# Permutability degrees of finite groups

D.E. Otera† and  F.G. Russo††

###### Abstract.

Given a finite group $G$, we introduce the permutability degree of $G$, as

$pd(G)=\frac{1}{|G|\
|\mathcal{L}(G)|}{\underset{X\in\mathcal{L}(G)}{\sum}}|P_{G}(X)|,$

where $\mathcal{L}(G)$ is the subgroup lattice of $G$ and $P_{G}(X)$ the
permutizer of the subgroup $X$ in $G$, that is, the subgroup generated by all
cyclic subgroups of $G$ that permute with $X\in\mathcal{L}(G)$. The number
$pd(G)$ allows us to find some structural restrictions on $G$. Successively,
we investigate the relations between $pd(G)$, the probability of commuting
subgroups $sd(G)$ of $G$ and the probability of commuting elements $d(G)$ of
$G$. Proving some inequalities between $pd(G)$, $sd(G)$ and $d(G)$, we
correlate these notions.

###### Key words and phrases:

Subgroup commutativity degree; commutativity degree; permutability degree;
dihedral groups; inequalities

###### 2010 Mathematics Subject Classification:

Primary 20P05; Secondary 06B23, 20D60

† Institute of Mathematics and Informatics, Vilnius University, Vilnius,
Lithuania; email<EMAIL_ADDRESS>
†† Department of Mathematics and Applied Mathematics, University of Cape Town,
Cape Town, South Africa; email<EMAIL_ADDRESS>
##  1\. Introduction

All the groups of the present paper are supposed to be finite. Given a group
$G$ and its subgroup lattice $\mathcal{L}(G)$, the subgroup commutativity
degree

$sd(G)=\frac{|\\{(H,K)\in\mathcal{L}(G)\times\mathcal{L}(G)\ |\
HK=KH\\}|}{|\mathcal{L}(G)|^{2}}$

of $G$ and the commutativity degree

$d(G)=\frac{|\\{(x,y)\in G\times G\ |\ xy=yx\\}|}{|G|^{2}}$

of $G$ have been largely studied in the last years. Fundamental properties and
interesting generalizations of $sd(G)$ can be found in [11, 12, 13, 22, 27,
28, 29], and for $d(G)$ in [1, 2, 6, 8, 9, 10, 14, 17, 18, 19, 20, 24]. To
study these notions, various perspectives have been considered in literature,
because both measure theory and combinatorial techniques may be applied in
order to get restrictions on the structure of a group.

The present paper investigates a similar concept, the $permutability$ $degree$
of $G$

$pd(G)=\frac{1}{|G|\
|\mathcal{L}(G)|}{\underset{X\in\mathcal{L}(G)}{\sum}}|P_{G}(X)|$

and its connections with $sd(G)$ and $d(G)$. In the previous formula, the
$permutizer$ $P_{G}(X)$ of a subgroup $X$ of $G$ is defined to be the subgroup
generated by all cyclic subgroups of $G$ that permute with $X$, that is,
$P_{G}(X)=\langle g\in G\ |\ \langle g\rangle X=X\langle g\rangle\rangle.$
This means that $X\in\mathcal{L}(P_{G}(X))$ and $X\not=P_{G}(X)$ if and only
if $X\langle g\rangle=\langle g\rangle X$ for some $g\in G-X$.

We concentrate on permutizers because several classifications are available in
literature on this topic. Recall that a group $G$ such that $X\not=P_{G}(X)$
for every proper subgroup $X$ of $G$ is said to satisfy the $permutizer$
$condition$ $\mathbf{P}$, or briefly $\mathbf{P}$–$group$. Therefore the
permutizer condition generalizes the well–known $normalizer$ $condition$ (see
[26]) and gives information on how the group is near to be supersolvable. The
study of permutizers is not new and it is based on a series of fundamental
contributions [3, 21, 23, 30] in the last 20 years. From [3, Corollary 2], we
know that for groups of odd order the permutizer condition is equivalent of
being supersolvable and actually, a complete classification of
$\mathbf{P}$–groups can be found in [3].

Now we may define the subgroup

$P(G)={\underset{H\in\mathcal{L}(G)}{\bigcap}P_{G}(H)}$

and correlate it with other subgroups of $G$. For instance, it is easy to
check that the norm $N(G)$ of $G$ (see [26] for the properties of $N(G)$)
satisfies the following relation

$Z(G)={\underset{x\in G}{\bigcap}C_{G}(x)}\subseteq
N(G)={\underset{H\in\mathcal{L}(G)}{\bigcap}N_{G}(H)}\subseteq{\underset{H\in\mathcal{L}(G)}{\bigcap}P_{G}(H)}=P(G).$

This relation emphasizes how $P(G)$ is connected with other subgroups, widely
investigated in literature, such as intersections of normalizers or of
centralizers. Note that for any $X\in\mathcal{L}(P(G))$ one has $P_{G}(X)=G$.

The subgroup $P(G)$ is also important because it allows us to “manipulate” the
expression of $pd(G)$, for getting some analogies with

$sd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\
{\underset{H\in\mathcal{L}(G)}{\sum}}|\mathcal{C}_{\mathcal{L}(G)}(H)|\ \
\mathrm{and}\ \ d(G)=\frac{1}{|G|^{2}}\sum_{x\in G}|C_{G}(x)|,$

where $P_{G}(X)$ is the natural substitute of
$\mathcal{C}_{\mathcal{L}(G)}(X)=\\{Y\in\mathcal{L}(G)\ |\ YX=XY\\}$ in [22,
27] and of $C_{G}(x)=\\{y\in G\ |\ xy=yx\\}$ in [2, 8]. This manipulation of
the expression of $pd(G)$ will allow us to detect whether $P(G)$ is cyclic or
not by looking only at the size of $pd(G)$.

##  2\. Basic properties and terminology

Some of the following observations will be useful later on.

######  Remark 2.1.

Since $P_{G}(X)$ is a subgroup of $G$, for all $X\in\mathcal{L}(G)$, and it
always contains the trivial subgroup, then $|P_{G}(X)|\leq|G|$ and also

$0<{\underset{X\in\mathcal{L}(G)}{\sum}}|P_{G}(X)|\leq|\mathcal{L}(G)|\ |G|$

so that $pd(G)\in]0,1]$.

######  Remark 2.2.

A group $G$ has $pd(G)=1$ if and only if the sum of all $|P_{G}(X)|$ for
$X\in\mathcal{L}(G)$ is equal to $|G||\mathcal{L}(G)|$. By default, a
quasihamiltonian group $G$, that is, a group in which every subgroup is
permutable, has $pd(G)=1$. A classification of quasihamiltonian groups can be
found in [26, Theorems 2.4.11 and 2.4.16] and, roughly speaking, these groups
are direct products of abelian groups by a copy of the quaternion group of
order 8. In particular, abelian groups have permutability degree equal to 1.

Another case in which the permutability degree reaches 1 is the following.

######  Remark 2.3.

A $\mathbf{P}$–group $G$ in which all proper subgroups are maximal has
$pd(G)=1$. In such a case for all proper subgroups $X$ of $G$ one has
$X\subset P_{G}(X)=G$ and so $pd(G)=1$. One might be tempted to think that all
$\mathbf{P}$–groups have permutability degree equal to 1, but Example 3.2
below shows this is false, and then the additional condition “in which all
proper subgroups are maximal” cannot be omitted.

Now we rewrite the original expression of permutability degree in the
following more useful form. Since $\mathcal{L}(P(G))$ is a sublattice of
$\mathcal{L}(G)$, it turns out that

(2.1) $pd(G)=\frac{1}{|G||\mathcal{L}(G)|}\
\left({\underset{X\in\mathcal{L}(P(G))}{\sum}}|P_{G}(X)|+{\underset{X\in\mathcal{L}(G)-\mathcal{L}(P(G))}{\sum}}|P_{G}(X)|\right).$

Note also that a cyclic group $C$ (or better a quasihamiltonian group $Q$) has
$sd(C)=pd(C)=d(C)=1$ (or better $sd(Q)=pd(Q)=1$). Therefore, relations between
$sd(G)$ and $pd(G)$ are meaningful when $G$ is noncylic and
nonquasihamiltonian (see Remark 2.2).

For the sake of completeness, we recall some results of Beidleman and Heineken
in [4]. The $quasicenter$ $Q(G)$ of $G$ is the subgroup of $G$ generated by
all elements $g\in G$ such that $\langle g\rangle K=K\langle g\rangle$, where
$K$ is an arbitrary subgroup of $G$. The subgroup $Q(G)$ was introduced by
Mukherjee and studied by several authors in the last years (see [4, 5, 25]),
who investigated chains of quasicenters and relations with supersolvable
groups. On the other hand, the $hyperquasicenter$ of $G$, denoted by
$Q_{\infty}(G)$, is the largest term of the chain $1=Q_{0}(G)\leq
Q_{1}(G)=Q(G)\leq\ldots\leq Q_{i}(G)\leq Q_{i+1}(G)\leq\ldots$ of normal
subgroups of $G$, where, for any $i\geq 0$,
$Q_{i+1}(G)/Q_{i}(G)=Q(G/Q_{i}(G))$ and $Q_{\infty}(G)={\underset{i\geq
0}{\bigcup}}Q_{i}(G).$

Recall that a normal subgroup $N$ of $G$ is said to be $hypercyclically$
$embedded$ in $G$ if it contains a $G$–invariant series whose factors are
cyclic. It is easy to see that $G$ contains a unique largest hypercyclically
embedded subgroup, which we denote $\Sigma(G)$. More precisely, [4, Theorem 1]
shows that $\Sigma(G)=Q_{\infty}(G)$ is true for any group $G$. Some
interesting connections hold between $\mathbf{P}$–groups, $P(G)$ and
$Q_{\infty}(G)$. For instance, [3, (3.1), p. 697] shows that a group $G$ is a
$\mathbf{P}$–group if and only if $G/\Sigma(G)$ is a $\mathbf{P}$–group. As a
first consequence, a group $G$ is a $\mathbf{P}$–group if and only if
$G/Q_{\infty}(G)$ is a $\mathbf{P}$–group. As a second consequence,
$Z(G)\subseteq Q(G)\subseteq P(G)$ is true for any group $G$. Furthermore,
$Q_{\infty}(G)=P(G)$ if and only if $P(G)=\Sigma(G)$.

##  3\. Examples

Now we specify some of the previous notions for the symmetric group $S_{3}$ on
3 objects. This will help us to visualize analogies and differences between
permutability degrees, subgroup commutativity degrees and commutativity
degrees.

