Extrinsic calibration for highly accurate trajectories reconstruction

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Maxime Vaidis1, William Dubois1, Alexandre Gu´

enette1, Johann Laconte1,

Vladim´

ır Kubelka2, Franc¸ois Pomerleau1

Abstract— In the context of robotics, accurate ground-truth

positioning is the cornerstone for the development of mapping

and localization algorithms. In outdoor environments and over

long distances, total stations provide accurate and precise mea-

surements, that are unaffected by the usual factors that deterio-

rate the accuracy of Global Navigation Satellite System (GNSS).

While a single robotic total station can track the position of a

target in three Degrees Of Freedom (DOF), three robotic total

stations and three targets are necessary to yield the full six DOF

pose reference. Since it is crucial to express the position of tar-

gets in a common coordinate frame, we present a novel extrinsic

calibration method of multiple robotic total stations with ﬁeld

deployment in mind. The proposed method does not require the

manual collection of ground control points during the system

setup, nor does it require tedious synchronous measurement

on each robotic total station. Based on extensive experimental

work, we compare our approach to the classical extrinsic cal-

ibration methods used in geomatics for surveying and demon-

strate that our approach brings substantial time savings during

the deployment. Tested on more than 30 km of trajectories, our

new method increases the precision of the extrinsic calibration

by 25 % compared to the best state-of-the-art method, which

is the one taking manually static ground control points.

I. INTRODUCTION

In mobile robotics, obtaining reference trajectories is vi-

tal for the development and evaluation of mapping and

control algorithms [1], while being critical to cornerstone

datasets [2]. In outdoor environments, total stations are the

preferred choice to obtain high-accuracy measurements in

the order of millimeters [3]. A total station is a precision

measurement instrument equipped with optics that allow it

to be precisely aimed at a given target. Two angles (i.e., el-

evation and azimuth) and the range between the total station

and the target are measured. This information is sufﬁcient to

express the position of the target in the coordinate system

of the total station. They are not affected by the factors that

may otherwise inhibit the usage of the Real Time Kinematics

(RTK) Global Navigation Satellite System (GNSS) localiza-

tion, such as dense foliage or urban and natural canyon envi-

ronments [4]. The only major requirement is the line-of-sight

between a total station and the tracked target. In the case of

a static robotic platform with several targets attached to it, it

is possible to obtain its six Degrees Of Freedom (DOF) pose

(i.e., position and orientation) as shown by Pomerleau et al.

[5]. This procedure requires a series of manual measurements

that capture each of these targets. If the robotic platform

1Northern Robotics Laboratory, Universit´

e Laval, Qu´

ebec City,

Canada, tmaxime.vaidis, francois.pomerleauu

@norlab.ulaval.ca

2Mobile Robotics and Olfaction lab of the AASS research center at

Orebro University, Sweden.

Fig. 1: Setup to record reference trajectories using three

robotic total stations to track three active prisms located on a

HD2 robotic platform to reconstruct its six DOFs in a 230 m

tunnel deployment at Universit´

e Laval.

moves during the measurement, a combination of a prismatic

retro-reﬂector (i.e., simply prism in the remaining of this

article) and a Robotic Total Station (RTS) is required [6].

The term robotic in RTS denotes the ability to automatically

track a prism in motion. Since a single RTS can continu-

ously track only one prism at a time, it is necessary to use

at least three RTSs and three prisms to achieve the six DOF

pose tracking of a dynamic robotic platform. In recent years,

manufacturers, such as Trimble, provide active prisms with

infrared signature insuring that a RTS will only track a given

prism without being disturbed by other prisms in proximity.

This new feature allowed us to investigate novel solutions to

reconstruct trajectories of moving vehicles.

As shown in Figure 1, each of the three RTSs is tracking

its assigned prism that is mounted on the robotic platform

and outputs the position of the associated prism expressed

in its coordinate system. To obtain the pose of a robot, there

are two necessary conditions: 1) All data must be expressed

in a common coordinate frame based on an extrinsic calibra-

tion, also called resection or free-stationing in the surveying

ﬁeld. In the literature, there are several extrinsic calibration

methods for multiple RTSs based on static Ground Control

Points (GCPs) [7], [8]. Yet, they require a signiﬁcant effort in

preparation and special equipment such as geodesic pillars to

achieve the desired accuracy. 2) The RTS measurements need

to be synchronized. Contrary to position measurements from

GNSS receivers, the measurements are not executed syn-

arXiv:2210.01048v1 [cs.RO] 3 Oct 2022

chronously between multiple RTSs by default since this func-

tionality is usually not required in surveying [9]. Therefore,

temporal interpolation is necessary to exploit the data from

these different RTSs to obtain the robotic platform’s pose.

In this paper, we propose an extrinsic calibration method

for three RTSs while a vehicle is in motion (i.e., without

having to set up static reference points in the environment as

done in our previous work [10]). This method does not re-

quire manual registration of additional reference points, and

the same conﬁguration is used both for the calibration and

the subsequent ground truth positioning, drastically reducing

the setup time. The method exploits the known distances

between the prisms attached to the robotic platform and only

requires the robot to be driven along a random trajectory

throughout the experimental area. We evaluate the proposed

approach against existing extrinsic calibration methods for

RTS using a dataset consisting of more than 30 km of indoor

and outdoor trajectories. These datasets and the code are

available online to the community.1

II. RELATED WORK

First, we describe extrinsic calibration methods found in

the surveying literature for multiple-RTS setups. Then, we

present works related to using multiple RTS together. Finally,

we list various robotic applications of RTS used for acquiring

reference trajectories of vehicles, and we put our work in

this context.

