Enhanced sidelink positioning with improved accuracy under UE mobility

Abstract

This specification describes systems, methods and apparatus for positioning user equipment and / or managing positioning resources. According to a first aspect of this specification, there is described an apparatus comprising: means for constructing an overdetermined system of equations for a user equipment position based at least on: position derivative data comprising an indication that the D-th derivative of the user equipment position is constant over a time interval, wherein D is a positive non-null integer; and a plurality of sequences of positioning data for the user equipment, each sequence of positioning data in the plurality of sequences of positioning data being associated with a respective anchor node in a plurality of anchor nodes and comprising positioning data for the user equipment at a plurality of times in the time interval. The apparatus further comprises means for determining, from the overdetermined system of equations, one or more estimates of the user equipment position at one or more respective times in the time interval.

Description

[0001]Enhanced Sidelink Positioning with Improved Accuracy Under UE Mobility Technical Field This specification describes systems, methods and apparatus for positioning user equipment and / or managing positioning resources. Background Positioning error due to user equipment (UE) mobility can be substantial even at low speeds because a single position estimate requires the collection of measurements from multiple anchors. This may be a relatively time-consuming process, during which the collected information ages. Often the decision-making node, such as a location management function (LMF), may also want to combine multiple independent measurement cycles to improve positioning accuracy, which exacerbates the input aging problem. Modest mobility can already have impact even during position estimate with respect to a single anchor, as such estimate typically consist of multiple time-independent position reference signal (PRS) measurements (especially the double-sided multi-RTT variant needed in sidelink positioning to offset clock drift error). At high UE speeds, even a single PRS measurement can be affected by non-negligible error due to UE displacement. Furthermore, positioning accuracy is proportional to the (sidelink, SL) PRS bandwidth as well as the (SL) PRS signal (and interference) to noise ratio S(I)NR. In other words, the wider is the reference signal bandwidth and the better is their S(I)NR, the better is the achievable accuracy of the entire localization process. The general practice is to use as frequency-wide (SL) PRSs as possible, typically spanning across the entire bandwidth part or SL resource pool. However, the usage of wideband SL PRS signals is problematic due the increased risk of transmission collisions and interference. In shared pools, wideband SL PRS cannot be easily avoided in frequency domain and so collisions and interference are difficult as time-domain multiplexing is the only collision / interference avoidance measure. Frequent SL PRS repetitions may be used to permit tracking of UE under mobility (especially high-speed UEs) which generally exacerbates the interference. Summary According to a first aspect of this specification, there is described an apparatus comprising: means for constructing an overdetermined system of equations for a user equipment position based at least on: position derivative data comprising an indication that the D-th derivative of the user equipment position is constant over a time interval, wherein D is a positive non-null integer; and a plurality of sequences of positioning data for the user equipment, each sequence of positioning data in the plurality of sequences of positioning data being associated with a respective anchor node in a plurality of anchor nodes and comprising positioning data for the user equipment at a plurality of times in the time interval. The apparatus further comprises means for determining, from the overdetermined system of equations, one or more estimates of the user equipment position at one or more respective times in the time interval. The apparatus may further comprise means for receiving, from the user equipment, the position derivative data. Constructing the overdetermined system of equations for the position of the user equipment may comprise: determining equations for the user equipment position at respective times in the plurality of times; and replacing the user equipment position at the respective times in the equations with a target position of the user equipment at a target time and a change in position of the user equipment with respect to the target time. The change in position may be based on the D-th derivative of the user equipment position being constant and a time difference between the target time and the respective times in the plurality of times. The change in position, Δ^(^), may be based on: where ^ denotes a respective time in the plurality of times, ^^denotes the target time, and X denotes the user equipment position. Determining, from the overdetermined system of equations, the target position of the user equipment at the target time may comprise determining one or more derivatives of the user equipment position at the target time from the overdetermined system of equations. The apparatus may further comprise means for determining a respective user equipment position at a respective time in the time interval based on the target position of the user equipment at the target time, the one or more position derivatives at the target time, and a difference between the target time and the respective time in the time interval. The apparatus may further comprise: means for forecasting a future user equipment position at a time outside the time interval based on the target position of the user equipment at the target time, the one or more derivatives of the user equipment position at the target time, and a difference between the target time and the time outside the time interval; and means for associating the forecast with a flag indicating that the forecast is outside the time interval. The positioning data for the user equipment associated with one or more anchor nodes in the plurality of anchor nodes may comprise one or more of measurements of positioning reference signals transmitted between the user equipment and the one or more anchor nodes in the plurality of anchor nodes, the one or more measurements comprising one or more of: a round trip time; a time of arrival; and / or an observed time difference of arrival. According to a further aspect of this specification, there is described apparatus comprising: means for determining that a D-th derivative of the apparatus position is constant or can be approximated as being constant over a time interval, wherein D is a positive non-null integer; means for communicating, to a network node, an indication that the D-th derivative of the apparatus position is constant over the time interval. The apparatus may be a user equipment. Determining