## Abstract

Leg segment rotations in human walking covary, so that the three-dimensional trajectory of temporal changes in the elevation angles lies close to a plane. Recently the role of central versus biomechanical constraints on the kinematics control of human locomotion has been questioned. Here we show, based on both modeling and experimental data, that the planar law of intersegmental coordination is not a simple consequence of biomechanics. First, the full limb behavior in various locomotion modes (walking on inclined surface, staircase stepping, air-stepping, crouched walking, hopping) can be expressed as 2 degrees of freedom planar motion even though the orientation of the plane and pairwise segment angle correlations may differ substantially. Second, planar covariation is not an inevitable outcome of any locomotor movement. It can be systematically violated in some conditions (e.g., when stooping and grasping an object on the floor during walking or in toddlers at the onset of independent walking) or transferred into a simple linear relationship in others (e.g., during stepping in place). Finally, all three major limb segments contribute importantly to planar covariation and its characteristics resulting in a certain endpoint trajectory defined by the limb axis length and orientation. Recent advances in the neural control of movement support the hypothesis about central representation of kinematics components.

## INTRODUCTION

Principal component analysis (PCA) is a powerful and elegant method of data analysis aimed at obtaining low-dimensional approximation of high-dimensional processes. It is very successful in capturing data redundancies and has been applied in human locomotion to discriminate different gaits (Courtine et al. 2005; Das et al. 2004; Ivanenko et al. 2007; Troje 2002), determine redundancies in kinematic and electromyographic data (Daffertshofer et al. 2004), and assess inter-segmental coordination (Borghese et al. 1996; Thomas et al. 2005). During human locomotion, limb segment rotations covary so that the three-dimensional (3D) trajectory of temporal changes in the elevation angles lies close to a plane. We have verified this planar covariation of leg segment angles in several independent studies (Bianchi et al. 1998a,b; Borghese et al. 1996; Cheron et al. 2001; Grasso et al. 1998, 2000; Ivanenko et al. 2002, 2005b; Nayer et al. 2002), and it has also been independently confirmed by other groups (Courtine and Schieppati 2004; Dan et al. 2000; Hicheur et al. 2006; Laroche et al. 2007; Noble and Prentice 2007). It has been postulated that planar covariation of limb segment motion might simplify the control of both posture (Lacquaniti and Maioli 1994; Lacquaniti et al. 1990) and locomotion (Lacquaniti et al. 1999, 2002) by reducing the effective degrees of freedom. Thus an understanding of this phenomenon might provide some basic understanding about how the CNS controls muscle activation (d'Avella et al. 2006; Ivanenko et al. 2004).

In a recent report, Hicheur et al. (2006) provided an interpretation of the nature and the origin of the planar covariation of the three leg-segment elevation angles under various walking conditions. In particular, they questioned the role of central constraints on the kinematics control of human locomotion. One could argue that *1*) planar covariance is due entirely to the high correlation between the shank and foot segment angles and the thigh angle is not essential; *2*) the thigh angle independently contributes to the pattern of intersegmental coordination; and *3*) planar covariance is an outcome of passive rather than active coupling between segment angles. These arguments raise a number of issues on the usage and interpretation of analytical tools such as covariance and correlation and of terms such as independence. Our first goal here is to review the basic mathematical framework and analytical tools used to investigate intersegmental coordination and to clarify some of the issues that, in our view, can lead to misleading interpretations. Moreover, the preceding arguments lead to specific predictions that can be tested experimentally. The first is that, if the elevation angles are close to a plane because of high shank-foot correlation, the plane must be oriented so that the thigh axis is parallel to it and the correlation between shank and foot must be high under all conditions. The second is that if biomechanical coupling always results in planar covariation, then one should see it in every mode of locomotion. We offer here a number of examples that fail to confirm those predictions.

