1
Tac ile E godic Co e age on Cu ed Su aces
Cem Bilaloglu∗, Tobias Löw∗, and Syl ain Calinon
Abs ac —In his a icle, we p esen a eedback con ol me hod
o ac ile co e age asks such as cleaning o su ace inspec ion.
Al hough hese asks a e challenging o plan due o he com-
plexi y o con inuous physical in e ac ions, he co e age a ge
and p og ess can be e ec i ely measu ed using a came a and
encoded in a poin cloud. We p opose an e godic co e age
me hod ha ope a es di ec ly on poin clouds, guiding he obo
o spend mo e ime on egions equi ing mo e co e age. Fo
obo con ol and con ac beha io , we use geome ic algeb a o
o mula e a ask-space impedance con olle ha acks a line
while simul aneously exe ing a desi ed o ce along ha line. We
e alua e he pe o mance o ou me hod in kinema ic simula ions
and demons a e i s applicabili y in eal-wo ld expe imen s on
ki chenwa e. Ou sou ce codes, expe imen ideos, and da a a e
a ailable a h ps://si es.google.com/ iew/ ac ile-e godic-con ol/.
Index Te ms—Tac ile Robo ics, E godic Co e age, Geome ic
Algeb a
I. INTRODUCTION
The long- e m ision o obo ics is o assis humans wi h
daily asks. The success o obo acuum cleane s and lawn-
mowe s as consume p oduc s highligh s he po en ial o
obo ic assis ance o common household cho es [1]. These
asks in ol e co e ing a egion in a epe i i e and exhaus-
i e manne . Cu en ly, hese obo s a e limi ed o ela i ely
la ge, plana su aces, and e en na iga ing slopes emains
challenging [2], [3]. O he daily asks, such as washing dishes
o g oce y i ems, p esen e en g ea e challenges due o he
complex physical in e ac ions wi h in ica e, cu ed su aces.
Simila ly, nume ous co e age asks on cu ed su aces a ise
in indus ial and medical applica ions. In indus ial se ings,
such asks include su ace ope a ions ha emo e ma e ial,
such as sanding [4], polishing [5], [6] o debu ing [7] as
well as su ace inspec ion asks le e aging con ac [8]. In
medical se ings, simila applica ions ange om mechanical
palpa ion [9], [10] and ul asound imaging [11], [12] o
massage [13], [14] and bed ba hing [15], [16]. Las bu no
leas , da ase s combining he ac ile p ope ies o objec s wi h
hei shape and isual appea ance emain sca ce and expensi e
o collec , as hey ely on eleope a ion [17]. Thus, ac ile
co e age is c i ical o au oma ing he collec ion o ac ile
da ase s ha complemen isual ones. The p oblem de ini ions
o his di e se ange o se ings and applica ions can be
dis illed in o wo key equi emen s: (i) ac ile in e ac ions wi h
This wo k was suppo ed by he S a e Sec e a ia o Educa ion, Resea ch
and Inno a ion in Swi ze land o pa icipa ion in he Eu opean Commis-
sion’s Ho izon Eu ope P og am h ough he INTELLIMAN p ojec (h ps:
//in elliman-p ojec .eu/, HORIZON-CL4-Digi al-Eme ging G an 101070136)
and he SESTOSENSO p ojec (h p://ses osenso.eu/, HORIZON-CL4-
Digi al-Eme ging G an 101070310).
∗Equal con ibu ion. The au ho s a e wi h he Idiap Resea ch Ins i u e,
Ma igny, Swi ze land and wi h he Ecole Poly echnique Fédé ale
de Lausanne (EPFL), Swi ze land. [email p o ec ed];
[email p o ec ed]; [email p o ec ed]
Di usion Co e age
G adien ield
Measu e co e age
Upda e a ge
Co e age Loop
Vision
Tac ile Con ol
Fo ce
Fig. 1: O e iew o ou eedback con ol me hod o ac ile
co e age. Le : We measu e he su ace and he ed a ge using
he came a and encode hem in a poin cloud. Bo om- igh :
We di use he a ge and use i s g adien ield o guide he
co e age. Then, we close he loop by measu ing he ac ual
co e age wi h he came a and use i as he nex a ge . Top-
igh : We measu e he ac ile in e ac ion o ces using he o ce
senso and he ool o ien a ion using he join posi ions. We
sol e he geome ic ask-space impedance con ol p oblem
using a line a ge and a o ce a ge along he line.
a possibly non-plana su ace and (ii) a con inuous ajec o y
o con ac poin s co e ing a egion o in e es on he su ace.
Acco dingly, his a icle add esses he o e a ching p oblem o
ac ile co e age on cu ed su aces.
Tac ile asks in ol e mul iple con ac in e ac ions wi h he
en i onmen , making hese sys ems no o iously di icul o
con ol [18]. While humans sol e hese asks e o lessly,
hey emain ex emely challenging o obo s. Fo ins ance,
when cleaning an objec , achie ing adequa e co e age depends
on ecognizing di , unde s anding he objec ’s ma e ial, and
assessing hei in e ac ion o de e mine he equi ed con ac
o ce o emo al. Consequen ly, he success o co e age de-
pends on unknown o di icul - o-measu e pa ame e s, making
i challenging o model all in e ac ions. Wi hou an accu a e
model, mo ion planning is p one o ailu e. By analyzing p e-
ious esea ch [19] and obse ing how humans add ess hese
challenges, we a gue ha humans bypass he complexi ies o
planning by sol ing he simple closed-loop con ol p oblem.
Humans le e age isual and ac ile eedback o online adap a-
ion. Simila ly, obo s can measu e p og ess in ac ile co e age
asks using ision, u ning he ask o iden i ying unco e ed
egions in o an image segmen a ion p oblem which has been
add essed using a ious model-based [20], [21] o lea ning-
based algo i hms [16], [22]. Howe e , de e mining how o
con ol a obo o co e hese a ge egions on cu ed su aces
emains an open challenge.
a Xi :2402.04862 3 [cs.RO] 31 Ma 2025
2
Exis ing esea ch on co e age has p ima ily ocused on
co e age pa h planning, which in ol es op imizing a pa h o
ensu e ha a speci ied egion o in e es is co e ed wi hin
a se ime ame. T adi ionally, he unde lying assump ion
is ha isi ing each poin in he egion o in e es only
once is su icien o ull co e age, an assump ion ha is
easonable o simple in e ac ions, bu no o many ac ile
asks. Tac ile in e ac ions a e o en oo complex o model
de e minis ically, making i challenging o ensu e ull co e age
a e a single isi . Ins ead, o a cleaning ask, a ela i ely
di y egion equi es mo e isi s compa ed o a less di y
egion. Simila ly, in a su ace inspec ion ask, egions equi -
ing highe p ecision demand mo e isi s o compensa e o
senso unce ain y. Fu he mo e, he obo is expec ed o keep
in con ac wi h he su ace while mo ing, which signi ican ly
inc eases he cos o mo emen . This cos depends on he
geodesic dis ance on he su ace a he han he Euclidean
dis ance. The e o e, nai e sampling s a egies ha ail o
accoun o he cos o cons ain s o mo emen and/o su ace
geome y a e unsui able o ac ile co e age asks. In con as ,
e godic co e age [23] con ols he ajec o ies o dynamical
sys ems by co ela ing he a e age ime spen in a egion
o he a ge spa ial dis ibu ion. The e o e, e godic co e age
inco po a es he mo ion model as he sys em dynamics and
di ec ly con ols he co e age ajec o ies by using he spa ial
dis ibu ion measu ed by he ision sys em.
