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Linear center-of-mass dynamics emerge from non-linear leg-spring properties in human hopping
Journal of Biomechanics 46 (2013) 2207–2212
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Linear center-of-mass dynamics emerge from non-linear leg-spring properties in human hopping Sebastian Riese n, Andre Seyfarth, Sten Grimmer Lauflabor Locomotion Laboratory, Institute for Sport Science, Technische Universität Darmstadt, Magdalenenstraße 27, D-64289 Darmstadt, Germany
art ic l e i nf o
a b s t r a c t
Article history: Accepted 28 June 2013
Given the almost linear relationship between ground-reaction force and leg length, bouncy gaits are commonly described using spring–mass models with constant leg-spring parameters. In biological systems, however, spring-like properties of limbs may change over time. Therefore, it was investigated how much variation of leg-spring parameters is present during vertical human hopping. In order to do so, rest-length and stiffness profiles were estimated from ground-reaction forces and center-of-mass dynamics measured in human hopping. Trials included five hopping frequencies ranging from 1.2 to 3.6 Hz. Results show that, even though stiffness and rest length vary during stance, for most frequencies the center-of-mass dynamics still resemble those of a linear spring–mass hopper. Rest-length and stiffness profiles differ for slow and fast hopping. Furthermore, at 1.2 Hz two distinct control schemes were observed. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Biomechanics Hopping Spring–mass model
1. Introduction For bouncy gaits the center-of-mass (CoM) movement can be approximated by a ballistic trajectory during flight and a spring law counteracting gravity during stance. This finding resulted in the theoretical concept of the spring-loaded inverted pendulum (SLIP) model for hopping and running (Blickhan, 1989; McMahon and Cheng, 1990). Here, the body is represented by a point mass and the leg is described by a massless linear spring. This approach is supported by the force–length function (FLF) of the leg, describing the relationship between ground-reaction force (GRF) and instantaneous leg length (Farley et al., 1991; Blickhan and Full, 1993). So far, it is unclear where global spring-like behavior of the leg originates. Some studies suggest that non-linear visco-elastic properties of the muscle–tendon complex, so-called “preflexes”, are the main contributors, especially during fast movements (Loeb Brown and Loeb, 1997). Others argue that muscle activation determines global leg behavior (Bobbert and Richard Casius, 2011). Also, combined control schemes incorporating preflexes and feed-forward patterns have been suggested (Cham et al., 2000). If leg segmentation is taken into account, linearity of leg behavior is lost on joint level. Even though the moment–angle relationship of the ankle joint is fairly linear for a variety of running patterns, this is not the case for the knee joint (Guenther and Blickhan, 2002). Results of Rapoport et al. (2003) support the loss of linearity on joint level. Using
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a segmented sagittal-plane hopping model for data analysis, joint stiffness was found to increase with joint flexion, resulting in bellshaped stiffness profiles over time with maximum stiffness near midstance. Leg compliance and its adaptation in response to changing environmental conditions are hypothesized to be crucial for successful locomotion (Grimmer et al., 2008). In contrast to serial-elastic actuators (SEAs), tunable compliant actuators allow to change mechanical stiffness on-the-fly. Hence, it was argued by Hurst et al. (2004) that this concept “could result in an effective actuation method for highly dynamic legged locomotion”. Following this argument, it was shown for a one-dimensional spring–mass model that simultaneous variations of leg-spring parameters (stiffness, rest length) during ground contact result in stable, robust and efficient hopping (Riese and Seyfarth, 2012a,b), motivating the variable-leg-spring (VLS) concept. So far, the results of previous studies indicate variations of leg stiffness and rest length during human hopping (Farley et al., 1991; Hobara et al., 2011), however without explicitly addressing them. Thus, in order to validate the VLS concept with experimental data, here the behavior of leg stiffness and rest length in vertical human hopping is investigated, assuming a tunable leg spring (Fig. 1). We hypothesize that the linear CoM dynamics observed in human hopping result from the interaction of non-linear leg-spring properties, namely non-constant leg stiffness and rest length, and that these parameter variations may be of considerable magnitude (4 10% of the touchdown value). According to Farley et al. (1991) human hopping patterns for frequencies below the preferred frequency differ from those above
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flight
flight
contact
F
CoM F ||
Table 1 Preferred frequency fp. The overall mean is in good agreement with results of Farley et al. (1991) (2.2 Hz). Subject
mean 7 s.d. (Hz)
1 2 3 4 5 6
2.017 0.07 2.50 70.13 2.42 70.06 1.98 7 0.04 2.76 70.14 2.22 70.05
Overall
2.337 0.30
F
Table 2 Marker set for calculation of kinematics. Marker placement
CoP
Forehead Right temple Left temple Light acromion Left acromion Right trochanter Left trochanter Right lateral elbow Left lateral elbow Right lateral wrist Left lateral wrist Right lateral knee Left lateral knee Right lateral malleolus Left lateral malleolus Right 5 metatarsals Left 5 metatarsals
Fig. 1. SLIP model during human hopping. All mass is located in the center of mass (CoM) and the leg length is assumed to be the distance between CoM and center of pressure (CoP). The misalignment of ground-reaction force (GRF) and leg spring was exaggerated to illustrate the GRF contributions parallel and perpendicular to the leg direction, F ∥ and F ⊥ , respectively.
the preferred one. Furthermore, it was shown that humans employ different control strategies to stabilize hopping depending on the rate of movement (Yen and Chang, 2010; Hobara et al., 2011). As the parameters accessible for control in the one-dimensional spring–mass model are stiffness and rest length, we expect slow hopping to exhibit different stiffness and rest-length profiles than fast hopping.
