^{1}

^{1}

^{*}

^{1}

^{2}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In human movement modeling, the problem of multi-link kinematics estimation by means of inertial measurement units has been investigated by several authors through efficient sensor fusion algorithms. In this perspective a single inertial measurement unit per link is required. This set-up is not cost-effective compared with a solution in which a single-axis accelerometer per link is used. In this paper, a novel fast technique is presented for the estimation of the sway angle in a multi-link chain by using a single-axis accelerometer per segment and by setting the boundary conditions through an

The inverted pendulum (IP) is a simple system that finds application in many disciplines of science. Despite its simple nature, it represents a non-linear, unstable and non-minimum phase system, finding several applications in control theory and biomechanics. Achieving stability of an IP has become a common engineering challenge for researchers and the problem has been discussed theoretically by several authors [

There are many examples of the IP model, both man made and found in the natural world. In control theory the challenge of control made the IP system a classic tool in control laboratories [

Arguably the most prevalent example of an IP is a human being. A standing human looks like an IP with the center of mass well above the ground [

In the last years, sensing hardware developments have made available on the market miniaturized inertial (accelerometers and/or gyroscopes) and magnetic sensors. These sensors have found applications in robotics and biomechanics because of their low cost, small size and weight, low power consumption, ease of use and portability. The problem of accurate tracking of orientation by means of these sensors has thus become important in several domains since these wearable sensors can be considered the most valuable opportunity to monitor kinematics and dynamics of human subjects outside specialized laboratories. The problem of solving orientation estimation by using efficient sensor fusion filtering algorithms has been investigated by several authors [

orientation estimated by time-integrating, from unknown initial conditions, the signals from a triad of mutually orthogonal uni-axial gyros is prone to errors that grow unbounded over time, due to low-frequency gyro bias drifts;

it is difficult give a simple interpretation to the accelerometer signals, where the component due to the gravity field (vertical reference) coexists with the component related to the motion of the object;

nearby ferromagnetic materials are critically disturbing sources when attempts are made to interpret the signals from a tri-axial magnetic sensor as the horizontal reference.

However, the set-up including one inertial measurement unit (IMU) per segment is not cost-effective compared with a solution in which a single-sensor is used. Recently, Bagalà

The aim of this study is to provide a novel technique, faster than that proposed previously [

An IP model in 2D is analyzed. A SAA is placed at height _{x}

Several authors [

Unlike the ideal condition of the mathematical model described by

In the discrete-time domain,

By using the first and second derivative approximations for the angular velocity,

For a given time window of _{1}, _{W}

The computational cost for the inversion of

In order to apply the Thomas algorithm for the evaluation of the sway angle,

In order to solve _{1}, _{W}

In the following, the vectors ^{(}^{k}^{)} and ^{(}^{k}^{)} represent the sliding window of

^{(1)} = [_{2}, …, _{W}_{−}_{1}^{T}_{1} = 0, _{W}_{k}^{(1)} = [_{x,2} …, _{x,W−1}]^{T}

^{(}^{k}^{)} is estimated from

^{(}^{k}^{)} is considered as correct estimate, under the assumption of adequately large time windows.

^{(}^{k}^{)} (left boundary) and as the linear interpolation of the last two elements of the vector ^{(}^{k}^{)} (right boundary) as follows:

The terms of the matrix

^{(}^{k}^{)} is updated by adding the new sample at the end of the vector and deleting the first one.

Steps (2–5) are iterated for the total length of the accelerometer output.

The algorithm's computational cost, _{T}_{i}

The computational cost is therefore reduced with respect to the offline method previously published [

To test the method, the same experimental setup used in [

The IMU's outputs were acquired at _{s}_{x}

In order to test the robustness of the method proposed in this paper, results in terms of RMSE were compared with:

those obtained by using the offline method proposed in [

those obtained by using an EKF applied to the outputs of the IMU (_{x}_{y}_{z}

The outputs of the accelerometers along the

The main equations of the EKF are shown in _{k}_{k}

The state vector of the EKF, _{[3×1]}, at every sampled instant of time

In the discrete-time domain the predicted state at the instant

The vector of the measurements _{[3×1]} was defined considering the outputs of the sensors as

According to (_{k}_{[3×3]} which relates the measurements _{[3×1]} to the state _{k}

The process covariance matrix _{[3×3]} was defined under the assumption that process noise affects the jerk only and there are no correlations between the jerk noise sequences. _{[3×3]} has therefore only one non-zero element ^{−3}. The measurement noise covariance matrix _{[3×3]} was defined considering the noise which affects the gyroscope and accelerometer outputs. Since correlations between noise of the sensors were assumed to be zero, the covariance matrix was put in the form 10^{−8}·I_{[3×3]}. All these tuning parameters were defined after an optimization procedure in which the encoder was assumed as validation standard. In order to start the filtering procedure, initial estimate of the state vector was zeroed whereas the initial estimate of the error covariance matrix _{[3×3]} was set equal to the identity matrix.

In order to compare the performance of the window-based algorithm proposed in this study with the EKF, seven different configurations of the EKF were implemented by considering the following measurement vectors:

_{x}

_{y}

_{z}

For each case, the output matrix

In order to show an application of the algorithm proposed in this paper and its extension to a multi-link model for human movement kinematics estimation, the knee flexion-extension angle of a subject performing a squat task was evaluated.

