# On Inertial Body Tracking in the Presence of Model Calibration Errors

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Sensor Fusion Methods

#### 1.2. Calibration Methods

#### 1.3. Contributions

- The development of two EKF-based methods with different state-space models, which use the free-segments representation, inspired by [17]. These are subsequently denoted Quattracker IMU and Quattracker segment. Here, rotations are represented through unit quaternions.
- The development of an online capable version of the optimization-based method in [19], based on sliding window optimization. The method is subsequently denoted Optitracker.
- A performance comparison of the new/adapted methods with an existing EKF-based method that uses the kinematic chain representation and DH coordinates to represent joint angles [16]. Performance is measured in terms of angular error statistics on complex (i.e., simultaneous variations in all joint DoFs) moderate and fast human motion (real and simulated) and on artificially simulated complex motion from a case study. In particular, the influence of the selected model calibration errors, i.e., I2S calibration and segment length errors, on the performances of the different methods and their dependence on magnetometer usage are assessed.

## 2. Materials and Methods

#### 2.1. Notation

#### 2.2. Biomechanical Model Representations

#### 2.3. EKF-Based Methods

#### 2.3.1. Measurement Models

#### 2.3.2. State Spaces

#### 2.3.3. Dynamic Models

#### 2.3.4. Constraints

#### 2.3.5. Initialization

#### 2.4. Sliding Window Optimization

#### 2.4.1. Biomechanical Constraints

#### 2.4.2. Initialization

#### 2.5. Summary and Overview

#### 2.6. Evaluation Setup

#### 2.6.1. Real Data Scenario

#### 2.6.2. Real Data Scenario: Discussion of Major Error Sources

#### 2.6.3. Simulation Scenario with Systematically Introduced Model Calibration Errors

- I2S position errors: $\Delta {p}_{I2S}^{along}\in [-0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m},0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}]$, i.e., position changes along the segment axis.
- I2S position errors: $\Delta {p}_{I2S}^{out}\in [-0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m},0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}]$, i.e., position changes perpendicular to the segment axis.
- Segment length variations: $\parallel \Delta {S}_{1}\parallel \in [-0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m},0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}]$.
- I2S orientation errors: $\Delta {q}_{I2S}^{along}\in [-{30}^{\circ},{30}^{\circ}]$ along the bone, i.e., rotations around the segment axis associated to the IMU.
- I2S orientation errors: $\Delta {q}_{I2S}^{out}\in [-{30}^{\circ},{30}^{\circ}]$ out of bone, i.e., rotations around the IMU axis perpendicular to the surface of the associated segment.

#### 2.6.4. Error Measures

## 3. Results

#### 3.1. Tracking Performances on Real Data

#### 3.2. Tracking Performances on Simulated Data with Model Calibration Errors

#### 3.3. Tracking Performances on Simulated Data without Calibration Errors

## 4. Discussion and Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DH | Denavit Hartenberg |

DoF(s) | Degree(s) of Freedom |

EKF | Extended Kalman Filter |

UKF | Unscented Kalman Filter |

IMU | Inertial Measurement Unit |

I2S | IMU-to-Segment |

WLS | Weighted Least Squares |

## Appendix A

## Appendix B

**Claim 1.**

**Proof of Claim 1.**

## Appendix C

## Appendix D

**Table D1.**Noise covariances used by all sensor fusion methods in all tests (cf. Table 1).

Chaintracker | Quattracker segment|IMU | Optitracker | |||
---|---|---|---|---|---|

${\Sigma}^{\ddot{\widehat{\theta}}}$ | $3e+6$ | ${\Sigma}^{\widehat{\omega}}$ | $1e+4\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{cq}$ | ${I}_{3\times 3}$ |

${\Sigma}^{a}$ | $0.8\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{\dot{\widehat{\omega}}}$ | $2e+4\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{cp}$ | ${I}_{3\times 3}$ |

${\Sigma}^{\omega}$ | $0.03\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{\widehat{p}}$ | $3e+6\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{\widehat{p}}$ | ${I}_{3\times 3}$ |

