Method of localising and tracking an object

Abstract

A method of detecting a location, a translation and / or an orientation of a remote object relative to an ego-object, each having a set of sensors arranged at respective outer surfaces or perimeters is presented. After determining pairwise inter-object distances between sensors of the ego-object and sensors of the remote object, the relative location, translation and / or orientation are determined using multi-dimensional scaling and double-centring operations. The method does not require knowledge of the sensor positions of the remote object, and the ego- and remote objects may have different sizes and numbers of sensors.

Description

[0001]202402559 -1- METHOD OF LOCALISING AND TRACKING AN OBJECT FIELD OF THE INVENTION The present invention relates to object localisation and tracking, in particular to localisation relative to an observer that also provides information about a shape and / or size of the object. BACKGROUND Wireless localization can be seen as a precursor of joint communication and sensing (JCAS), demonstrating how communication signals can also be used for sensing an environment, including localization of users. There are many types of information that can be extracted from radio signals for the purpose of localization, including finger- prints, received signal strength indicator (RSSI), angle of arrival (AoA), or delay- based estimates of radio range. Conventionally, such information needed for localization was generally assumed to be obtained by specialized equipment and protocols, requiring the transmission of dedicated signals, implicating in cost- and other constraints which in turn explains the predominance in related literature of methods to find the position of individual points. Recently, however, advances in JCAS technology has demonstrated that radar parameters, i.e., range, bearing and velocity, can be acquired by conventional communications signals not only actively, i.e., using signals transmitter by the target to the sensors, but also passively, i.e., using round-trip reflections of signals transmitted by the sensors themselves, which in turn implies a more abundant and richer availability of positioning information. A consequence of this development is an increasing interest in the rigid body (RBL) problem, described for example by Y. Wang, G. Wang, S. Chen, K. C. Ho, and L. Huang, in "An investigation and solution of angle based rigid body localization," IEEE Transactions on Signal Processing, vol.68, pp.5457-5472, 2020, or N. Führling, H. S. Rou, G. T. F. de Abreu, D. González González, and O. Gonsa, in "Soft-connected rigid body localization: State-of-the-art and research directions for 6G,"2023, whose objective is to determine not only the average location of targets, but their shape and orientation, based on a collection of points sufficient to define the object. This feature of RBL is particularly attractive to vehicle-to-anything (V2X) networks, where - unlike earlier 202402559 -2- applications of positioning technology such as asset management in industrial settings and people tracking in indoor settings - information on the size, shape, and orientation of vehicles are crucial to ensure the efficacy and safety of autonomous driving (AD) applications such as collision detection, navigation, and vehicle path prediction, to name only a few examples. It is important also to distinguish between the type of RBL system here addressed, which is based on radio signals, possibly under a JCAS paradigm, and conventional simultaneous localization and mapping (SLAM) technologies relying on dedicated equipment and massive amounts of data to function, which makes the latter less likely to be useful in AD applications envisioned for a future where autonomous vehicles (AVs) will be widely deployed. An example of the radio-based RBL approach which is the subject of the present invention is the method discussed by S. Brás, M. Izadi, C. Silvestre, A. Sanyal, and P. Oliveira, in "Nonlinear observer for 3D rigid body motion estimation using doppler measurements," IEEE Transactions on Automatic Control, vol.61, no.11, pp.3580- 3585, 2016, where the pose, angular velocity and trajectory of a rigid body is estimated using Lyapunov functions of Doppler measurements, obtained by a nonlinear observer. Another example is discussed by S. Chen and K. C. Ho, in "Accurate localization of a rigid body using multiple sensors and landmarks," IEEE Transactions on Signal Processing, vol.63, no.24, pp.6459-6472, 2015, in which a two-stage approach is used to estimate rotation, translation, angular velocity and translational velocity by range and Doppler measurements, making use of various weighted least square (WLS) minimization methods. And going beyond the problem of RBL involving a single object, the scheme discussed by A. Pizzo, S. P. Chepuri, and G. Leus, in "Towards multi-rigid body localization," 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2016, pp.3166- 3170, which, after an earlier presentation of an anchor-based scheme, propose a new relative multi-object RBL method in an anchorless scenario, where the relative translation and rotation between two rigid bodies is estimated by measuring the cross-body line-of-sight (LOS) distances between the points defining the two bodies. 