On Monte Carlo Smoothing in Multi Sensor Indoor Localisation

Frank Ebner, Toni Fetzer and Frank Deinzer

Faculty of Computer Science and Business Information Systems

University of Applied Sciences Würzburg-Schweinfurt

Würzburg, Germany

{frank.ebner, toni.fetzer, frank.deinzer}@fhws.de

Lukas Köping, Marcin Grzegorzek

Pattern Recognition Group University of Siegen Siegen, Germany {lukas.koeping, marcin.grzegorzek}@uni-siegen.de

Abstract—Indoor localisation continuous to be a topic of growing importance. Many different approaches for estimating the position of a pedestrian are presented year after year. Despite the advances made, several profound problems are still present. For example, estimating an accurate position from a multimodal distribution or recovering from the influence of faulty measurements. Within this work, we try do solve such problems with help of Monte Carlo smoothing methods, namely forwardbackward smoother and backward simulation. In contrast to normal filtering procedures like particle filtering, smoothing methods are able to incorporate future measurements instead of just using current and past data. This enables many possibilities for further improving the position estimation. Both smoothing techniques are deployed as fixed-lag and fixed-interval smoother and a novel approach for incorporating them easily within our localisation system is presented. All this is evaluated on four floors within our faculty building. The results show that smoothing methods offer a great tool for improving the localisation results. Especially fixed-lag smoothing provides a great runtime support by reducing timely errors and improving the overall estimation with affordable costs.

I. Introduction

Determining a position indoors is a challenging task. Besides the complex architecture of many buildings, a high accuracy needs to be achieved, especially for buildings with many small separated areas like shopping malls or office blocks. In recent years, many different systems were presented to meet those requirements. Especially Wi-Fi positioning and pedestrian dead reckoning (PDR) are very popular solutions. Approaches based on PDR try to estimate the current position given the previous position and thus require an initial state. However, this allows for cumulative errors and leads to an erroneous position estimation within a very short period. By incorporating the absolute position information of Wi-Fi this drift can be corrected. Additional improvements can be achieved by using environmental information about walls and obstacles provided by a floor map.

In most cases, probabilistic methods are used to incorporate those highly different sensor types. Here, a probability distribution describes the pedestrian’s possible whereabouts and therefore the uncertainty of the system. Drawing from a probability distribution and finding an analytical solution for densities is in most cases a difficult task, especially in case of time sequential, non-linear and non-Gaussian models. Due to the high complexity of the human movement, we consider indoor localisation as such. A broad class to obtain numerical results instead are the Monte Carlo (MC) methods. Here, a set of weighted random samples is used to solve any problem having a probabilistic interpretation. By applying the time sequential hidden Markov process of Bayes filtering, one of the most important MC techniques results: particle filtering. A particle filter updates the state estimation recursively in time with every new incoming measurement using the state transition and state evaluation step.

Based on this general methodology, many different approaches for estimating a position in indoor environments have been developed. All these approaches differ mainly in how the dynamics are modelled in the transition step and how a specific sensor measurement can be used for evaluation. For example, recent approaches are using a graph-based structure to consider environmental restrictions (walking through walls) and the characteristics of human movement (walking speed) within the transition model [1]–[3]. The evaluation model is mostly separated into any number of sensor models, each representing the probability for a noisy measurement in regard to the current position. For example, a barometer can be used to determine the probability of being on a certain floor [4].

Despite the many advances made in the last years, nearly all systems suffer from more or less the same problems. Of course, every sensor model brings its very own weaknesses. Like mentioned before, PDR suffers from an accumulating bias, the signal of Wi-Fi gets attenuated by walls

Frank: falls noch platz ist: noch mehr nachteile :P Wi-Fi location estimation strongly depends on the quality of the signal-strength estimation model (oder fingerprinting) and the way the smartphone is held

and the barometric pressure is highly af-

fected  by  weather  patterns  and  humidity

Frank: spontane fenster/tuer oeffnung

[4]. That is the reason for the use of statistical methods in the first place. Nevertheless, there are even more profound problems regarding the whole position estimation procedure.

