Method of estimating motion parameters of an object

Abstract

A method of estimating motion parameters of an object having sensors at its outer surfaces or perimeters located in an observed space comprises determining the sensors' respective velocities and / or determining the object's angular and translational velocity. Determining sensor velocities of the object or the object's angular and a translational velocity comprises performing a Gaussian Belief Propagation, GaBP, process, for iteratively determining, using respective observations associated with anchors in the observed space, soft replicas of elements of a respective vector that comprises velocity estimates for each spatial direction and the product of a position vector and a velocity vector, for each sensor of the object, or elements of sets of variables representing angular and translational velocities of the object around each spatial axis and in each spatial direction. Final estimates of the elements of each of each vector or each set, respectively, are obtained through consensus belief combination.

Description

[0001] 202407774 -1-

[0002] METHOD OF ESTIMATING MOTION PARAMETERS OF AN OBJECT

[0003] FIELD OF THE INVENTION

[0004] The present invention relates to object localisation and tracking, in particular to determining motion parameters of an object, i.e., translational and angular velocity, respectively.

[0005] NOTATIONS

[0006] Scalar values are denoted herein by lowercase or uppercase letters in italics, as in x or N, while vectors and matrices are denoted by boldface italic lowercase and uppercase letters, as in and X, respectively. The transpose of a vector or a matrix are denoted by superscript letter [-]T. ® is the Kronecker product operator, and IE{-} indicates the expectation operation.

[0007] BACKGROUND

[0008] Recent years have seen great advancements in wireless sensor technologies, which are capable of detecting environmental parameters such as temperature, pressure, luminosity, humidity and the strength of electric signals, and directly transmitting them to one or multiple receivers comprising a wireless sensor network (WSN), for applications ranging from monitoring and control, to smart factories, to Internet-of-Things (loT), to positioning systems, and more. In fact, many WSN applications either require or can be improved under the knowledge of accurate location information, such that the sensor location problem has been studied extensively.

[0009] More recent sensing-related applications such as virtual reality (VR), extended reality (XR), robotics and autonomous vehicles, however, require not only precise location information of individual sensors, but also the orientation of the various sensors associated to a given body or object, giving rise to a variation of the positioning problem known as rigid body localisation (RBL), where the relative position of sensors that form a set of sensors is fixed with regard to a rigid body (RB) in accordance with a rigid conformation, whose translation and rotation, i.e., location and orientation in space, is to be determined. 202407774 -2-

[0010] Several effective strategies exist for estimating the location and orientation of objects, including computer vision-based techniques, which focus on feature extraction and posture estimation that are usually based on image / video signals, and inertial measurement unit (IMU)-based techniques, which leverage information from accelerometers, gyroscopes, and magnetometers associated with the RB. A problem of computer-vision approaches is, however, that they typically require high volumes of data and rely on high-complexity methods, which limits their application. In turn, IMU-based methods are often unreliable or inaccurate, requiring frequent sensor recalibrations, not to mention the aid of external radio technology, which to some extent defeats the self-reliant idea behind the approach.

[0011] In contrast to the foregoing strategies, some conventional RBL concepts aim to generalize the traditional localisation paradigm and, to this end, exploit range measurements between body sensors associated with the RB and a set of anchor sensors at known positions, for estimating the location, orientation, and possibly the shape, too, of RBs.

[0012] Earlier conventional RBL is based on either algebraic methods leveraging multidimensional scaling (MDS) or semidefinite relaxation (SDR), both of which have a cubic computational complexity at best. Although least squares methods offer lower complexity alternatives to solve the RBL problem, all the aforementioned conventional approaches have in common that they only take into account stationary scenarios, in the sense that they seek to estimate only the translation and the orientation of the rigid body, respectively defined as the distance and the rotation matrix of the rigid body relative to a given reference.

[0013] In contrast, in RBL schemes suitable for moving RBs, in addition to the translation and orientation, also the angular and translational velocities must be determined. Unfortunately, existing solutions for moving RBs are either of low computational complexity but also low precision, as is the case of the alternating minimisationbased method proposed by Q. Yu, Y. Wang, Y. Shen, and X. Shi, in " Cooperative Multi-Rigid-Body Localization in Wireless Sensor Networks using Range and Doppler Measurements," IEEE Internet of Things Journal, vol. 10, no. 24, pp. 22748-22763, 2023, or of high precision but also of high computational complexity, such as the 202407774 -3-

[0014] SDR method discussed by J. Jiang, G. Wang, and K. C. Ho, in " Sensor Networkbased Rigid Body Localization via Semi-Definite Relaxation using Arrival Time and Doppler Measurements," IEEE Transactions on Wireless Communications, vol. 18, no. 2, pp. 1011-1025, 2019.

[0015] It is an object of the present invention to provide a method of determining parameters of a moving rigid body representing an RBs sensor’s translational and rotational velocity, as well as translational and angular velocity of the RB itself, which exhibits low complexity and high performance.

[0016] SUMMARY OF THE INVENTION

[0017] This object is attained by the method, and the apparatus implementing the method, presented in the claims. Advantageous embodiments and developments are provided in the respective dependent claims. A computer program product and a corresponding computer-readable medium, respectively, are likewise provided.

[0018] Prior to discussing the proposed method in greater detail, the underlying rigid body system model will be introduced.

[0019] 11 < Consi od»er a scenario where a rigid body comprising N body sensors is surrounded by a total of M reference sensors, hereafter referred to as anchor sensors, as illustrated in figure 1. Each body sensor and anchor sensor is described by a 3 x 1 vector comprising its x-, y-, and z-coordinates in the 3D Euclidean space, respectively denoted by cne IR3X1for n = {1,..., N} and ame IR3X1for m = {1,..., M}. The initial sensor structure in the rigid body is consequently defined by a conformation matrix C = [c1, c2,...,cN] ∈ ℝ3×Nat the reference frame, i.e., the local axis, of the rigid body.

[0020] A transformation of the rigid body in the 3D space, i.e., a change in its location and / or in its spatial orientation, can be fully defined by a translation and rotation, respectively described by the translation vector t [tx, ty, tz]Te IR3X1comprising the translation distances in each axis and a 3D rotation matrix Q e IR3x3given by =QZGIR3X3=QyGlR3x3=QXGIR3X3cos 6Z—sin 6Z0 cos 0y0 sin 0y1 0 0 sin 6Zcos 6Z0 0 1 0 0 cos 6X—sin 6X

[0021]

[0022] . 0 0 1. —sin 6y0 cos 6y.0 sin 6Xcos 6X. 202407774 -4-

[0023] where Qx, Qy, Qze IR3X3are the roll, pitch, and yaw rotation matrices about the x-, y-, and z-axes by rotation angles of 0x,0y,0ze [-180°, 180°] degrees, respectively. Note that the rotation matrix Q is part of the special orthogonal group such that

[0024] 50(3) = {Q Gx3: QTQ = I, det(Q) = 1}.