Conside ing hese challenges, we p esen a closed-loop
ac ile e godic con ol me hod ha ope a es on poin clouds o
ac ile co e age asks. Using poin clouds enables us o acqui e
he a ge objec and spa ial dis ibu ion a un ime using
ision, measu e co e age p og ess, and compensa e o un-
modeled dynamics in ac ile co e age asks. Ou me hod hen
cons ains he e godic con ol p oblem o a bi a y su aces o
co e a a ge spa ial dis ibu ion on he su ace. We p opaga e
co e age in o ma ion by sol ing he di usion equa ion on
poin clouds, which we compu e in eal- ime by exploi ing
he su ace’s in insic basis unc ions called Laplacian eigen-
unc ions. These eigen unc ions gene alize he Fou ie se ies
o mani olds (i.e., cu ed spaces). In o de o exe a desi ed
o ce on he su ace while mo ing, we o mula e a geome ic
ask-space impedance con olle using geome ic algeb a. This
con olle uses su ace in o ma ion o ack a line a ge ha
is o hogonal o he su ace while simul aneously exe ing
he desi ed o ce in he di ec ion o ha line. No ably, he
geome ic o mula ion ensu es ha hese wo objec i es do
no con lic wi h each o he and can he e o e be included in
he same con ol loop wi hou equi ing exhaus i e pa ame e
uning. In summa y, ou p oposed closed-loop ac ile e godic
con ol me hod o e s he ollowing con ibu ions:
• o mula ing he ac ile co e age as closed-loop e godic
con ol p oblem on cu ed su aces;
•closing he co e age loop by sol ing e godic con ol
p oblem on poin clouds using di usion;
•achie ing eal- ime equencies by compu ing he di u-
sion using Laplacian eigen unc ions;
•con ac line and o ce acking wi hou con lic ing objec-
i es.
The es o he a icle is o ganized as ollows. Sec ion II
desc ibes ela ed wo k. Sec ion III p o ides he ma hema ical
backg ound. Sec ion IV p esen s ou me hod. In Sec ion V,
we demons a e he e ec i eness o ou me hod in simula ed
and eal-wo ld expe imen s. Finally, we discuss ou esul s in
Sec ion VI.
II. RELATED WORK
The majo i y o he co e age me hods conside he p oblem
om a planning pe spec i e and a e gene ally known as
co e age pa h planning (CPP) algo i hms [24]–[26]. Al hough
hese me hods can handle plana egions wi h a ious bound-
a ies [27]–[29], hei ex ension o cu ed su aces imposes lim-
i ing assump ions, such as p ojec i ely plana [30] o pseudo-
ex uded su aces [31]. Addi ionally, CPP me hods assume
ha he co e age a ge is uni o mly dis ibu ed in space.
Ex ending CPP me hods o accoun o spa ial co ela ions in
he in o ma ion leads o in o ma i e pa h planning (IPP) [32].
Mos IPP and CPP app oaches add ess a a ian o he NP-ha d
a eling salesman p oblem [33], which limi s hei scalabili y
as domain complexi y inc eases. Consequen ly, exis ing me h-
ods a e ei he open-loop [34] o impose limi ing assump ions,
such as con exi y, o online planning upda es [32].
Closely ela ed o co e age is he p oblem o explo a ion,
whe e he en i onmen is ini ially unknown, and obo s ga he
in o ma ion using onboa d senso s [35], [36]. Tac ile ex-
plo a ion is pa icula ly necessa y o ga he ing in o ma ion
on su aces ha can only be acqui ed h ough con ac [37].
A no able example is non-in asi e p obing (palpa ion) o
issue s i ness, which aids in disease diagnosis o su ge y by
p o iding addi ional ana omical in o ma ion. Fo his pu pose,
Gaussian p ocesses (GP) ha e been used o disc e e [9] and
con inuous [38] p obing o map issue s i ness. While GP-
based app oaches e ec i ely guide sampling loca ions, hey
do no accoun o he obo ’s dynamics. This limi a ion was
la e add essed by using ajec o y op imiza ion o ac i ely
sea ch o issue abno mali ies [39]. Unlike o he sensing
modali ies ha depend solely on posi ion, ac ile in e ac ions
also depend on condi ions such as ela i e eloci y and con ac
p essu e [40]. To add ess his, me hods ha e been de eloped
o model o ces [41] and mo e complex in e ac ions be ween
obo ic ools and su aces [42].
The complexi y o he p oblem inc eases u he i we
conside scena ios wi h a obo physically in e ac ing wi h he
en i onmen . Fo example, in asks like su ace inishing (e.g.,
polishing, sanding, g inding), he su ace i sel changes, as ma-
e ial is emo ed [43]. Simila ly, in cleaning asks, he obo ’s
ac ions a ec he dis ibu ion o di on he su ace [44]. To
a oid complex modeling, he e a e app oaches ei he elying
on ein o cemen lea ning [45] o deep lea ning [46]. In a e y
simila se ing o ou s, a manipula o was used o clean he
s ains on a cu ed su ace by pe o ming mul iple passes [20].
Howe e , his wo k used a sampling-based planne , which
equi ed o p ede ine he maximum numbe o cleaning passes.
In con as , we ela e he a ge dis ibu ion (e.g., s ain) di-
ec ly o eedback con ol wi hou equi ing any ask-speci ic
assump ions.