### Basic mathematical framework and modeling of intersegmental coordination

To study the strategies that the CNS might use to coordinate different limb segments, it is useful to characterize the relationships between kinematic variables (e.g., joint or elevation angles). The time dependence of these variables is described mathematically by a trajectory in a vector space, i.e., a continuous mapping of an interval of real numbers into vectors of real values taken by the kinematic variables at each time. For a rhythmic behavior, such as locomotion, described by *M* variables the trajectory is a closed loop in an *M*-dimensional vector space where *T* is the period. We are interested in the relationship between the *M* variables characterized by the existence of a *N*-dimensional (with *N* < *M*) subspace of the vector space on which the mean subtracted trajectory lies, that is, the existence of *N* linearly independent *M*-dimensional vectors {**v**_{i}}_{i = 1,…,N} such that (1) where *f _{i}* are periodic scalar functions and

**v**

_{0}is the mean of

**x**over time. In the specific case of

*M*= 3 and

*n*= 2, the subspace is a plane in 3D space.

Given an experimentally observed trajectory, how is it possible to infer the existence of an *N*-dimensional subspace containing the trajectory? In absence of noise, the eigenvalues of the sample covariance matrix will allow such an inference. For *k* time samples along the trajectory, the sample covariance matrix (2) where **X**^{T} = [**x**(*t*_{1})…**x**(*t _{k}*)] is the data matrix and

**H**=

**I**− 1/

*k*

**11**

^{T}, with

**1**a column vector of

*M*ones, is the centering matrix, has

*M*−

*N*null eigenvalues if the trajectory lies on an

*N*-dimensional subspace. Thus PCA, which consists in the diagonalization of the covariance matrix (

**S**=

**UΣU**

^{T},

**UU**

^{T}=

**I**,

**Σ**= diag([σ

_{1}…σ

_{N}0…0])), can be used and the eigenvectors associated to the zero eigenvalues span the space orthogonal to the trajectory subspace. In simpler terms, the eigenvector matrix (

**U**) represents a rotation of the coordinate axes that aligns the first axis along the direction of maximum variance (σ

_{1}) and the second to the last axes along directions with decreasing amount of variance. For planarity of a 3D trajectory, because there is zero variance along the direction orthogonal to the plane, the eigenvector associated to the single null eigenvalue is the plane normal. In presence of noise, assuming noise with a zero mean multivariate normal distribution ε ∼

**N**(

**0**,

**U**

_{ε}diag([σ

_{ε}

^{max}…σ

_{ε}

^{min}])

**U**

_{ε}

^{T}) (3) If the noise amplitude is much smaller than the variance of the trajectory in the direction of smallest variability (σ

_{ε}

^{max}≪ σ

_{N}), then the smallest eigenvalues of the sample covariance matrix will not be zero but the eigenvalues associated only with noise are clearly distinguishable and thus the subspace can be identified. Practically, when a model of noise generation is not available, a heuristic criterion based on identifying a large difference between the smallest eigenvalues due to the data variability (σ

_{N}) and the eigenvalues due to noise is used. Such a difference is usually detected on a plot of the fraction of the total variation (V=Σ

_{i}σ

_{i}) explained by each principal component (a “scree plot”).

Because sometimes the relationship between variables captured by the existence of an embedding subspace may be confused with the existence of high correlations between pairs of individual variables, it is useful to describe clearly the relationship between subspace embedding and pairwise correlation. Consider for simplicity the planar covariation of three variables. As mentioned in the preceding text, the sample covariance, **S** = **UΣU**^{T}, **Σ** = diag([σ_{1} σ_{2} σ_{3}]), has two larger eigenvalues (σ_{1} σ_{2}), due to the variability of the trajectory on the plane and noise, and one smaller eigenvalue (σ_{3}), due only to noise. The correlation matrix, i.e., the covariance matrix obtained after normalizing each variable by its standard deviation (s_{i}) (4) where **D** = diag([*s*_{1}…*s _{M}*]) is a scaling matrix, has the three pairwise correlations as off-diagonal elements (5) These correlation values, for a given set of eigenvalues of the covariance matrix, i.e., for a given level of planarity of the trajectory, depend on the plane orientation, i.e., on the rotation matrix

**U**. The minimum correlation absolute value is always 0, and it is obtained when the eigenvectors of the covariance matrix are aligned with the coordinate axes. The maximum correlation value is obtained when the eigenvector associated to the largest eigenvalue (σ

