^{1}, Martin L. Tanaka

^{2}and Carmine Senatore

^{1}

### Abstract

Ridges in the state space distribution of finite-time Lyapunov exponents can be used to locate dynamical boundaries. We describe a method for obtaining dynamical boundaries using only trajectories reconstructed from time series, expanding on the current approach which requires a vector field in the phase space. We analyze problems in musculoskeletal biomechanics, considered as exemplars of a class of experimental systems that contain separatrix features. Particular focus is given to postural control and balance, considering both models and experimental data. Our success in determining the boundary between recovery and failure in human balance activities suggests this approach will provide new robust stability measures, as well as measures of fall risk, that currently are not available and may have benefits for the analysis and prevention of low back pain and falls leading to injury, both of which affect a significant portion of the population.

The identification of frontiers between qualitatively different kinds of behavior in a dynamical system is important in many applications. Increasingly, systems of interest are determined not by analytically defined model systems, but by noisy time series data sets resulting from experiments or complex simulations. Recently, the concept of Lagrangian coherent structures (LCSs), ridges in the state space distribution of finite-time Lyapunov exponents (FTLEs), has been used to locate these dynamical boundaries. The current approach to finding these structures requires a vector field in the state space at each instant in time. In this paper, we demonstrate a method that uses only trajectories reconstructed from time series. To demonstrate this approach, we analyze problems in musculoskeletal biomechanics, considered as exemplars of a class of experimental systems that contain boundaries which have gone largely ignored, despite their importance for determining possible behaviors and the relationship between them.

I. INTRODUCTION

A. Balance recovery: Standing and falling

B. Torso stability: Assessed with wobble chair

C. Stability in walking versus falling

D. Organization of paper

II. DETECTING DYNAMICAL BOUNDARIES

A. Sensitivity analysis and the finite-time Lyapunov exponent field

B. Generating the FTLE field from trajectories

1. State space reconstruction

2. Calculating the FTLE field from discrete trajectory data

3. Modifying factors

C. Extracting and characterizing boundaries from the FTLE field

1. LCS detection algorithm

III. SEPARATRICES IN MUSCULOSKELETAL BIOMECHANICS

A. Postural stability and the wobble chair

B. One degree of freedom model

1. Sagittal plane standing

2. Reduced order planar wobble chair

3. One segment, planar model

4. Dynamical boundaries for full and partial sampling with varying amounts of noise

C. Two degree of freedom model

1. Two segment, planar model

2. Portions of the basin of stability boundaries in two DOFS

D. Determining the basin of stability through wobble chair experiments

IV. CONCLUSIONS AND FUTURE WORK

A. Recovery envelopes: A new tool in the evaluation of fall risk?

### Key Topics

- Time series analysis
- 21.0
- Biomechanics
- 19.0
- Phase space methods
- 14.0
- Vector fields
- 10.0
- Kinematics
- 9.0

## Figures

Detecting dynamical boundaries from measured data. (a) Schematic showing two example trajectories, both of which start in the stable region and end in the unstable region. By analyzing the motion of points along the trajectory appropriately, we can find the boundary or recovery envelope between stable and unstable (i.e., failure) motions. The divergence between two points (b and c) on opposite sides of a recovery envelope separatrix is larger than the divergence of points on the same side [pairs (a, b) and (c, d)], which tend to move together over short times. (b) An observable separatrix forms from simulated experimental trials of the wobble chair. The separatrix is found as a LCS, i.e., a ridge in the FTLE field.

Detecting dynamical boundaries from measured data. (a) Schematic showing two example trajectories, both of which start in the stable region and end in the unstable region. By analyzing the motion of points along the trajectory appropriately, we can find the boundary or recovery envelope between stable and unstable (i.e., failure) motions. The divergence between two points (b and c) on opposite sides of a recovery envelope separatrix is larger than the divergence of points on the same side [pairs (a, b) and (c, d)], which tend to move together over short times. (b) An observable separatrix forms from simulated experimental trials of the wobble chair. The separatrix is found as a LCS, i.e., a ridge in the FTLE field.

