# Head Trajectory Diagrams for Gait Symmetry Analysis Using a Single Head-Worn IMU

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participtants and Data Colection

#### 2.2. Sensor Placement and Orientations of Coordination Systems

**O**

_{Global}) are defined as the north is the x-axis, the west is the y-axis, and the upward is the z-axis, which is given by XSENS system. The local orientations (origin:

**O**

_{Local}or

**O′**) are also defined for the proposed diagrams as x′ is the direction of the walking vector, y′ is the left of the walking direction, and z is the upward direction (Figure 1).

#### 2.3. Anatomical Planes in Observation

#### 2.4. Temporal Alignment: Step Time

#### 2.5. Spatial Alignment and Walking Vector

_{A}moves from (4.4, 9.9) to (6.4, 12.8) and P

_{B}moves from (−8.6, −33.1) to (−7.8, −36.9). These participants were located in different absolute positions and moved in different directions, and thereby data are not comparable. To compare movements in each step, the x- and y-positions must be aligned with the geometric transformation to a new x′-y′ coordinate system. For the new coordinate system, a walking vector is defined between positions (

**O**

_{n}) as shown in Figure 4. The walking vector is computed as below:

**W**

_{n}is the walking vector for the n-th step (n > 1). The position

**O**

_{n}and

**O**

_{n}

_{-1}are defined on the x-y coordinate system when the n-th and (n−1)-th head vertical peak are detected. After

**O**

_{n}is detected, n-th the head trajectory between

**O**

_{n}and

**O**

_{n}

_{+1}are defined on the transverse plane as

**D**

_{n}(Figure 4a). For the spatial alignment,

**W**

_{n}and

**D**

_{n}are transformed to

**W**′

_{n}and

**D**′

_{n}together on the new x′-y′ coordinate system as shown in Figure 4b. When it comes to six steps with six walking vectors (Figure 4c), unit trajectories are overlapped as shown in Figure 4d. We call the graph of Figure 4d gait eye diagram at the head type-I (GE-H I).

**D**

_{n}). Therefore, it is easily recognized which foot is stepping on the ground at HS, referring to the sign of the y′-values. From Figure 4d, the step lengths and widths are also compared. The length of

**O**′

_{n}(on the x′-axis) informs the step length, which can show the step length skew (differences between two feet) and step length jitter (differences of the same foot). In terms of the step width, the positive and negative peaks of the y′-value can indicate the head lateral movement, and thereby the symmetricity of left and right step widths can be estimated. In addition, the temporal skew and jitter are also measured and compared. The step time can be calculated by multiplying the frame interval time (16.7 ms at 60 fps) and the number of data points in a step.

#### 2.6. Parameters in the Concepts of the Eye Diagram

#### 2.6.1. Eye Height

_{n}) in Figure 4, and foot position data are also aligned after divided into unit trajectories. After these transformations, left foot motions have positive y′-values and right foot motions are on the negative part of the y′-axis. The step width was measured as a distance between positive and negative values in foot lateral motions at the head’s vertical valley (VV) event.

#### 2.6.2. Jitter

#### 2.6.3. Bandwidth

#### 2.6.4. Signal-to-Noise Ratio

#### 2.6.5. Vertical Movement

#### 2.7. Gait Symmetry Indices

_{L}and X

_{R}are the absolute values of parameters of the left and right side, respectively. For RI, the denominator (X

_{L}) is higher than the numerator (X

_{R}), meaning X

_{L}≥ X

_{R}. Otherwise, if X

_{L}< X

_{R}, X

_{L}and X

_{R}become the numerator and denominator, respectively. In terms of original GA [27], the sign of the results can be information about which side is bigger. In this study, however, the absolute value of GA was taken for comparison with two other coefficients which are always positive.

_{L}: ALP.L. y′; X

_{R}: ALP.R. y′). In the calculation of the jitter, X

_{L}is σ.event.L. y′, and X

_{R}is σ.event.R. y′. For SNR, each side of EH and jitter.

## 3. Results

#### 3.1. Concept: Eye Height

#### 3.2. Concept: Jitter

**O**

_{n}(see also Figure 4d). The other three methods are measured for comparison. In all four methods, step lengths are correlated to each other (α = 0.05) as shown in Figure 9a. No significant differences were observed from ANOVA analysis and Tukey’s HSD. In Figure 9b, gait symmetry indices of four methods are compared, and all indices of SL.VV and SL.FT are significantly correlated (α = 0.05). Other indices, however, resulted in no significant correlations and differences.