2. Methods 2.1. Experimental setup Six healthy, well-trained male subjects (76.5 7 8.4 kg) participated in the study. Prior to the measurements, the experiment was approved by the ethics review board of the University of Jena, as laid out in the Declaration of Helsinki, and all subjects gave their written informed consent. The subjects were asked to perform vertical jumps on both legs. Each subject was instructed to jump with self-selected frequency, subsequently referred to as fp (Table 1). Additionally, following Farley et al. (1991), the hopping frequencies 1.2 Hz, 1.8 Hz, 2.8 Hz and 3.6 Hz were prescribed with a metronome. The sequence of hopping frequencies was randomized for each subject. Each trial was of 30 s length. At the beginning and end of each trial, the subjects were asked to stand quiet for 5 s, leaving 20 s of vertical hopping, resulting in approximately 20–50 hopping cycles depending on subject and frequency.
2.2. Kinetics and kinematics GRFs were measured directly with 1 kHz using a Kistler force platform. Additionally, center-of-pressure (CoP) position was extracted from this data. In order to obtain kinematics, 17 reflective markers were placed on anatomical landmarks of each subject (Table 2). Marker positions were measured with 240 Hz using a ten-camera infrared system (Proflex MCU240, Qualisys, Gothenburg, Sweden) and interpolated to 1 kHz to match the GRF and CoP data. CoM position was then calculated in accordance with Dempster's body-segment parameter data (Dempster, 1955; Winter, 2009).
2.3. Estimation of stiffness and rest length In order to estimate global leg properties, the leg was approximated as a massless spring, connecting CoM and CoP (Fig. 1). GRFs and CoM movement during stance were projected into leg direction. Therefore, the data set is reduced to onedimensional (vertical) hopping. Additionally, GRFs were normalized to body weight (BW) and instantaneous leg length was normalized to initial CoM height linit, i.e. leg length while standing quiet. Thus, estimated stiffness K¼ klinit/BW and rest length L0 ¼ l0/linit are nondimensional. As the GRF and leg-length data are noisy, both data sets were filtered using a lowpass Butterworth of 5th order, with a cut-off frequency of 25 Hz.
For each trial, stance phases ðF ∥ ≥0:01 BWÞ were extracted and normalized to 100% stance time. F ∥ and leg length L were interpolated accordingly. Following previous studies (Rozendaal and van Soest, 2008; Peter et al., 2009) and assuming a piecewise-linear spring with stiffness K(i) and rest length L0 ðiÞ, the equation ! ! F ∥ ðiÞ L0 ðiÞLðiÞ ð1Þ ¼ KðiÞ F ∥ ði þ 1Þ L0 ðiÞLði þ 1Þ had to be solved for the time steps i¼ 1,3,…,99. As there are two unknowns per time step, K(i) and L0 ðiÞ, it was assumed that KðiÞ≡Kði þ 1Þ and L0 ðiÞ≡L0 ði þ 1Þ. Within this approach the spring is approximated as linear with constant parameters for two consecutive time steps. However, the resulting parameter profiles may be non-constant, allowing for a non-linear spring throughout stance. To ensure physically meaningful solutions, stiffness is constrained to values KðiÞ 40. Accordingly, during stance rest length has to satisfy L0 ðiÞ4 LðiÞ, as L0 ¼ L denotes the transition from flight to stance phases and vice versa. As a first approach, stiffness and rest length were calculated directly by solving Eq. (1) analytically for K(i) and L0 ðiÞ. However, the constraints for stiffness and rest length were violated for a considerable amount of time steps, especially for frequencies below the preferred frequency fp. Thus, Eq. (1) was solved numerically with the least-squares method lsqcurvefit implemented in MATLAB (R2010a, The MathWorks Inc., Natick, MA, USA) using the constraints for K(i) and L0 ðiÞ as lower boundaries.
3. Results Except at 1.2 Hz, results presented here for a given frequency are means over all trials of all six subjects at that frequency. At 1.2 Hz, behavior of half of the subjects distinctively differs from that of the other half, thus denoted in the figures as “1.2 Hz I” and “1.2 Hz II”, respectively. 3.1. Measured data For frequencies from fp to 3.6 Hz, instantaneous leg length L(i) corresponds to running-like single-minimum CoM trajectories (Fig. 2a). At lower frequencies, 1.2 and 1.8 Hz, also double-minimum
S. Riese et al. / Journal of Biomechanics 46 (2013) 2207–2212
leg-length profiles were measured, resembling walking patterns (Geyer et al., 2006). However, at 1.8 Hz the parameters estimated from the data exhibit a similar qualitative behavior for all subjects. Thus, at 1.8 Hz means over all trials of all subjects are displayed. Leg length at touchdown decreased with frequency (except for “1.2 Hz I”), so did CoM displacement and leg length at take-off (Fig. 2a). GRF profiles F ∥ ðiÞ for hopping ranging from fp up to 3.6 Hz feature only a single peak, as expected from spring–mass hopping (Fig. 2b). Landing impact forces were not observed; this is true for all trials and not an averaging effect. For 1.2 Hz walking-like double-peak GRF patterns were observed (even though half of the subjects exhibit running-like CoM trajectories). For 1.8 Hz two of the subjects mostly exhibit double-peak force profiles, while the others mostly feature single-peak GRF patterns. On average, however, single-peak hopping dominates. Resulting from GRFs and CoM movement, the FLF on leg level is fairly linear down to 1.8 Hz (Fig. 3). In contrast, at 1.2 Hz it is highly non-linear.
