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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/).

Many analyses of acoustic signals processing have been proposed for different applications over the last few years. When considering a bar-based structure, if the material through which the sound waves propagate is considered to be acoustically homogeneous and the sound speed is well known, then it is possible to determine the position and time of impact by a simple observation of the arrival times of the signals of all the transducers that are strategically disposed on the structure. This paper presents a generalized method for impact detection and location on a flat plate, together with a calibration procedure with which to obtain the sound speed from only one set of measurements. This propagation speed is not well known as a result of either imprecise material properties or the overlapping of longitudinal and transversal waves with different propagation velocities. The use of only three piezoelectric sensors allows the position and time of impact on the flat plate to be obtained when the sound speed is well known, while the use of additional sensors permits a larger detection area to be covered, helps to estimate the sound speed and/or avoids the wrong timing of difference measurements. Experimental results are presented using a robot with a specially designed knocking tool that produces impacts on a metallic flat plate.

Many analyses of acoustic signals measurement and processing have been carried out over the last few years for different and diverse purposes. Examples of experimental applications such as damage detection, failure prevention or interactive human-machine interfaces [

In other applications, when the estimation of impact forces on a structure or on a machine is required, the knowledge of the position in and time instant at which the impact was produced must be determined beforehand. Various methods have been proposed in literature. For example, Martin and Doyle [

The problem of time delay estimation of acoustic signals has been solved by using correlation methods [

However, if impacts are produced on a thin metallic plate (an acoustically isotropic material), the speed of propagation of the waves is not well determined because of the coupling of the longitudinal and transversal waves with different speeds during their propagation [

This paper presents a very simple algorithm with which to detect and locate an impact on a thin metallic plate based on simple differences of TOA, even when the speed of propagation is not well determined. It is based on triangulation techniques and consists of the evaluation of all the possible intersections between hyperbola branches with signs and the computation of the sum of the squared distances from which average speed of propagation can be obtained as an optimization algorithm with not too many iterations.

The paper is organized as follows: Section 2 details the algorithm for impact detection and location on a flat plate when the sound speed is known, and the algorithm for estimation of the speed of propagation is also presented. In Section 3, an experimental setup based on a robotized system is briefly described, and some experimental results are illustrated in Section 4 in order to validate the effectiveness of the proposed method. Finally, in Section 5, some conclusions and suggestions for further work are briefly outlined.

A flat plate of a homogeneous material as regards its acoustic properties (e.g., metallic) is considered. The flat plate is also considered to be isotropic. Piezoelectric sensors are located at known positions. Without any loss of generality, a sensor denoted as 1 is placed at the origin of a 2D Cartesian frame. The point at which the impact is produced is defined as P_{Im} = (x_{Im} y_{Im})^{T} and the position of the j-th sensor with regard to the absolute location of the i-th sensor P_{i}, i = 1, 2, … n − 1 is denoted as P_{ij} = (x_{ij} y_{ij})^{T} for i = 1, 2, … n − 1, i < j ≤ n, where P_{ii} = (x_{ii} y_{ii})^{T} = (0 0)^{T}. If the constant sound velocity C is known, then the propagation time of the fundamental acoustic wave from P_{Im} to each of the n sensors is computed as:
_{i Im}_{Im} and the position of the i-th sensor.

These times cannot be directly measured because the instant at which the impact was produced is not known. An analog electronic conditioner located near the piezoelectric sensors that was developed for up to 10 piezoelectric sensors allows the acoustic wave received by each of the piezoelectric sensors to be converted into a rising edge voltage signal. This first stage of each of the individual conditioners is responsible for impedance adaptation, current amplification, adjustable level detection and the differential transmissions of edge signals.

After this initial analog processing and wire transmission, an opto-isolation intermediate stage provides electric isolation and conversion into TTL digital signals with constant and deterministic delays. Differences in times of arrival (TOA) can then be obtained by simple time counting using digital counters with a high frequency clock signal. The digital stage of the conditioner was developed under the M-module standard [

After computing the time differences, any of the sensors may be used as a reference sensor. From any pair of sensors i and j, the time differences are signed values according to:

where ‖ ‖ denotes Cartesian norm and C, denotes the sound speed propagation which is considered to be well known. These time increments are signed times according to:

In the most general form, when an impact is detected by any of the sensors, a set of n_{H} differences of time is obtained, where n_{H} is:

These time differences can be converted into distance differences by using the following expression:

According to

A hyperbola, meanwhile, is a conic curve that can be defined as the locus of points where the differences of the distances to the two points called foci is a constant 2. a, where a retains the geometrical meaning of the semi-major axis. The hyperbola is thus composed of two branches, one of which corresponds with the positive constant differences and the other of which corresponds with the negative constant differences.

