# Center of Mass Estimation Using a Force Platform and Inertial Sensors for Balance Evaluation in Quiet Standing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Estimation Methods

#### 2.1. Modeling

_{b}is the COM displacement in the moving coordinate system; Ẍ

_{s}is the acceleration of the support surface in the stationary coordinate system; N

_{y}and R

_{x}are the moment around the y-axis and the horizontal force in the x-axis measured by the force platform, respectively; M is the weight of a subject; m

_{b}is the body weight; J

_{xb}is the moment of inertia around the body COM; l

_{b}is the length between the ankle joint and the body COM; L

_{f}is the height of the ankle; and g is the gravitational acceleration.

_{1}and x

_{2}are the COM displacements of the lower and upper bodies in the moving coordinate system; J

_{1}and J

_{2}are the moments of inertia around the COMs of the lower and upper bodies, respectively; L

_{1}is the length of the lower body; l

_{1}is the length from the ankle joint to the lower body COM; l

_{2}is the length from the hip joint to the upper body COM; and x

_{b}is the combined COM of the lower and upper bodies derived from x

_{b}= (m

_{1}x

_{1}+ m

_{2}x

_{2})/m

_{b}.

_{b}is the combined COM displacement except for feet; y

_{l}is the COM of both legs; y

_{u}is the upper body COM in the stationary coordinate system; ${\ddot{Y}}_{s}$ is the acceleration of the support surface in the moving coordinate system; N

_{x}and R

_{y}are the moments around the x-axis and the horizontal force to the y-axis measured by the force platform, respectively; m

_{l}is the mass of one leg; m

_{p}is mass of the pelvis; J

_{l}is the moment of inertia of one leg around the COM; J

_{u}is the moment of inertia of the upper body around the COM; l

_{l}is the length between the ankle joint and the leg COM; L

_{l}is the length between the ankle joint and the hip joint; l

_{p}is the length between the hip joint and pelvis COM; L

_{p}is the length of the pelvis; and l

_{u}is the length between the waist joint and the upper body COM.

#### 2.2. COM Estimation Method

_{x}and R

_{y}, horizontal head acceleration Ẍ

_{h}, Ÿ

_{h}, and support surface acceleration Ẍ

_{s}, Ÿ

_{s}. In the following section, we demonstrate COM estimation methods for two cases: with and without a head IMU.

_{b}and x

_{b}in the sagittal plane or by solving (9) and (10) for ${\ddot{y}}_{b}$ and y

_{b}in the frontal plane, respectively.

_{2}is the length between the hip joint and head IMU, and L

_{u}is the length between the waist joint and head IMU. In the sagittal plane, COM displacement of the body is estimated by solving the following equations derived from (3), (4), and (12) for ẍ

_{1}, ẍ

_{2}, and x

_{b},

_{b}:

**w**is the process noise, and v is the observation noise. When

**Q**=

_{w}**ww**

^{T}and Q

_{v}= v

^{2}, the COM position and velocity were estimated using the Kalman filter algorithm as follows:

**G**is the Kalman gain, and

**P**is the covariance matrix. The subscript ‘−’ means the prior prediction values.

## 3. Verification Methods

#### 3.1. Experimental Protocol

#### 3.2. Experimental Equipment

#### 3.3. Post-Processing

_{x}and R

_{y}from the force platforms because the horizontal forces in quiet standing are much smaller than the vertical forces, and drift occurred in some data. The attitude angles of the IMU were derived from the three-axis acceleration and angular velocity data using an extended Kalman filter, and the horizontal head acceleration Ẍ

_{h}and Ÿ

_{h}were obtained by coordinate transformation. These values were used to estimate the displacement and velocity of the body COM and the acceleration of the lower and upper body COM.

**Q**= diag(0.0025, 0.04), Q

_{w}_{v}= 1. Because only method (II) can estimate the COM accelerations of the lower and upper bodies, we evaluated the estimation accuracy of method (II).

_{C}) for the true values obtained from the optical motion capture system. The evaluation period was 30 s, ranging from 5 s to 35 s. The RMSE and C

_{C}were calculated by

_{f}is the estimated value obtained from the force platform measurements, q

_{m}is the true value obtained from the optical motion capture measurements, ${\overline{q}}_{f}$ is the mean value of q

_{f}, and ${\overline{q}}_{m}$ is the mean value of q

_{m}.

## 4. Results

_{C}are listed in Table 4 (AP) and Table 5 (ML). The COM displacement and velocity of the body and COM accelerations of the lower and upper bodies obtained from motion capture (black line) and method (II) (blue line) for one subject (180 cm, 83 kg) for six motions, once each for (A) to (F), are plotted in Figure 4. The RMSE and C

_{C}values for the same results are shown in the figures.