3
In ac ile co e age scena ios, isi ing a egion once can no
gua an ee ull co e age, and p edic ing how many imes he
obo should e isi a pa icula spo is challenging. Conse-
quen ly, de ining a ime ho izon o ajec o y op imiza ion is
di icul , as he quali y o he esul would be signi ican ly
a ec ed by his ha d- o-make choice. Ins ead, e godic con ol
ela es how o en he obo should e isi a pa icula spo
o he a ge densi y a ha spo . In his con ex , e godic
desc ibes a dynamical sys em in which he ime a e ages
o unc ions along i s ajec o ies a e equal o hei spa ial
a e ages [47]. The key ad an age o e godic con ol is i s
abili y o handle a bi a y spa ial a ge dis ibu ions wi hou
equi ing a p ede ined ime ho izon. When he spa ial a ge
dis ibu ion is measu able, he e godic con olle can use his
eedback o di ec he sys em o isi egions wi h highe
spa ial p obabili ies mo e equen ly. Recen indings ha e
demons a ed ha e godici y is no me ely a heu is ic [48]; i is
he op imal me hod o collec ing independen and iden ically
dis ibu ed da a while accoun ing o sys em dynamics.
E godic con ol was in oduced in he seminal wo k by
Ma hew and Mezi´
c [23], which p esen ed he spec al mul-
iscale co e age (SMC) algo i hm. SMC is a eedback con ol
law based on he Fou ie decomposi ion o he a ge dis ibu-
ion and obo ajec o ies, whe e mul iscale aspec p io i izes
low- equency componen s o e high- equency ones, co e-
sponding o s a ing wi h la ge-scale spa ial mo ions be o e
e ining ine de ails. Since his beha io is achie ed h ough
a myopic eedback con olle a he han an o line planne ,
he e godic con olle emains e ec i e e en when mo ion is
obs uc ed [49]. Recen wo ks ha e adap ed SMC’s objec i e
wi hin a ajec o y op imiza ion amewo k o inco po a e
addi ional objec i es, such as obs acle a oidance [50], ime-
op imali y [51] and ene gy-awa eness [52]. E godic con ol has
been used o ac ile co e age and explo a ion in applica ions
such as non-pa ame ic shape es ima ion [53] and able clean-
ing h ough lea ning om demons a ion [54]. Howe e , all
hese o mula ions, which ely on he Fou ie decomposi ion-
based e godic me ic, a e limi ed o ec angula domains in
Euclidean space.
The i s a emp o ex end he e godic con ol o Rie-
mannian mani olds [55] u ilized Laplacian eigen unc ions,
which gene alize he Fou ie se ies o cu ed spaces. Howe e ,
his app oach was es ic ed o homogeneous mani olds, such
as sphe es and o i, whe e closed- o m exp essions o he
Laplacian eigen unc ions a e a ailable. Mo e ecen ly, he
ke nel e godic me ic [56] was in oduced as an al e na i e
o SMC’s e godic me ic, enabling ex ensions o Lie g oups
and o e ing imp o ed compu a ional scalabili y. None heless,
a bi a y cu ed su aces collec ed using senso s, such as poin
clouds, lack bo h he g oup s uc u e and he homogeneous
mani old p ope ies, p esen ing addi ional challenges.
Ano he al e na i e o SMC is he hea equa ion-d i en
a ea co e age (HEDAC) algo i hm [57], which uses he di -
usion equa ion, a second-o de pa ial di e en ial equa ion
(PDE), o p opaga e in o ma ion abou unco e ed egions
o agen s ac oss he domain. Simila o SMC, he o iginal
HEDAC implemen a ion was es ic ed o ec angula domains
and lacked collision a oidance. Subsequen ex ensions ha e
adap ed HEDAC o plana meshes wi h obs acles [58], maze
explo a ion [59], and CPP on non-plana meshes [60]. How-
e e , i s applica ion on cu ed su aces emains limi ed o
meshes and o line planning due o he hea y p e-p ocessing
equi ed.
In addi ion o i s use in HEDAC, he di usion equa ion
is widely used in geome y p ocessing asks, anging om
geodesic compu a ion [61] o lea ning on su aces [62]. I s key
ad an age lies in i s abili y o accoun o su ace geome y
while emaining agnos ic o he unde lying ep esen a ion and
disc e iza ion [62]. The di usion equa ion is go e ned by a
second-o de di e en ial ope a o called he Laplacian which
can be compu ed o a bi a y su aces ep esen ed as meshes
o poin clouds using a ious disc e iza ion schemes [63]–[65].
In his wo k, we use a ecen app oach p oposed by Sha p e
al. which p o ides a obus and e icien implemen a ion [66],
capable o handling pa ial and noisy poin clouds.
III. BACKGROUND
A. E godic Con ol using Di usion
The e godic con ol objec i e co ela es he ime ha a
co e age agen spends in a egion o he p obabili y densi y
speci ied in ha egion. The HEDAC me hod [57] encodes
he co e age objec i e in he domain x∈Ωa ime using a
i ual sou ce e m
s(x, ) = max p(x)−c(x, ),02,(1)
whe e p(x)is he p obabili y dis ibu ion co esponding o he
co e age a ge and c(x, )is he no malized co e age o he
N i ual co e age agen s o e he domain
c(x, ) = ˜c(x, )
RΩ˜c(x, )dx.(2)
A single agen ’s co e age is he con olu ion o i s oo p in
φ( )wi h i s ajec o y xi( ′). Then, he o al co e age
becomes he ime-a e aged sum o hese con olu ions
˜c(x, ) = 1
N
N
X
i=1 Z
0
φx−xi( ′)d ′.(3)
HEDAC di uses he sou ce e m ac oss he domain Ωand
compu es he po en ial ield u(x, )using he s a iona y
(˙u(x, ) = 0) di usion
α∆u(x, )−u(x, ) + s(x, )=0,(4)
wi h he di usion coe icien α > 0and he Laplacian ope -
a o ∆. The Laplacian is a second-o de di e en ial ope a o
which educes o he sum o he second pa ial de i a i es in
Euclidean spaces
∆ =∇·∇ =
n
X
i=1
∂2
∂x2
i
,∀x∈Rn.(5)
The s a iona y di usion (4) go e ns he po en ial ield wi hin
he in e io o he domain Ω, while he beha io on he
bounda y ∂Ωis dic a ed by he ze o-Neumann bounda y
condi ion
n· ∇u(x, ) = 0,∀x∈∂Ω,(6)
4
whe e n ep esen s he ou wa d uni no mal ec o o he
bounda y ∂Ω. To guide he i- h co e age agen , HEDAC
u ilizes he smoo h g adien ield o he di used po en ial
u(x, )and simula es i s -o de dynamics [67]
˙
xi=∇u(xi, ).(7)
B. Con o mal Geome ic Algeb a
He e, we in oduce con o mal geome ic algeb a (CGA)
wi h a ocus on he ma hema ical backg ound necessa y o
unde s and he me hods used in his a icle. We will use he
ollowing no a ion h oughou he pape : x o deno e scala s,
x o ec o s, X o ma ices, X o mul i ec o s and X o
ma ices o mul i ec o s.