${\Sigma}^{m}$ | $0.01$ | ${\Sigma}^{a}$ | $0.8\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{\dot{\widehat{p}}}$ | ${I}_{3\times 3}$ |

${\Sigma}^{\omega}$ | $0.03\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{\widehat{q}}$ | ${I}_{3\times 3}$ | ||

${\Sigma}^{m}$ | $0.01$ | ${\Sigma}^{G}$ | ${I}_{3\times 3}$ | ||

${\Sigma}^{p}$ | $1e-7\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{m}$ | $0.001\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ||

${\Sigma}^{G}$ | $1e-7\phantom{\rule{3.33333pt}{0ex}}{I}_{3\times 3}$ | ${\Sigma}^{q0}$ | ${I}_{3\times 3}$ | ||

${\Sigma}^{p}$ | ${I}_{3\times 3}$ |

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**Figure 1.**Two different biomechanical model representations. Note the additional world coordinate system in the kinematic chain model.

**Figure 2.**Capturing setup for the real data scenario. In the picture on the left, the segment coordinate systems are associated to the proximal ends of the segments. Note, the axes are orthogonal and only roughly aligned with the anatomical axes of rotation through the skeleton fitting of the optical system as described in Section 2.6.1. Precise alignment with the anatomical axes was not in the focus of this study. In the N-pose, for the right arm, the x-axes are chosen perpendicular to the frontal plane pointing anterior, the y-axes are perpendicular to the transverse plane pointing along the segments in the direction from the distal to the proximal ends and the z-axes are perpendicular to the sagittal plane pointing lateral. The picture also indicates the initial arm configuration for real-slow and real-fast.

**Figure 3.**Real data scenario: Euler angle sequences ($z,{x}^{\prime},{y}^{\u2033}$ convention) and ranges of motion, $[\text{minimum}\phantom{\rule{4.pt}{0ex}}\text{angle},\text{maximum}\phantom{\rule{4.pt}{0ex}}\text{angle}]$ (each provided in degree), of real-slow (

**a**–

**c**) and real-fast (

**d**–

**f**). The segment axes and initial segment orientations are as shown in Figure 2. Note, the shoulder angles (left column) are represented w.r.t. to the initial upper arm configuration ${q}_{0,0}^{GS}$, rather than w.r.t. the global frame, in order to cancel out the unknown heading offset for easier interpretation.

**Figure 4.**Simulation scenario: angle sequence applied to each rotational DoF of the three segment kinematic chain model (cf. Table 5) used for simulating sim-fast-artificial.

**Figure 5.**Simulation scenario: Per segment mean angular error distributions on sim-fast for along-bone and out-of-bone I2S orientation calibration errors (cf. Section 2.6.3).

**Figure 6.**Simulation scenario: Per segment mean angular error distributions on sim-fast for along-bone and out-of-bone I2S position calibration errors (cf. Section 2.6.3).

**Figure 7.**Simulation scenario: The upper row shows the per segment mean angular error distributions on sim-fast for segment length errors. The lower row shows the errors w/o magnetometers splitted into yaw and pitch/roll errors.

**Table 1.**Characteristics of the different sensor fusion methods: n denotes the number of segments and w is the window size used by the Optitracker. All tuning parameters are given in Appendix D. Note, ${\theta}_{i}$ represent the joint angles estimated by the Chaintracker and ${\Sigma}^{\ddot{\theta}}$ refers to the process noise covariances used in the dynamic model [16].

Chaintracker (cf. [ 16]) | Quattracker segment | Quattracker IMU | Optitracker | |
---|---|---|---|---|