######  Example 3.1.

The smallest nonabelian group $S_{3}$ has
$\mathcal{L}(S_{3})=\\{\\{1\\},S_{3},A_{3},H,K,L\\}$, where
$A_{3}=\langle(123)\rangle=\langle a\rangle$, $H=\langle(12)\rangle=\langle
h\rangle$, $K=\langle(13)\rangle=\langle k\rangle$,
$L=\langle(23)\rangle=\langle l\rangle$. Noting that $HK\not=KH$, $HL\not=LH$,
$KL\not=LK$, one has

$P_{S_{3}}(\\{1\\})=P_{S_{3}}(A_{3})=P_{S_{3}}(S_{3})=S_{3};$ $\ \
P_{S_{3}}(H)=P_{S_{3}}(K)=P_{S_{3}}(L)=S_{3}\ ;\\\ P(S_{3})=S_{3}.$

The fact that $P_{S_{3}}(\\{1\\})=P_{S_{3}}(A_{3})=P_{S_{3}}(S_{3})=S_{3}$ is
clear, since we are dealing with permutizers of normal subgroups. On the other
hand, $A_{3}=\langle a\rangle\subseteq P_{S_{3}}(H)$ by definition. Now,
$H\subseteq P_{S_{3}}(H)$ is obvious. Then $HA_{3}$, which is a subgroup of
$S_{3}$, should be contained in $P_{S_{3}}(H)$ and $|HA_{3}|=\frac{|H|\
|A_{3}|}{|H\cap A_{3}|}=6$. This forces $P_{S_{3}}(H)$ to be equal to $S_{3}.$
Similarly we get $P_{S_{3}}(K)=P_{S_{3}}(L)=S_{3}$.

$S_{3}$$A_{3}$$\\{1\\}$$H$$K$$L$Fig.III.1.Hasse diagram of
$\mathcal{L}(S_{3})$.

It is interesting to note that an example as easy as this has a lot of
properties in our perspective of study. The group $S_{3}$ is supersovable by
looking at the series $\\{1\\}\triangleleft A_{3}\triangleleft S_{3}$, but is
not quasihamiltonian, due to $HK\not=KH$. At the same time, $S_{3}$ does not
satisfy the normalizer condition, since it is not nilpotent. Moreover, $S_{3}$
has $Z(S_{3})=\\{1\\}$, $\Sigma(S_{3})=Q_{2}(S_{3})=Q_{\infty}(S_{3})=S_{3}$,
$Q(S_{3})=A_{3}$ and it is a $\mathbf{P}$–group, since
$S_{3}/\Sigma(S_{3})=\\{1\\}$ is obviously a $\mathbf{P}$–group.

A direct calculation shows that

$6\cdot 6\
pd(S_{3})={\underset{X\in\mathcal{L}(S_{3})}{\sum}}|P_{S_{3}}(X)|=|P_{S_{3}}(H)|+|P_{S_{3}}(K)|$
$+|P_{S_{3}}(L)|+|P_{S_{3}}(A_{3})|+|P_{S_{3}}(S_{3})|+|P_{S_{3}}(\\{1\\})|=36,$

then $pd(S_{3})=1>sd(S_{3})=\frac{5}{6}>\frac{1}{2}=d(S_{3})$, agreeing with
the computations in [27, p.2510] and [8, 9].

Another easy (but interesting) example is the following.

######  Example 3.2.

The dihedral group of order 8 is $D_{8}=\langle a,b\ |\
a^{2}=b^{4}=1,a^{-1}ba=b^{-1}\rangle$ and has

$\mathcal{L}(D_{8})=\\{\\{1\\},\langle b\rangle,\langle b^{2}\rangle,\langle
a\rangle,\langle ba\rangle,\langle b^{2}a\rangle,\langle
b^{3}a\rangle,\\{1,b^{2},a,b^{2}a\\},\\{1,b^{2},ba,b^{3}a\\},D_{8}\\}.$

The normal subgroups are $D_{8}$, $\\{1\\}$, $B=\langle b\rangle$,
$Z(D_{8})=\langle b^{2}\rangle$, $M_{1}=\\{1,b^{2},a,b^{2}a\\}$ and
$M_{2}=\\{1,b^{2},ba,b^{3}a\\}$. Notice that $H=\langle b^{2}a\rangle$ and
$K=\langle a\rangle$ are contained in $M_{1}$, while $U=\langle ba\rangle$ and
$V=\langle b^{3}a\rangle$ in $M_{2}$.

$D_{8}$$M_{1}$$Z(D_{8})$$B$$H$$K$$U$$V$$M_{2}$$\\{1\\}$Fig.III.2.Hasse diagram
of $\mathcal{L}(D_{8})$.

Moreover

$8=|D_{8}|=|P_{D_{8}}(\\{1\\})|=|P_{D_{8}}(D_{8})|=|P_{D_{8}}(\langle
b\rangle)|=|P_{D_{8}}(Z(D_{8}))|$ $=|P_{D_{8}}(M_{1})|=|P_{D_{8}}(M_{2})|,$
$4=|M_{1}|=|P_{D_{8}}(H)|=|P_{D_{8}}(K)|,\ \
4=|M_{2}|=|P_{D_{8}}(V)|=|P_{D_{8}}(U)|$

and $Q(D_{8})=P(D_{8})=Z(D_{8})<\Sigma(D_{8})=Q_{\infty}(D_{8})=D_{8}.$ In
fact, $D_{8}=\Sigma(D_{8})$ is supersolvable and it is also a
$\mathbf{P}$–group, but nevertheless its permutability degree is different
from 1, because

$8\cdot 10\
pd(D_{8})={\underset{X\in\mathcal{L}(D_{8})}{\sum}}|P_{D_{8}}(X)|=6\cdot|D_{8}|+2\cdot|\\{1,b^{2},a,b^{2}a\\}|+2\cdot|\\{1,b^{2},ba,b^{3}a\\}|=64.$

More precisely,

$d(D_{8})=\frac{5}{8}<pd(D_{8})=\frac{64}{80}=\frac{4}{5}<sd(D_{8})=\frac{46}{55}.$

The value of $d(D_{8})$ can be found in [8] and that of $sd(D_{8})$ in [27].
This example shows that there exist $\mathbf{P}$–groups with permutability
degree different from 1. Note that $D_{8}$ satisfies
$8=|D_{8}|<|\mathcal{L}(D_{8})|=10$ but

$8=|\\{\\{1\\},D_{8},\langle
b\rangle,\\{1,b^{2},ba,b^{3}a\\},\\{1,b^{2},a,b^{2}a\\},\langle
b^{2}a\rangle,\langle b^{2}\rangle,\langle
a\rangle\\}|=|\mathcal{C}_{\mathcal{L}(D_{8})}(\langle a\rangle)|$
$\not\leq|\mathcal{Z}_{\mathcal{L}(D_{8})}(\langle
a\rangle)|=|\\{\\{1\\},\langle b^{2}a\rangle,\langle b^{2}\rangle,\langle
a\rangle\\}|=4.$

Examples 3.1 and 3.2 illustrate a series of problems for the computation of
the permutability degree, arising from the nature of the subgroup lattice of
the groups under consideration. We will come back to this point later on.

##  4\. General properties of the permutability degree

We note that [8, Theorems 2.5, 3.3] shows that the commutativity degree is
monotone. This is a well–known property, which is due to the fact that we are
dealing with a positive monotone measure of probability. Similar situations
can be found for $sd(G)$ in [27, Proposition 2.4, Corollaries 2.5, 2.6, 2.7,
Theorems 3.1.1, 3.1.5] and in [9, 14, 22]. For $pd(G)$ we have something
similar.

######  Theorem 4.1.

Let $H$ be a subgroup of a group $G$. Then

$\frac{|\mathcal{L}(H)|}{|\mathcal{L}(G)|\ |G:H|}\ pd(H)\leq\ pd(G).$

Moreover, if $|P_{G}(X):P_{H}(X)|\leq|G:H|$ for all $X\in\mathcal{L}(G)$,
$P(G)\leq H$ and $|\mathcal{L}(G)-\mathcal{L}(P(G))|\leq|\mathcal{L}(P(G))|$,
then $|\mathcal{L}(G)|\ pd(G)\leq 2|\mathcal{L}(H)|\ pd(H)$. In particular,

$\frac{|\mathcal{L}(H)|}{|\mathcal{L}(G)|\ |G:H|}\ pd(H)\leq\
pd(G)\leq\frac{2|\mathcal{L}(H)|}{|\mathcal{L}(G)|}pd(H).$

###### Proof.

We start by proving the first inequality. Since $|P_{H}(X)|\leq|P_{G}(X)|$ for
all $X\in\mathcal{L}(G)$, we have

$|G|\ |\mathcal{L}(G)|\ pd(G)=\sum_{X\in\mathcal{L}(G)}|P_{G}(X)|$
$\geq\sum_{X\in\mathcal{L}(G)}|P_{H}(X)|=\sum_{X\in\mathcal{L}(H)}|P_{H}(X)|=pd(H)\
|H|\ |\mathcal{L}(H)|$

and the result follows.

Now we prove the second inequality. Since by hypothesis,
$|P_{G}(X):P_{H}(X)|\leq|G:H|$, we have $|P_{G}(X)|\leq|G:H||P_{H}(X)|$ and so

$|G|\ |\mathcal{L}(G)|\
pd(G)={\underset{X\in\mathcal{L}(G)}{\sum}}|P_{G}(X)|\leq|G:H|{\underset{X\in\mathcal{L}(G)}{\sum}}|P_{H}(X)|$

but, from (2.1), this means

$=|G:H|\
\left({\underset{X\in\mathcal{L}(P(G))}{\sum}}|P_{H}(X)|+{\underset{X\in\mathcal{L}(G)-\mathcal{L}(P(G))}{\sum}}|P_{H}(X)|\right)$

and the inequality $|\mathcal{L}(G)-\mathcal{L}(P(G))|\leq|\mathcal{L}(P(G))|$
provided by hypothesis, implies

$\leq|G:H|\
\left({\underset{X\in\mathcal{L}(P(G))}{\sum}}|P_{H}(X)|+{\underset{X\in\mathcal{L}(P(G))}{\sum}}|P_{H}(X)|\right)=2|G:H|{\underset{X\in\mathcal{L}(P(G))}{\sum}}|P_{H}(X)|$
$\leq 2|G:H|{\underset{X\in\mathcal{L}(H)}{\sum}}|P_{H}(X)|=2\ |G:H|\ |H|\
|\mathcal{L}(H)|\ pd(H)$

from which we have $|\mathcal{L}(G)|\ pd(G)\leq 2|\mathcal{L}(H)|\ pd(H).$ ∎

A classic splitting result for the product probability of two independent
events is described by the following corollary. The proof may be generalized
to finitely many factors, whose orders are pairwise coprime.

######  Proposition 4.2.

Let $G$ and $H$ be two groups such that $\gcd(|G|,|H|)=1$. Then $pd(G\times
H)=pd(G)\ pd(H)$.