After computing these time differences, they can be grouped and sorted from lowest to highest absolute value in order to process the information with regard to the sensor that is nearest to the impact, as occurred in [_{H}. From this point onwards, the first sensor is considered to be the nearest to the point P_{Im} at which the impact is produced, and the first subscript of the sensor locations is avoided (P = P_{i}) for the sake of clarity.

In accordance with the same above _{2} = (0 100)^{T} distance units) determines the foci of the family of the hyperbolas, while the signed distance given by

If n sensors are located on a flat plate and all combinations are considered, the impact position is obtained with a simple computation of the intersections between n − 1 independent signed hyperbola branches. The maximum number of intersections is denoted by n_{I} and it is calculated as:

The minimum number of sensors needed for impact location on a plate is therefore n = 3, the number of independent signed hyperbola branches is n − 1 = 2, and the number of intersections is n_{I} = 1.

_{1} = (0 0)^{T}, P_{2} = (100 30)^{T} and P_{3} = (−20 100)^{T}, and the impact position is simulated at P_{Im} = (26 38)^{T} (denoted as a large black “+”).

Three segments of signed hyperbola branches, denoted as H_{ij} for i = 1, 2, i < j ≤ 3, have also been plotted in

The proof of this result is inmediate. In accordance with

Increment Δt_{23} can be expressed as a linear combination of increments Δt_{12} and Δt_{13} as:

Then, only two differences of times are independent and only one hyperbola branch intersection is produced.

If four sensors are used,

_{4} = (90 90)^{T}.

Once the impact position is known, if it is desired to calculate the absolute time instant t_{Im} when the impact occurred, this is easily computed from any of the edge signals received according to the following equation derived from

Several research groups have noticed that when a thin plate is considered for impact detection and location, the speed of propagation is not well determined [_{I} as before) increases from n_{I} = 1 to the value given in

When n = 4 and the speed of propagation C is not well determined either, the number of intersections increases from n = 1 to

_{I} = 3 hyperbola branches are plotted when they are computed, assuming a perfect knowledge of the sound propagation speed C (continuous red line) and when the sound of propagation speed C is not accurately determined (dashed green and magenta lines). Each of these sets of three hyperbola branches produces n_{I} = 3 intersection points which form a triangle. These intersections are shown in the box surrounded area in

Each of the three points of each set of intersections is obtained as a function of the unknown speed of propagation c, in accordance with the following notation:

A quadratic function S which computes the sum of the squared distances among the points that determine the triangle is therefore defined as:

This function has a global maximum when c ≈ 0, a global minimum when c = C, where S = 0, and different local maxima and minima values when c > C. Function S is monotone decreasing from c ≈ 0 to c = C, which provides a criterion with which to find the global minimum with any search algorithm such as the linear-dichotomy search with a computational cost of order O(N log N), the size of the searching problem being N.

If five sensors are used for an impact point detection and location on a plate, then the number of intersections increases from 1 to

Finally,

The proposed theoretical procedure was validated by developing an experimental setup based on a robotized system. A Stäubli RX-130 robot and a special knocking tool added to the robot's wrist (designed and manufactured by the research group) were used to carry out the experimental validation. The knocking tool developed is shown in the top left of

The most interesting features of the Stäubli RX130 in our experimental validation are summarized as follows: valid workspace with six degrees of freedom, programmable motions at 13.2 m/s maximum speed of linear motion, programming facilities with which to produce precise and repetitive impacts on pseudo random pre-stored impacts points and ±0.03 mm repeatability.

A steel flat rectangular plate that was 645 mm long, 310 mm wide and 3 mm thick was supported by four mounts with acoustic isolating rubber. Four sensors were also located at the following locations (given in mm): P_{1} = (0 0)^{T}, P_{2} = (450 0)^{T}, P_{3} = (0 200)^{T} and P_{4} = (450 200)^{T}.