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In Fig. 5 modified FLFs are shown for all measured frequencies, using leg compression ΔLðiÞ ¼ L0 ðiÞLðiÞ instead of actual leg length L(i). Due to the estimated rest-length profile L0 ðiÞ, non-linearity of the modified FLF at 1.2 Hz considerably increased with respect to
120
80
40
0
3.2. Estimation results
1.5
For hopping within fp to 3.6 Hz the stiffness profiles K(i) are roughly bell-shaped, with increasing maximum stiffness for increasing frequency (Fig. 4a). At 2.8 and 3.6 Hz an additional impact-like maximum is present in the stiffness pattern. However, in the GRF profiles no impact forces were observed. At 1.2 and 1.8 Hz there are considerable stiffness fluctuations over stance time. Most prominently, for “1.2 Hz I” there is a clear maximum during the second GRF peak. Rest-length profiles L0 ðiÞ at fp to 3.6 Hz resemble the instantaneous leg length L(i) (Fig. 2b and 4b). This is not true at lower frequencies. At 1.2 Hz, rest length L0 ðiÞ for half of the subjects exhibits a triple-peak pattern, “1.2 Hz I”, while the other half features a plateau with a drop around midstance, “1.2 Hz II”. At 1.8 Hz, a composite of patterns “1.2 Hz I” and “1.2 Hz II” can be seen.
1.4
0
20
40
60
80
100
0
20
40
60
80
100
1.3 1.2 1.1 1.0
Fig. 4. Estimated leg parameters, stiffness K and rest length L0, over stance time. Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
3.0
1.1
2.0
1.0
1.0 0.9 0
20
40
60
80
100
0
0
20
40
60
80
100
Fig. 2. Human hopping data. CoM movement projected into leg direction, i.e. instantaneous leg length (a) and the projected GRFs (b), is shown over stance time. Human hopping was investigated for five different hopping frequencies (ranging from 1.2 to 3.6 Hz). Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
3.0
2.0
1.0
0 0
0.1
Fig. 3. FLF on leg level. Human hopping was investigated for five different hopping frequencies (ranging from 1.2 to 3.6 Hz). Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
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3.0
2.0
1.0
0 0
0.1
0.2
Fig. 5. Modified FLF on leg level, using leg compression ΔL ¼ L0 L instead of leg length L. Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
the unmodified case. At 1.8 Hz the modified FLF now is non-linear, also due to the estimated L0 ðiÞ pattern. For all other measured hopping frequencies a linear approximation of the FLF is still applicable.
4. Discussion In this study, the behavior of leg stiffness and rest length in vertical human hopping was investigated. It was hypothesized that even though the global FLF is fairly linear, implying a constant linear spring, leg-spring parameters may change significantly during ground contact. Additionally, it was hypothesized that rest-length and leg-stiffness profiles show a clear distinction between slow and fast hopping.
4.1. Variable leg-spring properties In vertical human hopping, leg stiffness and rest length change during stance, up to 4KTD and 0.3lTD, respectively (Fig. 4). Thus, these findings support the primary hypothesis of this study, while contradicting the well-established assumption that, due to its linear FLF, hopping features approximately constant leg-spring parameters. However, the results support the theoretical findings of the VLS concept (Riese and Seyfarth, 2012a). There, for a one-dimensional spring–mass model simultaneous variations of rest length and leg stiffness during ground contact resulted in stable hopping, whereas the original spring–mass model is not stable. Additionally, GRF patterns and CoM trajectories were more human-like than for the spring–mass model with constant leg parameters. In Riese and Seyfarth (2012a), leg stiffness and rest length were assumed to change linearly with stance time. Leg-stiffness and restlength profiles estimated here are much more complex than this simple first-order approximation (Fig. 4a). Nevertheless, simultaneous variation of leg stiffness and rest length in the model could be validated based on experimental data. It is still an open question how global leg properties relate to properties on joint level. Applying a segmented sagittal-plane hopping model to data analysis Rapoport et al. (2003) found that joint stiffness increased with angular deflection, resulting in bell-shaped stiffness profiles over stance time. Interestingly, Fig. 4a corresponds directly to these results on leg level. For hopping frequencies coinciding with spring-like hopping ðf ≥f p Þ the estimated leg-stiffness profiles also are roughly bell-shaped over time. This unexpected result may suggest a simple correlation between leg and joint stiffness for spring–mass hopping and indicates further study.
4.2. Non-linear leg properties vs. linear dynamics There is considerable evidence that human legs behave like linear springs during bouncy gaits (Farley et al., 1991; Hobara et al., 2011). Even in the presence of severe perturbations, such as compliant surfaces (Moritz and Farley, 2005) or an elastic exoskeleton (Ferris et al., 2006; Grabowski and Herr, 2009), humans maintain spring-like CoM dynamics. As spring–mass systems have been proven to possess self-stabilizing properties (Seyfarth et al., 2002), it has been suggested that emulating linear spring–mass dynamics may ease control and thus “may be a primary neuromuscular control strategy during bouncing gait” (Grabowski and Herr, 2009). However, Ferris et al. (2006) and Bobbert and Richard Casius (2011) stressed that spring-like behavior should not be confused with actual mechanical springs. The results presented here support the latter argument. Linear spring-mass dynamics may be achieved by other means than linear springs with fixed parameters. Here, the relationship between GRF and instantaneous leg length is (in good approximation) linear for hopping frequencies down to 1.8 Hz (Fig. 3). Nevertheless, neither stiffness nor rest length of the underlying model is constant (Fig. 4). In accordance with the primary hypothesis of the present study, the actual variation of rest length and stiffness is masked by the interaction of both parameter profiles. Rest length and stiffness vary in a way that in the resulting GRFs and CoM trajectories the non-linearities compensate each other. This also holds if the modified FLF is used, the only major difference being that linear FLFs now are only found for frequencies ranging from fp to 3.6 Hz (Fig. 5). The non-linearity of the stiffness profile is also in agreement with findings of Karssen and Wisse (2011). In their simulation study, the effect of non-linear leg springs on disturbance rejection during running was investigated. It was shown that the optimal stiffness profile is highly non-linear, with considerably better disturbance rejection than the optimal linear stiffness. Interestingly, the predicted FLF resulting from the optimal stiffness profile for running became non-linear beyond a certain amount of leg compression, resembling the FLF for hopping with 1.2 Hz (Fig. 3). This may indicate that for slow hopping more emphasis is put on disturbance rejection, i.e. a different control scheme is applied for slow hopping (see Section 4.3).