## 5. Discussion

_{p}between the COP and estimated COM in method (I). Comparing the C

_{p}between (A) and (D) in the sagittal plane, (A) was 0.952 ± 0.032, and (D) was 0.129 ± 0.399. The correlation coefficient was definitively lower in (D), indicating a lower estimation accuracy in method (I). Figure 5 shows the COM estimation results in the sagittal plane for the worst-case correlation coefficient (C

_{P}= 0.800) during quiet standing. This result indicates that method (I) is applicable for C

_{p}> 0.8.

_{C}of the upper body was low during hip motion in the sagittal plane, as shown in Table 4. To understand the reason for the time series waveform of the hip motion shown in Figure 4, we found that the acceleration of the upper body was considerably smaller than that of the lower body. Because the scale was smaller than the COM acceleration of the lower body, the C

_{C}of the upper body was considered to decrease. Therefore, there is no problem with the estimation accuracy of the COM acceleration of the lower and upper bodies when using method (II).

## 6. Conclusions

## 7. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{x}+ Ẍ

_{s}/g as the input to the transfer function. The COM velocity was derived by numerically differentiating the COM displacement. Consequently, this method estimates the COM by applying the COP to a low-pass filter (approximately 0.5 Hz), which is empirical rather than theoretical. This concept is widely used to derive the extrapolated center of mass (XCOM) for evaluating gait stability [37].

_{x}(t) = 0 as the integration boundary. The COM acceleration was numerically integrated by the following equation, satisfying that COM and COP (x

_{p}) have the same initial values in each interval (t

_{i}→ t

_{i}

_{+1}),

_{p}(t) is the COP position. By numerical integration of (A3) and (A4) again using the initial COM velocity obtained from (A5), we can estimate the COM displacement and velocity. It should be noted that this method results in discontinuity in the COM velocity at the boundary points.

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**Figure 1.**Rigid body link models used in this study: (

**a**) single inverted pendulum model for the ankle joint strategy in the sagittal plane, (

**b**) double-inverted pendulum model for the ankle and hip joint strategy in the sagittal plane, and (

**c**) rigid link model for the balance motion in the frontal plane. All models allowed horizontal movement of the support surface.

**Figure 2.**Six motions in an experiment to validate the accuracy of center of mass (COM) estimation: (

**A**) quiet standing, (

**B**) ankle joint strategy motion in the AP direction at 0.25 Hz, (

**C**) ankle joint strategy motion in the ML direction at 0.25 Hz, (

**D**) hip joint strategy motion in the AP direction at 1 Hz, (

**E**) horizontal sway of the support surface in AP direction, and (

**F**) horizontal sway of the support surface in the ML direction.

**Figure 4.**Time series data of COM displacements and velocities of the body and COM accelerations of the lower and upper bodies for six motions (

**A**–

**F**) for a subject (180 cm, 83 kg). The black and blue lines show the estimated values from the motion capture measurement and method (II), respectively.

**Figure 5.**Time series data for the worst correlation coefficient C

_{p}in the sagittal plane (C

_{p}= 0.800). The black and red lines represent the estimates obtained from the motion capture system and method (I), respectively.

Sagittal Plane | Frontal Plane | ||||
---|---|---|---|---|---|

Segment | Symbol | Value | Segment | Symbol | Value |

Body | m_{b} | 0.978 M | Legs | m_{l} | 0.161 M |

J_{b} | 0.0425 MH^{2} | J_{l} | 0.00524 MH^{2} | ||

l_{b} | 0.531 H | l_{l} | 0.285 H | ||

Lower body | m_{1} | 0.322 M | L_{l} | 0.460 H | |

J_{1} | 0.00223 MH^{2} | Pelvis | m_{p} | 0.187 M | |

l_{1} | 0.285 H | l_{p} | 0.056 H | ||

L_{1} | 0.460 H | L_{p} | 0.144 H | ||

Upper body | m_{2} | 0.656 M | Upper body | m_{u} | 0.469 M |

J_{2} | 0.0114 MH^{2} | J_{u} | 0.00714 MH^{2} | ||

l_{2} | 0.191 H | l_{u} | 0.109 H | ||

L_{2} | 0.434 H | L_{u} | 0.290 H | ||

Foot | L_{f} | 0.038 H | Foot | L_{f} | 0.038 H |

**Table 2.**Root mean square error (RMSE) between the estimates of the four methods (I)–(IV) and the true values obtained from the optical motion capture system for the COM displacements and velocities of the body and COM accelerations of the lower and upper bodies in the sagittal plane.