[0025] In light of the above, the transformed coordinates of the n-th sensor after the rotation and translation is described by

[0026] s

[0027]

[0028] n= Qcn+ t E ^3x1, (2)

[0029] which is applied identically to all N sensors of the rigid body.

[0030] In the exemplary illustration shown in figure 2 the location of the moving rigid body has a 3D rotation Q and a translation t from the reference frame, as determined by equation (2). Note that the reference frame is generally not at the origin of the Euclidian space. However, since a rigid body rotation is only relative to its previous orientation, setting the reference frame to the origin, i.e., making

[0031]

[0032] I3x3, can be done without loss of generality.

[0033] The system model for stationary rigid bodies described above can be extended to moving rigid bodies by introducing the angular velocity vector to [t1,t2,t3]Te IR3X1, and the translational velocity vector t [ix, ty, iz]Te IR3X1, such that the velocity of the n-th sensor can be expressed as

[0034] sn= [to]xQcn+ t e IR3xl, (3)

[0035] where [■]xis the cross product operator for matrices, which maps the angular velocity vector to to a skew-symmetric matrix, given by

[0036]

[0037] 202407774 -5-

[0038] The velocity snof the n-th sensor as a function of the angular velocity to and translational velocity t, as determined by equation (3), are also illustrated in figure 2.

[0039] Next, the problem of performing RBL using pairwise range measurements between anchors and sensors, assumed to be available at the anchors, will be briefly discussed. The pairwise range measurements are described by

[0040]

[0041] T ^m,n II fyn II 2 T ^m,n (5)

[0042] with corresponding squared range measurements modelled as

[0043] d

[0044]

[0045] m,n II2 T "^dm,n^m,n T ^m,n (6)

[0046] In the above, dmn||am- sn||2is the true Euclidean distance between the m-th anchor and the n-th sensor, and wm,n~ N(0, < J1,) is the independent and identically distributed (i.i.d.) additive white Gaussian noise (AWGN) of variance a* affecting the range measurement, which can be reformulated in terms of a linear relation with a composite ranging noise e Kt given by

[0047]

[0048] m,n d"m,n II II 2 ll^n II2 T ~ 2d.m nWm n(7)

[0049] where the second-order noise term w^nis neglected.

[0050] Stacking equation (7) for all M anchor sensors and reformulating as a linear system on the n-th unknown body sensor variable yields

[0051] =xnelR4xldln~ Htillz -2dl, 1 ^l,n yne ^Mxl. (8)

[0052] . K ill d-Mji ~ II O-M II 2 -2a^, 1 ^M,n

[0053]

[0054] =GGIRMX4

[0055] In the foregoing representation of the observed data vector yne IRMX1, an effective channel matrix and G e IRMx4is implicitly defined, which is constructed from range measurements and anchor positions. Likewise, the vector xne IR4X1which contains 202407774 -6-

[0056] the unknown coordinates of the n-th sensor, as well as its distance to the origin, and a vector e IRMX1, which gathers the composite noise quantities defined in equation (7), are implicitly defined.

[0057] The linear system described by equation (8) can be leveraged for the estimation of the unknown sensor coordinate vector snand sensor position norm ||s|| in xn, from which equation (2) can be invoked for the translation and rotation extraction via Procrustes analysis or other conventional algorithms.

[0058] In addition to the sensor range measurement model relating to the sensor positions, under the assumption of moving rigid bodies and the associated body sensors, Doppler measurement information between the anchor sensors and body sensors can also be measured, whose relationship is given by

[0059] (sn— am)T

[0060] T T

[0061]

[0062] where e Kt is the true Doppler shift between m-th anchor sensor

[0063]

[0064] and n-th body sensor, and emn~ N(0, <

[0065]

[0066] J|) is the i.i.d. AWGN of the Doppler measurement with noise variance <

[0067]

[0068] J|.

[0069] Multiplying the Doppler measurements of equation (9) with the earlier range measurements of equation (5) yields

[0070]

[0071] T T

[0072] which can also be reformulated in terms of a

[0073]

[0074] composite Doppler noise e Kt, given by

[0075]

[0076] T T

[0077] where the second-order noise term w^nis considered negligible and is therefore omitted. 202407774 -7-

[0078] Finally, stacking equation (11) for all M anchors and reformulating as a linear system on the n-th unknown body sensor variable yields

[0079] =xneK4xl— a^, 1 e ^Mx1, (12) ~a~M> 1

[0080]

[0081] =GcIRMx4

[0082] where yne IRMX1and G e IRMx4are respectively the observed data vector and effective channel matrix constructed from the measured ranges, Doppler shifts and anchor sensor positions, xne IR4X1is the unknown RB sensor’s velocity vector, and e IRMX1is the vector of composite noise variables, whose elements are given by equation (11).

[0083] The linear system in equation (12) can be leveraged for the estimation of the unknown RB sensor’s velocity vector snand the multiplication of body sensor position and velocity shsnin xn, from which the unknown RB transformation variables can be estimated in a similar fashion to the range measurement-based approach.

[0084] In hand of the system models discussed above, the present invention proposes a low-complexity estimator for the translational and angular velocity of a RB that leverages the Gaussian Belief Propagation (GaBP) MP framework. Specifically, a first GaBP is derived for providing the RB sensors’ velocity vectors. Then, in possession of the sensors’ velocity vector estimates, a second GaBP is derived for obtaining the final estimate of the translational and angular velocity of the RB.

[0085] Starting with the fundamental system of equation (12), the system variables need to be expressed directly in terms of the moving RBL transformation parameters, i.e., the 3D angular velocity to [t1,t2,t3]Te IR3X1and the translational velocity vector t e IR3X1, in order to enable their estimation.

