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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/).

Stationary range laser sensors for intruder monitoring, restricted space violation detections and workspace determination are extensively used in risky environments. In this work we present a linear based approach for predicting the presence of moving agents before they trespass a laser-based restricted space. Our approach is based on the Taylor's series expansion of the detected objects' movements. The latter makes our proposal suitable for embedded applications. In the experimental results (carried out in different scenarios) presented herein, our proposal shows 100% of effectiveness in predicting trespassing situations. Several implementation results and statistics analysis showing the performance of our proposal are included in this work.

The tracking and prediction of objects or targets has several applications, such as traffic surveillance [

In general, a target tracking process can be divided into two main stages:

Several procedures are used for object detection in artificial vision based applications. In [

Range laser sensors are also used for target tracking applications, such as the case shown in [

In addition, range laser sensors are also used for intruders detection, trespassing situations and workspace determination, as pointed out by the manufacturers [

Despite the detection algorithm and the sensor used by the system, the

The Taylor's series expansion is also used as a tool for the object tracking and prediction problem. In [

The main contribution of this work is a workspace supervision application based on the prediction of trespassing situations by using multiple stationary range laser sensors. The last is accomplished by using the Taylor's series expansion of the motion of the detected targets as a tracking—and predicting— procedure

This work is organized as follows: Section 2 shows an overview of the proposed system, the sensors description, the problem's hypothesis and the mathematical formulation of the proposal; Section 3 shows the experimentation and statistical results of each proposed situation. Section 4 presents the pros and cons observed during the experimentation stage. Section 5 shows the conclusions of this work.

The abovementioned three stages form a standard supervision system [

It is worth mentioning that such a prediction of the object's movements can be used for the optimization of the sensed workspace by reducing its restricted region. Since the action execution is based on the prediction information, if the predicted object's movements do not compromise the process nor its integrity, then there is no need of an action execution. Nevertheless, the last statement is strongly related to the adopted horizon of prediction.

In the following sections, each stage of

In this work,

The restricted workspace determination, as shown in

In this work, the detection of moving objects within the sensed workspace shown in

From the set of 181 measurements acquired by the range laser sensor, the histogram method [

If two or more consecutive measurements are associated to a same point-based feature, then its

Each

The parameters of each detected feature are transformed according to a global Cartesian reference frame attached to the system (_{i}_{i}^{th}

If the same object is detected in two consecutive laser scans, then we are able to track it. In order to do so, a matching criterion must be adopted;

It is worth mentioning that the object detection method mentioned above allows for the detection of multiple objects. Further information regarding such a method can be found in [

The linear prediction formulation proposed in this work is based on the Taylor's series expansion [_{0} is the initial instant, _{0}) is the body's initial position, _{0}) is its velocity and _{0}) is its acceleration. The Taylor's expansion of

In _{m}_{m}_{m}

In order to estimate ^{2}, where ^{2} is the space of continuous functions with first and second differential also continuous.

If ^{3}, then _{m}

In general, if ^{n}^{th}

In addition, if we consider the Euler approximation:
_{t} = _{k}_{k−1} sufficiently small, we can apply such an approximation to ^{0}:

With the same insight, for ^{1}:

In addition, for ^{2} and considering that Δ_{t}_{i}_{i−1} for

Therefore, if the sampling time Δ_{t}_{k+1}) based on the Taylor's series expansion. The extension of the procedure shown in ^{n}

For the multi-dimensional case, let ^{b}^{l}

In _{k}^{p}_{k}^{th}_{k}_{i}_{i−1} for ^{0}, ^{1} and ^{2}):

Furthermore, for the two-dimensional case (^{2}) and taking into account the object detection procedure presented in Section 2.3, let [_{i,tk} y_{i,tk}^{T}^{th}_{k}_{m}^{2}. If we consider that the motion of the detected object falls within ^{2}, then _{m}

It is worth mentioning that, if more precision is required, the number of terms in ^{th}

By inspection we can see that, if ^{2}, then we need the previous knowledge of _{k}_{−1}) and _{k}_{−2}) in order to predict _{k}_{+1}). Therefore, the very first prediction of the process should consider _{k}_{−1}) and _{k}_{−2}) as a previously defined values (e.g., zero). In our implementations, due to the errors associated with the first predictions, we have discarded the first two predictions.

In addition, if an _{k}_{k}_{+}_{r}

The action execution, as shown in

Although several actions can be taken into account according to the application requirements, this work is focused on the

Several experimental results were carried out in order to show the performance of the proposal. They can be grouped as follows:

Single laser with single object prediction.

Single laser with multiple objects prediction.

Multiple lasers with single object prediction.

Multiple lasers with multiple objects prediction.

For each mentioned case, 50 trials were run for two different restricted workspace dispositions, see _{k}_{+}_{r}_{t}

_{laser} y_{laser}^{T}^{T}_{laser}

In addition, with

In

With the same insight,

As previously stated, we have implemented our supervision application to a system with multiple range lasers, as the one shown in _{laser,1} _{laser,1}]^{T}^{T}_{laser,1} = _{laser,2} _{laser,2}]^{T}^{T}_{laser,2} =

As mentioned in Section 3.3,

During the experimentation, a number of lessons were learned with regard to the supervision application strategy based on the Taylor's prediction criterion proposed in this work. First, the precision of the prediction is strongly related to the precision of the detection procedure. Thus, a noisy detection procedure can transform a movement that originally belongs to ^{2}—which is associated with a smooth movement—into a movement that belongs to ^{0}, such as an arbitrary movement.

Second, the effectiveness of the prediction is also affected by the horizon used in the implementation. As shown in the statistical results in Sections 3.1–3.4, for

Third, the multi-object case with both single or multi-laser situations has shown similar results than the single-object case. Despite the fact that each object was predicted independently, the objects detection procedure used has shown to be efficient for the application. In addition, it is worth mentioning that we have used subjects as objects for the execution of the experimentation.

Fourth, if the detected object moves following a straight line and the detection method is too noisy, then the system will face a situation in which a linear movement might be detected as a random one. Considering that the error in the detection is propagated to the prediction, it also conditions the order of the Taylor's expansion prediction to be used in the process. Therefore, it is recommendable to have previous information regarding the detection method's efficiency before choosing the order of the Taylor's expansion and its corresponding horizon of prediction.

This paper has presented a new prediction method based on the Taylor's series expansion of the motion of an object given its detection parameters. The accuracy of the method is related to the maximum order adopted for the Taylor's expansion. Considering the algebraic formulation of the prediction method, it is suitable for implementation in embedded systems. Also, it is scalable: it can be adapted to the number of objects whose motions are going to be predicted.

In addition, the proposed method was implemented in range laser-based supervision systems for the prediction of trespassing situations. Such situations are common when working in restricted environments and free motion is not allowed due to risky situations. Our proposal has shown to be effective to predict future trespassing situations. With this insight, two main cases were presented: the single-laser case with both single object and multi-object prediction; and the multi-laser case with both single object and multi-object prediction. For all the cases, our proposal had shown a 100% of effectiveness in predicting intended trespassing situations. However, the system had also predicted false trespassing situations—

The authors would like to thank to the Department of Electronics Engineering,

General system architecture.

Examples of object prediction. (

Range laser sensor used in this work.

Three examples of restricted workspace configuration.

Single object prediction approach: first case. (

Single object prediction approach: second case. (

First approach of the multi-object detection using a single laser sensor. (

Second approach of the multi-object detection using a single laser sensor.

Single object prediction by a multi-laser disposition. (

Multi-object prediction using a multi-laser disposition. (