_{1}) is oriented along the bisectrix of two coordinate axes and the eigenvector associated to the second largest eigenvalue (σ

_{2}) is aligned with the third coordinate axis. This maximum value is (6) From

*Eq. 6*it follows that if two variables have a correlation

*r*, the ratio of the third eigenvalue of the covariance matrix on the total variation,

*V*= trace(

**Σ**) = σ

_{1}+ σ

_{2}+ σ

_{3}, often used as an index of planarity, must satisfy (7) For example, if

*r*= 0.9 and σ

_{1}/

*V*= 0.6, then σ

_{3}/

*V*< 0.0316. Thus high correlation implies high planarity. However, the converse is not true. That is, for the specific plane orientation in which the eigenvectors of the covariance matrix coincide with the coordinate axes, all correlation coefficients are zero independently of σ

_{3}. Thus if σ

_{3}is 0, planarity does not imply high correlation coefficients. In sum, while the eigenvalues of the covariance matrix do not depend on the plane orientation, pairwise correlations depend both on those eigenvalues and on the plane orientation. Thus correlation is not an adequate measure of planarity in three dimensional space and, similarly, is not an adequate measure of subspace embedding in higher dimensional spaces.

Another source of possible confusion is the relationship between planarity (or subspace embedding) and dependence among the variables. A plane in a 3D vector space is defined as the set of vectors that are orthogonal to a given vector (the plane normal), but it can also be defined as the graph of a linear function of two variables (“independent variables”) into a third variable (“dependent variable”). However, in the special case in which the plane is orthogonal to the plane of the two independent variables, the plane cannot be defined as the graph of a function of those two variables. Thus in such a special case, the third variable cannot be determined once the other two are given. Importantly, this condition is not at all equivalent to *statistical* independence of the third variable from the first two. Statistical independence, i.e. (8) where *p* is the probability density function, requires additional assumptions on the probability density on the plane. For example, samples drawn from the probability density (9) (10) with **x**^{T} = [*x*_{1} *x*_{2} *x*_{3}], δ(*z*) a Dirac's delta function, and *A* a scalar normalization coefficient, are constrained to be on the plane *x*_{1} = *x*_{2}, orthogonal to the (*x*_{1}, *x*_{2}) plane, but *x*_{3} is not independent of *x*_{1} and *x*_{2} (Fig. 1*A*). Moreover, if the data are not scattered across the plane but they are restricted to a closed loop on the plane, for example a circle of radius 2 (11) only two values of the *x*_{3} variable are associated to each value of the *x*_{1} (or *x*_{2}) variable, *x*_{3} = ±, i.e., *x*_{3} is statistically dependent on the other variables, even if not functionally (Fig. 1*B*). These two values can be further disambiguated using the time dependence of the trajectory (Fig. 1*C*). In fact, as a function of time, the spatial coordinates of a trajectory are all dependent variables. In sum, even for a trajectory lying on a plane parallel to one of the coordinate axes, one variable is, in general, not independent of the other two and, in any case, never independent for a time-dependent trajectory.

## METHODS

### Protocols

To explore the issue of planarity and biomechanical factors in human gait, we recorded several conditions for which planar covariation might have been disrupted by various support-related constraints. Analyses are based on both new data and previously published data, as detailed in the following text. The following conditions have been included

Walking on a treadmill at different speeds (1, 2, 3, 5, 7, and 9 km/h). Ten healthy subjects participated in these experiments (age: 36 ± 7 yr, weight: 68 ± 11 kg). The data on eight of these subjects have been reported previously (Ivanenko et al. 2007).

Overground crouched walking. For the crouched walking, subjects were asked to walk knees and hips flexed at a natural speed (Grasso et al. 2000). The same 10 subjects of

*condition 1*participated in this experiment.Overground hopping with two legs at natural freely chosen speed. The same 10 subjects of

*conditions 1*and*2*participated.Uphill walking at 5 km/h on a treadmill (12% inclination of the support surface). Six new subjects volunteered for this experiment (age: 36 ± 7 yr, weight: 72 ± 7 kg).

