Inertial Pocket Navigation System: Unaided 3D Positioning

4.1. Horizontal Surfaces

If the inertial sensor is introduced in the pocket of the trousers, the estimated pitch angle describes the opening angle of the pedestrian's leg. Knowing how the leg moves is valuable information.

Figure 2b shows the estimation of the pitch angle during a walk for seven steps. The maximum elongation of the leg, when the foot is still in the air, is indicated as θ_Hmax, and the second positive peak occurs as a consequence of the foot hitting the floor. The lowest negative angle is indicated as θ_Hmin. Figure 2a shows that θ_Hmax is much larger than θ_Hmin. The reason for this difference is that the pedestrian bends the knee of the rear leg considerably when walking.

The vertical dashed line represents a pitch angle equal to zero. Therefore, if the pedestrian is standing, the legs are closed and the pitch angle is zero.

The amplitude of the opening angle of the leg or pitch amplitude is defined as:

Our step detector consists of the identification of the maximum of the pitch angle, because each maximum indicated as θ_Hmax corresponds to a step. As Figure 2 shows, the pitch angle is ideal for detecting steps, because its cyclic nature during the walk makes the maximum or the minimum occur clearly only once per step. No varying thresholds with the pedestrian speed or post-processing of the pitch signal are needed.

4.2. Stairs

This section aims at explaining the extension of the detection of steps in horizontal surfaces to steps up and down in stairs. The detection of steps in stairs will be explained through the study of the position and movement of the legs when walking up and down stairs.

Figure 3a represents the movement of legs while walking up stairs and Figure 3b shows seven steps, the first four occurred going up stairs and the following three walking on the landing zone of the staircase.

It is noticeable that θ_Umax is almost doubled compared to θ_Hmax. This is caused by the pedestrian raising his/her leg in order to reach the following step up of the staircase. In addition, |θ_Umin| is slightly smaller than θ_Hmin. It is convenient to use the absolute value because, depending on the height of the step of the staircase, way of walking of every pedestrian, his/her tiredness, hurry, etc. θ_Umin oscillates around 0 degrees, i.e., it can be slightly positive or slightly negative.

Additionally, it is observable that the pitch amplitude by walking up stairs Δθ_U, which is computed as in Equation (4), is notably larger than Δθ_H and the double maximum peak disappears appearing in contrast a double minimum peak.

Figure 4a represents the movement of legs while walking down stairs and Figure 4b shows seven steps, the first four occurred going down stairs and the following three walking on the landing zone of the staircase.

It is noticeable that θ_Dmin has a positive value in contrast to θ_Hmin. This is caused by the pedestrian moving his/her leg in order to reach the following step down of the staircase. In addition, θ_Dmax is slightly larger than θ_Hmax.

Additionally it is observable that the pitch amplitude by walking down stairs Δθ_D, which is computed as in Equation (4), is notably smaller than Δθ_H and the double maximum peak disappears appearing in contrast a double minimum peak.

Our step detector, as previously explained, consists of the identification of the maximum of the pitch angle because each maximum indicated as θ_Umax or θ_Dmax corresponds to a step. As shown, the pitch angle is adequate as well for detecting steps up and down in stairs.

4.3. Physical Activities Identification

This section presents the potential of the pitch angle to distinguish between different physical activities, particularly standing, walking, walking up stairs, walking down stairs and sitting.

Figure 5 shows the estimated pitch angle during a walk. In this walk, there are four clearly differentiated physical activities: standing is indicated by the cyan line; walking is indicated by the red line; walking up stairs is indicated by the green line; and lastly, walking down stairs is indicated by the blue line.

In this walk, the pedestrian started standing, and then, she walked through the corridor to the stairs. She walked up until the second floor of the building, and she walked until the end of the corridor and came back to the staircase. She descended again to the ground floor and walked to the starting point.

This figure shows how intuitive it is to detect activities by simply tracking changes in the pitch estimation. The algorithm for distinguishing walking on a horizontal surface, walking up stairs and walking down stairs proposed in this work is based on three parameters of the pitch angle estimation: θ_min, θ_max and Δθ.