202402559 -3- The latter case relates to a common scenario in AD where a vehicle is able to measure the distance between itself and vehicles in its surroundings, such that the corresponding RBL solution would find a large, direct and crucial application. Unfortunately, however, most state-of-the-art (SotA) RBL methods assume that the shape of the target rigid body is known, which is unrealistic in real life applications since vehicles vary greatly in shape and size. It is, therefore, desirable to obtain an anchorless RBL method that can estimate not only the distance, but also the shape and orientation of another object, e.g., another vehicle, possibly of different size and shape. SUMMARY OF THE INVENTION This object is attained by the method of claim 1 and the apparatus of claim 6. Advantageous embodiments and developments are provided in the respective dependent claims. A computer program product and a corresponding computer- readable medium are provided in claims 8 and 9, respectively. Prior to discussing the method in accordance with the invention, an underlying system model will be discussed, referring to a system as exemplarily shown in figure 1. Figure 1 shows an exemplary rigid body to be localised and tracked, here a vehicle, at two distinct locations ^ and ^(^). In the system model the rigid body is represented by a collection of ^ landmarkpoints ^ ^×^^ ∈ ℝ in the three-dimensional (3D) space, with ^ = {1, ⋯ , ^}, such thatthe shape of said body is well described by the corresponding conformation matrix ^ constructed by the column-wise collection of the vectors ^^with reference to the 3D space defined by the spatial dimensions q1, q2, and q3. Note that the landmark points are represented by the small antenna symbols located the vehicles in the figure. Landmark points are preferably located at extreme positions of the rigid body, e.g., corners and edges. Then, consider the representation of the location ^ of said rigid body relative to another location ^(^), e.g., a temporally earlier location in case the body is in motion, which without loss of generality can be set to be a "canonical"reference, i.e., centred at the absolute origin, such that ^ = ^ and thus one canwrite 202402559 -4- ^ where ^ ∈ ℝ^×^ is a translation vector given by the difference of the geometric centersof the body at the two locations, ^^is a column vector with ^ entries all equal to 1,and ^ ∈ ℝ^×^ is a rotation matrix determined by corresponding yaw, pitch and rollangles !, " and #, respectively, namely^ ≜ ^' = Note that, for the sake of simplicity, herein detecting the orientation of a rigid body will be interpreted as estimating of the 9 elements of the corresponding rotation matrix ^ as a whole. Next, consider a scenario as illustrated in figure 2, in which two rigid bodies,hereafter referred to by their indices 5 = {1,2}, have generally different shapes and / orare characterized by generally distinct numbers ^^and ^4of landmark points, respectively, such that under a common absolute reference, the bodies arerepresented by the corresponding distinct conformation matrices ^^ ∈ ℝ^×^6and^4 ∈ ℝ^×^7. The translation vector ^ between the bodies, depicted in a dash-dottedline, is defined by the difference between the geometric centers of the two bodies.Since ^^ ≠ ^4, it is obvious that in such a scenario the location of one body relative tothe other cannot be described in terms of equation (1). A common problem in V2X systems with relevance to AD applications is, however, that one rigid body - say, the truck in figure 2- is able to estimate not only its distance to the other - in this case, the 202402559 -5- car in figure 2 - but also its shape and orientation, based on a set of measurements of the distances between their corresponding landmark points. It will be considered, in what follows, that such measurements can be obtained by having deployed, at the landmark points of each rigid body, a set of wireless transceivers, hereafter referred to as "sensors", capable of performing such mutual distance estimates. As discussed further above, the setup with sensors deployed in each rigid body can be replaced by a setup in which each rigid body is equipped with radar or JCAS technology capable of measuring distances from a set of points in one body to a another set of points in the other. In this case, since corners and edges are most likely to reflect wireless signals, the corners and edges correspond to the sensors. Thus, the method presented herein mathematically applies to both setups. It will further be assumed that each body is only aware of its own shape, described bycorresponding conformation matrices ^ ^×^9 = :^9,^, ⋯ , ^9,^ ;;< ∈ ℝ , where ^9,^, is thelocation of the ^-th point of the 5-th body, with respect to its geometric center. When subject to unbiased estimation errors, the estimates of the distance between a pair of sensors =^,^on the first body, and =4,>on the second, can be described by where ?