MESON2016 - the 14${}^{\textrm{th}}$ International Workshop on Meson
Production, Properties and Interaction

11institutetext: Helmholtz-Institut für Strahlen- und Kernphysik, Universität
Bonn, Germany

# Double polarisation experiments in meson photoproduction

Jan Hartmann
for the CBELSA/TAPS Collaboration 11<EMAIL_ADDRESS>
###### Abstract

One of the remaining challenges within the standard model is to gain a good
understanding of QCD in the non-perturbative regime. A key step towards this
aim is baryon spectroscopy, investigating the spectrum and the properties of
baryon resonances. To gain access to resonances with small $\pi N$ partial
width, photoproduction experiments provide essential information. Partial wave
analyses need to be performed to extract the contributing resonances. Here, a
complete experiment is required to unambiguously determine the contributing
amplitudes. This involves the measurement of carefully chosen single and
double polarisation observables. In a joint endeavour by MAMI, ELSA, and
Jefferson Laboratory, a new generation of experiments with polarised beams,
polarised proton and neutron targets, and $4\pi$ particle detectors have been
performed in recent years. Many results of unprecedented quality were recently
published by all three experiments, and included by the various partial wave
analysis groups in their analyses, leading to substantial improvements, e.g. a
more precise determination of resonance parameters. An overview of recent
results is given, with an emphasis on results from the CBELSA/TAPS experiment,
and their impact on our understanding of the nucleon excitation spectrum is
discussed.

##  1 Introduction

The spectrum of excited nucleon states reflects the dynamics of QCD in the
non-perturbative regime. It has been studied for many years using pion beams.
However, the spectrum of known nucleon resonances is in conflict with
predictions from quark models isgur77 ; loring01 . Most obvious is the missing
resonance problem: the fact that more states are predicted by models at higher
masses than have been observed experimentally. But also the ordering of
excited states with positive and negative parity is partly in disagreement
with experiments, the most prominent example being the $N(1440)\,1/2^{+}$
which is predicted by most quark models to be heavier than the
$N(1535)\,1/2^{-}$. QCD calculations on the lattice edwards11 , though using
unphysically large quark masses, yield a similar pattern as the non-
relativistic quark model. Measuring the properties of the known resonances
more precisely and searching for new resonances is essential to understanding
the discrepancies between theory and experiment.

Photoproduction experiments allow access to resonances with small $\pi N$
couplings and therefore have great potential to observe the missing
resonances. The contributing resonances are extracted from the measured data
in a partial wave analysis (PWA). To determine the contributing amplitudes in
an unambiguous way, a complete experiment chiang97  is needed, which requires
the measurement of polarisation observables. In this paper, a selection of
recent measurements of single and double polarisation observables accessible
with linearly or circulary polarised photon beam and a longitudinally or
transversely polarised proton target is shown, and their impact on the PWA is
discussed.

##  2 Experimental setup

The data presented here were obtained with the CBELSA/TAPS experiment at ELSA
hillert06 . A linearly polarised photon beam was produced from the incident
$3.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ electron beam via coherent
bremsstrahlung off a carefully aligned diamond crystal elsner09 . For a
coherent edge at $E_{\gamma}=$950\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ a
maximum polarisation of $65\text{\,}\mathrm{\char 37\relax}$ was reached. A
circularly polarised photon beam was produced from a longitudinally polarised
$2.4\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ electron beam provided by ELSA.
The polarisation was measured with a Møller polarimeter, a maximum electron
polarisation of $64\text{\,}\mathrm{\char 37\relax}$ was reached. The
electrons, after hitting the radiator target, passed through a magnet onto a
tagging hodoscope which defined the energy of the bremsstrahlung photons. The
photon beam impinged on a frozen spin butanol target bradtke99  providing
transversely polarised protons with an average target polarisation degree of
$74\text{\,}\%$.

Figure 1: The experimental setup of the CBELSA/TAPS experiment.

The detector system, which is shown in Figure 1, consisted of two
electromagnetic calorimeters, the Crystal Barrel aker92  and the MiniTAPS
detector novotny91 , together covering the polar angle range from $1^{\circ}$
to $156^{\circ}$ and the full azimuthal angle. For charged particle
identification, a three-layer scintillating fibre detector suft05  surrounding
the target, and plastic scintillators in forward direction could be used. The
detector setup provided a high detection efficiency for photons and is
therefore ideally suited to measure single and double polarisation observables
in reactions with neutral mesons decaying into photons in the final state.

##  3 Results

The cross-section for photoproduction of single pseudoscaler mesons with a
polarised beam off a polarised target can be written in the form barker75

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}\varOmega}=\left(\frac{\mathrm{d}\sigma}{\mathrm{d}\varOmega}\right)_{0}\cdot\Big{[}$
$\displaystyle{1}-{\varSigma}{\delta_{\ell}\cos(2\phi)}+{T}{\varLambda_{y}}-{P}{\varLambda_{y}\delta_{\ell}\cos(2\phi)}$
(1)
$\displaystyle-{E}{\varLambda_{z}\delta_{\odot}}+{F}{\varLambda_{x}\delta_{\odot}}+{G}{\varLambda_{z}\delta_{\ell}\sin(2\phi)}+{H}{\varLambda_{x}\delta_{\ell}\sin(2\phi)}\Big{]},$

where $\left(\frac{\mathrm{d}\sigma}{\mathrm{d}\varOmega}\right)_{0}$ is the
unpolarised cross section, $\delta_{\ell}$ ($\delta_{\odot}$) denotes the
linear (circular) polarisation degree of the photon beam, and
$\varLambda_{x,y,z}$ is the polarisation degree of the target protons in the
centre-of-mass (CM) coordinate frame. In a polarised butanol (C4H9OH) target,
not only the polarised protons, but also unpolarised nucleons in the C and O
nuclei contribute to the count rate. This effectively reduces the target
polarisation and can be taken into account by the dilution factor $d$:
$\varLambda\to d\,\varLambda$. To determine $d$, the background contribution
from bound nucleons needs to be determined, e.g. using a background
measurement with a carbon target. For details, see e.g. gottschall14 ;
hartmann15 .

###  3.1 $\pi^{0}$ photoproduction

Figure 2: Results for the observables $E$ (left) gottschall14  and $G$ (right)
thiel12 ; thiel16  for four different angular bins, compared to different PWA
predictions: MAID (green, dotted) maid07 , SAID CM12 (red, dashed-dotted)
said12 , BnGa 2011-02 (black, solid) anisovich12 , and JüBo 13-01 (blue,
dashed) roenchen14 .

