# Impact-Aware Foot Motion Reconstruction and Ramp/Stair Detection Using One Foot-Mounted Inertial Measurement Unit

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## Abstract

**:**

## 1. Introduction

- (1)
- Propose a model to estimate acceleration bias and heel strike velocity error.
- (2)
- Propose a novel method to detect ramp and stair locomotion (vs. level walking) using only one foot-mounted IMU.
- (3)
- Propose an enhanced integration reconstruction method with improved accuracy in the vertical direction.

## 2. Related Work

## 3. Kinematic Model

#### 3.1. Linear and Impulsive Zero-Velocity Correction

#### 3.2. Zero Height Change Assumption

#### 3.3. Matrix Model

## 4. Terrain Determination

- (1)
- Strides on ramps have larger forward distance and less height change than strides on stairs.
- (2)
- Strides on ramps or strides on stairs are usually consecutive.

Algorithm 1: Terrain Determination. |

Input: acceleration data, orientation data, ZUPT index, heel strike index.for each stride j do:$\mathrm{double}\text{}\mathrm{integrate}\text{}\mathrm{world}-\mathrm{frame}\text{}\mathrm{acceleration}\text{}\mathrm{to}\text{}\mathrm{obtain}\text{}{v}_{residual}$$\text{}\mathrm{and}\text{}{h}_{residual}$ if$\text{}\mathrm{no}\text{}{k}_{hs}$ do$\mathrm{assign}\text{}\mathrm{an}\text{}\mathrm{average}\text{}\mathrm{value}\text{}\mathrm{to}\text{}{k}_{hs}$ end ifsolve Equation (27) to obtain ${\left[{b}_{x}\left(j\right),{b}_{y}\left(j\right),{b}_{z}\left(j\right),{\u2206\mathrm{v}}_{\mathrm{h}\mathrm{s},\mathrm{z}}\left(j\right)\right]}^{T}$ end forobtain mean $\u2206{v}_{mean}=$$\text{}\mathrm{mean}\text{}\mathrm{of}\text{}2\mathrm{nd}\text{}\mathrm{and}\text{}3\mathrm{rd}\text{}\mathrm{quartile}\text{}\mathrm{of}\text{}\u2206{v}_{hs,z}$ obtain mean ${b}_{mean}=$$\text{}\mathrm{mean}\text{}\mathrm{of}\text{}2\mathrm{nd}\text{}\mathrm{and}\text{}3\mathrm{rd}\text{}\mathrm{quartile}\text{}\mathrm{of}\text{}b$ for each stride j do:if $\left|{\u2206v}_{hs,z}\left(j\right)-\u2206{v}_{mean}\right|>$ V_THRESHOLD_RAMP doRampStairIndicator(j) = RAMP end ifif $\left|b\left(j\right)-{b}_{mean}\right|>$$\text{}\mathrm{B}\_\mathrm{THRESHOLD}\_\mathrm{STAIR}\text{}\mathrm{or}\text{}\left|{\u2206v}_{hs,z}\left(j\right)-\u2206{v}_{mean}\right|$ V_THRESHOLD_STAIR doRampStairIndicator(j) = STAIR end if$\mathrm{direction}=\mathrm{sign}\left({\u2206v}_{hs,z}\right(j)$) end forfor strides that are ramps/stairs doclean wrong determination by forward distance and height change criterion clean wrong determination by consecutive ramps/stairs criterion end if |

