arX

iv:2

010.

0431

3v1

[ee

ss.S

P] 9

Oct

202

01

Collision Prediction from UWB RangeMeasurements

Alemayehu Solomon Abrar, Anh Luong, Gregory Spencer, Nathan Genstein, Neal Patwari and

Mark Minor

Abstract—The ability to predict, and thus react to, oncoming collisions among a set of mobile agents is a fundamental requirement for

safe autonomous movement, both human and robotic. This paper addresses systems that use range measurements between mobile

agents for the purpose of collision prediction, which involves prediction of the agents’ future paths to know if they will collide at any

time. One straightforward system would use known-location static anchors to estimate agent coordinates over time, and use the track

to predict collision. Fundamentally, no fixed coordinate system is required for collision prediction, so using only the pairwise range

between two agents can be used to predict collision. We present lower bound analysis which shows the limitations of this pairwise

method. As an alternative anchor-free method, we propose the friend-based autonomous collision prediction and tracking (FACT)

method that uses all measured ranges between nearby (unknown location mobile) agents, in a distributed algorithm, to estimate their

relative locations and velocities and predict future collisions between agents. Using analysis and simulation, we show the potential for

FACT to achieve equal or better collision detection performance compared to other methods, while avoiding the need for anchors. We

then build a network of N ultra wideband (UWB) devices and an efficient multi-node protocol which allows all O(N2) pairwise ranges

to be measured in N slots. We run experiments with up to six independent robot agents moving and colliding in a 2D plane and up to

four anchor nodes to compare the performance of the collision prediction methods. We show that the FACT method can perform better

than either other method but without the need for a fixed infrastructure of anchor nodes.

Index Terms—Collision Prediction, Autonomous, Ultra wideband,

✦

1 INTRODUCTION

UNEXPECTED collisions that occur within a set of mobileagents, be it mid-air collisions or traffic collisions, canbe devastating. In contact sports like American football,head collisions are the main causes of millions of traumaticbrain injuries such as concussions every year [1]. We in-vestigate smart sensor-enabled devices which predict animpending collision between agents and take action to avoidor lessen the damage from collision, for example, smarthelmets that alert a player before a collision from behind,or autonomous drones that avoid colliding.

Such systems rely heavily on reliable sensing. Sensingmodalities for this purpose including global navigationsatellite system (GNSS) tracking [2], inertial measurements,vision sensors, and ultrasonic and millimeter wave radarsystems. Existing approaches have been used to enablekey safety features in particular applications, but do haveshortcomings. GNSS-based systems require significant bat-tery power, can be very inaccurate in urban areas, andunavailable indoors. Vision-based systems have shortcom-ings when the impending collision is between small or fastobjects, and in poor light conditions. Radar systems mustuse high transmit power to contend with d−4 losses.

Alemayehu Solomon Abrar was with the Preston M. Green Department ofElectrical & Systems Engineering at Washington University in St. LouisAnh Luong was with Carnegie Mellon UniversityGregory Spencer was with the University of UtahNathan Genstein was with the Preston M. Green Department of Electrical &Systems Engineering at Washington University in St. LouisNeal Patwari is with McKelvey School of Engineering at Washington Univer-sity in St. LouisMark Minor was with the University of Utah

In this paper, we investigate autonomous objects whichpredict (and react to) impending collisions among them-selves using pairwise range measurements. We note thatUWB transceivers provide low cost and energy-efficientrange measurements with fine resolution, on the order ofa few cm of standard deviation, due to their use of very nar-row pulses. Range-based collision sensing is complementaryto existing approaches because it is reliable in both indoorand outdoor environments, for small or large objects, andin the dark or in the light. Two-way ranging has d−2 (asopposed to d−4) losses, enabling fast moving objects topredict a collision even from a long distance. Such systemsrequire each object to have a transceiver, but as opposed toproposed FCC GNSS requirements for drones [2], pairwiseranging systems can autonomously avoid collision and canbe lower in power consumption.

1.0.0.1 Localization vs. Collision Prediction: Unlikelocalization, collision prediction does not require knowledgeof absolute coordinates. Whether or not a collision willoccur depends on only the relative kinematics of objects,including relative position and relative velocity. Thus, itis not necessary to have infrastructure nodes or a globalcoordinate reference. Compared to localization, however,collision prediction is significantly more challenging be-cause it must predict future positions for all time fromnow until a future time ∆t from now. To be clear aboutthis important point: predicting position at only the time∆t from now is not sufficient — an impending collisionshould be predicted if two trajectories will intersect for anytime between now and ∆t later. As such, collision predictiondepends critically on accurate estimation of relative velocity

http://arxiv.org/abs/2010.04313v1

2

(and relative acceleration), and it always involves two ormore nodes. In comparison, localization does not requirekinematics or multiple nodes.

Despite these differences, a naı̈ve approach for collisionprediction could simply first estimate absolute coordinatesand track for each individual agent, predict each agent’sfuture trajectory, and determine if these coordinate tra-jectories intersect. A typical approach in localization withUWB range measurements involves multilateration of a tagnode with respect to static reference nodes called anchors[4]. However, anchor-based localization entails the needfor deployed infrastructure of several anchors, each withlocation determined with some other method.

1.0.0.2 Relative Localization and Tracking: Basedon the fact that collision prediction requires only relativekinematics, our proposed method is more closely relatedto past methods in relative localization. A classical relativeapproach is to use multi-dimensional scaling (MDS) togenerate a relative map. However, such relative positionestimates cannot be directly used to determine relativevelocity since each successive position estimate has an ar-bitrary translation, rotation, and flip. Rajan et al. proposedto determine relative velocity with a modified MDS methodin which relative velocity is estimated from the secondorder derivative of squared distance measurements [5]. Yet,this method results in inaccurate velocity estimates in thepresence of noisy range measurements.

1.0.0.3 Questions on Performance: For some appli-cations, the requirement for anchors will be infeasible, andin others, pairwise range measurements will be infeasi-ble. Yet the question remains: Does anchor-based localiza-tion provide better collision prediction performance thana system using only relative range measurements betweennodes? This question is largely unanswered in the literature.This paper provides analysis and data that address thisquestion.