The parameters θ_Hmin, θ_Hmax and Δθ_H are learned during the walk under the assumption that the pedestrian starts walking on a horizontal surface. Therefore, these values are personalized for an optimal performance and continuously adapted to the pedestrian's mood, hurriedness, tiredness, etc.

In order to determine if the pedestrian is walking up stairs, down stairs or horizontally, the parameters θ_min, θ_max and Δθ are stepwise extracted and compared to the reference parameters $\overline{{\theta }_{\text{Hmin}}}$, $\overline{{\theta }_{\text{Hmax}}}$ and $\overline{\mathrm{\Delta }{\theta }_{\text{H}}}$, which are the result of averaging the last horizontal steps:

The thresholds needed in order to determine if the pedestrian is walking up stairs or down stairs can be universal or personalized. The more accurate option is to calibrate the system to correctly select them adapted to each pedestrian.

Regarding the standing and sitting activities, their main characteristic is the value of the pitch, which is almost constant. The pitch value lies around zero degrees when the pedestrian is standing and around 45-90 degrees if the pedestrian is sitting.

Figure 6 shows the pitch estimation of a pedestrian doing the following activities: standing, walking, sitting, walking and standing. Standing is indicated by the cyan line; walking is indicated by the red line and sitting is indicated by the green line.

The activities standing and sitting are only checked when no steps are detected. Then, the distinction between them only depends on the value of the pitch angle.

The identification of the physical activity distinguishes if the step taken was on a horizontal surface, was a step up or a step down. This distinction is of high interest for aiding the navigation, being the basis of the vertical displacement estimator that will be explained in Section 6. Furthermore, some applications, such as monitoring patients in hospitals or elderly care, usually need the fusion of the navigation and identification of physical activities.

5.1. Model Calibration

Fitting the data of the experiments with a regression line, as Equation (5) shows, is the simplest model assumption. Having the correct parameters a and b for each pedestrian is of crucial importance, but no calibration method has been proposed so far.

A closer investigation of the regression lines of different volunteers shows that the slope a is very similar among them. Figure 8 shows the regression lines in different colors of five volunteers who repeated the experiment without the running machine, walking directly on a horizontal surface.

The universal parameters are computed taking into account the complete set of steps of all volunteers. In Figure 8, the universal regression line overlaps the yellow line. This figure shows an overview of the error in the estimated step length for each volunteer if the universal parameters are used. The error has a range from almost 0 cm per step for the volunteers whose regression line is represented in yellow and blue to almost 14 cm per step for the volunteers whose regression lines are represented in green and violet.

From the experiments, it could not be concluded which pedestrian characteristics influence a and b. Biomechanical studies show that at any given walking speed, it is possible to select different combinations of step frequency and step length. However, individuals tend to constantly choose a specific step length for each walking speed [21]. Optimizing energy costs primarily determines the selection of a certain gait pattern.

A possible explanation of the similar slopes a of the regression lines is that all volunteers choose his/her step length at each speed basically regulated by the minimization of energy costs. This pattern is the same for all pedestrians.

There are, however, other characteristics, such as height, length of the legs, weight, gender, etc., that cause differences between pedestrians. This is a possible explanation of the different offsets b.

Based on our results, we suggest a calibration method using the universal parameter a. Therefore, the pedestrian only has to walk a known distance at his/her preferred comfortable speed to find the personalized parameter b. This is a simple and quick calibration, which does not require additional infrastructure or additional user information, such as height or length of the leg.

6.1. Experimental Setup

The experiment consists of recording the pitch information of steps taken walking up and down stairs. Almost 400 steps up and down have been recorded by one volunteer in order to study the relationship between pitch amplitude and the height of the steps of a staircase.

A wooden structure of adjustable steps, which varies between a height of 15 and 25 cm, has been built to carry out this experiment. The height of the steps has been chosen based on the document “Visual Interpretation of the International Residential Code” [22]. It specifies that the maximum height for the step of the staircase should be 19.7 cm, and for a spiral staircase, a maximum height of 24.1 cm is recommended. The minimum depth of a step of the staircase is 25.4 cm.

Figure 9 shows that a relationship between the pitch amplitude and the height of the steps of the staircase exists, indicating that the higher the step of the staircase is, the larger the pitch amplitude will be. The same behaviour is observed when the pedestrian walks up and down stairs.

