# Semicrossed products of operator algebras and their
$\mathrm{C}^{*}$-envelopes

Evgenios Kakariadis  Department of Mathematics
University of Athens
15784 Athens
GREECE<EMAIL_ADDRESS>and  Elias G. Katsoulis  Department of Mathematics
University of Athens
15784 Athens
GREECE  Alternate address: Department of Mathematics
East Carolina University
Greenville, NC 27858
USA<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{A}}$ be a unital operator algebra and let $\alpha$ be an
automorphism of ${\mathcal{A}}$ that extends to a $*$-automorphism of its
$\mathrm{C}^{*}$-envelope $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. In
this paper we introduce the isometric semicrossed product
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ and we
show that
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})\simeq\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$.
In contrast, the $\mathrm{C}^{*}$-envelope of the familiar contractive
semicrossed product ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ may not
equal $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$.
Our main tool for calculating $\mathrm{C}^{*}$-envelopes for semicrossed
products is the concept of a relative semicrossed product of an operator
algebra, which we explore in the more general context of injective
endomorphisms.

As an application, we extend the main result of [9] to tensor algebras of
$\mathrm{C}^{*}$-correspondences. We show that if
${\mathcal{T}}_{{\mathcal{X}}}^{+}$ is the tensor algebra of a
$\mathrm{C}^{*}$-correspondence $({\mathcal{X}},{\mathfrak{A}})$ and $\alpha$
a completely isometric automorphism of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
that fixes the diagonal elementwise, then the contractive semicrossed product
satisfies
$\mathrm{C}^{*}_{\text{env}}({\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+})\simeq{\mathcal{O}}_{{\mathcal{X}}}\times_{\alpha}{\mathbb{Z}}$,
where ${\mathcal{O}}_{{\mathcal{X}}}$ denotes the Cuntz-Pimsner algebra of
$({\mathcal{X}},{\mathfrak{A}})$.

2000 Mathematics Subject Classification. 47L55, 47L40, 46L05, 37B20

Key words and phrases: semicrossed product, crossed product

Second author was partially supported by a grant from ECU

##  1\. Introduction and preliminaries

In this paper, we offer three choices for defining the semicrossed product of
an operator algebra ${\mathcal{A}}$ by a unital, completely contractive
endomorphism $\alpha$ of ${\mathcal{A}}$ (Definitions 1.1 and 1.2.) In all
cases, the resulting algebras contain a completely isometric copy of
${\mathcal{A}}$ and a ”universal” operator that implements the covariance
relations. In the case where ${\mathcal{A}}$ is a $\mathrm{C}^{*}$-algebra and
$\alpha$ preserves adjoints, all three choices produce the same operator
algebra, Peters’ semicrossed product of a $\mathrm{C}^{*}$-algebra [25] by an
endomorphism. (Semicrossed products of $\mathrm{C}^{*}$-algebras have been
under investigation by various authors [1, 2, 4, 6, 7, 8, 14, 22, 26],
starting with the work of Arveson [2] in the late sixties.) In the general
(non-selfadjoint) case however, the semicrossed products we introduce here may
lead to non-isomorphic operator algebras. The main objective of this paper is
to clarify the relation between the three semicrossed products and calculate
their $\mathrm{C}^{*}$-envelope, whenever possible.

The present paper is a continuation of the recent work of Davidson and the
second named author in [9]. In the language of the present paper, the main
objective of [9] was to show that in the special case where ${\mathcal{A}}$ is
Popescu’s non-commutative disc algebra [27] and $\alpha$ a completely
isometric automorphism of ${\mathcal{A}}$, all three semicrossed products
coincide. One of the main results of this paper, Theorem 2.6, shows that two
of these semicrossed products, the isometric and the relative one, coincide
for any operator algebra ${\mathcal{A}}$ and any completely isometric
automorphism $\alpha$ of ${\mathcal{A}}$. To prove this, we had to abandon the
rather intricate but ad-hoc arguments of the second half of [9] and instead
adopt an abstract approach. Theorem 2.6 focuses now any further research on
semicrossed products to the study of the other two, the isometric and the
contractive semicrossed product. For these two, there are examples to show
that they do not coincide in general (Remark 2.8). Nevertheless, with Theorem
2.6 in hand, we show that they do coincide in the case of a tensor algebra of
a $\mathrm{C}^{*}$-correspondence and and a completely isometric isomorphism
of the algebra that fixes its diagonal (Corollary 2.6). This not only
generalizes the main result of [9] to a broader context but also paves the way
for additional results of this kind to come in the future.

The various semicrossed products we define in this paper are actually closed
images of the following universal semicrossed product under concrete
representations.

######  Definition 1.1.

Let $\alpha$ be a unital, completely contractive endomorphism of an operator
algebra ${\mathcal{A}}$. A contractive (isometric) covariant representation
$(\pi,K)$ of $({\mathcal{A}},\alpha)$ is a completely contractive
representation $\pi$ of ${\mathcal{A}}$ on a Hilbert space ${\mathcal{H}}$ and
a contraction (resp. isometry) $K\in{\mathcal{B}}({\mathcal{H}})$ so that

$\pi(A)K=K\pi(\alpha(A))\quad\text{for all}\,A\in{\mathcal{A}}.$

The contractive (resp. isometric) semicrossed product
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ (resp.
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$) for the
system $({\mathcal{A}},\alpha)$ is the universal operator algebra generated by
a copy of ${\mathcal{A}}$ and a contraction (resp. isometry) ${\mathfrak{V}}$
so that $A{\mathfrak{V}}={\mathfrak{V}}\alpha(A)$, for all
$A\in{\mathcal{A}}$.

The contractive semicrossed product has, by definition, a rich representation
theory which unfortunately makes it very intractable. This was first observed
in [9] based on the famous example of Varopoulos [28] regarding three
commuting contractions that do not satisfy the usual von Neumann inequality.
Nevertheless, there are significant cases where the contractive semicrossed
product has been completely identified. These include the case where
${\mathcal{A}}$ is a $\mathrm{C}^{*}$-algebra [17, 25] and the case where
${\mathcal{A}}$ is the non-commutative disc algebra ${\mathfrak{A}}_{n}$ and
$\alpha$ is an isometric automorphism [9].

The isometric semicrossed product is the (closed) image of
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ under the representation which
restricts to the entries where the contractions $K$ are actually isometries.
We believe that this is a more tractable object and as we shall see, in the
case where $\alpha$ is a completely isometric automorphism, i.e., it extends
to an automorphism of the $\mathrm{C}^{*}$-envelope
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$ of ${\mathcal{A}}$, we can
identify the $\mathrm{C}^{*}$-envelope of
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ as the crossed product
$\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$. The
main tool for establishing this result is the concept of a relative
semicrossed product.

Recall that a $\mathrm{C}^{*}$ algebra ${\mathcal{C}}$ is said to be a
$\mathrm{C}^{*}$-cover for a subalgebra ${\mathcal{A}}\subseteq{\mathcal{C}}$
provided that ${\mathcal{A}}$ generates ${\mathcal{C}}$ as a
$\mathrm{C}^{*}$-algebra, i.e., ${\mathcal{C}}=C^{*}({\mathcal{A}})$. If
${\mathcal{C}}$ is a $\mathrm{C}^{*}$-cover for ${\mathcal{A}}$, then
${\mathcal{J}}_{{\mathcal{A}}}$ will denote the Šilov ideal of ${\mathcal{A}}$
in ${\mathcal{C}}$. Therefore,
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})={\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
and the restriction of the natural projection
$q:{\mathcal{C}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ on
${\mathcal{A}}$ is a completely isometric representation of ${\mathcal{A}}$.
(Any ideal ${\mathcal{J}}\subseteq{\mathcal{C}}$, with the property that the
restriction of the natural projection
${\mathcal{C}}\rightarrow{\mathcal{C}}/{\mathcal{J}}$ on ${\mathcal{A}}$ is a
complete isometry, is called a boundary ideal and
${\mathcal{J}}_{{\mathcal{A}}}$ is the largest such ideal.)

######  Definition 1.2.

Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be an
$*$-endomorphism of ${\mathcal{C}}$ that leaves ${\mathcal{A}}$ invariant. The
subalgebra of Peters’ semicrossed product
${\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$, which is generated by
${\mathcal{A}}\subseteq{\mathcal{C}}\subseteq{\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$
and the universal isometry
${\mathfrak{V}}\in{\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$, is denoted by
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and is said to be
a relative semicrossed product for the system $({\mathcal{A}},\alpha)$.

Therefore, the relative semicrossed product
${\mathcal{A}}\times_{{\mathcal{C}}\,,\alpha}{\mathbb{Z}}^{+}$ comes from the
representation of ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ that
restricts to the entries where $\pi$ and $\alpha$ are $*$-extendable to
${\mathcal{C}}$ and the contraction $K$ satisfies the covariance relation with
these extensions. It seems plausible that non-isomorphic
$\mathrm{C}^{*}$-covers for ${\mathcal{A}}$ and varying extensions for the
endomorphism $\alpha$ could produce non-isomorphic relative semicrossed
products. It turns out that under a reasonable technical requirement, i.e.,
invariance of the Shilov ideal, all such relative semicrossed products are
completely isometrically isomorphic (Proposition 2.3). In particular, this
requirement is satisfied when $\alpha$ is a completely isometric automorphism
of ${\mathcal{A}}$; in that case all relative semicrossed products for
$({\mathcal{A}},\alpha)$ are completely isometrically isomorphic to each
other.

##  2\. The relative semicrossed product and its $\mathrm{C}^{*}$-envelope

We begin this section with some preliminary results. The first one is a
standard result that shows how to lift an injective $*$-endomorphism of a
$\mathrm{C}^{*}$-algebra to an automorphism of a possibly larger
$\mathrm{C}^{*}$-algebra.

######  Proposition 2.1.

If $\alpha$ is an injective endomorphism of a C*-algebra ${\mathfrak{A}}$,
then there is a unique triple $({\mathfrak{B}},\beta,j)$ (up to isomorphism)
where ${\mathfrak{B}}$ is a C*-algebra, $\beta$ is an automorphism of
${\mathfrak{B}}$ and $j$ is a $*$-monomorphism of ${\mathfrak{A}}$ into
${\mathfrak{B}}$ such that $\beta j=j\alpha$ and
${\mathfrak{B}}=\overline{\bigcup_{k\geq 0}\beta^{-k}j({\mathfrak{A}})}$.

To paraphrase, there is a unique minimal C*-algebra ${\mathfrak{B}}$
containing ${\mathfrak{A}}$ with an automorphism $\beta$ satisfying
$\beta|_{\mathfrak{A}}=\alpha$.

######  Proof..

Consider the inductive limit ${\mathfrak{B}}$ of the system

$\begin{CD}{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>{\mathfrak{A}}_{4}@>{\alpha_{4}}>{}>\cdots,\end{CD}$

where ${\mathfrak{A}}_{i}={\mathfrak{A}}$ and $\alpha_{i}=\alpha$, for all
$i\in{\mathbb{N}}$. Let $j_{i}$ be the associated $*$-monomorphism from
${\mathfrak{A}}={\mathfrak{A}}_{i}$ to ${\mathfrak{B}}$. This map is defined
as $j_{i}(A)=(0,0,\dots,0,A,\alpha(A),\alpha^{2}(A),\dots)$, with the
understanding that the infinite tuple in the definition signifies the
appropriate equivalence class. Define $j=j_{1}$.