^,>≜∥∥=^,^− =4,>∥∥4is the true pairwise distance between the sensors, whileA^,> denotes noise modelled as i.i.d. zero mean Gaussian random variables withvariance D4. Note that the notation E9,^is used herein as a reference to both, the sensor and its location. In order to avoid negative numbers and to linearize the relationship between the acquired squared distances and corresponding measurement errors, we shall also consider the equivalent model 4 202402559 -6- where the mean and variance of the squared-distance estimation error F^,>arerespectively given described by S. P. Chepuri, A. Simonetto, G. Leus, and A.-J. van der Veen, in "Tracking position and orientation of a mobile rigid body," 20135th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2013, pp.37-40. It proves convenient, to collect the true distances ?^,>from above into the euclidean distance matrix (EDM) NN = O ^ N^4 P (^6Q^7)×(^6Q^7)N^ ∈ ℝ (5)^4 N4With the system and measurement model introduced above, the problem to be solved solve can be clearly defined. To that end, first observe that assuming, without loss of generality, that the rigid body 1, i.e., the truck, attempts to egoistically locate body 2, i.e., the car, the system model conditions described earlier translate to the assumption that the self intra-distance matrix N^is known exactly, the target intra- distance matrix N4is unknown, and the squared cross-distance matrix N^4can bewritten as⊙4 ^^ ^ ^ ^^ where ^^and ^4are matrices containing the locations of the sensors in bodies 1 and2 , respectively, the auxiliary vectors ^9 ≜ ^9 ^9 carry the the squared norms of the corresponding individual sensor locations, and the symbol ⊙ indicates an element- wise matrix operation, e.g., multiplication or exponentiation. Next, consider an augmented sensor location matrix carrying the positions of all landmark points in both bodies, such that we may write, in similarity to equation (1) ^^^ ^ 202402559 -7-where ^9 and ^9 respectively denote the rotation matrix and translation vector of the5-th body, while W^×^ , W^ and ^^ denote an all-zero matrix and an all-zero / all-onecolumn vector, respectively.Under the egoistic assumptions that ^^ = ^^, and ^^ = W^, however, equation (7)reduces to ^^^ where the notation is simplified by omitting subscripts that can be inferred fromcontext, which includes relabelling ^ = ^4 and ^ = ^4.The problem addressed by the present invention is therefore to estimate, with basis on equations (6) and (8), the rotation matrix ^ and translation vector ^, given perfect knowledge of the conformation matrix ^^- which implies exact knowledge of N^- and possession of an estimate of the matrix N^4subject to noise, under the egoisticcondition that ^4 is unknown and for a general case where ^^ ≠ ^4.A known approach for estimating the rotation matrix ^ is discussed in "Towardsmulti-rigid body localization" (ibid.), however with the assumptions that ^^ = ^4 and^4 is also known. Also, the known approach is ineffective for the estimation of thetranslation vector ^. Nevertheless, the known approach is useful for understanding the estimation of the rotation matrix ^.First, consider the ^ × ^ classic Schönberg double-centring matrix (DCM), definedby W. S. Torgerson in "Multidimensional scaling: I. theory and method," Psychometrika, vol.17, no.4, pp.401-419, Dec.1952. [Online]. Available: https: / / doi.org / 10.1007 / BF02288916 Left- and right-multiplying a measured distance matrix by the DCM [, and scaling theresult by − ^4, yields 202402559 -8- In order to facilitate the formulation of a problem to estimate ^, it proves convenient to apply an orthogonal Procrustes problem (OPP) onto equation (10), which under the assumption of perfect knowledge of ^4can be achieved by defining N⊙̌4 ≜ ⊙4 ^^4 N^̂4 ^4̀ = ^^^, (11a)where^ ^^ ^ Then, the relative rotation ^ of body 2 with respect to the orientation of body 1 can be estimated by solving the problem 4^êff = arg which can be obtained in closed form via singular value decomposition (SVD) of the matrixIn particular, the solution of the problem represented by equation (12) is given by^̂ = ^eff mn (13a)with m and n such that^ N⊙̌4 = ^^ ^4 mon (13c)It is emphasized that although it was assumed in "Towards multi-rigid body localization" (ibid.) that both rigid bodies have the same number of landmark points,i.e., ^^ = ^4, the notion of a relative rotation between two bodies of different shapesand number of landmark points is geometrically well defined, as can be inferred from equation (8). In particular, by aligning the rotation matrix of the first rigid body with thecartesian coordinates, such that ^^ = Y, the orientation ^4 of the second body withrespect to the first, becomes simply the relative rotation itself. In other words,^ ^^ = Y ^ ^4 = ^, or more generally, ^ = ^^ ⋅ ^4. 