Calibration and Uncertainty Characterization for Ultra-Wideband Two-Way-Ranging Measurements Mohammed Ayman Shalaby Charles Champagne Cossette James Richard Forbes Jerome Le Ny

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Calibration and Uncertainty Characterization for Ultra-Wideband

Two-Way-Ranging Measurements

Mohammed Ayman Shalaby, Charles Champagne Cossette, James Richard Forbes, Jerome Le Ny

Abstract— Ultra-Wideband (UWB) systems are becoming

increasingly popular for indoor localization, where range mea-

surements are obtained by measuring the time-of-ﬂight of

radio signals. However, the range measurements typically suffer

from a systematic error or bias that must be corrected for

high-accuracy localization. In this paper, a ranging protocol

is proposed alongside a robust and scalable antenna-delay

calibration procedure to accurately and efﬁciently calibrate

antenna delays for many UWB tags. Additionally, the bias and

uncertainty of the measurements are modelled as a function

of the received-signal power. The full calibration procedure

is presented using experimental training data of 3 aerial

robots ﬁtted with 2 UWB tags each, and then evaluated

on 2 test experiments. A localization problem is then for-

mulated on the experimental test data, and the calibrated

measurements and their modelled uncertainty are fed into an

extended Kalman ﬁlter (EKF). The proposed calibration is

shown to yield an average of 46% improvement in localiza-

tion accuracy. Lastly, the paper is accompanied by an open-

source UWB-calibration Python library, which can be found at

https://github.com/decargroup/uwb calibration.

I. INTRODUCTION

Robotic localization and mapping applications typically

require a means of acquiring position information relative

to a reference point with known location. Global Naviga-

tion Satellite System (GNSS) provides accurate and pre-

cise positioning information outdoors; however, localization

performance degrades signiﬁcantly in obstructed or indoor

environments [1, 2]. An attractive indoor localization option

that has been increasingly gaining traction is the use of

ultra-wideband (UWB) radio signals between transceivers,

or tags, as a means of ranging. UWB transceivers, such as the

DWM1000 module provided by Decawave [3], are typically

inexpensive, consume little power, and provide a means for

data transfer between robots, thus deeming them particularly

useful for a variety of robotic applications [4–6].

UWB-based ranging typically relies on measuring the

time-of-ﬂight (ToF) of radio signals from one tag to another.

This requires estimating the offset between the clock on each

tag. Furthermore, the clocks often run at different rates due

to physical imperfections in the individual clock’s crystal

oscillator, causing the offset to be time-varying. The rate

This work was supported by the NSERC Alliance Grant program, by the

CFI JELF program, and by the FRQNT.

M. A. Shalaby, C. C. Cossette, and J. R. Forbes are with the department

of Mechanical Engineering, McGill University, Montreal, QC H3A

0C3, Canada. {mohammed.shalaby@mail.mcgill.ca,

charles.cossette@mail.mcgill.ca,

james.richard.forbes@mcgill.ca}.J. Le Ny is with the

department of Electrical Engineering, Polytechnique Montreal, Montreal,

QC H3T 1J4, Canada. {jerome.le-ny@polymtl.ca}.

Tag iTag i

Tag j

∆t41

∆t32

(a) SS-TWR.

Tag iTag i

Tag j

∆t41

∆t32

∆t64

∆t53

(b) Proposed DS-TWR.

Fig. 1: Timeline schematics for two tags iand jrepresenting the different

TWR ranging protocols, where t`represents the `th timestamp for a TWR

instance and ∆t`k ,t`−tk.

of change of the clock offset is referred to as the clock

skew. In order to negate the effect of the clock offset during

ranging, different ranging protocols have been proposed,

with the choice being dependent on the speciﬁc application

and availability of tags [7], [8, Section 7.1.4]. A commonly

used protocol is two-way ranging (TWR), which relies on

averaging out the measured ToF between two signals to

negate the clock offset. This form of TWR is referred to

as single-sided TWR (SS-TWR), and is shown in Figure 1a.

Nonetheless, even after correcting for clock offsets, UWB

range measurements typically suffer from a systematic error

or bias. A signiﬁcant contributor to this error is the skew

between the clocks of the two ranging tags, as the different

tags measure the passage of time in different units [9, 10].

This additional bias can be corrected by estimating the clock

skew between the tags and embedding a skew-dependent

correction factor when computing the range measurement,

as proposed in [9]. However, this necessitates estimating

the clock skew between all tags involved in ranging. Al-

ternatively, [10] proposes a form of computing the range

measurement utilizing double-sided TWR (DS-TWR), which

is shown to mitigate clock-skew-dependent bias.

Another source of ranging bias stems from relative-pose-

dependent antenna radiation pattern [11], where pose refers

to both position and attitude. The varying signal strength

can cause timestamping errors, and this effect is typically

addressed using data-driven models. In [12], a simple ex-

periment with pre-localized ﬁxed tags or anchors is used to

determine a relation between bias and the distance between

ranging tags, while in [13], models are trained using the

distance between the tags and 7 features extracted from the

channel impulse response (CIR). In [14] and [15], a robot

arXiv:2210.05888v3 [cs.RO] 16 Feb 2023

is ﬂown around in a room with UWB anchors to learn a

model of the range bias as a function of the robot’s pose.