## 5. Reconstruction Method

#### 5.1. Preprocessing

#### 5.2. Reconstruction

Algorithm 2: Trajectory Reconstruction. |

Input: acceleration data, orientation data, ZUPT index, heel strike index.Use Algorithm 1 to detect all ramps and stairs. $\mathrm{Convert}\text{}\mathrm{raw}\text{}\mathrm{acceleration}\text{}\mathrm{data}\text{}\mathrm{to}\text{}\mathrm{global}\text{}\mathrm{acceleration}\text{}\mathrm{data}\text{}accel\left(k\right)$ for each stride j do$\mathrm{double}\text{}\mathrm{integrate}\text{}\mathrm{the}\text{}\mathrm{raw}\text{}\mathrm{acceleration}\text{}\mathrm{to}\text{}\mathrm{obtain}\text{}{v}_{residual}$$\text{}\mathrm{and}\text{}{h}_{residual}$ $\mathrm{scale}\text{}{v}_{residual}$ with parameter C if RampStairIndicator(j) = false dosolve Equation (27) to obtain ${\left[{b}_{x}\left(j\right),{b}_{y}\left(j\right),{b}_{z}\left(j\right),\u2206{v}_{hs,z}\left(j\right)\right]}^{T}$ else dosolve Equation (15) to obtain ${\left[{b}_{x}\left(j\right),{b}_{y}\left(j\right),{b}_{z}\left(j\right)\right]}^{T}$ $\mathrm{set}\text{}\u2206{v}_{hs,z}=0$ end if$accel\left(1..N\right)$ = $accel\left(1..N\right)$ − ${\left[{b}_{x}\left(j\right),{b}_{y}\left(j\right),{b}_{z}\left(ij\right)\right]}^{T}$ for k = 1:N do$v\left(k\right)=v\left(k-1\right)+accel\left(k\right)\ast \u2206t$ if$k={k}_{hs}$ do$v\left(k\right)=v\left(k\right)+\u2206{v}_{hs,z}\left(j\right)$ end if$p\left(k\right)=p\left(k\right)+p\left(k\right)\ast \u2206t+\frac{1}{2}accel\left(k\right){\u2206t}^{2}$ end forend for |

## 6. Experiments

#### 6.1. Results: Terrain Determination

#### 6.2. Reduced Height Error

#### 6.3. Accuracy

## 7. Discussion

#### 7.1. Implementation of Terrain Determination

#### 7.2. Physically Meaningful De-Drifting Leads to Improved Accuracy

#### 7.3. Exploring and Quantifying Heel Strike Error

#### 7.4. Multiple Meanings of Acceleration Bias

#### 7.5. Limitations

## 8. Future directions

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Reconstruction Results with Orientation Computed Using a Non-KF Method

**Figure A1.**Total height error and mean height error per strides from reconstruction without Kalman Filter-based orientation estimation (t-test values: * p < 0.05; ** p < 0.01; *** p < 0.001).

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**Figure 1.**Three-axis velocity and horizontal trajectory in the sagittal plane ($x$ forward, $z$ up), with the instant of heel strike noted as the dashed line and circle. The proposed method de-drifts the residual non-zero velocity and non-zero height change by estimating acceleration bias and HS vertical velocity error.

**Figure 2.**Acceleration bias and HS vertical velocity error among all strides in an example trial. All large deviations in both figures belong to ramps/stairs strides. Additionally, the direction of deviation of HS vertical velocity error indicates the ascent/descent direction on ramps and stairs.

**Figure 3.**The example reconstruction results from one trial with 3 long stairways, 2 short stairways, and 4 ramps. The whole path includes a 0.6 m ramp with 3.7-degree inclination angle, 4.5 m stairs to the upper floor, and short 0.6 m stairs. The horizontal trajectory is plotted on the true floor plan of the Mechanical Engineering Building 3rd floor. The Height vs. Time figure labels 3 terrains with unique colors. All the strides on the turn between long stairs ascent and descent are correctly detected as level ground.

**Figure 4.**Histogram of acceleration bias (magnitude of $b$) and heel strike vertical velocity error (magnitude of the $\u2206{v}_{hs,z}$) among all 15 trials, including 4308 strides on level ground, 433 strides on ramps and 1268 strides on stairs. The histogram is normalized so that the probability of strides sums up to 1 for each terrain type, and the mean of the 2nd and 3rd quartiles of $\u2206{v}_{hs,z}$ is subtracted to remove base effect on level ground. A clear separation is shown on HS vertical velocity error between level ground vs. non-level ground strides.