202402559 -9- The assumption of pre-existing knowledge of the conformation matrix ^4, which is typical for the SotA RBL methods is hard to meet in practical conditions. In AD- related V2X applications, for instance, that would require that a vehicle attempting to locate other vehicles in its vicinity is aware of their shapes, an obviously impractical requirement given the enormous diversity in vehicle models, which are also are constantly updated. In order to mitigate this problem, the present invention proposes methods to estimate ^ and ^, respectively, without the requirement that ^4is known. For the estimation of the translation vector ^ it is crucial to acknowledge that not knowing ^4implicates not knowing the intra-distances matrix N4. And while thereverse implication is not logically true - i.e., in principle one could have knowledge ofN4 but not ^4 - the assumption that N4 is also not available to the rigid body 1 isconsistent with the egoistic principle assumed herein, as indeed, an assumption of knowledge of N4would require that the target vehicle broadcasts such information. Notice that an ^-point 3D conformation matrix contains 3^ entries, while thecorresponding intra-distance matrix contains ^(^ − 1) / 2 distinct entires, such thatthe intra-distances data is larger than the conformation data for ^ > 7, which is asmall number of points to define a rigid body in 3D. In what follows, therefore, it is assumed that N4is not known. Under such conditions, the first problem at hand is one of matrix completion, and although several methods to solve such a problem exist, a number of which could be used, here the simple and well-known Nyström approximation method as presented by C. K. I. Williams and M. Seeger, in "Using the Nyström method to speed up kernel machines," Proceedings of the 13th International Conference on Neural Information Processing Systems, ser. NIPS'00. Cambridge MA, USA: MIT Press, 2000, p.661- 667, is considered, which, applied to the EDM N from equation (5), yields thefollowing estimate of N4N4̂ ≈ ℍ[N^^4 Nb^^ N^4] (14)where ℍ[⋅] denotes a hollowing operator that enforces all elements of the diagonal matrix to be zero. Note that the Nyström approximation in general only works if therank (N^) ≥ rank (N4), which means that the first body must have at least the same 202402559 -10- amount of sensors as the second body. If that condition is not satisfied, alternative matrix completion methods may yield better results.In possession of the intra-distances matrix of the first body N^, the measurementsN^̃4corresponding to the distances between the two bodies, and the latter estimateN4̂ of the intra-distances matrix corresponding to the second rigid body, the fullsample EDM corresponding to all distances within and between the two rigid bodies can be reconstructed as such than a multi-dimensional scaling (MDS)-based first estimate of the position of all sensors from both rigid bodies can be obtained as described by W. S. Torgerson, in "Multidimensional scaling: I. theory and method" (ibid.), i.e., as :^∗̂ , ^∗4̂ < ^ / 4^ = nx (16)where n and x are the eigenvector and eigenvalue pairs of the correspondingdouble-centred EDM, that isN‾ = nxn^ (17)with 1N̂ = −2 [ ⊙4^6Q^7N̂ [^6Q^7 (18)where [^6Q^7 is + ^4)-point Schönberg DCM built as per equation (9). The initial MDS solution given by equation (16) can then be brought to the reference frame of the first rigid body via a Procrustes transformation by solving (^∗, ^∗) arg ∗ ^ ^^ from which 202402559 -11- ^̂ = :^ , ^̂< ∗ ∗ ∗ ^^ 4 = :^^, ^ ^4̂ + ^ ⊗ ^^7 < (20)can be obtained or, more explicitly ^∗^∗̂ ^∗ ^^ Substituting the latter result into equation (8) and using the relation ^4^4 = ^4̂ [^7yields ^^ Utilizing the latter expression, finally a quadratic program can be formulated to find the translation vector ^, namely 4 which can easily be solved by common optimization tools, such as gradient descent or interior point methods. With the estimate ^4̂ obtained via equation (21) in hand, a robust estimate of the rotation matrix ^ corresponding to the second rigid body can be obtained. In principle, ^̂ can be extracted by ^^ ^ ^ using ^4^4 = ^4̂ [^7 in the last equality.Notice, however, that the eigenvalue decomposition in equation (24) is such that the eigenvectors are ordered according to their corresponding eigenvalues, which in turn relate to the largest orthogonal dimensions of the body itself. It follows that the columns of the estimate obtained from equation (24) may be swapped for rigid 202402559 -12- bodies with approximately spherical shapes, leading to large overall estimation errors. In accordance with the invention a different method is used instead, starting from equation (10), but this time accounting for the fact that ^^and ^4have different numbers ^^and ^4of landmark points, such that Which, if left-multiplied with the pseudo-inverse of ^^^yields ^ where^ Then, squaring equation (26) yields ^^ ^^^^ from which the following optimization problem can be constructed 4^̂ = ^ It is emphasized that although the solution of the problem represented by equation (28) can be easily obtained via common optimization theory tools, the result can also be severely degraded by the order of the eigenvalues in x. Fortunately, however, in 3D there are only 6 distinct permutations of x, such that the solution with the permutation that yields the smallest objective can be assumed to be the correct one. In the following section simulation results illustrating the performance of the contributed egoistic MDS-based RBL technique are provided with reference to figures 5 and 6. 