A description of the experiment is illustrated in

The robot spent an average time of 38 seconds on any set of 30 impacts, while the signals directly measured from the piezoelectric sensors were considered to be fully damped after about 120 ms in the worst case. These experimental values of knocking and natural signal attenuation times allowed all the impacts caused on the plate to be detected and located through the application of the procedure explained in Section 2, they were subsequently compared with the random points generated that had been sent from the robot controller via serial link (RS-232) to the computer. The piezoelectric sensors were fixed and pressed against the plate (not shown in

^{©}, the Instrument Control Toolbox and Tektronix VISA software. The increments of time measured for this example were Δt_{12} = −185.6 μs, Δt_{13} = −58.4 μs and Δt_{14} = −202.8 μs.

The impact point generated by the robot controller and then produced for the exposed example is P_{Im} = (58.82 38.56)^{T} mm with an assumed error of ±0.03 mm (robot repeatability), while the speed of sound is considered to be completely unknown. _{min} = 1,300 m/s while the maximum speed is c_{MAX} = 7,000 m/s.

After 55 iterations, the algorithm is stopped because ^{T} mm, and exhibits an absolute error of 0.214 mm with regard to the real impact point P_{Im} = (58.82 38.56)^{T} mm which is considered to be a good correspondence.

_{Im} is the same as that shown in

Finally, in order to provide a better insight into the efficiency of the proposed methodology, forty tests were implemented (40 tests × 30 impacts/test = 1,200 impacts) showing the results illustrated in

It is well known that if the speed of propagation of acoustical waves on a flat plate is established, then only three piezoelectric sensors are necessary to determine the position and the time instant of an impact on the flat plate. The impacts on the flat plate produce acoustic waves on the material which are detected by the piezoelectric sensors. The propagation of the acoustic waves on a plate is usually modeled as a function of the vibration modes excited by the impact, and the superposition of both, longitudinal and transversal waves, with different propagation velocities causes inaccuracies in the achievement of the true value of the speed of propagation. In these cases, it is not possible to obtain the position in and the time instant at which the impact is produced from the signals collected from only three sensors.

An iterative algorithm based on a function which quantifies the quadratic distance between the intersection points of the hyperbola branches for different values of the unknown speed of propagation allows us to obtain the accurate value of the propagation velocity of the acoustic waves and to then determine the position in and the time instant at which the impact was produced. Only four piezoelectric sensors are required to compute this function, and the optimum number of sensors is consequently considered to be four. Moreover, if the number of sensors is increased, then the effect of incorrect time measurements or sensor failures can be solved after processing the information attained from the other piezoelectric sensors.

The proposed algorithm has been validated with an experimental setup based on an industrial robot with a specially designed knocking tool and a three-stage circuit with which to convert acoustic signals into edge voltage signals let us validate the proposed algorithm. The experimental results demonstrate the good correspondence between the randomly generated impact points and the estimated impact points with the proposed estimation method.

Finally, in the future it will be necessary to explore and resolve a mathematical convergence analysis of the proposed quadratic function in order to determine that the proposed function exhibits one and only one global minimum to allow more efficient search algorithms to estimate the speed of propagation to be developed, and these are proposed as topics for our future research.

This research has been partially supported by the JCCM (Spain) under contract no. PAI06-0075 and by the Spanish Ministerio de Economía y Competitividad under Research Grant DPI2011-24113. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper, and José A. Somolinos wishes to thank the Technical Staff at the UCLM for manufacturing the knocking tool.

The authors declare no conflict of interest.

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Family of hyperbolas branches for a given pair of sensors.

Sensor location, impact point and hyperbola intersection.

Sensor location, impact point and hyperbola intersection.

One set (correct) and two sets (not correct) of intersections with different computed speed of propagation C.

S as a function of c for 50 ≤ c ≤ 8,750 m/s in the above example. C = 3,000 m/s.

Temporal determination of C = c = 3,000 m/s for the above example.

Sequence of intersection points for 6, 9, 20 and 32 iterations.

Knocking tool and its integration into the robotized setup.

Experimental setup schematic view

_{1} and S_{2}; (_{1} and S_{3}; (_{1} and S_{4}.

Convergence of S function around P_{Im} (distance units in m).

Experimental impact detection and location for the given example (distance units in m).

Results of the proposed methodology after 1,200 impacts.