Current transition models, which aim to approximate the movement, are still very restrictive and unable to handle unforeseen events. Faulty sensor measurements, like a falsely detected turn, can cause the estimation to lose track. For example by taking

Time t  t − 1  t − 2

Fig. 1. An example of the occurrence of a multimodal distribution. At time t − 1 the floor is separated by a wall and the mode of the distribution (coloured circle), Frank: mode of the weglassen? einfach: distribution ... splits representing the current position, splits apart. The most likely position (green line) is estimated somewhere in-between. After a right turn, the distribution slowly starts to recover its unimodality.

Frank: by taking -¿ by recognising? a turn too soon and walking into a room instead of another big hallway. Due to this, the filter needs some time to recover, which again takes a while because of the restrictive model (e.g. no walking through walls and only realistic walking speed). This temporal delay worsens the estimate immensely. A solution to recover from such filter divergences faster, is using

Frank: is using -¿ involves?

methods to re-initialize the filtering procedure [2]. However, even this can not completely prevent delays. Another reason for possible time delays are slow sensor updates. For example, most mobile devices restrict the Wi-Fi module to update only every few seconds, to save on battery.

Further critical problems arise from multimodal distributions. Those are caused by multiple possible position estimates. Fig. 1 illustrates an example where a floor gets separated by a wall. Due to inaccurate measurements and a PDR approach for evaluating the movement, the distribution splits apart. Therefore, the most likely position Frank: wenn avg ueber alle particles, was ja default ist is somewhere in-between. Only after the pedestrian turns right, the distribution is again unimodal, since moving through walls is impossible. As one can imagine, this can lead to serious problems in big indoor environments. Such a situation can be improved by incorporating future measurements (e.g. the right turn) or predictive information (e.g. the most likely path) to the filtering procedure [5]. However, standard filtering methods are not able to use any future information and the possibilities to make a distant forecast are also limited [6]–[8].

One very promising way to deal with these problems is smoothing. Smoothing methods are able to make use of future measurements for computing their estimation. By running backwards in time, they are also able to remove multimodalities and improve the overall localization result. Since the problem of navigation, especially the representation of complex movement patterns, results in a non-linear and non-Gaussian state space, this work focuses mainly on smoothing techniques based on the broad class of MC methods. Namely, forward-backward smoothing [9] and backward simulation [10].

Within this work, we investigate the benefits and drawbacks of those techniques using our indoor localisation system presented in [5]. We provide both, fixed-lag and fixed-interval smoothing as well as a novel approach for incorporating them easily within the localisation procedure. The main goal is to solve above mentioned problems and to investigate new possibilities for even more advanced systems.

II. Related Work

Sequential MC filter, like aforementioned particle filter, use all observations o_{1:t} until the current time t for computing an estimation of the state q_{t}. In a Bayesian setting, this can be formalized as the computation of the posterior distribution p(q_{t} | o_{1:t}) using a sample of N independent random variables, X_{t}^{i} ∼ (q_{t} | o_{1:t}) for i = 1, ..., N for approximation. Due to importance sampling, a weight W_{t}^{i} is assigned to each sample X_{t}^{i}. In context of particle filtering {X_{1:t}^{i}, W_{1:t}^{i}}_{i=1}^{N} is a weighted set of samples, also called particles. Therefore a particle is a representation of one possible system state q. By considering a situation given all observations o_{1:T} until a time step T, where t ≪ T, standard filtering methods are not able to make use of this additional data for computing p(q_{t} | o_{1:T}). This problem can be solved with a smoothing algorithm.

Within this work we utilise two types of smoothing: fixedlag and fixed-interval smoothing. In fixed-lag smoothing, one tries to estimate the current state, given measurements up to a time t+τ, where τ is a predefined lag. This makes the fixed-lag smoother able to run online. On the other hand, fixed-interval smoothing requires all observations until time T and therefore only runs offline, after the filtering procedure is finished [8].

The origin of MC smoothing can be traced back to Genshiro Kitagawa. In his work [11] he presented the simplest form of smoothing as an extension to the particle filter. This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering. This approach uses the particle filter steps to update weighted paths {(X_{1:t}^{i}, W_{t}^{i})}_{i=1}^{N}, producing an accurate approximation of the filtering posterior p(q_{t} | o_{1:t}) with a computational complexity of only O(N). However, it gives a poor representation of previous states due a monotonic decrease of distinct particles caused by resampling of each weighted path [7]. Based on this, more advanced methods like the forward-backward smoother [9] and backward simulation [10] were developed. Both methods are running backwards in time to reweight a set of particles recursively by using future observations. Algorithmic details will be shown in section VII.