202402559 -13- Since no equivalent SotA method exists for the egoist set-up with unknown ^4that is considered herein, in figure 5 only results of the translation vector estimation via the non-egoistic method discussed in "Accurate localization of a rigid body using multiple sensors and landmarks" (ibid.) are compared with the translation estimation method proposed herein, but using the estimate of ^ from the SotA method in equation (22). For the sake of disambiguation, the corresponding results of the proposed method with an externally fed rotation matrix is referred to as the "Genie-Aided" (GA) scheme. The performance metric of choice is the RMSE, as a function of the ranging error D, namely ^ denotes an estimate obtained at a ^-th realization. It is emphasized that the dependence of } on D is due to the errors ^(̂^), not included explicit in the notation for simplicity. Note that the ranging error is not equivalent to the exact error in meters but rather the error used in the noise calculations given in equation (4). Each point in the figures is obtained by averaging ^ = 10^ Monte-Carlo realizations,using the system parameters described in Table I: The algorithms are implemented in MATLAB, with the minimization problems solved using the CVX optimization package. 202402559 -14- The results shown in figure 5 illustrate that in a non-egoistic scenario, the proposed method outperforms the SotA alternative if ranging errors are below 20 cm, which is well within the typical values of sensing technology used in the Automotive Industry. A comparison between the latter Genie-Aided method and the actually proposed egoistic method is shown in figure 6, the results shown therein confirming that the proposed egoistic method achieves a performance close to that of the GA non- egoistic alternative. In light of the discussion above and in accordance with a first aspect of the invention a method of detecting a location, a translation and / or an orientation of a remote object relative to an ego-object is presented. The ego-object and the remote object each have a set or a plurality of sensors arranged at respective outer surfaces or perimeters. The sensors of the remote object may be active sensors, transmitting and / or receiving electromagnetic signals, or may be features that reflect electromagnetic signals in a way that permits exploiting them for the purposes of the invention, e.g., edges or corners. The sensors of the ego-object are capable of at least receiving electromagnetic signals transmitted or reflected by the sensors of the remote object. The method comprises, for detecting the relative location, translation and / or orientation independent of the shape and / or size of the remote object, determining, using wireless signals, respective pairwise inter-object distances between sensors of the ego-object and sensors of the remote object, considering known intra-object distances between the sensors of the ego-object. The method further comprises receiving intra-object distances between the sensors of the remote object, or estimating such intra-object distances based on the determined pairwise inter-object distances and the known intra-object distances between the sensors of the ego- object. Receiving the intra-object distances between the sensors of the remote object may comprise receiving such information from a local or remote database, or obtaining the information from a transmission received from the remote object. Estimating the intra-object distances may comprise applying a Nyström approximation. Based on this available information, i.e., from the received or 202402559 -15- estimated intra-object distances and the determined pairwise inter-object distances, a full Euclidian distance matrix (EDM) is reconstructed. In a next step, a rotation matrix for the remote object is estimated. Estimating the rotation matrix may comprise mapping the positions of all sensors of the ego-object and the remote object represented in the reconstructed full EDM into a Cartesian space, and transforming the positions of all sensors in the Cartesian space to a reference frame of the ego- object. Next, respective central reference points for the ego-object are determined based on the known intra-object distances or a known underlying conformation matrix, and for the remote object based at least on the full EDM and the previously