ga o: Geome ic Algeb a o Robo ics
Tobias Löw∗†, Philip Abbe ∗and Syl ain Calinon∗†
∗Idiap Resea ch Ins i u e, Ma igny, Swi ze land
†EPFL, Lausanne, Swi ze land
Abs ac —Geome y is a undamen al pa o obo ics and
he e ha e been a ious amewo ks o ep esen a ion o e he
yea s. Recen ly, geome ic algeb a has gained a en ion o i s
p ope y o uni ying many o hose p e ious ideas in o one
algeb a. While he e a e al eady e icien open-sou ce implemen-
a ions o geome ic algeb a a ailable, none o hem is a ge ed a
obo ics applica ions. We wan o add ess his sho coming wi h
ou lib a y ga o. This a icle p esen s an o e iew o he imple-
men a ion de ails as well as a u o ial o ga o, an e icien c++
lib a y a ge ing obo ics applica ions using geome ic algeb a.
The lib a y ocuses on using con o mal geome ic algeb a. Hence,
a ious geome ic p imi i es a e a ailable o compu a ion as well
as igid body ans o ma ions. The modeling o obo ic sys ems
is also an impo an aspec o he lib a y. I implemen s a ious
algo i hms o calcula ing he kinema ics and dynamics o such
sys ems as well as objec i es o op imisa ion p oblems. The
so wa e s ack is comple ed by py hon bindings in pyga o and
a ROS in e ace in ga o_ os.
I. INTRODUCTION
Geome ic algeb a can be conside ed a high-le el ma he-
ma ical language o geome ic easoning. As such i is e y
well sui ed o gene al p oblems in obo ics.
Al hough geome ic algeb a has g ea po en ial o model-
ing, lea ning and con ol in obo ics, i has no been widely
adop ed in obo ics esea ch. One eason o his is he lack
o easy- o-use lib a ies o obo ics applica ions, while a he
same ime ools based on ma ix algeb a a e e y ma u e and
eadily a ailable. We aim o change ha by p o iding a eady-
o-use geome ic algeb a lib a y o obo ics ha can be used
wi h he mos popula p og amming amewo ks, namely c++,
py hon and ROS.
Fo obo modeling and con ol i is necessa y o compu e
he kinema ics and dynamics o obo s. These algo i hms a e
well s udied and ha e been implemen ed in a ious so wa e
amewo ks. In ou ga o lib a y we p o ide an implemen-
a ion o he geome ic algeb a a ian o hese algo i hms.
I is impo an o poin ou ha geome ic algeb a can be
used o compu e hese impo an quan i ies ha a e classically
compu ed using ma ix algeb a, while also o e ing a iche
oolse , i.e. i also includes ools o geome ic easoning ha
ma ix algeb a does no ha e.
In his a icle we wan o explain he implemen a ion de ails
o ou geome ic algeb a lib a y ga o and gi e a u o ial on
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
Commission’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).
how o use i o common obo ics p oblems such as in e se
kinema ics and op imal con ol. Ou aim is o make geome ic
algeb a mo e accessible o obo ics esea ch by p o iding his
eady- o-use lib a y. This should esul in a wide adop ion
and acili a e he esea ch on using his powe ul amewo k
o obo ics.
This a icle is o ganized as ollows: in Sec ion II we gi e an
o e iew o he a ailable p og amming in e aces, in Sec ion
III we explain he implemen a ion de ails o he algeb a, in
Sec ion IV we show how o model obo s using he lib a y, in
Sec ion V we compa e ga o o o he GA and obo modeling
lib a ies and inally in Sec ion VI we demons a e a ious
applica ions and gi e a u o ial on how o use he lib a y.
The documen a ion and he links o all eposi o ies can be
ound on ou websi e h ps://geome ic-algeb a.
obiloew.ch/ga o.
II. PROGRAMMING INTERFACES
In his sec ion we explain he p og amming in e aces ha
ga o o e s. The main lib a y is w i en in c++ o which we
p o ide py hon bindings called pyga o as well as he ROS
package ga o_ os. All men ioned eposi o ies can be ound
a h ps://gi lab.com/ga o.
A. c++ Lib a y
The co e implemen a ion o ga o is done in c++20 and
elies hea ily on empla es. Apa om he s anda d lib a y he
only manda o y dependency is he Eigen1lib a y. We use he
he yaml-cpp2lib a y as an op ional dependency o loading
o obo desc ip ions. Se e al obo desc ip ions a e al eady
a ailable and can be used ia he espec i e classes.