Estimation method | EKF | EKF | EKF | WLS |

State | ${\left({\{{\theta}_{i},{\dot{\theta}}_{i},{\ddot{\theta}}_{i}\}}_{i=0}^{n-1}\right)}_{t}^{T}$ | ${\left({\{{S}_{i,t}^{G},{\dot{S}}_{i,t}^{G}{\ddot{S}}_{i,t}^{G},{q}_{i,t}^{GS},{\omega}_{S,i,t}^{GS},{\dot{\omega}}_{S,i,t}^{GS}\}}_{i=0}^{n-1}\right)}^{T}$ | ${\left({\{{I}_{i,t}^{G},{\dot{I}}_{i,t}^{G}{\ddot{I}}_{i,t}^{G},{q}_{i,t}^{GI},{\omega}_{I,i,t}^{GI}\}}_{i=0}^{n-1}\right)}^{T}$ | ${\left({\{{\{{S}_{i,t}^{G},{q}_{i,t}^{GS},{I}_{i,t}^{G},{\dot{I}}_{i,t}^{G},{q}_{i,t}^{GI}\}}_{i=0}^{n-1}\}}_{t=0}^{w-1}\right)}^{T}$ |

Dimensions (state s, meas. vector k) | $x\in {\mathbb{R}}^{9n}$, $s=9n$, $k=7$ | $x\in {\mathbb{R}}^{19n}$, $s=19n$, $k=7$ | $x\in {\mathbb{R}}^{16n}$, $s=16n$, $k=7$ | $x\in {\mathbb{R}}^{17n\times w}$, $s=17nw$ |

Motion model | 1D const angular acc | 3D const angular & linear acc | 3D const angular vel; 3D const linear accel | IMU control input |

Tuning parameters | ${\Sigma}^{\ddot{\widehat{\theta}}},{\Sigma}^{a},{\Sigma}^{\omega},{\Sigma}^{m}$ | ${\Sigma}^{\dot{\widehat{\omega}}},{\Sigma}^{\widehat{p}},{\Sigma}^{a},{\Sigma}^{\omega},{\Sigma}^{m},{\Sigma}^{p},{\Sigma}^{G}$ | ${\Sigma}^{\widehat{\omega}},{\Sigma}^{\widehat{p}},{\Sigma}^{a},{\Sigma}^{\omega},{\Sigma}^{m},{\Sigma}^{p},{\Sigma}^{G}$ | ${\Sigma}^{cq}$, ${\Sigma}^{cp}$, ${\Sigma}^{\widehat{p}}$, ${\Sigma}^{\dot{\widehat{p}}}$, ${\Sigma}^{\widehat{q}}$, ${\Sigma}^{G},{\Sigma}^{m},{\Sigma}^{q0},{\Sigma}^{p}$ |

Complexity | $\mathcal{O}({k}^{2.4}+{s}^{2})$ [49] | $\mathcal{O}({k}^{2.4}+{s}^{2})$ | $\mathcal{O}({k}^{2.4}+{s}^{2})$ | $\mathcal{O}({s}^{3})$ (Gauss Newton method) |

Biomech. model | chain | free segments | free segments | free segments |

State coordinate system | segment centered | segment centered | IMU centered | IMU and segment centered |

**Table 2.**Measured/simulated instantaneous peak acceleration (Acc) and angular velocity (Gyr) 2-norms for real-slow, real-fast, sim-slow and sim-fast.

Mode | Sequence → | Slow | Fast | ||
---|---|---|---|---|---|

Sensor → | Acc(m/s${}^{2}$) | Gyr (${}^{\circ}/\mathrm{s}$) | Acc (m/s${}^{2}$) | Gyr (${}^{\circ}/\mathrm{s}$) | |

Real | ${I}_{0}$ | $13.44$ | $178.68$ | $26.88$ | $457.05$ |

${I}_{1}$ | $16.87$ | $394.68$ | $75.24$ | $957.29$ | |

${I}_{2}$ | $18.74$ | $468.55$ | $95.45$ | $1031.31$ | |

Re-simulated | ${I}_{0}$ | $18.05$ | $187.07$ | $25.70$ | $429.75$ |

${I}_{1}$ | $22.07$ | $395.27$ | $77.20$ | $968.35$ | |

${I}_{2}$ | $22.86$ | $440.23$ | $100.55$ | $1030.76$ |

**Table 3.**Mean (std,max) angular residual errors (cf. Equation (17)) for the hand-eye calibrations of each inertial measurement unit (IMU) ${I}_{i}$ as calculated on the data sequence used for calibration.

