Dynamic Determinants of the Uncontrolled Manifold during Human Quiet Stance

Human postural sway during stance arises from coordinated multi-joint movements. Thus, a sway trajectory represented by a time-varying postural vector in the multiple-joint-angle-space tends to be constrained to a low-dimensional subspace. It has been proposed that the subspace corresponds to a manifold defined by a kinematic constraint, such that the position of the center of mass (CoM) of the whole body is constant in time, referred to as the kinematic uncontrolled manifold (kinematic-UCM). A control strategy related to this hypothesis (CoM-control-strategy) claims that the central nervous system (CNS) aims to keep the posture close to the kinematic-UCM using a continuous feedback controller, leading to sway patterns that mostly occur within the kinematic-UCM, where no corrective control is exerted. An alternative strategy proposed by the authors (intermittent control-strategy) claims that the CNS stabilizes posture by intermittently suspending the active feedback controller, in such a way to allow the CNS to exploit a stable manifold of the saddle-type upright equilibrium in the state-space of the system, referred to as the dynamic-UCM, when the state point is on or near the manifold. Although the mathematical definitions of the kinematic- and dynamic-UCM are completely different, both UCMs play similar roles in the stabilization of multi-joint upright posture. The purpose of this study was to compare the dynamic performance of the two control strategies. In particular, we considered a double-inverted-pendulum-model of postural control, and analyzed the two UCMs defined above. We first showed that the geometric configurations of the two UCMs are almost identical. We then investigated whether the UCM-component of experimental sway could be considered as passive dynamics with no active control, and showed that such UCM-component mainly consists of high frequency oscillations above 1 Hz, corresponding to anti-phase coordination between the ankle and hip. We also showed that this result can be better characterized by an eigenfrequency associated with the dynamic-UCM. In summary, our analysis highlights the close relationship between the two control strategies, namely their ability to simultaneously establish small CoM variations and postural stability, but also make it clear that the intermittent control hypothesis better explains the spectral characteristics of sway.

where g represents gravitational acceleration. IL and IHAT represent the inertia moment of the lower link around the ankle joint and that of the upper link around the hip joint, respectively. τa and τh represent joint torques at ankle and hip joints. Joint torque Q = (τa, τh) T is described as the linear sum of the passive joint torque Qp = (τa passive , τh passive ) T without time delay and active joint torque Qa = (τa active , τh active ) T , which involves time delay (Asai et al. 2009;Suzuki et al. 2012). ). (A.5) The lower subscript Δ represents the time delay, e.g., xΔ(t) = x(t − Δ). For each joint, elastic and viscosity coefficients are denoted by Kj and Bj, respectively, for j = {a, h}. Pj and Dj are the proportional and derivative gains of the active neural feedback controller. Kj and Pj are defined by the total mass m of the double pendulum and the distance between the ankle joint to the center of mass position of the double pendulum under a fully extended hip joint condition, as follows.
where kj and pj are non-dimensional parameters, Kj and Pj.

Appendix B: Posture estimation of the upper body and lower limbs
In this study, lengths and postures of the upper body and lower extremities were estimated from the markers attached to the characteristic points on a subject's body using the following method.
According to Winter et al. 1998, we considered the human body as a twelve-link system (Pelvis, Trunk 1, Trunk 2, Trunk 3, Trunk 4, Head, L/R-Thigh, L/R-LowerLeg, L/R-Foot), and estimated the center of mass (CoM) position pi[n] for each of i = {pelvis, trunk-1, trunk-2, trunk-3, trunk-4, head, l-thigh, r-thigh, l-lower-leg, r-lower-leg, l-foot, r-foot}, with n being the discrete times from the spatiotemporal information of the markers. It should be noted that, in this study, the upper extremities were included in the trunk, and their spatial configuration was not considered. Marker information used for calculating the CoM position of each body segment is shown in Table 3.  where, mlower-leg and mthigh represent the masses of the lower legs and thighs, respectively (See Table   B.6 for details). We

Appendix C: Method for projecting the stable and unstable manifolds of the double inverted pendulum model in the θa-θh and ωa-ωh planes
The equilibrium point of the double inverted pendulum model without active feedback control (offmodel) is saddle-type and unstable, and there is a one-dimensional unstable manifold and onedimensional stable manifold (both corresponding to the in-phase mode), and a two-dimensional stable manifold corresponding to the anti-phase mode in the four-dimensional state space. These manifolds cannot be represented accurately in the θa-θh and ωa-ωh planes. In this study, we visualized these manifolds in the θa-θh and ωa-ωh planes as follows.

Visualization of the one-dimensional manifold
Here, the eigenvector v1d that spans the one-dimensional manifold is denoted as follows. .

(C.1)
For an arbitrary state point on the one-dimensional manifold, the ratio between the ankle joint angle θa and the hip joint angle θh always satisfies θa:θh = θa 1d :θh 1d , regardless of the values of the ankle joint angular velocity ωa and the hip joint angular velocity ωh. In the same way, for an arbitrary state point on the one-dimensional manifold, the ratio between the ankle joint angular velocity ωa and the hip joint angular velocity ωh always satisfies ωa:ωh = ωa 1d :ωh 1d , regardless of the values of the ankle joint angle θa and the hip joint angle θh. Therefore, the one-dimensional manifold as the straight line both on the θa-θh plane and the ωa-ωh plane can be visualized using the following equations.

Visualization of the two-dimensional manifold
Here, the eigenvectors v2d,a and v2d,b that span the two-dimensional manifold are denoted as follows. .

(C.4)
Using real value parameters α and β, the two-dimensional manifold can be described as follows. ( For a set of values of (ωa, ωh), α and β, which satisfy the equalities for the third and fourth elements in Eq. (C.5), are specified and then corresponding θa and θh values are determined according to the equalities for the first and second elements in Eq. (C.5) as follows. ( That is, the coordinate point (Eq. (C.6)) on the θa-θh plane corresponds to one point on the twodimensional manifold with the set of values of (ωa, ωh) (See Fig. 3). In this paper, each ωa and ωh is set to either −0.03 (rad/sec) or 0.03 (rad/sec), i.e., the four vertices of the square on the ωa-ωh plane

Appendix D: Analysis of the double inverted pendulum with continuous active feedback control
In this study, we analyzed the double inverted pendulum model with active feedback control (onmodel) using the same method used in the previous study (Suzuki et al. 2012). The on-model is described by the delay differential equation, which includes time delay due to signal processing (Eq. (8)). Here, we consider a solution of Eq. (8) described as follows.
( Alternatively, if λ has a positive real part, then the equilibrium is determined as unstable. For each λ, the dynamic mode is determined by using the corresponding eigenvector. We selected λ corresponding to the anti-phase mode, and calculated the eigenfrequency of the anti-phase mode from the imaginary part of λ.