B. ROS Package
The Robo Ope a ing Sys em (ROS) is he de ac o s anda d
o building obo applica ions. Hence, we p o ide a ROS
package called ga o_ os o use ou lib a y wi hin he ROS
amewo k. I allows passing geome ic p imi i es be ween
nodes ia messages. I also enables he isualiza ion o hese
geome ic p imi i es in R iz ia cus om plugins. Fu he mo e,
since mos obo desc ip ions a e a ailable in he Uni e sal
Robo Desc ip ion Fo ma (URDF), his package suppo s
loading his ile o ma .
1h ps://eigen. ux amily.o g
2h ps://gi hub.com/jbede /yaml-cpp
a Xi :2310.19090 1 [cs.RO] 29 Oc 2023
C. Py hon Bindings
Since he py hon p og amming language is a popula ool
o apid p o o yping and is gene ally mo e accessible han he
c++ language, we a e p o iding language bindings in py hon
o ga o using he pybind113lib a y.
III. IMPLEMENTATION OF CONFORMAL
GEOMETRIC ALGEBRA
In his sec ion we will explain in de ail ou implemen a ion
o con o mal geome ic algeb a (CGA). The aspec s ha a e
highligh ed a e he implemen a ion o a gene al mul i ec o
and he exp essions ha a e ac ing on i . The lib a y co e s
se e al poin s ha we e p oposed in [1] as a wishlis o
geome ic algeb a implemen a ions. We ha e designed he
lib a y in an objec -o ien ed way, so he classes also e-
lec he ma hema ical inhe i ance ela ionships. Fu he mo e
all classes, i.e. all specialized mul i ec o s a e ins an ia ed
as di e en ypes, which allows hem o be dis inguished
a compile- ime o ype-sa e y and ha e pe sis en s o age.
These specialized classes, also enable he compu a ion wi h
pa ial mul i ec o s, i.e. he lib a y exploi s he ac ha he
mos commonly muli ec o s a e spa se and only use ce ain
subspaces o he algeb a. We add ess nume ical imp ecisions
by ensu ing ha only elemen s o he esul ing mul i ec o s o
exp essions a e e alua ed ha a e known o be non-ze o. We
handle he ype explosion o bina y ope a o s by au oma ically
e alua ion pa ial exp essions when he ull exp essions ge oo
complex.
A. Gene al Mul i ec o
The co e elemen o compu a ion in geome ic algeb a is
he mul i ec o . Hence, i is e y impo an o hink abou he
design choices when implemen ing i s s uc u e, as his will
de e mine he memo y usage and compu a ional pe o mance.
The gene al s uc u e o a mul i ec o in CGA can be seen
in Figu e 1. I is composed o 32 basis blades, di ided in o
g ades ze o o i e. A gene al mul i ec o would he e o e be
qui e hea y in e ms o memo y and compu a ion. Howe e ,
g ade 0 1
g ade 1 e1e2e3e∞e0
g ade 2 e23 e13 e12 e1∞e2∞e3∞e01 e02 e03 e0∞
g ade 3 e123 e12∞e13∞e23∞e012 e013 e023 e01∞e02∞e03∞
g ade 4 e123∞e0123 e012∞e023∞e013∞
g ade 5 e0123∞
Fig. 1: S uc u e o con o mal geome ic algeb a wi h he 32
basis blades, di ided in o he di e en g ades. G ade 0 and 5
a e he scala and pseudo-scala , espec i ely. G ades 1 o 4
a e called bi-, i- and quad ec o s.
3h ps://gi hub.com/pybind/pybind11
Mo o
Twis
Poin
Line
Ci cle
Plane
Fig. 2: Non-ze o elemen s o a ious geome ic p imi i es in
hei p imal ep esen a ions in con o mal geome ic algeb a.
Boxes ep esen basis blades and colo ed boxes ep esen he
non-ze o blades o he geome ic p imi i e wi h he ma ching
colo . I can be seen ha o he 32 basis blades composing
mul i ec o s only a spa se numbe is used o he ep esen a-
ions. No e ha geome ic p imi i es a e single-g ade objec s,
while ans o ma ions a e mixed-g ade.
An impo an s uc u al aspec s o CGA ha acili a e he
design p ocess he e a e he spa si y o i s ep esen a ions and
he ac ha he s uc u e o mul i ec o exp essions is known
a compile- ime. Bo h o hese p ope ies mean ha we can
implemen he da a ec o o a mul i ec o by only s o ing i s
known non-ze o elemen s. This is achie ed by using a empla e
ha akes lis o blade indices as inpu :
empla e <class T, in ... blades>
class Mul i ec o
{
public:
cons exp s a ic in size = sizeo ...(blades);
{...}
p i a e:
Eigen::Ma ix<T, size, 1> da a_;
};
The lis o indices is hen s o ed in e nally as a bi se ha
acili a es he compa ison o he subspaces o wo muli ec o s.