A second key question addressed in this paper is: Formethods using only relative range measurements, does in-creasing the N used by a collision predictor improve itsperformance? Minimally, N = 2 nodes i and j can measuretheir relative distance over time, and use the measurementsto predict an impending collision between i and j, whichwe call the pairwise method. We propose to additionally usemeasurements between pairs of nodes in the set includingi, j, and other “friend” nodes. We are unaware of anypublished relative collision prediction method using N > 2.Although no node has absolute coordinate information, weshow that the additional relative range measurements canimprove the prediction of collision between i and j, viaanalysis and experiment.

1.0.0.4 Contribution 1: FACT Algorithm: In thispaper, we propose and evaluate a novel collision predic-tion method that uses only relative range measurementsand distributed computation for mobile nodes, which wecall the Friend-based Autonomous Collision prediction andTracking (FACT) method. The FACT method assumes eachnode is equipped with a UWB transceiver for ranging,that N mobile neighboring nodes cooperate to compute allpairwise range measurements, and use them all to predictthe impending collision of each pair of nodes. We evaluateand compare the method with pairwise and anchor-based

approaches via theoretical bound analysis, simulation, andexperiments involving robotic vehicles.

1.0.0.5 Contribution 2: O(N) Multi-node RangingProtocol & Implementation: We found, however, that ex-isting UWB protocols did not allow us to measure

(

N2

)

pairwise range measurements at a high enough rate whenN is large. In existing protocols, many UWB packets areused solely to estimate clock frequency offset (which hasmore significant effects in multi-node protocols). Existingprotocols use O(N2) UWB packets to separately measurethe pairwise ranges [6]. Note that the UWB time stamp dataincreases as O(N2) when sharing N time stamps for eachof N nodes. In combination, the range measurement periodis long and rises as O(N2).

Our key insight is that using a secondary transceiver,capable of handling both frequency synchronization anddata sharing, allows us to implement a dramatically fastermulti-node ranging protocol. As narrowband (as comparedto UWB) transceivers can use much higher transmit powers,they can communicate data with much greater bandwidthefficiency. Further, they can be used to synchronize localoscillators among all nodes. In our implementation, wedevelop and test our multi-node ranging protocol which,using a narrowband transceiver for synchronization andtime stamp data communication, dramatically shortens themeasurement period. In one round of our multi-node rang-ing protocol, each node exchanges a single short UWBpacket, which enables each node to calculate

(

N2

)

ranges.

Thus the measurement period for calculating(

N2

)

rangesreduces to O(N) from the O(N2) of prior protocols.

2 PROBLEM STATEMENT

The objective of collision prediction is to detect a futureintersection between the trajectories of a pair of mobileagents. We assume that, for a short period of time, nodevelocities can be well approximated as constant.

Fig. 1: Agents with linear relative motion

Consider, for example, a simple scenario where twonodes A and B move in uniform rectilinear motion. Weassume that node A and node B, initially located at xA andxB move at constant velocities vA and vB respectively asshown in Figure 1. For the given setup problem, we areinterested in the relative position xBA and relative velocityvBA. For convenience, we drop subscripts for all subsequentrelative vectors with respect to node A. Their relative initialdistance and speed are denoted by x and v respectively.

We determine the possibility of collision between a pairof non-stationary agents based on three scalar collision pre-diction (CP) parameters, namely:

3

• v: relative speed• dm: the minimum passing distance between two nodes.• tm: the time at which the relative distance reaches its

minimum.

For a given initial relative position and relative velocity,the collision prediction parameters can be written as:

v = ‖v‖tm = −

x · vv2

dm =1

v

(

‖x‖2‖v‖2− (x · v)2)

1

2

(1)

In practical settings, a node has a non-zero volume, andcan be modelled by a spherical volume with some radiusr. In this case, collisions could occur even prior to reachingthe minimum passing distance. Assuming all nodes with thesame radius r moving in a 2-D plane, then for positive tmand dm < 2r, the time of collision tc is calculated as:

tc = tm −1

v

√

4r2 − d2m (2)The values of CP parameters estimated for a pair of

mobile agents can be used to determine if there is a futurecollision. For example, a system may declare an inboundcollision when two nodes are coming close to less thana meter (dm < 1m) within half a second (tm < 0.5s)Furthermore, the accuracy of a collision prediction methodcan be evaluated in terms of its performance in accuratelyestimating of these parameters.

In this article, we present methods to estimate CP pa-rameters solely from distance measurements to predict col-lisions.

3 ESTIMATION AND DETECTION

In this section, we present methods to estimate collision pre-diction parameters and to detect oncoming collisions usingrange measurements. We develop statistical bounds basedon Cramér-Rao lower bound (CRLB) analysis to evaluatethe methods.

Range measurements between a set of mobile agents canbe used in distributed or an infrastructure-based manner toestimate CP parameters. Distributed collision prediction isperformed using range measurements from either a singleor multiple pairs simultaneously. We refer to methods thatuse range measurements from a single pair as pairwisemethods and those employing multiple pairs for collisionprediction of every pair as Friend based methods. On theother hand, for ranging involving nodes taking measure-ments with respect to fixed reference nodes, we employanchor-based collision prediction.

3.1 Cramér-Rao Lower Bound Analysis

We apply CRLB analysis in order to determine theoreticallymaximum accuracy for methods used in collision predic-tion. The CRLB is the most common variance bound due toits simplicity [7]. It provides the lowest possible estimationvariance achieved by any unbiased estimator. To the bestof our knowledge, we present the first bound analysis inrelation to collision prediction from range measurements.We compute the bound on estimation variance for a CP

parameter vector given by θ := [dm, tm, v]T

under multiplerange measurement models.

3.1.1 Bounds for Pairwise Ranging

For pairwise methods, the CP parameters are estimatedusing range measurements from a single pair of nodes. Weconsider a noisy distance δ(tn) between the nodes measuredat time tn, which is given by

δ(tn) = ‖x + vtn‖+w(tn)

=(

d2m + v2 (tm − tn)2

)1

2

+ w(tn)(3)

where w represents zero-mean, additive white Gaussiannoise, w ∼ N

(

0, σ2)

For a set of N i.i.d. distance measure-ments δ = {δ(t1), · · · , δ(tN )}, the log-likelihood becomes

L(θ|δ) = −12log(

(2π)Nσ2)

− 1σ2

N∑

n=1

(

δ(tn)−√

d2m + v2 (tm − tn)2

)2

.(4)

Defining I(θ) = −E[

∂2L(θ|δ)/∂θ2]

, we have

I(θ) = 1σ2

N∑

n=1

Fn

d2m + v2 (tm − tn)2

(5)

where

Fn =

v4(tm − tn)2 v2dm(tm − tn) v3(tm − tn)3v2dm(tm − tn) d2m vdm(tm − tn)2v3(tm − tn)3 vdm(tm − tn)2 v2(tm − tn)4