4.3. Frequency-dependent control In accordance with the secondary hypothesis of this study, different leg-stiffness and rest-length profiles were found for slow and fast hopping.
S. Riese et al. / Journal of Biomechanics 46 (2013) 2207–2212
At hopping frequencies of fp and above, leg stiffness and rest length exhibit a similar qualitative behavior. Stiffness and restlength profiles are smooth, even for rapid movements (up to 3.6 Hz). This uniform parameter variation across different hopping frequencies may be due to preflexes, i.e. zero-time-delay control based on mechanical properties of the system (Loeb Brown and Loeb, 1997; Haeufle et al., 2010). The parameter profiles themselves are bell-shaped, what might reflect the bell-shaped stiffness profiles found on joint level (Rapoport et al., 2003). The impactlike peaks at 2.8 and 3.6 Hz may be due to pre-activation of the muscles in anticipation of touchdown (Seyfarth et al., 2000; Moritz and Farley, 2004). At 1.2 Hz however, stiffness and rest-length profiles are quite unsteady (Fig. 4), which may suggest neural control, such as muscle activation. Two distinct control schemes can be observed at this frequency. While for “1.2 Hz II” control is mainly reflected by rest length, for “1.2 Hz I” stiffness and rest length oscillate phase-shiftedly, with a dramatic stiffness increase in preparation of take-off. In accordance with Farley et al. (1991), this suggests an active push-off to compensate for energy dissipated in the first half of contact. Similar actuation schemes with stiffness increases during the second half of contact have been proposed and shown to be quite effective for control of artificial legged systems (Koditschek and Buehler, 1991; Kalveram et al., 2010). At 1.8, a transitory behavior between slow and fast hopping is found. These findings support the statement of Yen and Chang (2010) that humans employ different control strategies to stabilize bouncy gaits depending on the rate of movement. Surprisingly, the results presented here resemble a gait transition from walking to running, without locomoting forward (Fig. 2). For “1.2 Hz II” even hybrid gaits with running-like CoM trajectories and walkinglike GRFs were observed. In accordance with Bobbert and Richard Casius (2011) as well as Hobara et al. (2011), linear spring-like behavior was found for hopping with a smaller frequency than fp (Fig. 3), contradicting findings of Farley et al. (1991). However, this is only true if the FLF on leg level is based on instantaneous leg length. This approach implicitly assumes a constant rest length. As substantial changes in rest length were found here (up to 0.3LTD for “1.2 Hz II”), using the unmodified FLF may limit the interpretation of leg-spring behavior. Changes in rest length were not taken into account by the aforementioned studies. However, doing so significantly changes the patterns of the FLF on leg level during hopping, cf. Figs. 3 and 5, and facilitates interpretation of leg function and control.
5. Conclusion As biological locomotory systems are more than simple spring– mass setups, leg stiffness and (virtual) leg rest-length may be adapted by control of muscle activation or by mechanical properties. In the commonly used approach of constant leg stiffness during contact the rest length of the leg spring is assumed to be fixed (e.g. Farley et al., 1991; Hobara et al., 2011). The present analysis shows that by keeping both leg stiffness and leg restlength (and corresponding leg compression) a function of time, interpretation of the FLF becomes more reasonable. Variability of the leg-spring parameters is shown to be an important system property rather than being caused by perturbations, thus validating the VLS concept. Currently, it is not known how these two parameters are controlled by the neuro-musculoskeletal system. However, this study provides an experimental analysis along with insights, which may be used to identify appropriate control schemes based on more detailed biomechanical and neuromechanical models.
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Conflict of interest statement We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In doing so we confirm that we have followed the regulations of our institutions concerning intellectual property. We further confirm that any aspect of the work covered in this manuscript that has involved either experimental animals or human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected].
Acknowledgments This work was partially funded by a grant of jenarbeit (City Council of Jena) to SR. References Blickhan, R., 1989. The spring–mass model for running and hopping. Journal of Biomechanics 22, 1217–1227. Blickhan, R., Full, R.J., 1993. Similarity in multilegged locomotion: bouncing like a monopode. Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioral Physiology 173, 509–517. Bobbert, M.F., Richard Casius, L.J., 2011. Spring-like leg behaviour, musculoskeletal mechanics and control in maximum and submaximum height human hopping. Philosophical Transactions of the Royal Society B: Biological Sciences 366, 1516–1529. Cham, J.G., Bailey, S.A., Cutkosky, M.R., 2000. Robust dynamic locomotion through feedforward–preflex interaction. In: ASME IMECE Proceedings, pp. 5–10. Dempster., W., 1955. Chapter: Space Requirements of the Seated Operator. WADC Technical Report 55159 (WADC-55-159, AD-087-892), pp. 55–159. Farley, C.T., Blickhan, R., Saito, J., Taylor, C.R., 1991. Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits. Journal of Applied Physiology 71, 2127–2132. Ferris, D.P., Bohra, Z.A., Lukos, J.R., Kinnaird, C.R., 2006. Neuromechanical adaptation to hopping with an elastic ankle–foot orthosis. Journal of Applied Physiology 100, 163–170. Geyer, H., Seyfarth, A., Blickhan, R., 2006. Compliant leg behaviour explains basic dynamics of walking and running. Proceedings of the Royal Society B: Biological Sciences 273, 2861–2867. Grabowski, A.M., Herr, H.M., 2009. Leg exoskeleton reduces the metabolic cost of human hopping. Journal of Applied Physiology 107, 670–678. Grimmer, S., Ernst, M., Guenther, M., Blickhan, R., 2008. Running on uneven ground: leg adjustment to vertical steps and self-stability. Journal of Experimental Biology 211, 2989–3000. Guenther, R., Blickhan, R., 2002. Joint stiffness of the ankle and the knee in running. Journal of Biomechanics 35, 1459–1474. Haeufle, D.F.B., Grimmer, S., Seyfarth, A., 2010. The role of intrinsic muscle properties for stable hopping: stability is achieved by the force-velocity relation. Bioinspiration & Biomimetics 5, 016004. Hobara, H., Inoue, K., Omuro, K., Muraoka, T., Kanosue, K., 2011. Determinant of leg stiffness during hopping is frequency-dependent. European Journal of Applied Physiology 111, 2195–2201.