With circularly polarised photons and longitudinally polarised protons, the
helicity asymmetry $E$ is accessible. It can be determined from the asymmetry
of events with parallel ($N_{3/2}$) and anti-parallell ($N_{1/2}$) spins of
the initial-state particles as

$E=\frac{N_{1/2}-N_{3/2}}{N_{1/2}+N_{3/2}}\cdot\frac{1}{d\,\varLambda_{z}\,\delta_{\odot}}.$
(2)

With linearly polarised photons and logitudinally polarised protons, the
double polarisation observable $G$ can be deduced from the correlation between
the photon polarisation plane and the scattering plane. The number of events
$N$ as a function of the azimuthal angle $\phi$ between the two planes is
given by

$N(\phi)/N_{0}=1-\delta_{\ell}\,\varSigma_{\mathrm{eff}}\cos(2\phi)+d\,\varLambda_{z}\,\delta_{\ell}\,G\sin(2\phi),$
(3)

where $N_{0}$ is given by averaging $N(\phi)$ over $\phi$.
$\varSigma_{\mathrm{eff}}$ mixes the beam asymmetry from free and bound
nucleons. Results for both $E$ and $G$ are shown in Fig. 2 in comparison to
different PWA predictions.

With linearly polarised photons and transversely polarised protons, the
observables $T$, $P$, and $H$ can be measured simultaneously. In this case,
the azimuthal distribution of events is given by

$\displaystyle N(\phi)/N_{0}=1$
$\displaystyle-\delta_{\ell}\,\varSigma_{\mathrm{eff}}\cos(2\phi)+d\,\varLambda_{T}\,T\sin(\phi-\varphi)$
(4)
$\displaystyle-d\,\varLambda_{T}\,\delta_{\ell}P\cos(2\phi)\sin(\phi-\varphi)+d\,\varLambda_{T}\,\delta_{\ell}H\sin(2\phi)\cos(\phi-\varphi),$

where $\varphi$ is the azimuthal angle between the target polarisation vector
and the photon polarisation plane. Results are shown in Fig. 3.

Figure 3: Results for the observables $T$, $P$, and $H$ hartmann14 ;
hartmann15  as a function of the scattering angle and the $\gamma p$ invariant
mass W (in GeV, only every third bin is shown here). References to earlier
data (red points) are given in anisovich12 , Refs. [49-71] therein. The solid
black line represents the BnGa2014 fit hartmann15 . The data are compared to
predictions (dashed curves) from BnGa2011-02 (red) anisovich12 , MAID (green)
maid07 , SAID CM12 (blue) said12 , and JüBo 2015 (magenta) roenchen15 .

Our data up to $E_{\gamma}=$930\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ were
used as a basis for an energy-independent PWA hartmann14 , allowing for the
determination of the $N(1520)\,3/2^{-}$ helicity amplitudes with minimal model
dependence. All the data were included in the BnGa multi-channel PWA, together
with further data on other channels.111For a complete list, see hartmann15 ,
Ref. [25] therein. Starting from the previous solutions BnGa2011-01 and
BnGa2011-02 anisovich12  all parameters were re-optimised. The newly
determined multipoles are compatible with the previous ones at the $2\sigma$
level over the full mass range. The errors are significantly reduced, on
average by a factor of 2.25 hartmann15 . The impact of the new data on the
BnGa, SAID, and JüBo analyses was recently investigated in a joint effort of
the analysis groups pwa16 . To quantify the discrepancy between different
analyses, the variance of the 16 (complex) multipole amplitudes $\mathcal{M}$
up to $L=4$ can be used, defined as

$\mathrm{var}(1,2)=\frac{1}{2}\sum\limits_{i=1}^{16}\bigl{(}\mathcal{M}_{1}(i)-\mathcal{M}_{2}(i)\bigr{)}\bigl{(}\mathcal{M}^{*}_{1}(i)-\mathcal{M}^{*}_{2}(i)\bigr{)}.$
(5)

Once the new data are included, the discrepancy between all three analyses is
significantly reduced, as can be seen in Fig. 4.

Figure 4: The variance of all three PWAs summed over all $\gamma p\to\pi^{0}p$
multipoles up to $L=4$ pwa16 . The range covered by the new double-
polarisation observables is indicated by shaded areas. Over the largest part
of the energy range the new data have enforced an improvement of the overall
consistency. The improvement is displayed as a light green area and,
separately, as difference of the variance. Ranges with an overall
deterioration are marked in red.

###  3.2 $\eta$ photoproduction

Figure 5: Preliminary results for the observables $T$, $P$, and $H$ as a
function of the scattering angle and the $\gamma p$ invariant mass W (in GeV,
only every second bin is shown here). Earlier ELSA data (red) bock98  and
recent MAMI results (green) akondi14  are shown for comparison. The solid
black line represents a preliminary BnGa fit. The data are compared to
predictions (dashed curves) from BnGa2011-02 (red) anisovich12 , MAID (green)
maid07 , SAID GE09 (blue) mcnicoll10 , and JüBo 2015 (magenta) roenchen15 .

Preliminary results for the polarisation observables $T$, $P$, and $H$ are
shown in Figure 5. Large deviations from the data are observed for the
predictions from MAID maid07 , SAID mcnicoll10 , BnGa2011 anisovich12 , Gießen
shklyar13 , and the JüBo model roenchen15 , emphasising how important these
new data are to constrain the amplitudes for $\eta$ photoproduction.

The analysis by the BnGa group of our new data on $T$, $P$, and $H$, together
with not yet published data on $E$ and $G$ and further data from Mainz ($T$,
$F$) akondi14  and Jefferson Laboratory senderovich15  ($E$) is presently
ongoing. The results will be published in the near future mueller16 .

##  4 Summary and outlook

In $\pi^{0}$ photoproduction, the unprecedented precision of the data
significantly reduces the errors of the PWAs, leading to a more precise
determination of resonance parameters. In $\eta$ photoproduction, where
several observables are now measured for the first time, the new data are
crucial to constrain the photoproduction amplitudes. Further reaction channels
are also being investigated. In particular multi-meson final states like
$p\pi^{0}\pi^{0}$ or $p\pi^{0}\eta$ are sensitive to cascade decays of higher-
mass resonances via intermediate $N^{*}$ and $\varDelta^{*}$ states
sokhoyan15a ; sokhoyan15b ; mahlberg16 .