The theoretical bounds on variance for tm, dm and v esti-mates are computed as:

var(t̂m) ≥{

I(θ)−1}

11

var(d̂m) ≥{

I(θ)−1}

22

var(v̂) ≥{

I(θ)−1}

33

(6)

3.1.2 Bounds for Anchor-Based Ranging

In anchor-based ranging, every node takes measurementswith respect to nodes with fixed known location. we assumethat there are K anchor nodes with the kth anchor nodelocated at xk. For a pair having i

th and jth mobile nodes,the distances with respect anchor k are denoted by xik andxjk respectively. The measured distance vector at anchor kis given by:

δk(tn) =

[

δik(tn)δjk(tn)

]

=

[

xik(tn)xjk(tn)

]

+ wk(tn)

= δ̄k(tn) + wk(tn)

(7)

Where the additive noise is assumed to be Gaussian (i.e,wk ∼ N (0, σ2I)

Since there is no direct relationship between the param-eters in θ and the range measurements δk, We define initialpositions and velocities of a pair of nodes as the initialparameters. Hence, the initial parameter vector is given

by α := [xi, xj , vi, vj ]T

. The log-likelihood distribution formeasurement vector δk at the k

th anchor becomes

log p(δk(tn) = dk|α) = −((

‖dk − δ̄k(tn)‖22σ2

)

+ log(

2πσ2)

)

4

Assuming i.i.d. distance measurements, the log likelihoodfor all measurements from K anchors for N samples iscalculated as

L(α|δ) =K∑

k=1

N∑

n=1

log p(δk(tn) = dk|α) (8)

If we define Rik(tn) :=xik(tn)xik(tn)

T

‖xik(tn)‖2, then I(α)

becomes

I(α) = 1σ2

K∑

k=1

N∑

n=1

Γk(tn).

where

Γk(tn) =

Rik(tn) 0 tnRik(tn) 00 Rjk(tn) 0 tnRjk(tn)

tnRik(tn) 0 t2nRik(tn) 0

0 tnRjk(tn) 0 t2nRjk(tn)

.

The desired parameter vector θ = [dm, tm, v]T is a

function of the initial parameter vector g(α) as given by (1).Then, we perform FIM transformation [7] of I(α) to I(θ):

(I(θ))−1 =(

∂

∂αg(α)

)

I(α)−1(

∂

∂αg(α)

)

(9)

Which leads to the theoretical bounds on estimation vari-ance for tm, dm and v as given in (6).

3.1.3 Bounds for Friend-based Ranging

When there are no anchors, measurements between a givenpair and those with respect to other mobile nodes canbe used together to improve estimation of CP parameters.In this setup, the measurement vector corresponding tothe pair having the ith and jth nodes has two parts: thefirst part involves range measurements of the pair withrespect to each friend node. In the analysis, we ignorerange measurements between friend nodes other than ith

and jth node as the the CP parameters are not relatedwith these measurements and the FIM resulted from thesemeasurements is zero. For the ith and jth mobile nodes, thedistances with respect kth friend are given by xik and xjkrespectively. Then, the distance vector corresponding the kth

friend node at time tn is modeled as

δk(tn) =

[

δik(tn)δjk(tn)

]

=

[

xik(tn)xjk(tn)

]

+ wk(tn)

= δ̄k(tn) + wk(tn)

(10)

Where the additive noise is assumed to be i.i.d. Gaussian(i.e., wk ∼ N (0, σ2I) The second part of the measurementinclude the range measurements δ(tn) between the pair ofinterest (i.e. the ith and jth nodes), and this is given in (3).

We assume that all measurements are i.i.d. Hence theFisher information matrix of CP parameters for friend-basedranging is given as a sum of two matrices correspondingto the two parts of the measurement. The first matrix iscomputed in the same as Section 3.1.2 except that the anchorpositions are replaced by the unknown friend locations. Thesecond matrix is based on pairwise measurements and givenin (5).

I(θ) = I1(θ) + I2(θ) (11)

Fig. 2: Simulation Setup

The resulting FIM leads to bounds on estimation vari-ance of the CP parameters using (1).

3.1.4 Numerical results

Next, we compare the three different approaches: pairwise,anchor-based, and friend-based, in terms of estimation vari-ance bounds on the CP parameters. We use simulation toprovide a random set of node geometries. We first considernodes initially positioned randomly within a circle of radius50 m from the origin. Further, the nodes move at randomconstant velocity with speed in [0,10] m/s. The anchornodes are randomly positioned on a circle of radius 50 m.We run 100 trials and compute the bounds for the CPparameters under different measurement models.

The bounds on standard deviation for estimated CPparameters are shown in Figure 3 considering the samenumber of anchor and friend nodes. For a given initialposition and speed of the nodes, we compute CRLB forvarying angles of incidence. The ranging noise is assumed tobe white Gaussian with standard deviation of 0.1 m. We use4 anchor and 4 friend nodes in the simulation. We note fromFigure 3, the pairwise ranging generally results in higherestimation standard deviation due to relatively few numberof measurements. Particularly, for pairwise measurement,

the standard deviation bound on d̂m becomes large as therelative angle of incidence . On the other hand, friend-basedranging provides the lowest estimation variance of all theCP parameters. Anchor based ranging results in the secondlowest estimation variance.

Compared to anchor based methods which use 2K mea-surements for each pair of nodes, friend-based methods arebased on

(

N2

)

measurements for N mobile and K anchornodes. Friend-based ranging can be shown to improve esti-mation by lowering the theoretical bound on the varianceof tm, dm, and v estimates despite the fact that friendnode positions are unknown. Friend-based methods couldthus improve collision prediction performance in anchorinfrastructure-free systems. Furthermore, it can be shownthat estimation variance increases with ranging noise anddecreases with the number of friends and anchors in friend-based and anchor-based range respectively.

We also compare the estimation bounds for randomlygenerated geometries. In Figure 4, we show the boundon estimation standard deviation for collision predictionparameters as a function of standard deviation of rangingnoise. Every point in the plots is derived from 150 differentgeometries in 100 trials. The results show that for the estima-tion variances of the CP parameters dm, tm and v all increasewith ranging noise, and friend-based approaches provide

5

Fig. 3: Bound on standard deviation of estimation for CPparameters

the lowest estimation bound in all cases. For example, for aranging noise with standard deviation of 0.2m, the boundon dm for friend-based method is only 0.09m while anchor-based and pairwise methods provide bounds of 0.11m and0.25m respectively.