The inhe en algeb aic p oduc o geome ic algeb a is
called he geome ic p oduc
ab =a·b+a∧b,(8)
which ( o ec o s) is he sum o an inne ·and an ou e ∧
p oduc . The inne p oduc is he me ic p oduc and he e o e
depends on he me ic o he unde lying ec o space o e
which he geome ic algeb a is buil . The unde lying ec o
space o CGA is R4,1, which means he e a e ou basis ec o s
squa ing o 1 and one o -1. The ou e p oduc , on he o he
hand, is a spanning ope a ion ha e ec i ely makes subspaces
o he ec o space elemen s o compu a ion. These subspaces
a e called blades. In he case o CGA, he e a e 32 basis blades
o g ades 0 o 5. The e m g ade e e s o he numbe o basis
ec o s in a blade ha a e ac o izable unde he ou e p oduc .
Vec o s, consequen ly, a e o g ade 1 and he ou e p oduc o
wo independen ec o s, called bi ec o s, a e o g ade 2. A
gene al elemen o geome ic algeb a is called a mul i ec o .
In p ac ice, CGA ac ually applies a change o basis by
in oducing he wo null ec o s e0and e∞, which can be
hough o as a poin a he o igin and a in ini y, espec i ely.
Since he Euclidean space is embedded in CGA, we can embed
Euclidean poin s x o con o mal poin s P ia he con o mal
embedding
P=C(x) = e0+x+1
2x2e∞.(9)
In gene al, geome ic p imi i es in geome ic algeb a a e
de ined as nullspaces o ei he he inne o he ou e p oduc ,
which a e dual o each o he . The ou e p oduc nullspace
(OPNS) is de ined as
NOG(X) = x∈R3:C(x)∧X= 0.(10)
A simila exp ession can be ound o he inne p oduc
nullspace. The con o mal poin s a e he basic building blocks
o cons uc o he geome ic p imi i es in hei OPNS ep e-
sen a ion. The ele an p imi i es o his wo k a e lines
L=P1∧P2∧e∞,(11)
which can be cons uc ed om wo poin s and a poin a
in ini y, planes
E=P1∧P2∧P3∧e∞,(12)
which can be cons uc ed om h ee poin s and a poin a
in ini y and sphe es
S=P1∧P2∧P3∧P4,(13)
which can be cons uc ed om ou poin s.
Rigid body ans o ma ions in CGA a e achie ed using
mo o s M, which a e exponen ial mappings o dual lines, i.e.
bi ec o s (essen ially, he sc ew axis o he mo ion). No e ha
mo o s can be used o ans o m any objec in he algeb a,
i.e. hey can di ec ly be used o ans o m he p e iously
in oduced poin s, lines, planes and sphe es, by a sandwiching
ope a ion
X′=MX
M, (14)
whe e is
Mis he e e se o a mo o .
The o wa d kinema ics o se ial kinema ic chains can be
ound as he p oduc o mo o s, i.e.
M(q) =
N
Y
i=1
Mi(qi) =
N
Y
i=1
exp(qiBi),(15)
whe e qis he cu en join con igu a ion and Bia e sc ew
axes o he join s. The geome ic Jacobian JG(q)∈B1×N⊂
G1×N
4,1is a bi ec o alued mul i ec o ma ix and can be ound
as
JG=B′
1. . . B′
N,(16)
whe e he bi ec o elemen s can be ound as
B′
i=
i
Y
j=1
Mj(qj)Bi
i
Y
j=1
Mj(qj).(17)
Twis s Vand w enches Wa e also pa o he algeb a
and hence bo h can be ans o med in he same manne
as he geome ic p imi i es using (14). No e ha , con a y
o classic ma ix Lie algeb a, no dual adjoin ope a ion is
needed o ans o m w enches. The e is, howe e , s ill a
duali y ela ionship be ween wis s and w enches, which can
be ound ia mul iplica ion wi h he conjuga e pseudoscala
Ic=Ie0[68]. Bo h wis s and w enches a e bi ec o s and
he space o w enches can be ound as
W ∈ span{e23,e13,e12,e01,e02,e03}.(18)
The inne p oduc o wis s and w enches V · W =−pyields
a scala , whe e pis he powe o he mo ion. Simila ly, he
inne p oduc o a sc ew axis and a w ench B·W =−τyields
a o que τ, which we will use o he ask-space impedance
con ol in his a icle.
IV. METHOD
We p esen ou closed-loop ac ile e godic co e age me hod
in h ee pa s: (i) su ace p ep ocessing; (ii) ac ile co e age;
and (iii) obo con ol. The su ace p ep ocessing compu es
he quan i ies ha need o be calcula ed only once when he
su ace is cap u ed. Tac ile co e age gene a es he mo ion
commands o he i ual co e age agen using he p ecom-
pu ed quan i ies om he su ace p ep ocessing and he obo
con olle acks he gene a ed mo ion commands wi h a
manipula o using impedance con ol.
5
A. P oblem S a emen
We o mula e a ac ile e godic con olle ha co e s a -
ge spa ial dis ibu ions on a bi a y su aces. Simila ly o
HEDAC, we p opaga e he in o ma ion encoding he co e age
objec i e by di using he sou ce e m. Howe e , we u ilize
he non-s a iona y ( ˙u= 0) di usion equa ion
˙u(x, τ)=∆Mu(x, τ),(19)
as i allows con ol o e he desi ed smoo hness [69]. Since he
di usion equa ion depends on ime, we in oduce an addi ional
ime a iable τ. The di usion ime τis independen o he
co e age ime used by he HEDAC algo i hm and unlike
HEDAC, we equi e an ini ial condi ion u(x,0). We se he
ini ial condi ion using he sou ce e m gi en in Equa ion (1)
which encodes he co e age objec i e a he - h imes ep o
he co e age, i.e., u(x,0) = s(x, τ). Addi ionally, he e we use
∆M, which gene alizes he Laplacian o Euclidean spaces ∆
o non-Euclidean mani olds M. This ope a o ∆Mis also
known as Laplace-Bel ami ope a o bu o conciseness we
will use he e m Laplacian.