Upstairs stepping (2 steps, 17 cm height and 30 cm long each). The same six subjects of

*condition 4*were recorded in this condition. Different protocols during walking up and down stairs and on inclined surfaces have previously been published in abstract form (Nayer et al. 2002; Noble and Prentice 2007).Stooping and grasping an object on the floor while walking (

*stoop*condition). For the stoop condition, eight subjects (age: 35 ± 5 yr, weight: 58 ± 11 kg) were asked to walk at a natural speed and grasp a small object (soft ball; diameter, 8 cm) located on the floor with the right hand while the right leg was the weight-bearing leg. These data have been reported previously (Ivanenko et al. 2005a).Air-stepping with body-weight support. In air-stepping, subjects were supported in a harness pulled upward by a force equal to the body weight (Ivanenko et al. 2002) and stepped in the air at a comfortable cadence. Air-stepping was examined in 22 subjects (age: 30 ± 9 yr, weight: 70 ± 10 kg). The data on eight of these subjects were reported in a previous study (Ivanenko et al. 2002). However, now we expanded our previous analysis by recording 14 new volunteers. The rationale for including a larger population of subjects is related to a greater interindividual variability in the covariance plane orientation during air-stepping relative to all other conditions studied (see results).

Unsupported walking in toddlers. We recorded surface locomotion in eight toddlers (4 males, 4 females, 11–14 mo of age), who were just beginning to walk independently. Six of our toddlers were the same as in our previous paper (Dominici et al. 2007), but now we recorded them a second time 0.2–3 yr after the first recording session. For the toddlers, daily recording sessions were programmed around the parents' expectation of the very first day of independent walking until unsupported locomotion was recorded. For the recording of the very first steps, one parent initially held the toddler by hand. Then the parent started to move forward, leaving the toddler's hand and encouraging her or him to walk unsupported on the floor. Only sequences of steps executed naturally by the toddler (e.g., no stop between steps) and while looking forward, were retained to avoid initiation and braking phases and head movements due to looking in other directions. Typically, we recorded two to five consecutive step cycles in each trial.

Before each experiment, adult subjects practiced for 1–2 min to perform the specific condition at a preferred speed (or at a constant speed in the case of treadmill walking). The studies conformed to the Declaration of Helsinki, and informed consent was obtained from all participants (or the parents of the children) according to the procedures of the Ethics Committee of the Santa Lucia Foundation.

### Data recording and analysis

Because the method has been thoroughly documented in our previous papers (Bianchi et al. 1998b; Borghese et al. 1996; Grasso et al. 2000; Ivanenko et al. 2002, 2007), here we describe only briefly the experimental measurements and analysis. We recorded kinematic data at 100 Hz by means of either the Optotrak system (Northern Digital, Waterloo, Ontario, Canada) or 9-TV cameras Vicon-612 system (Oxford, UK). Infrared markers were attached to the skin overlying the following landmarks: gleno-humeral joint (GH), the midpoint between the anterior and the posterior superior iliac spine (ilium, IL), greater trochanter (GT), lateral femur epicondyle (LE), lateral malleolus (LM), heel (HE) and fifth metatarso-phalangeal joint (VM). The gait cycle was defined with respect to the right leg movement, beginning with right foot contact with the surface (Ivanenko et al. 2007). The body was modeled as an interconnected chain of rigid segments: IL-GT for the pelvis, GT-LE for the thigh, LE-LM for the shank, and LM-VM for the foot. The gait cycle was defined as the time between two successive foot contacts of the right leg corresponding to the local minima of the HE marker (Ivanenko et al. 2007). The timing of the lift-off was determined analogously (when the VM marker elevated by 3 cm). For hopping and upstairs stepping, we used the VM marker to evaluate the timing of both the touch-down and lift-off events. For air-stepping, the gait cycle was determined using the time between two successive maxima in the whole limb (GT-VM) elevation angle (Ivanenko et al. 2002). The touch-down and lift-off events were also verified from the force plate recordings, and we found that the kinematic criteria we used predicted the onset and end of stance phase with an error smaller than 2% of the gait cycle duration (Borghese et al. 1996; Ivanenko et al. 2005b). The data were time interpolated over individual gait cycles on a time base with 200 points.