A bi se is simply a lis o 32 bi s, ha a e ei he 0 o 1,
depending on whe he he co esponding blade is p esen in
he mul i ec o o no .
The memo y ha is alloca ed co esponds o he numbe
o blade indices ha is gi en o he empla e. I uses an
Eigen::Ma ix o s o e he da a, ha is exposed ia an
accesso unc ion called ec o (). This makes i possible
o di ec ly use he pa ame e ec o o any mul i ec o , which
is use ul o e.g. op imisa ion sol e s.
The unde lying da a ype Tis a empla e a gumen , which
makes i possible o ei he use e.g. loa o double,
depending on he sys em a chi ec u e. Fu he mo e, i allows
he usage o gene al pu pose au oma ic di e en ia ion lib a ies
such as au odi 4. This helps when o mulizing op imisa ion
p oblems in geome ic algeb a using ga o since i acili a es
he coding o complex objec i e unc ions and hus accele a es
p o o yping.
The Mul i ec o class and all i s de i a es (including
he exp essions) ha e a me hod called ge ha is empla ed
on he blade index and p o ec ed by he concep s lib a y o
c++20.
4h ps://au odi .gi hub.io/
TABLE I: Una y exp essions ha a e implemen ed as membe
unc ions o he Mul i ec o class.
membe unc ion una y exp ession ma hema ical symbol
e e se Re e se
e
X
in e se In e se X−1
dual Dual X∗
B. Algeb aic Compu a ions using Exp ession Templa es
When implemen ing geome ic algeb a, he e a e some
co e bina y exp essions ha need o be a ailable o gene al
algeb aic compu a ions. These ope a o s a e lis ed in Table II.
TABLE II: Bina y ope a o s.
ope a o symbol
addi ion + +
subs ac ion - −
ou e p oduc ˆ∧
inne p oduc | ·
geome ic p oduc ∗
All exp essions a e implemen ed as exp ession empla es.
The exp ession empla es de e mine he esul ype o he
exp ession a compile ime. This is achie ed ia accompanying
e alua ion classes ha a e hidden in he de ail namespace.
The challenge in he implemen a ion he e is he ac ha he
esul ing mul i ec o s only a ely ha e he same blades as
he inpu ope ands. No e ha he exp essions a e e alua ed
in a lazy ashion, which means ha he blades a e e alua ed
on demand. This makes i possible o e.g. only e alua e a
single blade o he esul ing mul i ec o , depending on he
equi emen s.
empla e <class De i ed, class Resul >
class Exp ession
{
public:
empla e <in blade>
equi es(Resul ::has(blade))
ypename Resul ::V ype ge () cons
{
e u n s a ic_cas <cons De i ed &>(* his). empla e ge <blade>();
}
};
A i s example o his can be seen in he Sum exp ession in
Figu e 3. The co esponding ype e alua ion class cons uc s
he ype o he esul ing mul i ec o a compile- ime. In he
case o a summa ion his amoun s o a simple bi wise OR
ope a ion compa ing he bi se s o he inpu mul i ec o s.
The inne and ou e p oduc s wo k essen ially in he
same way and hus he co esponding exp ession bo h in-
he i om a base P oduc class, i.e. Inne P oduc and
Ou e P oduc . The P oduc class akes a class s uc u e
implemen ing he co esponding Cayley able as empla e
a gumen . This Cayley able de ines he esul ing blades o
a blade by blade mul iplica ion unde he inne and ou e
p oduc , espec i ely. Thus, in he case o CGA, i de ines
1024 ope a ions. In o de o de e mine he ype o he esul ing
mul i ec o , we employ old exp essions ha allow us o i e a e
o e he blades o bo h inpu muli ec o s a compile- ime. In
his loop, we ob ain he esul ing blade pe pai o blades
P1+P2=P3
+=
(a) The summa ion o wo muli ec o s wi h he same blades esul s
in ano he mul i ec o wi h he same blades.
P1+P2=P3
+=
(b) The summa ion o wo muli ec o s wi h he di e en blades
esul s in a mul i ec o wi h he blades o bo h inpu muli ec o s.
Fig. 3: Summa ion
using he espec i e Cayley able and hen assemble hem in o
he esul ing mul i ec o again using OR ope a ions. Figu e 4
shows an example o each he inne and he ou e p oduc .