3.2 Collision Prediction Algorithms

We develop three different methods to determine the possi-bility of collision between a pair of mobile nodes using onlyrange measurements. The objective of the algorithms is toestimate CP parameters.

3.2.1 Pairwise collision prediction

In pairwise collision prediction, every pair uses range mea-surements from a single pair. Assuming nodes with lin-ear motion, the squared distance between a pair d2 is aquadratic function of time and can be expressed in termsof CP parameters as

d2(t) = d2m + v2 (tm − t)2

= a0 + a1t+ a2t2

(12)

Fig. 4: Bound on standard deviation of estimation as afunction of ranging noise

We apply quadratic regression on a set of squared pair-wise range measurements to determine CP parameters asfollows:

tm = −a12a2

dm =

√

a0 −a214a2

v =√a2

(13)

3.2.2 Anchor-based Collision Prediction

Anchor-based collision prediction involves two steps. First,we use time-difference-of-arrival (TDOA) measurements tofind the absolute positions of each node. Second, we applylinear regression on estimated position coordinates to de-termine velocity. The CP parameters are estimated from theposition and velocity estimates using 1.

6

We adopt TDoA-based localization as used in pastwork [8]. Consider a tag positioned at xe and M anchornodes with fixed locations X = {x1, x2, · · · , xM}. We de-fine the distances between the tag and the anchor nodesas {D1, D2, · · · , DM} and the range difference measure-ment at anchor node i with respect to anchor node 1 by∆i,1 = Di − D1. Then, the position of the a mobile tagxe can be determined using range and range differencemeasurements [9] by solving the equation given by (14):

xe = (AT A)−1AT (b + cD1) (14)

where A = [x1 − x2, · · · , x1 − xM ], c = [∆2,1, · · · ,∆M,1]T ,D1 = ‖xe − x1‖, and

b =

‖x1‖2 − ‖x2‖2 +∆22,1...

‖x1‖2 − ‖xM‖2 +∆2M,1

. (15)

Assuming rectilinear motion for small fixed durationT , the position coordinates estimated from (14) are usedto determine the average velocity. We apply batch linearregression to position coordinates and the correspondingtime on a window size of T seconds. Then, the computedslopes become the velocities of the nodes.

Once the velocity is estimated, the initial position andvelocity estimated for each pair can be used to estimatecollision prediction parameters using (1).

In this scheme, the system includes a receiving anchornode and synchronization node all with fixed known loca-tions. [9]

3.2.3 Friend based Collision Prediction

In this approach, no anchor nodes or infrastructure exists.Nodes can only measure range with respect to other mobilenodes called friend nodes.

One intuitive approach is to determine relative positionusing classical multidimensional scaling (MDS) which mapspairwise dissimilarities to high dimensional coordinates [5].However, using MDS for collision prediction is difficult formultiple reasons. First, it does not provide relative velocityinfo which is critical in collision prediction. Second, succes-sive position estimates of a node can be translated, rotated,or flipped since there is no frame of reference.

Past work has made some effort to estimate relativevelocity within an MDS framework [10]. However, thisapproach results in less accurate velocity estimates for noisyrange measurements because it relies on the second ordertime difference of distance measurements. Further, the rela-tive velocity estimates are, in general, from a different frameof reference from the MDS position estimates.

In this article, we propose the friend-based autonomouscollision prediction and tracking (FACT) algorithm to esti-mate relative position and velocities in a distributed manner.Consider N mobile nodes with their position at time t givenby Xt = {xt1, xt2, · · · , xtN}. The pairwise distance measure-ments {δtij}Ni,j=1 between the nodes correspond to the actualEuclidean distances {dtij}Ni,j=1, i.e.

dtij = d(xti, x

tj) = ‖xti − xtj‖ =

√

(xti − xtj)T (xti − xtj). (16)

FACT estimates relative positions by minimizing a costfunction given by:

S =N∑

i=1

∑

j 6=i

T∑

t=1

wtij(

δtij − dtij)2

+N∑

i=1

T∑

t=1

ri‖xt−1i + xt+1i − 2xti‖2 (17)

The first term in the cost function ensures that the estimatedpositions {xti}i are at the correct relative distances fromeach other at time t. The second term is introduced topenalize changes in velocity of node i by counting thesquared magnitude of the double difference in position,‖(xti − xt−1i ) − (xt+1i − xti)‖2, as error. The objective is tominimize (17) with respect to {xti}. Assuming i.i.d. distancemeasurements, the cost function can be expressed as a sumof local cost functions:

S =N∑

i=1

Si + c (18)

where c is a constant, and with local cost function definedas:

Si =N∑

j=1j 6=i

T∑

t=1

wtij(

δtij − ‖xti − xtj‖)2

+T∑

t=1

ri‖xt−1i + xt+1i − 2xti‖2.

We note that the local cost function Si depends only onmeasurements at node i and positions of friend nodes. Min-imizing the cost function can be done iteratively by usinga quadratic majorizing function. We find a majorizationfunction Ti(xi, yi) ≤ Si(xi) for all yi [11]:

Ti(xi, yi) =N∑

j=1j 6=i

T∑

t=1

wtij(

(δtij)2 + (dtij)

2)

(19)

+T∑

t=1

ri‖x(t−1)i + x(t+1)i − 2xti‖2

+N∑

j=1j 6=i

T∑

t=1

wtijδtijdtij

(xti − xtj)T (yti − ytj).

Since Ti(xi, yi) is quadratic function of xi, the position canbe determined by finding {xti} to minimize Ti iteratively.The distributed optimization is given below in Algorithm 1.The overall algorithm is summarized by in Figure 5.

7

Fig. 5: Collision detection based on FACT algorithm

Algorithm 1 FACT distributed optimization

1: Inputs: {δtij},{wtij}, ǫ, {ri}, {xti},initial condition X(0)2: Initialize: k = 0, S(0), compute ai from equation (20)3: repeat4: k ← k + 15: for t = 1, . . . , T do6: for i = 1, . . . , N do7: compute b

(k−1)i from (21)

8: xti(k)

= ai(

rixti(k−1)

+ X(k−1)b(k−1)i

)

9: S(k) ← S(k) − S(k−1)i + S(k)i

10: communicate xti(k)

to friend nodes11: communicate S(k) to node (i+ 1) mod N

12: until S(k−1) − S(k) < ǫ

Where

a−1i =N∑

j=1j 6=i

wij + ri (20)

and b(k)i = [b1, b2, · · · bN ]T is a vector given by

bj = wij [1− δtij/dtij(X(k))], for j 6= i

bi =N∑

j=1j 6=i

wijδtij/d

tij(X

(k)). (21)

Successive estimated coordinates in the algorithm arethen used to determine the relative velocity of each nodeby applying linear regression on them with respect to time.