that the D-th derivative of the apparatus position is constant or can be approximated as being constant over the time interval may be based on one or more of: motion sensor data from one or more motion sensors of the apparatus; sidelink positioning protocol data; map data; relative positioning data with respect to one or more further apparatus; and / or satellite positioning data. The apparatus may further comprise: means for determining one or more values of one or more respective derivatives of the apparatus position; and means for communicating, to the network node, the one or more values. The apparatus may further comprise means for communicating, to the network node, information indicative of the time interval. The information indicative of the time interval may comprise a start time and an end time of the time interval. The information indicative of the time interval may comprise information indicative of a length of the time interval with respect to a reference time. The network node may host a location management function. According to a further aspect of this specification, there is described a method performed by a network element, the method comprising: constructing an overdetermined system of equations for a user equipment position based on at least; position derivative data comprising an indication that the D-th derivative of the user equipment position is constant over a time interval, wherein D is a positive non-null integer; a plurality of sequences of positioning data for the user equipment, each sequence of positioning data being associated with a respective anchor node in a plurality of anchor nodes and comprising positioning data for the user equipment at a plurality of times in the time interval; and determining, from the overdetermined system of equations, one or more estimates of the user equipment position at one or more respective times in the time interval. According to a further aspect of this specification, there is described a method performed by a user equipment, the method comprising: determining that a D-th derivative of the user equipment position is constant or can be approximated as being constant over a time interval, wherein D is a positive non-null integer; and communicating, to a network node, an indication that the D-th derivative of the apparatus position is constant over the time interval. According to a further aspect of this specification, there is described an apparatus comprising at least one processor, and at least one memory storing instructions that, when executed by the at least one processor, cause the apparatus to perform any of the methods described herein. According to a further aspect of this specification, there is described a computer program product / non-transitory computer readable medium storing computer- readable instructions that, when executed by a computer (such as a network node), cause the computer to perform any one or more of the methods described herein. Brief Description of the Drawings Example embodiments will now be described by way of non-limiting example, with reference to the accompanying drawings, in which: FIG.1 shows an overview of an example of a method for constructing an overdetermined system of equations for a UE position; FIG.2 shows an overview of an example of a method constructing an overdetermined system of equations for a UE position in a multi-UE environment; FIG.3 shows an example of a signalling diagram for adapting positioning resources in a multi-UE environment; FIG.4 shows a flow diagram of an example method for positioning a UE. The method is, in some examples, performed by a network node. The network node may implement a LMF; FIG.5 shows a flow diagram of an example method of adapting positioning resources used of positioning a UE; FIG.6 shows a flow diagram of an example method performed by a UE; FIG.7 shows an apparatus according to some example embodiments, which may form at least a part of a user equipment or network node; and FIG.8 shows a non-transitory media 800 according to some embodiments. Detailed Description This specification describes systems, methods and apparatus for accurate positioning of user equipment (UE) under UE mobility, as well as systems, methods and apparatus for allocating / distributing UE positioning resources. The subject matter of this specification utilises knowledge that a D-th derivative of the UE position of a mobile UE is constant to construct an overdetermined system of equations for a UE position. The overdetermined system of equations can be used to determine the UE position (or positions of multiple UEs) with a higher degree of localisation accuracy and / or to adapt positioning resource usage assigned to the UE (or the multiple UEs). FIG.1 shows an overview of an example of a method 100 for constructing an overdetermined system of equations for a UE position. The example shown in FIG.1 will be described with reference to a UE 102 with a constant acceleration 112 over a time interval. However, it will be appreciated that the method 100 may be applied to a UE 102 in which any position derivative is constant (to within a threshold value) over a time interval, e.g. the velocity 110, acceleration 112, jerk etc. A UE 102 measures position reference signals 106A-C (PRSs) from a plurality of anchor nodes 104A-C at a sequence of times, ti, in a time interval, and reports the measurements to a location management function (LMF – not shown). The UE 102 also provides to the LMF an indication that the D-th derivative of the UE position is constant over the time interval, or can be approximated as being constant, where D>0. The LMF constructs an overdetermined set of equations for the UE position at a target time in the time interval (also referred to herein as a “target UE position”) based on the measurements and the knowledge that the D-th derivative of the user position is constant. The overdetermined system of equations can be solved to determine a UE position, X(ttarg), at the target time, from which the UE positions at other times in the time interval can be determined / reconstructed. Alternatively or additionally, the degree of overdetermination of the overdetermined system of equations can be used to adapt the amount of network resources assigned to positioning the UE 102. In out-of-coverage scenarios, the LMF role can be replaced by implementing server UE functionality in, for example, an on-board UE of a vehicle and / or the target UE. In such cases, information about the UE velocity and acceleration can be directly fed into the UE by using on-board UE hardware and / or software, e.g. the acceleration and velocity of a vehicle. The UE position 108 at a time ti in the time interval is denoted X(ti), with position derivatives dnX(ti) / dtn. The first time derivative provides the UE velocity 110, v(ti), the second time derivative provides the UE acceleration 112, a(ti), the third time derivative provides the jerk, and so on. A series of UE positions at a sequence of times, (t1, t2, … tY), provides a trajectory 114 for the UE. The UE 102 may be any device that can wirelessly