[0086] To that end, first a small-angle approximation, as discussed by J. Diebel, in " Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors," Matrix, vol. 58, no. 15-16, pp. 1-35, 2006, is applied onto the rotation matrix of equation (1), obtained by leveraging cos 0 1 and sin 0 « 0, which yields 202407774 -8-

[0087] 1 0ZQ -0Z1 e IR3X3,

[0088]

[0089] 6y — 6X1

[0090] which in turn can be vectorized into a linear system directly in terms of the Euler angles, namely

[0091] ^yeK9xlvec(Q) = y 1 - LO = [1 0 0 0 1 0 0 0

[0092] 0 1 0 -1 0 0 0 0 0 + 0 0 -1 0 0 0 1 0 0 (14).0 0 0 0 0 1 0 -1 0.

[0093]

[0094] il 6 K9x3

[0095] Note that for practical rigid body tracking applications subsequent transformation estimations are assumed to be performed within a sufficiently short time period, such that the respective change in rotation angle is small.

[0096] Utilizing the small-angle approximation, the angular velocity matrix [< »]xin equation (4) can be vectorized and decomposed into a linear system directly in terms of the individual velocities, yielding

[0097] 0 0 0 0 0 1 0 -1 0Tiw vec([to]x) = <£>0) G IR9xl= 0 0 -1 0 0 0 1 0 0 • &>2

[0098] .0 1 0 -1 0 0 0 0 0..<^3.

[0099]

[0100] ^<t>eIR9x3

[0101] Then, substituting equation (15) into equations (3) and (11) and rearranging the terms, the following alternative representation of the composite noise is obtained

[0102] (m,?i T ®7n([^] Q<-n T

[0103]

[0104] = dmnvmn— s^sn+ ((Qcn)T® )to + a^t e IR, (16)

[0105] where the matrix product vectorisation identity vec(XYZ) = (ZT® X)vec(Y) has been used. 202407774 -9-

[0106] In light of the above the fundamental system can be rewritten leveraging the linearization of equation (15), leading to

[0107] with

[0108] (17b)

[0109]

[0110] and

[0111] e ^Mx3, (17c)

[0112]

[0113] where une IRMX1is the effective observed data vector, andωandte IRMx3and

[0114] Bte IRMx3are respectively the effective channel matrices for the unknown angular velocity and translational velocity variables, while

[0115]

[0116] e IRMX1is the vector of composite noise variables from equation (11).

[0117] It is noted that parameters of the stationary RBL that may be used in the various equations may be obtained using any suitable conventional method, e.g., as described by Volodymyr Vizitiv, Hyeon Seok Rou, Niclas Führling, and Giuseppe Thadeu Freitas de Abreu, in “ Belief Propagation-based Rotation and Translation Estimation for Rigid Body Localization", arXiv preprint arXiv:2407.09232, 2024.

[0118] Available online: https: / / arxiv.org / abs / 2407.09232.

[0119] Next, the GaBP rules for moving RBL will be derived, based on the linear model of equation (12). Since the derivation is identical for each n-th sensor node the subscriptnis omitted herein for convenience. Accordingly, the estimation of the velocity vector xncorresponding to any given n-th sensor node in equation (12), here temporarily denoted b

[0120]

[0121] y x = [x1(...,x / <+1]Te K^A’+1>X1, with k e {1,..., K + 1} and K being the dimension of the space, is given by the collection of soft replicas for each of its fc-th element xkfor each m-th observation, which are denoted by x L whose mean-squared-error MSE is defined as 202407774 -10-

[0122] ( 2 A ^m.k = Exfc(|xfc- X^k| J = ExJ^l - Xm,k6 K(18)

[0123]

[0124] where the superscript (-)W denotes the variable at the j-th iteration of the GaBP process.

[0125] The first step of the GaBP process is the soft-interference cancellation, or soft-IC, on the m-th element of the observed data ymfor the estimation of the fc-th estimated variable, described by

[0126] ymk Vm / 9m,k^k T / 9m, i (%i %m i) + ’ hk ' hk, (19)

[0127]

[0128] where $^ke Kt is the soft-IC symbol corresponding to the m-th element of the observed data vector and the fc-th element of the velocity vector and gm ke Kt is the (m, fc)-th element of the effective channel matrix G in equation (11 ).

[0129] Assuming that the interference-plus-noise term p^ke Kt in equation (19) follows a normal distribution under the scalar Gaussian approximation, the conditional probability density function (PDF) of ^mkis given by

[0130] Cfc - 9m,k*k?Bt(1 k ex(20)

[0131] 0 x. m,k,

[0132]

[0133] whose conditional variance is given by

[0134] a,2“ = ^ l^rC +Wo e lR. (21)

[0135]

[0136] i*k

[0137] where Nois the power of the composite noise described in equation (16). 202407774 -11-

[0138] In hand of the conditional PDFs of the soft-IC symbols, the extrinsic PDF of ẋkis written as

[0139] n (22 i±m1**)k exp)

[0140]

[0141] with the extrinsic mean and variance, respectively, given by

[0142] (23 a)

[0143] (23b)

[0144]

[0145] The extrinsic mean and variance are subsequently denoised, such that the denoised mean and variance are given by

[0146] e

[0147]

[0148] = (24) (b 'xir + v m,k, “and® (b^ + v m,k,e“■

[0149]

[0150] where φẋ∈ ℝ is the variance of the prior distribution of the velocity variable. Note that a classic zero-mean Gaussian denoiser may be used in many cases. However, if an alternative prior distribution of the position variables is assumed, i.e., uniform distribution, a different Bayes-optimal denoiser can be used.

[0151] The ( / + l)-th soft-replica is obtained via damped update to prevent error floors caused by early erroneous convergence to a local optimum:

[0152] *^k]= P*m,k+C1- P^m,k> <25a) C

[0153]

[0154] fc1]= P^m,k+C1- P^m,k’ (25&)

[0155] where p e [0,1] is a selected damping parameter. 202407774 -12-

[0156] At the end of the GaBP iterations, with the last iteration denoted by, the final estimate of the velocity variable is obtained via a consensus belief combination, given by

[0157] / MI ■ 12\ / MD’max] \ *k=( I y / i _2[jmax] \ ( I y / , 2[jmax] ) \e R (26)

[0158]

[0159] 'm=lum,k ' '771=1um,k '

[0160] The proposed method for the sensor velocity estimation without rigid body conformation, solely based on the range and Doppler information between the sensors and anchors, and ultimately yielding (K + 1) elements in xncomposed of K velocity estimates of snand the multiplication of sensor position and velocity s

[0161]

[0162] ^snis summarised below with reference to figure 3.