IMU | Residual Error |
---|---|

${I}_{0}$ | $1.{14}^{\circ}(0.{57}^{\circ},\phantom{\rule{0.166667em}{0ex}}3.{89}^{\circ})$ |

${I}_{1}$ | $2.{28}^{\circ}(0.{90}^{\circ},\phantom{\rule{0.166667em}{0ex}}6.{10}^{\circ})$ |

${I}_{2}$ | $1.{51}^{\circ}(0.{77}^{\circ},\phantom{\rule{0.166667em}{0ex}}5.{07}^{\circ})$ |

**Table 4.**Mean (std) of magnetic field vector 2-norms (upper values) and global angular deviations (lower values) for each IMU as calculated from the real data sequences.

${I}_{0}$ | ${I}_{1}$ | ${I}_{2}$ | |
---|---|---|---|

real-slow | $0.92\phantom{\rule{3.33333pt}{0ex}}(0.00)$ | $0.90\phantom{\rule{3.33333pt}{0ex}}(0.01)$ | $0.90\phantom{\rule{3.33333pt}{0ex}}(0.02)$ |

$1.{79}^{\circ}(1.{18}^{\circ})$ | $3.{66}^{\circ}(2.{41}^{\circ})$ | $4.{36}^{\circ}(2.{89}^{\circ})$ | |

real-fast | $0.92\phantom{\rule{3.33333pt}{0ex}}(0.00)$ | $0.91\phantom{\rule{3.33333pt}{0ex}}(0.01)$ | $0.91\phantom{\rule{3.33333pt}{0ex}}(0.02)$ |

$2.{63}^{\circ}(2.{38}^{\circ})$ | $4.{45}^{\circ}(5.{60}^{\circ})$ | $4.{59}^{\circ}(6.{04}^{\circ})$ |

**Table 5.**Denavit-Hartenberg (DH) coordinates for the three segment kinematic chain model used for simulating the sim-fast-artificial data sequence. The angles ${\alpha}_{[0:5]}(t)$ and ${\theta}_{[0:2]}(t)$ are the Degrees of Freedom (DoFs) that are controlled via Equation (20). The Inertial Measurement Unit (IMU)-to-Segment (I2S) positions are given by translations along the bone, in z-direction relative to the segment origins (i.e., DH$(z,0,0,0)$). The initial chain configuration (pointing up opposite gravity) is illustrated on the right.

Segment (${S}_{i}$) | d | a | ${\alpha}_{j}(t)$ | ${\theta}_{j}(t)$ | IMU | Image |
---|---|---|---|---|---|---|

${S}_{0}$ | 0 | 0 | ${\alpha}_{0}(t)$ | $-\frac{pi}{2}$ | None | |

0 | 0 | ${\alpha}_{1}(t)$ | $\frac{pi}{2}$ | None | ||

0 | 0 | 0 | ${\theta}_{0}(t)$ | None | ||

$0.4$ | 0 | 0 | 0 | $z=0.3$ | ||

${S}_{1}$ | 0 | 0 | ${\alpha}_{2}(t)$ | $-\frac{pi}{2}$ | None | |

0 | 0 | ${\alpha}_{3}(t)$ | $\frac{pi}{2}$ | None | ||

0 | 0 | 0 | ${\theta}_{1}(t)$ | None | ||

$0.4$ | 0 | 0 | 0 | $z=0.3$ | ||

${S}_{2}$ | 0 | 0 | ${\alpha}_{4}(t)$ | $-\frac{pi}{2}$ | None | |

0 | 0 | ${\alpha}_{5}(t)$ | $\frac{pi}{2}$ | None | ||

0 | 0 | 0 | ${\theta}_{2}(t)$ | None | ||

$0.2$ | 0 | 0 | 0 | $z=0.1$ |

**Table 6.**Peak acceleration (Acc) and angular velocity (Gyr) 2-norms for sim-fast-artificial. For all sensors, the values vary smoothly between 0 and the peak values shown in the table.