We thank the technical staff of ELSA, the polarised target group, and the
participating institutions for their invaluable contributions to the success
of the experiment. We acknowledge support from the Deutsche
Forschungsgemeinschaft (SFB/TR16) and Schweizerischer Nationalfonds.

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MnLargeSymbols’164 MnLargeSymbols’171

# On the isomorphism problem for power semigroups

Salvatore Tringali  (S. Tringali) School of Mathematical Sciences, Hebei Normal University | Shijiazhuang, Hebei province, 050024 China<EMAIL_ADDRESS>http://imsc.uni-graz.at/tringali

###### Abstract.

Denote by $\mathcal{P}(S)$ the semigroup obtained by equipping the family of
all non-empty subsets of a (multiplicatively written) semigroup $S$ with the
operation of setwise multiplication induced by $S$ itself. We call a
subsemigroup $P$ of $\mathcal{P}(S)$ downward complete if any element of $S$
lies in at least one set $X\in P$ and any non-empty subset of a set in $P$ is
again in $P$.

We prove, for a commutative semigroup $S$, a characterization of the
cancellative elements of a downward complete subsemigroup of $\mathcal{P}(S)$
in terms of the cancellative elements of $S$. Consequently, we show that, if
$H$ and $K$ are cancellative semigroups and either of them is commutative,
then every isomorphism from a downward complete subsemigroup of
$\mathcal{P}(H)$ to a downward complete subsemigroup of $\mathcal{P}(K)$
restricts to an isomorphism from $H$ to $K$. This solves an open case of a
problem of Tamura and Shafer from the late 1960s and generalizes a recent
result by Bienvenu and Geroldinger, where it is assumed, among other
conditions, that $H$ and $K$ are numerical monoids.

###### Key words and phrases:

Isomorphism problems, numerical monoids, power algebras, cancellative
semigroups, setwise sums and products.

###### 2020 Mathematics Subject Classification:

Primary 20M10. Secondary 11B30, 20M13, 20M35.

##  1\. Introduction

Let $S$ be a semigroup (see the end of the section for notation and
terminology). Equipped with the binary operation of setwise multiplication
induced by $S$ and defined by

$(X,Y)\mapsto\\{xy\colon x\in X,\kern 1.00006pty\in Y\\},$

the _non-empty_ subsets of $S$ form a semigroup, herein denoted by
$\mathcal{P}(S)$ and called the large power semigroup of $S$. Furthermore, the
non-empty _finite_ subsets of $S$ make up a subsemigroup of $\mathcal{P}(S)$,
which we denote by $\mathcal{P}_{\mathrm{fin}}(S)$ and call the finitary power
semigroup of $S$ (see Example 1.1(1) for a generalization of the same idea).
Below, we will refer to either of these structures as a power semigroup of
$S$.

Power semigroups offer a natural algebraic framework for many important
problems in additive combinatorics and related fields, from Ostmann’s
conjecture [18, p. 13] on the “asymptotic additive irreducibility” of the set
$\mathbb{P}$ of (positive rational) primes to Sárközy’s conjecture [21,
Conjecture 1.6] on the “additive irreducibility” of the set of quadratic
residues modulo $p$ for all large $p\in\mathbb{P}$. More broadly, the
arithmetic of power semigroups — with a focus on questions related to the
possibility or impossibility of factoring certain sets as a product of other
sets that are, in a way, “irreducible” — is a rich topic in itself [9, 2, 3,
29] and has been serving as a guiding example in the development of a
“unifying theory of factorization” [27, 5, 6, 7, 28] whose primary aim is to
extend the classical theory [11, 12] beyond its current boundaries (so as to
make it usable in non-traditional settings).

As it turns out, $\mathcal{P}(S)$ was first _systematically_ studied by Tamura
and Shafer [23, 13] in the 1960s (its definition can at least be traced back
to the early literature on power algebras [4, Sect. 2]) and gained substantial
attention in the 1980s and 1990s due to its role in the theory of automata and
formal languages [1, 20]. This led to the following question, arisen from
Tamura and Shafer’s work and commonly known as the isomorphism problem for
power semigroups:

######  Question 1.

Suppose that a semigroup $H$ is globally isomorphic to a semigroup $K$,
meaning that $\mathcal{P}(H)$ is (semigroup-)isomorphic to $\mathcal{P}(K)$.
Is it necessarily true that $H$ is isomorphic to $K$?

The question was quickly answered in the negative by Mogiljanskaja [17], but
remains open for _finite_ semigroups [13, p. 5] (despite some authors claiming
differently based on results that Tamura himself announced, _without proof_ ,
in [25]). More generally, one can ask:

######  Question 2.

Given a class $\mathcal{C}$ of semigroups, prove or disprove that $H$ is
globally isomorphic to $K$, for some $H,K\in\mathcal{C}$, if and only if $H$
is isomorphic to $K$.

It is easy to check (see Remark 2.4(2) and cf. [29, Remark 1.1]) that, if
$f\colon H\to K$ is a semigroup isomorphism, then the same is true of the
function

$\mathcal{P}(H)\to\mathcal{P}(K)\colon\allowbreak X\mapsto\\{f(x)\colon x\in
X\\}.$

Therefore, the interesting aspect of Question 2 lies entirely in the “only if”
condition. E.g., the answer to the question is positive for groups [22],
completely $0$-simple semigroups [24, 26], and Clifford semigroups [10]; is
negative for involution semigroups [8]; and appears to be unknown for
cancellative semigroups, no matter whether commutative or not (cf.
https://mathoverflow.net/questions/456604/).

The present paper adds to this line of research. More precisely, we say that a
subsemigroup $P$ of the large power semigroup of a semigroup $S$ is downward
complete if each element of $S$ lies in at least one set $X\in P$ and every
non-empty subset of a set in $P$ is itself in $P$. The following examples
illustrate the property of downward completeness and will come in handy later.

######  Examples 1.1.

(1) $\mathcal{P}(S)$and $\mathcal{P}_{\mathrm{fin}}(S)$ are downward complete
subsemigroups of $\mathcal{P}(S)$ for any semigroup $S$; and so is the set of
all one-element subsets of $S$, which is, by the way, isomorphic to $S$
through the embedding $S\to\mathcal{P}(S)\colon x\mapsto\\{x\\}$. More
generally, let $\kappa$ be an _infinite_ cardinal number. By basic principles
of set theory (in any of the standard foundations of mathematics), each of the
families  $\mathcal{P}_{<\kappa}(S):=\\{X\subseteq S\colon
1\leq|X|<\kappa\\}\qquad\text{and}\qquad\mathcal{P}_{\leq\kappa}(S):=\\{X\subseteq
X\colon 1\leq|X|\leq\kappa\\}$ is a downward complete subsemigroup of
$\mathcal{P}(S)$. In particular, we have that
$\mathcal{P}_{<\kappa}(S)=\mathcal{P}_{\mathrm{fin}}(S)$ when $\kappa$ is the
cardinality of $\mathbb{N}$, and $\mathcal{P}_{\leq\kappa}(S)=\mathcal{P}(S)$
when $\kappa$ is the maximum between $|\mathbb{N}|$ and $|S|$.