Flowchart depicting how separatrices (as LCSs) are traditionally determined. The underlying vector fields are unavailable in musculoskeletal experiments; only the resulting time series (i.e., trajectories) are available. Therefore we must consider an algorithm that starts with trajectories (the boxed steps) while assuming that trajectories were generated by a, perhaps unknown, vector field.

Flowchart depicting how separatrices (as LCSs) are traditionally determined. The underlying vector fields are unavailable in musculoskeletal experiments; only the resulting time series (i.e., trajectories) are available. Therefore we must consider an algorithm that starts with trajectories (the boxed steps) while assuming that trajectories were generated by a, perhaps unknown, vector field.

The wobble chair is an unstable sitting apparatus designed to isolate the movement of the low back to determine torso stability. (Figure is adapted from Fig. 1 in Ref. 6 and Fig. 1a in Ref. 57.) The angle is the angle the seat makes with the horizontal and is the angle the torso makes with the vertical.

The wobble chair is an unstable sitting apparatus designed to isolate the movement of the low back to determine torso stability. (Figure is adapted from Fig. 1 in Ref. 6 and Fig. 1a in Ref. 57.) The angle is the angle the seat makes with the horizontal and is the angle the torso makes with the vertical.

Recovery envelopes: a new tool in the evaluation of fall risk. Shown schematically, the boundary between the states corresponding to stable walking (recovery) and falling (failure) is given by a recovery envelope (thick line). We assume the state space is equipped with a metric and suggest that the minimum state space distance between a subject’s kinematic variability and their recovery envelope can be used as a measure of their falling risk.

Recovery envelopes: a new tool in the evaluation of fall risk. Shown schematically, the boundary between the states corresponding to stable walking (recovery) and falling (failure) is given by a recovery envelope (thick line). We assume the state space is equipped with a metric and suggest that the minimum state space distance between a subject’s kinematic variability and their recovery envelope can be used as a measure of their falling risk.

(a) A trajectory and a neighboring trajectory . (b) The state transition matrix is a deformation gradient about the reference trajectory describing how an initially spherical blob of surrounding states deforms into an ellipsoid.

(a) A trajectory and a neighboring trajectory . (b) The state transition matrix is a deformation gradient about the reference trajectory describing how an initially spherical blob of surrounding states deforms into an ellipsoid.

Estimating the maximum FTLE at a location in phase space by evaluating the growth of perturbation vectors in multiple state space directions. We make the assumption that the maximum FTLE dominates the evolution of the perturbation vectors.

Estimating the maximum FTLE at a location in phase space by evaluating the growth of perturbation vectors in multiple state space directions. We make the assumption that the maximum FTLE dominates the evolution of the perturbation vectors.

Main steps of the algorithm. From the FTLE field (left panel) the ridge points are calculated and aggregated toward the ridge axis (central panel). Thereafter, the points are appropriately connected (right panel) in order to create continuous curves.

Main steps of the algorithm. From the FTLE field (left panel) the ridge points are calculated and aggregated toward the ridge axis (central panel). Thereafter, the points are appropriately connected (right panel) in order to create continuous curves.

Subject balances on the wobble chair moving the lumbar region of her torso to maintain stability.

Subject balances on the wobble chair moving the lumbar region of her torso to maintain stability.

One DOF model for the wobble chair. In the simplified model the angle between the seat and the torso is fixed and control is applied at the base. This is an approximation to more complex system where torso flexion causes rear spring compression and extension causes front spring compression for effective base applied control.

One DOF model for the wobble chair. In the simplified model the angle between the seat and the torso is fixed and control is applied at the base. This is an approximation to more complex system where torso flexion causes rear spring compression and extension causes front spring compression for effective base applied control.