Ou co e age domains a e cu ed su aces (i.e. 2-mani olds)
and we cap u e he unde lying mani old Mas a poin cloud
Pcomposed o nPpoin s using an RGB-D came a
P:= ((xi,ci)
xi∈R3,ci∈ {0,...,255}3
o i= 1, . . . , nP),(20)
whe e xiis he posi ion o he i- h su ace poin in Euclidean
space and ciis he ec o o RGB colo in ensi ies. We assume
he e is a p ocessing pipeline (i.e., such as [20], [62], [70])
which maps he poin posi ions and colo s o he p obabili y
mass pio he spa ial dis ibu ion encoding he co e age
objec i e. Acco dingly, ou co e age a ge becomes a disc e e
spa ial dis ibu ion p(xi) = pion he poin cloud P.
In o de o sol e (19) on i egula and disc e e domains,
such as poin clouds, we disc e ize he p oblem in space
and ime. Hence, we use ui,τ o deno e he alue o he
po en ial ield a he i- h poin a he τ- h imes ep. We omi
he subsc ip ii we e e o all poin s.
B. Su ace P ep ocessing
Fi s , we compu e he spa ial disc e iza ion o he Laplacian
∆M. No e ha he e a e a ious app oaches o disc e izing
he Laplacian on poin clouds [63]–[66]. In his wo k, we
ollow he app oach p esen ed in [66] and show a simpli ied
e sion o i he e, bu e e he eade s o he o iginal wo k
o mo e de ails. Using his me hod, he disc e e Laplacian is
ep esen ed by he ma ix L∈RnP×nP
L=M−1C,(21)
whe e Mis he diagonal mass ma ix and Cis a spa se
symme ic ma ix called he weak Laplacian. The en ies o
Mco espond o he Vo onoi cell a eas in he local angen
plane a ound each poin o P. Simila ly, he en ies o C
a e de e mined by he connec i i y o he poin s on he local
angen space and he dis ance be ween he connec ed poin s.
No e ha he local angen space s uc u e also iden i ies he
bounda y poin s. Fo a gi en poin , he lines be ween he
o iginal poin and i s neighbo s a e cons uc ed. I he angle
be ween wo consecu i e lines is g ea e han π/2, he poin is
a bounda y and i s bounda y condi ion is se as ze o-Neumann.
Nex , we disc e ize he di usion equa ion (19) in ime and
inco po a e he disc e e Laplacian L. Using he backwa d
Eule me hod, we de i e he implici ime-s epping equa ion,
which emains s able o any imes ep τ
1
τ(uτ−u0) = Luτ,(22)
whe e u0and uτa e column ec o s con aining he po en ial
ield alues a he e ices o he poin cloud a he ini ial and
inal imes, espec i ely. Then, combining (21) and (22) and
sol ing o uτwe ob ain he linea sys em
uτ= (M−τC)−1Mu0.(23)
No e ha sol ing (23) equi es in e ing a la ge spa se ma ix,
which migh be compu a ionally expensi e depending on he
size o he poin cloud and equi es he imes ep o be se
be o e he in e sion. Al e na i ely, we can sol e he p oblem
in he spec al domain by p ojec ing he o iginal p oblem
and ep ojec ing he solu ion back o he poin cloud. This
p ocedu e gene alizes using he Fou ie ans o m o sol -
ing he di usion equa ion on a ec angula domain in Rn
o a bi a y mani olds. No e ha he Fou ie se ies a e he
eigen unc ions o he Laplacian ∆in Rn. The e o e, we can
use he eigen ec o s o he disc e e Laplacian L o sol ing
he di usion equa ion on poin clouds.
We can w i e he gene alized (i.e., M=I) eigen alue
p oblem o he Laplacian as
Cϕm=λmMϕm,(24)
whe e {λm,ϕm}a e he eigen alue/eigen ec o pai s. Since
Mis diagonal and Cis symme ic posi i e de ini e, by he
spec al heo em, we know ha he eigen alues a e eal, non-
nega i e, and in ascending o de analogous o he equency.
The e o e, we can use he i s nMeigen alue/eigen ec o
pai s as a low- equency app oxima ion o he whole spec um.
Fu he mo e, he eigen ec o s a e o hono mal wi h espec o
he inne p oduc de ined by he mass ma ix M. Acco dingly,
we can s ack he i s nMeigen ec o s ϕmas column ec o s
o cons uc he ma ix Φ∈RnP×nMencoding an o hono -
mal ans o ma ion Φ⊤MΦ=I. Then, we can ans o m he
coo dina es (shown wi h supe sc ip s) om he poin cloud o
he spec al domain
uϕ=Φ⊤Mux.(25)
No e ha his s ep is equi alen o compu ing he Fou ie
se ies coe icien s o a a ge dis ibu ion in SMC. Due o
he o hono mal ans o ma ion, he PDE on he poin cloud
becomes a sys em o decoupled ODEs in he spec al domain.
I is well known ha he solu ion o a i s -o de linea ODE
˙x(τ) = −cx(τ)is gi en by x(τ) = e−cτ x(0), whe e cis
a cons an and x(0) is he ini ial condi ion. The e o e, he
solu ion o he sys em o ODEs in he spec al domain is gi en
in ma ix o m as
uϕ
τ=e−λ1τ. . . e−λmτ⊤⊙uϕ
0,(26)

6
whe e ⊙deno es he Hadama d p oduc . We obse e om (26)
ha he exponen ial e ms wi h la ge eigen alues (i.e., highe
equencies) will decay as e . The e o e, app oxima ing he
di usion using he i s nMcomponen s in oduces minimal
e o . Secondly, simila o he mixed no m used in SMC,
he low- equency spa ial ea u es a e p io i ized. Nex , we
ans o m he solu ion back o he poin cloud o ge he
di used po en ial ield
ux=Φuϕ.(27)
We can combine (25), (26) and (27) in o a uni ied spec al
scheme
uτ= Φ e−λ1τ. . . e−λmτ⊤⊙(Φ⊤Mu0).(28)
We omi he supe sc ip s when wo king on he poin cloud o
b e i y. No e ha τis he only ee pa ame e in he di usion
compu a ion. Howe e , i s alue should be adap ed acco ding
o he mean spacing be ween he adjacen poin s hon he
poin cloud. Fo ha pu pose, we in oduce he hype pa ame e
α > 0and embed i in o he imes ep calcula ion
τ=αh2.(29)
Acco dingly, we can con ol he di usion beha io indepen-
den ly o he poin cloud size. Inc easing α esul s in longe
di usion imes and a enua es he high- equency spa ial ea-
u es (see (26) o de ails). This co esponds o a mo e global
co e age [69]. Con e sely, dec easing α esul s in sho e
di usion imes, which leads o p ese ing he high- equency
spa ial ea u es, hence mo e local co e age beha io .