The pitch angle has been measured using a precise fiber optic gyroscope (FOG), the DSP-1750 FOG from KVH. It has been mounted externally in the pocket with the help of a solid wood base, because the size of the FOG makes it impossible to introduce it in the pocket. The Allan variance analysis of a 10 h's turn rate recording reveals that the white noise of the fiber optic gyroscopes is two orders of magnitude smaller than the MTw gyroscopes' white noise. Additionally, the biases of the fiber optic gyroscopes can be considered stable within the length of the aforementioned measurements. Therefore, the pitch amplitude measured for this experiment will be considered the ground truth.

Figure 9 shows that a relationship between the pitch amplitude and the height of the steps of the staircase exists, indicating that the higher the step of the staircase is, the larger the pitch amplitude will be. The same behavior is observed when the pedestrian walks up and down stairs.

For assessing the results of the experiment of the wooden adjustable structure, the same has been done using the regular stairs of the DLR building office, whose height of the steps of the staircase is 17 cm.

Around 180 steps up and down have been recorded by the same volunteer walking on the five-floor staircase. The value of the estimated pitch amplitude for this particular height of steps is 53.07 ° and 26.62 ° for up and down stairs, respectively. These values are used to assess the experiments carried out with the wooden structure, whose results are shown in Figure 9.

6.2. Model Derivation

The simplest assumption for the behavior shown in Figure 9 is a linear model:

It is important to note that the pitch amplitude walking up and down stairs is due to the vertical and horizontal displacement of the pedestrian. If the stairs depth is larger than the standard, for example in parks outdoors, the model parameters should be re-calibrated, even for the same pedestrian.

The presented model is valid to estimate the height of the steps up and down of the vast majority of the staircases that can be found in every building, because it assumes a standard depth and focuses on the height of the step. Thus, the horizontal displacement is assumed. This is a reasonable assumption, because, as shown in [22], the staircases of any building are standardized and adapted to the length of the human foot. Additionally, the most important characteristic of the stairs is their height, which allows estimating the height between floors.

The algorithm gathers data when the pedestrian walks up and down stairs. There are small differences in the pitch amplitude, as shown in Figure 9. However, the vertical displacement is the same, because the height of the steps is the same for the whole staircase. The height of the steps of the staircase is estimated as the average of the pitch amplitude of all steps up and down using the models presented in Equations (6) and (7), respectively.

Even if the model of Equations (6) and (7) will be tested with different volunteers, an extension of the experiment described in Section 6.1 to more volunteers of different ages, genders and heights is desirable.

7.1. Step Detector Assessment

To assess the performance of the proposed step detector, four different scenarios have been chosen. The scenario Horizontal 1 describes a rectangle of 100 × 80 m walking at a constant comfortable speed of 3.5 km/h. For the scenario Horizontal 2, the pedestrian walks the long rectangle side at a slow speed and the short side at a speed of 4.8 km/h. The goal of scenarios Upstairs and Downstairs is to test the detection of steps up and down in stairs, walking at a constant and comfortable speed. After 40 m of a straight line, the pedestrian enters the staircase and walks four floors up and then down. Finally, after leaving the staircase zone, the volunteer walks back to the starting point. The experiment log files and more information of these walks is available online [23].

The methods under study are the norm of the complete acceleration and its low-pass filtered version. The complete acceleration instead of the vertical acceleration is used to be robust to different sensor orientations. Regarding the pitch signal, both methods are studied, the zero-crossing detection (Z-C) and the method proposed in this article. The detection thresholds are adapted to the comfortable volunteer walking speed. Therefore, these parameters are constant during the experiments, thus not adapted to different scenarios. The results are shown in Table 1, where both the undetected steps (UD) and the false detections (FD) are detailed.

The comparison between the aforementioned algorithms is limited to the detection of steps, because it is not possible with the literature step detectors under study to distinguish between horizontal steps and steps up or down.

The experiments summarized in this table show that it is possible to successfully detect steps at a constant and comfortable speed, as shown in scenario Horizontal 1, with the four different algorithms under study.

We have applied a low-pass filter of 15 samples to the norm of the acceleration signal to detect steps with the second proposed method. The filter smoothing makes the signal contain less peaks that could yield false step detections. Therefore, it is recommended to filter the norm of the acceleration to avoid false detections. The smoothing helps in all scenarios.