#### 3.3. Concept: Bandwidth

#### 3.4. Concept: Signal-to-Noise Ratio

#### 3.5. Vertical Movement in Gait W-Diagram

#### 3.6. Validation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview of the proposed real-time feedback system, including the data accusation, the sensor placement, and the orientation of the coordinate systems.

**Figure 2.**Head trajectories on the y-z coordinate systems of (

**a**) participant A (P

_{A}) in the walking direction A (

**W**

_{A}) and (

**b**) participant B (P

_{B}) in the walking direction B (

**W**

_{B}). Their z-positions are comparable because the positions vary near the walker’s height, independently from walking directions or the global positions.

**Figure 3.**Head trajectories on the x-y coordinate systems of (

**a**) the participant A (P

_{A}) in the walking direction A (

**W**

_{A}, the north-west) and (

**b**) the participant B (P

_{B}) in the walking direction B (

**W**

_{B}, the north-east). Their x- and y-positions at peaks are incomparable because the positions have no common reference and also vary depending on walking directions and the global positions.

**Figure 4.**Head trajectories of the participant A (P

_{A}) for a walking vector (

**a**) on the x-y coordinate systems and (

**b**) on the new x′-y′ coordinate systems, as well as trajectories of six walking vectors of P

_{A}(

**c**) on the x-y and (

**d**) x′-y′ coordinate systems (gait eye diagram at the head type-I, GE-H I).

**Figure 5.**Normalized head unit trajectories of six steps on the transverse plane (x″-y′ coordinate system; x″: 0.0–1.0), also called the gait eye diagram at the head type-II (GE-H II).

**Figure 6.**Head trajectory of six steps on the frontal plane (y′-z coordinate system), also called the gait W-diagram at the head (GW-H).

**Figure 7.**The gait eye diagrams at the head (x′-y′, GE-H I), normalized eye diagrams (x′′-y′, GE-H II) and the gait W-diagrams (y′-z, GW-H) of 12 participants (P1–P12).

**Figure 8.**Box and Whisker diagrams of (

**a**) the average eye height at three events: lateral peak (EH.LP), vertical valley (EH.VV), and heel strike (EH.HS), and the step width measured by foot sensors (SW.FT), as well as (

**b**) their three gait symmetry indices (RI, SI, and GA).

**Figure 9.**Box and Whisker diagrams of (

**a**) the step length in four different measurement sources and (

**b**) their three gait symmetry indices.

**Figure 10.**Box and Whisker diagrams of (

**a**) the step time in four different measurement sources and (

**b**) and their three gait symmetry indices.

**Figure 11.**Box and Whisker diagrams of gait velocity (m/s) in (

**a**) the average and (

**b**) gait symmetry index values in four different measurement methods.

**Figure 12.**Box and Whisker diagrams of signal to noise ratio (SNR) with (

**a**) the average values and (

**b**) gait symmetry indices in four different measurement methods.

**Figure 13.**Box and Whisker diagrams of (

**a**) the average vertical displacements of the head and (

**b**) their gait symmetry indices.

Side | Parameters | VP | VV | HS | Ground Truth |
---|---|---|---|---|---|

Both | Steps (Mean ± Std.) | 85.6 ± 2.9 | 85.3 ± 3.0 | 85.0 ± 2.7 | 85.6 ± 3.1 |

^{1} MAPE (%) | 0.19 | 0.48 | 0.65 | ||

Left | Steps (Mean ± Std.) | 42.8 ± 1.8 | 42.7 ± 1.9 | 42.3 ± 1.3 | 42.7 ± 1.4 |

^{1} MAPE (%) | 0.99 | 1.59 | 0.93 | ||

Right | Steps (Mean ± Std.) | 42.8 ± 1.2 | 42.7 ± 1.2 | 42.8 ± 1.6 | 42.9 ± 1.8 |

^{1} MAPE (%) | 1.36 | 2.13 | 0.37 |

^{1}Mean absolute percentage error.

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**MDPI and ACS Style**

Hwang, T.-H.; Effenberg, A.O.
Head Trajectory Diagrams for Gait Symmetry Analysis Using a Single Head-Worn IMU. *Sensors* **2021**, *21*, 6621.
https://doi.org/10.3390/s21196621

**AMA Style**

Hwang T-H, Effenberg AO.
Head Trajectory Diagrams for Gait Symmetry Analysis Using a Single Head-Worn IMU. *Sensors*. 2021; 21(19):6621.
https://doi.org/10.3390/s21196621

**Chicago/Turabian Style**

Hwang, Tong-Hun, and Alfred O. Effenberg.
2021. "Head Trajectory Diagrams for Gait Symmetry Analysis Using a Single Head-Worn IMU" *Sensors* 21, no. 19: 6621.
https://doi.org/10.3390/s21196621