The intersegmental coordination of the thigh, shank, and foot elevation angles in the sagittal plane was evaluated in position space as previously described using the principal component analysis (Borghese et al. 1996). For each eigenvector of the covariance matrix, the parameters *u _{it}*,

*u*, and

_{is}*u*correspond to the direction cosines (range: [−1 1]) with the positive semi-axis of the thigh, shank, and foot angular coordinates, respectively. We specifically analyzed and plotted the

_{if}*u*

_{3t}parameter for different conditions (it should be close to 0 if the thigh angle is not essential for planar covariation) (Hicheur et al. 2006). The planarity of the trajectories was quantified by the percentage of total variation (PV) accounted for by the first two eigenvectors of the data covariance matrix (for ideal planarity PV = 100% and the 3rd eigenvalue = 0).

## RESULTS

### Planar covariation of elevation angles in different gaits

Figures 2 and 3 illustrate the basic results. The full limb behavior in all cases shown in Fig. 2 can be expressed as the 2 degrees of freedom (dof) planar motion (2 PCs accounted for 98.2 ± 1.2% of the variation in the 3 angular waveforms across all conditions, range: 97.0–99.3%) although the orientation of the plane might differ significantly, thus rejecting the hypothesis that planar covariation is fully determined by high correlations between shank and foot angles. Indeed for perfectly correlated shank and foot angles, the normal to the covariance plane (*u*_{3}) should always be orthogonal to the thigh axis, i.e., the *u*_{3t} parameter should be equal to 0. However, this parameter varies considerably both across speeds during walking (Fig. 2*B*) and across gaits (*C*). In fact, in walking u_{3t} generally differs significantly (*P* < 0.05, 1-tailed *t*-test) from zero at all typical speeds (1–7 km/h), except for the fastest and unusual speeds >8 km/h. Moreover, *u*_{3t} does not remain constant but changes systematically with increasing speed. On average, the normal to the covariance plane rotates by ∼20° for speed changes between 3 and 9 km/h (Bianchi et al. 1998b) and varies relatively little between 1 and 3 km/h (Fig. 2*B*). *u*_{3t} deviates from 0 even more strongly with other styles of locomotion (Fig. 2*C*).

The correlation between shank and foot is high in most conditions (e.g., during upstairs stepping, it was ∼0.90). However, a high correlation is not obligatory to generate a gait planar covariance. For instance, in hopping, the correlation between shank and foot angular motion was close to zero (0.09 ± 0.23) and could even be negative (it varied from −0.45 to 0.44 across subjects) while planar covariation was observed in all subjects (PV = 98.8 ± 0.7%).

Furthermore, planar covariation is not an inevitable outcome of any locomotor movement because it could be violated during the “non-steady-state ” stoop condition (limb and trunk kinematics in the stoop cycle were obviously different from those in the preceding or following gait cycles, as well as the body configuration at the onset of the stoop cycle was different from that at the end of it, see stick diagram in Fig. 3*B*) or it was reduced to a line during stepping in place (Fig. 3*C*). For this reason, we did not indicate the *u*_{3t} parameter (and best regression planes in Fig. 3, *B* and *C*) because it is not informative. It is also worth noting that the correlation between the shank and foot segment angles in the example shown in Fig. 3*B* is relatively high (*r* = 0.90).

### Air-stepping

In the absence of support (air-stepping), we have observed higher interindividual variability than in all other conditions, likely because the subjects were free to choose their own kinematics pattern. Most subjects showed the plane orientation similar to that of normal walking (*u*_{3t} ranged from 0.15 to −0.15), whereas in some subjects, the orientation of the covariance plane differed substantially (cf. the 2 subjects in Fig. 3*A*). However, we never observed the orientation of the covariance plane with large positive *u*_{3t} values (>0.15). In general, *u*_{3t} ranged from 0.15 to −0.85 across all subjects. While the nature of these individual differences requires further investigations, in the context of the present study, it is important to emphasize that despite different phase relationships, amplitudes of the ankle joint motion and different plane orientations, all subjects demonstrated nearly perfect planar covariation in air-stepping (2 PCs accounted for >99% of variation in all subjects, on average 99.5 ± 0.3%). These results suggest that the 2 dof strategy may reflect the basic kinematic control strategy for leg movements (Lacquaniti et al. 1999, 2002) even in the absence of support-related constraints.