Figure 10 shows the pitch angle estimation for the scenario Downstairs. The green dots represent the truth of the taken steps. The pedestrian starts walking on the landing zone of the stairs of the upper floor. Then, the pedestrian takes the stairs to reach the ground floor. In the figure, the landing zones of the stairs are highlighted with a red line, and the stairs are highlighted with a cyan line. Finally, the pedestrian walks approximately 40 m in a straight line through the corridor. This is marked in the figure with a magenta line.

The zero-crossing-based pitch detector performs similarly to our algorithm; however, it is not able to detect all steps on the stairs, because zero-crossing does not occur, as shown in Figure 10 for the scenario Downstairs. The zero-crossing for walking up stairs, however, depends on the pedestrian, his/her mood, tiredness, etc. Therefore, the minimum of the pitch signal by walking up stairs can be slightly positive or slightly negative, as mentioned in Section 4.2. For the scenario Upstairs, the minimum pitch particularly for the volunteer has a slightly negative value; thus, the zero-crossing detects every step up.

As Figure 10 shows, the pedestrian walking speed is lower in the landing zones than in the corridor; therefore, the pitch amplitude is smaller, and the steps take more time. The vast majority of the steps taken in the landing zones are not detectable with the acceleration-based algorithms. The same problem appears in the scenario Horizontal 2, where the pedestrian was asked to walk the long rectangle side at a slow speed and the short side at a higher speed.

Pedestrians naturally adapt their walking speed and step length to the scenario or to their circumstances. Therefore, the changes of speed during the same walk represent a realistic scenario. In order to avoid losing steps in such situations, a possible solution consists of an adaptable threshold. Such an adaptable threshold, however, is not necessary with the detection based on the pitch angle. This is one of the greatest advantages of the pitch-based step detection.

Our step detector based on the pitch angle successfully detects every step of the presented scenarios and does not incur the false detections that are likely to happen, for example, when the pedestrian opens or closes doors, as Figure 10 shows for Second 185.

7.2. Step Length Estimator Assessment

In [9], an assessment of the proposed step length estimator based on the pitch angle was shown. Additionally, in this article, a comparison between our step length estimator and the well-known step length estimator based on the step frequency is analyzed.

To better and fairly compare the frequency step length model, its regression line has been extracted from the same data of the experiments with the running machine described in [9]. Figure 11 shows the steps from the same pedestrian shown in Figure 7a represented against the step frequency instead of the pitch amplitude. The shape of the clouds of points of each walking speed results from the restriction of the constant speed imposed by the running machine.

For this model, the following equation will be used:

We have chosen to use the horizontal scenario described in Section 7.1, because the step length estimator based on the frequency does not cope with the vertical displacement of the pedestrian. The scenarios Horizontals 1 and 2 have been evaluated in this section. The scenario Horizontal 3 simulates a walk in a museum where the pedestrian walks slowly, at approximately 2 km/h. The step detector for this evaluation is the one proposed in this article, and it is used with both step length estimators.

Table 2 presents the errors of each method for the proposed scenarios. The parameters for both step length estimators are personalized for the volunteer and constant for all scenarios, thus not adapted to different waking speeds. The experiment log files and more information of these walks is available online [23].

The results in Table 2 for the scenario Horizontal 1 show that both methods perform similarly at a constant comfortable speed, 3.5 km/h. Such an accurate estimation of the length of the trajectory, with errors below 1%, is possible because the sensor is attached to the pedestrian's body. This is a clear advantage of the pocket location: the movements of the sensor are fully coupled with the movements of the pedestrian. This is, for example, not the case when the sensor is held in the hand.

The scenario Horizontal 2 shows less accurate results, which are caused by the change of walking speed. The speed changes are clearly the worst case for any step length estimator, because these algorithms are usually adapted to work optimally at a comfortable and quasi-constant walking speed.

The scenario Horizontal 3 represents an extreme situation for both step length estimators under study, because the walking speed is 2 km/h and the walk includes some stops. This speed is far from the optimal working zone of the regression line for both step length estimator models. The results show however higher error for the step length estimator based on the step frequency. A possible explanation is that the frequency was computed as the inverse of the time between steps. A more sophisticated approach uses the Fourier transform to find out the pedestrian walking frequency. However, this method requires periodicity, which is broken by the changes of speed by turning or stopping.