The system

$\begin{CD}{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>\cdots\\\
@V{\alpha}V{}V@V{\alpha}V{}V@V{\alpha}V{}V\\\
{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>\cdots\end{CD}$

gives rise to an $*$-automorphism $\beta$ of ${\mathfrak{B}}$ defined as

$\beta(A_{1},A_{2},A_{3},\dots)=(\alpha(A_{1}),\alpha(A_{2}),\alpha(A_{3}),\dots),\quad
A_{i}\in{\mathfrak{A}}_{i},i\in{\mathbb{N}}.$

Clearly, $\beta j=j\alpha$. The inverse of $\beta$ on $\bigcup_{k\geq
1}j_{k}({\mathfrak{A}}_{k})$ satisfies

$\beta^{-1}(A_{1},A_{2},A_{3},\dots)=(0,\alpha(A_{1}),\alpha(A_{2}),\alpha(A_{3}),\dots),\quad
A_{i}\in{\mathfrak{A}}_{i},i\in{\mathbb{N}}.$

and so, if $A\in{\mathfrak{A}}$, then

$\beta^{-k}(A,0,0,\dots)=(0,0,\dots,0,A,\alpha(A),\alpha^{2}(A),\dots).$

Therefore, $\bigcup_{k\geq 0}\beta^{-k}j({\mathfrak{A}})$ is dense in
${\mathfrak{B}}$. ∎

We now fix some notation and use the previous result to construct a useful
embedding of ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.

Let ${\mathcal{A}}$ be an operator algebra and let $\alpha$ be a completely
contractive endomorphism of ${\mathcal{A}}$. If $\pi$ is a completely
contractive representation of ${\mathcal{A}}$ on a Hilbert space
${\mathcal{H}}$, we define

$\widetilde{\pi}:{\mathcal{A}}\longrightarrow{\mathcal{B}}({\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}^{+}))$

so that

(1)
$\widetilde{\pi}(A)\equiv(\pi(A),\pi(\alpha(A)),\pi(\alpha^{2}(A)),\dots),\quad
A\in{\mathcal{A}}.$

Let $V_{{\mathcal{H}}}\equiv I\otimes V$, where $V$ denotes the unilateral
shift on $\ell^{2}({\mathbb{Z}}^{+})$. The pair
$(\widetilde{\pi},V_{{\mathcal{H}}})$ forms a contractive covariant
representation of $({\mathcal{A}},\alpha)$ and the associated representation
of ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ is denoted as
$\widetilde{\pi}\times V_{{\mathcal{H}}}$. If $\alpha$ happens to be a
completely isometric automorphism of ${\mathcal{A}}$, we also have the
representation
$\widehat{\pi}:{\mathcal{A}}\rightarrow{\mathcal{B}}(H\otimes\ell^{2}({\mathbb{Z}}))$,
such that

$\widehat{\pi}(A)\equiv(\dots,\pi(\alpha^{-1}(A)),\pi(A),\pi(\alpha(A)),\pi(\alpha^{2}(A)),\dots),\quad
A\in{\mathcal{A}},$

the unitary $U_{{\mathcal{H}}}=I\otimes U$, where $U$ is the bilateral shift
on $\ell^{2}({\mathbb{Z}})$ and the associated representation
$\widehat{\pi}\times U_{{\mathcal{H}}}$ of
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.

######  Proposition 2.2.

Let $\alpha$ is an injective endomorphism of a C*-algebra ${\mathfrak{A}}$ and
let $({\mathfrak{B}},\beta,j)$ be the triple of Proposition 2.1. Then
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ embeds completely
isometrically in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Furthermore,
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ becomes a $\mathrm{C}^{*}$-cover
for ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.

######  Proof..

Let $\pi$ be a faithful representation of ${\mathfrak{A}}$ on a Hilbert space
${\mathcal{H}}$. Since every representation of ${\mathfrak{A}}$ is a direct
sum of cyclic representations, the GNS construction implies that there exists
a representation $\pi_{\beta}$ of ${\mathfrak{B}}$ on a Hilbert space
${\mathcal{H}}_{\beta}\supseteq{\mathcal{H}}$ so that ${\mathcal{H}}$ is
reducing for $\pi_{\beta}(j({\mathfrak{A}}))$ and
$\pi_{\beta}(j(A))\mid_{{\mathcal{H}}}=\pi(A)$ for all $A\in{\mathfrak{A}}$.

By gauge invariance, $\widetilde{\pi}\times V_{{\mathcal{H}}}$ is a completely
isometric representation for ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$;
therefore the same is true for the representation
$\widehat{\pi_{\beta}}\,j\times U_{{\mathcal{H}}_{\beta}}$. Now notice that
the representation $\widehat{\pi_{\beta}}$ is faithful on $\bigcup_{k\geq
0}\beta^{-k}j({\mathfrak{A}})$ and so, by inductivity, on all of
${\mathfrak{B}}$. By gauge invariance, the representation
$\widehat{\pi_{\beta}}\times U_{{\mathcal{H}}_{\beta}}$ is also faithful on
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. The proposition now follows by
comparing the ranges of $\widehat{\pi_{\beta}}\,j\times
U_{{\mathcal{H}}_{\beta}}$ and $\widehat{\pi_{\beta}}\times
U_{{\mathcal{H}}_{\beta}}$. ∎

As we shall see in Theorem 2.5, ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ is
actually the $\mathrm{C}^{*}$-envelope of
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.

Let ${\mathcal{A}}$ be an operator algebra and let ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$. Let $\alpha$ be a $*$-endomorphism
of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$ and
${\mathcal{J}}_{{\mathcal{A}}}$ and let
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
be defined as
$\dot{\alpha}(X+{\mathcal{J}}_{{\mathcal{A}}})=\alpha(X)+{\mathcal{J}}_{{\mathcal{A}}}$,
$X\in{\mathcal{C}}$. In this context, there are two relative semicrossed
products to be considered,
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
The following proposition, which clarifies the relation between these two
semicrossed product, is an application of two significant recent results in
the theory of maximal dilations for completely contractive maps. First,
Dritschel and McCullough [11] have recently proven that every completely
contractive map $\varphi:{\mathcal{A}}\rightarrow B({\mathcal{H}})$ admits a
maximal dilation $(\Phi,{\mathcal{K}})$, i.e., a dilation
$\Phi:{\mathcal{A}}\rightarrow B({\mathcal{K}})$ so that any further dilation
of $\Phi$ has $\Phi$ as a direct summand. Furthermore, Muhly and Solel [24]
have shown that any such maximal dilation $\Phi$ extends (uniquely) to a
$*$-representation of any $\mathrm{C}^{*}$-cover of ${\mathcal{A}}$.

######  Proposition 2.3.

Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be a
$*$-endomorphism of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$
and ${\mathcal{J}}_{{\mathcal{A}}}$. Then the relative semicrossed products
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$
are completely isometrically isomorphic.

######  Proof..

Let
$F=\sum_{n=0}^{k}\,{\mathfrak{V}}^{n}A_{n}\in{\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$
and
$F^{\prime}=\sum_{n=0}^{k}\,{\mathfrak{V}}^{n}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\in{\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
We have to show that the homomorphism $F\mapsto F^{\prime}$ is a completely
isometric map.

Let $\pi$ be a faithful representation of ${\mathcal{C}}$ on a Hilbert space
${\mathcal{H}}$ and let $(\widetilde{\pi},V_{{\mathcal{H}}})$ be as in the
beginning of the section (see (1)). Consider the completely isometric map

$\varphi:{\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\longrightarrow{\mathcal{B}}({\mathcal{H}}):A+{\mathcal{J}}_{{\mathcal{A}}}\longmapsto\pi(A),\quad
A\in{\mathcal{A}}.$

According to our earlier discussion, there is a maximal dilation
$(\Phi,{\mathcal{K}})$ of $\varphi$ which extends uniquely to a representation
of ${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ such that

$P_{{\mathcal{H}}}\Phi(A+{\mathcal{J}}_{{\mathcal{A}}})|_{{\mathcal{H}}}=\varphi(A+{\mathcal{J}}_{{\mathcal{A}}})=\pi(A),$

for all $A\in A$. Since
$P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=P_{{\mathcal{H}}}\otimes
I$, we have that

$P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\widetilde{\Phi}(A+{\mathcal{J}}_{{\mathcal{A}}}))|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=\widetilde{\pi}(A+{\mathcal{J}}_{{\mathcal{A}}}),$

for all $A\in A$. Also,
$V_{{\mathcal{K}}}|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=V_{{\mathcal{H}}}$
and so

$\displaystyle\|F\|$
$\displaystyle=\|\sum_{n=0}^{k}\,V^{n}_{{\mathcal{H}}}\widetilde{\pi}(A_{n})\|$
$\displaystyle=\|P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\left(\sum_{n=0}^{k}\,V^{n}_{{\mathcal{K}}}\widetilde{\Phi}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\right)|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\|$
$\displaystyle\leq\|\sum_{n=0}^{k}\,V^{n}_{{\mathcal{K}}}\widetilde{\Phi}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\|\leq\|F^{\prime}\|.$

The same is also true for all the matrix norms and so the map
$F^{\prime}\mapsto F$ is well defined and completely contractive. By reversing
the roles of ${\mathcal{A}}$ and ${\mathcal{A}}/{\mathcal{J}}({\mathcal{A}})$
in the previous arguments, we can also prove that $F\mapsto F^{\prime}$ is
completely contractive, and the conclusion follows. ∎

Now we wish to identify the $C^{*}$-envelope of
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$. From the
previous result we know that it coincides with the $C^{*}$-envelope of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
In the following we consider the case where
$\alpha:{\mathcal{C}}\rightarrow{\mathcal{C}}$ is injective. This is easily
seen to imply that
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
is an injective $*$-homomorphism. Indeed,

######  Lemma 2.4.

Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be an injective
$*$-endomorphism of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$
and ${\mathcal{J}}_{{\mathcal{A}}}$. Then
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
is an injective $*$-homomorphism.

######  Proof..

In that case, $\alpha$ is a completely isometric map. Therefore,

$\left\|q(A)+\ker\dot{\alpha}\right\|=\left\|\dot{\alpha}(A+{\mathcal{J}}_{{\mathcal{A}}})\right\|=\left\|\alpha(A)+{\mathcal{J}}_{{\mathcal{A}}}\right\|=\left\|\alpha(A)\right\|=\left\|A\right\|,$

since ${\mathcal{J}}_{{\mathcal{A}}}$ is a boundary ideal and
$\alpha({\mathcal{A}})\subseteq{\mathcal{A}}$. The same argument holds for all
the matrix norms. Thus $\ker\dot{\alpha}$ is a boundary ideal of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$. However,
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ is the $\mathrm{C}^{*}$-envelope
of ${\mathcal{A}}$ and so it contains no non-trivial boundary ideals for
${\mathcal{A}}$. Thus $\ker\dot{\alpha}=(0)$. ∎

The following is the main technical result of the section.

######  Theorem 2.5.

Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let
${\mathcal{J}}_{{\mathcal{A}}}$ be the Šilov ideal of ${\mathcal{A}}$ in
${\mathcal{C}}$. Let $\alpha$ be an injective $*$-endomorphism of
${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$ and
${\mathcal{J}}_{{\mathcal{A}}}$. Then

$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+})\simeq{\mathfrak{B}}\times_{\beta}{\mathbb{Z}}\,,$

where $({\mathfrak{B}},\beta,j)$ is the unique triple of Proposition 2.1
associated with the injective $*$-endomorphism $\dot{\alpha}$ of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$.

######  Proof..

Proposition 2.3 shows that it suffices to identify the $C^{*}$-envelope of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.

If $({\mathfrak{B}},\beta,j)$ is the unique triple of Proposition 2.1
associated with the injective $*$-endomorphism $\dot{\alpha}$ of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ then Proposition 2.2 shows that
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{\dot{\alpha}}{\mathbb{Z}}^{+}$,
and therefore
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$,
embeds completely isometrically in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$.
Moreover, ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ is a
$\mathrm{C}^{*}$-cover for
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
Let ${\mathcal{J}}$ be the Šilov ideal of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$
in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. We are to show that
${\mathcal{J}}=\\{0\\}$.

