Sprint diagnostic with GPS and inertial sensor fusion

Abstract

The purpose of this research was to develop a wearable, low-cost prototype based on real-time kinematic GPS and a microelectromechanical inertial measurement unit to measure the sprinting velocity of an athlete. The software package RTKLIB was used to calculate the RTK-GPS positions and different Kalman filters were implemented to provide a loosely coupled sensor fusion. With this setup, we performed empirical studies to determine whether the velocities obtained by this novel approach are sufficiently accurate for a performance orientated training. Therefore, field tests for 30- to 400-m sprint distance were conducted with simultaneous measurements with different reference systems, such as a laser device or timing gates. The evaluation revealed a correspondence between prototype and reference systems with distance and timing errors of $\pm \, 2\,\%$ and high correlations for the velocities (R = 0.996, P <0.001) for 68 % of the trials. However, for remaining 32 % of the trials no acceptable performance parameters could be obtained due to GPS problems. Overall, the developed prototype showed great potential and might allow closing the gap between the accuracy and flexibility of the established reference systems as soon as its susceptibility to GPS problems is lowered.

Appendix

This appendix provides the mathematical descriptions of the recursive Kalman filters KF2 and KF5, which are based on the alternating execution of the prediction (Eq. 1) and correction steps (Eq. 2).

$$\begin{aligned}&\mathrm Prediction \nonumber \\ \hat{x}_{t}^*&= A \cdot \hat{x}_{t-1} \nonumber \\ P_{t}^*&= A \cdot P_{t-1} \cdot A^T + Q, \end{aligned}$$

(1)

$$\begin{aligned}&\mathrm Correction \nonumber \\ K_t&= P_{t}^* \cdot H^T \cdot ( H \cdot P_{t}^* \cdot H^T + R)^{-1} \nonumber \\ \hat{x}_{t}&= \hat{x}_{t}^* + K_t \cdot (z_t - H \cdot \hat{x}_{t}^*) \nonumber \\ P_{t}&= P_{t}^* - K_t \cdot H \cdot P_{t}^*. \end{aligned}$$

(2)

The mainly used movement model is the constant-jerk model (CJ) with the state vector $\hat{x}$ (Eq. 3) and the dynamic matrix A (Eq. 4).

$$\begin{aligned} \hat{x}&= \begin{pmatrix} {\text{ d }isplacement} \ d \ in \ m\\ \mathrm{velocity} \ v \ in \ \frac{m}{s}\\ \mathrm{acceleration} \ a \ in \ \frac{m}{s^2}\\ \mathrm{jerk} \ j \ in \ \frac{m}{s^3}, \end{pmatrix} \end{aligned}$$

(3)

$$\begin{aligned} A&= \begin{pmatrix} 1 &{} \Delta t &{} \frac{\Delta t^2}{2} &{} \frac{\Delta t^3}{6} \\ 0 &{} 1 &{} \Delta t &{} \frac{\Delta t^2}{2} \\ 0 &{} 0 &{} 1 &{} \Delta t \\ 0 &{} 0 &{} 0 &{} 1. \end{pmatrix} \end{aligned}$$

(4)

Removing the jerk tate leads to the constant-acceleration model (CA). The uncertainty matrix P is initialised with the identity matrix and the measurement vector H is set to $H = (1~0~1~0)$.

1.1 KF2

For KF2, the CJ model is used, taking the horizontal displacement and the IMU’s forward acceleration as inputs. The measurement covariance matrix R (Eq. 5) depends on the current GPS-error $\sigma GPS$ (provided by the GPS receiver) and the IMU’s standard deviation. As the upper body rotation causes additional noise, it is increased from 0.01 $\frac{m}{s^2}$ to 0.1 $\frac{m}{s^2}$. The process–noise–variance matrix Q (Eq. 6) was determined experimentally below.

$$\begin{aligned} R&= \begin{pmatrix} \sigma GPS &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0.1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0, \end{pmatrix} \end{aligned}$$