C·S=X
·=
(a) The inne p oduc is a g ade lowe ing ope a ion, i.e. he esul ing
mul i ec o will be o lowe g ade han he inpu s. The example shows
ha he inne p oduc o a ci cle Cwi h a sphe e S esul s in a poin
P.
P P ∧e∞=L
∧=
(b) The ou e p oduc is a g ade aising ope a ion, i.e. he esul ing
mul i ec o will be o highe g ade han he inpu s. The example
shows ha he ou e p oduc o a poin pai P P and e∞ esul s in a
line L
Fig. 4: The esul ing mul i ec o o he inne and ou e p oduc
ope a ions has a di e en g ade han he inpu s.
The geome ic p oduc class Geome icP oduc also
inhe i s om he base P oduc class and comes wi h i s
own Cayley able. So implemen a ion wise i is he same as
he inne and ou e p oduc s. The main di e ence is ha wo
blades can esul om a blade mul iplica ion, which causes
he esul ing mul i ec o o po en ially ha e bo h a lowe and
a highe g ade han he inpu s, as can be seen in Figu e 5a.
Fu he mo e, we ha e p oduc s ha a e based on he geome ic
p oduc such as he sandwich p oduc , which is also ea ed
as a bina y exp ession and shown in Figu e 5b.
M P =X
=
(a) Using he geome ic p oduc esul s in bo h blades o lowe and
highe g ade.
M C
M=C′
=
(b) The sandwich p oduc is a g ade-p ese ing ope a ion. Nume ical
issues migh lead o esiduals in o he blades, which we a oid by
simply no e alua ing hem in he exp essions.
Fig. 5: The geome ic p oduc is a combina ion o he inne
and ou e p oduc .
C. Geome ic P imi i es
Since we know he subspaces o all he geome ic p imi-
i es, we chose o implemen hem in an objec -o ien ed way
by inhe i ing om he base Mul i ec o class. Hence,
he a ailable classes a e Vec o ,Di ec ionVec o ,
Tangen Vec o ,Poin ,Poin Pai ,Line,Ci cle,
Plane and Sphe e. Thei co esponding subspaces wi hin
he geome ic algeb a can be seen in Figu e 2. Ha ing he
geome ic p imi i es as explici classes allows he implemen-
a ion o commonly used equa ions as membe s unc ions,
which acili a es he usage. Fo example he cons uc o s o he
geome ic p imi i e classes implemen he a ious ways hey
can be de ined. The explici classes a e mean o acili a e he
use and cons uc ion, bu o cou se, using compu a ion wi h
base mul i ec o s is also possible. This p ese es he p ope y
o co a ian compu a ion wi hin he algeb a.
D. Rigid Body T ans o ma ions
The igid body ans o ma ions ha a e cu en ly a ail-
able a e implemen ed in he classes Ro o ,T ansla o ,
Mo o and Dila o . They all inhe i om he base
class Ve so . Since all h ee classes a e exponen ial map-
pings o bi ec o s hey a e accompanied by he exp essions
*Loga i hm and *Exponen ial, espec i ely. The main
me hod o he igid body ans o ma ions is apply, which
implemen s he sandwich p oduc X′=M X
Mand ensu es
ype sa e y. While Xcan echnically be any mul i ec o , he
in ended usage is wi h he geome ic p imi i es ha we e
p esen ed in Sec ion III-C. Hence, in his con ex , ype sa e y
means ha X′s ays he same geome ic p imi i e as X, e.g. a
Poin s ays a Poin . This ensu es ha he exp ession only
e alua es blades ha a e pa o he geome ic p imi i e, which
no only educes he numbe o loa ing poin ope a ions, i
also deals wi h nume ical imp ecisions in he compu a ion ha
a e known o occu in geome ic algeb a implemen a ions.
IV. ROBOT MODELING
The p e ious sec ions in oduced he ea u es ela ed o he
unde lying geome ic algeb a implemen a ion o ga o. This
sec ion will now in oduce he highe le el ea u es o he
lib a y ela ed o obo modeling, ha dis inguish i om o he
geome ic algeb a lib a ies.
The main aspec s o obo modeling a e he compu a ion
o he kinema ics and dynamics o obo ic sys ems. While
he o wa d kinema ics and o wa d/in e se dynamics can
be compu ed using e icien ecu si e algo i hms, he in e se
kinema ics p oblem o edundan manipula o s is an op imi-
sa ion p oblem. Hence, he obo modeling sec ion also co e s
he necessa y ools o sol ing such op imisa ion p oblems.
