to assume that practicing walking at low velocities would be more beneficial for prosthetic walkers than at higher velocities because, intuitively, one might feel that walking at low velocities allows for greater control and stability. The results of the present study, however, suggest otherwise in that the instability due to the prosthesis clearly diminished with increasing walking velocity. There are several biomechanical factors that may underlie this result, among which are the following. At relatively high walking velocities an ideal mechanism for balance control is to lean forward in order to bring the center of mass forward by gravitation and to have the legs follow as a result. Whereas at relatively high velocities walking actually consists of a series of controlled falls, walking at low velocities poses the risk of becoming a sequence of standing postures with less dynamic coherence and stability. At Chapter 5
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low walking velocities, it is important to keep the body’s center of gravity continuously above the supporting leg, which requires a continuous, active regulation of the location of the body’s center of gravity relative to the support surface. Especially for prosthetic walkers such active control of balance is often problematic due to the instability caused by the prosthetic leg (Geurts & Mulder, 1994; Geurts, Mulder, Nienhuis, & Rijken, 1991). Thus, prosthetic walkers may benefit from walking or learning to walk at higher velocities. However, an important question that remains for future research on prosthetic walking is whether the resulting increase in coordinative stability also reduces the actual risk of falling.
Acknowledgments
We are grateful to Bart Nienhuis for his technical assistance in setting up the experiment. We are further indebted to Theo Evers, Harmen van de Linde, Rein van der Ploeg, and all participating volunteers for their cooperation and help in conducting this study. Finally, we would like to thank Theo Mulder and Jaques Duysens for their helpful comments on an earlier draft of this article. This research was carried out at the Sint Maartens Kliniek - Research with financial support of the Foundation for Behavioral Sciences (SGW; grant number 575-23-005) of the Netherlands Organization for Scientific Research (NWO).
Notes
1. In spite of this large variation all participants were considered to be experienced walkers as they were able to walk independently up to at least 3.5 km/h. 2.This cut-off frequency eliminated the noise from the data while leaving the characteristic frequencies of the movement, which were all lower than 2 Hz, unaffected. 3. Group analysis was performed by means of linear statistics (GLM, SPSS), which does not account for differences in leading legs across participants, resulting in phase jumps. We therefore collapsed Mφ of each participant between 0º and 180°. As a result, the variation around 180° points in one direction, resulting in a systematic deviation from 180° of about 2° in the control group. 120
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