Assume to the contrary that ${\mathcal{J}}\neq\\{0\\}$. Since ${\mathcal{J}}$
is invariant by automorphisms of the $\mathrm{C}^{*}$-cover, it remains
invariant by the natural gauge action on
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Therefore it has non-trivial
intersection with the fixed point algebra of the natural gauge action, i.e.,
${\mathcal{J}}\cap{\mathfrak{B}}\neq\\{0\\}$. However

${\mathfrak{B}}=\overline{\bigcup_{k\geq
0}\beta^{-k}j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})}$

and therefore by inductivity there exists $k\in{\mathbb{N}}$ so that

${\mathcal{J}}\cap\beta^{-k}j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\neq\\{0\\}.$

However, $\beta$ acts by conjugating with a unitary in
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Since ${\mathcal{J}}$ is an ideal
of ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$, the above implies that

${\mathcal{J}}\cap j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\neq\\{0\\}.$

But then $j^{-1}\left({\mathcal{J}}\cap
j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\right)$ is a non-zero boundary
ideal for ${\mathcal{A}}$ in ${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$, a
contradiction. ∎

In [9] Davidson and the second named author proved that

$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}_{n}\times_{\alpha}{\mathbb{Z}}^{+})={\mathcal{O}}_{n}\times_{\alpha}{\mathbb{Z}},$

where ${\mathcal{A}}_{n}$ is Popescu’s non-commutative disc algebra [27],
$\alpha$ a is a (completely) isometric automorphism of ${\mathcal{A}}_{n}$ and
${\mathcal{O}}_{n}$ denotes the Cuntz algebra generated by $n$ isometries. A
dilation result in the first half of [9] reduces the problem of calculating
the $\mathrm{C}^{*}$-envelope of
${\mathcal{A}}_{n}\times_{\alpha}{\mathbb{Z}}^{+}$ to essentially verifying
that
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}_{n}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})={\mathcal{O}}_{n}\times_{\alpha}{\mathbb{Z}}$.
It takes the second half of [9] and intricate use of the representation theory
for ${\mathcal{O}}_{n}$ to verify that claim. The next result establishes the
same claim for arbitrary operator algebras using only abstract arguments.

######  Theorem 2.6.

Let ${\mathcal{A}}$ be an operator algebra and $\alpha$ be an automorphism of
${\mathcal{A}}$ that extends to a $*$-automorphism of
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. Then, any relative semicrossed
product for $({\mathcal{A}},\alpha)$ is completely isometrically isomorphic to
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$. Hence,

$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})\simeq\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}.$

######  Proof..

In light of Theorem 2.5, it suffices to show that
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ dilates
to a relative semicrossed product. This is done as follows.

Let
${\mathfrak{V}}\in{\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$
be the universal isometry acting on a Hilbert space ${\mathfrak{H}}$ and let
${\mathcal{H}}$ be the direct limit Hilbert space of the inductive system

$\begin{CD}{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}<EMAIL_ADDRESS>
For each $A\in{\mathcal{A}}$, the commutative diagram

$\begin{CD}{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>\cdots\\\
@V{A}V{}V@V{\alpha^{-1}(A)}V{}V@V{\alpha^{-2}(A)}V{}V\\\
{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>\cdots\end{CD}$

defines an operator $\pi(A)\in{\mathcal{B}}({\mathcal{H}})$. It is easily seen
that $\pi$ defines a completely isometric representation of ${\mathcal{A}}$ on
${\mathcal{H}}$. Consider now the unitary $U\in{\mathcal{B}}({\mathcal{H}})$
defined as

$U(h_{1},h_{2},h_{3},\dots)=(h_{2},h_{3},\dots),\quad
h_{i}\in{\mathfrak{H}},i\in{\mathbb{N}}.$

and notice that $\pi(\alpha(A))=U^{*}\pi(A)U$, $A\in{\mathcal{A}}$. Therefore,
the conjugation by $U$ defines a $*$-automorphism of
${\mathcal{C}}\equiv\mathrm{C}^{*}(\pi({\mathcal{A}}))$, which extends
$\alpha$ and is denoted by the same symbol as well. Therefore,

${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}\simeq{\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$

and the conclusion follows from Theorem 2.5. ∎

######  Remark 2.7.

In light of Theorem 2.6, we wonder whether one can compute the
$\mathrm{C}^{*}$-envelope of
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ in the
case where $\alpha$ is an endomorphism of ${\mathcal{A}}$ that extends to an
injective $*$-endomorphism of $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. To
do this, one will have to prove an analogue of Theorem 2.5 in the case where
$\alpha$ may not preserve the Šilov ideal ${\mathcal{J}}_{{\mathcal{A}}}$ of
${\mathcal{A}}$ in ${\mathcal{C}}$.

######  Remark 2.8.

An observation from [9] shows that Theorem 2.6 fails for the contractive
semicrossed product, thus showing that the isometric and the contractive
semicrossed product are not completely isometrically isomorphic in general.

Indeed, the bidisk algebra $A({\mathbb{D}}^{2})$ sits inside
${\mathrm{C}}({\mathbb{T}}^{2})$, which is its C*-envelope by Ando’s theorem.
Consider the identity automorphism ${\operatorname{id}}$. Ando’s theorem also
shows that the completely contractive representations of $A({\mathbb{D}}^{2})$
are determined by an arbitrary pair $T_{1},T_{2}$ of commuting contractions. A
covariant representation of $(A({\mathbb{D}}^{2}),{\operatorname{id}})$ is
given by such a pair and a third contraction $T_{3}$ which commutes with
$T_{1}$ and $T_{2}$. If it were true that the C*-envelope of this system was
${\mathrm{C}}({\mathbb{T}}^{2})\times_{\operatorname{id}}{\mathbb{Z}}\simeq{\mathrm{C}}({\mathbb{T}}^{3})$,
then it would be true that every commuting triple of contractions satisfies
the usual von Neumann inequality. This has been disproved by Varopoulos [28].

##  3\. An application to tensor algebras

In spite of Remark 2.8, there are special cases where the contractive and
isometric semicrossed products coincide. The purpose of this section is to
verify this in the case where ${\mathcal{A}}$ is the tensor algebra of a
$\mathrm{C}^{*}$-correspondence and $\alpha$ a completely isometric
isomorphism of ${\mathcal{A}}$ that fixes its diagonal elementwise (Corollary
2.6).

The tensor algebras for $\mathrm{C}^{*}$-correspondences were introduced by
Muhly and Solel in [23]. This is a broad class of non-selfadjoint operator
algebras which includes as special cases Peters’ semicrossed products [26],
Popescu’s non-commutative disc algebras [27], the tensor algebras of graphs
(introduced in [23] and further studied in [19, 20]) and the tensor algebras
for multivariable dynamics [10], to mention but a few.

Let ${\mathfrak{A}}$ be a $\mathrm{C}^{*}$-algebra and ${\mathcal{X}}$ be a
(right) Hilbert ${\mathfrak{A}}$-module, whose inner product is denoted as
$\langle\,.\mid.\,\rangle$. Let ${\mathcal{L}}({\mathcal{X}})$ be the
adjointable operators on ${\mathcal{X}}$ and let
${\mathcal{K}}({\mathcal{X}})$ be the norm closed subalgebra of
${\mathcal{L}}({\mathcal{X}})$ generated by the operators $\theta_{\xi,\eta}$,
$\xi,\eta\in{\mathcal{X}}$, where
$\theta_{\xi,\eta}(\zeta)=\xi\langle\eta|\zeta\rangle$,
$\zeta\in{\mathcal{X}}$.

A Hilbert ${\mathfrak{A}}$-module ${\mathcal{X}}$ is said to be a
$\mathrm{C}^{*}$-correspondence over ${\mathfrak{A}}$ provided that there
exists a $*$-homomorphism
$\varphi_{{\mathcal{X}}}:{\mathfrak{A}}\rightarrow{\mathcal{L}}({\mathcal{X}})$.
We refer to $\varphi_{{\mathcal{X}}}$ as the left action of ${\mathfrak{A}}$
on ${\mathcal{X}}$. From a given $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ over ${\mathfrak{A}}$, one can form new
$\mathrm{C}^{*}$-correspondences over ${\mathfrak{A}}$, such as the $n$-fold
ampliation or direct sum ${\mathcal{X}}^{(n)}$ ([21, page 5]) and the $n$-fold
interior tensor product ${\mathcal{X}}^{\otimes
n}\equiv{\mathcal{X}}\otimes_{\varphi_{{\mathcal{X}}}}{\mathcal{X}}\otimes_{\varphi_{{\mathcal{X}}}}\dots\otimes_{\varphi_{{\mathcal{X}}}}{\mathcal{X}}$
([21, page 39], $n\in{\mathbb{N}}$, (${\mathcal{X}}^{\otimes
0}\equiv{\mathfrak{A}}$). These operation are defined within the category of
$\mathrm{C}^{*}$-correspondences over ${\mathfrak{A}}$. (See [21] for more
details.)

A representation $(\pi,t)$ of a $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ over ${\mathfrak{A}}$ on a $\mathrm{C}^{*}$-algebra
${\mathcal{B}}$ consists of a $*$-homomorphism
$\pi:{\mathfrak{A}}\rightarrow{\mathcal{B}}$ and a linear map
$t:{\mathcal{X}}\rightarrow{\mathcal{B}}$ so that

  * (i)

$t(\xi)^{*}t(\eta)=\pi(\langle\xi|\,\eta\rangle)$, for
$\xi,\eta\in{\mathcal{X}}$,

  * (ii)

$\pi(A)t(\xi)=t(\varphi_{{\mathcal{X}}}(A)\xi)$, for $A\in{\mathfrak{A}}$,
$\xi\in{\mathcal{X}}$.

For a representation $(\pi,t)$ of a $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ there exists a $*$-homomorphism
$\psi_{t}:{\mathcal{K}}({\mathcal{X}})\rightarrow{\mathcal{B}}$ so that
$\psi_{t}(\theta_{\xi,\eta})=t(\xi)t(\eta)^{*}$, for
$\xi,\eta\in{\mathcal{X}}$. Following Katsura [18], we say that the
representation $(\pi,t)$ is covariant iff
$\psi_{t}(\varphi_{{\mathcal{X}}}(A))=\pi(A)$, for all
$A\in{\mathcal{J}}_{{\mathcal{X}}}$, where

${\mathcal{J}}_{{\mathcal{X}}}\equiv\varphi_{{\mathcal{X}}}^{-1}({\mathcal{K}}({\mathcal{X}}))\cap(\ker\varphi_{{\mathcal{X}}})^{\perp}.$

If $(\pi,t)$ is a representation of ${\mathcal{X}}$ then the
$\mathrm{C}^{*}$-algebra (resp. norm closed algebra) generated by the images
of $\pi$ and $t$ is denoted as $\mathrm{C}^{*}(\pi,t)$ (resp.
$\operatorname{Alg}((\pi,t)$). There is a universal representation
$(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$ for
${\mathcal{X}}$ and the $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
is the Toeplitz-Cuntz-Pimsner algebra ${\mathcal{T}}_{{\mathcal{X}}}$.
Similarly, the Cuntz-Pimsner algebra ${\mathcal{O}}_{{\mathcal{X}}}$ is the
$\mathrm{C}^{*}$-algebra generated by the image of the universal covariant
representation $(\pi_{{\mathfrak{A}}},t_{{\mathcal{X}}})$ for ${\mathcal{X}}$.