### Planar covariation in toddlers

In toddlers, at the onset of independent walking, the gait loop departs significantly from planarity and the mature pattern. Figure 4 illustrates a typical example of planar covariation in one toddler during first unsupported steps (*top*), 7 wk after (*middle*), and 13 mo after (*bottom*). Three consecutive gait cycles were analyzed and plotted for each case. Although the plane resembles that of the adults, the percentage of total variation was significantly lower in toddlers (96.1 ± 2.7%) than in older (2–4 yr) children and adults (98.9 ± 0.2 and 99.1 ± 0.2%, respectively, *P* < 0.0001 Student's unpaired *t*-test). This agrees with previous findings (Cheron et al. 2001; Ivanenko et al. 2004) and suggests that independent walking experience is essential for emergence of mature covariation pattern. Moreover, the step-by-step variability of plane orientation (estimated as the angular dispersion of the plane normal) (Dominici et al. 2007) was considerably higher in toddlers (16.1 ± 7.1°) than in adults (2.9 ± 1.0). When the gait loop was averaged across consecutive steps, the index of planarity (PV) slightly increased in toddlers (97.1 ± 2.2 vs. 96.1 ± 2.7%) likely because fluctuations across steps canceled in part, whereas in adults, it remained almost unchanged (99.1 ± 0.2 vs. 99.0 ± 0.2%).

### On the relationship between PCA and covariation plane

In this section, we deal with some methodological caveats related to the usage of PCA and planar covariation. Planar covariation is directly related to the dimensionality of the original data set that is often assessed in various biological applications using PCA (Daffertshofer et al. 2004). The aim of PCA is to represent the original waveforms as a linear combination of a few PCs (12) where *W* are weighting coefficients (3 × 2 matrix) and PCs are the first two principal components that explain most variation in the limb angle space (Fig. 5*A*). It is worth stressing that the core of PCA and estimation of planarity is based on orthogonal projections of the original data onto the covariance plane along principal component axes (Chatterjee 2000; Pearson 1901) rather than on “oblique” projections along *x*, *y*, *z* axes (Hicheur et al. 2006). This is illustrated in Fig. 5*B* with simulated planar data (PV = 99%). Indeed, for perfect planar covariation all three variables are interrelated according to the formula (13) where *A*, *B*, and *C* define the normal to the plane. However, one should not mix up the ideal plane (*Eq. 13*) with planar approximation that implies also some residual error (*Eq. 12*). Accordingly, a particular variable cannot be assessed using other variables as predictors (*Eq. 13*, corresponding to the oblique projection in Fig. 5*B*). This may introduce very large errors (see Fig. 5, *B* and *C*), especially when one of the coefficients (*A*, *B*, *C*) is small. In addition, the quantitative criterion for estimation of planarity used by Hicheur et al. (2006; appendix b), based on an estimation error defined as e_{z} = |(z – *ẑ*)/z|, is problematic because the error diverges to infinity as the variable in the denominator (*z*) approaches zero even with very small errors (z –*ẑ*), and such condition occurs in the data. The reconstruction should be done using PCs (*Eq. 12*) rather than other original variables (*Eq. 13*). In essence, this is shown in Fig. 5*A*. One can see that the reconstruction is nearly perfect (very high *r*).

## DISCUSSION

Planar covariation of leg segment angles holds for different locomotion conditions and the orientation of the covariance plane normal is generally not orthogonal to the thigh axis and varies widely (Figs. 2 and 3*A*). The thigh angle contributes in an important way to planar covariation and its characteristics. If the planar covariance were an artifact of high shank-foot correlations or an obligatory outcome of biomechanical coupling between segment angles, one would see it in every mode of locomotion. Instead we found systematic violations (Figs. 3*B* and 4) or reductions to a simple line (Fig. 3*C*).