(5)

$$\begin{aligned} Q&= \begin{pmatrix} 0.0001 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0.01 &{} 0 &{} 0\\ 0 &{} 0 &{} 0.01 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0.01. \end{pmatrix} \end{aligned}$$

(6)

1.2 KF5

KF5 is a combination of six Kalman filters. The first one utilizes a CA model with state vector $\hat{x}_Y$ (Eq. 7) to maintain a drift-free yaw estimation, using the yaw angle provided by the GPS and the yaw-angle velocity provided by the IMU as inputs. The measurement noise for the GPS heading and the process–covariance matrix $Q_Y$ (Eq. 9) had to be determined experimentally. However, the gyroscope has a stated variance of $0.1^\circ$/s with a bias of ± 2.5 $^\circ$. A selected measurement variance of $5^\circ$/s in $R_Y$ (Eq. 8) is, therefore, appropriate for around 2 h of measuring.

$$\begin{aligned} \hat{x}_Y&= \begin{pmatrix} yaw\\ \dot{yaw} \\ \ddot{yaw} \end{pmatrix}, \end{aligned}$$

(7)

$$\begin{aligned} R_Y&= \begin{pmatrix} 15 &{} 0 &{} 0 \\ 0 &{} 5 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}, \end{aligned}$$

(8)

$$\begin{aligned} Q_Y&= \begin{pmatrix} 0.005 &{} 0 &{} 0 \\ 0 &{} 0.005 &{} 0 \\ 0 &{} 0 &{} 0.05 \end{pmatrix}. \end{aligned}$$

(9)

The second Kalman filter utilizes again a CA model with state vector $\hat{x}_B$ (Eq. 11) to smoothen the current yaw bias of the IMU. The only input is the bias itself (Eq. 10), and $R_B$ (Eq. 12) and $Q_B$ (Eq. 13) were determined experimentally.

$$\begin{aligned} bias&= \hat{x}_{Y_1} - yaw_{IMU}\nonumber \\ \hat{x}_B&= \begin{pmatrix} bias\\ \dot{bias} \\ \ddot{bias} \end{pmatrix}, \end{aligned}$$

(10)

$$\begin{aligned} R_B&= \begin{pmatrix} 30 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}, \end{aligned}$$

(11)

$$\begin{aligned} Q_B&= G \cdot G_B^T \cdot \sigma _{qB}, \end{aligned}$$

(12)

$$\begin{aligned} G_B&= \begin{pmatrix} \frac{\Delta t^2}{2}&\Delta t&1 \end{pmatrix} \qquad \sigma _{qB} = 0.5. \end{aligned}$$

(13)

In a next step, the bias $\hat{x}_{B_1}$ is used in a quaternion rotation to nullify the IMU’s yaw drift and align the accelerations provided by the IMU with the north–east–up axis. With three further Kalman filters utilizing the CJ model, the GPS northing, easting and height are then separately fused with the corresponding acceleration measurement of the IMU. Therefore, the same process–noise–variance matrix Q is used as in KF2 (Eq. 6), the measurement covariance matrix R (Eq. 5) is adapted to the GPS-error on the specific axis and the IMU variance is lowered to the original value of 0.01 $\frac{m}{s^2}$. These three Kalman filters provide the displacement and velocity in every three directions, but not on the desired horizontal plane. Thus, one last Kalman filter with a CA model and the velocity as only input state is required in KF5. The covariance matrix $R_V$ (Eq. 14) and sigma $\sigma _{qV} = 0.05$ for $Q_V$ (Eq. 15) had to be determined experimentally and the horizontal velocity, as input parameter, is the combined velocity towards north and east (Eq. 16).

$$\begin{aligned} R_V&= \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 20 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}, \end{aligned}$$

(14)

$$\begin{aligned} G_V&= \begin{pmatrix} \frac{\Delta t^2}{2}&\Delta t&1 \end{pmatrix} \nonumber \\ Q_V&= G_V \cdot G_V^T \cdot \sigma _{qV} \end{aligned}$$

(15)

$$\begin{aligned} v_{Horizontal,t}&= \sqrt{v_{North,t}^2 + v_{East,t}^2}. \end{aligned}$$

(16)