(2) Let $S$ be a semigroup and $\sim$ be an equivalence relation on (the
carrier set of) $S$ with the additional property that $x_{1}\sim y_{1}$ and
$x_{2}\sim y_{2}$ imply $x_{1}x_{2}\sim y_{1}y_{2}$, i.e., $\sim$ is a
(semigroup) congruence on $S$. Denote by $\mathcal{P}_{\sim}(S)$ the family of
all non-empty sets $X\subseteq S$ such that $x\sim y$ for any $x,y\in X$.
Every non-empty subset of a set in $\mathcal{P}_{\sim}(S)$ is, of course,
itself in $\mathcal{P}_{\sim}(S)$; and since equivalence relations are
reflexive, it is also clear that $\\{x\\}\in\mathcal{P}_{\sim}(S)$ for each
$x\in S$. On the other hand, we are guaranteed by the general properties of
congruences that $XY\in\mathcal{P}_{\sim}(S)$ for all
$X,Y\in\mathcal{P}_{\sim}(S)$, because $XY\subseteq[xy]_{\sim}$ for every
$x\in X$ and $y\in Y$, where $[z]_{\sim}$ is the equivalence class of an
element $z\in S$ in the quotient set $S/{\sim}$. Therefore,
$\mathcal{P}_{\sim}(S)$ is a downward complete subsemigroup of
$\mathcal{P}(S)$.

(3) Given a semigroup $S$, the intersection of any indexed family of downward
complete subsemigroups of $\mathcal{P}(S)$ is itself a downward complete
subsemigroup of $\mathcal{P}(S)$. The proof is straightforward, on account of
the fact that a downward complete subsemigroup of $\mathcal{P}(S)$ contains, a
fortiori, every one-element subset of $S$ (further details are left to the
reader).

That being said, we obtain, for a _commutative_ semigroup $S$, a
characterization of the cancellative elements of a downward complete
subsemigroup of $\mathcal{P}(S)$ in terms of the the cancellative elements of
$S$ (Proposition 2.2); and we use it in Theorem 2.5 to prove that, if $H$ and
$K$ are cancellative semigroups and either of them is commutative, then any
isomorphism from a downward complete subsemigroup of $\mathcal{P}(H)$ to a
downward complete subsemigroup of $\mathcal{P}(K)$ maps singletons to
singletons and hence restricts to an isomorphism from $H$ to $K$ (up to the
embedding mentioned in Example 1.1(1)). It follows (Corollary 2.6) that $H$
and $K$ are globally isomorphic if and only if they are isomorphic, if and
only if $\mathcal{P}_{\mathrm{fin}}(H)$ is isomorphic to
$\mathcal{P}_{\mathrm{fin}}(K)$. We can thus give a positive answer to
Question 2 for the class of cancellative commutative semigroups and extend a
recent result of Bienvenu and Geroldinger [3, Theorem 3.2(3)], where $H$ and
$K$ are assumed to be numerical monoids (i.e., submonoids of the additive
monoid of non-negative integers with finite complement in $\mathbb{N}$) with
$\mathcal{P}_{\mathrm{fin}}(H)$ isomorphic to $\mathcal{P}_{\mathrm{fin}}(K)$
(see the comments at the end of Sect. 2).

### Generalities

We denote by $\mathbb{N}$ the (set of) non-negative integers, by
$\mathbb{N}^{+}$ the positive integers, and by $|\cdot|$ the cardinality of a
set. Unless otherwise specified, we write all semigroups multiplicatively. We
refer to Howie’s monograph [15] for the basic theory of semigroups and
monoids. In particular, we recall that, given a semigroup $S$, an element
$a\in S$ is left (resp., right) cancellative if the map $S\to S\colon x\mapsto
ax$ (resp., $S\to S\colon x\mapsto xa$) is injective; and is cancellative if
it is left and right cancellative. The semigroup itself is then said to be
cancellative if each of its elements is cancellative. Further notation and
terminology, if not explained upon first use, are standard or should be clear
from the context.

##  2\. Results

The main result of the paper (namely, Theorem 2.5) is actually a direct
consequence of Proposition 2.2 below, which may be of independent interest.
Let us start with a basic remark.

######  Remark 2.1.

Let $u$ be a cancellative element in a semigroup $S$, and assume $uX=uY$
(resp., $Xu=Yu$) for some $X,Y\in\allowbreak\mathcal{P}(S)$. Given $x\in X$,
there then exists $y\in Y$ such that $ux=uy$ (resp., $xu=\allowbreak yu$). It
follows (by the cancellativity of $u$ in $S$) that $x=\allowbreak y$ and hence
$X\subseteq\allowbreak Y$. By symmetry, we can thus conclude that $X=Y$, which
ultimately shows that $\\{u\\}$ is a cancellative element in $\mathcal{P}(S)$.

The key idea, simple as it may be, is to establish that, in a downward
complete subsemigroup of the power semigroup of a cancellative _commutative_
semigroup $S$, the cancellative elements are all and only the one-element
subsets of $S$. In fact, we will prove something slightly more general.

######  Proposition 2.2.

Let $S$ be a commutative semigroup. The cancellative elements of a downward
complete subsemigroup of $\mathcal{P}(S)$ are all and only the singletons
$\\{u\\}$ such that $u$ is a cancellative element in $S$.

###### Proof.

Let $P$ be a downward complete subsemigroup of $\mathcal{P}(S)$. If $ux=uy$
for some $u,x,y\in S$ with $x\neq\allowbreak y$, then
$\\{u\\}\allowbreak\\{x\\}=\\{u\\}\\{y\\}$ in $\mathcal{P}(S)$. Since all one-
element subsets of $S$ are in $P$, it is thus clear from Remark 2.1 that
$\\{u\\}$ is a cancellative element in $P$ if and only if $u$ is cancellative
in $S$.

It remains to check that every cancellative element of $P$ is a singleton.
For, fix $A\in\allowbreak P$ with $|A|\geq\allowbreak 2$. We have to prove
that the function $f\colon P\to P\colon X\mapsto AX$ is non-injective. We
distinguish two cases.