Motion | Method | COM Position | COM Velocity | Lower Body Acceleration | Upper Body Acceleration |
---|---|---|---|---|---|

mm | mm/s | mm/s^{2} | mm/s^{2} | ||

(A) Quiet Standing | RMS | 3.25 ± 1.09 | 2.36 ± 0.59 | 10.2 ± 4.2 | 11.0 ± 4.1 |

(I) | 0.65 ± 0.21 | 1.83 ± 0.56 | - | - | |

(II) | 0.59 ± 0.18 | 1.75 ± 0.55 | 9.6 ± 4.0 | 8.3 ± 4.1 | |

(III) | 0.31 ± 0.12 | 0.72 ± 0.44 | - | - | |

(IV) | 1.03 ± 0.67 | 3.71 ± 1.18 | - | - | |

(B) Ankle Motion (AP) | RMS | 26.89 ± 7.70 | 33.70 ± 9.81 | 48.2 ± 13.0 | 85.6 ± 24.2 |

(I) | 2.89 ± 1.45 | 9.73 ± 3.86 | - | - | |

(II) | 2.62 ± 1.17 | 8.86 ± 3.77 | 33.7 ± 8.3 | 27.9 ± 8.9 | |

(III) | 1.50 ± 0.43 | 2.29 ± 0.52 | - | - | |

(IV) | 9.32 ± 6.96 | 10.81 ± 3.04 | - | - | |

(D) Hip Motion (AP) | RMS | 9.03 ± 2.64 | 35.65 ± 14.50 | 421.5 ± 202.9 | 119.3 ± 66.5 |

(I) | 15.10 ± 6.86 | 32.59 ± 15.28 | - | - | |

(II) | 3.47 ± 1.67 | 12.16 ±5.36 | 100.0 ± 37.4 | 103.9 ± 57.7 | |

(III) | 4.44 ± 2.06 | 24.01 ± 11.28 | - | - | |

(IV) | 5.17 ± 3.34 | 10.22 ± 6.27 | - | - | |

(E) Horizontal Sway (AP) | RMS | 10.89 ± 2.44 | 16.36 ± 1.76 | 50.3 ± 11.6 | 99.2 ± 11.3 |

(I) | 1.99 ± 0.44 | 7.44 ± 0.81 | - | - | |

(II) | 1.60 ± 0.34 | 6.90 ± 0.79 | 30.6 ± 8.4 | 32.2 ± 8.8 | |

(III) | 1.21 ± 0.24 | 2.33 ± 0.50 | - | - | |

(IV) | 12.15 ± 7.24 | 15.09 ± 3.92 | - | - |

**Table 3.**RMSE between the estimates of the four methods (I)–(IV) and the true values obtained from the optical motion capture system for the COM displacements and velocities of the body and COM accelerations of the lower and upper bodies in the frontal plane.

Motion | Method | COM Position | COM Velocity | Lower Body Acceleration | Upper Body Acceleration |
---|---|---|---|---|---|

mm | mm/s | mm/s^{2} | mm/s^{2} | ||

(A) Quiet Standing | RMS | 1.51 ± 0.62 | 1.55 ± 0.53 | 9.4 ± 4.9 | 10.4 ± 6.1 |

(I) | 0.56 ± 0.15 | 1.76 ± 0.46 | - | - | |

(II) | 0.56 ± 0.15 | 1.75 ± 0.45 | 8.7 ±4.9 | 9.3 ± 6.3 | |

(III) | 0.17 ± 0.07 | 0.66 ± 0.44 | - | - | |

(IV) | 2.48 ± 2.78 | 5.23 ± 2.20 | - | - | |

(C) Ankle Motion (ML) | RMS | 34.47 ± 8.77 | 42.86 ± 10.47 | 53.1 ±11.9 | 93.7 ± 21.4 |

(I) | 3.60 ± 2.50 | 20.93 ± 5.38 | - | - | |

(II) | 3.33 ± 1.96 | 19.67 ± 5.11 | 34.6 ±8.1 | 32.2 ± 8.4 | |

(III) | 1.41 ± 0.80 | 2.16 ± 1.05 | - | - | |

(IV) | 25.95 ± 22.62 | 33.36 ± 9.14 | - | - | |

(F) Horizontal Sway (ML) | RMS | 10.08 ± 1.98 | 19.10 ± 1.95 | 61.0 ±8.0 | 100.0 ± 11.9 |

(I) | 2.32 ± 0.41 | 7.95 ± 1.16 | - | - | |

(II) | 1.79 ± 0.36 | 7.38 ± 1.28 | 23.8 ±3.4 | 29.9 ± 5.8 | |

(III) | 1.10 ± 0.24 | 1.91 ± 0.28 | - | - | |

(IV) | 12.36 ± 7.71 | 16.39 ± 3.78 | - | - |

**Table 4.**Correlation coefficients (Cc) between the estimated values from methods (I)–(IV) and the true values obtained from the optical motion capture system for COM displacement and velocity of the body and COM acceleration of the lower and upper bodies in the sagittal plane.