To answer the question about the origin or functional significance of the planar law of inter-segmental coordination, one should endeavor to find functional interpretation of PCs. While PCA adequately captures the space spanned by the basic waveforms, there is no unique expansion for a set of waveforms, and one usually examines several solutions (rotations) and chooses the one that makes the best “sense” (Glaser and Ruchkin 1976). We recently performed such an analysis and showed that the two PCs can represent limb length- and orientation-related angular covariances (Ivanenko et al. 2007). This representation held for different gaits and speeds and might be indicative of differences in the distribution of yielding across limb segments. Our hypothesis predicted that leg movements that were confined along only one component axis, like limb length for example, would reduce the total degrees of freedom further to one. This prediction was realized for in-place movements (running in place, hopping in place, etc.), where the motion cycle was made by varying only the limb axis length (see, e.g., stepping in place, Fig. 3*C*). Furthermore, a linear combination of in-place angular covariances with limb orientation covariance could predict actual intersegmental coordination in different gaits (Ivanenko et al. 2007). Thus we propose that the limb segments are controlled jointly providing an appropriate “telescopic” limb behavior (Kuo et al. 2005).

Emergence and stability of planar covariation may be consistent with both energetic and control optimizations of rhythmic movements. For instance, in toddlers, despite its approximation to a single plane the gait loop departs significantly from planarity and the mature pattern, because the data are not well fitted by a plane (Fig. 4) (Cheron et al. 2001; Ivanenko et al. 2005b). Moreover, the step-by-step variability of plane orientation is considerably higher in toddlers, reflecting a high degree of instability in the phase relationship between the angular motion of different limb segments (Fig. 4). Thus developmental studies suggest that planar covariation may reflect a coordinated, centrally controlled behavior in addition to biomechanical constraints. Besides, in adults, locomotion conditions such as hopping or crouched walking are less stereotyped than walking or running (Ivanenko et al. 2007) and show somewhat larger intersubject and inter-stop variability. Likely these conditions require some training to achieve a more stereotyped or reproducible performance. Various gait optimization criteria have been proposed and discussed (Collins 1995; Zajac et al. 2002). As for planar covariation, the plane orientation in different subjects correlates with the net mechanical power output during the gait cycle (Bianchi et al. 1998a). Finally, planar covariation of elevation angles holds not only for human gait but also for quadrupedal monkey locomotion (Courtine et al. 2005) although the orientation of the covariance plane differs substantially (Fig. 6). Hence, the 2 dof strategy may be generalized to various gaits and animal species. Even though biomechanical factors contribute to emergence of planar covariation, the CNS must take these factors into consideration. Thus the planar covariation “strategy” emerges from both biomechanical and central constraints.

In sum, the argument that thigh angle independently contributes to the pattern of intersegmental coordination is not supported by our data. It is important to note that a low correlation coefficient does not imply that the thigh angle is independent of the other segment angles. Rather planar covariation implies the existence of intersegmental coordination rules that may reflect a way in which the so-called “telescopic” limb behavior is realized in human locomotion, likely by controlling the distribution of joint stiffness and thus the relative rotation of limb segments. Indeed telescopic limb control in terms of limb length and orientation requires coordination among all three segments. According to these rules, the full limb behavior in all gaits can be expressed as the 2 dof planar motion for each gait, plus the rotation of the planes about a defined axis (1 dof) (Ivanenko et al. 2007). This extends the analysis to a full 3 dof spatial control of locomotion where the third dimension may determine the gait pattern. However, the issue of the origin of these rules is certainly still open and requires further investigation. Recent studies on animals suggest that in the absence of any input from supraspinal structures, the lumbar spinal cord is capable of correcting kinematic errors in hindlimb coordination through practice (Heng and de Leon 2007). In fact, the way in which the limb kinematics is encoded centrally may be a fascinating area of research (Poppele and Bosco 2003).

## GRANTS

The financial support of Italian Health Ministry, Italian University Ministry (PRIN and FIRB projects), and Italian Space Agency (DCMC grant) is gratefully acknowledged.

## Footnotes

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- Copyright © 2008 by the American Physiological Society