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Hurst, J.W., Chestnutt, J.E., Rizzi, A.A., 2004. An actuator with physically variable stiffness for highly dynamic legged locomotion. In: Proceedings of the IEEE International Conference on Robotics and Automation. New Orleans, LA, pp. 4662–4667. Kalveram, K.T., Haeufle, D.F.B., Grimmer, S., Seyfarth, A., 2010. Energy management that generates hopping: comparison of virtual, robotic and human bouncing. In: Proceedings of the International Conference on Simulation, Modeling and Programming for Autonomous Robots. Darmstadt, Germany, pp. 147–156. Karssen, J.G.D., Wisse, M., 2011. Running with improved disturbance rejection by using non-linear leg springs. The International Journal of Robotics Research 30, 1585–1595. Koditschek, D.E., Buehler, M., 1991. Analysis of a simplified hopping robot. The International Journal of Robotics Research 10, 587–605. Loeb Brown, I.E., Loeb, G.E., 1997. A reductionist approach to creating and using neuromusculoskeletal models. In: Winters, J.M., Crago, P.E. (Eds.), Biomechanics and Neural Control of Posture and Movement. Springer. McMahon, T.A., Cheng, G.C., 1990. The mechanics of running: how does stiffness couple with speed?. Journal of Biomechanics 23 (Suppl. 1), 65–78. Moritz, C.T., Farley, C.T., 2004. Passive dynamics change leg mechanics for an unexpected surface during human hopping. Journal of Applied Physiology 97, 1313–1322. Moritz, C.T., Farley, C.T., 2005. Human hopping on very soft elastic surfaces: implications for muscle pre-stretch and elastic energy storage in locomotion. Journal of Experimental Biology 208, 939–949.
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Linear center-of-mass dynamics emerge from non-linear leg-spring properties in human hopping Sebastian Riese n, Andre Seyfarth, Sten Grimmer Lauflabor Locomotion Laboratory, Institute for Sport Science, Technische Universität Darmstadt, Magdalenenstraße 27, D-64289 Darmstadt, Germany
art ic l e i nf o
a b s t r a c t
Article history: Accepted 28 June 2013
Given the almost linear relationship between ground-reaction force and leg length, bouncy gaits are commonly described using spring–mass models with constant leg-spring parameters. In biological systems, however, spring-like properties of limbs may change over time. Therefore, it was investigated how much variation of leg-spring parameters is present during vertical human hopping. In order to do so, rest-length and stiffness profiles were estimated from ground-reaction forces and center-of-mass dynamics measured in human hopping. Trials included five hopping frequencies ranging from 1.2 to 3.6 Hz. Results show that, even though stiffness and rest length vary during stance, for most frequencies the center-of-mass dynamics still resemble those of a linear spring–mass hopper. Rest-length and stiffness profiles differ for slow and fast hopping. Furthermore, at 1.2 Hz two distinct control schemes were observed. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Biomechanics Hopping Spring–mass model
1. Introduction For bouncy gaits the center-of-mass (CoM) movement can be approximated by a ballistic trajectory during flight and a spring law counteracting gravity during stance. This finding resulted in the theoretical concept of the spring-loaded inverted pendulum (SLIP) model for hopping and running (Blickhan, 1989; McMahon and Cheng, 1990). Here, the body is represented by a point mass and the leg is described by a massless linear spring. This approach is supported by the force–length function (FLF) of the leg, describing the relationship between ground-reaction force (GRF) and instantaneous leg length (Farley et al., 1991; Blickhan and Full, 1993). So far, it is unclear where global spring-like behavior of the leg originates. Some studies suggest that non-linear visco-elastic properties of the muscle–tendon complex, so-called “preflexes”, are the main contributors, especially during fast movements (Loeb Brown and Loeb, 1997). Others argue that muscle activation determines global leg behavior (Bobbert and Richard Casius, 2011). Also, combined control schemes incorporating preflexes and feed-forward patterns have been suggested (Cham et al., 2000). If leg segmentation is taken into account, linearity of leg behavior is lost on joint level. Even though the moment–angle relationship of the ankle joint is fairly linear for a variety of running patterns, this is not the case for the knee joint (Guenther and Blickhan, 2002). Results of Rapoport et al. (2003) support the loss of linearity on joint level. Using
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a segmented sagittal-plane hopping model for data analysis, joint stiffness was found to increase with joint flexion, resulting in bellshaped stiffness profiles over time with maximum stiffness near midstance. Leg compliance and its adaptation in response to changing environmental conditions are hypothesized to be crucial for successful locomotion (Grimmer et al., 2008). In contrast to serial-elastic actuators (SEAs), tunable compliant actuators allow to change mechanical stiffness on-the-fly. Hence, it was argued by Hurst et al. (2004) that this concept “could result in an effective actuation method for highly dynamic legged locomotion”. Following this argument, it was shown for a one-dimensional spring–mass model that simultaneous variations of leg-spring parameters (stiffness, rest length) during ground contact result in stable, robust and efficient hopping (Riese and Seyfarth, 2012a,b), motivating the variable-leg-spring (VLS) concept. So far, the results of previous studies indicate variations of leg stiffness and rest length during human hopping (Farley et al., 1991; Hobara et al., 2011), however without explicitly addressing them. Thus, in order to validate the VLS concept with experimental data, here the behavior of leg stiffness and rest length in vertical human hopping is investigated, assuming a tunable leg spring (Fig. 1). We hypothesize that the linear CoM dynamics observed in human hopping result from the interaction of non-linear leg-spring properties, namely non-constant leg stiffness and rest length, and that these parameter variations may be of considerable magnitude (4 10% of the touchdown value). According to Farley et al. (1991) human hopping patterns for frequencies below the preferred frequency differ from those above
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flight
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contact
F
CoM F ||
Table 1 Preferred frequency fp. The overall mean is in good agreement with results of Farley et al. (1991) (2.2 Hz). Subject
mean 7 s.d. (Hz)
1 2 3 4 5 6
2.017 0.07 2.50 70.13 2.42 70.06 1.98 7 0.04 2.76 70.14 2.22 70.05
Overall
2.337 0.30
F
Table 2 Marker set for calculation of kinematics. Marker placement
CoP
Forehead Right temple Left temple Light acromion Left acromion Right trochanter Left trochanter Right lateral elbow Left lateral elbow Right lateral wrist Left lateral wrist Right lateral knee Left lateral knee Right lateral malleolus Left lateral malleolus Right 5 metatarsals Left 5 metatarsals
Fig. 1. SLIP model during human hopping. All mass is located in the center of mass (CoM) and the leg length is assumed to be the distance between CoM and center of pressure (CoP). The misalignment of ground-reaction force (GRF) and leg spring was exaggerated to illustrate the GRF contributions parallel and perpendicular to the leg direction, F ∥ and F ⊥ , respectively.