Case 1: There exist $a,b\in A$ with $a^{2}=ab$ and $a\neq b$. Set
$B:=A\smallsetminus\\{a\\}$. If $a\neq x\in A$, then $x\in B$ and hence
$xa=ax\in AB$ (by the commutativity of $S$). Moreover, $a^{2}=ab=\allowbreak
ba\in AB$ (since $a\neq b$ yields $a\in B$). It follows that $Aa\subseteq AB$
and hence $A^{2}=AB\cup Aa\subseteq AB\subseteq\allowbreak A^{2}$, namely,
$f(A)=f(B)$. But this means that $f$ is not an injection (as wished), because
$B$ is a non-empty proper subset of $A$ and $P$ is a downward complete
subsemigroup of $\mathcal{P}(S)$ (with the result that $A,B\in P$ and $A\neq
B$).

Case 2: We are not in Case 1. Pick $a,b\in A$ with $a\neq b$ (recall that
$|A|\geq 2$) and set $B:=A^{2}\smallsetminus\allowbreak\\{ab\\}$. Since
$b^{2}\neq ab$ (or else we would fall back to Case 1), we have $b^{2}\in B$
and hence $\varnothing\neq B\subsetneq A^{2}$. Given that $P$ is a downward
complete subsemigroup of $\mathcal{P}(S)$, it follows that $A^{2},B\in P$ and
$A^{2}\neq B$.

Now, let $x\in A$. If $xa=ab$, then it is obvious that $xab=ab^{2}\in AB$ (by
the fact that $ab\neq\allowbreak b^{2}\in\allowbreak B$). Otherwise, $xa\in B$
and again $xab=bxa\in AB$ (by the commutativity of $S$). It follows that
$Aab\subseteq AB$ and hence $A^{3}=AB\cup Aab\subseteq AB\subseteq A^{3}$,
namely, $f(A^{2})=f(B)$. So, $A$ is a non-cancellative element of $P$ (as we
have already observed that $A^{2},B\in P$ and $A^{2}\neq B$). ∎

The next example demonstrates that Proposition 2.2 does not hold for non-
commutative cancellative semigroups, which suggests that addressing Question 2
in the cancellative case (assuming the answer is still affirmative) may
necessitate a somewhat different approach than the one employed in this work
for the cancellative commutative case.

######  Example 2.3.

Let $V$ be a non-empty set and $\mathscr{F}^{+}(V)$ be the free semigroup over
$V$. The underlying set of $\mathscr{F}^{+}(V)$ consists of all the non-empty
finite tuples with components in $V$; and the semigroup operation, hereby
denoted by the symbol $\ast\kern 1.00006pt$, is the (ordered) concatenation of
such tuples.

As a matter of fact, $\mathscr{F}^{+}(V)$ is a cancellative semigroup. We
claim that every non-empty subset $X$ of $V$ is a cancellative element in the
large power semigroup of $\mathscr{F}^{+}(V)$, whose operation we continue to
denote by $\ast\kern 1.00006pt$. In particular, this will prove (taking
$|V|\geq 2$) that a cancellative element of the large power semigroup of a
cancellative semigroup need not be a singleton (in contrast with Proposition
2.2).

For the claim, assume $X\ast Y_{1}=X\ast Y_{2}$ for some non-empty sets
$Y_{1},Y_{2}\subseteq\mathscr{F}^{+}(V)$ (the case when $Y_{1}\ast\allowbreak
X=\allowbreak Y_{2}\ast X$ is symmetric and we leave it to the reader). We
need to check that $Y_{1}=Y_{2}$. To this end, it is clear that, since the
elements of $\mathscr{F}^{\ast}(V)$ can be _uniquely_ represented as a (non-
empty, finite, ordered) concatenation of elements of $V$, the sets $x_{1}\ast
Y_{1}$ and $x_{2}\ast Y_{2}$ are disjoint for all $x_{1},x_{2}\in X$ with
$x_{1}\neq x_{2}$. It follows that $X\ast Y_{1}=X\ast Y_{2}$ if and only if
$x\ast Y_{1}=x\ast Y_{2}$ for each $x\in\allowbreak X$, which, by Remark 2.1
and the non-emptiness of $X$, implies $Y_{1}=Y_{2}$ (as wished).

We continue with a few additional remarks that will prove useful shortly
afterwards.

######  Remarks 2.4.

(1) Let $f\colon H\to K$ be a semigroup isomorphism and $x$ be a left
cancellative element of $H$, and pick $u,v\in K$. Since $f$ is onto, there
exist $y,z\in H$ such that $u=f(y)$ and $v=f(z)$. Consequently, $f(x)u=f(x)v$
if and only if $f(xy)=f(xz)$, which implies, by the injectivity of $f$, that
$xy=xz$. It is thus obvious from the left cancellativity of $x$ in $H$ that
$y=z$ and hence $u=f(y)=f(z)=v$.
By the left-right symmetry inherent in the previous argument, we can therefore
conclude that $f$ maps left (resp., right) cancellative elements of $H$ to
left (resp., right) cancellative elements of $K$. In particular, it maps
cancellative elements to cancellative elements.

(2) Let $H$ and $K$ be semigroups, and let $f$ be an isomorphism from a
downward complete subsemigroup $P$ of $\mathcal{P}(H)$ to a downward complete
subsemigroup $Q$ of $\mathcal{P}(K)$. In addition, assume $H$ is commutative.
Given $u,v\in K$, there then exist $X,Y\in P$ with $f(X)=\\{u\\}$ and
$f(Y)=\\{v\\}$, since $f$ is surjective and $Q$ contains every one-element
subset of $K$. On the other hand, it is straightforward from the commutativity
of $H$ that $XY=YX$. As a result, we have
$\\{u\\}\\{v\\}=f(XY)=f(YX)=\\{v\\}\\{u\\}$ and hence $uv=vu$, which
ultimately proves that $K$ is itself a commutative semigroup (cf. Proposition
1.3(a) in [19]).