communicate with a network. For example, the UE 102 may be a mobile phone, a laptop, a wearable device (such as a smartwatch, smart glasses or the like), an Internet of things (IoT) device, a virtual reality (VR) or augmented reality (AR) headset, a car, a truck, a boat, a drone, an airplane, etc. The UE 102 comprises, in some examples, means for determining properties of the UE position 108 and / or its derivatives, such as one or more sensors (e.g. gyroscopes, accelerometers, compasses or the like). The properties of the UE position 108 and / or its derivatives include a determination of whether derivatives of the UE position are constant, or approximately constant, over a time interval, e.g. determining the lowest order of the UE position derivatives, D, if any, that is constant over the time interval. The properties of the UE position 108 and / or its derivatives, in some examples, further comprises values of the position derivatives of the UE position 108. The UE 102 further comprises means for communicating, to a network node (e.g. a network node hosting an LMF), an indication that the D-th derivative of the UE position 108 is constant over the time interval. Additional information relating to properties of the UE position 108 and / or its derivatives may also be reported to the network node, e.g. values of one or more of the derivatives. The time interval over which the D-th derivative is constant is, in some examples, also provided to the LMF by the UE 102. The time interval may be provided explicitly or implicitly. For example, standard motion sensor reporting, e.g. as per TS 37.355, may be used to deliver time-stamped information on travelled distances, angular directions, velocity and / or acceleration etc., from on-board sensors such as gyroscopes, accelerometers and / or compasses. Alternatively or additionally, the start and end time of the time interval (e.g. t1 and tY) may be signalled explicitly from the UE 102 to the LMF. Explicit 3GPP-compatible messaging can be used to indicate the starts and ends of time intervals. Information elements in the Sidelink Positioning Protocol (SLPP) and / or LTE positioning protocol (LPP) messages, or any other communication channel from the UE to the LMF, may be used to signal the start and end of time interval. For example, a one-bit mobility flag, M, may be used to indicate a mobility change of the UE 102, e.g. with a ‘1’ indicating a change in mobility from the last report and a ‘0’ indicating no change in mobility since the last report. Additional flags may be used to indicate whether constant derivatives are present, e.g. a one-bit speed flag, C, used to indicate whether the velocity 110 is constant or not (e.g. with a ‘1’ indicating a constant speed and a ‘0’ indicating a changing speed), a one-bit speed flag, A, used to indicate whether the acceleration 112 is constant or not (e.g. with a ‘1’ indicating a constant acceleration and a ‘0’ indicating a changing acceleration), etc. As an example, these combinations of MC flags may be used to indicate the following: MC 00 = still under same non-constant mobility pattern (e.g., constant acceleration) MC 01 = still same speed as before MC 10 = still under same non-constant mobility pattern (e.g., constant acceleration) MC 11 = new but still constant speed Alternatively, explicit non-3GPP indications may be used to determine / signal the periods over which the D-th derivative is constant. For example, direct inputs from other hardware / software (e.g. car hardware / software or other positioning systems, such as satellite positioning systems) may be used. In some examples, knowledge of the UE 102 trajectory and / or local area map (e.g., a highway lane) may be used. As described in further detail with respect to FIG.2, group and / or relative positioning (e.g., platoon members following platoon lead whereby relative positioning is used to confirm relative speed) may alternatively or additionally be used. Each anchor 104A-C is a network node with a known position that transmits (e.g. broadcasts) positioning signals 106A-C (e.g. PRSs or the like). The anchors 104A-C may, for example, comprise transmission and reception points (TRPs) of a 5G NR network. The anchors 104A-C may, for example, comprise further UEs (i.e. assisting UEs). The known positions of the anchors 104A-C may be static (i.e. fixed in space). Alternatively, one or more of the anchors 104A-C may have a time dependent position that is known a-priori or determined by the LMF. In the example shown, three anchors 104A-C are shown, though in general a plurality of anchors 104 are used. The positioning signals 106A-C transmitted by the anchors 104A-C are measured by the UE 102. The UE 102 may transmit measurement reports to the LMF. Alternatively or additionally, in response to measuring the PRSs 106A-C, the UE 102 transmits a response to the anchors 104A-C. The anchors 104A-C perform measurements on the response, and report the results to the LMF. Based on the measurements, whether from the UE 102 or the anchors 104A-C, the LMF may determine timing information for the PRSs. For example, the LMF may determine a time of flight, observed time difference of arrival and / or round trip time measurements. Based on the PRS measurements and the known positions of the anchors 104, a system of equations for the position of the UE 102 at the times in the time interval is given by: Where N is the number of anchors, and Di(tj) is the distance between anchor i and the UE at time tj determined from the PRS measurements. For example, the Di(tj) may be based on the time of flight of a PRS between anchor i and the UE, i.e. ^^^^^^ = ^ ^^^^^(^^), where c is the speed of transmission of the PRS signal through the medium between the anchor and the UE, e.g. the speed of light. In the above, and throughout this specification, it is assumed that the same anchors are used at each timestep. However, in some alternative examples the set of anchors used at each timestep do not need to be the same, e.g. because the UE moves out of range of one or more of the anchors and / or into range of one or more further anchors. This set of equations has 3Y unknown variables corresponding to the three components of the UE position at each time. If the D-th position derivative of the UE is constant over time, this can be reduced to at most 3(D+1) unknown values, corresponding to the three components of the UE position at a target time, and the three components of each of the D derivatives of the UE position. If additional derivative data, such as values of any of the D derivative of the UE position, is provided, the number of variables can be reduced further. To obtain this reduction, the position of the UE at any time in the time interval can be Taylor expanded around the UE position at the target time, ttarg. Since position derivatives higher than the D-th derivative are null, or can be approximated as such, the expansion terminates at the D-th derivative of the UE position and is approximately exact. The UE position at a time, t, in the time interval thus given by: Where dnX(ttarg) / dtnis the