The base class o he modeling o obo ic sys ems is
called Sys em. I implemen s he main unc ionali y o
he compu a ion o he kinema ics and dynamics o obo ic
sys ems and s o es he join s and links. Those a e o ganized
in he classes Join ,FixedJoin ,Re olu eJoin ,
P isma icJoin and Link. A cus om obo sys em can
hen be c ea ed using he unc ions addJoin and addLink.
No e ha hose unc ions only add he links and join s o
he sys em, he p ope ela ionships be ween hem need o
be speci ied indi idually by se ing he espec i e child/pa en
links/join s. Usually, obo ic sys ems consis o one o mo e
kinema ic chains o in e es s, e.g. he chain om he base link
o he end-e ec o ame o se ial manipula o s. This concep
is implemen ed in he Kinema icChain helpe class and
can be added o a Sys em ia he addKinema icChain
unc ion.
The o wa d and in e se dynamics a e implemen ed in he
membe unc ions compu eJoin Accele a ions and
compu eJoin To ques, espec i ely. These unc ions a e
using he con o mal geome ic algeb a e sions o he Recu -
si e New on-Eule Algo i hm (RNEA) and A icula ed Body
Algo i hm (ABA).
The Manipula o class ha implemen s a se ial ma-
nipula o wi h Ndeg ees o eedom is a specializa ion o
he base Sys em class. The pa ame e Nis gi en as a
empla e a gumen , which makes i possible o use ixed-
size Eigen::Ma ix ypes o he implemen a ion o e.g.
he join con igu a ion ec o s and Jacobian ma ices. The
Manipula o class implemen s unc ions ha a e mos
commonly o in e es when dealing wi h such ypes o sys-
ems. Mo e speci ically, hese a e unc ions dealing wi h he
end-e ec o o wa d kinema ics and Jacobian ma ices. Hence,
hese unc ions a e
•ge EEMo o
•ge EEAnaly icJacobian
•ge EEGeome icJacobian
•ge EEF ameJacobian
These unc ions use he a o emen ioned Kinema icChain
class in o de o de ine he end-e ec o kinema ic chain
and p ecompu e ce ain ela ed quan i ies. Hence, loading a
Manipula o om a ile addionally equi es he name o
he end-e ec o join .
The ga o lib a y includes se e al p ede ined se ial manip-
ula o s ha inhe i om he Manipula o class, hose a e
implemen ed in he ollowing classes:
•F ankaEmikaRobo
•UR5
These classes a e loading he de ini ion om yaml iles and
a e only a ailable i yaml-cpp is ins alled on he sys em. The
ROS package con ains a p og am ha con e s URDF iles o
he yaml o ma ha is being used by his lib a y.
V. COMPARISON TO OTHER LIBRARIES
The e ha e been a ious wo ks ha published implemen a-
ions o geome ic algeb a. These lib a ies all ha e in common
ha hey a e mean o be gene ic geome ic algeb a imple-
men a ions ocusing on he compu a ional and ma hema ical
aspec s o he algeb a i sel . In con as o ha , ou implemen-
a ion is a ge ed speci ically a obo ics applica ions and hus
no only implemen s he low-le el algeb aic compu a ions bu
also ea u es he compu a ion o he kinema ics and dynamics
o se ial manipula o s as well as gene ic cos unc ions o
op imal con ol.
A. Algeb aic Ope a ions Benchma ks
In o de o compa e he pe o mance o ou ga o li-
b a y o o he geome ic algeb a lib a ies we o ked he
ga_benchma k5 eposi o y in o de o in eg a e ga o. Ou
o k can be ound a h ps://gi hub.com/loew /
ga-benchma k.
Dualiza ion
012345
0
5.126
10.25
15.38
20.5
25.63
30.76
35.88
41.01
46.14
51.26
G ade
Time [ns]
Re e se
012345
0
10.97
21.94
32.91
43.87
54.84
65.81
76.78
87.75
98.72
109.7
G ade
In e sion
012345
0
18.62
37.25
55.87
74.49
93.11
111.7
130.4
149
167.6
186.2
G ade
GATL
Ga amon
Gaale
Ve so
ga o
(a) Benchma ks o una y algeb aic ope a ions.