A concrete presentation of both ${\mathcal{T}}_{{\mathcal{X}}}$ and
${\mathcal{O}}_{{\mathcal{X}}}$ can be given in terms of the generalized Fock
space ${\mathcal{F}}_{{\mathcal{X}}}$ which we now describe. The Fock space
${\mathcal{F}}_{{\mathcal{X}}}$ over the correspondence ${\mathcal{X}}$ is
defined to be the direct sum of the ${\mathcal{X}}^{\otimes n}$ with the
structure of a direct sum of $\mathrm{C}^{*}$-correspondences over
${\mathfrak{A}}$,

${\mathcal{F}}_{{\mathcal{X}}}={\mathfrak{A}}\oplus{\mathcal{X}}\oplus{\mathcal{X}}^{\otimes
2}\oplus\dots.$

Given $\xi\in{\mathcal{X}}$, the (left) creation operator
$t_{\infty}(\xi)\in{\mathcal{L}}({\mathcal{F}}_{{\mathcal{X}}})$ is defined by
the formula

$t_{\infty}(\xi)(A,\zeta_{1},\zeta_{2},\dots)=(0,\xi
A,\xi\otimes\zeta_{1},\xi\otimes\zeta_{2},\dots),$

where $\zeta_{n}\in{\mathcal{X}}^{\otimes n}$, $n\in{\mathbb{N}}$, and
$A\in{\mathfrak{A}}$. Also, for $A\in{\mathfrak{A}}$, we define
$\pi_{\infty}(A)\in{\mathcal{L}}({\mathcal{F}}_{{\mathcal{X}}})$ to be the
diagonal operator with $\varphi_{{\mathcal{X}}}(A)\otimes id_{n-1}$ at its
${\mathcal{X}}^{\otimes n}$-th entry. It is easy to verify that
$(\pi_{\infty},t_{\infty})$ is a representation of ${\mathcal{X}}$ which is
called the Fock representation of ${\mathcal{X}}$. Fowler and Raeburn [13]
(resp. Katsura [18]) have shown that the $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\pi_{\infty},t_{\infty})$ (resp
$\mathrm{C}^{*}(\pi_{\infty},t_{\infty})/{\mathcal{K}}({\mathcal{F}}_{{\mathcal{X}}{\mathcal{J}}_{{\mathcal{X}}}})$)
is isomorphic to ${\mathcal{T}}_{{\mathcal{X}}}$ (resp.
${\mathcal{O}}_{{\mathcal{X}}}$).

######  Definition 3.1.

The tensor algebra of a $\mathrm{C}^{*}$-correspondence ${\mathcal{X}}$ over
${\mathfrak{A}}$ is the norm-closed algebra
$\operatorname{alg}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
and is denoted as ${\mathcal{T}}_{{\mathcal{X}}}^{+}$.

According to [13, 18], the algebras
${\mathcal{T}}_{{\mathcal{X}}}^{+}\equiv\operatorname{alg}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
and $\operatorname{alg}(\pi_{\infty},t_{\infty})$ are completely isometrically
isomorphic and we will therefore identify them.

In order to prove the main results of this section, we follow the strategy of
the first half of [9]. However, the generality in which we are working with,
presents new difficulties and requires innovation. One such innovation is the
following.

######  Lemma 3.2.

Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,

$\pi(L)X=X\pi(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

Then there exist isometric co-extensions $\pi^{\prime}$ and
$(\pi\circ\alpha)^{\prime}$, of $\pi$ and $\pi\circ\alpha$ respectively, and
an isometric co-extension $X^{\prime}$ of $X$, all acting on some Hilbert
space ${\mathcal{H}}^{\prime}$ and satisfying

(2) $\pi^{\prime}(L)X^{\prime}=X^{\prime}(\pi\circ\alpha)^{\prime}(L),\mbox{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$

and

(3) $\pi^{\prime}(A)=(\pi\circ\alpha)^{\prime}(A),\mbox{ for all
}A\in{\mathfrak{A}}.$

######  Proof..

First we construct isometric co-extensions $\widehat{\pi}_{1}$ and
$\widehat{\pi}_{2}$, of $\pi$ and $\pi\circ\alpha$ respectively, and an
isometric co-extension $\widehat{X}$ of $X$, with the property that

(4) $\widehat{\pi}_{1}(A)=\widehat{\pi}_{2}(A)$

and

(5)
$\phantom{XXiXXX}\widehat{X}\widehat{\pi}_{i}(A)=\widehat{\pi}_{i}(A)\widehat{X},\quad
i=1,2,$

for all $A\in{\mathfrak{A}}$.

To do this, notice that $X$ commutes with $\pi({\mathfrak{A}})$. co-extend $X$
to its Schaeffer dilation

$S_{X}\simeq\begin{bmatrix}K&0&0&0&\dots\\\ D_{K}&0&0&0&\dots\\\
0&I&0&0&\dots\\\ 0&0&I&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}\in
B({\mathcal{H}}^{(\infty)}),$

where $D_{K}=(I-K^{*}K)^{1/2}$. Let $\pi^{(\infty)}$ be the infinite
ampliation of $\pi$ and notice that $S_{X}$ commutes with
$\pi^{(\infty)}({\mathfrak{A}})$. Subsequently, using [23, Theorem 3.3], we
obtain some isometric co-extension $\widehat{\pi}$ of $\pi^{(\infty)}$, on
some Hilbert space
${\mathcal{K}}={\mathcal{H}}^{(\infty)}\oplus{\mathcal{M}}$, and let
$\widehat{\pi}_{1}=\widehat{\pi}$,
$\widehat{\pi}_{2}=\widehat{\pi}\circ\alpha$ and $\widehat{X}=S_{X}\oplus
I_{{\mathcal{M}}}$. These $\widehat{\pi}_{1},\widehat{\pi}_{2}$ and
$\widehat{X}$ satisfy (4) and (5).

Since $\widehat{X}$ satisfies (5), the pairs
$(\widehat{t}_{i},\widehat{\pi}_{i}|_{{\mathfrak{A}}})$, $i=1,2$, where,

$\displaystyle\widehat{t}_{1}(\xi)$
$\displaystyle=\widehat{\pi}_{1}(t_{\infty}(\xi))\widehat{X},$
$\displaystyle\widehat{t}_{2}(\xi)$
$\displaystyle=\widehat{X}\widehat{\pi}_{2}(t_{\infty}(\xi)),\quad\xi\in{\mathcal{X}},$

define isometric representations of $({\mathcal{X}},{\mathfrak{A}})$ and so
there exist $*$-representations
$\rho_{i}:{\mathcal{T}}_{{\mathcal{X}}}\rightarrow B({\mathcal{K}})$ which
integrate $(\widehat{t}_{i},\widehat{\pi}_{i}|_{{\mathfrak{A}}})$, $i=1,2$.
Since,

(6)
$P_{{\mathcal{H}}}\rho_{1}(L)|_{{\mathcal{H}}}=P_{{\mathcal{H}}}\rho_{2}(L)|_{{\mathcal{H}}},\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$

the representations $\rho_{i}$ co-extend the same contractive representation
of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ (appearing in (6)). By the uniqueness
of the minimal isometric co-extension [23, Proposition 3.2], there exist
projections $Q_{i}$ commuting with
$\rho_{i}({\mathcal{T}}_{{\mathcal{X}}}^{+})$, $i=1,2$, (hence commuting with
$\widehat{\pi}_{i}({\mathfrak{A}})$, $i=1,2$) and a unitary
$W:Q_{1}({\mathcal{K}})\rightarrow Q_{2}({\mathcal{K}})$, so that

(7)
$W\rho_{1}(L)|_{Q_{1}({\mathcal{K}})}W^{*}=\rho_{2}(L)|_{Q_{2}({\mathcal{K}})},\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

Furthermore, for each $i=1,2$, ${\mathcal{H}}\subseteq Q_{i}({\mathcal{K}})$
and $W$ fixes ${\mathcal{H}}$, because both $\rho_{1}$ and $\rho_{2}$ co-
extend the same completely contractive representation of
${\mathcal{T}}_{{\mathcal{X}}}^{+}$, which acts on ${\mathcal{H}}$.

For $i=1,2$, let

$\widetilde{\pi}_{i}(L)=\widehat{\pi}_{i}(L)\oplus\left(\rho_{1}(L)|_{Q_{1}^{\perp}({\mathcal{K}})}\right)^{(\infty)}\oplus\left(\rho_{2}(L)|_{Q_{2}^{\perp}({\mathcal{K}})}\right)^{(\infty)},\,\,L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$

and let

$\widetilde{X}=\widehat{X}\oplus
I_{Q_{1}^{\perp}({\mathcal{K}})}^{(\infty)}\oplus
I_{Q_{2}^{\perp}({\mathcal{K}})}^{(\infty)},$

all of them acting on

${\mathcal{H}}^{\prime}={\mathcal{K}}\oplus
Q_{1}^{\perp}({\mathcal{K}})^{(\infty)}\oplus
Q_{2}^{\perp}({\mathcal{K}})^{(\infty)}.$

Because of (7), there exists a unitary $U\in B({\mathcal{H}}^{\prime})$ which
fixes ${\mathcal{H}}$, commutes with
$\widetilde{\pi}_{1}({\mathfrak{A}})=\widetilde{\pi}_{2}({\mathfrak{A}})$ and
satisfies

$U\widetilde{\pi}_{1}(L)\widetilde{X}U^{*}=\widetilde{X}\widetilde{\pi}_{2}(L),\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

Consider the isometric representations
$(t_{i},\widetilde{\pi}_{i}|_{{\mathfrak{A}}})$ of
$({\mathcal{X}},{\mathfrak{A}})$, where,

$\displaystyle t_{1}(\xi)$
$\displaystyle=\widetilde{\pi}_{1}(t_{\infty}(\xi))U,\mbox{ and,}$
$\displaystyle t_{2}(\xi)$
$\displaystyle=\widetilde{\pi}_{2}(t_{\infty}(\xi))U,\xi\in{\mathcal{X}},$

and let $\pi_{i}^{\prime}$, $i=1,2$, be the $*$-representations of
${\mathcal{T}}_{{\mathcal{X}}}$ which integrate them. Let
$X^{\prime}=U^{*}\widetilde{X}$ and notice that for any
$L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}$ we have

$\pi_{1}^{\prime}(L)X^{\prime}=\widehat{\pi}_{1}(L)UU^{*}\widetilde{X}=\widehat{\pi}_{1}(L)\widetilde{X}$

while

$X^{\prime}\pi_{2}^{\prime}(L)=U^{*}\widetilde{X}\widehat{\pi}_{2}(L)U=\widehat{\pi}_{1}(L)\widetilde{X},$

and the conclusion follows. ∎

######  Remark 3.3.

In the previous Lemma, one may take the isometric co-extension $X^{\prime}$ to
be the minimal isometric co-extension $X_{m}$ of $X$. In that case however,
the co-extensions $\pi^{\prime}$ and $(\pi\circ\alpha)^{\prime}$ can only be
considered completely contractive and not necessarily isometric.