n-th derivative of the UE position evaluated at the target time. This equation can also be obtained by integrating the constant D-th derivative D times. This expansion (2) can be used to replace the UE position at the times in the system of equations (1) for the UE position. For simplicity, in the following examples it is assumed that ttarg is t1, though in general it can be any time in the time interval over which the D-th derivative of the UE position is constant. After this replacement, an overdetermined system of equations for the UE position is given by:^^(^^)− ^^^(^^)^^ ^ ^ ^ + ^^(^^)− ^^^(^^)^ + ^^(^^)− ^^^(^^)^ = ^^^(^^)^ This set of overdetermined equations has an overdetermination factor of YN / (3D+3) when the values for the derivatives are all unknown. In general, the degree of overdetermination is equal to the number of equations in the overdetermined system of equations divided by the number of unknown variables. This will depend on the number of timesteps, the number of anchors used at each timestep, the value of D and whether any values of the time derivatives are known (e.g. measured and communicated to the LMF by the UE). The overdetermined set of equations (3) can be solved to provide an estimate of the UE position at the target time, X(ttarg). Any method known in the art for solving overdetermined systems of equations may be used to determine the estimate of the position at the target time, for example Gauss-Newton iteration or other quasi- Newtonian methods. In some examples, values of the unknown position derivatives are also determined. From the position at the target time, X(ttarg), and its derivatives up to order D, the position of the UE at any time, t, in the time interval can be reconstructed using equation (2). This time need not correspond to any of the times in the sequence of timesteps; it can be any time in the time interval. In some examples, the UE position may be projected to times outside of time interval based on equation (2) to estimate the UE position at future / past times. This can allow an estimation of the UE position if there is no positioning data available outside the time interval. Such estimates may, in some examples, be tagged with a flag indicating that the estimate is outside of the time interval over which the D-th derivative is constant. The flag may comprise an integrity report for the estimate, e.g. indicating an error range or a probabilistic quantification of validity. The network may use such estimates in the absence of any up-to-date positioning data or position estimation. Alternatively or additionally, the degree of overdetermination of the overdetermined system of equations is used to adjust the amount of positioning resources allocated to positioning the UE 102. The LMF can reduce the total amount of network / wireless resources in use for tracking the positioning of this UE based on the degree of overdetermination, while keeping the accuracy of the UE position estimate within a threshold accuracy, e.g. such that the achievable positioning accuracy remains constant or not worse than prior to resource usage reduction. The network / wireless resources used for positioning the UE 102 are, in some examples, determined from a look-up table that is based on previous positioning performance using the anchors 104A-C and / or UEs with similar capabilities to the UE 102. The look- up table may be updated dynamically based on current positioning performance in order to maintain positioning accuracy. For example, the PRS bandwidth, repetition and / or power can be adjusted, e.g. reduced. Alternatively or additionally, the number of measurements and reports may be adjusted. In some examples, the min / max bounds for dynamic SL PRS bandwidth allocation are adapted. Alternatively, the server UE function ensures that, within a given permissible range of SL PRS bandwidth, lower or higher frequency resources are allocated for actual SL PRS transmissions. Similarly, in some examples, PRS spatial reuse / interference / collisions are controlled. Resource adjustment is, in some examples, performed via an assistance data update of the target UE 102 and / or anchor nodes 104A-C. In some examples, the resource adaptation reverts to the default resource allocation when one or more threshold conditions are satisfied. The one or more threshold conditions, for example, comprise a predetermined amount of time elapsing. Alternatively or additionally, the one or more threshold conditions comprise a change in the mobility pattern of the UE being reported, e.g. D-th derivative of the UE position changing, or the value of D (i.e. the lowest constant position derivative) changing. FIG.2 shows an overview of an example of a method 200 constructing an overdetermined system of equations for a UE position in a multi-UE environment. In the example shown, two UEs 202A, 202B are present, though in general a plurality of UEs may be used with the method 200. In the example shown, a first UE 202A (e.g. a UE associated with a car) is overtaking a second UE 202B (e.g. a UE associated with a truck). For example, the truck may be moving with a constant speed (D=1), while being overtaken by a car with (piecewise) constant acceleration (D=2). The plurality of UEs 202A, 202B have a known mobility patterns 208A, 208B. Knowledge of a UE 202A, 202B mobility pattern can be indicated to said UE 202A, 202B, for example, by the LMF or a third-party device (not shown). In the former case, the LMF can communicate, for example, the information of [vX, vY, vZ], [aX, aY, aZ] and / or higher order position derivative as determined during individual UE localization (e.g., as described above in relation to FIG.1). In the latter case, a related device, such as an on-board computer in a car, can accurately provide the mobility pattern info directly and in real time, e.g. to the on-board 5G NR UE. In such a case, it would be the on-board UE informing the LMF or server UE about its mobility pattern. Once provided with their respective mobility pattern information, each UE 202A, 202B can advertise it (e.g., via discovery messages or other broadcast mechanisms). UEs 202A, 202B with identical or similar mobility pattern, (e.g., cars sharing the same road lane or passengers in the same vehicle) are then be grouped together by the LMF for joint positioning and / or resource adaptation. The term “similar mobility pattern” is preferably used to connote a scenario where the mobility patterns of the plurality of UEs 202A, 202B are not identical, but are related by some known relationship or transformation. For example, an overtaking UE 202A can have a known relative velocity and / or acceleration compared to the overtaken UE 202B. A plurality of anchors 204A-C broadcast respective PRS 206A-C that are measured by the plurality of UEs 202A, 202B. Based on the measurements, a joint overdetermined set of equations for the UE positions of the plurality of UEs 202A, 202B is constructed. One of the UEs in the plurality of UEs 202, 202B can be designated as a “target UE”. Based on the known mobility pattern between the UEs, positions of