Addi ion
0 6 12 18 24 30 36
0
10.65
21.31
31.96
42.62
53.27
63.93
74.58
85.24
95.89
106.5
G ade
Time [ns]
Inne P oduc
0 6 12 18 24 30 36
0
7.832
15.66
23.5
31.33
39.16
46.99
54.82
62.65
70.49
78.32
G ade
Ou e P oduc
0 6 12 18 24 30 36
0
7.286
14.57
21.86
29.14
36.43
43.71
51
58.29
65.57
72.86
G ade
Geome ic P oduc
0 6 12 18 24 30 36
0
43.55
87.09
130.6
174.2
217.7
261.3
304.8
348.4
391.9
435.5
G ade
GATL
Ga amon
Gaale
Ve so
ga o
(b) Benchma ks o bina y algeb aic ope a ions.
Fig. 6: Benchma ks o di e en geome ic algeb a lib a ies. All
ope a ions a e compu ed using con o mal geome ic algeb a.
We omi ed TbGAL om he plo s o he benchma k esul s,
since i is by a he slowes lib a y. The benchma ks show
ha ga o can compe e in e ms o pe o mance wi h GATL
and Ve so , which we e p e iously epo ed o be he as es
GA lib a ies. We belie e, howe e , ha he API o ga o
is a lo mo e app oachable and easy o use. Fu he mo e,
i addi ionally implemen s specialized algo i hms o obo
modeling and op imisa ion p oblems.
5h ps://gi hub.com/ga-de elope s/ga-benchma k
TABLE III: Compa ison o di e en lib a ies.
(a) O e iew o o he geome ic algeb a lib a ies.
Ga amon [2] a gene a o o C++ lib a ies dedica ed o geome ic
algeb a
GATL [3] C++ lib a y o Euclidean, homogeneous/p ojec i e,
Minkowski/space ime, con o mal, and a bi a y geo-
me ic algeb as using empla e me a-p og amming
Ve so [4] ( as ) gene ic C++ lib a y o geome ic algeb as
GAL [5] C++17 exp ession compile and engine o compu ing
wi h geome ic algeb a
Gaigen [6] code gene a o o geome ic algeb a
Gaale [7] C++ lib a y o e alua ion o geome ic algeb a ex-
p essions o e ing com o able implemen a ion and
easonable speed by using exp ession empla es and
me ap og amming echniques
Gaalop [8] so wa e o op imize geome ic algeb a iles
TbGAL [9] C++/Py hon lib a y o Euclidean, homogeneous/p o-
jec i e, Minkowski/space ime, con o mal, and a bi-
a y geome ic algeb as ep esen ing blades (and e -
so s) in hei decomposed s a e o scale o scale high
dimensions
(b) O e iew o o he lib a ies o obo modeling.
DQ Robo ics [10] lib a y o obo modeling and con ol based on
dual qua e nion algeb a
Pinocchio [11] s a e-o - he-a igid body algo i hms o poly-
a icula ed sys ems
Raisim [12] mul i-body physics engine o obo ics and AI
KDL [13] applica ion independen amewo k o modeling
and compu a ion o kinema ic chains
Mujoco [14] physics engine o model-based op imisa ion
RBDL [15] highly e icien code o bo h o wa d and in e se
dynamics o kinema ic chains and b anched
models
B. Robo ics Algo i hms Benchma ks
Since his lib a y implemen s obo kinema ics and dynam-
ics algo i hms, we a e compa ing and benchma king ga o
agains se e al lib a ies ha a e commonly used in obo ics
applica ions. This eposi o y ac ually uses a gi pipeline in
o de o con inuously pull he la es changes o he lib a ies
and upda e he benchma k esul s. The cu en benchma king
esul s on ou sys em can be ound in Figu e 7. As can be seen,
ga o is e y compe i i e when i comes o he compu a ion
o he kinema ics o a obo ic sys em. The compu a ion o
he dynamics, howe e , especially he o wa d dynamics, is
s ill slowe a his poin . The eason o his is an ou da ed
implemen a ion choice in he lowe le els o he lib a y. This
issue will be add essed and ixed in a u u e ealease o ga o,
which should make he compu a ion o he dynamics also
compe i i e compa ed o he es ablished lib a ies.
VI. APPLICATIONS AND TUTORIAL
In his sec ion we p o ide some example applica ions o
how he lib a y can be used. Fo ha pu pose, we p o ide

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(d) Fo wa d Dynamics.
Fig. 7: Benchma ks o obo ics algo i hms. The benchma k
was un on an AMD Ryzen 7 4800U CPU. All lib a ies
we e compiled using gcc 13.1.1 wi h he compile lags -O3
-ma ch=na i e. The e e ence sys em is he F anka Emika
Robo .
an accompanying eposi o y ga o_examples ha explains he
usage o ga o,pyga o and ga o_ os. No e ha in he ex we
a e always e e encing he c++ iles, bu he same examples
can also be ound in py hon in he co esponding olde . These
examples a e using he same naming scheme.