The above analyzed experiments show the benefits of using the proposed step length estimator when the sensor is in the pocket. The optimal working zone of the regression line of the pitch-based model is larger than the one based on the step frequency. Thus, the proposed model can cope with a wider range of walking speeds.

Additionally, it is interesting to study the results using a more complex model than a regression line. The representation of the step length against the pitch amplitude for a set of steps recorded at speeds, from 1 km/h to 10 km/h for one volunteer, is available online [23]. Such a wide range of speeds requires two different models, as mentioned in Section 5: walking and running.

7.3. Vertical Displacement Estimator Assessment

In order to assess the vertical displacement estimator, we have tested the inertial pocket navigation system with a multi-story walk in a real-world scenario.

Figure 12 shows the 3D view of the estimated odometry of a 10 min's walk at the Deutsches Museum, which is a well-attended and well-known museum in Munich, Germany. The video, the experiment log files and more information is available online [23].

This odometry was estimated following the diagram of Figure 1, but using only a three-axis accelerometer and a three-axis gyroscope. We did not use the magnetometer in order to have unaided inertial positioning. No aid from other sensors, such as a barometer, WiFi access points, GNSSs, maps, etc., was used.

The step detector used is the one proposed in Section 4, and the step length estimator used is the one proposed in Section 5 with the universal parameter a and the personalized parameter b for the volunteer. The drift in the heading has been compensated in order to evaluate only the influence in the odometry of the errors in the pitch angle that may cause errors in the step detector, step length estimator and vertical displacement estimator and the errors of the proposed algorithms themselves.

Based on the definition of a step (see Figure 2a), a step is counted between consecutive hits on the floor of the same foot. Therefore, the depth of each step of the stairs shown in Figure 12 has a length of 0.52 m, twice the standard depth of 0.26 m.

The main part of the trajectory is on the ground floor and is depicted in blue, dark blue for the initial part starting at (0,0,0) and light blue for the final part for walking back to the starting point. This area of the museum is filled with real ships and aircraft. The stairs are depicted in green.

First, the pedestrian visited the mezzanine, which is a narrow area depicted in magenta. The pedestrian went around it and took the stairs again to go up to the first floor, which is depicted in red. She walked around for a few minutes, repeating part of the trajectory, and finally, she again took the stairs to go back to the ground floor again. The way back to the main entrance, i.e., the selected starting point of the walk, is depicted in light blue. On the way back, she did not choose to repeat the previous trajectory, but she surrounded two big ships, which are located between 10 and 50 m in the main exhibition hall.

Figure 13 shows the vertical displacement estimation of the pedestrian over time for the aforementioned walk. For clarity, the color code is the same as Figure 12.

This figure assesses that the vast majority of steps up and down are detected. However, the fact that the steps of the staircase are detected in pairs causes that, if a single step up or down is taken with the leg that does not have the sensor in the pocket, it may not be detected. This happens with the platform situated in the ground floor. The detector only detected the step down for Second 150, but not the step up. This platform was, however, correctly detected on the way back, as the light blue line represents. One extra step up is falsely detected in the first part of the staircase due to the same reason. The correct step up and down detection depends on the number of steps of the staircase (odd or even) and on the foot the pedestrian starts walking on for the stairs.

The estimation of the height of the steps of the staircase based on the amplitude of the pitch angle for up and down stairs is summarized in Table 3.

The error of V_U and V_D corresponding to the vertical displacement up stairs and down stairs, respectively, is below 1%. This implies an error of one meter every 100 m of vertical displacement of the pedestrian. We consider that the error is not acceptable if a confusion between consecutive floors is possible. Considering a separation of floors of three meters, an error of 1% in the vertical displacement implies that the pedestrian has to walk 300 m up or down in order to mistake one floor, and this is highly unlikely to happen.

Therefore, we conclude that it is possible and recommended to use the pitch angle, not only for detecting steps and estimating the step length, but also for estimating the height of the steps of the staircase and, consequently, the height between floors.