For all applications using multiple RTS, measurements

need to be expressed in a common coordinate frame. The

process of ﬁnding appropriate transformations is called ex-

trinsic calibration. The most common extrinsic calibration

methods use multiple static GCPs. The minimum number of

required static GCPs is two, and this method is called two-

point resection in surveying [7], [11]. This method requires

the knowledge of the relative position of two GCPs with

millimeter accuracy. This requirement can be very difﬁcult

to comply with during ﬁeld deployment. Therefore, in most

applications, three or more static GCPs with unknown global

coordinates are used [8]. In outdoor environments with good

GNSS coverage, the GNSS can be used to obtain the GCP

coordinates [12]. In that case, the extrinsic calibration ex-

presses the pose of RTS in the global frame of the GNSS.

Although all of these methods with static GCPs are accurate

in the order of a few millimeters, they can take hours to be

carried out to achieve the desired accuracy [13]. To address

this issue, a new extrinsic calibration method, which dynam-

ically captured GCPs, was implemented by Zhang et al. [14].

The GCPs were generated by two RTSs tracking one prism

carried by an Unmanned Aerial Vehicle (UAV). Although

such measurements are less accurate than the static ones, the

large number of GCPs obtained allows to compensate for the

inaccuracy and provides a ﬁve-millimeter-accurate result in

two minutes. In this paper, a new dynamic extrinsic calibra-

tion that uses multiple prisms is presented, which does not

need GCPs.

1https://github.com/norlab-ulaval/RTS_Extrinsic_

Calibration

To properly analyze the results obtained by RTSs, it is

necessary to take into account the different types of mea-

surement noise. The ﬁrst type of noise originates in extrinsic

calibration. The works of Horemuˇ

zet al. [15] and Amin

Alizadeh-Khameneh et al. [16] searched for the optimal

number of GCPs to minimize the uncertainty of the extrinsic

calibration. The method we propose removes the requirement

of GCPs altogether while still providing a precise extrin-

sic calibration. Another source of noise is the measurement

equipment itself. The contributing factors are the alignment

of the prism with respect to the line of sight of the instru-

ment and the type of electronic distance measurement unit

inside the instrument. Errors of two to four millimeters can

occur [17]. Weather conditions also have a signiﬁcant impact

on range accuracy [18]. The differences in temperature and

pressure need to be compensated as well [19]. In multiple-

RTS conﬁgurations, the temporal synchronization of the in-

strument clocks signiﬁcantly affects the accuracy [20]. An er-

ror of a one millisecond in the synchronization can lead to in-

accuracy of one millimeter in the resulting measured position

for a speed of 1 m s−1. The usage of the GNSS’s clock can

mitigate this problem [9]. Moreover, RTS conﬁgurations that

require communication over large distances can beneﬁt from

the long-range radio protocol with time synchronization [10].

Finally, the last type of noise to be considered is interlinked

with the application of tracking mobile robots. The motion

of the robotic platform can lead to outlier measurements.

Kalman ﬁltering can be applied to the raw data to increase

the precision as demonstrated by Zhang et al. [14]. Some

applications may require interpolation of the RTS measure-

ments, which adds another source of errors. A simple linear

interpolation can be used to process the data and synchronize

them [10]. Although not used in surveying for interpolation,

Gaussian Processes (GPs) are widely used in robotics to ob-

tain continuous trajectories of robotic platforms and can be

applied to prism trajectories [21]. In this paper, a new pre-

processing pipeline applied to the raw RTS data is introduced

to increase the precision of the proposed extrinsic calibration.

In mobile robotics, a wide variety of position-referencing

systems are based on RTSs, but their use remains overall

atypical. An application of these systems is to register many

robots in the same global frame before beginning swarm ex-

ploration of extreme environments [22]. The design of these

position-referencing systems also depends on the number of

DOF required, and also on the payload capacity of the plat-

form carrying prisms. Most of the referencing systems use

only one RTS to track the position of the robotic platform,

being a skid steered robot [23], a tracked robot [24], a teth-

ered wheeled robot [25], a planetary rover [26], an unmanned

surface vessel [27], a UAV [28] or a walking robot [29].

Adding a second RTS leads to a reduction in the uncertainty

of the position as shown by Gabriel Kerekes et al. [30]. In

the work of Reitbauer et al. [31], a compost turner with two

different prisms attached to it was tracked by two RTSs. This

conﬁguration provided ground truth measurement on four

DOF, namely the position and the yaw angle. To obtain the

full position and orientation reference of a robotic platform,

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ExtrinsiccalibrationforhighlyaccuratetrajectoriesreconstructionMaximeVaidis1,WilliamDubois1,AlexandreGu´enette1,JohannLaconte1,Vladim´rKubelka2,Franc¸oisPomerleau1AbstractInthecontextofrobotics,accurateground-truthpositioningisthecornerstoneforthedevelopmentofmappingandlocalizationalgorithms.Inout...

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Extrinsic calibration for highly accurate trajectories reconstruction

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