(3) Assume $f\colon H\to K$ is a semigroup homomorphism and let $F$ be the
function $\mathcal{P}(H)\to\mathcal{P}(K)\colon X\mapsto f[X]$, where
$f[X]:=\\{f(x)\colon x\in X\\}\subseteq K$ is the (direct) image of $X$ under
$f$. It is immediate that  $F(XY)=\\{f(xy)\colon x\in X,\kern 1.00006pty\in
Y\\}=\\{f(x)f(y)\colon x\in X,\kern 1.00006pty\in Y\\}=F(X)F(Y),$ for all
$X,Y\in\mathcal{P}(H)$. Consequently, $F$ is a semigroup homomorphism
$\mathcal{P}(H)\to\mathcal{P}(K)$.
On the other hand, if $f$ is injective and $F(X_{1})=F(X_{2})$ for some
$X_{1},X_{2}\in\mathcal{P}(H)$, then $X_{1}=X_{2}$ (and hence $F$ is itself
injective); otherwise, there would exist either $x\in X_{1}$ with $f(x)\in
F(X_{1})\smallsetminus F(X_{2})$, or $x\in X_{2}$ with $f(x)\in
F(X_{2})\smallsetminus F(X_{1})$. Moreover, if $f$ is surjective and $Y$ is a
non-empty subset of $K$, then $X:=\\{x\in X\colon f(x)\in Y\\}$ is a non-empty
subset of $H$ with $F(X)=Y$ (so that also $F$ is surjective).
Now, suppose that $f$ is an isomorphism. It then follows from the above that
$F$ is an isomorphism from $\mathcal{P}(H)$ to $\mathcal{P}(K)$. In addition,
$f$ being a bijection implies that a subset of $H$ and its image under $f$
have the same cardinality. So, $F$ restricts to both an isomorphism from
$\mathcal{P}_{<\kappa}(H)$ to $\mathcal{P}_{<\kappa}(K)$ and an isomorphism
from $\mathcal{P}_{\leq\kappa}(H)$ to $\mathcal{P}_{\leq\kappa}(K)$ for any
infinite cardinal number $\kappa$ (see Example 1.1(1) for the notation). Most
notably, $\mathcal{P}_{\mathrm{fin}}(H)$ is isomorphic to
$\mathcal{P}_{\mathrm{fin}}(K)$.

We are finally ready to prove our main result and show, in the subsequent
corollary, that Question 2 has a positive answer in the class of cancellative
commutative semigroups.

######  Theorem 2.5.

If $H$ and $K$ are cancellative semigroups and either of them is commutative,
then every isomorphism from a downward complete subsemigroup of
$\mathcal{P}(H)$ to a downward complete subsemigroup of $\mathcal{P}(K)$
restricts to an isomorphism $H\to K$.

###### Proof.

Let $P$ be a downward complete subsemigroup of $\mathcal{P}(H)$ and $Q$ be a
downward complete subsemigroup of $\mathcal{P}(K)$, and suppose that there is
an isomorphism $f$ from $P$ to $Q$. Since either of $H$ and $K$ is commutative
(by hypothesis), we have from Remark 2.4(2) that both $P$ and $Q$ are. On the
other hand, each of $H$ and $K$ is cancellative. Therefore, we gather from
Proposition 2.2 that the cancellative elements of $\mathcal{P}(H)$ (resp., of
$\mathcal{P}(K)$) are all and only the one-element subsets of $H$ (resp., of
$K$). It then follows by Remark 2.4(1) that, for every $x\in H$, there is a
uniquely determined $y\in K$ such that $f(\\{x\\})=\allowbreak\\{y\\}$.
Considering that the same argument also applies to the functional inverse of
$f$, this suffices to conclude that $f$ restricts to an isomorphism from $H$
to $K$ (as wished). ∎

######  Corollary 2.6.

If $H$ and $K$ are cancellative semigroups and either of them is commutative,
then the following conditions are equivalent:

  1. (a)

$H$ and $K$ are globally isomorphic, i.e., $\mathcal{P}(H)$ is isomorphic to
$\mathcal{P}(K)$.

  2. (b)

$\mathcal{P}_{\mathrm{fin}}(H)$ is isomorphic to
$\mathcal{P}_{\mathrm{fin}}(K)$.

  3. (c)

$H$ is isomorphic to $K$.

###### Proof.

We noted in Example 1.1(1) that $\mathcal{P}(S)$ and
$\mathcal{P}_{\mathrm{fin}}(S)$ are both downward complete subsemigroups of
$\mathcal{P}(S)$ for every semigroup $S$. So, it is immediate from Theorem 2.5
that (a) $\Rightarrow$ (c) and (b) $\Rightarrow$ (c). This concludes the
proof, as we know from Remark 2.4(3) that (c) $\Rightarrow$ (a) and (c)
$\Rightarrow$ (b). ∎

Corollary 2.6 is a generalization of a theorem of Bienvenu and Geroldinger [3,
Theorem 3.2(3)] according to which the finitary power semigroups of two
numerical monoids $H$ and $K$ are isomorphic if and only if $H=\allowbreak K$.
But there is a subtle detail here to work out.

In fact, Bienvenu and Geroldinger’s result is actually about _monoid_
isomorphisms (note that the finitary power semigroup of a monoid $S$ with
identity $1_{S}$ is a monoid with identity $\\{1_{S}\\}$). However, two
numerical monoids are isomorphic if and only if they are equal [14, Theorem
3]; and it is an easy to show that a surjective _semigroup_ homomorphism $f$
from a monoid $H$ to a monoid $K$ is, a fortiori, a _monoid_ homomorphism,
namely, it maps the identity $1_{H}$ of $H$ to the identity $1_{K}$ of $K$.
(Set $e:=f(1_{H})$. Each $y\in\allowbreak K$ is the image under $f$ of an
element $x\in\allowbreak H$, which implies $ye=\allowbreak
f(x)f(1_{H})=\allowbreak f(x1_{H})=\allowbreak f(x)=\allowbreak y$. It
follows, by taking $y=\allowbreak 1_{K}$, that $f(1_{H})=\allowbreak
e=\allowbreak e1_{K}=\allowbreak 1_{K}$.)

## Acknowledgements

I am thankful to Paolo Leonetti (Insubria University, Italy) for the idea
underlying Example 1.1(2), and to Daniel Smertnig (University of Ljubljana,
Slovenia) for pointing out that two monoids are isomorphic as monoids if and
only if they are isomorphic as semigroups (cf. the last lines of Sect. 2).

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# REDUCED-ORDER-MODEL-BASED FEEDBACK DESIGN FOR THRUSTER-ASSISTED LEGGED
LOCOMOTION

Pravin Dangol  Alireza Ramezani

###### Abstract

Real-time constraint satisfaction for robots can be quite challenging due to
the high computational complexity that arises when accounting for the system
dynamics and environmental interactions, often requiring simplification in
modelling that might not necessarily account for all performance criteria.