other UEs in the plurality of UEs can be written in terms of the position of the target UE at a target time in a similar manner to that described in relation to FIG.1. The overdetermined set of equations can be solved to determine the positions of the plurality of UEs 202A, 202B, as described in relation to FIG.1. Alternatively or additionally, the degree of over-determination of the overdetermined system of equations can be then exploited to reduce the joint PRS measurement / reporting overhead for the plurality of UEs 202A, 202B. For example, in the case of two UEs 202A, 202B, the two UEs can be configured to measure in odd and even PRS instances respectively. As a further example of beneficial resource usage reduction, reduced-capability UEs can be configured to lower the amount of conducted PRS measurements to save power, or the LMF can compensate for bandwidth limitations of the reduced capability UE using the measurements of a full-bandwidth UE in the plurality of UEs. The network / wireless resources assigned for positioning can also be adjusted, as described in relation to FIG.1. By monitoring how the over-determination bounds vary as function of SL PRS bandwidth / power / repetitions, the LMF / server UE can adaptively optimize these parameters, e.g., to reduce bandwidth when speed and acceleration is constant for both / either of the UEs 202A, 202B. The achievable accuracy can remain constant / within a threshold accuracy despite the adaptation (e.g. reduction) of SL PRS resources. If interference and collisions are reduced as function of bandwidth reduction, further improvement of positioning accuracy and / or SL PRS resource usage is possible. In some examples, the resource adaptation reverts to the default resource allocation / individual positioning of the UEs 202A, 202B when one or more threshold conditions are satisfied. The one or more threshold conditions may comprise the threshold conditions described in relation to FIG.1. Alternatively or additionally, the threshold conditions comprise detecting a divergent mobility pattern between the plurality of UEs 202A, 202B, which can trigger fall back to individual positioning of individual UEs based on their own reports. For example, the car associated with the first UE 202A may leave a shared road. FIG.3 shows an example of a signalling diagram for adapting positioning resources in a multi-UE environment. FulCap UE 302A refers to a standard full-capability UE (e.g. a platoon lead) while RedCap UE 302B refers to a reduced-capability UE (e.g., a platoon member). The RedCap UE 302B may have a limited transmission (TX) and / or reception (RX) bandwidth, e.g. with TX / RX bandwidth limited to, for example, 5MHz. This can significantly reduce positioning accuracy of the RedCap UE 302B. Initially, the LMF (or the server UE) 306 sets up a collective (group) positioning session 308. For example, each UE 302A, 302B determines its individual mobility pattern, e.g. based on single UE positioning and / or local mobility sensors. Data indicative if the mobility pattern can be transmitted to the LMF, which uses said mobility pattern data to group the UEs 302A, 302B for joint positioning based on the UEs 302A, 302B having similar mobility patterns. The RedCap 302B and FulCap UE 302A then provide the LMF 306 with motion sensor reporting 310A, 310B, for example as described in TS 37.355. This may include, for example, time-stamped information on distance travelled, compass direction, speed and / or acceleration. The UE motion sensor reporting 310A, 310B may comprise an indication that a respective D-th derivative for each UE is constant. In examples in which the anchor node positions are not fixed, the anchor nodes 304 may also provide motion sensor reporting 310C to allow known positions of the anchor nodes to be determined. The motion sensor reporting information 310 permits the LMF 306 to implement joint multilateration 312 for the plurality of UEs 302A, 302B, where overlapping intervals characterized by constant time-derivative of location of order “D” for the UEs are used for joint localization and positioning resource adaption, as described in relation to FIG. 2. Following determination of the positioning resource adjustments by the LMF 306, the LMF 306 transmits assistance data updates 314 to the anchors 304 (and / or UEs transmitting PRSs, if present). The assistance data updates 314 contain configuration information for the anchors 304 that instruct the anchors 304 how to adapt PRSs for the joint positioning of the UEs 302A, 302B. For example, the assistance data updates 314 may provide a maximum and / or minimum bandwidth for SL PRSs. The anchors 304 then broadcast PRSs with the adjusted configuration 316, e.g. with a reduced bandwidth. The RedCap UE 302A and FulCap UE 302B perform measurements on the reduced bandwidth SL PRSs 314 and report the measurements 316, 318 to the LMF 306. Based on the measurement reports, the LMF 306 performs collective localisation 320 of the UEs 302A, 302B, for example as described in relation to FIG.2. FIG.4 shows a flow diagram 400 of an example method for positioning a UE. The method is, in some examples, performed by a network node. The network node may implement a LMF. At operation 402, an overdetermined system of equations for a UE position is constructed. The overdetermined system of equations is based on position derivative data and a plurality of sequences of positioning data for the user equipment. The positioning derivative data comprises an indication that the D-th derivative of the user equipment position is constant, and / or approximately constant, over a time interval, wherein D is a positive non-null integer (i.e. D>0). In some examples, the positioning derivative data further comprises a value of one or more derivatives of the user equipment position, e.g. as determined by sensors of the UE. The position derivative data is, in some examples, received by the LMF from the UE. Each sequence of positioning data for the user equipment in the plurality of sequences of positioning data is associated with a respective anchor node in a plurality of anchor nodes. A sequence of positioning data comprises positioning data for the user equipment at a plurality of times in the time interval. Examples of such positioning data comprise measurement data obtained from measurements of PRSs transmitted between the UE and one or more anchor nodes in the plurality of anchor nodes, such as time of flight, observed time difference of arrival and / or round trip time measurements. A general set of equations for the UE position at respective times in the plurality of times may be constructed. The general set of equations may be based on differences between each anchor position and the UE position at respective times in the plurality of times, and a distance determined from the positioning data. For example, for each anchor, i, at each time, tj, in the time interval, an equation for the UE position may be given by: where Xi = (Xi, Yi, Zi) is the position of the i-th anchor, X(tj) = (X(tj), Y(tj), Z(tj)) is the UE position at time tj, Ti(tj) is