The method is tested on one subject (male, 29 years old, weight 74 kg, height 176 cm) who participated after giving his informed consent. In order to estimate the thigh and shank sway in the sagittal plane during squat tasks [_{1}_{1}_{1}

Four reflective markers were placed on the vertices of each plate, and a stereo-photogrammetric system (SMART eMOTION, BTS) was used for calibration of the geometric parameters (_{1}, _{2}, _{1}, _{1}, _{2}). The two reference angles with respect to the vertical were evaluated through the 2D Singular Value Decomposition (SVD) method [

The outputs of the sensor placed on the thigh can be evaluated by modifying _{t}_{c}

These two contributions can be evaluated considering the second derivative of the position of the knee joint with respect to the ankle joint:

The accelerometer and gyroscope outputs placed on the thigh can thus be expressed as follows:

The accelerometer and gyroscope outputs placed on the shank segment are expressed as

The accelerometers outputs
^{−1}(

In addition, the IMU's outputs

The knee flexion-extension angle was evaluated from the shank and thigh angles, _{shank}_{thigh}

RMSEs between the flex-extension angles estimated by (1) the algorithm proposed in this paper, (2) the EKF in the seven different configurations described above, were evaluated with reference to the stereo-photogrammetric measurements.

Five trials were performed. For each of them, the calibration parameters were estimated (mean ± std): the distance

In

The time required by the algorithm was about 170 ms to process 50 s of acquisition with a window of 100 samples, by using a DELL Studio 1,535 computer (processor Intel Core 2 Duo T9300, 2.50 GHz, 4 GB Memory, OS: Windows Vista 32 bit).

A good trade-off between the algorithm's speed and accuracy may be obtained by using

It is trivial to note that high values of

The RMSE shows an exponential dependence on the window size with a significant reduction for a window duration defined by the time constant of the system in _{T}_{i}_{i}

The mean values and the standard deviations of the calibration parameters for the five trials, obtained by minimizing sway angle errors between stereo-photogrammetric and inertial sensor data, are (mean ± std): _{1} = 0.20 ± 0.01 m, _{2} = 0.22 ± 0.00 m, _{1} = 0.40 ± 0.06 m, and _{1} = −8.98 ± 0.24°, _{2} = −2.25 ± 0.71°. These parameters are then used to predict angular sway by using the two accelerometer outputs,

RMSEs reported in

This paper suggests a novel method for the quasi-real time estimation of the sway angle in an IP model by using one SAA per segment. The relationship between the sway angle and the SAA output is described by a second order differential equation which is solved by applying the Thomas algorithm [

It is worth noting that the geometric parameters (position of the sensor with respect the pivot point, misalignment angle, segment lengths) are estimated by using encoder outputs or stereo-photogrammetric system as validation instruments only in a preliminary calibration phase. After this calibration the parameters are used to predict the sway angle with the algorithm proposed in the paper using inertial sensor only. As shown in the Results section, the estimated segment lengths and the position of the sensor with respect to the pivot are quite similar to those measured by hand. The misalignment angles are lower than 9° and neglecting the misalignment does not significantly increase the RMSE.

The method presented in this paper improves the previous method [_{s}

The performance of the estimation method depends on the window size.

The results suggest possible applications in various fields, from automatic control to bioengineering. Mainly in biomechanics, the algorithm proposed in this paper may speed up the procedure for the kinematics evaluation of human movement with a cost-effective set-up. An IMU integrating accelerometers and gyroscopes is more expensive and requires the use of a more complex HW/SW architecture for efficient signal processing. These costs increase even more with a multilink chain where one sensor per segment is required. One of the limitations of this study is the quasi-real time implementation due to the intrinsic delay imposed by the window size. Future studies will be addressed to this issue.

We would like to thank the reviewers for providing valuable suggestions that helped us to improve the manuscript.

For the sake of clarity, _{F}_{F}

In the discrete-time domain, assuming

By summing up and subtracting the two filtered signals of

By manipulating

By substituting the two terms of _{i}x_{i}_{−1} + _{i}x_{i}_{i}x_{i}_{+1} = _{i}

(

Algorithm structure.

Mechanical Inverted Pendulum.

Two-links model for the knee flexion-extension angle estimation.

Residual error between the encoder output and sway angle estimated from the accelerometer output in the mechanical inverted pendulum.

(

Residual error of the knee flexion-extension angle estimated by the quasi-real time algorithm and the EKF fusing accelerometers and gyroscope outputs. The angle estimated by stereo-photogrammetry was used as reference.

Averaged RMSEs for the mechanical IP.

0.40 ± 0.02 | |

0.39 ± 0.05 | |

0.45 ± 0.05 | |

0.46 ± 0.05 | |

2.12 ± 0.15 | |

20.64 ± 12.48 | |

_{x} |
26.71 ± 0.70 |

_{y} |
41.83 ± 7.50 |

_{z} |
8.87 ± 6.92 |

Averaged RMSEs for the knee flex-extension angle.

1.01 ± 0.11 | |

0.95 ± 0.48 | |

2.43 ± 0.76 | |

2.46 ± 0.62 | |

2.83 ± 0.24 | |

37.13 ± 18.28 | |

_{x} |
19.31 ± 0.64 |

_{y} |
43.27 ± 5.05 |

_{z} |
3.95 ± 2.59 |