Sequence → | sim-fast-artificial | |
---|---|---|

Sensor → | Acc (m/s${}^{2}$) | Gyr (${}^{\circ}/\mathbf{s}$) |

${I}_{0}$ | $14.03$ | $356.90$ |

${I}_{1}$ | $38.16$ | $705.20$ |

${I}_{2}$ | $61.90$ | $1047.99$ |

**Table 7.**Real data scenario: mean (std; max) angular errors for each segment. Note, the color represents a linear interpolation of the mean error over all segments between red (maximum error) and green (minimum error). This helps comparing the performances of the different sensor fusion methods. Also note, w/mag refers to using the real magnetometer measurements, w/sim. mag refers to using the simulated magnetometer measurements (cf. Section 2.6.2) and w/o mag refers to dropping the magnetometer information.

**Table 8.**Simulation scenario: the normalized range error ${E}_{quot}^{tracker}$ (cf. Equation (23)) is shown for the different simulated model calibration errors and sensor fusion methods. The latter have the following shortcuts: Chaintracker (chain), Quattracker segment (Seg. q.), Quattracker IMU (IMU q.), Optitracker (opti.). Note, for each table separately, the color represents a linear interpolation of the error from red (maximum error) to green (minimum error).

**Table 9.**Simulation scenario without calibration errors (sim-fast): mean (std; max) angular errors over all segments on the two test configurations (w/mag, w/o mag), both on noise-free (perfect) and noisy IMU data.

Method | Chaintracker | Quattracker segment | Quattracker IMU | Optitracker |
---|---|---|---|---|

Noise-free w/mag | 1.42${}^{\circ}$ (1.40${}^{\circ}$; 8.04${}^{\circ}$) | 1.19${}^{\circ}$ (1.23${}^{\circ}$; 6.83${}^{\circ}$) | 0.66${}^{\circ}$ (0.64${}^{\circ}$; 3.30${}^{\circ}$) | 0.01${}^{\circ}$ (0.01${}^{\circ}$; 0.06${}^{\circ}$) |

Noise-free w/o mag | 3.50${}^{\circ}$ (2.57${}^{\circ}$; 9.45${}^{\circ}$) | 1.57${}^{\circ}$ (1.45${}^{\circ}$; 7.52${}^{\circ}$) | 0.97${}^{\circ}$ (0.65${}^{\circ}$; 3.36${}^{\circ}$) | 0.01${}^{\circ}$ (0.01${}^{\circ}$; 0.06${}^{\circ}$) |

Noise w/mag | 1.46${}^{\circ}$ (1.39${}^{\circ}$; 8.09${}^{\circ}$) | 1.22${}^{\circ}$ (1.21${}^{\circ}$; 6.83${}^{\circ}$) | 0.69${}^{\circ}$ (0.62${}^{\circ}$; 3.28${}^{\circ}$) | 0.40${}^{\circ}$ (0.05${}^{\circ}$; 0.49${}^{\circ}$) |

Noise, w/o mag | 3.73${}^{\circ}$ (2.68${}^{\circ}$; 9.76${}^{\circ}$) | 1.55${}^{\circ}$ (1.40${}^{\circ}$; 7.42${}^{\circ}$) | 0.95${}^{\circ}$ (0.65${}^{\circ}$; 3.31${}^{\circ}$) | 0.40${}^{\circ}$ (0.05${}^{\circ}$; 0.49${}^{\circ}$) |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Miezal, M.; Taetz, B.; Bleser, G. On Inertial Body Tracking in the Presence of Model Calibration Errors. *Sensors* **2016**, *16*, 1132.
https://doi.org/10.3390/s16071132

**AMA Style**

Miezal M, Taetz B, Bleser G. On Inertial Body Tracking in the Presence of Model Calibration Errors. *Sensors*. 2016; 16(7):1132.
https://doi.org/10.3390/s16071132

**Chicago/Turabian Style**

Miezal, Markus, Bertram Taetz, and Gabriele Bleser. 2016. "On Inertial Body Tracking in the Presence of Model Calibration Errors" *Sensors* 16, no. 7: 1132.
https://doi.org/10.3390/s16071132