Indeed, using Lemma 3.2, we obtain isometric co-extensions
$\pi^{\prime},(\pi\circ\alpha)^{\prime}$ and $X^{\prime}$ on some Hilbert
space ${\mathcal{H}}^{\prime}$ that satisfy (2) and (3). Let $Q$ be the
reducing projection for $X^{\prime}$ so that
$QX^{\prime}|_{Q({\mathcal{H}}^{\prime})}\simeq X_{m}$. By (3), the projection
$Q$ commutes with $\pi^{\prime}({\mathfrak{A}})$ and
$(\pi\circ\alpha)^{\prime}({\mathfrak{A}})$ and so the completely contractive
representations of $({\mathcal{X}},{\mathfrak{A}})$, determined by the
representations $\pi^{\prime}$ and $(\pi\circ\alpha)^{\prime}$ of
${\mathfrak{A}}$ and the mappings

$\displaystyle{\mathcal{X}}\ni\xi$ $\displaystyle\longmapsto
Q\pi^{\prime}(t_{\infty}(\xi))|_{Q({\mathcal{H}}^{\prime})}$
$\displaystyle{\mathcal{X}}\ni\xi$ $\displaystyle\longmapsto
Q(\pi\circ\alpha)^{\prime}(t_{\infty}(\xi))|_{Q({\mathcal{H}}^{\prime})},$

can be integrated to the desired contractive representations of
${\mathcal{T}}_{{\mathcal{X}}}^{+}$, satisfying the analogues of (2) and (3)
with $X_{m}$ instead of $X^{\prime}$.

If we take $\alpha={\operatorname{id}}$ in Lemma 3.2, then we obtain the
commutant lifting Theorem of Muhly and Solel [23], without using the ”one-
step” extension in the proof. (In [23, page 418] the authors ask for such a
proof.) Indeed

######  Corollary 3.4.

Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,

$\pi(L)X=X\pi(L),\mbox{ for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

If $\pi_{m}$ is the minimal isometric co-extension of $\pi$, then there exists
a contraction $X^{\prime}$ co-extending $X$ and satisfying

$\pi_{m}(L)X^{\prime}=X^{\prime}\pi_{m}(L),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

######  Proof..

Use Lemma 3.2 to obtain isometric co-extensions
$\pi^{\prime},X^{\prime\prime}$ on some Hilbert space ${\mathcal{H}}^{\prime}$
that do the job. (Note however that $\pi^{\prime}$ may be ”larger” than the
minimal isometric co-extension.) There exists now a reducing subspace
${\mathcal{K}}\subseteq{\mathcal{H}}^{\prime}$ for $\pi^{\prime}$ so that
$\pi^{\prime}|_{{\mathcal{K}}}\equiv\pi_{m}$. Letting $X^{\prime}$ be the
compression of $X^{\prime\prime}$ on ${\mathcal{K}}$, the conclusion follows.
∎

A familiar $2\times 2$ matrix trick also establishes the intertwining form of
the commutant lifting theorem for minimal isometric co-extensions.

######  Theorem 3.5.

Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,

$\pi(L)X=X\pi(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

Then there exist an isometric co-extension $\pi_{1}$ of $\pi$ and an isometric
co-extension $Z$ of $X$, so that

$\pi_{1}(L)Z=Z\pi_{1}(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$

######  Proof..

Notice that if $\pi_{m}$ is the minimal isometric dilation of $\pi$, then
$\pi_{m}\circ\alpha$ is the minimal isometric dilation of $\pi\circ\alpha$.
Therefore, by applying commutant lifting to the covariance relations, we
obtain a contraction $X_{1}$ on a Hilbert space ${\mathcal{H}}_{1}$,
satisfying

$\pi_{m}(L)X_{1}=X_{1}(\pi_{m}\circ\alpha)(L),\text{ for all
}L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$

Let $X_{1,m}$ be the minimal dilation of $X_{1}$, i.e.,

(8) $X_{1,m}\simeq\begin{bmatrix}X_{1}&0&0&0&\dots\\\ D_{X_{1}}&0&0&0&\dots\\\
0&I&0&0&\dots\\\ 0&0&I&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}$

where $D_{X_{1}}=(I-X_{1}^{*}X_{1})^{1/2}$. We apply now Remark 3.3 to obtain
completely contractive representations $\widehat{\pi}_{m}$ and
$\widehat{\pi_{m}\circ\alpha}$, which co-extend $\pi_{m}$ and
$\pi_{m}\circ\alpha$, coincide on ${\mathfrak{A}}$, and satisfy

(9)
$\widehat{\pi}_{m}(L)X_{1,m}=X_{1,m}\widehat{\pi_{m}\circ\alpha}(L)\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$

Assume that these dilations have the form

$\widehat{\pi}_{m}(L)=\begin{bmatrix}\pi_{m}(L)&0\\\
Y^{(L)}&[Y^{(L)}_{jk}]_{j,k\geq 1}\end{bmatrix}\text{ and \,
}\widehat{\pi_{m}\circ\alpha}(L)=\begin{bmatrix}\pi_{m}\circ\alpha(L)&0\\\
Z^{(L)}&[Z^{(L)}_{jk}]_{j,k\geq 1}\end{bmatrix}$

with regards to the decomposition of the Hilbert space that corresponds to the
matricial form of $X_{1,m}$ in (8).

Claim: $Y^{(L)}=Z^{(L)}=0$, for all $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$.

Indeed, the claim is true in the case where $L=\pi_{\infty}(A)$,
$A\in{\mathfrak{A}}$, since the restrictions of $\widehat{\pi}_{m}$ and
$\widehat{\pi_{m}\circ\alpha}$ on ${\mathfrak{A}}$ are $*$-homomorphisms
dilating the $*$-homomorphisms $\pi_{m}$ and $\pi_{m}\circ\alpha$
respectively. Hence it suffices to prove the claim in the case where
$L=t_{\infty}(\xi)$, $\xi\in{\mathcal{X}}$. We show that $Y^{(\xi)}=0$; a
similar argument will show that $Z^{(t_{\infty}(\xi))}=0$. By the Schwarz
inequality for completely contractive maps on unital operator algebras we have

$\widehat{\pi}_{m}\left(t_{\infty}(\xi)\right)^{*}\widehat{\pi}_{m}(t_{\infty}(\xi))\leq\widehat{\pi}_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$

By taking into account the matricial form of $\widehat{\pi}_{m}$ and comparing
$(1,1)$-entries in the above inequality, we obtain

(10)
$\pi_{m}\left(t_{\infty}(\xi)\right)^{*}\pi_{m}(t_{\infty}(\xi))+(Y^{(t_{\infty}(\xi))})^{*}Y^{(t_{\infty}(\xi))}\leq\pi_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$

However the map $\pi_{m}$ is an isometric representation and so

$\pi_{m}\left(t_{\infty}(\xi)\right)^{*}\pi_{m}(t_{\infty}(\xi))=\pi_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$

and so (10) obtains

$(Y^{(t_{\infty}(\xi))})^{*}Y^{(t_{\infty}(\xi))}\leq 0,$

which proves the claim.

By comparing $(2,i)$-entries, $i=2,3,\dots$, in the covariance relation (9) we
also obtain

$Y_{2,i}^{(L)}=0,\quad\text{for all }i\geq 2.$

In addition, by comparing $(i,1)$-entries $i=3,4,\dots$, in (9) we obtain

$Y_{i,2}^{(L)}D_{X_{1}}=0,\quad\text{for all }i\geq 3,$

and so $Y_{i,2}^{(L)}=0,\text{for all }i\geq 3$. This combined with the Claim
implies that the second row and column of $\widehat{\pi}_{m}(L)$,
$L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, are equal to zero, except perhaps
from $Y_{2,2}^{(L)}$. Therefore, the map

$\rho:{\mathcal{T}}^{+}_{{\mathcal{G}}}\longrightarrow
B(D_{X_{1}}({\mathcal{H}}_{1})),\,\,L\longmapsto Y_{2,2}^{(L)}$

is a completely contractive representation of
${\mathcal{T}}^{+}_{{\mathcal{X}}}$. By comparing $(2,1)$-entries in the
covariance relation (9), we now obtain

(11) $\rho(L)D_{X_{1}}=D_{X_{1}}\pi_{m}\circ\alpha(L),\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$

For any $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, we now define

$\pi_{m}^{\prime}(L)=\begin{bmatrix}\pi_{m}(L)&0&0&0&\dots\\\
0&\rho(L)&0&0&\dots\\\ 0&0&\rho(\alpha(L))&0&\dots\\\
0&0&0&\rho(\alpha^{(2)}(L))&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}$

By (11), this is a completely contractive representation $\pi_{m}^{\prime}$ on
a Hilbert space ${\mathcal{H}}_{2}$ so that

$\pi_{m}^{\prime}(L)X_{1,m}=X_{1,m}\pi_{m}^{\prime}\circ\alpha(L)\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$

Continuing in this fashion, we obtain a sequence

$(\pi,X),\,(\pi_{m},X_{1}),\,(\pi_{m}^{\prime},X_{1,m}),\,((\pi_{m}^{\prime})_{m},X_{2}),\,(((\pi_{m}^{\prime})_{m})^{\prime},X_{2,m})\dots$

of pairs of operators and representations acting on Hilbert spaces
${\mathcal{H}}\subseteq{\mathcal{H}}_{1}\subseteq{\mathcal{H}}_{2}\dots,$ co-
extending $\pi$ and $X$ and satisfying the covariance relations. Let
${\mathcal{H}}=\bigvee_{j}{\mathcal{H}}_{j}$, and consider these pairs as
acting on ${\mathcal{H}}$ by extending them to be zero on the complement. Let

$Z=\operatorname*{\textsc{sot}--lim}X_{j}=\operatorname*{\textsc{sot}--lim}X_{m,j}$

and define $\pi_{1}(L)$, $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, as a strong
limit in a similar fashion. These limits evidently exist as in both cases the
sequences consist of either isometries or isometric representations that
decompose as infinite direct sums. Multiplication is sot-continuous on the
ball, hence the covariance relations hold in the limit. ∎

Combining the Proposition above with Theorem 2.6 we obtain the main result of
the section,

######  Corollary 3.6.

Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Then
${\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$
are completely isometrically isomorphic and

$\mathrm{C}^{*}_{\text{env}}({\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+})\simeq{\mathcal{O}}_{{\mathcal{X}}}\times_{\alpha}{\mathbb{Z}}.$

In particular, the above corollary recaptures the main result of [9] with a
different proof.

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# Whole-Body Control of a Mobile Manipulator for Passive Collaborative
Transportation

F. Benzi  C. Mancus  C. Secchi  Department of Sciences and Methods of
Engineering, University of Modena and Reggio Emilia, Italy (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Human-robot collaborative tasks foresee interactions between humans and robots
with various degrees of complexity. Specifically, for tasks which involve
physical contact among the agents, challenges arise in the modelling and
control of such interaction. In this paper we propose a control architecture
capable of ensuring a flexible and robustly stable physical human-robot
interaction, focusing on a collaborative transportation task. The architecture
is deployed onto a mobile manipulator, modelled as a whole-body structure,
which aids the operator during the transportation of an unwieldy load. Thanks
to passivity techniques, the controller adapts its interaction parameters
online while preserving robust stability for the overall system, thus
experimentally validating the architecture.

###### keywords:

Force and Compliance Control, Adaptive Robot Control, Robust Robot Control,
Haptic Interaction, Physical Human-Robot Interaction.

##  1 Introduction

The modern industrial paradigm foresees an increasingly close interaction
among humans and robots. This is most strongly manifested in collaborative
applications, in which the robot assists or cooperates with the human in order
to accomplish a common goal.

During some collaborative tasks, a physical contact between the robot and its
surroundings, these being the human or the environment, might take place. This
contact can take place either directly onto the surface of the robot, or via
the manipulation of a collaboratively held object. In this paper, we focus
onto these types of contact, aiming at rendering such interactions both
flexible and robust to different environments and parameter variations.