No e ha he Laplacian is de e mined comple ely by he
connec i i y on he local angen space and he dis ance
be ween hese connec ed poin s. The e o e, i is in a ian o
dis ance p ese ing (i.e., isome ic) ans o ma ions such as
igid body mo ion o de o ma ion wi hou s e ching. Acco d-
ingly, we compu e C,Mand de i ed quan i ies only once
in he p ep ocessing s ep o a gi en su ace. Recompu a ion
is no necessa y i he objec s ays s ill, mo es igidly, o he
a ge dis ibu ion pichanges.
C. Tac ile E godic Co e age
We model he ac ual co e age ool/senso as a complian
i ual co e age agen shaped as a disk wi h adius a.
No ably, one can ep esen a bi a y ool/senso oo p in s as
a combina ion o disks [69]. We posi ion ou agen a he end-
e ec o o ou manipula o . Thus, o a gi en kinema ic chain
and join con igu a ion q, we can use he o wa d kinema ics
o compu e he posi ion o ou agen as a con o mal poin Pa
Pa=M(q)e0
M(q).(30)
Since he poin cloud is disc e e and he agen should mo e
con inuously on he su ace, we p ojec ou agen Paand i s
oo p in o he closes local angen space on he poin cloud.
1) Local Tangen Space and Co e age Compu a ion: Gi en
he agen ’s posi ion Pa, we i s compu e he closes angen
space on he poin cloud. Fo ha , we que y a K-D ee T(P)
o he poin s xi∈ P ha a e wi hin he adius ao he
agen . Then, we compu e he con o mal embeddings Pio he
neighbo ing Euclidean poin s xiusing (9). We e e o he se
composed o poin s Pias he local neighbo hood. Then, we
i a angen space o he local neighbo hood by minimizing
he classical leas squa es objec i e
min
nN
X
i=1
(Pi·X∗)2,(31)
whe e X∗is he dual ep esen a ion o ei he a plane o
a sphe e and he inne p oduc ·is a dis ance measu e. In
CGA, planes can be seen as limi cases o sphe es, i.e. planes
a e sphe es wi h in ini e adius. This is also easy o obse e
by looking a Equa ions (12) and (13) which cons uc hese
geome ic p imi i es. No e ha i ing a local angen sphe e
wi h he adius de e mined by he local cu a u e would always
esul in smalle o equal esiduals han i ing a plane.
I has been shown in [71] ha he solu ion o he leas
squa es p oblem gi en in (31) is he eigen ec o co esponding
o he smalles eigen alue o he 5×5ma ix
bj,k =
nN
X
i=1
wi,jwi,k,(32)
whe e
wi,k =




pi,k i k∈ {1,2,3}
−1i k= 4
−1
2p2
ii k= 5.
(33)
Using he i e componen s io his eigen ec o we can ind
he geome ic p imi i e as
X= ( 0e0+ 1e1+ 2e2+ 3e3+ 4e∞)∗.(34)
No e ha i Xis a plane hen 0= 0, o he wise Xis a
sphe e. Nex , we wan o p ojec Pa o Xby using he gene al
subspace p ojec ion o mula o CGA
Ppai =(Pa∧e∞)·XX−1.(35)
He e we i s cons uc he poin pai Pa∧e∞, whe e e∞
co esponds o he poin a in ini y. Pa∧e∞is also called a
la poin . No e ha he p ojec ion essen ially amoun s o i s
cons uc ing he dual line (Pa∧e∞)·X ha passes h ough
he poin Paand is o hogonal o X, hen in e sec ing his line
wi h he p imi i e X.
I Xis a sphe e, hen he in e sec ion o he line and he
sphe e will esul in wo poin s on he sphe e. I Xis a plane,
i will esul in ano he la poin , i.e. one poin on he plane
and one a in ini y. In any case, we can e ie e he close one
o he agen posi ion Pausing he spli ope a ion
P′
a=spli [Pp].(36)
He e, P′
ais he p ojec ed agen posi ion on he angen space
X. Nex , we compu e ou agen ’s oo p in (i.e., ins an aneous
co e age) by p ojec ing i s su ace o he poin cloud. I he
a ge su ace was la , all he poin s wi hin he adius ao
7
ou agen P′
awould be co e ed by he oo p in . Howe e ,
in he gene al case, bo h he ool and he su ace can be
cu ed and de o mable. Fo simplici y, we assume ha he
su ace is igid, and i de o ms he ool wi h a cons an bending
adius. We use he adius o he local angen sphe e ha we
compu ed using CGA as an app oxima ion o he bending
adius. Acco dingly, we can quan i y he e o o he local
angen space app oxima ion o he i- h neighbo Piby he
no malized esiduals eio he leas squa es compu a ion (31).
We encode his app oxima ion e o in o he oo p in by
weigh ing he i- h neighbo by he Gaussian ke nel φ( )using
he no malized esiduals i=ei/max(e)
φ( i) = exp −ε2 2
i,(37)
whe e he hype pa ame e ε > 0con ols he co e age allo .
Nex , we subs i u e he Gaussian ke nel weigh ed oo p in
in o (3) o compu e he co e age c , which is hen used o
calcula e he i ual sou ce e m s ia (1). As men ioned
ea lie , his i ual sou ce e m se es as he ini ial condi ion
o he di usion equa ion (19) a each i e a ion o he ac ile
co e age loop, i.e., u0=s .
2) G adien o he Di used Po en ial Field: We guide he
co e age agen using he g adien o he di used po en ial ield
as he accele a ion command
¨
P′
a=∇uP′
a,τ ,(38)
whe e ∇uP′
a,τ deno es he g adien o he di used po en ial
ield a he p ojec ed agen posi ion P′
a. Howe e , compu ing
he g adien on he poin cloud is mo e in ol ed han a
egula g id o a mesh. Recall ha in Sec ion IV-C1, we
al eady compu ed he p ojec ed agen posi ion P′
a, he local
neighbo hood and he angen space X∗. As he i s s ep, we
compu e he angen plane Ea,τ a P′
a, namely
Ea,τ =L∗
a,⊥∧P′
a∧e∞,(39)
using he line La,⊥, which is o hogonal o he su ace and
passes h ough P′
a. I is ound by wedging he dual p imi i e
Xwi h P′
a o in ini y wi h
La,⊥=X∗∧P′
a∧e∞.(40)
Then, we p ojec he poin s Piin he local neighbo hood o
he angen plane Ea,τ using (35) and (36), by se ing Ea,τ as
he p imi i e X. Nex , we use he alues o he po en ial ield
a he neighbo loca ions as he heigh hi=ui,τ o a second
su ace om he angen plane. Then, we i a 3- d deg ee
polynomial o his su ace as shown by using he weigh ed
leas squa es objec i e
ˆ
A= a g min
A (Y−XA)⊤W(Y−XA),(41)
wi h he diagonal weigh ma ix W
W=diagφ( 1), φ( 1),...φ( m),(42)
whose en ies a e gi en by he Gaussian ke nel (37). One can
e e o [72] o he de ails. Las ly, we calcula e he g adien a
he p ojec ed agen ’s posi ion using he analy ical g adien s o
he polynomial. We depic he app oach isually in Figu e 2.