[0163] Inputs to the method 500, received in step 505, are the observed data vector ẏn∀n constructed from the measured ranges, Doppler shifts and anchor positions, the effective channel matrix G, the maximum number jmax of iterations, the variance

[0164]

[0165] of the prior distribution of the velocity variable, the number No of sensors nodes, and a damping parameter p. The output of the method 500 are the estimates

[0166]

[0167] of the velocity variable, for all sensor nodes ∀n.

[0168] In step 510 the initial soft replicas x^kand corresponding MSE

[0169]

[0170] are initialised, before the iteration loop is started with step 520, in which the soft-IC symbol

[0171]

[0172] is computed via equation (19). In step 530 the corresponding conditional variance a^kis computed via equation (21). Next, in step 540, the extrinsic mean x^,k is computed via equation (23a), and in step 550 the corresponding extrinsic variance v^kis computed via equation (23b). The extrinsic mean x^kand variance

[0173]

[0174] are subsequently input to a denoising step 560 that uses equation (24), yielding the corresponding denoised beliefs x^ and

[0175]

[0176] MSEs The soft-replicas

[0177]

[0178] and MSEs 'mfc11 arethen obtained via damped update as per equation (25) in step 570. Step 580 checks if a termination criterion is met. Termination criteria may comprise reaching the predetermined maximum number of iterations jmax, and / or the convergence of the soft replicas below a predetermined threshold, whichever 202407774 -13-

[0179] criterion is first met. The iterative process steps described before are performed ∀n, m, k. If the termination criterion is not met, “no” -branch of step 580, the iteration is repeated from step 520, using the most recent soft replicas

[0180]

[0181] and MSEs ip[{+k^ as inputs. If the termination criterion is met, “yes”-branch of step 580, the final estimate of the velocity variable x̃kis obtained in step 590 via a consensus belief combination as per equation (26), which is output in step 595.

[0182] Subsequently, the estimated multiplication of sensor position and velocity can be used to construct the second linear equation of equation (17a) incorporating the rigid body conformation in terms of the angular velocity and translational velocity, as will be leveraged in the following subsection to derive the second GaBP algorithm for directly estimating the moving rigid body transformation parameters.

[0183] While the linear formulation and the MP rules for the GaBP iterations are very similar to method 500 described above, the velocity parameter estimation from equation (17a) comprises two sets of variables,kwith k e {1,..., K} and i with ■£ e {1,..., K}, such that the GaBP rules are elaborated separately.

[0184] First, soft-IC for the angular and translational velocity variables is performed on the observed information respectively as

[0185] K

[0186] r. D] _ Vh AD] _ Vh2U]uaj:m,k “-tn / uor.

[0187] i±k i=l K bar.mji^k T (j'i (27 Cl) i=l

[0188] i±k K

[0189] j^D] — Ti _ \ ' / xD] _ \ '?[f],ut.m.i'' m,i i=l K

[0190] "h (^i (276) i=l

[0191]

[0192] In turn, the soft-IC symbol conditional PDFs are given by 202407774 -14-

[0193] \M:m,k bM.m ka)k

[0194] Pfi cWm,k J'Wfck (v^m,k1<*>*)7 K eXP < J c2^m,k.

[0195] Pu tp:m] (um) oc exp,f O~^

[0196] :J\

[0197]

[0198] t m,t

[0199] with the corresponding conditional variances defined as

[0200] K

[0201] a a2>^:m,k = / , 'r a)-.m,i ■ + 3 / "1, i 'r t-.m,i. + Nou(29a) i*k i=l K

[0202] y Ib^.jnil2! / ^. + y ■ (29b) i = l

[0203] and the corresponding MSEs as ip^m k= EWfcK-e'i'j and

[0204]

[0205] = ^,[1^- er ■ With the conditional PDFs in hand, the extrinsic PDF is obtained as

[0206] n i±m (fl"«1"*)k exp

[0207] 1(30)

[0208]

[0209] n i^mp^<1 tf)x expwhere the corresponding extrinsic means and variances are given by 202407774 -15-

[0210] (31a)

[0211] (31b)

[0212] (32a)

[0213] (32b)

[0214]

[0215] Finally, the denoisers with a Gaussian prior are given by

[0216] <^m,k - e e (33a)

[0217] !m,k &+

[0218] co-.m,k - e K, e Kt, (33b) 0- + v'7'

[0219]

[0220] ' a>.m,k ' ( rbtt + v tD\m], H,

[0221]

[0222] where and (ptare the element-wise variances of to and t.

[0223] Subsequently, the soft-replicas are iteratively updated similarly to equation (25) until a termination criterion is met, e.g., for jmaxiterations of the MP algorithm or meeting a convergence criterion. The (j + l)-th soft-replica is obtained via a damped update with a damping factor p, described by

[0224] Cfc1]= P*m,k + (i - p^m,k> (34a)

[0225]

[0226] = P^m,k+~ P^m,k- (34^)

[0227] Finally, after jmaxiterations or any other termination criterion being met the consensus estimates are obtained from 202407774 -16-

[0228] (35a)

[0229] (35b)

[0230]

[0231] The MP rules elaborated by equations (27)-(35) are complete to yield the angular and translational velocity vectors. However, due to significant differences in the effective channel powers ofωandtand Btin equation (17), with the latter typically being much larger than the former, the estimation performance of the translational velocity vector elements may be better than that of the angular velocity vector in a joint estimation described by the GaBP procedure as described hereinbefore. Note that the effective channel powers are highly dependent on the sensor and anchor deployment structure. For typical indoor sensing scenarios as schematically illustrated in figure 1, the anchor coordinates have larger absolute values than the rigid body sensor coordinates, leading to such large power difference.

[0232] This behaviour can also be intuitively understood by considering the illustration in figure 2, where a small rotation of the rigid body is expected to have a less prominent effect on the absolute sensor positions then the translation.

[0233] In order to address the aforementioned poorer estimation performance for the estimation of the angular velocity parameters to, the present invention proposes an interference cancellation-based approach to remove the components corresponding to the translation of the sensors, and perform the GaBP again only on the angular velocity parameters.

[0234] Using the estimated consensus translational velocity vector t [?i,

[0235]

[0236] eIR3X1obtained at the end of the GaBP process via equation (33b), the interference-cancelled system is given by

[0237] u

[0238]

[0239] n' = un— Bft = B^a) + CneIRMxl- (36) 202407774 -17-

[0240] The GaBP procedure for estimating the angular velocity parameters to is identical to the linear GaBP in method 500, where the post-interference cancellation (IC) factor node equations are given by

[0241] Uar.m,k ~Urn. E IR, (37) i*k + «o £ «. (38)

[0242]

[0243] i*k

[0244] which is then again concatenated with the previous bivariate GaBP process to describe the complete estimation process for the rigid body transformation parameters to and t.