###### Proof.

Given three groups $A$, $B$ and $C$ such that $A\times B\subseteq C$, we know
that $N_{C}(A\times B)=N_{C}(A)\times N_{C}(B)$. This holds similarly for the
permutizers and it is easy to see that $P_{C}(A\times B)=P_{C}(A)\times
P_{C}(B).$ Now this fact and the assumption $\gcd(|G|,|H|)=1$ allow us to
conclude that

$\frac{1}{|G\times H|\ \ |\mathcal{L}(G\times H)|}{\underset{X\times
Y\in\mathcal{L}(G\times H)}{\sum}}|P_{G\times H}(X\times Y)|$ $=\frac{1}{|G|\
|\mathcal{L}(G)|}\ \frac{1}{|H|\ |\mathcal{L}(H)|}\
{\underset{X\in\mathcal{L}(G)}{\sum}}|P_{G}(X)|\
{\underset{Y\in\mathcal{L}(H)}{\sum}}|P_{H}(Y)|.$

∎

The underlying problem we deal with is the order of the subgroup lattices,
which is hard to predict in general. If we concentrate on some groups arising
from finite geometries, then the situation is more clear (dihedral groups,
semidihedral groups and generalized quaternion groups were studied in [2, 8,
11, 22, 27, 28, 29] from a similar perspective). Having in mind Examples 3.1
and 3.2, we observe from [26, pp. 26–29] and [22, 27] that the dihedral group

(4.1) $D_{2n}=\langle x,y\ |\
x^{2}=y^{n}=1,x^{-1}yx=y^{-1}\rangle=C_{2}\ltimes C_{n}=\langle
x\rangle\ltimes\langle y\rangle$

of symmetries of a regular polygon with $n\geq 1$ edges has order $2n$ and
splits in the semidirect product of a cyclic group $\langle y\rangle\simeq
C_{n}$ of order $n$ by a cyclic group $\langle x\rangle\simeq C_{2}$ of order
2 acting by inversion on $C_{n}$. In particular, $S_{3}\simeq D_{6}$ for $n=3$
and one can note that the Hasse diagram of
$\mathcal{L}(D_{6})=\mathcal{L}(S_{3})$ forms a diamond in which there are
only 4 atomic elements (see [26] for this terminology) in between $\\{1\\}$
and $D_{6}$ and their number can be easily computed. The following Fig.III.3
summarizes the information of Example 3.1 in a more general situation.

$D_{2p}$$C_{p}$$\\{1\\}$$H_{1}$$H_{2}$$H_{p}$…Fig.III.3.Hasse diagram of
$\mathcal{L}(D_{2p})$ with odd prime $p\geq 3$;$H_{1}\simeq
H_{2}\simeq\ldots\simeq H_{p}\simeq C_{2}$.

From Figs.III.1, III.2 and III.3, it is clear that $\mathcal{L}(D_{2p})$ has
$p+1$ proper subgroups, and this fact comes out from the following formula

(4.2) $|\mathcal{L}(D_{2n})|=\sigma(n)+\tau(n),$

where $\sigma(n)$ and $\tau(n)$ are the sum and the number of all divisors of
$n$ (here $n$ is arbitrary, not necessarily an odd prime), respectively.

In particular, if $n=p^{m}$ is a power of a prime $p$ (possibly $p=2$) for
some $m\geq 0$, then the set of all divisors of $p^{m}$ is
$\mathrm{Div}(p^{m})=\\{1,p,p^{2},\ldots,p^{m}\\}$ so that

(4.3) $\sigma(p^{m})=\sum^{m}_{j=0}p^{j}=\frac{1-p^{m+1}}{1-p}\ \
\mathrm{and}\ \ \tau(p^{m})=|\mathrm{Div}(p^{m})|=m+1.$

The reader has probably noted that we used the formula for the sum of a
geometric series in the previous expression for $\sigma(p^{m})$. Then we may
conclude that

(4.4)
$|\mathcal{L}(D_{2p^{m}})|=1+m+\frac{1-p^{m+1}}{1-p}=m+\frac{p^{m+1}+p-2}{p-1}.$

The next result shows an upper bound for $pd(G)$, when $|\mathcal{L}(G)|$ is
of type (4.4).

######  Theorem 4.3.

Let $G$ be a noncyclic group and $p$ the smallest prime divisor of $|G|$. If
$|P(G)|=p$ and $|\mathcal{L}(G)|=m+\frac{p^{m+1}+p-2}{p-1}$ for some $m\geq
0$, then

$pd(G)\leq\frac{p^{m+1}+2p^{2}+(m-3)p-m}{p^{m+2}+(m+1)p^{2}-(m+2)p}.$

###### Proof.

If $P_{G}(X)=G$ for some $X\in\mathcal{L}(G)$, then $|G|=|P_{G}(X)|=|P(G)|$
and $G$ would be cyclic, contradicting our assumption. Without loss of
generality we may assume $P_{G}(X)\neq G$ for all $X\in\mathcal{L}(G)$. The
minimality of $p$ implies that $|P_{G}(X)|\leq\frac{|G|}{p}$ for all
$X\in\mathcal{L}(G)$. Of course, $|\mathcal{L}(P(G))|=2$ and (2.1) becomes

$|G|\cdot\left(\frac{mp-m+p^{m+1}+p-2}{p-1}\right)\cdot
pd(G)=2|G|+{\underset{X\in\mathcal{L}(G)-\mathcal{L}(P(G))}{\sum}}|P_{G}(X)|$
$\leq 2|G|+\left(|\mathcal{L}(G)-\mathcal{L}(P(G))|\right)\frac{|G|}{p}$
$=2|G|+\frac{|G|}{p}(|\mathcal{L}(G)|-2)=2|G|+\frac{|G|}{p}\left(\frac{mp-m+p^{m+1}+p-2}{p-1}-2\right)$
$=\frac{|G|}{p}\left(\frac{mp-m+p^{m+1}+p-2-2p+2+2p^{2}-2p}{p-1}\right)$
$=\frac{|G|}{p}\left(\frac{p^{m+1}+2p^{2}+(m-3)p-m}{p-1}\right).$

This gives, as claimed. ∎

##  5\. Some theorems of structure

The present section is devoted to prove restrictions on $P(G)$, arising from
exact bounds for $pd(G)$, when $G$ is an arbitrary group. The evidences of
Examples 3.1 and 3.2 motivated most of the following results.

######  Theorem 5.1.

Let $G$ be a group (with $pd(G)\not=1$) and $p$ the smallest prime divisor of
$|G|$. Then

$\left(1-\frac{p}{|G|}\right)\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}+\frac{p}{|G|}\leq
pd(G).$

Moreover, if $P_{G}(X)$ is a proper subgroup of $G$ for all
$X\in\mathcal{L}(G)-\mathcal{L}(P(G))$, then

$pd(G)\leq\frac{1}{p}+\frac{(p-1)|\mathcal{L}(P(G))|}{p\ |\mathcal{L}(G)|}.$

###### Proof.

In order to prove the lower bound, it is enough to note from (2.1) that

(5.1)
$|\mathcal{L}(G)||G|pd(G)=|\mathcal{L}(P(G))||G|+{\underset{X\in\mathcal{L}(G)-\mathcal{L}(P(G))}{\sum}}|P_{G}(X)|$
$\geq|\mathcal{L}(P(G))||G|+\Big{(}|\mathcal{L}(G)|-|\mathcal{L}(P(G))|\Big{)}p,$

where we have used in the last step that $|P_{G}(X)|\geq p$. Then we continue

$=(|G|-p)|\mathcal{L}(P(G))|+p|\mathcal{L}(G)|$

from which we get

$pd(G)\geq\frac{(|G|-p)|\mathcal{L}(P(G))|+p|\mathcal{L}(G)|}{|\mathcal{L}(G)||G|}=\frac{(|G|-p)|\mathcal{L}(P(G))|}{|\mathcal{L}(G)||G|}+\frac{p}{|G|}.$

Now we prove the upper bound. Formula (2.1) becomes again (5.1)

$|G||\mathcal{L}(G)|\
pd(G)=|\mathcal{L}(P(G))||G|+{\underset{X\in\mathcal{L}(G)-\mathcal{L}(P(G))}{\sum}}|P_{G}(X)|$

once one uses the fact that $P_{G}(X)=G$ for every $X\in\mathcal{L}(P(G))$.
Now, since $|G:P_{G}(X)|\not=1$ for every
$X\in\mathcal{L}(G)-\mathcal{L}(P(G))$, we get $|G:P_{G}(X)|\geq p$, that is,
$|P_{G}(X)|\leq\frac{|G|}{p}$. Therefore (5.1) is upper bounded by

$\leq|\mathcal{L}(P(G))||G|+\Big{(}|\mathcal{L}(G)|-|\mathcal{L}(P(G))|\Big{)}\frac{|G|}{p}=\frac{(p-1)|\mathcal{L}(P(G))||G|}{p}+\frac{|\mathcal{L}(G)||G|}{p}$

and the result follows. ∎

Of course, $D_{8}$ satisfies the lower bound, but not the upper bound, of
Theorem 5.1. Details can be deduced from the information of Example 3.2. This
is to justify that Theorem 5.1 originates from evidences of computational
nature. On the other hand, Example 3.2 shows also that
$Z(D_{8})=P(D_{8})\simeq C_{2}$. Then, when can we say that $P(D_{8})$ is
noncyclic? The next two results concern this question.

######  Theorem 5.2.

If $P(G)$ is a nontrivial proper subgroup of a group $G$ and
$pd(G)=\frac{1}{2}+\frac{|\mathcal{L}(P(G))|}{2\ |\mathcal{L}(G)|}$, then
$P(G)$ is noncyclic.