Fig. 2: Blue- ed poin s show he alue o he po en ial ield uτ
on he poin cloud Pand he yellow poin is he p ojec ed agen
posi ion P′
a. We also p ojec he agen ’s neighbo s Pi o he
angen plane Ea,τ , shown in g een. Nex , we use he heigh
unc ion hi=ui,τ which uses he alues o he po en ial
ield o li he p ojec ed poin s in he no mal di ec ion o he
angen plane. We show he li ed poin s wi h la ge blue- ed
poin s. We i a polynomial o his li ed su ace and compu e
i s analy ical g adien s a he neighbo loca ions ∇ui,τ , as
shown wi h a ows in he de ail iew.
D. Robo Con ol
The e a e se e al aspec s ha he con ol o he physical
obo needs o achie e. The i s is o ack he i ual co e age
agen on he a ge su ace, while keeping he end-e ec o no -
mal o he su ace. The second is o exe a desi ed o ce on he
su ace. To do so, we design a ask-space impedance con olle
while u he exploi ing geome ic algeb a o e iciency and
compac ness. The con ol law is o he ollowing o m
τ=−J⊤· W,(43)
whe e J∈B1×N⊂G1×N
4,1is he Jacobian mul i ec o
ma ix wi h elemen s co esponding o bi ec o s, Wis he
desi ed ask-space w ench and τa e he esul ing join o ques.
Be o e composing he inal con ol law, we will explain i s
componen s indi idually.
1) Su ace O ien a ion: F om Equa ion (40), we ob ained
a line La,⊥ ha is o hogonal o he su ace ha we wish o
ack. In [73], i was shown how he mo o be ween con o mal
objec s can be ob ained. We use his o mula ion o ind he
mo o be ween he a ge o hogonal line and he line ha
co esponds o he z-axis o he end-e ec o o he obo in
i s cu en con igu a ion, which is ound as
Lee =M(q)(e0∧e3∧e∞)
M(q).(44)
Then, he mo o MLeeLa,⊥, which ans o ms Lee in o La,⊥
can be ound as
MLeeLa,⊥=1
C(1 + La,⊥Lee),(45)
whe e Cis a no maliza ion cons an . No e ha Cdoes no
simply co espond o he no m o 1 + La,⊥Lee, bu equi es
a mo e in ol ed compu a ion. We he e o e omi i s exac
compu a ion he e o b e i y and e e eade s o [73].
We can now use he mo o MLeeLa,⊥in o de o ind a
con ol command o he obo ia he loga i hmic map o
mo o s, i.e.
VLa,⊥= log MLeeLa,⊥.(46)
8
O cou se, i he lines a e equal, MLeeLa,⊥= 1 and conse-
quen ly VLa,⊥= 0. No e ha VLa,⊥is s ill a command in ask
space (we will explain how o ans o m i o a join o que
command once we ha e de i ed all he necessa y componen s).
Ano he issue is ha algeb aically, VLa,⊥co esponds o
a wis , no a w ench. Hence, we need o ans o m i
acco dingly. F om physics, we know ha wis s ans o m
o w enches ia an ine ial map, which we could use he e
as well. In he con ex o con ol, his ine ia enso is,
howe e , a uning pa ame e and does no ac ually co espond
o a physical quan i y. Thus, in o de o simpli y he inal
exp ession, we will use a scala ma ix alued ine ia, ins ead
o a geome ic algeb a ine ia enso and choose o ans o m
he wis command o w ench command pu ely algeb aically.
As i has been shown be o e, his can be achie ed by he
conjuga e pseudoscala Ic=Ie0[68]. I ollows ha
WLa,⊥=VLa,⊥Ic,(47)
and WLa,⊥now algeb aically co esponds o a w ench.
2) Ta ge Su ace Fo ce: Since his a icle desc ibes a
me hod o ac ile su ace co e age, he goal o he obo
con ol is o no simply s ay in con ac wi h he su ace, bu
o ac i ely exe a desi ed o ce on he su ace. Fi s o all, we
deno e he cu en measu ed w ench as Wm( )and he desi ed
w ench as Wd. Bo h a e bi ec o s as de ined by Equa ion (18).
We use Wdw. . . end-e ec o in o de o make i mo e in ui i e
o de ine. Hence, we need o ans o m Wm( ) o he same
coo dina e ame, i.e.
W′
m( ) =
M(q)Wm( )M(q).(48)
In o de o achie e he desi ed in e ac ion o ce, we simply
apply a s anda d PID con olle in w ench space, i.e.
WC=Kp,WWe+Ki,WZ⊤
0
We(τ)dτ+Kd,W
d
d We( ),
(49)
whe e he w ench e o is
We( ) = Wd− W′
m( ),(50)
whe e Kp,W,Ki,Wand Kd,Wa e he co esponding gain
ma ices, and WCis he esul ing con ol w ench.
Since he desi ed w ench is de ined in end-e ec o coo di-
na es, i usually amoun s o a linea o ce in he z-di ec ion o
he end-e ec o ame, i.e. Wd= de03. Addi ionally, o an
imp o ed cleaning beha io one could also se a desi ed o que
a ound ha axis by adding τde12. The pa e n o how o se
his o que, howe e , would be subjec o u he in es iga ion.
3) Task-Space Impedance Con ol: Recalling he con ol
law om Equa ion (43), we now collec he e ms om he
p e ious subsec ions in o a uni ied ask-space impedance con-
ol law. We s a by looking in mo e de ail a he Jacobian J.
P e iously, we men ioned ha we a e using he cu en end-
e ec o mo o as he e e ence, hence, we equi e he Jacobian
o be compu ed w. . . ha e e ence. This is he e o e no he
geome ic Jacobian ha was p esen ed in Equa ion (16), bu
a a ia ion o i . The end-e ec o ame geome ic Jacobian
Jee
Gcan be ound as
Jee
G=Bee
1. . . Bee
N,(51)
whe e he bi ec o elemen s can be ound as
Bee
i=
Mee
i(q)BiMee
i,(52)
wi h
Mee
i=
i
Y
j=N
Mi(qi).(53)
Hence, he ela ionship be ween JGand Jee
Gcan be ound
as
Jee
G=
M(q)JGM(q).(54)
The w ench in he con ol law is composed o he h ee
w enches ha we de ined in he p e ious subsec ions. As
commonly done, we add a damping e m ha co esponds o
he cu en end-e ec o wis and as be o e, we ans o m i
o an algeb aic w ench, i.e.