[0245] The proposed method for estimating the parameters for the moving RBL is summarised below with reference to figure 4.

[0246] Inputs to the method 600, received in step 605, are the vectors un,sn,snVn respectively representing the effective observed data, the sensor’s positions and the sensor’s velocities, the effective channel matricesωandt

[0247]

[0248] and for the angular and translational velocities, respectively, the variance

[0249]

[0250] of the angular velocity, the variance of the translational velocity, the maximum number jmax of iterations, the number No of sensor nodes, and a damping parameter p. The output of the method 600 are the estimates &kof the angular velocity and

[0251]

[0252] of the translational velocity, Yk,£ for all sensor nodes ∀n.

[0253] In step 610 the variables for the angular velocity,

[0254]

[0255] the translational velocity

[0256]

[0257] and the corresponding MSEs

[0258]

[0259] kand are initialised, before the iteration loop is started with step 620, in which the soft-IC symbols for the angular and translator velocity variables,

[0260]

[0261] u^mkand respectively, are computed via equation (27). In step 630 the corresponding extrinsic means a)^kand

[0262]

[0263] are computed via equation (31). Next, in step 640, the extrinsic variances v^mkand

[0264] „ are computed via equation (32). The extrinsic means

[0265]

[0266] and and the extrinsic variances v^mkand are subsequently input to a denoising step 650 202407774 -18-

[0267] that uses equation (33a), yielding the corresponding denoised beliefs )mkand tmi. The associated error variances are denoised in step 660, using equation (33b), yielding the denoised error variancekfor the rotational velocity and

[0268]

[0269] for the translational velocity. The soft-replicas

[0270]

[0271] and ip^+k1]are then obtained via damped update as per equation (34) in step 670. Step 690 checks if a termination criterion is met. Termination criteria may comprise reaching the predetermined maximum number of iterations jmax, and / or the convergence of the soft replicas below a predetermined threshold, whichever criterion is first met. If the termination criterion is not met, “no”-branch of step 690, the iteration is repeated from step 620, using the most recent soft replicas

[0272]

[0273] and 'as inputs. If the termination criterion is met, “yes”-branch of step 690, the final estimate of the rotation and the translation variables a>kand t respectively, are obtained in step 700 via a consensus belief combination as per equation (35a), and in step 710 an angular velocity-only input data vector un' is generated in accordance with equation (36).

[0274] The previously generated rotational velocity-only input data vector un' is subjected to a further iterative GaBP process, for determining the rotational velocity parameters to with higher estimation accuracy. To this end, in step 720, the soft-IC symbols u'^m kfor the vector containing the angular velocity parameters are computed via equation (37), and in step 730 the corresponding conditional variances

[0275]

[0276] are computed via equation (38). Next, in steps 740 and 750, respectively, the extrinsic means t

[0277]

[0278] o^]kand extrinsic variance v^m kare computed via equations (31a) and (32a), respectively. The so-obtained parameters are input to denoising steps 760 and 770 that use equations (33a) and (33b), yielding the corresponding denoised beliefs d)m kand denoised error variances

[0279]

[0280] Finally, the soft-replicas

[0281]

[0282] are obtained via damped update as per equation (34) in step 780. Step 790 checks if a termination criterion is met. Termination criteria may comprise reaching the predetermined maximum number of iterations jmax, and / or the convergence of the soft replicas below a predetermined threshold, whichever criterion is first met. If the termination criterion is not met, “no”-branch of step 790, the iteration is repeated from step 720, using the most recent soft replicas

[0283]

[0284] as inputs. If the termination criterion is met, “yes”-branch of step 790, the final estimate of the rotational velocity variables a>kare 202407774 -19-

[0285] obtained in step 800 via a consensus belief combination as per equation (35a), which is output in step 805. The iterative process steps of the two iteration loops are performed ∀n, m, k, ℓ.

[0286] In the following section the effectiveness of the proposed multi-stage GaBP-based approach for moving RBL will be demonstrated using simulations. Specifically, the estimation performance of the proposed approach of determining the sensor velocity estimation via method 500, and the RBL velocity parameter estimation via method 600 for a moving rigid body are compared.

[0287] In the simulations the full set of RBL parameters for the stationary and moving case is considered. The stationary RBL parameters provided along with the moving RBL parameters determined in accordance with the methods presented herein are derived using the methods discussed in “ Belief Propagation-based Rotation and Translation Estimation for Rigid Body Localization" (full citation further above).

[0288] The performance of the moving RBL methods presented herein is compared against the relevant conventional RBL solution, where the time of arrival (TOA)-based sensor position estimation is performed via the approach presented by Z. Ma and K. Ho, in " TO A Localization in the Presence of Random Sensor Position Errors," IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011, and where the joint time difference of arrival (TDOA) and frequency difference of arrival (FDOA)-based sensor velocity estimation is performed via the approach discussed by K. Ho and W. Xu, in " An Accurate Algebraic Solution for Moving Source Location using TDOA and FDOA Measurements," in IEEE Transactions on Signal Processing, vol. 52, no. 9, pp. 2453-2463, 2004. The RBL parameter estimation is performed as proposed 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.

[0289] The simulation setup corresponds to the exemplary scenario illustrated in figure 1, where the rigid body comprises N = 8 sensors positioned at the vertices of a unit cube at the origin, with sensor positions described by the conformation matrix given by 202407774 -20-

[0290] 0.5 0.5 -0.5 -0.5 0.5 -0.5 0.5

[0291] C = -0.5 -0.5 0.5 0.5 -0.5 -0.5 0.5 0.5 e IR3X8,

[0292]

[0293] -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5.

[0294] and where the M = 8 anchors are positioned at the vertices of a larger cube, e.g., a room, where the anchor conformation matrix A e IR3x8is given by

[0295] -10 10 10 -10 -10 10 -10 10

[0296] A = -10 -10 10 10 -10 -10 10 io e IR3X8.

[0297] -10 -10 -10 -10 10 10 10 10.

[0298] The RBL rotation angles 0x,0y,0zfollow a zero-mean Gaussian distribution of variance (pg= 10, and the RBL translation vector elements also follow a zero-mean Gaussian distribution of variance (pt= 5. The RBL parameters in the moving scenario follow a zero-mean Gaussian distribution of variance

[0299]

[0300] = 10 for the angular velocity and a variance of <>j = 5 for the translational velocity.