###### Proof.

By assumption we exclude the cases $P(G)=G$ and $P(G)=\\{1\\}$, which are the
extremal situations already known. Assume that $P(G)$ is cyclic of prime order
$q\geq p\geq 2$, where $p$ is the smallest prime dividing $|G|$. We may apply
the arguments of the proof of Theorem 4.3 and, noting that
$|\mathcal{L}(P(G))|=2$, we find that

$pd(G)=\frac{1}{2}+\frac{|\mathcal{L}(P(G))|}{2\
|\mathcal{L}(G)|}=\frac{1}{2}+\frac{1}{|\mathcal{L}(G)|}=\frac{|\mathcal{L}(G)|+2}{2|\mathcal{L}(G)|}\leq\frac{2\
|G|}{|G|\ |\mathcal{L}(G)|}+\frac{|G|\ (|\mathcal{L}(G)|-2)}{p\ |G|\
|\mathcal{L}(G)|}$ $=\frac{(2p+|\mathcal{L}(G)|-2)\ |G|}{p\ |G|\
|\mathcal{L}(G)|}$

and then the inequality

$\frac{|\mathcal{L}(G)|+2}{2}=\frac{|\mathcal{L}(G)|}{2}+1\leq\frac{(2p-2)}{p}+\frac{|\mathcal{L}(G)|}{p}$

which means

$\frac{|\mathcal{L}(G)|}{2}-\frac{|\mathcal{L}(G)}{p}=\frac{(p-2)|\mathcal{L}(G)|}{2p}\leq
1-\frac{2}{p}=\frac{p-2}{p}.$

From this we derive the contradiction $\frac{|\mathcal{L}(G)|}{2}\leq 1$, as
at least $\\{1\\}$ and $G$ are contained in $\mathcal{L}(G)$. Therefore,
$P(G)$ cannot be cyclic of prime order and we may assume that $P(G)$ is cyclic
of order $k\geq 2$. Now, we note that $|\mathcal{L}(P(G))|=|\mathrm{Div}(k)|$,
where $\mathrm{Div}(k)$ is the set of all divisors of $k$. Here the argument
we just used for $q$ may still be applied. In fact we have

$\frac{1}{2}+\frac{|\mathrm{Div}(k)|}{2\
|\mathcal{L}(G)|}=\frac{|\mathcal{L}(G)|+|\mathrm{Div}(k)|}{2|\mathcal{L}(G)|}\leq\frac{|\mathrm{Div}(k)|\
|G|}{|G|\ |\mathcal{L}(G)|}+\frac{|G|\
(|\mathcal{L}(G)|-|\mathrm{Div}(k)|)}{p\ |G|\ |\mathcal{L}(G)|}$
$=\frac{(|\mathrm{Div}(k)|\ p+|\mathcal{L}(G)|-|\mathrm{Div}(k)|)\ |G|}{p\
|G|\ |\mathcal{L}(G)|}$

then

$\frac{|\mathcal{L}(G)|+|\mathrm{Div}(k)|}{2}=\frac{|\mathcal{L}(G)|}{2}+\frac{|\mathrm{Div}(k)|}{2}\leq\frac{(p-1)|\mathrm{Div}(k)|}{p}+\frac{|\mathcal{L}(G)|}{p}$

which means

$\frac{|\mathcal{L}(G)|}{2}-\frac{|\mathcal{L}(G)}{p}=\frac{(p-2)|\mathcal{L}(G)|}{2p}\leq\frac{(p-1)|\mathrm{Div}(k)|}{p}$

and this would imply that
$|\mathcal{L}(G)|\leq|\mathrm{Div}(k)|=|\mathcal{L}(P(G))|$, that is,
$\mathcal{L}(G)\subseteq\mathcal{L}(P(G))$ and then
$\mathcal{L}(G)=\mathcal{L}(P(G))$. This condition implies $G=P(G)$, a
contradiction. ∎

The reader may note that Theorem 5.2 describes a very general situation, which
cannot be reduced to those in Examples 3.1 and 3.2. In fact, looking at
Example 3.1, $P(D_{6})=D_{6}$ and so $P(D_{6})$ is not a proper subgroup of
$D_{6}$, and this means that one of the assumptions of Theorem 5.2 is not
satisfied. On the other hand, $P(D_{8})=Z(D_{8})$ is a nontrivial proper
subgroup of $D_{8}$ (see Example 3.2), but
$\frac{4}{5}=pd(D_{8})\not=\frac{1}{2}+\frac{1}{10}=\frac{3}{5}$, and in fact
$P(D_{8})$ is cyclic. Once again, Theorem 5.2 can not be applied. These two
examples show that we cannot strengthen further Theorem 5.2.

However, we may detect groups $G$ with cyclic $P(G)$. The following result
shows this circumstance.

######  Theorem 5.3.

Let $P(G)$ be a nontrivial proper subgroup of a group $G$ with
$pd(G)=\frac{4}{5}$ and $p$ be the smallest prime divisor of $|G|$. If
$\frac{4|G|-5p}{5|G|-5p}\leq\frac{2}{|\mathcal{L}(G)|},$ then $P(G)$ is cyclic
of prime order.

###### Proof.

From the lower bound of Theorem 5.1, we have

$pd(G)=\frac{4}{5}\geq\left(1-\frac{p}{|G|}\right)\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}+\frac{p}{|G|}\Leftrightarrow\frac{\frac{4}{5}}{1-\frac{p}{|G|}}-\frac{\frac{p}{|G|}}{1-\frac{p}{|G|}}\geq\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}$
$\Leftrightarrow\frac{\frac{4|G|-5p}{5|G|}}{\frac{|G|-p}{|G|}}\geq\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}\Leftrightarrow\frac{4|G|-5p}{5|G|-5p}\geq\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}.$

We conclude that
$\frac{|\mathcal{L}(P(G))|}{|\mathcal{L}(G)|}\leq\frac{2}{|\mathcal{L}(G)|},$
hence $|\mathcal{L}(P(G))|\leq 2$. This forces $P(G)$ to be cyclic of prime
order. ∎

##  6\. Computations for dihedral groups

We describe an instructive example, which correlates most of the notions which
we have seen until now.

######  Proposition 6.1.

Let $p$ be an odd prime. Then

$1=pd(D_{2p})>sd(D_{2p})=\frac{7p^{3}-5p^{2}-11p+9}{p^{4}+4p^{3}-2p^{2}-12p+9}>\frac{p+3}{4p}=d(D_{2p}).$

###### Proof.

Noting that $D_{2p}=C_{2}\ltimes C_{p}$ (see (4.1)) and that
$\mathcal{L}(D_{2p})$ forms a diamond (as in Fig.III.3), we conclude that
$Z(D_{2p})$ is trivial and $C_{D_{2p}}(C_{p})=C_{p}$ is the unique maximal
normal subgroup of $D_{2p}$. Moreover $D_{2p}$ is a $\mathbf{P}$–group,
because $Q_{\infty}(D_{2p})=D_{2p}$. Therefore a proper subgroup $H$ of
$D_{2p}$ should be properly contained in $P_{D_{2p}}(H)$ and necessarily
$P_{D_{2p}}(H)=D_{2p}$. Thus, we find that $pd(D_{2p})=1$.

On the other hand, we may specialize the formula

$sd(D_{2p})=\frac{\tau(p)^{2}+2\tau(p)\sigma(p)+g(p)}{(\tau(p)+\sigma(p))^{2}},$

given in [27, Theorem 3.1.1], where

$g(p)=\frac{3p^{3}-5p^{2}+p+1}{p^{2}-2p+1}$

is the arithmetic function in [27, Eq. 10, p.2514]. From (4.3) we deduce
$\tau(p)=2$ and $\sigma(p)=p+1$ so that

$sd(D_{2p})=\frac{4+2\cdot
2\cdot(p+1)+\frac{3p^{3}-5p^{2}+p+1}{p^{2}-2p+1}}{p^{2}+6p+9}=\frac{8+4p+\frac{3p^{3}-5p^{2}+p+1}{p^{2}-2p+1}}{p^{2}+6p+9}$
$=\frac{\frac{8p^{2}-16p+8+4p^{3}-8p^{2}+4p+3p^{3}-5p^{2}+p+1}{p^{2}-2p+1}}{p^{2}+6p+9}$
$=\frac{7p^{3}-5p^{2}-11p+9}{(p^{2}+6p+9)(p^{2}-2p+1)}=\frac{7p^{3}-5p^{2}-11p+9}{p^{4}+4p^{3}-2p^{2}-12p+9}.$

Now [20, Remark 4.2] shows that $d(D_{2p})=\frac{p+3}{4p}$ and for all odd
primes we have

$0>p^{5}-21p^{4}+30p^{3}+26p^{2}-63p+27$ $\Leftrightarrow
28p^{4}-20p^{3}-44p^{2}+36p>p^{5}+7p^{4}+10p^{3}-18p^{2}-27p+27$
$\Leftrightarrow
28p^{4}-20p^{3}-44p^{2}+36p>p^{5}+4p^{4}-2p^{3}-12p^{2}+9p+3p^{4}+12p^{3}-6p^{2}-36p+27$
$\Leftrightarrow 4p(7p^{3}-5p^{2}-11p+9)>(p+3)(p^{4}+4p^{3}-2p^{2}-12p+9)$
$\Leftrightarrow\frac{7p^{3}-5p^{2}-11p+9}{p^{4}+4p^{3}-2p^{2}-12p+9}>\frac{p+3}{4p}.$

The result follows. ∎

From Proposition 6.1, $pd(D_{2p})$ has a constant value for all odd primes,
while $sd(D_{2p})$ and $d(D_{2p})$ are functions of $p$. This is an important
difference of the permutability degree with respect to the subgroup
commutativity degree and the commutativity degree. This reflects the fact that
we are looking at permutizers in a group, and not at centralizers.

##  7\. Some open questions

We end with a series of open questions. They arise naturally from the study of
the present subject. The first is motivated by the families of dihedral
groups, analyzed in Section 6 (and in other parts of the present paper). Most
of these groups may be described in terms of product of groups.

######  Open Question 7.1.

Let $G=NH$ be a product of a normal subgroup $N$ by a subgroup $H$. What can
be said about the permutability degree of $G$ ?

It is well known that most of dihedral groups, generalized quaternion groups
and semidihedral groups has this structure. They present further analogies in
terms of central quotients: one, for instance, is that $D_{8}/Z(D_{8})\simeq
Q_{8}$ and, roughly speaking, one can generalize this isomorphism to
$Q_{2^{4}}=Q_{16}$, $Q_{2^{5}}=Q_{32}$ and so on. The reader may refer to [15,
16] for recent studies on these groups. Therefore:

######  Open Question 7.2.

What is the permutability degree of generalized quaternion groups and
semidihedral groups ?

There is also another question, which is more general and may require some
computational efforts. A classical result of Cayley allows us to embedd a
group in a suitable symmetric group. The knowledge of symmetric groups plays
in fact a fundamental role in several aspects of the theory of groups.
Therefore:

######  Open Question 7.3.

What is the permutability degree of the symmetric group $S_{n}$ on $n$
objects?

Finally, the classification of (finite) simple groups may provide interesting
aspects of study. For the (finite) simple (and almost simple) groups of
sporadic type a lot is known about their subgroup lattices, see [7]. Therefore

######  Open Question 7.4.

What is the permutability degree of (finite) simple groups ?

## Acknowledgements

The first author was partially supported by GNSAGA of INdAM (Italy). The
research of the second author is supported in part by NRF (South Africa) for
the Grant No. 93652 and in part from the Launching Grant No. 459235 of the
University of Cape Town (South Africa). He also thanks Universidade Federal do
Rio de Janeiro (Rio de Janeiro, Brazil), where the present subject was
originally investigated via the project CAPES No. 061/2013. Finally, we thank
the editor for his valuable work and the referee for the questions in Section
7.

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##  8\. ERRATA CORRIGE – 10TH OF SEPTEMBER 2017

After the publication of the previous results in:

D.E. Otera and F.G. Russo, Permutability degrees of finite groups, Filomat 30
(2016), 2165–2175.

some errors have been reported to our attention, thanks to the communication
of some colleagues (in particular we thank Prof. Paz Jiménez Seral). As noted
below, the main results are not directly involved, but we cannot avoid to give
more details on some points which might be ambiguous if we don’t do it.

  * (1)

Definition of $P(G)$ in Introduction. Two lines below. Replace the sentence
“Note that for any $X\in\mathcal{L}(P(G))$ one has $P_{G}(X)=G$” by “Note that
when $X=P(G)$ one has $P_{G}(X)=P(G)$”.