WV=Jee
G˙qe0∞.(55)
Wi h his, we now ha e e e y hing in place o compose ou
inal con ol law as
τ=−Jee,⊤
G·KLa,⊥WLa,⊥−DVWV+WC,(56)
whe e KLa,⊥is a s i ness and DVa damping gain.
V. EXPERIMENTS
Ou expe imen al se up comp ises a Bo aSys SensOne 6-
axis o ce o que (F/T) senso a ached o he w is o a 7-
axis F anka obo manipula o and a cus om 3-D p in ed pa
a ached o he F/T senso . The cus om pa in e aces an
In el Realsense D415 dep h came a and a sponge a i s ip.
We conside he sponge’s cen e poin o be he co e age
agen ’s posi ion Pa. Be o e he ope a ion, we pe o m ex insic
calib a ion o he came a o combine he dep h and RGB eeds
om he came a and o ob ain i s ans o ma ion wi h espec
o he obo join s. Addi ionally, we calib a e he F/T senso
o compensa e o he weigh o he 3-D p in ed pa and
he came a. We show he expe imen al se up on he le o
Figu e 1.
A. Implemen a ion De ails
The pipeline o ou ac ile e godic co e age me hod consis s
o h ee modules: (i) su ace acquisi ion, (ii) su ace co e age
and (iii) obo con ol. Figu e 3 summa izes he in o ma ion
low be ween he componen s.
1) Su ace Acquisi ion: The su ace acquisi ion node is
esponsible o collec ing he poin cloud and pe o ming
p ep ocessing ope a ions desc ibed in Sec ion IV-B. We use
scipy1 o he nea es neighbo que ies and o sol ing he
eigenp oblem in (24). The ma ices Cand Mcomposing
he disc e e Laplacian in (21) a e compu ed wi h he o-
bus _laplacian package2[66].
2) Su ace Co e age: The su ace co e age node pe -
o ms he compu a ions based on he p ocedu e gi en in
Sec ion IV-C. I uses he in o ma ion p o ided by he su ace
acquisi ion node and p oduces he a ge line o he obo
con ol node.
1h ps://scipy.o g
2h ps://gi hub.com/nmwsha p/ obus -laplacians-py
9
Su ace Acquisi ion Su ace Co e age Robo Con ol
Inne loop
Ou e loop
Ta ge , K-D T ee, Laplacian eigenbasis
Ac ual co e age
Line objec
Ac ual agen posi ion
Fo ce a ge 
Fig. 3: In o ma ion low be ween he h ee componen s. The pipeline is composed o an ou e loop esponsible o con olling
he co e age p og ess wi h he eedback om he came a, whe eas he inne loop compensa es o he misma ch due o he
obo dynamics.
3) Robo con ol: On a high le el, he obo con ol can
be seen as a s a e machine wi h h ee disc e e s a es. The
i s wo s a es a e essen ially wo p e- eco ded join posi ions
in which he obo is wai ing o o he pa s o he pipeline
o be comple ed. One o hese posi ions co esponds o he
pic u e- aking posi ion, i.e., a join posi ion whe e he came a
has he ull objec in i s ame and he poin cloud can be
ob ained. The obo is wai ing in his posi ion un il he poin
cloud has been ob ained, a e wa ds i changes i s posi ion
o ho e sho ly o e he objec . In his second posi ion, i is
wai ing o he compu a ion o he Laplacian eigen unc ions o
be comple ed, such ha he co e age can s a . The swi ching
be ween hose wo posi ions is achie ed using a simple join
impedance con olle .
The hi d and mos impo an s a e is when obo is
ac ually con olled o be in con ac wi h he su ace and
o ollow he a ge co esponding o he co e age agen .
This beha iou is achie ed using he con olle ha we
desc ibed in Sec ion IV-D. The ele an pa ame e s, ha
we e chosen empi ically o he eal-wo ld expe imen s, a e
he s i ness and damping o he line acking con olle ,
i.e. KLa,⊥=diag(30,30,30,750,750,300) and DV=
diag(10,10,10,150,150,50), as well as he gains o he
w ench PID con olle , i.e. Kp,W= 0.5,Ki,W= 5 and
Kd,W= 0.5. The con olle has been implemen ed using ou
open-sou ce geome ic algeb a o obo ics lib a y ga o3 ha
we i s p esen ed in [74]. No e ha in some cases, ma ix-
ec o p oduc s o geome ic algeb a quan i ies ha e been used
o he implemen a ion, whe e he ma hema ical s uc u e o
he geome ic p oduc ac ually simpli ies o his, which can be
exploi ed o mo e e icien compu a ion.
B. Simula ed Expe imen s
1) Compu a ion Pe o mance: In o de o assess he com-
pu a ional pe o mance, we in es iga ed he wo main ope a-
ions o ou me hod: (i) p ep ocessing by sol ing ei he he
eigenp oblem (24) o ma ix in e sion in (23) (ii) in eg a ing
he di usion a un ime using ei he he spec al (28) o
implici (23) o mula ions. In his expe imen , we used he
S an o d Bunny as he e e ence poin cloud and pe o med
oxel il e ing o se he poin cloud esolu ion. We p esen he
3h ps://gi lab.com/ga o
esul s o he p ep ocessing in Figu e 4 and o he un ime
in Figu e 5.
3k 6k 9k 12k 15k 18k
5
0.1
2
5
1
2
5
10
2
5
25
50
100
200
implici
Fig. 4: Compu a ional complexi y o he p ep ocessing s ep
o di e en nPand nM. Legend shows nM alues. The ime
axis is loga i hmic and he legend shows nM alues.
3k 6k 9k 12k 15k 18k
0.1
2
5
1
2
5
10
2
5
100
2
25
50
100
200
implici
Fig. 5: Compu a ional complexi y o in eg a ing he di usion
equa ion a un ime o di e en nPand nM. The ime axis
is loga i hmic and he legend shows nM alues.
2) Co e age Pe o mance: We es ed he co e age pe o -
mance in a se ies o kinema ic simula ions. As he co e age
me ic, we used he no malized e godici y o e he a ge
dis ibu ion, which compa es he ime-a e aged s a is ics o
agen ajec o ies o he a ge dis ibu ion
ε =∥max (p−c ,0) ∥2
PnP
i=1 pi
.(57)