[0301] The performance is assessed using the RMSE defined as

[0302] RMSE = ||x̂[i]- x||2(39)

[0303]

[0304] where x̂[i]is the RBL parameter vector, i.e., position, angle, or translation, estimated during the i-th Monte-Carlo simulation, x is the true RBL parameter vector, and E = 104is the total number of independent Monte-Carlo experiments used for the analysis, and is evaluated for different noise standard deviations a of equation (5) and (9), defined as a = aw= 0.1<je.

[0305] Figure 5 illustrates the RMSE of the sensor velocity estimates of method 500 compared to the conventional solution discussed in " An Accurate Algebraic Solution for Moving Source Location using TDOA and FDOA Measurements" (full citation further above). It can be observed that the proposed method outperforms the conventional approach in all velocity errors with a large performance gain. 202407774 -21-

[0306] Figures 6 and 7 show the performance of the estimation of the velocity parameters, i.e., the angular velocity and the translational velocity. In figure 6, the translational velocity estimate of the proposed method 600 is compared to the conventional method discussed in in " Accurate Localization of a Rigid Body Using Multiple Sensors and Landmarks" (full citation further above). It can be observed that the proposed method outperforms the conventional method again for all range errors.

[0307] Figure 7 illustrates the performance of the angular velocity estimation, where the proposed method outperforms the conventional approach for all range errors, showing a larger gain for high noise and a smaller gain for low noise, and reaching the ideal behaviour of the proposed GaBP method, illustrated via the matched filter bound (MFB) for higher noise values.

[0308] In the following the performance of the proposed RBL parameter estimation algorithms are further assessed in terms of the computational complexity and convergence behaviour, and compared against conventional methods.

[0309] Figure 8 illustrates the convergence behaviour of the proposed methods 500 and 600 for a noise level of = 10-1. For comparison purposes, the convergence of the conventional methods for stationary RBL discussed in “ Belief Propagation-based Rotation and Translation Estimation for Rigid Body Localization" (full citation further above) are provided. It can be observed that the two methods converge very efficiently, with the bivariate methods SotA2 and 600 converging slightly faster and to a higher estimation accuracy than the two linear algorithms, SotA1 and 500. Note that the curves for methods 500 and 600 are almost identical in the figure.

[0310] Finally, the computational complexities of the proposed methods are outlined in Table I in terms of the complexity order on the system size parameters using the well-known Big-0 notation, as well as a convenient measure of practical runtimes in seconds simulated via an ordinary computer. 202407774 -22-

[0311] Method Complexity Runtime conv. stationary linear GaBP (SotA1) O(NMK) 0.15 ms prop, moving linear GaBP (method 500) O(NMK) 0.14 ms SotA position / velocity est. (" An Accurate Algebraic

[0312] Solution for Moving Source Location using TDOA and O(MNK2) 1.6 ms FDOA Measurements")

[0313] conv. stationary double GaBP (SotA2) O(NMK2) 1.1 ms conv. stationary parameter est. " Accurate Localization

[0314] O(K3

[0315] of a Rigid Body Using Multiple Sensors and 0.24 ms + MN)

[0316] Landmarks"

[0317] prop, moving double GaBP (method 600) O(NMK2) 1 ms conv. moving parameter est. " Accurate Localization of O(K3

[0318] 0.39 ms a Rigid Body Using Multiple Sensors and Landmarks" + MN)

[0319]

[0320] Table I: Complexity and runtime

[0321] In light of the foregoing discussion and in accordance with a first aspect of the invention a method of estimating motion parameters of an object in an observed space is presented. The object, which may also be referred to as rigid body, or RB, in this specification, has a set of sensors arranged at respective outer surfaces or perimeters. A plurality of anchors is arranged in the observed space. In this context the expression sensors refers to specific points or landmarks that are fixed relative to the object, and the expression anchors refers to specific points or landmarks of the observed space or observer that are fixed relative to the observed space or observer. Each of the anchors is configured for performing, using wireless signals and in cooperation with corresponding signal transmitting, signal receiving and signal processing circuitry, for performing range and Doppler measurements with regard to at least a respective first subset of the set of sensors of the object. The motion 202407774 -23-

[0322] parameters may relate to respective velocities of the object’s sensors and / or to an angular and a translational velocity of the object as a whole.

[0323] The method comprises receiving, for at least a second subset of the object’s sensors that comprises one or more first subsets from one or more respective anchors, respective observed data vectors constructed based on measured ranges, Doppler shifts and anchor positions, an effective channel matrix, a variance of the prior distribution of a velocity variable. Alternatively or additionally, the method may comprise receiving, for at least the second subset of the object’s sensors, vectors representing the effective observed data, the sensors’ respective position and the sensors’ respective velocity, effective channel matrices for the angular and translational velocities, values representing a variance of the angular velocity and a variance of the translational velocity, respectively. The method or its variants further comprise receiving a termination criterion, and the number of sensor nodes of the object. Based on respective subsets of the received information the sensors’ respective velocities are and / or an angular and a translational velocity of the object are determined.

[0324] Determining sensor velocities comprises performing a Gaussian Belief Propagation, GaBP, process on at least a first subset of the received information, for iteratively determining, using respective observations associated with corresponding anchors, soft replicas of elements of a respective vector that comprises velocity estimates for each spatial direction and the product of a position vector and a velocity vector. Determining is carried out for each sensor of the second subset of sensors of the object, yielding final estimates of the elements of each vector through consensus belief combination. The so-determined final estimates are output for further processing or other purposes.

[0325] Determining an angular and a translational velocity of the object comprises performing a GaBP process on at least a second subset of the received information, for iteratively determining, using respective observations associated with corresponding anchors, soft replicas of elements of sets of variables representing angular and translational velocities of the object around each spatial axis and in each spatial direction, respectively, yielding final estimates of the elements of each set 202407774 -24-

[0326] through consensus belief combination. The so-determined final estimates are output for further processing or other purposes.

[0327] Embodiments of the method further comprise determining, for each sensor of the object, an angular velocity-only input data vector by cancelling the previously determined translational velocity of the object from the received effective observed data. The angular velocity-only input data vector is then submitted to a refining process whose output is a refined estimate of the object’s angular velocity. The refining process may comprise an iterative GaBP process for iteratively determining, using respective observations associated with corresponding anchors, soft replicas of elements of a set of variables representing angular velocities of the object around each spatial axis, yielding refined estimates of the elements of the set through consensus belief combination. The so-determined refined estimate is output for further processing or other purposes.