Note that the algorithm starts with initialized positions X(0)

of nodes. We use MDS to initialize these positions at thestart of tracking. For following rounds, we use projectedcoordinates, using the prior time coordinate and velocityestimates, to initialize the algorithm. We set the the firstnode at the origin, i.e xtij = 0 and the weight matrix isgiven as:

wtij =

{

1 if i 6= j0 otherwise

. (22)

Although it may be possible to improve the algorithm byadaptively setting wtij , we leave such approaches to futurework.

4 MULTI-NODE RANGING

Global navigation satellite systems (GNSS) are the mostwidely used positioning systems for outdoor applications.

These systems fail in indoor and urban settings due to highattenuation and multipath fading. In recent years, time-of-flight measurements from impulse radio UWB (IR-UWB)transceivers have demonstrated to have cm-level rangingaccuracy, even in severe multipath environments. IR-UWBsystems use pulse widths on the order of nanoseconds andan RF bandwidth on the order of GHz in order to achieveprecise timestamping, which enables accurate ranging. Pastwork in IR-UWB ranging and positioning is mostly limitedto either single pair ranging or ranging between individualtag and an infrastructure of anchor nodes. In this section,we introduce a system that can measure IR-UWB rangemeasurements between multiple mobile nodes using theminimal number UWB message exchanges, which workswith or without infrastructure nodes.

4.1 Hardware

We implement multi-node ranging using commercial IR-UWB transceivers. Every node is equipped with an ARMCortex-M4 processor connected to an IR-UWB transceiver,a narrowband transceiver, and a shared voltage controlledtemperature compensated crystal oscillator (VCTCXO). Weuse a Decawave DW1000 IR-UWB radio which supportsIEEE 802.15.4a, and provide the timestamp of the arrivalof the first arriving path [12]. For clock synchronization,we adopt radio frequency synchronization (RFS) methodproposed in [13] where one node periodically broadcastsa reference unmodulated signal using a TI CC1200 nar-rowband transceiver and the other nodes compute carrierfrequency offset and adjust their VCTCXO frequency. TheCC1200 transceiver is also used to exchange data betweennodes. Frequency synchronization is performed less fre-quently compared to UWB message exchanges, and thenarrowband transceiver is programmed to communicate theUWB message timestamps.

4.2 Multi-node Ranging Protocol

Typically, two-way ranging requires at least two mes-sage exchanges for every pair of nodes with frequency-synchronized clocks [12]. For N nodes, this would requireN(N − 1) message exchanges to compute ranges for allpairs. In this article, we present a novel multi-node rangingprotocol which requires only N messages per ranging cycle.

Figure 6 illustrates our multi-node ranging protocol.At the start of the protocol, node 1 transmits a referenceunmodulated signal over a narrowband channel for clockfrequency synchronization. Up on receiving the narrowband

8

Fig. 6: Multi node two-way ranging protocol

Fig. 7: UWB packet format

signal, every other node applies RFS [13] to synchronizeits local VCTCXO. Next, starting from node 1, each nodetakes a turn transmitting a UWB packet, until the last nodetransmits its packet, completing the cycle. The UWB packetformat, shown in Figure 7, includes an exchange numberand the transmitter id. Each node stores its most recenttransmit and receive timestamps, and broadcasts them overthe secondary narrowband radio. Frequency synchroniza-tion is performed once each 100 cycles.

To compute two-way range rij between nodes i and j(when i > j) in cycle n is given as:

rij [n] =c

2

{(

tij [n]− tjj [n− 1])

−(

tji [n]− tji [n− 1])}

(23)where tjj is the transmit timestamp at node j, t

ij is the

receive timestamp at node j for a message sent from nodei, and c is the speed of light. Note that tj and tj refer totimestamps recorded on different frequency-synchronizedclocks, and thus may differ by a constant offset. However,(4.2) uses only the differences in timestamps at j and i, andthus any constant offset is removed. Without frequency syn-chronization, (4.2) would be inaccurate because the offsetwould be changing with time.

We can use the same multinode ranging protocol toimplement time-difference-of-arrival (TDOA) localization,which we do in this article for comparison. In this case,we assign some static nodes to be the anchor nodes, whichare set to receive-only, i.e., recording receive time stampsbut never transmitting. In TDOA, full time synchronizationbetween anchor nodes is necessary [12]. We adopt theTDOA synchronization method of [8] which adds a nodewith known location called a synch node which broadcasts aUWB packet at the start of every cycle as shown in Figure 8.Unlike past work, we additionally use achieve frequencysynchronization without wires by configuring the synchnode to transmit an unmodulated narrowband signal afterevery 100 cycles, to which the anchors synchronize theirTCVCXO.

For anchors a and b, and a synch node with coordinatesxa, xb and xs, respectively, the range difference ∆r

iab be-

Fig. 8: Anchor-based one-way ranging protocol to measurerange differences

tween anchor a and anchor b with respect to a given tag i atcycle n is given by:

∆riab[n] = c{

(tia[n]− tsa[n])− (tib[n]− tsb[n]) +‖xa − xs‖ − ‖xb − xs‖} . (24)

The range difference measurements are then used to locatea tag based on TDOA-based localization as given by (14).

5 EXPERIMENTS

To validate FACT and to compare it to other collision pre-diction. methods, we conduct experiments involving floorrobots in a research laboratory.

We show the experimental environment in Figure 9, atotal of six iRobot Create robots are deployed in 7 m by 7 mempty area in a research laboratory. Our hardware is placedon top of the robot, with IR-UWB antenna in the center ofthe iRobot. The area surrounding the robots has significantRF clutter, including desks, desktop computers, solderingstations, and RF and digital measurement equipment. Theempty part of the lab where the experiments are conductedare lined with a set of OptiTrack optical tracking cameras.PVC pipe is used at the edges as a “wall” to prevent therobots from leaving the 7 m square area.