The FTLE field and detected boundaries (ridges) are shown for a full sampling of phase space for the reduced order wobble chair model at four values of the noise. The boundary separating the stable and unstable regions of phase space is shown. The boundary has two parts (roughly an upper and a lower part), and we give the average flux across each. Each boundary is made up of elements color coded according to the amount of flux across the element: low flux (higher strength) is lighter, high flux (lower strength) is darker. The solid closed curve in (a) is the theoretical boundary for zero noise.

The FTLE field and detected boundaries (ridges) are shown for a full sampling of phase space for the reduced order wobble chair model at four values of the noise. The boundary separating the stable and unstable regions of phase space is shown. The boundary has two parts (roughly an upper and a lower part), and we give the average flux across each. Each boundary is made up of elements color coded according to the amount of flux across the element: low flux (higher strength) is lighter, high flux (lower strength) is darker. The solid closed curve in (a) is the theoretical boundary for zero noise.

A time-series data set generated from 20 simulated experimental trials. (a) Time plot shows each trial beginning from the state space origin. (b) The trials shown as trajectories in phase space.

A time-series data set generated from 20 simulated experimental trials. (a) Time plot shows each trial beginning from the state space origin. (b) The trials shown as trajectories in phase space.

(a) A portion of a dynamical boundary found by analyzing the FTLE field resulting from the trajectories of Fig. 11. Only portions of the complete boundary are accessible by this method. The flux for the upper-right and lower-left portions is also shown. (b) A comparison of the dynamical boundaries from the partial sample (solid) and full-grid data (dotted).

(a) A portion of a dynamical boundary found by analyzing the FTLE field resulting from the trajectories of Fig. 11. Only portions of the complete boundary are accessible by this method. The flux for the upper-right and lower-left portions is also shown. (b) A comparison of the dynamical boundaries from the partial sample (solid) and full-grid data (dotted).

(a) Model of a person sitting on the wobble chair. Components of the lower body contribute to segment one, while components of the upper body make up segment two. (b) Simplified model of a person sitting on the wobble chair. Vectors are from joint to the COM of segment , while is the vector from joint 1 to joint 2. The deflection of the lower body and seat from the balanced configuration is and the deflection of the upper body from the vertical position is . Forces are applied to the stabilizing springs providing effective base control of the combined COM as the body pivots at the lumbar spine (L4-L5) during flexion and extension of the torso.

(a) Model of a person sitting on the wobble chair. Components of the lower body contribute to segment one, while components of the upper body make up segment two. (b) Simplified model of a person sitting on the wobble chair. Vectors are from joint to the COM of segment , while is the vector from joint 1 to joint 2. The deflection of the lower body and seat from the balanced configuration is and the deflection of the upper body from the vertical position is . Forces are applied to the stabilizing springs providing effective base control of the combined COM as the body pivots at the lumbar spine (L4-L5) during flexion and extension of the torso.

Recovery envelopes and basin of stability from 4D simulated data. FTLE field for simulated wobble chair data seen in two different two-dimensional slices of the full 4D state space (the slice is at the origin in the two other dimensions): (a) and (b) . Notice the ridge of high FTLE surrounding a region of low FTLE. This ridge is a slice of a LCS, a recovery envelope which bounds—and from which we can measure the size of—the basin of stability. Although only two dimensions are shown for visualization purposes, the basin volume can be computed in the full 4D state space.

Recovery envelopes and basin of stability from 4D simulated data. FTLE field for simulated wobble chair data seen in two different two-dimensional slices of the full 4D state space (the slice is at the origin in the two other dimensions): (a) and (b) . Notice the ridge of high FTLE surrounding a region of low FTLE. This ridge is a slice of a LCS, a recovery envelope which bounds—and from which we can measure the size of—the basin of stability. Although only two dimensions are shown for visualization purposes, the basin volume can be computed in the full 4D state space.

## Tables

Parameter table for one DOF model.

Parameter table for one DOF model.

Areas and flux of the basin of stability.

Areas and flux of the basin of stability.

Parameter table for two DOF model.

Parameter table for two DOF model.

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