  * (2)

The value of $pd(D_{8})$ in Example 3.2 is wrong. In fact $pd(D_{8})=1$, in
agreement with Proposition 6.1 when we show that $pd(D_{2p})=1$ for dihedral
groups (of order $2p$ with $p$ odd). The error of Example 3.2 is here:

$4=|M_{1}|=|P_{D_{8}}(H)|=|P_{D_{8}}(K)|=|P_{D_{8}}(V)|=|P_{D_{8}}(U)|,$

in fact we must replace this computation with

$8=|P_{D_{8}}(H)|=|P_{D_{8}}(K)|=|P_{D_{8}}(V)|=|P_{D_{8}}(U)|.$

Consequently, $P(D_{8})=D_{8}$ and $P(D_{8})\neq Q(D_{8})$. In particular, the
final two sentences from “This example shows …” until the end of Example 3.2
must be removed. In addition,

    * (a)

the final sentence of Remark 2.3 must be removed;

    * (b)

the sentence just before Theorem 5.1 must be removed;

    * (c)

the sentences before Theorem 5.2 “Of course, $D_{8}$ … $P(D_{8})\simeq C_{2}$
” must be removed.

    * (d)

the sentences “On the other hand… further Theorem 5.2” before Theorem 5.3 must
be removed.

  * (3)

Proof of the lower bound in Theorem 4.1. There is a misprint: it must be

$\sum_{X\in\mathcal{L}(G)}|P_{H}(X)|\geq\sum_{X\in\mathcal{L}(H)}|P_{H}(X)|$

  * (4)

In Theorem 4.3 we use an assumption, not justified properly in the proof, but
this may be motivated by evidences of computational nature. Therefore Theorem
4.3 must be reformulated in the following way:

######  Theorem 8.1.

Let $G$ be a noncyclic group and $p$ the smallest prime divisor of $|G|$. If
$P_{G}(X)$ is a proper subgroup of $G$ for all $X\in\mathcal{L}(G)$,
$|P(G)|=p$ and $|\mathcal{L}(G)|=m+\frac{p^{m+1}+p-2}{p-1}$ for some $m\geq
0$, then

$pd(G)\leq\frac{p^{m+1}+2p^{2}+(m-3)p-m}{p^{m+2}+(m+1)p^{2}-(m+2)p}.$

The proof is that of Theorem 4.3 without the first two sentences.


# Two-layer adaptive trajectory tracking controller for quadruped robots on
slippery terrains

Despina-Ekaterini Argiropoulos1,3, Dimitrios Papageorgiou1,2, Michael
Maravgakis1,3,
Drosakis Drosakis1 and Panos Trahanias1,3 The research leading to these
results has received partial funding from the European Community’s HORIZON.1.2
- Marie Skłodowska-Curie Actions (MSCA) under Grant agreement No. 101072634,
project RAICAM.1Authors are with the Institute of Computer Science, Foundation
for Research and Technology–Hellas, Heraklion 700 13, Greece. {dimpapag,
despinar, maravgakis, drosakis<EMAIL_ADDRESS>Papageorgiou
is also with the Department of Electrical and Computer Engineering, Hellenic
Mediterranean University, 714 10, Heraklion, Crete, Greece. 3Despina-Ekaterini
Argiropoulos, Michael Maravgakis and Panos Trahanias are also with the
Department of Computer Science, University of Crete, 714 09 Heraklion, Crete,
Greece.

###### Abstract

Task space trajectory tracking for quadruped robots plays a crucial role on
achieving dexterous maneuvers in unstructured environments. To fulfill the
control objective, the robot should apply forces through the contact of the
legs with the supporting surface, while maintaining its stability and
controllability. In order to ensure the operation of the robot under these
conditions, one has to account for the possibility of unstable contact of the
legs that arises when the robot operates on partially or globally slippery
terrains. In this work, we propose an adaptive trajectory tracking controller
for quadruped robots, which involves two prioritized layers of adaptation for
avoiding possible slippage of one or multiple legs. The adaptive framework is
evaluated through simulations and validated through experiments.

This paper is a pre-print version under review.

##  I Introduction

One of the main advantages of legged robots is their capability to transverse
unstructured environments, such as sewers or construction sites, which may
involve a variety of challenging terrain types. This capability enables the
utilization of legged robots in applications potentially dangerous for the
human, such as search and rescue operations or missions for inspection and
maintenance in critical asset facilities. Except their structural complexity,
such difficult-to-transverse environments also impose dynamic challenges, with
the most dominant being the variation of the friction coefficient of the
terrain. Partially or globally slippery terrains are considered to be one of
the most frequent problems faced by legged robots, which may arise in case of
mud, wet floor, oil or ice [1]. Slippage of any leg with respect to the
supporting surface could trigger unknown and unmodelled dynamics which would
in-turn worsen the trajectory tracking performance or even lead to robot’s
instability; e.g. it could lead to singular configurations or to
configurations in which the contact with the supporting surface is lost.

Slippage is occurred when the applied force is not within the static friction
cone, defined by a threshold to the ratio between the tangential and the
orthogonal force magnitudes; the so-called static friction coefficient. This
coefficient, however, is difficult to model, varies in space and therefore, in
most of the cases, it is considered to be unknown a priori. To tackle the
problem of identifying the slippage phenomenon, current methods in literature
propose the utilization of machine learning [2, 3, 4], classical estimators
such as Extended/Unscented Kalman Filters [5, 6], model-based estimators [7],
most of the times utilizing proprioceptive information to yield an estimate of
the probability of stable contact, as proposed in our recent work [8].

After identifying the slippage of a single or multiple legs, one has to define
a reactive behavior for ensuring the stability and controllability of the
system. Common practices involve the utilization of estimators for slippage
alongside with an on-line trajectory generation mechanism for slippage
recovery. For instance, in [9], to recover from the slippage of one leg, the
slipping leg is commanded to move towards its initial configuration, which is
executed within a layer of a hierarchical framework that also involve the
task-space trajectory. That means that the system may sacrifice the task
performance for recovering from slippage, as it is not guaranteed that the
joint-space recovery motion for the legs will not affect the overall tracking
performance; e.g. in case of globally slippery terrains. In [10], the slippage
recovery strategy is handled also by a hierarchical framework which ensures
that the reactive ground forces stay within an estimate of the friction cone.
However, in case of infeasibility of solution (e.g. when all four legs slip)
the motion is stopped (frozen state). In [11], a Kalman Filter is utilized in
a finite state machine framework to identify slippage and react. However, the
paper focuses mainly on the generalized-momentum disturbance observation,
while the reactive behavior refers to the high-level control without
considering the tracking of a specific trajectory. From a machine learning
perspective, some works employ the notion of reinforcement learning [12, 13]
for tackling the problem of slippage during locomotion. Most specifically, in
[12] reinforcement learning is employed based on proprioceptive sensorial
information, while in [13] the concurrent training of a state estimator and a
reactive policy is considered; both works consider the training of the system
in a simulated environment, exploring as much as possible the variety of cases
that the robot may face during its operation.

Trajectory tracking can be utilized as the barebone for locomotion and
facilitate walking over unstructured and slippery terrains. To the best of our
knowledge, although some works tackle the problem of trajectory tracking for
quadruped robots in general, such as [14] in joint space and [15] in task-
space, the control problem is not explicitly addressed considering the
operation on a partially or globally slippery terrain. On the other hand,
works that make such a consideration (e.g. [9, 10, 11, 12, 13]), are mainly
tackling the locomotion problem and they cannot guarantee that the task-space
control will remain unaffected, while most of them require the extensive
training of the system in a simulated environment.

In this work, a novel trajectory tracking control scheme is proposed for
quadruped robots, incorporating two prioritized layers of adaptation for
minimizing the possible slippage of one or multiple legs. The proposed control
framework builds upon our previous work on contact state estimation, in which
the probability of stable contact for each foot is estimated independently
based solely on proprioceptive data, namely IMU sensors attached to the
robot’s feet. In the proposed control framework, the estimated probability is
utilized as an input for two layers of adaptation. In particular, in the first
layer, the adaptation of the distribution of the control effort among the legs
is considered, without affecting the task performance, exploiting the
redundancy of the quadruped robot. In the second layer, which is enabled only
if the problem cannot be solved by the first layer, the time-scaling of the
trajectory is considered, which affects only the temporal properties of the
task, without distorting the path followed by the robot. In other words, when
the control effort distribution is not enough for tackling the problem of
minimizing the slippage, the temporal scaling of the trajectory (slowing down
the motion) will lead to an overall reduction of the control effort that has
to be applied by the robot’s legs.

The main contributions of the proposed method are:

  * •

A task-space trajectory tracking controller is proposed, which is
theoretically proven to yield an asymptotically stable behavior.

  * •

The proposed control scheme involves two layers of adaptation, as it accounts
for partially or globally slippery terrains. Simulations show the ability of
the proposed scheme to maintain stability and controllability of the system.

  * •

The first layer of adaptation exploits the redundancy of the quadruped robot
and dynamically distributes the control effort to the legs, based on the
estimated slippage probability of each leg, while the second layer of
adaptation involves the time-scaling of the trajectory and it is enabled only
if the dynamic distribution cannot solve the problem of avoiding the slippage.

Finally, to facilitate and promote research, the control scheme has been
released as an open-source module in ROS/C++ coined as Maestro [16] while the
probabilistic contact estimator can be found in [17].