**Figure 5.**Height trajectory and error analysis of the example reconstruction result. Left: height–time plots reconstructed using three methods: ESKF, ZUPT, and proposed. Right: Total height error and mean height error per strides on 3 terrains. The proposed method removes 99% of the error on level ground while achieving similar performance on ramps and stairs.

**Figure 6.**Overall error analysis for all 15 trials. The proposed method removes 99% of the error on level ground while achieving slightly improved performance on ramps and stairs (t-test values: * p < 0.05; ** p < 0.01; *** p < 0.001).

**Figure 7.**Stride lengths and stride heights of proposed method compared with the ZUPT method and ESKF method, in one example trial. (

**Left**): sagittal view of reconstructed strides for level walking using each method. (

**Center-left**): distribution of stride length. (

**Center-right**): distribution of maximum IMU height during each stride. (

**Right**): sagittal view of reconstructed strides for ramps (green) and stairs (purple), dash line indicating the kinematic boundary for separating ramps vs. stairs by trajectory. Further separation is achieved by the consecutive ramps/stairs criterion.

**Figure 8.**Reconstruction of an example stride with large heel strike impact, comparing trajectory reconstructed from Raw integration, traditional ZUPT, ZUPT combined with linear height de-drifting (modeled as a constant bias in velocity z), and the Proposed method. (

**Top left**) plot is the raw acceleration data in the IMU local frame. (

**Top right**) plot is the vertical velocity in the world frame. (

**Bottom plot**) is the IMU trajectory in the sagittal plane ($x$ forward, $z$ up), with the instance of heel strike noted as a dashed line and circle. Note the flaws in the comparison methods: Raw integration leaves drift during the following stance phase; traditional ZUPT generates substantial height error; and ZUPT with linear height de-drifting yields negative height at the beginning of the movement and reduced height during forward swing.

**Figure 9.**The magnitude of heel strike is quantified by the maximum peak acceleration magnitude and acceleration energy percentage above 40 Hz in accelerometer signals. (

**Top**): median peak and energy percentage with respect to mean height error on stairs for all 15 trials. As an observation, the mean height error on stairs increases as both peak acceleration and energy percentage increases. (

**Bottom**): peak and energy percentage for all strides on level ground and stairs. The distribution of both acceleration peak and energy percentage differs on the two terrains.

**Figure 10.**The change of height from foot strike to foot-flat phase vs. pitch angle change from foot strike to foot-flat. The angular motion of the foot and the linear motion of the IMU should be related through the kinematics of the foot lever. Both positive values of pitch angle (typical heel strike) and negative values (forefoot strike, mainly from down-stairs strides) show evidence of the expected linear scaling.

Terrain Types | Predicted | |||
---|---|---|---|---|

Level Ground | Ramps | Stairs | ||

Ground Truth | Level ground | 4290 | 1 | 0 |

Ramps | 18 ^{1} | 432 | 0 | |

Stairs | 0 | 0 | 1268 |

^{1}Seventeen of eighteen strides are transition strides on the edge of ramps.

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Fehr, K.H.; Adamczyk, P.G.
Impact-Aware Foot Motion Reconstruction and Ramp/Stair Detection Using One Foot-Mounted Inertial Measurement Unit. *Sensors* **2024**, *24*, 1480.
https://doi.org/10.3390/s24051480

**AMA Style**

Wang Y, Fehr KH, Adamczyk PG.
Impact-Aware Foot Motion Reconstruction and Ramp/Stair Detection Using One Foot-Mounted Inertial Measurement Unit. *Sensors*. 2024; 24(5):1480.
https://doi.org/10.3390/s24051480

**Chicago/Turabian Style**

Wang, Yisen, Katherine H. Fehr, and Peter G. Adamczyk.
2024. "Impact-Aware Foot Motion Reconstruction and Ramp/Stair Detection Using One Foot-Mounted Inertial Measurement Unit" *Sensors* 24, no. 5: 1480.
https://doi.org/10.3390/s24051480