Different approaches in literature have been proposed for a safe and reliable
collaborative load transportation. A vastly employed one is to teach the robot
the collaborative task by means of programming by demonstration (PbD) Calinon
et al. (2009),Gribovskaya et al. (2011), in which the robot learns by encoding
the human demonstrations as a task. The encoding process of the task model can
be carried out via probabilistic frameworks based on Gaussian Mixture Model
(GMM) Calinon et al. (2007) or Hidden Markov Model Hersch et al. (2008).
However, except for a certain degree of adaptation during the task execution,
the performance of these controllers drop significantly whenever the
interaction varies strongly from the demonstrated data, since their principle
of operation is based on the local knowledge of the task. Thus, their
generalization capabilities and consequently their flexibility are limited,
making those non-ideal for dynamic environments. Additionally, learning-based
methods also do not generally include the possibility of additional tasks,
such as safety and joint limits control. Finally, processes as task model
training and parameters tuning can result time consuming.

Other approaches directly integrate haptic information into an impedance or
admittance controller Agravante et al. (2013); Tagliabue et al. (2017) in
order for the robot to comply with the external forces acting onto the
manipulated object. Impedance and admittance control Siciliano and Villani
(2012) are two commonly deployed strategies in interaction control. These
controllers ensure compliance of the robot during the interaction phase by
enforcing a dynamical behaviour in the form of a mechanical impedance,
characterised by desired stiffness, damping and inertia parameters. In this
way, the robot can adapt its behaviour according to haptic information and
guarantee a stable interaction. In Tagliabue et al. (2017) two Micro Aerial
Vehicles (MAVs) perform a collaborative transportation task based on the
master-slave paradigm, in which the slave complies to the external force
applied by the master to the payload via an admittance controller. The
inherent passivity of the dynamics ensures a robustly stable interaction
Secchi et al. (2007).

Adapting the dynamic parameters during the interaction can be greatly
beneficial for the application, since it allows for an higher flexibility of
the behaviour, of peculiar interest for collaborative applications Dimeas and
Aspragathos (2016). This is carried out e.g. in Gribovskaya et al. (2011), in
which an adaptive impedance is utilised to compensate for unmodeled
uncertainties during the collaborative task execution due to the human
behaviour. In here, however, there is no formal stability analysis of the
adaptive impedance. In fact, varying online the dynamic parameters may lead to
the loss of passivity Ferraguti et al. (2015) and unstable behaviours might be
implemented during the human-robot interaction Ferraguti et al. (2019).

The formulation of a provably stable variable admittance controller has been
successfully carried out in Secchi and Ferraguti (2019) by means of an
optimization framework. The exploitation of energy tanks Ferraguti et al.
(2015); Franken et al. (2011) allows to separate passivity from set dynamics,
treating the energy flow in the tank as a requirement for the passivation of
any desired behaviour.

The architecture was successively augmented in Benzi and Secchi (2021),
allowing for the simultaneous execution of multiple additional tasks alongside
the passivisation of the behaviour. In here, the tasks are encoded by means of
Control Barrier Functions (CBFs) Ames et al. (2019) following a procedure akin
to Notomista et al. (2020). CBFs have been deployed in robotics in order to
constraint the system inside a subset of its state space, thus enforcing
constraints onto the behaviour of the robot. Via time-varying CBFs Notomista
and Egerstedt (2020) it is also possible to enforce time-varying constraint,
which are well suited for safety applications. The control input satisfying
the set of constraint can be found as the solution of a convex optimization
problem, ensuring the real-time capabilities of the architecture.
Nevertheless, only simple robotic arms have been considered so far.

Subsequent evolutions of the architecture are presented in Benzi et al.
(2022), in which the control structure is deployed for passively implementing
shared autonomy in a multi-robot teleoperation scenario, and in Ferrari et al.
(2022) for an effective and seamless human-robot collaboration via a
bidirectional communication channel.

In this paper, we aim at deploying the architecture presented in Benzi and
Secchi (2021), further augmenting it for achieving the whole body control of
the robot, in order to reproduce a collaborative human-robot transportation
task in a robustly stable way, while performing additional tasks. This novel
approach allows us to overcome the rigidity of standard PbD approaches,
ensuring a flexible and adaptable behaviour at all times, while preserving the
safety of all the parties involved.

The robot chosen for the application is a custom mobile manipulator,
consisting of a differential drive mobile platform with a robotic arm mounted
on top, modeled as a whole body system. A similar task was implemented in
Nozaki and Murakami (2009), but the arm and the mobile base were modelled
separately and the impedance dynamics was fixed.

Thus, the contributions of this paper are:

  * •

A control architecture capable of performing a collaborative human-robot
transportation task in a flexible and robust way

  * •

A whole-body kinematic model and control of the robot, capable of considering
non-holonomic constraints introduced by the mobile base

  * •

The possibility of implementing multiple tasks alongside the main
transportation one

The paper is organised as follows: in Sec. 2 the main problem addressed in
this paper is formulated. In Sec. 3 the whole-body model of the mobile
manipulator is laid out. In Sec. 4 the collaborative constraint-oriented
control architecture is presented. In Sec. 5 the architecture is
experimentally validated and finally in Sec. 6 conclusions are drawn and
future work is discussed.

##  2 Problem Formulation

Figure 1: Whole-body admittance control architecture. We indicate as
$\dot{\textbf{q}}_{c1}$ and $\dot{\textbf{q}}_{c2}$ the setpoint for the
velocity controllers of the manipulator and the mobile base respectively.

Let us consider a velocity controlled mobile manipulator composed of a fully
actuated $n_{a}$-DOF manipulator mounted onto a differential drive mobile
base, assuming that the low-level controllers of both robots ensure accurate
reproduction of a desired velocity profile. We consider the following extended
joint space for the robot:

q $\displaystyle=\begin{bmatrix}\textbf{q}_{b}\\\ \textbf{q}_{a}\end{bmatrix}$
(1)

in which $\textbf{q}_{a}\in\mathbb{R}^{n_{a}}$ are the generalized coordinates
of the manipulator, while $\textbf{q}_{b}\in\mathbb{R}^{n_{b}}$ are the
angular position of the wheels of the mobile base. We define as
$N=n_{a}+n_{b}$ the dimension of the augmented configuration space.

We construct the augmented mapping for the robot

${}^{W}\dot{\textbf{x}}_{E}=\textbf{J}_{m}\dot{\textbf{q}}$ (2)

relating the end tip velocity w.r.t the world frame
${}^{W}\dot{\textbf{x}}_{E}\in\mathbb{R}^{m}$ with the robot velocity in the
augmented joint space. We refer to $\textbf{J}_{m}\in\mathbb{R}^{m\times N}$
as the augmented Jacobian.

We can obtain the desired velocity input in the joint space
$\textbf{u}\in\mathbb{R}^{N}$ which implements a desired task space velocity
$\dot{\textbf{x}}_{des}$ as
$\textbf{u}=\dot{\textbf{q}}_{des}=J_{m}^{+}\dot{\textbf{x}}_{des}$ in which
$J_{m}^{+}\in\mathbb{R}^{N\times m}$ is the Moore-Penrose pseudo-inverse of
the augmented Jacobian.

We finally assume that the human-robot contact takes place via a commonly
manipulated object, whose interaction force $\textbf{F}_{e}\in\mathbb{R}^{m}$
is sensed by an haptic interface. We model the interaction using the whole
body admittance controller in Fig. 1, i.e. by integrating the interaction
force via the admittance dynamics, synthesizing the admittance velocity
$\dot{\textbf{x}}_{a}\in\mathbb{R}^{m}$. We can then ensure the reproduction
of the admittance dynamics by setting
${}^{W}\dot{\textbf{x}}_{E}\approx\dot{\textbf{x}}_{des}\approx\dot{\textbf{x}}_{a}$.

In order to enhance the performance of the controller, we want to vary the
admittance parameters according to the perceived intention of the human
operator, assisting him during the manipulation of the object at the best of
the robot capabilities (see e.g. Ferraguti et al. (2019)). Passivity is
however lost if we consider a variable admittance controller, i.e. whose
dynamic parameters vary over time. Consider the following time-varying
admittance model

$\textbf{M}(\textbf{x}_{a},t)\ddot{\textbf{x}}_{a}+\textbf{C}(\textbf{x}_{a},\dot{\textbf{x}}_{a},t)\dot{\textbf{x}}_{a}+\textbf{D}(\textbf{x}_{a},t)\dot{\textbf{x}}_{a}+\frac{\partial
P}{\partial\textbf{x}_{a}}(t)=\textbf{F}_{e}$ (3)

where $\textbf{M}(\textbf{x}_{a})=\textbf{M}^{T}(\textbf{x}_{a})\geq 0$ is the
inertia matrix, $\textbf{C}(\textbf{x}_{a},\dot{\textbf{x}}_{a})$ represents
the Coriolis term, $\textbf{D}(\textbf{x}_{a})\geq 0$ is a damping matrix and
$P:\mathbb{R}^{m}\rightarrow\mathbb{R}$ is a potential field acting on the
system. Here, the variation of the dynamic parameters might end up introducing
energy into the system, thus threatening passivity (see e.g. Secchi and
Ferraguti (2019); Ferraguti et al. (2015)). The standard admittance controller
cannot in fact ensure a robustly stable interaction when reproducing a time-
varying dynamics.

Alongside passivity, other conditions need to be satisfied for ensuring safety
during the transportation task, both for the human, the robot and the
manipulated object. These conditions require a dedicated proper formulation.

We aim at realising a control architecture capable of performing a
collaborative human-robot transportation task in a safe and flexible way. The
architecture must be capable of passively implementing a variable admittance
dynamics as in (3), whose parameters must adapt to properly assist the human
operator, alongside accomplishing multiple safety-related and application-
oriented tasks, while treating the mobile manipulator as a single whole-body
system.

##  3 Whole Body Kinematic Model

In this section we present the construction of the augmented kinematic mapping
in (2).

First, we separately compute the kinematic contribution of the arm to the end-
effector velocity ${}^{W}\dot{\textbf{x}}_{E(a)}$ and the contribution of the
mobile base ${}^{W}\dot{\textbf{x}}_{E(b)}$. Considering the kinematic mapping
of the arm, this is readily available as:

${}^{0}\dot{\textbf{x}}_{a}=\textbf{J}_{a}\dot{\textbf{q}}_{a}$ (4)

being $\textbf{J}_{a}$ the geometric Jacobian of the arm, relating the
velocity of the terminating link $\mathcal{F}_{a}$ w.r.t the initial link
frame $\mathcal{F}_{0}$ expressed in $\mathcal{F}_{0}$ with the joint
velocities $\dot{\textbf{q}}_{a}$. We assume that the tool frame is related to
the end tip one via a constant homogeneous transformation matrix
${}^{a}\textbf{T}_{E}$, as the initial frame is related to the world frame via
${}^{W}\textbf{T}_{0}$. The contribution of the manipulator to the end-
effector velocity, expressed in the world frame, is obtained as:

$\displaystyle\begin{split}{}^{W}\dot{\textbf{x}}_{E(a)}=\textbf{J}_{A}\dot{\textbf{q}}_{a}\\\
\textbf{J}_{A}={}^{W}\mathcal{T}_{0}{}^{E}\mathcal{T}_{a}\textbf{J}_{a}\end{split}$
(5)

in which ${}^{W}\mathcal{T}_{0}$ and ${}^{E}\mathcal{T}_{a}$ are the adjoint
matrices:

$\displaystyle{}^{E}\mathcal{T}_{a}$
$\displaystyle=\begin{bmatrix}{}^{a}\textbf{R}_{E}^{T}&\quad-{}^{a}\textbf{R}_{E}^{T}\times{}^{a}\textbf{r}_{E}\\\
\quad\textbf{0}_{3}&\quad{}^{a}\textbf{R}_{E}^{T}\end{bmatrix}$ (6)

$\displaystyle{}^{W}\mathcal{T}_{0}$
$\displaystyle=\begin{bmatrix}{}^{W}\textbf{R}_{0}&\quad{}^{W}\textbf{r}_{0}\times{}^{W}\textbf{R}_{0}\\\
\quad\textbf{0}_{3}&\quad{}^{W}\textbf{R}_{0}\end{bmatrix}$ (7)

in which ${}^{a}\textbf{R}_{E}\in\mathbb{SO}(3)$ is the rotation matrix
obtained from ${}^{a}\textbf{T}_{E}$, with
${}^{a}\textbf{r}_{E}\in\mathbb{R}^{3}$ being the position vector of the
transformation. Similarly, we can extract
${}^{W}\textbf{R}_{0}\in\mathbb{SO}(3)$ and ${}^{W}\textbf{r}_{0}$ from
${}^{W}\textbf{T}_{0}$.