[0328] Embodiments of the method further comprise denoising intermediate results in iterative GaBP processes before updating soft replicas in iterative GaBP processes. Denoising may comprise using a zero-mean Gaussian denoiser or Bayes-optimal denoiser.

[0329] Embodiments of the method further comprise receiving a damping parameter and updating soft replicas in iterative GaBP processes the via damping.

[0330] Embodiments of the method further comprise applying a matrix completion process on subsets comprising observed data vectors, for obtaining complete sets. These embodiments may be used in case observed data vectors or similar information vectors or sets are not available from all anchors and / or in relation to all of the object’s sensors. Performing the method on complete matrices may improve the performance and accuracy of the method. A variety of matrix completion processes may be used including, e.g., the OptSpace algorithm described by R. H. Keshavan, A. Montanari and S. Oh, in " Matrix Completion From a Few Entries," IEEE Transactions on Information Theory, vol. 56, no. 6, pp. 2980-2998, June 2010, or the methods described by E. J. Candes and B. Recht, in " Exact matrix completion via convex optimization," Foundations of Computational Mathematics, vol. 9, no. 6, pp. 202407774 -25-

[0331] 717-772, 2009, by E. J. Candes and Y. Plan, in " Matrix completion with noise," Proceedings of the IEEE, vol. 98, no. 6, pp. 925-936, 2010, or by E. J. Candes and T. Tao, in " The power of convex relaxation: Near-optimal matrix completion," IEEE Transactions on Information Theory, vol. 56, no. 5, pp. 2053-2080, 2010, which mainly serve to improve the performance and reduce complexity by replacing the non-convex rank objective with corresponding convex relaxations, i.e. the nuclear norm (NN).

[0332] In accordance with a second aspect of the invention an apparatus for estimating motion parameters of an object in an observed space is presented. The apparatus comprises one or more microprocessors, associated volatile and non-volatile memory, and one or more communication interfaces configured for receiving data from, or generated in relation to, a plurality of anchors that are arranged in the observed space, the one or more communication interfaces and the memory being communicatively coupled to the one or more microprocessors via one or more signal and / or data lines and / or buses, wherein 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 as described hereinbefore.

[0333] In accordance with a further aspect of the invention a system for determining and / or tracking movements of an object in an observed space is presented. The system comprises a plurality of anchors each configured for providing and / or generating, based on wireless signals, data relating to range and / or Doppler measurements between sensors of the object relative and the respective anchor. The system further comprises one or more apparatus in accordance with the second aspect of the invention.

[0334] As will be appreciated by one skilled in the art, aspects of the embodiments may be embodied as a system, apparatus, method, or program product. Accordingly, embodiments may take the form of an entirely hardware embodiment, an entirely software-implemented embodiment, including firmware, resident software, microcode, etc., or an embodiment combining software and hardware aspects. 202407774 -26-

[0335] For example, the disclosed embodiments may be implemented as a hardware circuit comprising custom very-large-scale integration (VLSI) circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. The disclosed embodiments may also be implemented in programmable hardware devices such as field programmable gate arrays, programmable array logic, programmable logic devices, or the like. As another example, the disclosed embodiments may include one or more physical or logical blocks of executable code which may, for instance, be organised as an object, procedure, or function.

[0336] The method presented hereinbefore may be represented by computer program instructions. Accordingly, in accordance with a further aspect of the invention, a computer program product comprises computer program instructions which, when executed by a microprocessor of a wireless transmitter in accordance with the third aspect of the invention, cause the microprocessor to execute the method in accordance with the first aspect of the present invention, and to accordingly control hardware and / or software blocks or modules of the wireless transmitter or, when executed by a microprocessor of a wireless receiver in accordance with the fourth aspect of the invention, cause the microprocessor to execute the method in accordance with the second aspect of the present invention, and to accordingly control hardware and / or software blocks or modules of the wireless receiver.

[0337] Computer program instructions, or code, for carrying out operations for embodiments may be any number of lines and may be written in any combination of one or more programming languages including an object- oriented programming language such as Python, Ruby, Java, Smalltalk, C++, or the like, and conventional procedural programming languages, such as the “C” programming language, or the like, and / or machine languages such as assembly languages. The code may execute entirely on the user’s computer, partly on the user’s computer, as a stand-alone software package, partly on the user’s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user’s computer through any type of network, including a local area network (LAN), wireless LAN (WLAN), or a wide area network (WAN), or the connection may be made to an external computer, for example, through the Internet using an Internet Service Provider (ISP). 202407774 -27-

[0338] 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 tangibly or 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.

[0339] The described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In this description, numerous specific details are provided, such as examples of programming, software modules, user selections, network transactions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments. One skilled in the relevant art will recognize, however, that embodiments may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment. Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, appearances of the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment, but mean “one or more but not all embodiments” unless expressly specified otherwise. The terms “including,” “comprising,” “having,” and variations thereof mean “including but not limited to,” unless expressly specified otherwise. An enumerated listing of items does not imply that any or all of the items are mutually exclusive, unless expressly specified otherwise. The terms “a,” “an,” and “the” also refer to “one or more” unless expressly specified otherwise.

[0340] Where aspects of the embodiments are described in this specification with reference to schematic flowchart diagrams and / or schematic block diagrams of methods, apparatuses, systems, and program products according to embodiments it will be 202407774 -28-

[0341] understood that each block of the schematic flowchart diagrams and / or schematic block diagrams, and combinations of blocks in the schematic flowchart diagrams and / or schematic block diagrams, can be implemented by code. This code may be provided to a processor of a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart diagrams and / or block diagrams.

[0342] It should be noted that, in some implementations or embodiments, the functions noted in the exemplary embodiments shown in the figures may occur out of the order shown in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more blocks, or portions thereof, shown in the figures.

[0343] The framework presented herein permits efficiently solving the RBL problem for moving rigid bodies, i.e., estimation of the RB sensor’s velocity, and the RB’s angular velocity and translational velocity, via a series of tailored GaBP message passing estimators.

[0344] The proposed RBL methods are shown to outperform conventional methods in all of the sensor velocity, angular velocity and translational velocity estimation performance, in addition to having a significantly lower computational complexity.