In recent years, smoothing gets attention mainly in other areas as indoor localisation. The early work of [12] demonstrates the possibilities of smoothing for visual tracking. They used a combination of the CONDENSATION particle filter with a forward-backward smoother. Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed [13]. For example, in [14] a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or [15] uses a smoother to overcoming the problem of particle impoverishment while predicting the Remaining Useful Life (RUL) of equipment (e.g. a Lithium-ion battery).

Nevertheless, their are some promising approaches for indoor localisation systems as well. For example [2] deployed a fixed-interval forward-backward smoother to improve the position estimation for non-real-time applications. They combined Wi-Fi, step and turn detection, a simple line-of-sight model for floor plan restrictions and the barometric change within a particle filter. The state transition samples a new state based on the heading change, altitude change and a fixed step length. The experiments of [2] clearly emphasize the benefits of smoothing techniques. The estimation error could be decreased significantly. However, a fixed-lag smoother was treated only in theory.

In the work of [16] both fixed-interval and fixed-lag smoothing were presented. They implemented Wi-Fi, binary infra-red motion sensors, binary foot-switches and a potential field for floor plan restrictions. Those sensors were incorporated using a sigma-point Kalman filter in combination with a forward-backward smoother. It was also proven by [16], that the fixed-lag smoother is slightly less accurate then the fixed-interval smoother. As one would expect from the theoretical foundation. Unfortunately, even a sigma-point Kalman filters is after all just a linearisation and therefore not as flexible and suited for the complex problem of indoor localisation as a non-linear estimator like a particle filter.

Toni: Kann das jemand nochmal verifizieren? Das mit dem Kalman Filter. Danke.

Additionally, the Wi-Fi RSSI model requires known calibration points and is deployed using a remarkable number of access points for very small spaces. In our opinion this is not practical and we would further recommend adding a PDR-based transition instead of a random one.

In contrast, the here presented approach is able to use two different smoothing algorithm, both implemented as fixedinterval and fixed-lag versions. Further, our localisation system presented in [5] enables us to walk stairs and therefore going into the third dimension. Therefore, a regularly tessellated graph is utilized to avoid walls, detecting doors and recognizing stairs. Within this work, this is additionally supported by a simple classification that detects the activities unknown, standing, walking and walking stairs. Finally, we incorporate prior navigation knowledge by using syntactically calculated realistic human walking paths [5]. This method makes use of the given destination and thereby provides a more targeted movement.

III. Recursive State Estimation

As mentioned before, most smoothing methods require a preceding filtering. In our previous work [5], we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem. Therefore, a Bayes filter that satisfies the Markov property is used to calculate the posterior, which is given by

p(q_{t} | o_{1:t}) ∝

\underbrace{p(o_{t} | q_{t})}_{evaluation}∫ \underbrace{p(q_{t} | q_{t−1}, o_{t−1})}_{transition} \underbrace{p(q_{t−1} | o_{1:t−1})dq_{t−1}}_{recursion} .

(1)

Here, the previous observation o_{t−1} is included into the state transition [17]. For approximating eq. (1) by means of MC methods, the transition is used as proposal distribution, also known as CONDENSATION algorithm [12].

In context of indoor localisation, the hidden state q is defined as follows:

q = (x, y, z, θ, ρ̂_{rel}), x, y, z, θ, ρ̂_{rel} ∈ ℝ ,  (2)

where x, y, z represent the position in 3D space, θ the user’s heading and ρ̂_{rel} the relative atmospheric pressure prediction in hectopascal (hPa). Further, the observation is given by

o = (s_{wifi}, s_{ib}, Δθ, n_{steps}, ρ_{rel}, x) ,  (3)

covering all relevant sensor measurements. Here, s_{wifi} and s_{ib} contain the measurements of all nearby access-points (AP) and iBeacons, respectively. Δθ and n_{steps} describe the relative angular change and the number of steps detected for the pedestrian. ρ_{rel} is the relative barometric pressure with respect to a fixed reference. Finally, x contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or walking stairs.