A. Geome ic Algeb a
Since many po en ial use s o ga o a e likely o be un a-
milia wi h he concep o geome ic algeb a we a e p o iding
some examples on how o do compu a ions using his algeb a.
1) Mul i ec o Ope a ions: In a i s example we a e show-
ing how o c ea e di e en gene al mul i ec o s and use hem
o algeb aic compu a ions. This example can be ound in
mul i ec o .cpp. In pa icula , he e we show he usage o he
di e en p oduc s in geome ic algeb a and how hey a e used
wi h hei co esponding ope a o s ha we e lis ed in Table II.
2) Geome ic P imi i es: An example on how o use he
lib a y o cons uc geome ic p imi i es is shown in geo-
me ic_p imi i es.cpp. Since hese geome ic p imi i es a e
essen ially specializa ions o he base Mul i ec o class
all ope a ions ha we e p esen ed in he example in Sec ion
VI-A1 a e also alid o he geome ic p imi i es. Hence, we
a e showing he cons uc ion o he geome ic p imi i es in
di e en ways. Fi s , using hei algeb aic de ini ion based
on he ou e p oduc and second, using hei cons uc o s.
Essen ially, he cons uc o s implemen he algeb aic de ini-
ion, and se e o simpli y he usage o he lib a y. Bu he
example should show he equi alence o he cons uc ion.
Since geome ic algeb a acili a es geome ic easoning, we a e
also p o iding an example o how o p ojec ions, e lec ions
and in e sec ions using he geome ic p imi i es, which can be
ound in geome ic_p imi i es_incidence.cpp.
3) Con o mal T ans o ma ions: This pa o he u o ial
is dedica ed o showing he usage o he di e en Ve so
classes, i.e. i shows how o apply con o mal ans o ma-
ions o geome ic p imi i es. The e a e ac ually wo pa s
o his example. The i s one, which is implemen ed in
e so _log_exp_map.cpp, shows how de ine he di e en e -
so s using he exponen ial mappings and eco e he gen-
e a o bi ec o s using he co esponding loga i hmic map-
pings. In he second example, which is implemen ed in e -
so s_ ans o ming_geome ic_p imi i es.cpp, we hen show
how o apply he e so s o he geome ic p imi i es.
B. Robo Modeling
One o he a ge ed use cases o he ga o lib a y is he
modeling o obo ic sys ems. In his pa o he u o ial we
will show how o ha in p ac ice.
1) Gene ic Sys em: A gene ic sys em can be c ea ed using
he Sys em class and i s addJoin and addLink unc ions
as men ioned in Sec ion IV. No e ha in his case he pa en-
/child ela ionships be ween he join s/links need o handled
as well. An easie way is o load he sys em desc ip ion om
a yaml ile, whe e he lib a y will d op an e o message
i no all necessa y ela ionships a e de ined. These wo
ways o c ea ing a sys em a e shown in he examples c e-
a e_sys em_ om_code.cpp and c ea e_sys em_ om_ ile.cpp.
2) Manipula o Sys em: Manipul o sys ems a e specialized
obo ic sys ems, which can be seen om he Manipula o
class inhe i ing om he Sys em class. So in o de o
de ined a manipula o , a no mal sys em can be c ea ed and
hen mo ed o he manipula o sys em wi h he addi ional
in o ma ion abou he end-e ec o join in o de o c e-
a e he kinema ic chain. We a e again p o iding he wo
examples o c ea ing a manipula o om code o om a
ile in examples c ea e_manipula o _ om_code.cpp and c e-
a e_manipula o _ om_ ile.cpp, espec i ely.
3) Robo Kinema ics/Dynamics: A e c ea ing he model
o he sys em and in o de o use he lib a y in p ac ical
applica ions, he compu a ion o he kinema ics and dynamics
o he sys ems a e necessa y. We show how he compu e all
ele an quan i ies using he F ankaEmikaRobo class in
he example anka_emika_kinema ics_and_dynamics.cpp.
4) Robo Kinema ics and Geome ic P imi i es: In con-
as o o he lib a ies, ga o uses geome ic algeb a o
he modeling o obo ic sys ems, which allows he usage
o a ious geome ic p imi i es a a kinema ic le el. In
anka_emika_geome ic_p im i es.cpp we show an example
using he F ankaEmikaRobo o how o mo e hose ge-
ome ic p imi i es o he end-e ec o , which will la e be
exploi ed in he o mula ion o he op imisa ion p oblems.