the preferred one. Furthermore, it was shown that humans employ different control strategies to stabilize hopping depending on the rate of movement (Yen and Chang, 2010; Hobara et al., 2011). As the parameters accessible for control in the one-dimensional spring–mass model are stiffness and rest length, we expect slow hopping to exhibit different stiffness and rest-length profiles than fast hopping.
2. Methods 2.1. Experimental setup Six healthy, well-trained male subjects (76.5 7 8.4 kg) participated in the study. Prior to the measurements, the experiment was approved by the ethics review board of the University of Jena, as laid out in the Declaration of Helsinki, and all subjects gave their written informed consent. The subjects were asked to perform vertical jumps on both legs. Each subject was instructed to jump with self-selected frequency, subsequently referred to as fp (Table 1). Additionally, following Farley et al. (1991), the hopping frequencies 1.2 Hz, 1.8 Hz, 2.8 Hz and 3.6 Hz were prescribed with a metronome. The sequence of hopping frequencies was randomized for each subject. Each trial was of 30 s length. At the beginning and end of each trial, the subjects were asked to stand quiet for 5 s, leaving 20 s of vertical hopping, resulting in approximately 20–50 hopping cycles depending on subject and frequency.
2.2. Kinetics and kinematics GRFs were measured directly with 1 kHz using a Kistler force platform. Additionally, center-of-pressure (CoP) position was extracted from this data. In order to obtain kinematics, 17 reflective markers were placed on anatomical landmarks of each subject (Table 2). Marker positions were measured with 240 Hz using a ten-camera infrared system (Proflex MCU240, Qualisys, Gothenburg, Sweden) and interpolated to 1 kHz to match the GRF and CoP data. CoM position was then calculated in accordance with Dempster's body-segment parameter data (Dempster, 1955; Winter, 2009).
2.3. Estimation of stiffness and rest length In order to estimate global leg properties, the leg was approximated as a massless spring, connecting CoM and CoP (Fig. 1). GRFs and CoM movement during stance were projected into leg direction. Therefore, the data set is reduced to onedimensional (vertical) hopping. Additionally, GRFs were normalized to body weight (BW) and instantaneous leg length was normalized to initial CoM height linit, i.e. leg length while standing quiet. Thus, estimated stiffness K¼ klinit/BW and rest length L0 ¼ l0/linit are nondimensional. As the GRF and leg-length data are noisy, both data sets were filtered using a lowpass Butterworth of 5th order, with a cut-off frequency of 25 Hz.
For each trial, stance phases ðF ∥ ≥0:01 BWÞ were extracted and normalized to 100% stance time. F ∥ and leg length L were interpolated accordingly. Following previous studies (Rozendaal and van Soest, 2008; Peter et al., 2009) and assuming a piecewise-linear spring with stiffness K(i) and rest length L0 ðiÞ, the equation ! ! F ∥ ðiÞ L0 ðiÞLðiÞ ð1Þ ¼ KðiÞ F ∥ ði þ 1Þ L0 ðiÞLði þ 1Þ had to be solved for the time steps i¼ 1,3,…,99. As there are two unknowns per time step, K(i) and L0 ðiÞ, it was assumed that KðiÞ≡Kði þ 1Þ and L0 ðiÞ≡L0 ði þ 1Þ. Within this approach the spring is approximated as linear with constant parameters for two consecutive time steps. However, the resulting parameter profiles may be non-constant, allowing for a non-linear spring throughout stance. To ensure physically meaningful solutions, stiffness is constrained to values KðiÞ 40. Accordingly, during stance rest length has to satisfy L0 ðiÞ4 LðiÞ, as L0 ¼ L denotes the transition from flight to stance phases and vice versa. As a first approach, stiffness and rest length were calculated directly by solving Eq. (1) analytically for K(i) and L0 ðiÞ. However, the constraints for stiffness and rest length were violated for a considerable amount of time steps, especially for frequencies below the preferred frequency fp. Thus, Eq. (1) was solved numerically with the least-squares method lsqcurvefit implemented in MATLAB (R2010a, The MathWorks Inc., Natick, MA, USA) using the constraints for K(i) and L0 ðiÞ as lower boundaries.
