# A Kalman Filter Approach for Estimating Tendon Wave Speed from Skin-Mounted Accelerometers

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Shear Wave Tensiometer

#### 2.2. Wave Travel Times

#### 2.3. Kalman Filter

#### 2.4. Shear Wave Tensiometry Simulations

#### 2.5. Noise Covariance

#### 2.6. Tensiometry Simulations during Gait

#### 2.7. Experimental Protocol

## 3. Results

#### 3.1. Sensor Noise

#### 3.2. Sensor Position

#### 3.3. In Vivo Tensiometry

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A shear wave tensiometer fundamentally consists of an impulsive excitation which induces a shear wave in a tendon or other tissue. The wave travels longitudinally along the tissue. The wave is detected by an array of accelerometers that are adhered to the skin over a tendon. (

**b**) An example of the wave measured at three accelerometers. Relevant times are defined for each accelerometer and pair: the variable $T$ is the arrival time of the wave at each accelerometer, $\tau $ denotes the wave travel time between consecutive accelerometers, and $\Delta T$ indicates the change in wave arrival time between consecutive excitations at a single accelerometer.

**Figure 2.**(

**a**) An example of the Kalman filter measurements, inputs, and outputs for a tensiometer with 3 accelerometers. The wave travel time (${\tau}_{ij}$) between successive pairs of equally spaced accelerometers was generally similar, with slight differences due to noise in the system. The change in wave arrival ($\Delta {T}_{i}$) at an accelerometer between tap events was centered around zero, with larger changes in arrival for accelerometers further from the tapper. (

**b**) The estimated arrival time of the wave at each accelerometer (${T}_{i}$) as computed by the Kalman filter.

**Figure 3.**(

**a**) Flowchart depicting the iterative Kalman algorithm used to compute wave arrival times at each accelerometer location. (

**b**) A weighted least squares regression was used to determine the best wave speed based on the Kalman arrival times. The weights of each arrival time were based on a secondary output of the Kalman filter—the estimated state uncertainty.

**Figure 4.**Depiction of the model used to numerically simulate tensiometer data. The tapper-induced wave was represented by the impulse response of an underdamped mass-spring-damper system. The propagating shear wave (traveling with wave speed $c$) was detected by skin-mounted accelerometers. Sensor noise, represented by band-pass filtered white noise process, was introduced to generate the simulated accelerometer data.

**Figure 5.**The variance of $dt$ ($\Delta T$ or $\tau $ in application) with respect to the mean correlation coefficient $r$ is well-approximated with a quadratic fit.

**Figure 6.**The participant walked (3.0 mph) and ran (7.0 mph) on a treadmill. The shear wave tensiometer was placed over the right Achilles tendon. The tensiometer consisted of a tapper driven by an electrodynamic surface transducer (SparkFun Electronics, Niwot, CO, USA) and two miniature accelerometers (PCB Piezotronics, Depew, NY, USA) affixed in a silicone holder.

**Figure 7.**Both the Kalman filter and the use of additional accelerometers reduced errors in wave speed estimates for signal-to-noise ratios (SNR) tested. Plotted is the mean coefficient of variation (CoV—standard deviation of error normalized to prescribed wave speed) for simulated tensiometer data over a walking gait cycle. SNR represents the signal to noise ratio of the simulated accelerometer signals.

**Figure 8.**The coefficient of variation (CoV) at peak wave speed was reduced dramatically by using a Kalman filter to process simulated tensiometry data with two accelerometers. Increasing the number of accelerometers further reduced the errors in wave speed estimates. The data shown were from simulated gait cycles with a signal to noise ratio (SNR) of 4.

**Figure 9.**The Kalman filter had no effect on the mean percent error when accelerometer position was randomly varied. However, increasing the number of accelerometers in the array did substantially reduce the wave speed estimate errors.

**Figure 10.**(

**a**) Raw tensiometer data (two accelerometers) from the Achilles tendon for an example stride of walking and running. The amplitude of the measured accelerations varies in a regular patter with fluctuations of the muscle state and ankle angle throughout the gait cycle. (

**b**) Stride-average estimates of the wave travel time ($\tau $) between accelerometers. (

**c**) Stride averaged variations in the change in wave arrival ($\Delta T$) for each accelerometer during walking and running. The line colors in (

**b**,

**c**) indicate the magnitude of the correlation coefficient (r) at that point in the gait cycle. Note that the cross-correlation template for event $k$ was defined using a 2 ms window centered at the arrival time $T$ computed for the previous Kalman iteration $k-1$.

**Figure 11.**Walking and running data for one subject processed with only the cross-correlation method (blue) and then with the Kalman filter method (green), with one standard deviation above and below highlighted. Application of the Kalman filter resulted in overall lower measured variability both between strides and across the stance and swing phases of gait.

Amplitude of measurement | 20 m/s^{2} |

Sample rate | 50,000 Hz |

Accelerometer locations(relative to excitation) | [15, 25, 35, 45] mm |

Damping ratio ^{1} | 0.5 (0.05) |

Natural frequency of oscillation ^{1} | 1600 (100) Hz |

^{1}Mean (Standard Deviation).

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**MDPI and ACS Style**

Schmitz, D.G.; Thelen, D.G.; Cone, S.G.
A Kalman Filter Approach for Estimating Tendon Wave Speed from Skin-Mounted Accelerometers. *Sensors* **2022**, *22*, 2283.
https://doi.org/10.3390/s22062283

**AMA Style**

Schmitz DG, Thelen DG, Cone SG.
A Kalman Filter Approach for Estimating Tendon Wave Speed from Skin-Mounted Accelerometers. *Sensors*. 2022; 22(6):2283.
https://doi.org/10.3390/s22062283

**Chicago/Turabian Style**

Schmitz, Dylan G., Darryl G. Thelen, and Stephanie G. Cone.
2022. "A Kalman Filter Approach for Estimating Tendon Wave Speed from Skin-Mounted Accelerometers" *Sensors* 22, no. 6: 2283.
https://doi.org/10.3390/s22062283