Motion | Method | COM Position | COM Velocity | Lower Body Acceleration | Upper Body Acceleration |
---|---|---|---|---|---|

(A) Quiet Standing | (I) | 0.978 ± 0.023 | 0.778 ± 0.122 | - | - |

(II) | 0.981 ± 0.020 | 0.781 ± 0.126 | 0.377 ± 0.136 | 0.446 ± 0.189 | |

(III) | 0.997 ± 0.006 | 0.951 ± 0.073 | - | - | |

(IV) | 0.954 ± 0.031 | 0.655 ± 0.152 | - | - | |

(B) Ankle Motion (AP) | (I) | 0.998 ± 0.002 | 0.980 ± 0.024 | - | - |

(II) | 0.998 ± 0.001 | 0.980 ± 0.025 | 0.689 ± 0.126 | 0.953 ± 0.031 | |

(III) | 0.999 ± 0.001 | 0.999 ± 0.001 | - | - | |

(IV) | 0.943 ± 0.060 | 0.970 ± 0.026 | - | - | |

(D) Hip Motion (AP) | (I) | 0.102 ± 0.363 | 0.767 ± 0.071 | - | - |

(II) | 0.938 ± 0.046 | 0.949 ± 0.043 | 0.958 ± 0.074 | 0.377 ± 0.429 | |

(III) | 0.894 ± 0.063 | 0.893 ± 0.129 | - | - | |

(IV) | 0.844 ± 0.183 | 0.942 ± 0.082 | - | - | |

(E) Horizontal Sway (AP) | (I) | 0.984 ± 0.008 | 0.934 ± 0.013 | - | - |

(II) | 0.989 ± 0.006 | 0.945 ± 0.011 | 0.813 ± 0.056 | 0.935 ± 0.028 | |

(III) | 0.995 ± 0.002 | 0.992 ± 0.003 | - | - | |

(IV) | 0.711 ± 0.196 | 0.840 ± 0.082 | - | - |

**Table 5.**Cc between the estimated values from methods (I)–(IV) and the true values obtained from the optical motion capture system for COM displacement and velocity of the body and COM acceleration of the lower and upper bodies in the frontal plane.

Motion | Method | COM Position | COM Velocity | Lower Body Acceleration | Upper Body Acceleration |
---|---|---|---|---|---|

(A) Quiet Standing | (I) | 0.922 ± 0.062 | 0.636 ± 0.131 | - | - |

(II) | 0.920 ± 0.068 | 0.623 ± 0.135 | 0.252 ± 0.125 | 0.339 ± 0.151 | |

(III) | 0.991 ± 0.010 | 0.889 ± 0.100 | - | - | |

(IV) | 0.713 ± 0.267 | 0.452 ± 0.159 | - | - | |

(C) Ankle Motion (ML) | (I) | 0.998 ± 0.001 | 0.974 ± 0.017 | - | - |

(II) | 0.998 ± 0.001 | 0.971 ± 0.019 | 0.872 ± 0.075 | 0.962 ± 0.034 | |

(III) | 1.000 ± 0.000 | 0.999 ± 0.001 | - | - | |

(IV) | 0.919 ± 0.120 | 0.975 ± 0.022 | - | - | |

(E) Horizontal Sway (ML) | (I) | 0.974 ± 0.011 | 0.946 ± 0.019 | - | - |

(II) | 0.985 ± 0.006 | 0.951 ± 0.022 | 0.924 ± 0.023 | 0.957 ± 0.022 | |

(III) | 0.995 ± 0.002 | 0.996 ± 0.002 | - | - | |

(IV) | 0.710 ± 0.195 | 0.873 ± 0.073 | - | - |

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**MDPI and ACS Style**

Sonobe, M.; Inoue, Y.
Center of Mass Estimation Using a Force Platform and Inertial Sensors for Balance Evaluation in Quiet Standing. *Sensors* **2023**, *23*, 4933.
https://doi.org/10.3390/s23104933

**AMA Style**

Sonobe M, Inoue Y.
Center of Mass Estimation Using a Force Platform and Inertial Sensors for Balance Evaluation in Quiet Standing. *Sensors*. 2023; 23(10):4933.
https://doi.org/10.3390/s23104933

**Chicago/Turabian Style**

Sonobe, Motomichi, and Yoshio Inoue.
2023. "Center of Mass Estimation Using a Force Platform and Inertial Sensors for Balance Evaluation in Quiet Standing" *Sensors* 23, no. 10: 4933.
https://doi.org/10.3390/s23104933