3. Results Except at 1.2 Hz, results presented here for a given frequency are means over all trials of all six subjects at that frequency. At 1.2 Hz, behavior of half of the subjects distinctively differs from that of the other half, thus denoted in the figures as “1.2 Hz I” and “1.2 Hz II”, respectively. 3.1. Measured data For frequencies from fp to 3.6 Hz, instantaneous leg length L(i) corresponds to running-like single-minimum CoM trajectories (Fig. 2a). At lower frequencies, 1.2 and 1.8 Hz, also double-minimum
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leg-length profiles were measured, resembling walking patterns (Geyer et al., 2006). However, at 1.8 Hz the parameters estimated from the data exhibit a similar qualitative behavior for all subjects. Thus, at 1.8 Hz means over all trials of all subjects are displayed. Leg length at touchdown decreased with frequency (except for “1.2 Hz I”), so did CoM displacement and leg length at take-off (Fig. 2a). GRF profiles F ∥ ðiÞ for hopping ranging from fp up to 3.6 Hz feature only a single peak, as expected from spring–mass hopping (Fig. 2b). Landing impact forces were not observed; this is true for all trials and not an averaging effect. For 1.2 Hz walking-like double-peak GRF patterns were observed (even though half of the subjects exhibit running-like CoM trajectories). For 1.8 Hz two of the subjects mostly exhibit double-peak force profiles, while the others mostly feature single-peak GRF patterns. On average, however, single-peak hopping dominates. Resulting from GRFs and CoM movement, the FLF on leg level is fairly linear down to 1.8 Hz (Fig. 3). In contrast, at 1.2 Hz it is highly non-linear.
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In Fig. 5 modified FLFs are shown for all measured frequencies, using leg compression ΔLðiÞ ¼ L0 ðiÞLðiÞ instead of actual leg length L(i). Due to the estimated rest-length profile L0 ðiÞ, non-linearity of the modified FLF at 1.2 Hz considerably increased with respect to
120
80
40
0
3.2. Estimation results
1.5
For hopping within fp to 3.6 Hz the stiffness profiles K(i) are roughly bell-shaped, with increasing maximum stiffness for increasing frequency (Fig. 4a). At 2.8 and 3.6 Hz an additional impact-like maximum is present in the stiffness pattern. However, in the GRF profiles no impact forces were observed. At 1.2 and 1.8 Hz there are considerable stiffness fluctuations over stance time. Most prominently, for “1.2 Hz I” there is a clear maximum during the second GRF peak. Rest-length profiles L0 ðiÞ at fp to 3.6 Hz resemble the instantaneous leg length L(i) (Fig. 2b and 4b). This is not true at lower frequencies. At 1.2 Hz, rest length L0 ðiÞ for half of the subjects exhibits a triple-peak pattern, “1.2 Hz I”, while the other half features a plateau with a drop around midstance, “1.2 Hz II”. At 1.8 Hz, a composite of patterns “1.2 Hz I” and “1.2 Hz II” can be seen.
1.4
0
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100
0
20
40
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100
1.3 1.2 1.1 1.0
Fig. 4. Estimated leg parameters, stiffness K and rest length L0, over stance time. Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
3.0
1.1
2.0
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1.0 0.9 0
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0
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Fig. 2. Human hopping data. CoM movement projected into leg direction, i.e. instantaneous leg length (a) and the projected GRFs (b), is shown over stance time. Human hopping was investigated for five different hopping frequencies (ranging from 1.2 to 3.6 Hz). Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
3.0
2.0
1.0
0 0
0.1
Fig. 3. FLF on leg level. Human hopping was investigated for five different hopping frequencies (ranging from 1.2 to 3.6 Hz). Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
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3.0
2.0
1.0
0 0
0.1
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Fig. 5. Modified FLF on leg level, using leg compression ΔL ¼ L0 L instead of leg length L. Results are means over all trials of all subjects at a given frequency (at 1.2 Hz there are two distinct subsets consisting of half of the subjects each).
the unmodified case. At 1.8 Hz the modified FLF now is non-linear, also due to the estimated L0 ðiÞ pattern. For all other measured hopping frequencies a linear approximation of the FLF is still applicable.
4. Discussion In this study, the behavior of leg stiffness and rest length in vertical human hopping was investigated. It was hypothesized that even though the global FLF is fairly linear, implying a constant linear spring, leg-spring parameters may change significantly during ground contact. Additionally, it was hypothesized that rest-length and leg-stiffness profiles show a clear distinction between slow and fast hopping.
4.1. Variable leg-spring properties In vertical human hopping, leg stiffness and rest length change during stance, up to 4KTD and 0.3lTD, respectively (Fig. 4). Thus, these findings support the primary hypothesis of this study, while contradicting the well-established assumption that, due to its linear FLF, hopping features approximately constant leg-spring parameters. However, the results support the theoretical findings of the VLS concept (Riese and Seyfarth, 2012a). There, for a one-dimensional spring–mass model simultaneous variations of rest length and leg stiffness during ground contact resulted in stable hopping, whereas the original spring–mass model is not stable. Additionally, GRF patterns and CoM trajectories were more human-like than for the spring–mass model with constant leg parameters. In Riese and Seyfarth (2012a), leg stiffness and rest length were assumed to change linearly with stance time. Leg-stiffness and restlength profiles estimated here are much more complex than this simple first-order approximation (Fig. 4a). Nevertheless, simultaneous variation of leg stiffness and rest length in the model could be validated based on experimental data. It is still an open question how global leg properties relate to properties on joint level. Applying a segmented sagittal-plane hopping model to data analysis Rapoport et al. (2003) found that joint stiffness increased with angular deflection, resulting in bell-shaped stiffness profiles over stance time. Interestingly, Fig. 4a corresponds directly to these results on leg level. For hopping frequencies coinciding with spring-like hopping ðf ≥f p Þ the estimated leg-stiffness profiles also are roughly bell-shaped over time. This unexpected result may suggest a simple correlation between leg and joint stiffness for spring–mass hopping and indicates further study.