the time of flight of a PRS between the i-th anchor and the UE at time tj(for example, measured directly or derived from other measurements), and c is the speed of transmission of the PRS signal (i.e. the speed of light). In some examples, every anchor node provides positioning data at each time in the plurality of times. In other examples, one or more of the anchors may not provide positioning data at one or more of the times in the plurality of times, i.e. there is “missing” positioning data in one or more of the sequences of positioning data. Within this general set of equations, the UE positions at respective time can be substituted with a UE position at a target time, ttarg, (i.e. the target position) and a change in position of the user equipment with respect to the target time, ΔX(tj). The target time may be one of the plurality of times in the time interval. Alternatively, the target time may be at time in the time interval that is not one of the plurality of times. The change in position is based on the D-th derivative of the UE position being constant and a time difference between the target time and the respective times in the plurality of times, (ttarg- tj). The change in position of the UE with respect to the target time may be determined based on the D-th integral of the constant D-th derivative. For example, the change in position may be given by: Where dnX / dtnis the n-th derivative of the UE position with respect to time evaluated at the target time, e.g. n=1 is the velocity, V, n=2 is the acceleration, A, n=3 is the jerk etc. In some examples, the resulting overdetermined system of equations comprises YN equations for 3D+1 variables (the three components of the UE position at the target time and its derivatives), where Y is the number of times in the plurality of times in the time interval and N is the number of anchor nodes. In some examples, where one or more times in the plurality of times is associated with positioning data from only a subset of the anchor nodes, fewer equations will be present in the overdetermined system of equations. In some examples, the overdetermined system of equations is constructed based on positioning data from a plurality of UEs with a similar mobility pattern, i.e. a known relationship between their positions. One UE may be assigned as a target UE, and positions of other UEs written in terms of the target UE position using the relationship between them. The overdetermined set of equations may be solved for the position of the target UE at a target time, as described in the single UE case. The target UE position at the target time can then be used to determine the positions of the other UEs at times in the time interval using the known relationship between the UE positions. At operation 404, one or more estimates of the UE position are determined based on the overdetermined set of equations for the UE position. The overdetermined set of equations may be used to determine the UE position at the target time. The derivatives of the UE position at the target time may also be determined. Any method known in the art for solving overdetermined (linear or non-linear) systems of equations may be used to determine the UE position (and it time derivatives) at the target time. For example, Gauss-Newton iteration may be used to estimate the UE position at the target time. In some examples, the UE position at each time in the plurality of times is reconstructed based on the target position of the user equipment at the target time, the one or more position derivatives at the target time, and a difference between the target time and the respective time in the time interval, e.g. using: A UE position at a time outside the time interval may also be forecast based on the target position of the user equipment at the target time, the one or more derivatives of the user equipment position at the target time, and a difference between the target time and the time outside the time interval, e.g. using the equation above. In such examples, the forecast UE position may be associated with a flag indicating that the forecast is outside of the time window. FIG.5 shows a flow diagram 500 of an example method of adapting positioning resources used of positioning a UE. The method is, in some examples, performed by a network node. The network node may implement a LMF. At operation 502, an overdetermined system of equations for a UE position is constructed. The overdetermined system of equations is based on position derivative data and a plurality of sequences of positioning data for the user equipment. Operation 502 may correspond to operation 402 of FIG.4. At operation 504, a degree of overdetermination of the overdetermined system of equations is determined. The degree of overdetermination is equal to the number of equations in the system of equations divided by the number of unknown variables. In some examples, determining the degree of overdetermination of the system of equations is performed without explicitly constructing the overdetermined set of equations, i.e. without performing operation 502. The degree of overdetermination may be determined from the number of timesteps for the positioning, the number of anchors used at each timestep and the value of D, i.e. the order of the constant position derivative directly. For example, when the same number of anchors, N, is used for each of Y timesteps, and none of the values of the UE position derivatives are known, the degree of overdetermination is given by YN / (3D+3). In general, the degree of overdetermination will be given by a sum of the number of anchors at each timestep divided by the number of unknowns in the system, i.e. the number of components of the target position plus a product of the number of components of the unknown position derivatives with the number of unknown position derivatives. At operation 506, one or more adjustments to one or more positioning resources used to determine the user equipment position are made based on the degree of overdetermination. The adjustments are, in some examples, made while maintaining at least a threshold accuracy for positioning of the UE. In some examples, one or more properties of positioning reference signals transmitted between the plurality of anchor nodes and the UE are adjusted, e.g. a number of positioning reference signals; a bandwidth of the positioning reference signals; a repetition pattern of the positioning reference signals; and / or a power of the positioning reference signals. The adjusted properties may be determined from a lookup table. The lookup table may be based on previous positioning performance of UEs with similar capabilities as the UE. Adjusting the positioning resources may reduce the degree of overdetermination of the system for future measurements. In examples where multiple UEs are jointly positioned, the one or more adjustments to one or more positioning resources used to determine the user equipment position may comprise distributing positioning resources between the UE and the one or more further UEs. For example, in a system of two UEs, one may measure