estimated rotation matrix. Finally, a location or translation vector between the ego- object and the remote object based on the respective central reference points for the ego-object and the remote object is determined. Alternatively or in addition a distance between the ego-object and the remote object is determined based on the respective central reference points and the respective known or estimated intra-object distances. In the alternative determination, a minimum distance between a sensor pair of the ego-object and the remote may be of particular interest for autonomous driving, notably for collision avoidance. In one or more embodiments of the method mapping the positions of all sensors of the ego-object and the remote object represented in the reconstructed full EDM into a Cartesian space comprises applying multi-dimensional scaling (MDS). In one or more embodiments of the method transforming the positions of all sensors in the Cartesian space to a reference frame of the ego-object comprises applying a Procrustes transformation. In one or more embodiments of the method determining respective central reference points for the ego-object and / or the remote object comprises applying a double- centring relation. In accordance with a second aspect of the invention an apparatus for detecting a location, a translation and / or an orientation of a remote object relative to an ego- object is presented. The apparatus comprises a plurality of sensors arranged at outer surfaces or a perimeter of the ego-object. The sensors are configured for at least 202402559 -16- receiving electromagnetic signals, in particular communication or radar signals, and comprise corresponding radio frequency processing components and circuits. The sensors may further comprise analogue-to-digital converters, for converting the received electromagnetic signals into the digital domain for further processing and / or transmission to processing blocks of the apparatus. Some or all sensors may also be configured for transmitting electromagnetic signals. The apparatus may also comprise one or more a transmitters for transmitting electromagnetic signals, which are not sensors. The apparatus further comprises one or more microprocessors, and associated volatile and non-volatile memory. The sensors, the and the memory are communicatively coupled to the one or more microprocessors via one or more signal and / or data lines and / or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure the apparatus to execute embodiments of the method in accordance with the first aspect of the invention, and to accordingly control hardware components of the apparatus 300. The methods described hereinbefore may be represented by computer program instructions. Accordingly, a computer program product comprises computer program instructions which, when executed by a microprocessor of a transmitter, cause the microprocessor to execute the method in accordance with first or the second aspect of the invention as presented above and to accordingly control hardware components of the apparatus in accordance with the second aspect of the invention. The computer program instructions may be retrievably stored or transmitted on a computer-readable medium or data carrier. The medium or the data carrier may by physically embodied, e.g., in the form of a hard disk, solid state disk, flash memory device or the like. However, the medium or the data carrier may also comprise a modulated electro-magnetic, electrical, or optical signal that is received by the computer by means of a corresponding receiver, and that is transferred to and stored in a memory of the computer. While in the conventional methods anchors are required for obtaining the full set of location, size , orientation – or rotation –, translation, and tracking, of a rigid body, the method in accordance with the present invention dispenses with the need for such 202402559 -17- anchors. Further, the method in accordance with the present invention does not mandate identical numbers of sensors for relative localisation, and provides the desired information solely based on cross-body or cross-object sensor-to-sensor range measurements. The method in accordance with the present invention addresses two distinct but related problems. The first problem considers two different vehicles that can have a different number of sensors on them, which can perform cross-body distance measurements knowing the reference topology of each other. By assuming that vehicle two was at the position of vehicle one are some specific time, the rotation and translation of vehicle two can be estimated assuming that at the specific time, the rotation of vehicle one is zero, i.e., an identity matrix and the translation is zero. The second problem assumes that the two vehicles are the same at two different instants in time, where the distance is also known, with the same assumptions as before, such that the translation and rotation can again be estimated. Thus, the anchorless RBL