##  II Problem formulation and concept solution

Consider the quadruped robot depicted in Fig.1, having $n\in\mathbb{N}$ joints
in each leg and let $q_{i,j}\in\mathbb{R},i=1,...,4,j=1,...,n$ be the joint
position variables of the $i$-th leg. Let
$\mathbf{q}\triangleq[q_{1,1}\;q_{1,2}\;...\;q_{4,n-1}\;q_{4,n}]^{\intercal}\in\mathbb{R}^{4n}$
be the vector of the total joint variables of the robot. Furthermore, let
$\\{C\\}$ be the frame placed at the Center of Mass (CoM) of the robot (as
depicted in Fig.1) and
${}^{c}\mathbf{p}_{i}(q_{i,1},...,q_{i,n})\in\mathbb{R}^{3}$ be the position
of the tip of each leg with respect to $\\{C\\}$. The position and orientation
of $\\{C\\}$ with respect to the world frame $\\{0\\}$ is denoted by
$\mathbf{p}_{c}\in\mathbb{R}^{3}$ and $\mathbf{R}_{c}\in SO(3)$ respectively.
World frame $\\{0\\}$ could refer to a known inertial frame, or (in most of
the cases) to the initial pose of the robot (i.e. $\\{C\\}$ at $t=0$). When
all four tips are in contact with the supporting surface the mapping between
the forces $\mathbf{f}_{i}\in\mathbb{R}^{3},i=1,...,4$ applied to the tips of
the legs (e.g. reactive forces from the supporting surface) and the
corresponding generalized force
$\mathbf{F}_{c}\triangleq[\mathbf{f}_{c}^{\intercal}\;\boldsymbol{\tau}_{c}^{\intercal}]^{\intercal}\in\mathbb{R}^{6}$
at the CoM, with $\mathbf{f}_{c}\in\mathbb{R}^{3}$ and
$\boldsymbol{\tau}_{c}\in\mathbb{R}^{3}$ being the force and torque at the CoM
respectively, is the following:

$\mathbf{F}_{c}=\mathbf{G}(\mathbf{q})\mathbf{F}_{a},$ (1)

where

$\mathbf{G}(\mathbf{q})\triangleq\begin{bmatrix}\mathbf{I}_{3}&\mathbf{I}_{3}&\mathbf{I}_{3}&\mathbf{I}_{3}\\\
\mathbf{S}(\mathbf{p}_{c1})&\mathbf{S}(\mathbf{p}_{c2})&\mathbf{S}(\mathbf{p}_{c3})&\mathbf{S}(\mathbf{p}_{c4})\end{bmatrix}$
(2)

and

$\mathbf{F}_{a}\triangleq\begin{bmatrix}\mathbf{f}_{1}\\\ \mathbf{f}_{2}\\\
\mathbf{f}_{3}\\\ \mathbf{f}_{4}\end{bmatrix}\in\mathbb{R}^{12},$ (3)

with
$\mathbf{p}_{ci}(q_{i,1},...,q_{i,n})\triangleq\mathbf{R}_{c}{}^{c}\mathbf{p}_{i}(q_{i,1},...,q_{i,n}),i=1,...,4$,
$\mathbf{I}_{3}\in\mathbb{R}^{3\times 3}$ the identity matrix and
$\mathbf{S}(.):\mathbb{R}^{3}\rightarrow\mathbb{R}^{3\times 3}$ the skew
symmetric mapping. Notice that $\mathbf{G}(\mathbf{q})$ belongs to
$\mathbb{R}^{6\times 12}$, and therefore the problem of solving (1) with
respect to $\mathbf{F}_{a}$, i.e. finding $\mathbf{F}_{a}$ for a given
$\mathbf{F}_{c}$, is redundant. The mapping between the force applied to the
tip of the $i$-th leg and the corresponding torques at the joints of the leg,
is given by:

$\boldsymbol{\tau}_{i}=\left(\mathbf{R}_{c}{}^{c}\mathbf{J}_{i}(q_{i,1},...,q_{i,n})\right)^{\intercal}\mathbf{f}_{i},$
(4)

where ${}^{c}\mathbf{J}_{i}(q_{i,1},...,q_{i,n})\in\mathbb{R}^{3\times n}$ is
the position part of the Jacobian of the leg with respect to the CoM and
$\boldsymbol{\tau}_{i}\in\mathbb{R}^{n}$ the torques at the joints of the leg.

Figure 1: Force distribution among the legs of the quadruped robot.

###### Remark 1

When less than four tips are in contact with the environment,
$\mathbf{G}(\mathbf{q})$ has to be modified accordingly to involve only the
legs that are in contact with the supporting surface. A representative example
is in case of locomotion, in which one or more legs are in swing phase.

The dynamic model of the system, assuming that the inertia of the legs is
negligible as compared to the inertia of the rest of the body, is given by:

$\mathbf{H}_{c}\dot{\mathbf{V}}_{c}+\mathbf{C}_{c}\mathbf{V}_{c}+\mathbf{g}_{c}=\mathbf{F}_{c},$
(5)

where
$\mathbf{V}_{c}\triangleq[\dot{\mathbf{p}}_{c}^{\intercal}\;\boldsymbol{\omega}_{c}^{\intercal}]^{\intercal}\in\mathbb{R}^{6}$
is the generalized velocity of $\\{C\\}$,
$\boldsymbol{\omega}_{c}\in\mathbb{R}^{3}$ being its angular velocity,
$\mathbf{H}_{c}\triangleq\text{diag}(m\mathbf{I}_{3},\mathscr{I}_{c})\in\mathbb{R}^{6\times
6}$ is the positive definite inertia matrix of the robot,
$\mathbf{g}_{c}\in\mathbb{R}^{6}$ the gravity vector and
$\mathbf{C}_{c}\triangleq\text{diag}(\mathbf{0}_{3\times
3},\mathbf{S}(\mathscr{I}_{c}\boldsymbol{\omega}_{c}))\in\mathbb{R}^{6\times
6}$ the Coriolis-centrifugal matrix, with
$\mathscr{I}_{c}\triangleq\mathbf{R}_{c}\mathscr{I}\mathbf{R}_{c}^{\intercal}\in\mathbb{R}^{3\times
3}$ and $\mathscr{I}\in\mathbb{R}^{3\times 3}$ the inertia tensor of the main
body of the robot. In case in which the $z$-axis of the inertial frame is
aligned to the gravity direction,
$\mathbf{g}_{c}=[0\;0\;m_{r}g\;0\;0\;0]^{\intercal}$, with
$m_{r}\in\mathbb{R}^{+}$ being the mass of the robot and $g\in\mathbb{R}^{+}$
the constant acceleration due to gravity.

Consider a torque controlled robot, i.e. accepting joint torque commands
$\boldsymbol{\tau}_{i}(t)\in\mathbb{R}^{n},i=1,...,4$ (joint torques of the
$i$-th leg). For solving the task-space trajectory tracking problem, one has
to solve (1) with respect to $\mathbf{F}_{a}$, i.e. compute the inverse
mapping, to calculate the forces that each leg should apply in order to render
the commanded force in the task-space, $\mathbf{F}_{c}$, as follows:

$\mathbf{F}_{a}=\mathbf{G}^{\dagger}(\mathbf{q})\mathbf{F}_{c},$ (6)

where $\mathbf{G}^{\dagger}\in\mathbb{R}^{12\times 6}$ is a right pseudo-
inverse of $\mathbf{G}$. In this point, there are multiple options regarding
the pseudo-inverse. Some of them are the right Moore-Penrose pseudo-inverse,
given by
$\mathbf{G}^{\dagger}\triangleq\mathbf{G}^{\intercal}(\mathbf{G}\mathbf{G}^{\intercal})^{-1}$,
which will result in an equal distribution of control effort among the four
legs (minimum norm solution), or the right weighted pseudo-inverse, given by:

$\mathbf{G}^{\dagger}\triangleq\mathbf{W}^{-1}\mathbf{G}^{\intercal}(\mathbf{G}\mathbf{W}^{-1}\mathbf{G}^{\intercal})^{-1},$
(7)

with $\mathbf{W}\in\mathbb{R}^{12\times 12}$ being a positive definite weight
matrix. The latter will result in distributing the control effort based on the
selected weight matrix $\mathbf{W}$. More specifically, in this case, by
selecting a positive definite diagonal matrix
$\mathbf{W}\triangleq\text{diag}(w_{1,1},w_{1,2},w_{1,3},...,w_{4,3})$, the
higher the $w_{i,m}$, the less the force appended to the $m$-th direction of
the $i$-th leg’s tip; for instance a high value of $w_{3,2}$ as compared to
the other $w_{i,m}$-s, will result in appending less force along the
$y$-direction ($m=2$) of the third leg ($i=3$). After computing the
corresponding force in each leg, i.e. $\mathbf{f}_{i}$ which is included in
$\mathbf{F}_{a}$, one can compute the commanded torques from (4).

###  II-A The problem of slippage

From a robot control perspective, slippage of the tip with respect to the
supporting surface could potentially result in losing the controllability of
the system. In particular, slippage could lead a) to singular configurations,
in which the rank of ${}^{c}\mathbf{J}_{i}(q_{i,1},...,q_{i,n})$ will be
decreased, b) reaching the joint limits or c) losing contact in one or
multiple legs without accounting for it, which is required for the validity of
(6). As long as the force applied by each leg, i.e. $\mathbf{f}_{i}$, is
within the friction cone, no slippage of the tip (with respect to the
supporting surface) is occurred. More specifically, for a given terrain with
constant static friction coefficient $\mu\in\mathbb{R}^{+}$, the static
friction cone (the area in which there is not slippage) is expressed by:

$\mathscr{C}\triangleq\\{\mathbf{f}_{i}\in\mathbb{R}^{3}:\mu|\mathbf{n}^{\intercal}\mathbf{f}_{i}|>\|(\mathbf{I}_{3}-\mathbf{n}\mathbf{n}^{\intercal})\mathbf{f}_{i}\|\\},$
(8)

where $\mathbf{n}\in\mathbb{S}^{2}$ (with
$\mathbb{S}^{2}\triangleq\\{\mathbf{x}\in\mathbb{R}^{3}:\|\mathbf{x}\|=1\\}$)
is the normal to the supporting surface vector. However, due to the fact that
$\mu$ is not easy to measure or estimate and most of the times it is
considered to be unknown, one cannot assess a priori, i.e. before commanding
$\mathbf{f}_{i}$ to the leg, whether the leg’s tip would slip or not.

###  II-B Concept solution

Building on our previous work [8], in which we introduced a probabilistic
approach for estimating the stable contact probability (i.e. not slipping), we
propose a novel trajectory tracking control scheme incorporating a two-layer
on-line adaptation. In particular, in [8] the probability of a contact being
stable is estimated in real-time, based on a set of IMU sensors mounted on the
robot’s feet. This probability is counter proportional to the probability of
slippage. Based on this estimate, we firstly propose an adaptation law for the
weights of distribution of the control effort among all the directions of
forces that should be applied by each leg of the robot. The rationale behind
the adaptation law is to append less tangential to the surface forces to the
legs for which the slippage probability is high, attracting in this way the
appended force towards the friction cone. Furthermore, when the aforementioned
force distribution cannot guarantee the elimination of slippage, we propose
the dynamic time-scaling of the trajectory (e.g. to slow down the motion),
which will consequently yield a reduced control effort magnitude in general.
The complete adaptive control scheme is shown in Fig.2.

Figure 2: Block diagram of the proposed adaptive scheme.

##  III Proposed scheme

Given a reference trajectory $\mathbf{p}_{d}(t)\in\mathbb{R}^{3}$ an
$\mathbf{R}_{d}(t)\in SO(3)$ for frame $\\{C\\}$ in position and orientation
respectively, we consider the trajectory tracking problem, i.e. the problem of
minimizing the Euclidean norms of the following errors in time:

$\displaystyle\mathbf{e}_{p}$
$\displaystyle\triangleq\mathbf{p}_{c}(t)-\mathbf{p}_{d}(t),$ (9)
$\displaystyle\mathbf{e}_{o}$
$\displaystyle\triangleq\text{log}\left(\mathbf{R}_{c}(t)\mathbf{R}_{d}^{\intercal}(t)\right),$
(10)

where $\text{log}(\mathbf{R})\triangleq\mathbf{k}\theta\in\mathbb{R}^{3}$ the
logarithmic mapping, with $\theta\in[0,\pi]$ being the angle and
$\mathbf{k}\in\mathbb{S}^{2}$ being the axis of rotation of a given
$\mathbf{R}$. The trajectory $\mathbf{p}_{d}(t)$, $\mathbf{R}_{d}(t)$ could
represent the motion of the main robot’s body during its locomotion, or even
dexterous motions for avoiding collisions in unstructured environments, e.g.
the case of passing through a narrow opening. Notice that in case of
locomotion one should re-define the matrix $\mathbf{G}$ on-line, based on the
feet that are in contact with the terrain.