odd PRSs, and one measure even PRSs. Alternatively, one or more reduced capability UEs may be assigned fewer positioning resources (e.g. lower bandwidths, fewer frequencies) than a related full capability UE. In some examples, the adjustments to the positioning resources may be reverted to a default level in response to one or more threshold conditions being satisfied. The one or more threshold conditions may comprise one or more of: a threshold time elapsing; the end time of the time interval being reached; a change in mobility condition of the UE (e.g. the D-th derivative of the position of the user equipment changing); and / or a positioning accuracy falling below a threshold value. In multi-UE examples, the one or more threshold conditions may alternatively or additionally comprise detecting a diverging mobility pattern between two or more UEs in the plurality of UEs. FIG.6 shows a flow diagram of an example method performed by a UE in communication with a network node. The network node may host a location management function. The method may correspond to the method performed by a UE in FIGs.1 to 3. At operation 602, the UE determines that that a D-th derivative of the UE position is constant, or can be approximated as being constant, over a time interval, wherein D is a positive non-null integer. The UE may determine that the D-th derivative of the UE position can be approximated as being constant when variations in the D-th derivative of the UE position over the time interval are below a threshold value. The value of D may be determined by the UE as the smallest value of D for which the derivative of the UE position is constant or can be approximated as being constant. Determining that the D-th derivative of the apparatus position is constant, or can be approximated as being constant, over the time interval may be based on sensor data collected by sensors of the UE. For example, the sensors may include one or more of: motion sensor data from one or more motion sensors of the apparatus; sidelink positioning protocol data; map data; relative positioning data with respect to one or more further apparatus; and / or satellite positioning data. In some examples, the data collected by the sensors is processed to determine values for one or more derivatives of the UE position. At operation 604, the UE communicates, to a network node, an indication that the D-th derivative of the UE position is constant over the time interval. Values of one or more derivatives of the UE position may also be communicated to the network node. In some examples, the method may further comprise communicating, to the network node, information indicative of the time interval. The information indicative of the time interval may comprise a start time and an end time of the time interval. For example. a flag in a measurement report may indicate whether a time derivative of the UE position has changed from the last measurement (a “change flag”) or is constant (a “constant flag”. A change flag may indicate a start of the time interval, with the next change flag indicating the end of the time interval. Alternatively, the information indicative of the time interval comprises information indicative of a length of the time interval with respect to a reference time. The reference time may, for example, be the first PRS transmission from the anchor nodes received by the UE, a time of receipt of the indication by the LMF, or a time of transmission of the indication by the UE. Many other examples of a reference time are possible. FIG.7 shows an apparatus according to some example embodiments, which may form at least a part of a user equipment or network node. The apparatus may be configured to perform the operations described herein, for example operations described with reference to any disclosed process. The apparatus comprises at least one processor 700 and at least one memory 701 directly or closely connected to the processor. The memory 701 includes at least one random access memory (RAM) 701A and at least one read-only memory (ROM) 701B. Processor instructions (software) 705 is stored in the ROM 701B. The apparatus may be connected to a transmitter (TX) and a receiver (RX). The apparatus may, optionally, be connected with a user interface (UI) for instructing the apparatus and / or for outputting data. The at least one processor 700, with the at least one memory 701 and the instructions 705 are arranged to cause the apparatus to at least perform at least the method, or a part of the method, according to any preceding process, for example as disclosed in relation to the flow diagrams of FIGs.4, 5 and / or 6 and related features thereof. FIG.8 shows a non-transitory media 800 according to some embodiments. The non- transitory media 800 is a computer readable storage medium. It may be e.g., a CD, a DVD, a USB stick, a blue ray disk, etc. The non-transitory media 800 stores instructions that, when executed by at least one processor, cause an apparatus to perform the method of any preceding process, for example as disclosed in relation to the flow diagrams and related features thereof. Names of network elements, protocols, and methods are based on current standards. In other versions or other technologies, the names of these network elements and / or protocols and / or methods may be different, as long as they provide a corresponding functionality. For example, embodiments may be deployed in 2G / 3G / 4G / 5G networks and further generations of 3GPP but also in non-3GPP radio networks such as Wi-Fi. A memory may be volatile or non-volatile. It may be e.g., a RAM, a SRAM, a flash memory, a FPGA block ram, a DCD, a CD, a USB stick, and a blue ray disk. If not otherwise stated or otherwise made clear from the context, the statement that two entities are different means that they perform different functions. It does not necessarily mean that they are based on different hardware. That is, each of the entities described in the present description may be based on a different hardware, or some or all of the entities may be based on the same hardware. It does not necessarily mean that they are based on different software. That is, each of the entities described in the present description may be based on different software, or some or all of the entities may be based on the same software. Each of the entities described in the present description may be embodied in the cloud. Implementations of any of the above-described blocks, apparatuses, systems, techniques or methods include, as non-limiting examples, implementations as hardware, software, firmware, special purpose circuits or logic, general purpose hardware or controller or other computing devices, or some combination thereof. Some embodiments may be implemented in the cloud. It is to be understood that what is described above is what is presently considered the preferred embodiments. However, it should be noted that the description of the preferred embodiments is given by way of example only and that various modifications may be made without departing from the scope as defined by the appended claims.