The hardware we use on the iRobots include a UWBnode for performing the ranging measurements, and aRaspberry Pi 3 processor that controls the robot motion.The robots are also tagged with four reflective markers sothat the optical tracking system can track its position andorientation in the space. Our code controls a robot to move,in a random direction, at a constant velocity of 0.5 m/s.The robot continues until it hits an obstruction, either robotor wall, at which point it changes direction randomly andstarts a new constant velocity path.

The OptiTrack motion capture system records groundtruth data. It provides 3D node coordinates with mm-levelaccuracy using a set of 16 infra-red cameras at a 60 Hz rate.The experiment involves two setups:

1) Setup 1: Six mobile nodes are programmed to measuretwo-way UWB ranges between each other at rate of18 ranges per second for use in distributed collisionprediction.

2) Setup 2: The second setup involves five mobile nodes,three anchor nodes, and one synch node each posi-tioned 30 cm above the floor at each corner of the ex-periment area. We measure one way range differencesat each anchor node at a rate of 18 samples per second.

9

Fig. 9: Measurement experiment environment with six mo-bile nodes moving within our 7 m by 7 m area, with bluetape used to define coordinate axis in meters. One PVC pipe“wall” is visible at top left.

Fig. 10: Ranging accuracy

The experiment is run for 18 minutes under each setup.While our proposed method uses a distributed algorithm

for collision prediction, in order to compare the results withother centralized algorithms, we capture ranging data ata receiver node directly connected via USB to a Dell XPStower which performs the offline processing.

6 EXPERIMENTAL RESULTS

We experimentally evaluate the performance of the pro-posed system both in ranging accuracy and in collisionprediction performance. Ground truth coordinates obtainedfrom OptiTrack system are used to generate collision labelsand validate the results.

6.1 Ranging Accuracy

The performance of range-based collision prediction relieson accurate range estimates. First, we show the rangingaccuracy of the system proposed in this paper. The sys-tem provides one-way range difference (24) and two-wayrange (4.2) measurements. In Figure 10, we compare rangeestimates with with actual distances between nodes. Oursystem provides accurate range estimates with standarddeviation of only 0.08 m. For one way range differencemeasurements, the system has a standard deviation 0.17 m.

Figure 11 shows the cumulative distribution function ofabsolute range error for range and range difference mea-surements using our system. We note that higher accuracy is

0.0 0.2 0.4 0.6 0.8 1.0Absolute Error(m)

0.0

0.2

0.4

0.6

0.8

1.0

CD

F

range diff

range

−1.0 −0.5 0.0 0.5 1.0Error(m)

0.000

0.001

0.002

0.003

0.004

Pro

bab

ilit

yD

istr

ibu

tio

n

range diff

range

Fig. 11: Ranging error distribution

obtained with range estimates compared to one-way rangedifferences. The root mean squared error (RMSE) for rangeestimation is 0.13 m whereas the RMSE for range differencesis 0.21 m. The overall ranging accuracy of the system issummarized in Table 1.

TABLE 1: Ranging accuracy for two-way ranges and one-way range differences.

Error Abs. Error

Measurement Bias(m) RMSE(m) Median(m) 90th Perc. (m)2-way range 0.022 0.13 0.085 0.21-way range diff -0.045 0.21 0.093 0.24

6.2 Collision Prediction

To evaluate collision prediction performance of each pro-posed algorithm, we identify collision times using opticaltracking data and and validate them using video capturedduring the experiment. We identify a total of 70 collisionsfrom anchor-free and 60 collisions from anchor-based mea-surements. Assuming a player with radius r and reactiontime τ to detected collision, the system is determines on-coming collisions based on estimated values of the CPparameters (i.e, dm, tm, and v).

d̂m ≶ Tt = 2r + ǫt,

t̂m ≶ Td = τ +

√

4r2 − d̂2mv̂2

+ ǫd,(25)

where ǫd (m) and ǫt (s) represent real-valued constantswhich parameterize a “buffer” in range and time for the

10

(a)

zoom−−−→

(b)

Fig. 12: Receiver operating characteristic for collision prediction on experiment data where the solid lines are Lowess-smoothed representations. FACT achieves the highest probability of detection while TDoA method provides the lowestprobability of detection at a given probability of false alarm.

collision detector, allowing it to be more or less conservative.Then, the probability of false alarm PFA(Tt, Td) is controlledwith the threshold values {Tt, Td}. For our analysis, we usea reaction time τ = 1 s and the diameter of the each robot2r = 0.34 m.

We evaluate the methods proposed in this paper with ex-periment data. Figure 12 shows the receiver operating char-acteristic (ROC) curve in which the probability of detectingoncoming collision is given as the function the probability offalse alarm (PFA). We compare four different methods: ourposition and velocity adaptation of the multi-dimensionalscaling (MDS) method, the pairwise detector, the time differ-ence of arrival (TDOA) positioning-based detector, and theproposed FACT method. We note that distributed methods,including MDS [5], pairwise and FACT methods, generallyprovide better collision prediction performance followed bythe centralized TDOA method.

It is shown that collision prediction based on TDOAprovides the worst ROC curve with lowest probability ofdetection (PD) for any probability of false alarm. It achieveda PD of 90% at a higher PFA of 5% compared to thesame PD achieved by the other methods only with lessthan 2% PFA. We note that the accuracy of TDOA basedposition estimates decreases when agents are positionednear the boundary of the area defined by the convex hullof the anchors. In addition, as shown in Table 1, one-wayrange difference measurements used in TDOA approach hashigher errors compared to two-way range measurementsused in distributed collision prediction algorithms.

In our evaluation, distributed methods including pair-wise, MDS-based and FACT methods achieved higher prob-ability of collision of prediction at any given PFA. FACToutperforms all other methods, attaining a PD of 90% at1.5% PFA. The MDS-based collision prediction depends onfirst finding relative position and relative velocity fromsquared ranges and their second order time derivatives.For noisy measurements, this sometimes leads to impreciseposition and velocity estimates and hence lower PD. Thepairwise regression-based method [14], although achieveshigher probability of collision prediction, its performancedeclines severely for with highly noisy measurements.