method proposed herein permits tracking the translation of a vehicle from its previous position, or a different vehicle, in an anchorless scenario by only knowing the topologies of the rigid bodies and the cross-body distance measurements. The method can be used with great advantage in autonomous driving, enabling a rigid body, e.g., a vehicle, to egoistically detect the relative translation, i.e., the effective distance, and orientation, i.e., the relative rotation, of another body, e.g., another vehicle, based only on a set of measurements of the distances between sensors of one rigid body to the other and without knowledge of the shape of the latter. A key point of the method proposed herein is that the translation vector between the two-bodies is modelled using the MDS double- centring operator, enabling its applicability between rigid bodies of different shapes, in contrast to conventional approaches which require both bodies to have the same shape. Simulation results illustrate the good performance of the proposed method in terms of RMSE as a function of the ranging error, in the desired moderate to low ranging errors regime. BRIEF DESCRIPTION OF THE DRAWING 202402559 -18- Fig.1 exemplarily shows localising and tracking the same rigid body at two distinct locations ^(^)and ^(^),Fig. 2 exemplarily shows localising two different rigid bodies at two distinct locations^(^) and ^(^),Fig.3 shows an exemplary schematic flow diagram of a first embodiment of the method in accordance with the invention, Fig.4 shows an exemplary schematic flow diagram of a second embodiment of the method in accordance with the invention, Fig.5 shows a comparison of the RMSE of the translation estimate of the GA proposed method and the SotA, over the range error D, Fig.6 shows a comparison of the RMSE of the translation estimate of the proposed method compared to the GA variation, over the range error D, and Fig.7 shows an exemplary block diagram of an apparatus in accordance with embodiments of the second aspect of the present invention In the figures identical and similar elements may be referenced using the same reference designators. DETAILED DESCRIPTION OF EMBODIMENTS Figures 1, 2, 5 and 6 have been described further above and will not be discussed again. Figure 3 shows an exemplary schematic flow diagram of a first embodiment of the method in accordance with the invention. In this embodiment of the method, the cross-object pair-wise distance measurements are provided to a conventional method for estimating a rotation matrix ^4for the remote object solely based on the cross-object pair-wise distance measurements. The cross-object pair-wise distance measurements are further provided to a matrix reconstruction process that yields a full Euclidian distance matrix (EDM), based further on the intra-object sensor distance measurements. The full EDM and the estimated rotation matrix ^4are then submitted to a double-centring process, and the output thereof is optimised for the translation vector. 202402559 -19- Figure 4 shows an exemplary schematic flow diagram of a second embodiment of the method in accordance with the invention. In this embodiment, the cross-object pair-wise distance measurements are provided to a matrix reconstruction process that yields a full Euclidian distance matrix (EDM), based further on the known intra- object sensor distances for the ego-object, and the approximated intra-object sensor distances for the remote object, which approximation is based on the cross-object pair-wise distance measurements and the known the intra-object sensor distances for the ego-object. The full EDM is then submitted to a process branch that determines or estimates a rotation matrix ^4for the remote object thereon, using mapping and transformation processes, including multi-dimensional scaling (MDS) and mean centring, further being based on the known sensor positions ^^of the ego- object. Translation vectors are then determined based on the full EDM and the determined or estimated rotation matrix ^4for the remote object, using a double- centring process, and the output thereof is optimised for the translation vector. Figure 7 shows an exemplary block diagram of an apparatus 300 in accordance with embodiments of the second aspect of the present invention. The apparatus 300 comprises a plurality of sensors 356 arranged at its outer surfaces or its perimeter, the sensors being at least configured for receiving electromagnetic signals. The apparatus 300 further comprises one or more microprocessors 350, volatile memory 352, and non-volatile memory 354. The aforementioned elements are communicatively connected via one or more signal or data connections or buses 358. The non-volatile memory 354 stores computer program instructions which, when executed by the microprocessor 350, cause the transmitter 300 to execute the method according to the first aspect of the present invention as presented herein, and to accordingly control hardware components of the apparatus 300.