[0345] BRIEF DESCRIPTION OF THE DRAWING

[0346] In the following section the invention will be described with reference to the drawings, in which

[0347] Fig. 1 shows an exemplary schematic scenario of a rigid body comprising N body sensors that is surrounded by a total of M reference sensors,

[0348] Fig. 2 shows an exemplary schematic representation of a translation and a rotation of a rigid body, 202407774 -29-

[0349] Fig. 3 shows an exemplary flow diagram of a method in accordance with a first aspect of the invention,

[0350] Fig. 4 shows an exemplary flow diagram of a method in accordance with a second aspect of the invention,

[0351] Fig. 5 shows a performance comparison between a conventional method of determining sensor velocity of a rigid body and the method in accordance with the first aspect of the invention,

[0352] Fig. 6 shows a performance comparison between a conventional method of determining a translational velocity of a rigid body and the method in accordance with the second aspect of the invention,

[0353] Fig. 7 shows a performance comparison between a conventional method of determining an angular velocity of a rigid body and the method in accordance with the fourth second of the invention,

[0354] Fig. 8 shows a performance comparison of the convergence between the methods in accordance with the invention, and with conventional methods,

[0355] Fig. 9 shows an exemplary block diagram of an apparatus configured for executing one or more methods, or embodiments thereof, in accordance with the invention.

[0356] In the figures identical or similar elements may be referenced using the same reference designators.

[0357] DETAILED DESCRIPTION OF EMBODIMENTS

[0358] Figures 1 to 8 have been described further above and will not be discussed again.

[0359] Figure 9 shows an exemplary block diagram of an apparatus 900 in accordance with embodiments of the second aspect of the present invention. The apparatus 900 comprises one or more microprocessors 950, volatile memory 952, non-volatile memory 954, and one or more communication interfaces 956 configured for receiving data from, or generated in relation to, a plurality of anchors that are arranged in the observed space. The aforementioned elements are communicatively connected via one or more signal or data connections or buses 958. The non-volatile memory 954 stores computer program instructions which, when executed by the microprocessor 202407774 -30-

[0360] 950, cause the apparatus 900 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 900.

Claims

202407774 -31-CLAIMS1. A method (500; 600) of estimating motion parameters of an object having a set of sensors arranged at respective outer surfaces or perimeters, in an observed space in which a plurality of anchors is arranged, each of said anchors being configured for performing, using wireless signals, range and Doppler measurements with regard to at least a respective first subset of the set of sensors of the object, the method comprising:- receiving (505), for at least a second subset of the object’s sensors that comprises one or more first subsets from one or more respective anchors, respective observed data vectors (yn) constructed based on measured ranges, Doppler shifts and anchor positions, an effective channel matrix (G), a variance (φ̇ₓ) of the prior distribution of a velocity variable, and / or receiving (605), for at least the second subset of the object’s sensors, vectors representing effective observed data (un), the sensors’ respective position (sn) and the sensors’ respective velocity (sn), effective channel matricesωandtand for the angular and translational velocities, values representing a variance φωof the angular velocity and a variance φtof the translational velocity, respectively, and further receiving (505, 605) a termination criterion, and the number (No) of sensor nodes of the object,- determining (510-590) the sensors’ respective velocities and / or determining (610-700) an angular and a translational velocity of the object,wherein determining sensor velocities of the object comprises:- performing (510-580) a Gaussian Belief Propagation, GaBP, process on at least a first subset of the received information, for iteratively determining, using respective observations associated with corresponding anchors, soft replicas (x̃k) of elements of a respective vector (xn) that comprises velocity estimates for each spatial direction and the product of a position vector and a velocity vector, for each sensor of the second subset of sensors of the object, yielding final estimates (k) of the elements of each vector (xn) through consensus belief combination (590), and- outputting (595) the previously determined final estimates (x̃k)wherein determining an angular and a translational velocity of the object comprises:202407774 -32-- performing (610-690) a GaBP process on at least a second subset of the received information, for iteratively determining, using respective observations associated with corresponding anchors, soft replicast^) of elements of sets of variables representing angular and translational velocities of the object around each spatial axis and in each spatial direction, respectively, yielding final estimates (ω̃k, t̃ℓ) of the elements of each set through consensus belief combination (700), and- outputting (705) the previously determined final estimates (ω̃k,2. The method (600) of claim 1, further comprising:- determining (710), for each sensor of the object, an angular velocity-only input data vector (un' ) by cancelling the previously determined translational velocity (t̃) of the object from the received effective observed data (un), - submitting the angular velocity-only input data vector (un' ) to a refining process (720-790) whose output is a refined estimate of the object’s angular velocity (ω̃k).

3. The method (600) of claim 2, wherein the refining process (720-790) comprises an iterative GaBP process for iteratively determining, using respective observations associated with corresponding anchors, soft replicas °f elements of a set of variables representing angular velocities of the object around each spatial axis, yielding refined estimates (ω̃k) of the elements of the set through consensus belief combination (800), and- outputting (805) the previously determined refined estimate (ω̃k).

4. The method (500; 600) of any one or more of the preceding claims, further comprising:- denoising (650, 660; 760, 770) intermediate results in iterative GaBP processes before updating (670; 780) soft replicas in iterative GaBP processes.

5. The method (500; 600) of any one or more of the preceding claims, further comprising:202407774 -33-- receiving a damping parameter ( ) in step 505 and updating (670; 780) soft replicas in iterative GaBP processes the via damping.

6. The method (500; 600) of any one or more of the preceding claims, further comprising:- applying a matrix completion process on subsets comprising observed data vectors.

7. Apparatus (900) for estimating motion parameters of an object in an observed space, the apparatus comprising one or more microprocessors (950), associated volatile (952) and non-volatile memory (954), and one or more communication interfaces (956) configured for receiving data from, or generated in relation to, a plurality of anchors that are arranged in the observed space, the one or more communication interfaces (956) and the memory (952, 954) being communicatively coupled to the one or more microprocessors (950) via one or more signal and / or data lines and / or buses (958), wherein the non-volatile memory (954) stores computer program instructions which, when executed by the one or more microprocessors (950), configure the apparatus (900) to execute the method of one or more of claims 1 to 6.

8. A system for determining and / or tracking movements of an object in an observed space, the system comprising a plurality of anchors each configured for providing and / or generating, based on wireless signals, data relating to range and / or Doppler measurements between sensors of the object relative and the respective anchor, and further comprising one or more apparatus in accordance with claim 7.

9. 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 to accordingly control hardware components of the apparatus (300) in accordance with the method of one or more of claims 1 to 6.202407774 -34-10. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 9.

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