FACT leverages measurements from multiple pairs toenhance PD at lowest PFA. FACT has the potential to

achieve even better performance with extended test scenariowith more number of mobile agents. Since the maximumspeed of the robots used in the experiments is limited to0.5 m/s, we also evaluate the methods using simulated dataat a maximum speed of 10 m/s. The simulation involves150 different geometries in 100 trials with 4 anchors and 6sensor nodes positioned in 2D as used in Section 3. Figure13 compares the RMSE of estimates of the CP parameters, i.etm and dm, as a function of ranging noise for the methodsproposed in this paper. The results show that FACT providesthe lowest error for tm whereas TDOA based approachprovides the lowest error for dm. It is shown that the es-timation error for the pairwise regression-based approach isconsiderably higher than all other methods. FACT maintainslower error for the CP parameters that is comparable tothe CRB, further proving its potential in accurate collisionprediction with noisy measurements while avoiding theneed for infrastructure.

We believe that FACT can be further enhanced with moresensor data including inertial measurements to extend linearmotion assumption. However, this is beyond the scope ofthis article.

7 RELATED WORK

Several sensors and algorithms have been proposed to pre-dict collisions between autonomous vehicles, for example,swarms of drones or autonomous cars. Various sensors suchas lidar, radar, and visual sensors, are used to obtain rawinformation about events in the surrounding environment,which is followed by prediction of impending collisions.

Collision Detection Sensors: include visual and acousticsensors [15], Doppler radar [16], tactile sensors [17], andUltra Wideband (UWB) radar [18]. While some systemsallow periodic exchange of position and velocity informa-tion between agents [19], [20], [21], other collision predic-tion systems are non-cooperative. Active sensors includingradar and lidar systems generally rely on continuous orpulsed electromagnetic wave transmissions to measure theranging and Doppler information. Modern cars include aradar-based collision warning system called adaptive cruisecontrol (ACC) [22]. Passive optical sensors or cameras areusually affected by changes in illumination and complex

11

(a) RMSE for tm (b) RMSE for dm

Fig. 13: RMSE for CP parameters as a function of ranging noise based on simulation 150 different geometry and 100iterations.

outdoor settings [23], [24]. With the exception of depth cam-eras, vision-based systems do not provide direct range in-formation. Compared to optical/vision based collision pre-diction sensors, radar systems such as frequency-modulatedcontinuous wave (FMCW) radars perform well under vari-ous light and weather conditions.

Collision Detection Algorithms: use sensor data to calcu-late trajectory and determine distances between cooperatingmoving agents. Vision-based systems typically employ avideo feed from one or more cameras to track the trajectoryof a moving agent using computer vision techniques [24].However, tracking many objects using video analysis can becomputationally inefficient [25]. In robotics, moving agentscan be tasked with path planning in robot swarms, wherecollisions can be avoided by employing motion planning inwhich each agent determines a preferred path that avoidscollisions [26]. For autonomous agents that are not part ofthe same “swarm”, path planning may not be sufficientand collisions can be predicted by estimating one’s trajec-tory from relative kinematics such as relative position andrelative velocity with respect to other nearby autonomousagents.

Several positioning methods have been proposed in theliterature. Depending on the type of sensors used, relativeposition and relative velocity are determined. In radar sys-tems, range and Doppler information is used to estimate rel-ative position and relative velocity between moving agents.Compared to most radar based measurements, UWB basedranging has proven to provide cost effective yet accurateranging at higher update rate [27].

Range based localization has been extensively studied.Common anchor-based localization methods such as thosebased on Time Difference of Arrival (TDoA) and Time ofArrival (TOA) measurements require careful infrastructuresetup. When there are no nodes with known locations,anchor-free solutions like multi-dimensional scaling (MDS)solutions are used to map pairwise distance measurementsinto a geometry of nodes which generated them [28], [29],[30]. However, tracking nodes with MDS to predict col-lision is difficult as MDS solutions are only up to rota-tion, reflection and translation. In [5], joint relative positionand relative velocity estimation method is proposed inwhich relative velocity is determined from second-order

time derivative of simulated range measurements and itsaccuracy is highly sensitive to ranging noise. Moreover, thismethod is centralized and unscalable which limits its usefor practical applications. In addition, collision predictionusing UWB ranging has not completely addressed in priorwork. This article provides a distributed solution for estima-tion of relative position and relative velocity from rangingmeasurements and applies them in collision prediction.

8 CONCLUSION

This article presents a novel approach to predict collisionbetween moving agents, such as autonomous vehicles, thatcan measure and share distance measurements with eachother and use them to predict that the two vehicles will col-lide. We develop a ranging system for measuring the rangebetween every pair of N nodes based on UWB transceivermeasurements which requires only N transmissions. Wepropose and evaluate a novel distributed collision predic-tion method that does not require infrastructure. We test theperformance of the proposed method in theoretical analysisand experiments involving autonomous floor robots. Wecompare performance to that of three alternate collisiondetection methods. We show that our FACT algorithm out-performs infrastructure based methods and two alternateinfrastructure-free methods. The FACT method allows col-lision detection between vehicles with low-cost hardware,without infrastructure, while simultaneously providing re-liable collision prediction among mobile agents.

ACKNOWLEDGMENT

This material is based upon work supported by the USNational Science Foundation under Grant No. #1622741.

REFERENCES

[1] D. H. Daneshvar, C. J. Nowinski, A. C. McKee, and R. C. Cantu,“The epidemiology of sport-related concussion,” Clinics in sportsmedicine, vol. 30, no. 1, pp. 1–17, 2011.

[2] Remote Identification of Unmanned AircraftSystems. National Archives and RecordsAdministration, Dec 2019. [Online]. Available:https://www.federalregister.gov/documents/2019/12/31/2019-28100/remote-identification-of-unmanned-aircraft-systems

[3] F. A. Administration, “Remote identification of unmanned aircraftsystems,” 2019.

https://www.federalregister.gov/documents/2019/12/31/2019-28100/remote-identification-of-unmanned-aircraft-systems

12

[4] A. W. Weiser, Y. Orchan, R. Nathan, M. Charter, A. J. Weiss, andS. Toledo, “Characterizing the accuracy of a self-synchronizedreverse-gps wildlife localization system,” in 2016 15th ACM/IEEEInternational Conference on Information Processing in Sensor Networks(IPSN). IEEE, 2016, pp. 1–12.

[5] R. T. Rajan, G. Leus, and A.-J. van der Veen, “Relative kinematicsof an anchorless network,” Signal Processing, vol. 157, pp. 266–279,2019.

[6] B. Kempke, P. Pannuto, and P. Dutta, “Polypoint: Guiding indoorquadrotors with ultra-wideband localization,” in Proceedings of the2nd International Workshop on Hot Topics in Wireless, 2015, pp. 16–20.