Considering the system dynamics given in (5), the control objective can be
achieved by applying the following state-feedback control law with gravity
compensation, representing the commanded generalized force that should be
applied to the CoM:

$\begin{split}\mathbf{F}_{c}\triangleq&\mathbf{H}_{c}\begin{bmatrix}\ddot{\mathbf{p}}_{d}\\\
\frac{d}{dt}\left(\mathbf{R}_{c}\mathbf{R}_{d}^{\intercal}\boldsymbol{\omega}_{d}\right)\end{bmatrix}+\mathbf{C}_{c}\begin{bmatrix}\dot{\mathbf{p}}_{d}\\\
\mathbf{R}_{c}\mathbf{R}_{d}^{\intercal}\boldsymbol{\omega}_{d}\end{bmatrix}\\\
&-\begin{bmatrix}k_{p}\mathbf{e}_{p}\\\
k_{o}\mathbf{e}_{o}\end{bmatrix}-\mathbf{K}_{v}\mathbf{e}_{v}+\mathbf{g}_{c},\end{split}$
(11)

where

$\mathbf{e}_{v}\triangleq\mathbf{V}_{c}-\begin{bmatrix}\dot{\mathbf{p}}_{d}\\\
\mathbf{R}_{c}\mathbf{R}_{d}^{\intercal}\boldsymbol{\omega}_{d}\end{bmatrix},$
(12)

$k_{p},k_{o}\in\mathbb{R}^{+}$, $\mathbf{K}_{v}\in\mathbb{R}^{6\times 6}$ are
constant positive control gains and $\boldsymbol{\omega}_{d}\in\mathbb{R}^{3}$
is the reference angular velocity which can be calculated by
$\mathbf{S}(\boldsymbol{\omega}_{d})=\dot{\mathbf{R}}_{d}\mathbf{R}_{d}^{\intercal}$.
The proof of global asymptotic stability of the origin of the state-space
(corresponding to zero error in position and velocity), under the application
of the control law (11), is proven in Appendix A.

###  III-A Slippage detection

For the slippage detection mechanism, stable contact is considered to be the
state in which the robot’s foot is in touch with the ground whilst there is no
relative motion between them. To estimate the stable contact probability, a 6D
IMU sensor is mounted on each foot of the robot. By exploiting the uncertainty
of the inertial measurements, we employ Kernel Density Estimation (KDE) to
approximate the Probability Density Function for each axis of the IMU and
consequently the per axis stable probability over a small interval, as
dictated in [8]. The method, practically estimates the probability that the
inertial measurements are close to zero and finally, since they are
independent, the total stable probability is acquired via multiplication.
Finally, in order to detect when the foot touches the ground, one can utilize
force measurements, haptic or dedicated contact sensors. To this end, in this
work we exploited the vertical force measurement $(f_{i,z}>0)$.

###  III-B First layer of adaptation: Adaptive effort distribution

Based on the above slippage detection mechanism, we propose the following
adaptive law for the weights $w_{i,m},m=1,2$ (the $x-y$ coefficients) of the
tangential force directions of the $i$-th leg:

$\begin{split}\dot{w}_{i,1}=\dot{w}_{i,2}\triangleq\alpha P_{i},\\\
{w}_{i,1}(0)={w}_{i,2}(0)\triangleq w_{0}\end{split}$ (13)

where $\alpha\in\mathbb{R}^{+}$ is a tunable constant adaptation gain,
$w_{0}\in\mathbb{R}^{+}$ the initial value of the weights in $x-y$ direction
and $P_{i}\in[0,1]$ the probability of slippage of the $i$-th leg. Notice that
the normalization of $w_{i,m},i=1,...,4,m=1,2,3$ is not required, as (6), (7)
do not assume constraints for the values of $\mathbf{W}$.

###### Remark 2

Notice that (13) assumes the orthogonality between the supporting surface and
the gravity direction, for the sake of simplicity of presentation. However,
the generalization to inclined surfaces can be easily done by considering a
non-diagonal $\mathbf{W}$ matrix.

###### Remark 3

Given (13), the weights will increase only as long as slippage is estimated,
which means that the weights will eventually reach the value in which the
control effort appended to the specific leg does not yield any slippage. The
increase of these weights (i.e. the weights corresponding only to the
tangential forces) will result in decreasing the magnitude of forces appended
towards these directions. Therefore, the appended force $\mathbf{f}_{i}$ will
converge to the friction cone $\mathscr{C}$, as graphically depicted in Fig.3.

Figure 3: Convergence of the control effort of each leg towards the friction
cone.

###  III-C Second layer of adaptation: Trajectory time-scaling

The first layer of adaptation could fail when all the legs of the robot are
contacting a terrain with a relatively low static friction coefficient. Hence,
to handle this type of occasions, we propose the time-scaling of the
trajectory, i.e. to sacrifice the temporal accuracy of the task for
guaranteeing stability and controllability, maintaining however accuracy with
respect to the spatial properties of the path.

Let $\mathbf{p}_{d}(t_{v}),\mathbf{R}_{d}(t_{v})$ be the time-parametric
trajectory, with $t_{v}(t)\in\mathbb{R}^{+}$ being the scaled time parameter
and $\beta\in[0,1]$ the time-scaling coefficient. Hence, the evolution of the
scaled time parameter is characterized by $\dot{t}_{v}(t)=\beta(t)$. For
instance, setting a constant $\beta=1$ would result in $t_{v}=t$ and
consequently would lead to the execution of the trajectory on a nominal speed,
while setting $\beta<1$ would slow down the motion. To tackle the problem of
global slippage, we propose the utilization of the following time-scaling
coefficient:

$\beta(t)\triangleq\frac{w_{0}}{\text{min}(w_{1,1},w_{2,1},w_{3,1},w_{4,1})}.$
(14)

The rationale behind (14) is to reduce the speed (reflected by $\beta$) when
slippage has occurred in all four legs, an occasion which is signified by the
increase of the weights of all four legs due to (13). For instance, if at
least one of the legs does not face any slippage, then
$\text{min}(w_{1,1}(t),w_{2,1}(t),w_{3,1}(t),w_{4,1}(t))$ will be equal to
$w_{0}$ and therefore $\beta$ will be 1, which means that no time scaling
would occur.

As the on-line time scaling is considered, the trajectory should be generated
on-line from $t_{v}$, which is calculated by the integration of
$\dot{t}_{v}=\beta(t)$ in real-time. Notice that, in such a case
$\dot{\mathbf{p}}_{d}(t)=\beta(t)\frac{d\mathbf{p}_{d}(t_{v})}{dt_{v}},\dot{\mathbf{R}}_{d}(t)=\beta(t)\frac{d\mathbf{R}_{d}(t_{v})}{dt_{v}}$.

The complete algorithm of the proposed control scheme is given in Algorithm 1.

###### Remark 4

Notice that if the trajectory is generated online by a dynamical system (e.g.
a Dynamic Movement Primitives model [18]), the application of the
aforementioned idea is straight forward, as in that case $\beta$ would
correspond to the time scaling parameter of the dynamical system.

Algorithm 1 Implementation of the control loop

1:Select values for: $k_{p},k_{o},\mathbf{K}_{v},w_{0}$

2:$\mathbf{W}:=w_{0}\mathbf{I}_{12},\beta:=1,t_{v}:=0$ $\triangleright$
Initialization

3:while control is enabled do

4: Get current state of the robot
$\mathbf{p}_{c},\mathbf{R}_{c},\dot{\mathbf{p}}_{c},\boldsymbol{\omega}_{c}$

5: Estimate $P_{i},\forall i=1,...,4$ $\triangleright$ Slip. prob. estimator

6: Compute $\dot{w}_{i,1},\dot{w}_{i,2},\forall i=1,...,4$ from (13)

7: Integrate $\dot{w}_{i,1},\dot{w}_{i,2},\forall i=1,...,4$ to update
$\mathbf{W}$

8: Compute $\beta$ from (14)

9: Integrate $\dot{t}_{v}=\beta$ to update $t_{v}$

10: Compute
$\mathbf{p}_{d}(t_{v}),\dot{\mathbf{p}}_{d}(t_{v}),\ddot{\mathbf{p}}_{d}(t_{v}),\mathbf{R}_{d}(t_{v}),\boldsymbol{\omega}_{d}(t_{v}),\dot{\boldsymbol{\omega}}_{d}(t_{v})$

11: Compute $\mathbf{F}_{c}$ from (11)

12: Compute $\mathbf{F}_{a}$ (includes the $\mathbf{f}_{i}$-s) from (6), (7)

13: Compute $\boldsymbol{\tau}_{i},\forall i=1,...,4$ from (4)

14: Command $\boldsymbol{\tau}_{i},\forall i=1,...,4$ to the joints

15:end while

##  IV Simulation study

To assess the performance of the proposed adaptive control scheme we consider
three simulation scenarios: a) A simple point-to-point motion to evaluate the
trajectory tracking performance, b) a scenario involving the tracking of a
periodic motion with the rear right foot contacting a slippery surface and c)
a scenario involving the tracking of a periodic motion with global slippage,
i.e. all four legs are contacting a slippery surface. For the simulations, the
model of a Unitree Go1 robot is utilized in the Gazebo environment and a
control cycle of 2ms is considered. The parameters utilized are
$k_{p}=3000,k_{o}=150,\mathbf{K}_{v}=\text{diag}(550\mathbf{I}_{3},55\mathbf{I}_{3}),w_{0}=35,\alpha=150$.
In Fig.4 the simulation environment is shown, with the yellow area
representing the slippery terrain considered in the second scenario (slippage
of the rear right foot).

Figure 4: The initial configuration of the simulations. Yellow area: The
slippery area considered for the second scenario.

###  IV-A Scenario 1: Point to point motion

For this scenario, a terrain with a static friction coefficient of $1.4$ is
considered, representing a non-slippery terrain. The desired trajectory is
generated online by the following first order dynamical system:
$\dot{\mathbf{p}}_{d}(t)=\mathbf{p}_{d}(t)-\mathbf{p}_{T}$, with
$\mathbf{p}_{T}=\mathbf{p}_{d}(0)+[0.1\;0.05\;-0,005]^{\intercal}$ being the
constant target. The initial actual and desired values are
$\mathbf{p}(0)=[-0.043\;-0.0037\;0.356]^{\intercal}$m and
$\mathbf{p}_{d}(0)=[-0.023\;0.0063\;0.355]^{\intercal}$m in order to impose an
initial position error of $\mathbf{e}_{p}=[-2\;\;-1\;\;0.1]^{\intercal}$cm. In
Fig.5 the actual position evolution is compared to the desired trajectory, in
which one can notice the tracking performance. Notice that the tracking
performance is affected by the unmodelled joint friction that acts as a
disturbance to the system, with the $z$-direction being the most disturbed
direction, due to the manipulability ellipsoid of the given robot’s
configuration. One could possibly reduce this steady state error by further
tuning the control gains (as no extensive tuning was performed), or by
incorporating an additional integral term to the controller.