We instead propose an optimization-free approach where reference trajectories
are manipulated to satisfy constraints brought on by ground contact as well as
those prescribed for states and inputs. Unintended changes to trajectories
especially ones optimized to produce periodic gaits can adversely affect gait
stability, however we will show our approach can still guarantee stability of
a gait by employing the use of coaxial thrusters that are unique to our robot.

## 1 Introduction

We have categorized constraint satisfaction in legged robots in three broad
categories. Namely: i) trajectory optimization (TO), ii) optimization-based
controls and iii) reference trajectory manipulation. i) The goal of the TO
problem is to generate optimal trajectory which satisfy constraints on states,
inputs and ground reaction forces (GRF) while ensuring that the trajectories
lead to stable walking gaits.

TO problems for legged robot are difficult to solve due to their nonlinear
dynamics, high degrees of freedom and the hybrid nature of the system brought
on by ground impact [1, 2, 3, 4, 5]. Previous works such as [6], [7], [8] have
proposed methods to transcribe this as a non-linear programming (NLP) problem
through direct collocation methods where polynomial splines are used to
approximate the continuous dynamics and thus reducing computational complexity
without needing to account for the actual dynamics. While [9] has instead
proposed utilizing multiple shooting method to break the original problem down
into smaller steps without approximations.

In both cases, however, the dynamics of the system needs to be considered
along with contact dynamics to generate the trajectories. To alleviate the
need of explicitly defining ground contact dynamics [10], [11] have employed
null space projection methods, whereas zero acceleration constraint are
enforced on feet ends in [7]. The issue still remains that these methods are
extremely computationally expensive and cannot be implemented in real time,
taking a few minutes to solve the TO problem. For works that use reduced order
models such as the centroidal dynamics or utilize zero moment point (ZMP)
based methods as in [12] and [13], experimental results of online-optimization
are available however they are restricted to pseudo-static gaits rather than
dynamic ones.

ii) In optimization-based control schemes, the goal is to compute constraint-
aware feedback stabilizing control loops. This is most commonly achieved
through a predictive framework, usually by creating a linearized model over
finite time horizon. For instance in [14], [15], [16] reduced-order model
around the center of mass are used in a hierarchical framework to generate
reference acceleration for the lower level controller to track. The downfall
of these options are the need to linearize and/or simplify the underlying
dynamics of the robot in order to make it feasible in real-time, and as a
result not all constraints can be accounted for. In [17], [11], [18], [19] the
desired control inputs are computed taking into account the full dynamics, and
then optimization is carried out on a simplified least square or QP problem
for tractability.

iii) A different approach is to remove optimization from the control strategy
and instead modify the reference trajectories to obey desired constraints.
This idea was popularized through reference governors (RG) [20], where an
efficient online optimization method is employed. The core of the idea being
that the reference trajectory that the controller must follow can be
manipulated while keeping it close to the original trajectory in the event
that boundaries created by constraints are to be violated. Since its inception
this idea has spawned many variations [21], [22] including an optimization
free approach known as explicit reference governor (ERG) [23]. Besides the
possibility of utilizing an optimization-free form, the other major advantage
with RG is that it acts as a add-on scheme to an existing controller without
the need of any modification on the control scheme.

While gait re-design in a reactive fashion is widely used in quasi-static
walkers, one major reason that within-strides gait adjustment is less common
in dynamic walkers is that their small support polygons, which can be as small
as a point contact, leave small to no stability margins to avoid fall-overs.
With this observation, we aim to apply a new control action, i.e., thrusters,
during each gait cycle.

We will employ thruster actions during a small part of a gait cycle to secure
hybrid invariance. In this paper we extend our previous works [24, 25] by
offering a systematic method, and modify the RG formulation to work with a
popular framework known as hybrid zero dynamics (HZD) used to create stable
gaits in bipeds.

## 2 Overview of Control Design Idea

In this section, we outline an overview of our approach to satisfy state,
control and GRF constraints during gait cycles by deforming the Zero Dynamics
Manifolds describing the gaits which will be described in brevity here.

The governing equations of motion are derived using the methods of Lagrange by
taking the kinetic and potential energies of the system. The resulting
equations of motion are expressed in vector form as following

$D_{s}(q_{s})\ddot{q}_{s}+H_{s}(q_{s},\dot{q}_{s})=B_{s}(q_{s})u,$ (1)

Figure 1: A) Stick figure diagrams which represents the thruster-assisted
legged model, here the green boxes are the planes where each legs lies in.
B,C,D) reduced order models, starting from 5-link, 3-link and VLIP models
respectively.

where $q_{s}$ is the configuration variable vector, $D_{s}(q_{s})$ is the
symmetric inertial matrix and is only dependent on $q_{s}$, the
$H_{s}(q_{s},\dot{q}_{s})$ matrix contains Coriolis and gravity terms, and the
control matrix $B_{s}(q_{s})$ maps the inputs to the generalized coordinate
accelerations $\ddot{q}_{s}$. Consequently, the full model can be written in
state-space form as

$\dot{x}_{s}=\begin{bmatrix}\dot{q}_{s}\\\
D_{s}^{-1}(-H_{s}+B_{s}u)\end{bmatrix}=f_{s}(x_{s})+g_{s}(x_{s})u,$ (2)

where the state vector is denoted by
$x_{s}=[q^{\top}_{s},\dot{q}^{\top}_{s}]^{\top}$. The notion of directional
derivatives $L_{f}y=\frac{\partial y}{\partial x}f$ and holonomic constraints
$y=h_{s}(x_{s})=h_{s}(q_{s})$ will be adopted in a similar way as mainstream
publications in this field. The holonomic constraints ($y$) are widely known
as Virtual Constraints (VCs) since they are enforced by closed-loop feedback.
Based on these constraints celebrated control invariant sets of the form
$\Gamma=\\{[q_{s}^{\top},\dot{q}_{s}^{\top}]^{\top}\in\mathbb{R}^{2n}|F(q_{s},\dot{q}_{s})=0\\}$
where $F=[h_{s}(x_{s})^{\top},L_{f_{s}}h_{s}(x_{s})^{\top}]^{\top}$ and
$rank\\{L_{f_{s}}h_{s}(x_{s})\\}=n-1$ can be defined. The rank condition
guarantees that $\Gamma$, i.e., $q_{s}=h_{s}^{-1}(0)$, is $1$-dimensional.
Since there is only one degree of under-actuation (DOU) in our model,
therefore, $h_{s}^{-1}(0)$ takes the form of a closed curve. We consider the
following parametric descriptions
$q_{1}=r_{1}(q_{n}),\dots,q_{n-1}=r_{n-1}(q_{n})$ where $q_{n}$ in our planar
model is the last entry of $q_{s}$. As a result, the output function takes the
following form:

$h_{s}(q_{s})=Hq_{s}-r(q_{n})=q_{Act}-r(q_{n})$ (3)

Note that $r=[r_{1},\dots,r_{n-1}]^{\top}$ and the matrix
$H\in\mathbb{R}^{(n-1)\times n}$ can take a trivial form if each joint
independently is derived with a single actuator only, here $q_{Act}$ denotes
the actuated variables.