[7] S. M. Kay, Fundamentals of Statistical Signal Processing. Prentice-Hall, 1993.

[8] J. Tiemann, F. Eckermann, and C. Wietfeld, “Atlas-an open-sourcetdoa-based ultra-wideband localization system,” in 2016 Interna-tional Conference on Indoor Positioning and Indoor Navigation (IPIN),2016, pp. 1–6.

[9] A. Bensky, Wireless positioning technologies and applications. ArtechHouse, 2016.

[10] R. T. Rajan, G. Leus, and A.-J. van der Veen, “Joint relativeposition and velocity estimation for an anchorless network ofmobile nodes,” Signal Processing, vol. 115, pp. 66–78, 2015.

[11] J. A. Costa, N. Patwari, and A. O. Hero III, “Distributed weighted-multidimensional scaling for node localization in sensor net-works,” ACM Transactions on Sensor Networks (TOSN), vol. 2, no. 1,pp. 39–64, 2006.

[12] M. Yavari and B. G. Nickerson, “Ultra wideband wireless posi-tioning systems,” Dept. Faculty Comput. Sci., Univ. New Brunswick,Fredericton, NB, Canada, Tech. Rep. TR14-230, 2014.

[13] A. Luong, P. Hillyard, A. Abrar, C. Che, A. Rowe, T. Schmid, andN. Patwari, “A stitch in time and frequency synchronization savesbandwidth,” in 2018 17th ACM/IEEE International Conference onInformation Processing in Sensor Networks (IPSN), 2018, pp. 96–107.

[14] A. S. Abrar, N. Patwari, and J. Decavel-Bueff, “Demo abstract: Col-lision prediction from pairwise ranging,” in 2020 19th ACM/IEEEInternational Conference on Information Processing in Sensor Networks(IPSN), 2020, pp. 359–360.

[15] R. Chellappa, G. Qian, and Q. Zheng, “Vehicle detection and track-ing using acoustic and video sensors,” in 2004 IEEE InternationalConference on Acoustics, Speech, and Signal Processing, vol. 3. IEEE,2004, pp. iii–793.

[16] A. Viquerat, L. Blackhall, A. Reid, S. Sukkarieh, and G. Brooker,“Reactive collision avoidance for unmanned aerial vehicles usingdoppler radar,” in Field and Service Robotics. Springer, 2008, pp.245–254.

[17] Z. Ji, H. Zhu, H. Liu, T. Chen, and L. Sun, “A flexible capacitive

tactile sensor for robot skin,” in 2016 International Conference onAdvanced Robotics and Mechatronics (ICARM), 2016, pp. 207–212.

[18] I. Gresham, A. Jenkins, R. Egri, C. Eswarappa, N. Kinayman,N. Jain, R. Anderson, F. Kolak, R. Wohlert, S. P. Bawell et al., “Ultra-wideband radar sensors for short-range vehicular applications,”IEEE transactions on microwave theory and techniques, vol. 52, no. 9,pp. 2105–2122, 2004.

[19] E. Boivin, A. Desbiens, and E. Gagnon, “Uav collision avoidanceusing cooperative predictive control,” in 2008 16th MediterraneanConference on Control and Automation. IEEE, 2008, pp. 682–688.

[20] R. Parker and S. Valaee, “Cooperative vehicle position estimation,”in 2007 IEEE International Conference on Communications. IEEE,2007, pp. 5837–5842.

[21] X. Xiang, W. Qin, and B. Xiang, “Research on a dsrc-based rear-end collision warning model,” IEEE Transactions on IntelligentTransportation Systems, vol. 15, no. 3, pp. 1054–1065, 2014.

[22] G. Marsden, M. McDonald, and M. Brackstone, “Towards anunderstanding of adaptive cruise control,” Transportation ResearchPart C: Emerging Technologies, vol. 9, no. 1, pp. 33–51, 2001.

[23] M. B. Van Leeuwen and F. C. Groen, “Vehicle detection with amobile camera: spotting midrange, distant, and passing cars,”IEEE robotics & automation magazine, vol. 12, no. 1, pp. 37–43, 2005.

[24] Z. Sun, G. Bebis, and R. Miller, “On-road vehicle detection usingoptical sensors: A review,” in Proceedings. The 7th InternationalIEEE Conference on Intelligent Transportation Systems (IEEE Cat. No.04TH8749). IEEE, 2004, pp. 585–590.

[25] M. C. Lin, D. Manocha, J. Cohen, and S. Gottschalk, “Collisiondetection: Algorithms and applications,” Algorithms for roboticmotion and manipulation, pp. 129–142, 1997.

[26] D. Hennes, D. Claes, W. Meeussen, and K. Tuyls, “Multi-robotcollision avoidance with localization uncertainty.” in AAMAS,2012, pp. 147–154.

[27] F. Zafari, I. Papapanagiotou, and K. Christidis, “Microlocationfor internet-of-things-equipped smart buildings,” IEEE Internet ofThings Journal, vol. 3, no. 1, pp. 96–112, 2015.

[28] Y. Zhao, Z. Zhang, T. Feng, W.-C. Wong, and H. K. Garg,“Graphips: Calibration-free and map-free indoor positioning us-ing smartphone crowdsourced data,” IEEE Internet of Things Jour-nal, 2020.

[29] Y. Wang, T. Sun, G. Rao, and D. Li, “Formation tracking in sparseairborne networks,” IEEE Journal on Selected Areas in Communica-tions, vol. 36, no. 9, pp. 2000–2014, 2018.

[30] B. Beck and R. Baxley, “Anchor free node tracking using ranges,odometry, and multidimensional scaling,” in 2014 IEEE Interna-tional Conference on Acoustics, Speech and Signal Processing (ICASSP),2014, pp. 2209–2213.

1 Introduction2 Problem Statement3 Estimation and Detection3.1 Cramér-Rao Lower Bound Analysis3.1.1 Bounds for Pairwise Ranging3.1.2 Bounds for Anchor-Based Ranging3.1.3 Bounds for Friend-based Ranging3.1.4 Numerical results

3.2 Collision Prediction Algorithms3.2.1 Pairwise collision prediction3.2.2 Anchor-based Collision Prediction3.2.3 Friend based Collision Prediction

4 Multi-node Ranging4.1 Hardware4.2 Multi-node Ranging Protocol

5 Experiments6 Experimental Results6.1 Ranging Accuracy6.2 Collision Prediction

7 Related Work8 ConclusionReferences