The probability density of the state evaluation is given by

p(o_{t} | q_{t}) = p(o_{t} | q_{t})_{baro} p(o_{t} | q_{t})_{ib} p(o_{t} | q_{t})_{wifi} (4)

and therefore similar to [5]. Here, we assume a statistical independence of all sensors and every single component refers to a probabilistic sensor model. The barometer information is evaluated using p(o_{t} | q_{t})_{baro}, whereby absolute position information is given by p(o_{t} | q_{t})_{ib} for iBeacons and by p(o_{t} | q_{t})_{wifi} for Wi-Fi.

IV. Filtering

Toni: Bin mir nicht sicher ob wir diese Section berhaupt brauchen. Knnte man bestimmt auch einfach unter Section 3 packen. Aber dann knnen wir ungestrt voneinander schreiben.

V. Evaluation

A. Barometer

The probability of currently residing on a given floor is evaluated using the smartphone’s barometer. Environmental influences are circumvented by using relative pressure readings instead of absolute ones. To reduce the impact of noisy sensors, we calculate the average of several sensor reading, carried out while the pedestrian chooses his destination. This \overline{ρ_{rel}} serves as relative base. Likewise, we estimate the sensor’s uncertainty σ_{baro} for later use within the evaluation step.

In order to evaluate relative pressure readings, we need a prediction to compare them with. Therefore, each transition from q_{t−1} to q_{t} estimates the state’s relative pressure prediction ρ̂_{rel} by examining every height-change (z-axis):

qρ̂_{rel} ; t = qρ̂_{rel} ; t−1 + Δz · b, Δz = q_{t−1}^{z} − q_{t}^{z}, b ∈ ℝ . (5)

In eq. (5), b denotes the common pressure change in \frac{hPa}{m}. The evaluation step compares the predicted relative pressure with the observed one using a normal distribution with the previously estimated σ_{baro}:

p(o_{t} | q_{t})_{baro} = N (o_{t}^{ρ_{rel}} | qρ̂_{rel} ; t, σ_{baro}^{2}) .  (6)

B. Wi-Fi & iBeacons

The smartphone’s Wi-Fi and iBeacon component provides absolute location estimation by measuring the signal-strengths of nearby transmitters. The positions of detected access-points (AP) and iBeacons are known beforehand. This allows a comparison of each measurement with a corresponding estimation using the wall-attenuation-factor signal strength prediction model [1]. This model uses the 3D distance d and the number of floors Δf between transmitter and the state-in-question:

P_{r}(d, Δf) = P_{0} − 10γ log_{10} \frac{d}{d_{0}} + Δfβ ,  (7)

As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is:

p(o_{t} | q_{t})_{wifi} = ∏_{i=1}^{n} N (s_{wifi}^{i} | P_{r}(d_{i}, Δf_{i}), σ_{wifi}^{2}) . (8)

The prediction model itself needs three parameters per AP: P_{0} measured at a distance d_{0} (usually 1 m), the path-loss exponent γ describing the environment and the attenuation per floor β. Frank: aufs andere paper beziehen zum kuerzen? To reduce the system’s setup time, we use the same values for all access-points at the cost of accuracy. All parameters are chosen empirically. Further details on how to determine this parameters exactly, can be found in [18].

The same holds for the iBeacon component, except P_{0}, which is broadcasted by each beacon. As iBeacons cover only a small area, γ is usually much smaller compared to the one needed for Wi-Fi.

VI. Transition

The distribution p(q_{t} | q_{t−1}) is sampled via random walks on a graph G = (V, E), which is generated from the buildings floorplan TODO FUSION2016 . p(q_{t} | q_{t−1}) is determined by walking along adjacent edges e_{i,j} connecting two vertices v_{i}, v_{j} ∈ V until a certain distance d_{step} is reached. Thus, the position of any q is represented by the position pos(v_{i}) of the corresponding vertex. This approach draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.

While sampling, to-be-walked edges are not chosen uniformly, but depending on a probability p(e_{i,j}). The latter depends on several constraints and recent sensor-readings from the smartphone. Using sensors directly within the transition step provides a more robust posterior distribution. Adding them to the evaluation instead, would lead to sample impoverishment due to the used Monte Carlo methods.

A. Pedestrian’s Destination

We assume the pedestrian’s desired destination to be known beforehand. This prior knowledge is incorporated during the random walk using p(e_{i,j})_{path}, which is a simple heuristic, favouring movements (edges) approaching the chosen destination with a ratio of 0.9 : 0.1 over those, departing from the destination [?]. The underlying shortest-path is based on a special distance metric, considering special architectural facts:

B. Architectural Facts

To ensure realistic path estimations, we include additional architectural knowledge. Each vertex’s distance from the nearest wall is used to prevent the shortest-path from clinging to walls. Likewise, his distance from the nearest door (favour ).