Figure 5: [Scenario 1: Point-to-point motion] Time evolution of the actual and
desired position.

###  IV-B Scenario 2: One-foot slippage

For the second scenario, the rear right foot of the robot ($i=3$) is
considered to contact a slippery surface having a static friction coefficient
of $0.4$, which is considered to be unknown for the controller. For the rest
of the feet a non-slippery surface is considered. For comparison, two tests
are performed, namely one with the adaptive mechanism and the other without
it. The desired trajectory involves a periodic sinusoidal motion, executing an
ellipse on the $x-z$ plane, for position and a periodic rotation around the
$x$-axis for orientation. The frequency of the periodic trajectory is $0.7$Hz
and $0.2$Hz in position and orientation respectively. The weights of
distribution along the $x$ direction of each leg (which is equal to the ones
along the $y$ direction), i.e. $w_{i,1}$, are depicted in Fig.6, alongside
with the slippage probability provided by the estimator, i.e. $1-P_{i}$.
Notice the rise of the value of $w_{3,1}$ (the leg that slips), which results
in appending less force along the $x-y$ directions of the third leg. Further
notice that the third leg stops slipping after the adaptation which means that
the force converged to a value within the friction cone and the system reaches
a stable steady-state condition. In Fig.7, the position and orientation errors
are depicted, with and without the proposed adaptive scheme for comparison
purposes. Notice that without the proposed adaptation mechanism, the system is
not able to maintain its stability, as the robot loses contact with the
environment at $t\approx 4.5$s. Last, notice that $\beta=1$ during the whole
simulation, due to equation (14) and the fact that $w_{i,1}=35,\forall
i=1,2,4$, which means that the first adaptation layer can sufficiently provide
a solution by dynamically distributing the control effort.

Figure 6: [Scenario 2: One-foot slippage] Weight adaptation due to the first
layer (the second layer is not enabled). Figure 7: [Scenario 2: One-foot
slippage] Position and orientation error norms with and without adaptation.

###  IV-C Scenario 3: Global slippage

For this scenario, all four legs of the robot are considered to contact the
slippery surface having a static friction coefficient of $0.4$. For
comparison, we performed two tests, namely one with the adaptive control
scheme and one without it and the same trajectory with that of the second
scenario is considered. The weights of distribution along the $x$ direction of
each leg (which is equal to the ones along the $y$ direction), i.e. $w_{i,1}$,
as well as the time-scaling parameter $\beta(t)$ are depicted in Fig.8,
alongside with the slippage probability provided from the estimator, i.e.
$1-P_{i}$. Notice the rise of the values of all $w_{i,1}$, $i=1,...,4$, which
results in slowing down the motion, which is reflected by the reduction of
$\beta(t)$ which converges to the value of $\beta\approx 0.77$ after $t\approx
5$s. In Fig.9, the evolution of the position of the CoM in time is depicted
both with and without the proposed control scheme. Notice that without the
proposed adaptation mechanism, the system is, also in this case, not able to
maintain the stability of the system as the robot, signified by the drop of
the CoM in Fig.9. Last, notice the smooth on-line time-scaling of the
trajectory occurred after $t\approx 5$s.

Figure 8: [Scenario 3: Global slippage] Weight adaptation due to the first and
second layer. Figure 9: [Scenario 3: Global slippage] Evolution of the
position of the CoM in time, with and without adaptation.

##  V Experimental validation

The implementation of the is done utilizing a real Unitree Go1 robot, to
validate the adaptation performed by the first layer of the adaptation
mechanism. In particular, a 6DOF IMU is attached to the second leg of the
robot, as shown in Fig.10, which is in contact with a slippery surface, (i.e.
lubricant is utilized to emulate the slippery area below the second leg),
while the pose of the robot is found on-line via an external camera with an
off-the-shelf visual odometry
system111https://github.com/IntelRealSense/librealsense/blob/master/doc/t265.md
and therefore initial robot’s pose is considered as the world frame for the
experiment. The proposed adaptive scheme parameters are set to
$k_{p}=2400,k_{o}=,\mathbf{K}_{v}=\text{diag}(280\mathbf{I}_{3},28\mathbf{I}_{3}),w_{0}=35,\alpha=1000$.
The robot was commanded to move along the $x$-axis with a similar to the
second simulation periodic trajectory for the axis of motion, having a
frequency of $0.4$Hz. Fig.10 depicts the experimental setup with the robot
being in the initial configuration, while in Fig. 11, the weight corresponding
to the $x-y$ directions of the second leg is given, alongside with the
slippage probability estimate; the weights of the rest of the legs remained
unaltered during the experiment. In Fig.12 the evolution of position in time
is depicted utilizing the adaptive scheme and without its utilization, for
comparison. Notice that the activation of the first adaptation layer results
in maintaining stability, while when executing the same scenario without
enabling the adaptation mechanism the robot is not able to maintain stability
at $t\approx 14.6$s. Further, notice that without the adaptation mechanism the
tracking performance is affected by the slippage of the second leg, as it
triggers unmodelled dynamics.

Figure 10: The experimental setup and initial configuration of the robot.
Figure 11: The second weight during the experimental validation. Figure 12:
Evolution of the position of the CoM in time, with and without adaptation.

##  VI Conclusions and future work

In this work, an adaptive trajectory tracking controller is proposed for
quadruped robots, which involves two prioritized layers of adaptation for
minimizing the slippage of one or multiple legs. The first adaptation layer
considers the dynamic distribution of the control effort among the legs, given
the slippage probability for each leg. The second layer, which is enabled only
if the problem cannot be solved by the dynamic distribution of the effort,
which may occur when all for legs slip, acts on the time-scaling of the
trajectory by dynamically and smoothly slowing down the motion, without
affecting the spatial properties of the task. The proposed method is proven to
be asymptotically stable. Furthermore, it is shown through simulations and
experiments that the method equips the system with robustness, as it is able
to minimize the slippage of the legs and it ensures the stability and
controllability of the robot. This is accomplished without sacrificing the
task space trajectory, this feature is particularly useful in both visual and
depth based Simultaneous Localization and Mapping applications where stable
and precise movement is essential for the performance of the approach. Future
work will mainly focus on employing the proposed adaptation control scheme
coupled with our contact estimation approach as the foundation for the
development of a model-based dexterous and slippery-robust dynamic locomotion
algorithm. By capitalizing on the real-time adjustment of the weights for each
leg we aim at developing algorithms for walking and running in terrain
agnostic environments while maintaining the task space desired trajectory.

## Appendix A

After substituting (11) in (5), we get the following closed loop system
dynamics:

$\displaystyle\mathbf{H}_{c}\dot{\mathbf{e}}_{v}$
$\displaystyle=-(\mathbf{C}_{c}+\mathbf{K}_{v})\mathbf{e}_{v}-\begin{bmatrix}k_{p}\mathbf{e}_{p}\\\
k_{o}\mathbf{e}_{o}\end{bmatrix},$ (15)
$\displaystyle\begin{bmatrix}\dot{\mathbf{e}}_{p}\\\
\dot{\mathbf{e}}_{o}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\mathbf{I}_{3}&\mathbf{0}_{3}\\\
\mathbf{0}_{3}&\mathbf{J}_{l}(\mathbf{e}_{o})\end{bmatrix}\mathbf{e}_{v},$
(16)

where $\mathbf{J}_{l}(\mathbf{e}_{o})\in\mathbb{R}^{3\times 3}$ the matrix
mapping the orientation part of $\mathbf{e}_{v}$ to $\dot{\mathbf{e}}_{o}$, as
detailed in [19], for which the following holds:
$\mathbf{J}_{l}^{\intercal}\mathbf{e}_{o}=\mathbf{J}_{l}\mathbf{e}_{o}=\mathbf{e}_{o}$
(as shown in [19]).

###### Theorem 1

The origin of the state-space of the system (15), i.e.
$(\mathbf{e}_{p},\mathbf{e}_{o},\mathbf{e}_{v})=(\mathbf{0},\mathbf{0},\mathbf{0})$,
is globally asymptotically stable.

###### Proof:

Consider the following candidate Lyapunov function:

$L=\frac{k_{p}}{2}\|\mathbf{e}_{p}\|^{2}+\frac{k_{o}}{2}\|\mathbf{e}_{o}\|^{2}+\frac{1}{2}\mathbf{e}_{v}^{\intercal}\mathbf{H}_{c}\mathbf{e}_{v}.$
(17)

By taking its time derivative, we get:

$\dot{L}=k_{p}\mathbf{e}_{p}^{\intercal}\dot{\mathbf{e}}_{p}+k_{o}\mathbf{e}_{o}^{\intercal}\dot{\mathbf{e}}_{o}+\frac{1}{2}\mathbf{e}_{v}^{\intercal}\dot{\mathbf{H}}_{c}\mathbf{e}_{v}+\mathbf{e}_{v}^{\intercal}\mathbf{H}_{c}\dot{\mathbf{e}}_{v}.$
(18)

After substituting $\mathbf{H}_{c}\dot{\mathbf{e}}_{v}$ from (15) to (18) and
utilizing the skew symmetric property, i.e.
$\mathbf{e}_{v}^{\intercal}\left(\dot{\mathbf{H}}_{c}-2\mathbf{C}_{c}\right)\mathbf{e}_{v}=0$,
we get:

$\dot{L}=k_{p}\mathbf{e}_{p}^{\intercal}\dot{\mathbf{e}}_{p}+k_{o}\mathbf{e}_{o}^{\intercal}\dot{\mathbf{e}}_{o}+\mathbf{e}_{v}^{\intercal}\left(-\mathbf{K}_{v}\mathbf{e}_{v}-\begin{bmatrix}k_{p}\mathbf{e}_{p}\\\
k_{o}\mathbf{e}_{o}\end{bmatrix}\right).$ (19)

By utilizing (16) and the property
$\mathbf{J}_{l}^{\intercal}\mathbf{e}_{o}=\mathbf{J}_{l}\mathbf{e}_{o}=\mathbf{e}_{o}$,
(19) becomes:

$\dot{L}=-\mathbf{K}_{v}\mathbf{e}_{v}^{\intercal}\mathbf{e}_{v},$ (20)

which is less or equal to zero for all
$\mathbf{e}_{p}\neq\mathbf{0},\mathbf{e}_{o}\neq\mathbf{0},\mathbf{e}_{v}\neq\mathbf{0}$.
Hence, by invoking the LaSalle theorem we can conclude that the origin is
globally asymptotically stable. ∎

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