C. Op imisa ion P oblems
Many p oblems in impo an domains o obo ics, such as
lea ning and con ol, can be cas as op imiza ion p oblems.
Hence, in his sec ion we a e p o iding some examples on how
ga o can be used o simpli y he modeling o op imisa ion
p oblems using geome ic algeb a.
1) In e se Kinema ics: A common example in obo ics
is he compu a ion o he in e se kinema ics. Since we a e
o en dealing wi h edundan manipula o s, his becomes an
op imisa ion p oblem. We a e modeling his p oblem using he
Mo o class o ga o, which ep esen s poses in Euclidean
space. An in e se kinema ics p oblem in geome ic algeb a,
o mula ed as an op imiza ion p oblem, hus minimizes he
e o be ween wo mo o s, which is exp essed ia he log-
a i hmic map. The co esponding class ha implemen s his
cos unc ion is called SingleManipula o Mo o Cos .
I compu es he alue o he cos unc ion, as well as i s
g adien and hessian, o allow o i s - o second-o de
op imisa ion. The example o compu ing he in e se kine-
ma ics using he Gauss-New on algo i hm can be ound in
in e se_kinema ics.cpp.
2) Reaching Geome ic P imi i es: Geome ic algeb a ex-
ends he cos unc ion o be uni o mly applicable ac oss
he di e en geome ic p imi i es. I is implemen ed in he
class SingleManipula o Ta ge . The empla e a gu-
men s Tool and Ta ge can be di e en combina ions o
geome ic p imi i es. He e, we cas he op imisa ion p oblem
again as an in e se kinema ics p oblem o simplici y, so we
a e op imizing o he join angle con igu a ion in which he
end-e ec o eaches a ce ain geome ic p imi i e. In p e ious
wo k, howe e , we ha e shown he applica ion o CGA o
modeling manipula ion asks in an op imal con ol amewo k
o model p edic i e con ol [16], which can o cou se be
achie ed using he same cos unc ion. The di e en examples
a e lis ed in Table IV.
P imi i es File
Poin &Poin in e se_kinema ics_poin _poin .cpp
Poin &Line in e se_kinema ics_poin _line.cpp
Poin &Poin Pai in e se_kinema ics_poin _poin pai .cpp
Poin &Ci cle in e se_kinema ics_poin _ci cle.cpp
Poin &Plane in e se_kinema ics_poin _plane.cpp
Poin &Sphe e in e se_kinema ics_poin _sphe e.cpp
Line &Poin in e se_kinema ics_line_poin .cpp
Line &Line in e se_kinema ics_line_line.cpp
TABLE IV: Example o op imisa ion p oblems using di e en
geome ic p imi i es. These examples a e loca ed in he olde
ga o_examples/s c/cpp/. No e ha wi h he excep ion o he
de ini ion o he cos unc ion, all hese iles a e iden ical.
D. ROS Visualiza ion
The main pu pose o he ga o_ os package is isualiza ion
o he geome ic p imi i es in R iz. Hence, we a e p o iding
an example ha isualizes he F anka Emika obo eaching
a ious geome ic p imi es. The example is implemen ed
isualizing_geome ic_p imi i es.cpp and equi es o be com-
piled in a ROS wo kspace. The R iz ou pu o he example is
shown in Figu e 8.
VII. CONCLUSION
In his a icle we p esen ed he implemen a ion de ails as
well as a u o ial o ou so wa e s ack a ound ga o, which
is a c++ lib a y ha implemen s con o mal geome ic algeb a
Poin
acking
Plane
acking
Op ion
Poin 1
Poin ing
Line
acking
Ci cle
acking
Op ion
Poin 2
Cons ain s
Fig. 8: R iz isualiza ions o he F anka Emika eaching
a ious geome ic p imi es.
o obo ics. The so wa e s ack also includes py hon bindings
in pyga o as well as a ROS package in ga o_ os. Tu o ial
ma e ial and oy examples can be ound in ga o_examples.
While showing compa able pe o mance o he obo mod-
eling, geome ic algeb a also o e s an easy and in ui i e way
o model a ious geome ic ela ionships. Hence i p o ides a
iche oolse han s anda d ma ix algeb a wi hou loosing any
o he exis ing ools. Ou lib a y ga o p o ides hese s anda d
ools o obo modeling and augmen s hem wi h concep s
ha a e exclusi e o geome ic algeb a. P o iding his lib a y
ha makes geome ic algeb a easily accessible o obo ics
esea ch should allow o a wide adop ion and acili a e he
esea ch on using his powe ul amewo k o obo ics.
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