C. Step- & Turn-Detection

Steps and turns are detected using the smartphone’s IMU, implemented as described in [1]. The number of steps detected since the last transition is used to estimate the to-be-walked distance d_{step} assuming a fixed step-size with some deviation:

d_{step} = o_{t−1}^{n_{steps}} · s_{step} + N (0, σ_{d_{step}}^{2}) .  (9)

Turn-Detection supplies the magnitude of the detected heading change by integrating the gyroscope’s change since the last transition. Together with some deviation and the state’s previous heading, the magnitude is used to estimate the state’s current heading:

θ_{walk} = q_{t}^{θ} = q_{t−1}^{θ} + o_{t−1}^{Δθ} + N (0, σ_{θ_{walk}}^{2})  (10)

During the random walk, edges should satisfy the heading:

p(e_{i,j})_{head} = p(e_{i,j} | θ_{walk}) = N (∠e_{i,j} | θ_{walk}, σ_{head}^{2}) . (11)

While the distribution eq. (11) does not integrate to 1.0 due to circularity of angular data, in our case, the normal distribution can be assumed as sufficient for small enough σ^{2}.

D. Activity-Detection

Additionally we perform a simple activity detection for the pedestrian, able to distinguish between standing, walking, walking stairs upwards and downwards. Likewise, this knowledge is evaluated when walking the grid: Edges e_{i,j} matching the currently detected activity are favoured using p(e_{i,j})_{act} = 0.8 and 0.2 otherwise.

p(e_{i,j})_{act} = \begin{cases}0.8 & stairs up, pos(v_{j})_{z} > pos(v_{i})_{z} \\ 0.2 & stairs up, pos(v_{j})_{z} ≤ pos(v_{i})_{z} \\ ⋯ \end{cases} (12)

Frank: das switch ist wahrscheinlich unnoetig und der text reicht Frank: hier passen die sachen vom lukas. kurze beschreibung der beiden geschaetzten verteilungen etc

The transition model includes a simple recognizer of different locomotion modes like normal walking or ascending/descending stairs. The reasoning behind this is to favour paths that correspond with the detected locomotion mode.

We use a Naives Bayes classifier with two features. For this, the sensor signals are split in sliding windows. Each window has a length of one second and overlaps 500 ms with its prior window. The first feature is the variance of the accelerometer’s magnitude during a window and the second feature is the difference between the last and first barometer measurement of the particular window. Based on these features the classifier assigns an activity to each sliding window.

TODO Was passiert wenn ein berlappendes Fenster zwei verschiedene Aktivitten zugewiesen bekommt? Sliding windows evtl. weglassen?

VII. Smoothing

The main purpose of this work is to provide MC smoothing methods in context of indoor localisation. As mentioned before, those algorithm are able to compute probability distributions in the form of p(q_{t} | o_{1:T}) and are therefore able to make use of future observations between t and T.

In the following we discuss the algorithmic details of the forward-backward smoother and the backward simulation. Further, a novel approaches for incorporating them into the localisation system is shown.

A. Forward-backward Smoother

The forward-backward smoother (FBS) of [19] is a well established alternative to the simple filter-smoother. The foundation of this algorithm was again laid by Kitagawa in [20]. An approximation is given by

p(q_{t} | o_{1:T}) ≈ ∑_{i=1}^{N} W_{t|T}^{i} δ_{X}_{t}^{i}(q_{t}) ,  (13)

where p(q_{t} | o_{1:T}) has the same support as the filtering distribution p(q_{t} | o_{1:t}), but the weights are different. This means, that the FBS maintains the original particle locations and just reweights the particles to obtain a smoothed density. The complete FBS can be seen in algorithm 1 in pseudoalgorithmic form. At first, the algorithm obtains the filtered distribution (particles) by deploying a forward step at each time t. Then the backward step for determining the smoothing distribution is carried out. The weights are obtained through the backward recursion in line 9.