For the mobile base, we utilise the kinematic mapping commonly found in
literature for a differential drive robot (see e.g. LaValle (2006)). Given the
choice of $\textbf{q}_{b}$, we indicate as
$\dot{\textbf{q}}_{b}=[\omega_{l}\,\,\omega_{r}]^{T}$ the angular velocity of
the left and right wheel respectively. We can then define the kinematic
mapping of the mobile base as:

${}^{W}\dot{\textbf{x}}_{b}=\textbf{J}_{b}\dot{\textbf{q}}_{b}$ (8)

which relates the velocity of the wheels with the Cartesian velocity of the
mobile base w.r.t the world frame $\mathcal{F}_{W}$ expressed in
$\mathcal{F}_{W}$. The Jacobian $\textbf{J}_{b}\in\mathbb{R}^{m\times n_{b}}$
stems from the kinematic model of the unicycle together with the geometric
parameters of the differential drive system:

$\displaystyle\textbf{J}_{b}=\begin{bmatrix}[c]\frac{r\,\cos(\theta)}{2}&\frac{r\,\cos(\theta)}{2}\\\
\frac{r\,\sin(\theta)}{2}&\frac{r\,\sin(\theta)}{2}\\\
\lx@intercol\hfil\textbf{0}_{3\times 2}\hfil\lx@intercol\\\
\frac{r}{L}&-\frac{r}{L}\end{bmatrix}$ (9)

in which $L$ is the distance between the two wheels, $r$ is the radius of the
wheels and $\theta$ is the steering angle of the mobile base w.r.t. the world
frame $\mathcal{F}_{W}$.

The kinematic model in (8) considers as a reference point for the motion the
midpoint of the wheel axis. This is however subject to non-holonomic
constraints, which limit the instantaneous mobility of the robot. In order to
overcome this limitation, we deploy a I-O SFL controller d’Andréa Novel et al.
(1995), which allows us to linearize static state feedback laws. By shifting
to a reference point located at a distance $b$ from the wheel axis w.r.t. the
base frame $\mathcal{F}_{b}$, the translational velocity of the reference
control point is unrestricted. The kinematic model in (8) becomes:

${}^{W}\dot{\textbf{x}}_{B}=\textbf{J}_{B}\dot{\textbf{q}}_{b}$ (10)

in which ${}^{W}\dot{\textbf{x}}_{B}$ is the Cartesian velocity of the new
control point w.r.t the world frame $\mathcal{F}_{W}$, while the corresponding
Jacobian $\textbf{J}_{B}$ is defined as:

$\displaystyle\textbf{J}_{B}=\begin{bmatrix}[c](\frac{r\,\cos\theta}{2}-\frac{br\,\sin(\theta)}{L})&(\frac{r\,\cos\theta}{2}+\frac{br\,\sin(\theta)}{L})\\\
(\frac{r\,\sin\theta}{2}+\frac{br\,\cos(\theta)}{L})&(\frac{r\,\sin\theta}{2}-\frac{br\,\cos(\theta)}{L})\\\
\lx@intercol\hfil\textbf{0}_{3\times 2}\hfil\lx@intercol\\\
\frac{r}{L}&-\frac{r}{L}\end{bmatrix}$ (11)

Finally, we introduce a mapping term which allows us to compute the
contribution to the end effector velocity due to the mobile base as:

${}^{W}\dot{\textbf{x}}_{E(b)}=\mathbb{H}\,\textbf{J}_{B}\dot{\textbf{q}}_{b}$
(12)

in which $\mathbb{H}\in\mathbb{R}^{m\times m}$ is defined as:

$\displaystyle\mathbb{H}=\begin{bmatrix}[c]1&0&0&0&0&-{}^{B}y_{E}\\\
0&1&0&0&0&\phantom{-}{}^{B}x_{E}\\\ \lx@intercol\hfil\textbf{0}_{3\times
6}\hfil\lx@intercol\\\ 0&0&0&0&0&\phantom{-}1\end{bmatrix}$ (13)

with $({}^{B}x_{E},{}^{B}y_{E})$ being the Cartesian coordinates of the end
effector with respect to the frame $\mathcal{F}_{B}$.

Merging (5) and (12) we can achieve the mapping in (2) using the augmented
generalized velocities
$\dot{\textbf{q}}=[\dot{\textbf{q}}_{b}^{T}\,\,\dot{\textbf{q}}_{a}^{T}]^{T}$
and by constructing the augmented Jacobian $\textbf{J}_{m}$ as:

$\displaystyle\textbf{J}_{m}=\begin{bmatrix}\mathbb{H}\,\textbf{J}_{B}\quad\textbf{J}_{A}\end{bmatrix}$
(14)

We can thus track a desired reference velocity for the robot
$\dot{\textbf{x}}_{des}={}^{W}\dot{\textbf{x}}_{E(a)}+{}^{W}\dot{\textbf{x}}_{E(b)}$
via the generalized velocities:

$\dot{\textbf{q}}_{des}=\textbf{J}_{m}^{+}\dot{\textbf{x}}_{des}$ (15)

##  4 Collaborative Constraint-Oriented Control Architecture

In this section we provide the necessary tools for building the constraint-
oriented control architecture proposed in Benzi and Secchi (2021), which is
based on energy tanks Franken et al. (2011) and CBFs for task encoding
Notomista and Egerstedt (2019). We finally present the policy for adapting the
admittance parameters online.

###  4.1 Energy Tanks

Energy tanks are energy reservoirs modelled as:

$\begin{cases}\dot{x}_{t}=u_{t}\\\
y_{t}=\frac{\partial{T}}{\partial{x_{t}}}=x_{t}(t)\end{cases}$ (16)

where $x_{t}\in\mathbb{R}$ is the state of the tank, the pair
$(u_{t},y_{t})\in\mathbb{R}\times\mathbb{R}$ represents the power port through
which the tank exchanges energy and

$T(x_{t})=\frac{1}{2}x^{2}_{t}$ (17)

is the storage function of the energy tank.

The energy stored inside tanks is not affected by any specific dynamics,
meaning that it can be purposefully utilised at any stage for performing any
desired behaviour. This is carried out, in the case of the standard admittance
controller in (3), by interconnecting the power port of the tank
$(u_{t},y_{t})$ with the power port $(\textbf{F}_{e},\dot{\textbf{x}}_{des})$
of the implemented admittance dynamics, via:

$\begin{cases}u_{t}(t)=\dot{x}_{t}(t)=\textbf{A}^{T}(t)\textbf{F}_{e}(t)\\\
\dot{\textbf{x}}_{des}(t)=\textbf{A}(t)y_{t}(t)=\bm{\gamma}(t)\end{cases}$
(18)

where $\textbf{A}(t)\in\mathbb{R}^{m}$ is defined as

$\textbf{A}(t)=\frac{\bm{\gamma}(t)}{x_{t}(t)}$ (19)

and $\bm{\gamma}(t)\in\mathbb{R}^{n}$ is the desired value for the output
$\dot{\textbf{x}}_{des}(t)$, i.e. the velocity resulting from the admittance
dynamics. This means that by a proper modulation of the energy flow it is
possible to implement any desired port behaviour. From (16),(17) and (18) we
obtain

$\dot{T}=u_{t}y_{t}=\textbf{A}^{T}(t)\textbf{F}_{e}y_{t}=\bm{\gamma}^{T}\textbf{F}_{e}$
(20)

showing how the energy flow necessary for implementing a desired admittance
dynamics is provided by the tank.

A singularity occurs in (18) whenever $x_{t}(t)=0$ due to the definition of
$\textbf{A}(t)$, meaning that the tank is depleted and no behavior can be
performed. This issue can be faced by initializing $x_{t}$ such that
$T(x_{t}(0))\geq\underline{\varepsilon}>0$ and by guaranteeing that
$T(x_{t}(t))\geq\underline{\varepsilon}$ $\forall t>0$, with
$\underline{\varepsilon}$ being an arbitrarily set lower bound.

In Secchi and Ferraguti (2019), it was shown that if
$T(x_{t})\geq\underline{\varepsilon}\,\,\forall t\geq 0$, then the modulated
tank (18) remains passive independently of the desired output
$\bm{\gamma}(t)$. Thus, as long as the tank is not depleted, any desired port
behavior can be passively implemented by modulating the energy flow.
Specifically, it is possible to reproduce any passive dynamics Giordano et al.
(2013); Riggio et al. (2018).

For reproducing a non-passive dynamics, such as the time varying admittance in
(3), we can exploit the previous results for passivizing the desired
behaviour. The passivity of the modulated tank can be encoded as a constraint:

$T(x_{t})\geq\underline{\varepsilon}\quad\forall t\geq 0$ (21)

Then, as in Secchi and Ferraguti (2019) and Benzi and Secchi (2021), the
appropriate control input can be synthesised via the following optimization
problem

$\displaystyle\underset{\dot{\textbf{x}}_{des}}{\text{minimize}}$
$\displaystyle||\dot{\textbf{x}}_{des}-\dot{\textbf{x}}_{a}||^{2}$ (22)
subject to
$\displaystyle\int_{0}^{t}\textbf{F}_{e}^{T}(\tau)\dot{\textbf{x}}_{des}(\tau)d\tau\geq-T(x_{t}(0))+\underline{\varepsilon}$

The solution of (22) provides the best passive approximation of the desired
behavior $\dot{x}_{a}$, by keeping track of the energy stored in the tank. We
then set $\dot{\textbf{x}}_{des}=\bm{\gamma}$ and utilize it to tune the
modulation matrix $A(t)$ in (18). This ensures a passive energy balance even
if the variation of the parameters would inject energy in the system.

A discrete-time version of (22) was proposed in Secchi and Ferraguti (2019),
thus obtaining a convex formulation, for a computationally fast and simple
optimization problem.

###  4.2 Control Barrier Functions

CBFs can be formulated for enforcing multiple time varying constraints on a
robot described by (2). A specific procedure was described in Notomista et al.
(2020), by which both a set of tasks, both kinematic limits are modelled as
dynamic constraints onto the input of the systems. These can then be inserted
into an optimization problem, whose solution is the input satisfying all the
constraints, i.e. leading to the simultaneous execution of all the tasks.