4.2. Non-linear leg properties vs. linear dynamics There is considerable evidence that human legs behave like linear springs during bouncy gaits (Farley et al., 1991; Hobara et al., 2011). Even in the presence of severe perturbations, such as compliant surfaces (Moritz and Farley, 2005) or an elastic exoskeleton (Ferris et al., 2006; Grabowski and Herr, 2009), humans maintain spring-like CoM dynamics. As spring–mass systems have been proven to possess self-stabilizing properties (Seyfarth et al., 2002), it has been suggested that emulating linear spring–mass dynamics may ease control and thus “may be a primary neuromuscular control strategy during bouncing gait” (Grabowski and Herr, 2009). However, Ferris et al. (2006) and Bobbert and Richard Casius (2011) stressed that spring-like behavior should not be confused with actual mechanical springs. The results presented here support the latter argument. Linear spring-mass dynamics may be achieved by other means than linear springs with fixed parameters. Here, the relationship between GRF and instantaneous leg length is (in good approximation) linear for hopping frequencies down to 1.8 Hz (Fig. 3). Nevertheless, neither stiffness nor rest length of the underlying model is constant (Fig. 4). In accordance with the primary hypothesis of the present study, the actual variation of rest length and stiffness is masked by the interaction of both parameter profiles. Rest length and stiffness vary in a way that in the resulting GRFs and CoM trajectories the non-linearities compensate each other. This also holds if the modified FLF is used, the only major difference being that linear FLFs now are only found for frequencies ranging from fp to 3.6 Hz (Fig. 5). The non-linearity of the stiffness profile is also in agreement with findings of Karssen and Wisse (2011). In their simulation study, the effect of non-linear leg springs on disturbance rejection during running was investigated. It was shown that the optimal stiffness profile is highly non-linear, with considerably better disturbance rejection than the optimal linear stiffness. Interestingly, the predicted FLF resulting from the optimal stiffness profile for running became non-linear beyond a certain amount of leg compression, resembling the FLF for hopping with 1.2 Hz (Fig. 3). This may indicate that for slow hopping more emphasis is put on disturbance rejection, i.e. a different control scheme is applied for slow hopping (see Section 4.3).
4.3. Frequency-dependent control In accordance with the secondary hypothesis of this study, different leg-stiffness and rest-length profiles were found for slow and fast hopping.
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At hopping frequencies of fp and above, leg stiffness and rest length exhibit a similar qualitative behavior. Stiffness and restlength profiles are smooth, even for rapid movements (up to 3.6 Hz). This uniform parameter variation across different hopping frequencies may be due to preflexes, i.e. zero-time-delay control based on mechanical properties of the system (Loeb Brown and Loeb, 1997; Haeufle et al., 2010). The parameter profiles themselves are bell-shaped, what might reflect the bell-shaped stiffness profiles found on joint level (Rapoport et al., 2003). The impactlike peaks at 2.8 and 3.6 Hz may be due to pre-activation of the muscles in anticipation of touchdown (Seyfarth et al., 2000; Moritz and Farley, 2004). At 1.2 Hz however, stiffness and rest-length profiles are quite unsteady (Fig. 4), which may suggest neural control, such as muscle activation. Two distinct control schemes can be observed at this frequency. While for “1.2 Hz II” control is mainly reflected by rest length, for “1.2 Hz I” stiffness and rest length oscillate phase-shiftedly, with a dramatic stiffness increase in preparation of take-off. In accordance with Farley et al. (1991), this suggests an active push-off to compensate for energy dissipated in the first half of contact. Similar actuation schemes with stiffness increases during the second half of contact have been proposed and shown to be quite effective for control of artificial legged systems (Koditschek and Buehler, 1991; Kalveram et al., 2010). At 1.8, a transitory behavior between slow and fast hopping is found. These findings support the statement of Yen and Chang (2010) that humans employ different control strategies to stabilize bouncy gaits depending on the rate of movement. Surprisingly, the results presented here resemble a gait transition from walking to running, without locomoting forward (Fig. 2). For “1.2 Hz II” even hybrid gaits with running-like CoM trajectories and walkinglike GRFs were observed. In accordance with Bobbert and Richard Casius (2011) as well as Hobara et al. (2011), linear spring-like behavior was found for hopping with a smaller frequency than fp (Fig. 3), contradicting findings of Farley et al. (1991). However, this is only true if the FLF on leg level is based on instantaneous leg length. This approach implicitly assumes a constant rest length. As substantial changes in rest length were found here (up to 0.3LTD for “1.2 Hz II”), using the unmodified FLF may limit the interpretation of leg-spring behavior. Changes in rest length were not taken into account by the aforementioned studies. However, doing so significantly changes the patterns of the FLF on leg level during hopping, cf. Figs. 3 and 5, and facilitates interpretation of leg function and control.
5. Conclusion As biological locomotory systems are more than simple spring– mass setups, leg stiffness and (virtual) leg rest-length may be adapted by control of muscle activation or by mechanical properties. In the commonly used approach of constant leg stiffness during contact the rest length of the leg spring is assumed to be fixed (e.g. Farley et al., 1991; Hobara et al., 2011). The present analysis shows that by keeping both leg stiffness and leg restlength (and corresponding leg compression) a function of time, interpretation of the FLF becomes more reasonable. Variability of the leg-spring parameters is shown to be an important system property rather than being caused by perturbations, thus validating the VLS concept. Currently, it is not known how these two parameters are controlled by the neuro-musculoskeletal system. However, this study provides an experimental analysis along with insights, which may be used to identify appropriate control schemes based on more detailed biomechanical and neuromechanical models.
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Conflict of interest statement We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In doing so we confirm that we have followed the regulations of our institutions concerning intellectual property. We further confirm that any aspect of the work covered in this manuscript that has involved either experimental animals or human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected].
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