The main drawback of these methods is that the learned

model is dependent on the relative poses of the ranging tags,

which are typically unknown in real-time without the bias-

corrected measurements in the ﬁrst place. Additionally, the

learnt models are trained and tested on the same anchor

formations and are therefore not necessarily generalizable;

calibration must occur for every new anchor formation. In

[9], the former issue is addressed by ﬁnding a relation

between the bias and the received ﬁrst-path power (FPP)

in line-of-sight (LOS) conditions with 2D motion.

Delays in communication between the embedded mi-

crochip and the UWB antenna are another source of bias

[16]. This antenna delay is roughly the same for different

UWB tags with the same physical design and is at least a

few hundreds of nanoseconds [16], but can vary tenths of a

nanosecond or more from tag to tag due to manufacturing

inaccuracies. Given that a one-nanosecond timestamping

error corresponds to 30 cm in ranging error, the need to

perform antenna-delay calibration for every tag is critical. In

[16], a basic TWR-based calibration procedure is suggested

for calibrating antenna delays. However, the lack of motion

introduces a risk of learning the aforementioned relative-

pose-dependent bias as antenna delays. In [9], experiments

involving a pair of tags at a time ranging with each other is

used to ﬁt what is referred to as a “pair-dependent constant”.

Therefore, the calibration procedure involves calibrating the

relative delay between one pair at a time, which does not

scale well to systems with many UWB tags.

This paper addresses the problem of calibrating UWB tags,

and the main contributions are as follows.

•An alternative DS-TWR protocol is proposed and is

shown to mitigate the clock-skew-induced bias.

•A scalable antenna-delay calibration algorithm is pre-

sented that is robust to outliers and pose-dependent bias.

•The bias-versus-FPP ﬁt presented in [9] is extended to

also address the uncertainty of the measurements as a

function of FPP, and DS-TWR is utilized to overcome

the need to estimate the clock skew.

•The proposed antenna-delay and bias-FPP calibration

are evaluated on an aerial experiment with no anchors,

where all the tags are ﬁtted to moving robots.

•The code for the full calibration procedure

is attached to this paper as an open-access

online repository, which can be found at

https://github.com/decargroup/uwb calibration.

The remainder of this paper is organized as follows. The

proposed DS-TWR is discussed in Section II, alongside a

theoretical analysis of the clock-skew-dependent bias. In

Section III, a robust antenna-delay calibration algorithm is

presented, followed by the bias and uncertainty calibration

as a function of FPP in Section IV. The calibration methods

presented in Sections III and IV are introduced on the same

experimental training data, and are then evaluated on 2

testing experiments in Section V.

II. THE RANGING PROTOCOL

UWB ranging relies on the time-of-ﬂight (ToF) of signals

between two tags in order to compute range measurements.

The simplest way to do this is using SS-TWR, shown in

Figure 1a, where the ToF measurement can be computed as

tf=1

2(∆t41 −∆t32).(1)

However, different UWB tags have differrent clocks that

are typically running at different rates, and this clock skew

results in additional bias in the computed ToF measurement.

In [10], an alternative DS-TWR-based ranging protocol is

proposed to mitigate clock-skew-dependent bias. In this

paper, the DS-TWR protocol shown in Figure 1b is proposed,

which differs from [10] by having the responding tag instead

of the initiating tag transmit the third signal. The ToF

measurement can then be computed as

tf=1

2∆t41 −∆t64

∆t53

∆t32.(2)

This protocol is motivated by the intuitive understanding that

the additional correcting factor in (2) transforms ∆t32 from

time units of the receiver tag’s clock to time units of the ini-

tiator tag’s clock. Additionally, the proposed ranging protocol

allows the initiating tag to process the range measurement

by computing (2), without requiring additional signals for

the responding tag to send ∆t32 and ∆t53.

A. Analytical Bias Model

To demonstrate clock-skew-dependent bias, consider in

SS-TWR the clock-skew-corrupted ToF measurement,

tss

f=1

2((1 + γi)(∆t41 +η41)−(1 + γj)(∆t32 +η32)) ,

(3)

where γiis the skew of Tag i’s clock relative to real

time, ηk` =ηk−η`, and ηk, η`∼ N(0, R)are mutually-

independent timestamping white noise associated with times-

tamps tkand t`, respectively. The ToF error is thus

ess ,˜

tss

f−tf

2(γi∆t41 + (1 + γi)η41 −γj∆t32 −(1 + γj)η32),

(4)

and the expected value of ess is

E[ess] = 1

2(γi∆t41 −γj∆t32)

(1)

2(γi(2tf+ ∆t32)−γj∆t32)

=γitf+1

2(γi−γj) ∆t32.(5)

The ﬁrst component of (5) is negligible as skew is in the

order of parts-per-million and ToF in nanoseconds. However,

∆t32 is typically in hundreds of microseconds, meaning that

clock-skew-dependent bias is not negligible.

Negating the second component of (5) is the motivation

behind the proposed ranging protocol. Rewriting (2) as

tf=∆t41∆t53 −∆t64∆t32

2∆t53

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CalibrationandUncertaintyCharacterizationforUltra-WidebandTwo-Way-RangingMeasurementsMohammedAymanShalaby,CharlesChampagneCossette,JamesRichardForbes,JeromeLeNyAbstractUltra-Wideband(UWB)systemsarebecomingincreasinglypopularforindoorlocalization,whererangemea-surementsareobtainedbymeasuringthetime-...

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Calibration and Uncertainty Characterization for Ultra-Wideband Two-Way-Ranging Measurements Mohammed Ayman Shalaby Charles Champagne Cossette James Richard Forbes Jerome Le Ny

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