Claims

Claims 1. An apparatus comprising: means for constructing an overdetermined system of equations for a user equipment position based at least on; position derivative data comprising an indication that the D-th derivative of the user equipment position is constant over a time interval, wherein D is a positive non-null integer; and a plurality of sequences of positioning data for the user equipment, each sequence of positioning data in the plurality of sequences of positioning data being associated with a respective anchor node in a plurality of anchor nodes and comprising positioning data for the user equipment at a plurality of times in the time interval; and means for determining, from the overdetermined system of equations, one or more estimates of the user equipment position at one or more respective times in the time interval.

2. The apparatus of claim 1, further comprising: means for receiving, from the user equipment, the position derivative data.

3. The apparatus of any of claims 1 or 2, wherein constructing the overdetermined system of equations for the position of the user equipment comprises: determining equations for the user equipment position at respective times in the plurality of times; and replacing the user equipment position at the respective times in the equations with a target position of the user equipment at a target time and a change in position of the user equipment with respect to the target time, wherein the change in position is based on the D-th derivative of the user equipment position being constant and a time difference between the target time and the respective times in the plurality of times.

4. The apparatus of claim 3, wherein the change in position, Δ^(^), is based on:where ^ denotes a respective time in the plurality of times, ^^denotes the target time, and X denotes the user equipment position.

5. The apparatus of any of claims 3 or 4, wherein determining, from the overdetermined system of equations, the target position of the user equipment at the target time comprises determining one or more derivatives of the user equipment position at the target time from the overdetermined system of equations.

6. The apparatus of claim 5, further comprising means for determining a respective user equipment position at a respective time in the time interval based on the target position of the user equipment at the target time, the one or more position derivatives at the target time, and a difference between the target time and the respective time in the time interval.

7. The apparatus of any of claims 5 or 6, further comprising: means for forecasting a future user equipment position at a time outside the time interval based on the target position of the user equipment at the target time, the one or more derivatives of the user equipment position at the target time, and a difference between the target time and the time outside the time interval; and means for associating the forecast with a flag indicating that the forecast is outside the time interval.

8. The apparatus of any preceding claim, wherein the positioning data for the user equipment associated with one or more anchor nodes in the plurality of anchor nodes comprises one or more of measurements of positioning reference signals transmitted between the user equipment and the one or more anchor nodes in the plurality of anchor nodes, the one or more measurements comprising one or more of: a round trip time; a time of arrival; and / or an observed time difference of arrival.

9. An apparatus comprising: means for determining that a D-th derivative of the apparatus position is constant or can be approximated as being constant over a time interval, wherein D is a positive non-null integer; means for communicating, to a network node, an indication that the D-th derivative of the apparatus position is constant over the time interval.

10. The apparatus of claim 9, wherein determining that the D-th derivative of the apparatus position is constant or can be approximated as being constant over the timeinterval is based on one or more of: motion sensor data from one or more motion sensors of the apparatus; sidelink positioning protocol data; map data; relative positioning data with respect to one or more further apparatus; and / or satellite positioning data.

11. The apparatus of any of claims 9 or 10, further comprising means for: determining one or more values of one or more respective derivatives of the apparatus position; and communicating, to the network node, the one or more values.

12. The apparatus of any of claims 9 to 11, further comprising means for communicating, to the network node, information indicative of the time interval.

13. The apparatus of claim 12, wherein the information indicative of the time interval comprises a start time and an end time of the time interval.

14. The apparatus of claim 12, wherein the information indicative of the time interval comprises information indicative of a length of the time interval with respect to a reference time.

15. The apparatus of any of claims 9 to 13, wherein the network node hosts a location management function.

16. A method performed by a network element, the method comprising: constructing an overdetermined system of equations for a user equipment position based on at least; position derivative data comprising an indication that the D-th derivative of the user equipment position is constant over a time interval, wherein D is a positive non-null integer; a plurality of sequences of positioning data for the user equipment, each sequence of positioning data being associated with a respective anchor node in a plurality of anchor nodes and comprising positioning data for the user equipment at a plurality of times in the time interval; and determining, from the overdetermined system of equations, one or more estimates of the user equipment position at one or more respective times in the time interval.

17. A method performed by a user equipment, the method comprising: determining that a D-th derivative of the user equipment position is constant or can be approximated as being constant over a time interval, wherein D is a positive non-null integer; and communicating, to a network node, an indication that the D-th derivative of the apparatus position is constant over the time interval.

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