We represent the desired tasks as the minimization of a non-negative, time-
varying, continuously-differentiable cost function
$C:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}$. Considering a robot
modelled as (2) and the time-varying task variable
$\bm{\sigma}\in\mathbb{R}^{n}$ as an output variable, the task execution can
be encoded via the following optimization problem:

$\displaystyle\underset{u}{\text{minimize}}$ $\displaystyle C(\bm{\sigma},t)$
(23) subject to
$\displaystyle{}^{W}\dot{\textbf{x}}_{E}=\textbf{J}_{m}(\textbf{q})\,\textbf{u}$
$\displaystyle\bm{\sigma}=\textbf{k}({}^{W}\textbf{x}_{E},t)$

A convex formulation of the problem can be obtained by leveraging Control
Barrier Functions. Let $\mathcal{C}\subset\mathbb{R}^{n}$ be the subset in
which the task is considered to be executed, i.e. $C(\bm{\sigma},t)=0$. Let
then $h:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}$ be a control
barrier function defined as $h(\bm{\sigma},t)=-C(\bm{\sigma},t)$. We have that
$h$ is non negative only whenever $C(\bm{\sigma},t)=0$, i.e. the region of
satisfaction of the task. By enforcing the non-negativity of $h$ we can then
achieve the execution of the task $\bm{\sigma}$. This results in the following
convex optimization problem Notomista and Egerstedt (2019):

$\displaystyle\underset{\dot{\textbf{q}}}{\text{minimize}}$
$\displaystyle||\dot{\textbf{q}}||^{2}$ (24) subject to
$\displaystyle\frac{\partial h}{\partial t}+\frac{\partial
h}{\partial\bm{\sigma}}\frac{\partial\bm{\sigma}}{\partial{}^{W}\textbf{x}_{E}}\textbf{J}_{m}(\textbf{q})\dot{\textbf{q}}+\alpha(h(\bm{\sigma},t))\geq
0$

where $\alpha(\cdot)$ is an extended class $\mathcal{K}$ function111An
extended class $\mathcal{K}$ is a function
$\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi$ is strictly increasing
and $\phi(0)=0$ and where we have chosen as an input for (2)
$\textbf{u}=\dot{\textbf{q}}$.

We can straightfowardly extended the formulation in order to implement the
execution of multiple M different tasks at the same time, each respectively
encoded by the cost functions $C_{1},\dots,C_{M}$. At the same time, we can
add the passivity constraint in (22) to the stack, thus obtaining a single
convex optimization problem as in Benzi and Secchi (2021). The final
formulation is:

$\displaystyle\underset{\dot{\textbf{q}},\bm{\delta}}{\text{minimize}}$
$\displaystyle||\dot{\textbf{q}}-\dot{\textbf{q}}_{adm}||^{2}+l||\bm{\delta}||^{2}$
(25) subject to $\displaystyle\frac{\partial h_{m}}{\partial t}+\frac{\partial
h_{m}}{\partial\bm{\sigma}}\frac{\partial\bm{\sigma}}{\partial{}^{W}\textbf{x}_{E}}\textbf{J}_{m}(\textbf{q})\dot{\textbf{q}}$
$\displaystyle+\alpha(h_{m}(\bm{\sigma},t))\geq-\bm{\delta}_{m}\quad
m\in\\{1,\dots,M\\}$
$\displaystyle\int_{0}^{t}\textbf{F}_{e}^{T}(\tau)\textbf{J}_{m}(\textbf{q})\dot{\textbf{q}}(\tau)d\tau\geq-T(x_{t}(0))+\underline{\varepsilon}$

in which
$\dot{\textbf{q}}_{adm}=\textbf{J}_{m}(\textbf{q})^{+}\dot{\textbf{x}}_{a}$ is
the desired admittance expressed in the joint space,
$h_{m}(\bm{\sigma},t)=-C_{m}(\bm{\sigma},t)$ and
$\bm{\delta}=[\delta_{1},\dots,\delta_{M}]^{T}$ is the vector of slack
variables dedicated to relaxing each constraint, while $l\geq 0$ is a scaling
factor. The slack variables $\delta_{m}$ are required for ensuring the
feasibility of the problem at all times, even if conflicting constraints are
simultaneously active.

We have thus built an architecture capable of implementing both a passive
time-varying admittance controller, both the execution of multiple tasks at
the same time.

###  4.3 Admittance Parameters Adaptation

The choice of the dynamic parameters for a given admittance directly affects
how the human perceives the interaction and the overall quality of the
collaboration Duchaine and Gosselin (2007).

Our goal is to adapt online the admittance parameters according to his
perceived intention of the operator, in order to properly assist him during
the collaborative transportation task. We utilize the adaptation policy
proposed in Lecours et al. (2012), where the human intention is inferred by
monitoring the magnitude and the direction of the desired acceleration,
alongside the current velocity value. Two possible intentions are taken into
account: acceleration and deceleration. The damping and inertia parameters are
updated at each cycle according to the perceived intention, starting from
default values of $\textbf{D}_{f}$ and $\textbf{M}_{f}$.  The damping term D
is tuned as follows:

$\begin{cases}\textbf{D}=\textbf{D}_{f}-\mathbb{I}_{m}\odot(\alpha_{a}|\ddot{\textbf{x}}_{a}|\textbf{1}_{m}^{T})\quad\text{if
acceleration}\\\
\textbf{D}=\textbf{D}_{f}+\mathbb{I}_{m}\odot(\alpha_{d}|\ddot{\textbf{x}}_{a}|\textbf{1}_{m}^{T})\quad\text{if
deceleration}\end{cases}$ (26)

in which $\alpha_{a}$ and $\alpha_{d}$ are two gains to be tuned, $|\cdot|$
indicates the magnitude of the vector, $\mathbb{I}_{m}\in\mathbb{R}^{m\times
m}$ is the identity matrix, $\textbf{1}_{m}\in\mathbb{R}^{m}$ is the vector of
ones, while $\odot$ indicates the Hadamard (or element-wise) product.  The
inertia values are computed according to the damping ones, by ensuring that
the ratio $\frac{\textbf{M}(\textbf{x}_{a})}{\textbf{D}(\textbf{x}_{a})}$
remains proportional,  resulting in a more intuitive interaction for the
operator:

$\begin{cases}\textbf{M}=\frac{\textbf{M}\,\textbf{D}}{\textbf{D}_{f}}\quad&\text{if
acceleration}\\\
\textbf{M}=\frac{\textbf{M}}{\textbf{D}_{f}}(1-\beta(1-e^{-\eta(\textbf{D}-\textbf{D}_{f})}))\textbf{D}\quad&\text{if
deceleration}\end{cases}$ (27)

in which $\beta\in(0,1)$ serves to tune the steady-state value of
$\frac{\textbf{M}(\textbf{x}_{a})}{\textbf{D}(\textbf{x}_{a})}$, while $\eta$
is a parameter defining the smoothness with which the ratio varies (see
Lecours et al. (2012)).

##  5 Experiments

Figure 2: Setup and structure of the mobile manipulator and frames utilized
for the kinematic computations

An experimental evaluation has been conducted, in order to validate each
individual component of the architecture. The mobile manipulator used in the
validation is composed of a MiR 100 mobile base, with a 6-DOF collaborative
manipulator UR10e mounted on top. In Fig. 2 the overall setup, as well as the
utilized frames can be observed.

###  5.1 Passive Time-Varying Admittance Controller

Figure 3: Distribution over time of the force exerted by the human using the
two admittance controllers Figure 4: Norm of the variation of the admittance
parameters and evolution of the energy in the tank over time

First, the collaborative transportation task is carried out, highlighting the
advantages of the passive time-varying admittance controller. During the
experiment, a wooden plank of $2.5m$ in length is attached to the tool of the
manipulator (see Fig. 2), while the human is holding the opposite side. As the
human pulls his side of the plank, the interaction force $\textbf{F}_{e}$ is
sensed by the on-board F/T sensor of the UR10e. After being transformed by
means of ${}^{W}\mathcal{T}_{E}$ into the world frame, the force is integrated
via the time-varying admittance dynamics (3) for producing a desired
operational velocity set-point $\dot{\textbf{x}}_{des}$. By means of the whole
body kinematic mapping in (15), a desired velocity for both the arm joints and
the mobile base wheels is then synthesized. In this way, the robot can
actively assist the human in the transportation of the load.

During the transportation task, an experienced user has to navigate in a
constricting environment, with frequent stops and turns. In this scenario, the
online adaptation of the admittance parameters allows to visibly enhance the
quality of the interaction. In fact, by dynamically varying the mass and
damping terms as shown in Sec. 4, the human is assisted while decelerating and
accelerating the robot, resulting in reduced physical effort from his side.
This can be observed in Fig. 3, in which the distribution of the force exerted
by the operator over time using the standard and the variable admittance
controller is portrayed. For the standard case, the values of inertia and
damping were fixed at the default ones $\textbf{M}_{f}$ and $\textbf{D}_{f}$,
namely 4$kg$ and 20$\frac{Ns}{m}$.

As shown in Sec. 4, the overall passivity is ensured by the energy tank and
the energetic condition in (22), thus ensuring a robustly stable interaction.
The variation of the dynamic parameters during the task, as well as the
evolution of the energy in the tank, are portrayed in Fig. 4.

###  5.2 CBF-based tasks

Figure 5: Evolution of the safety distance over time. Figure 6: Value of the
augmented joint velocities and evolution of $h_{cyl2}$ over time. Whenever
$h_{cyl2}=0$, only the mobile base contributes to the motion.

Alongside the main transportation task, a set of additional secondary task is
implemented. These are encoded via CBFs and inserted into the optimization
problem in (25).

First, the safety of the transported object is ensured. The load is wrapped
with a capsule Lin et al. (2017), a virtual object composed by two semi-
spheres, centred in the two mid-points of the short edges of the plank, and a
cylinder whose longitudinal axis links the two points. Then, a minimum safety
distance $D_{min}=0.3\,m$ between the capsule and a detectable object in the
scene is guaranteed by means of the following CBF

$h_{safe}=d^{2}-D^{2}_{min}$ (28)

in which the safety distance $d\in\mathbb{R}$ is defined as

$d=||\textbf{x}_{obs}-\textbf{x}_{c}||-r_{c}$ (29)

where $\textbf{x}_{obs}\in\mathbb{R}^{m}$ is the position of the obstacle,
while $\textbf{x}_{c}\in\mathbb{R}^{m}$ is the position on the capsule which
is closest to the obstacle and $r_{c}\in\mathbb{R}$ is the radius of the
capsule. Given that the manipulator is significantly more dexterous than the
mobile base and that sudden rotations of the base can be negatively perceived
by the human, we decided to only employ the manipulator for this safety task.
Thus, its execution can performed by adding in (25) the constraint:

$2(\textbf{x}_{obs}-\textbf{x}_{c})^{T}\textbf{J}_{lim}\dot{\textbf{q}}\geq-\alpha(h_{safe})+\delta_{safe}$
(30)

with $\textbf{J}_{lim}=[\textbf{0}_{6\times 2}^{T}\,\,\textbf{J}_{a}^{T}]^{T}$
being the restriction of the augmented Jacobian to the manipulator alone. The
plot in Fig 5 portrays the evolution of $d$ over time, showing how the safety
distance is always respected.

This procedure can be extended for guaranteeing the safety of the overall
structure by wrapping with capsules the entire system.

Additionally, we restrict the workspace of the manipulator to a suitable
subset for the application. First, in order to avoid self collision between
the tool mounted onto the manipulator and its joints, we define the following
CBF

$h_{cyl1}=d_{1}^{2}-R_{1}^{2}$ (31)

with $d_{1}=\sqrt{{}^{0}x_{E1}^{2}+{}^{0}x_{E2}^{2}}$ and $R_{1}$ being the
radius which defines a safety cylinder centred in the manipulator base.

Additionally, we want to limit the outreach of the manipulator, in order to
avoid singularities during the transportation, as they could lead to
unexpected behaviours. We then deploy the following CBF for forcing the tip of
the manipulator to stay inside a set cylinder of radius $R_{2}$

$h_{cyl2}=-d_{1}^{2}+R_{2}^{2}$ (32)

Fig. 6 shows the contribution of this CBF; as the manipulator reaches the edge
of the cylinder, its joint velocities halt, restarting only as it recoils in
the desired area.

##  6 Conclusions and Future Works

In this paper we developed a control architecture capable of performing a
flexible and robust collaborative transportation task, treating the mobile
manipulator as a whole-body system and considering non-holonomic constraints.
In future works, we aim at further exploiting the redundancy of the robot, for
maximizing performance and compliance.

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