Claims

202402559 -20- CLAIMS 1. Method of detecting a location, a translation and / or an orientation of a remote object relative to an ego-object, each having a set of sensors arranged at respective outer surfaces or perimeters, the method comprising, for detecting the relative location, translation and / or orientation independent of the shape and / or size of the remote object: - determining, using wireless signals, respective pairwise inter-object distances (N^4) between sensors of the ego-object and sensors of the remote object, considering known intra-object distances (N^) between the sensors of the ego- object, - receiving intra-object distances (N4) between the sensors of the remote object, or estimating the intra-object distances (N4̂) based on the determined pairwise inter-object distances and the known intra-object distances (N^) between the sensors of the ego-object, - reconstructing a full Euclidian distance matrix (EDM) (N)̂ from the received or estimated intra-object distances (N^, N4 / N4̂) and the determined pairwise inter-object distances (N^4), - estimating a rotation matrix (^4) for the remote object, - determining respective central reference points for the ego-object based on the known intra-object distances (N^) or a known underlying conformation matrix (^^), and for the remote object based at least on the full EDM and the estimated rotation matrix (^4), and - determining a location or translation vector (^) between the ego-object and the remote object based on the respective central reference points for the ego- object and the remote object and / or determining a distance between the ego- object and the remote object based on the respective central reference points and the respective known or estimated intra-object distances.

2. The method of claim 1, wherein estimating the intra-object distances (N4̂) between the sensors of the remote object comprises applying a Nyström approximation.202402559 -21- 3. The method of claim 1 or 2, wherein estimating a rotation matrix (^4) for the remote object comprises: - mapping the positions of all sensors of the ego-object and the remote object represented in the reconstructed full EDM into a Cartesian space, and - transforming the positions of all sensors in the Cartesian space to a reference frame of the ego-object.

4. The method of claim 3, wherein mapping the positions of all sensors of the ego-object and the remote object represented in the reconstructed full EDM into a Cartesian space comprises applying multi-dimensional scaling (MDS).

5. The method of claim 3 or 4, wherein transforming the positions of all sensors in the Cartesian space to a reference frame of the ego-object comprises applying a Procrustes transformation.

6. The method of any one or more of claims 1 to 5, wherein determining respective central reference points for the ego-object and / or the remote object comprises applying a double-centring relation.

7. Apparatus (300) for detecting a location, a translation and / or an orientation of a remote object relative to an ego-object, the apparatus comprising a plurality of sensors (356) arranged at outer surfaces or a perimeter of the ego-object, the sensors being at least configured for receiving electromagnetic signals, further comprising one or more microprocessors (350), and associated volatile (352) and non-volatile memory (354), the sensors (356) and the memory (352, 354) being communicatively coupled to the one or more microprocessors (350) via one or more signal and / or data lines and / or buses (358), wherein the non- volatile memory (354) stores computer program instructions which, when executed by the one or more microprocessors (350), configure the apparatus (300) to execute the method of one or more of claims 1 to 6.

8. Computer program product comprising computer program instructions which, when executed by a microprocessor (352) of an apparatus (300) according to claim 7, cause the microprocessor (352) to execute the method and to202402559 -22- accordingly control hardware components of the apparatus (300) in accordance with the method of one or more of claims 1 to 6.

9. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 8.

10. A vehicle comprising an apparatus (300) as claimed in claim 7.

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