By reweighting the filter particles, the FBS improves the simple filter-smoother by removing its dependence on the inheritance (smoothed) paths [21]. However, by looking at algorithm 1 it can easily be seen that this approach computes in O(N^{2}), where the calculation of each particle’s weight is an O(N) operation. To reduce this computational bottleneck,

[22] introduced a solution using algorithms from N-body simulation. By integrating dual tree recursions and fast multipole techniques with the FBS a run-time cost of O(N log N) can be achieved.

Algorithm 1 Forward-Backward Smoother

1: for t = 1 to T do ▷ Filtering 2: Obtain the weighted trajectories {W_{t}^{i}, X_{t}^{i}}_{i=1}^{N} 3: end for 4: for i = 1 to N do ▷ Initialization 5: Set W_{T |T}^{i} = W_{T}^{i} 6: end for 7: for t = T − 1 to 1 do ▷ Smoothing 8: for i = 1 to N do

9: W_{t|T}^{i} = W_{t}^{i} [∑_{j=1}^{N} W_{t+1|T}^{j} \frac{p(X_{t+1}^{j}|X_{t}^{i})}{∑_{k=1}^{N} W_{t}^{k} p(X_{t+1}^{j}|X_{t}^{k})}]

10: end for 11: end for

Algorithm 2 Backward Simulation Smoothing

1: for t = 1 to T do ▷ Filtering 2: Obtain the weighted trajectories {W_{t}^{i}, X_{t}^{i}}_{i=1}^{N} 3: end for 4: for k = 1 to N_{sample} do 5: Choose q̃_{T}^{k} = X_{T}^{i} with probability W_{T}^{i} ▷ Initialize 6: for t = T − 1 to 1 do ▷ Smoothing 7: for j = 1 to N do 8: W_{t|t+1}^{j} = W_{t}^{j} p(q̃_{t+1} | X_{t}^{j}) 9: end for 10: Choose q̃_{t}^{k} = X_{t}^{j} with probability W_{t|t+1}^{j} 11: end for 12: q̃_{1:T}^{k} = (q̃_{1}^{k}, q̃_{2}^{k}, ..., q̃_{T}^{k}) is one approximate realization

from p(q_{1:T} | o_{1:T}) 13: end for

B. Backward Simulation

For smoothing applications with a high number of particles, it is often not necessary to use all particles for smoothing. This decision can for example be made due to a high sample impoverishment and/or highly certain sensors. By choosing a good sub-set for representing the posterior distribution, it is theoretically possible to further improve the estimation.

Therefore, [10] presented the backward simulation (BS). Where a number of independent sample realizations from the entire smoothing density are used to approximate the smoothing distribution. This method can be seen in algorithm 2 in pseudo-algorithmic form. Again, a particle filter is performed at first and then the smoothing procedure gets applied. Here, q̃_{t} is a random sample drawn approximately from p(q_{t} | q̃_{t+1}, o_{1:T}). Therefore q̃_{1:T} = (q̃_{1}, q̃_{2}, ..., q̃_{T}) is one particular sample realization from p(q_{1:T} | o_{1:T}). Further independent realizations are obtained by repeating the algorithm until the desired number N_{sample} is reached. The computational complexity for one particular realization is O(N). However, the computations are then repeated for each realization drawn [10].

C. Transition for Smoothing

As seen above, both algorithms are reweighting particles based on a state transition model. Unlike the transition presented in section VI, it is not possible to just draw a set of new samples. Here, p(q_{t+1} | q_{t}) needs to provide the probability of the known future state q_{t+1} under the condition of the current state q_{t}. In case of indoor localisation using particle filtering, it is necessary to not only provide the probability of moving to a particle’s position under the condition of its ancestor, but also of all other particles at time t. The smoothing transition model therefore calculates the probability of being in a state q_{t+1} in regard to previous states and the pedestrian’s walking behaviour. This means that a state q_{t} is more likely if it is a proper ancestor (realistic previous position) of a future state q_{t+1}. In the following a simple and inexpensive approach for receiving this information will be described.

By writing

p(q_{t+1} | q_{t}, o_{t})_{step} = N (Δd_{t} | μ_{step}, σ_{d_{step}}^{2})  (14)

we receive a statement about how likely it is to cover a distance Δd_{t} between two states q_{t+1} and q_{t}. In the easiest case, Δd_{t} is the linear distance between two states. Of course, based on the graph structure, one could calculate the shortest path between both and summarize the respective edge lengths. However, this requires tremendous calculation time for negligible improvements. Therefore this is not further discussed within this work. The average step length μ_{step} is based on the pedestrian’s walking speed and σ_{d_{step}}^{2} denotes the step length’s variance. Both values are chosen depending on the activity x recognized at time t. For example μ_{step} gets smaller while a pedestrian is walking upstairs, then just walking straight. This requires to extend the smoothing transition by the current observation o_{t}. Since q is hidden and the Markov property is satisfied, we are able to do so.

The heading information is incorporated using

p(q_{t+1} | q_{t}, o_{t})_{turn} = N (Δα_{t} | Δθ, σ_{turn}^{2}) ,  (15)

where Δα_{t} is the absolute angle between q_{t+1} and q_{t} in the range of [0, π]. The relative angular change Δθ is then used to receive a statement about how likely it is to walk in that particular direction. Again the normal distribution of eq. (15) does not integrate to 1.0. Therefore the same assumption as in eq. (11) has to be made.

To further improve the results, especially in 3D environments, the vertical (non-absolute) distance Δz between two successive states is used as follows:

p(q_{t+1} | q_{t}, o_{t})_{baro} = N (Δz | μ_{z}, σ_{z}^{2}) .  (16)

This assigns a low probability to false detected or misguided floor changes. Similar to eq. (14) we set μ_{z} and σ_{z}^{2} based on the activity recognized at time t. Therefore, μ_{z} is the expected change in z-direction between two time steps. This means, if the pedestrian is walking alongside a corridor, we set μ_{z} = 0. In contrast, μ_{z} is positive while walking downstairs or otherwise negative for moving upstairs. The size of μ_{z} and also μ_{step} could be a predefined value or set dynamically based on the measured vertical and linear acceleration.

Looking at eq. (14) to eq. (16), obvious similarities to a sensor fusion process can be seen. By assuming statistical independence between those three, the probability density of the smoothing transition is given by

p(q_{t+1} | q_{t}) = p(q_{t+1} | q_{t}, o_{t})_{step} p(q_{t+1} | q_{t}, o_{t})_{turn} p(q_{t+1} | q_{t}, o_{t})_{baro} .  (17)

It is important to notice, that all particles at each time step t of the forward filtering need to be saved. Therefore, the memory requirement increasing proportional to the processing time.

VIII. Experiments

ddd [23] dddd Evaluation:

• Filter ist immer der gleiche mit MultiPathPrediction und Importance Factors

• FBS Interval mit 500 und 7500 Partikeln auf 4 Pfaden mit SimpleSmoothingTrans

• BS Interval mit 500 zu 50 und 7500 zu 500 Partikeln auf auf 4 Pfaden mit SimpleSmoothingTrans

• FBS Lag = 5 mit 500 und 7500 Partikeln auf 4 Pfaden mit SimpleSmoothingTrans

• BS Lag = 5 mit 500 zu 50 und 7500 zu 500 Partikeln auf auf 4 Pfaden mit SimpleSmoothingTrans

• BS Lag zu Error Plot. Lag von 0 bis 100, wie verhlt sich der Error. Am besten auf Pfad 4 mit SimpleSmoothing-Trans.

• BS Lag = 5 mit 500 Partikeln auf einem Pfad der manuell angepasst ist (mach ich) mit DijkstraTrans.

• Vorwrtsschritt die Ergebnisse und Probleme beschreiben. Zeitlicher Verzug etc.

• Fixed-Interval Smoothing mit beiden Methoden. Was

wird besser? Warum?

AUf die schlechte Performance eingehen fr was knnte man es nutzen?

• Fixed-lag Smoothing der beiden Methoden

erstmal ergebnisse mit 3 unterschiedlichen lags. 5, 30, 50 oder so

Fehlerplot wie beim fusion paper. wo ist der error besonders hoch, warum? warum wurde er besser? welche anwendungen gibt es da jeweils? plot mit kurven die den fehler bei den unter-

schiedlichen lag lngen angibt.

• Fixed-lag gap

einen offset (gap) im smoothing. was bringt es? sinnvoll?

IX. Conclusion

map information into smoothing. better way and faster then just dijkstra. compensate big jumps caused by wifi. better method for estimation and drawing of particles in backward simulation. more advanced smoothing transition. not used evaluating using the observations, but using the given information for more advanced approaches.

References

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