In a nutshell, the idea is to continuously deform $q_{s}=h_{s}^{-1}(0)$ such
that the following conditions are satisfied. First, we want $h^{-1}_{s}(0)$ to
remain continuous and closed curves, i.e., $r(q_{n}(t))=r(q_{n}(t+T))$ where
$T$ is the gait period. Second, we want $\Gamma$ remain stabilizable at all
times otherwise enforcing VCs will be impossible. Last, we want gait
feasibility conditions including equality $C_{eq}(q_{n},\dot{q}_{n})=0$ and
inequality constraints $0\leq C_{ineq}(q_{n},\dot{q}_{n})$ to be satisfied.
This problem can take the following constrained ordinary differential equation
form:

$\left\\{\begin{array}[]{ll}\ddot{q}_{n}&=a_{1}(q_{n})+a_{2}(q_{n})\dot{q}^{2}_{n}+\mathrm{effects~{}of~{}deforming}~{}h^{-1}_{s}(0)\\\
0&\neq\frac{\partial h_{s}}{\partial
x_{s}}\left[0^{\top},\left[D_{s}^{-1}(q_{n})B_{s}(q_{n})\right]^{\top}\right]^{\top}~{}\mathrm{for~{}deformed~{}h^{-1}_{s}(0)}\\\
0&=C_{eq}(q_{n},\dot{q}_{n})\\\ 0&\leq
C_{ineq}(q_{n},\dot{q}_{n})\end{array}\right.$ (4)

where, the first line governs restriction dynamics and the second line is the
condition for the stabilizability of $\Gamma$. Widely considered gait
feasibility constraints such as $q_{n}(t+T)=q_{n}(t)$,
$\dot{q}_{n}(t+T)=\dot{q}_{n}(t)$, $|x_{s}|<x_{max}$, $|u|<u_{max}$, $0<F_{N}$
and $|\frac{F_{\tau}}{F_{N}}|<\mu$ can potentially form the equality and
inequality constraints, where $F_{\tau},F_{N}$ are the tangential and normal
GRF respectively.

This problem can be looked upon as a classical time-invariant, trajectory
tracking problem, i.e., $q_{Act}$ parameterized in terms of $q_{n}$. It can be
easily resolved using optimization. In our approach, to solve the problem in
an optimization-free fashion (or minimally use optimization) positive
invariance property of $\Gamma$ – i.e., being able to find the control input
$u$ such that particularly when $q_{s}(0)$ is on $q_{s}=h_{s}^{-1}(0)$,
$\dot{q}_{s}(t)$ remains tangent to $q_{s}=h^{-1}_{s}(0)$ yielding the
solutions $q_{s}(t)$ remain on $\Gamma$ for all $t>0$ – plays a key role and
is closely dependent on how $q_{s}=h_{s}^{-1}(0)$ is deformed.

In the case that this property is guaranteed, the constraint satisfaction
problem can be transformed into a motion planning problem in the state space
of the internal dynamics which can be conveniently tackled using simple path
planning tools. This particularly becomes important and handy when fast and
reactive gait planing is in need in dynamic walkers. We will further elaborate
this concept with a simple example.

## 3 Valid Deformations of $q_{s}=h^{-1}_{s}(0)$

Consider the state-space representation of the system dynamics given by (2).
Here, we will explore our options and choices in order to manipulate (3). Our
possible options are: i) scaling, i.e., $w_{i}q_{n}$, and ii) shifting, i.e.,
$q_{n}+w_{i}$, of the equilibrium point of the error function $y(q_{s})$.

Evidently, constant priming terms yield a discrete collections of
parameterized systems, i.e., $x_{s}=f_{s}(x_{s})+g_{s}(x_{s})u_{\omega}$,
which is not desired here. Considering the numerical parameters $\omega$,
which are used to parametereize $r(q_{n},\omega)$, as auxiliary control input
in discrete maps (e.g., Poincare maps) can only provide discontinuous means of
priming $\Gamma_{\omega}$ at the boundaries. The continuous manipulation of
these parameters can violate the transversality condition. Consider
$y=q_{Act}-r(q_{n},\omega)$. For a fixed $q_{n}$, it is possible to show that

$B_{s}^{*}(q_{n})D_{s}(q_{n})\left[r^{\prime}(\omega(t),q_{n})^{\top},1\right]^{\top}$
(5)

can vanish on $\Gamma_{\omega}$. Here, $B_{s}^{*}(q_{n})=[0^{1\times(n-1)},1]$
and the annihilation of (5) implies $g_{s}(x_{s})$ and $\frac{\partial
h_{s}(x_{s})}{\partial x_{s}}$ are orthogonal.

On the other hand, it is straightforward to show that
$\dot{y}=\dot{q}_{Act}-r^{\prime}(q_{n})\dot{q}_{n}+\omega(t)$ yields relative
degree $\left\\{2,\dots,2\right\\}$ on $\Gamma_{\omega}$. It is also
noticeable that this choice of manipulating $y$ has no effects on
$null\left\\{\frac{\partial h_{s}}{\partial
q_{s}}\right\\}=\left[r^{\prime}(q_{n})^{\top},1\right]$ which means that at
least the primer has no influence over the transversality condition as long as
$y=q_{Act}-r(q_{n})$ is itself is valid VC.

Notice that the role of the primer $\omega(t)$ in this form is comparable to
the role of a disturbance term in the system given below

$\left[\begin{array}[]{c}\dot{y}\\\
\ddot{y}\end{array}\right]=\left[\begin{array}[]{l}\dot{q}_{Act}-r^{\prime}(q_{n})\dot{q}_{n}\\\
\gamma_{1}\end{array}\right]+\left[\begin{array}[]{l}0^{n-1}\\\
\gamma_{2}\end{array}\right]u+B_{\omega}\omega(t)$ (6)

where $\gamma_{1}$ contains the feed-forward terms [26],
$\gamma_{2}=-\left[I^{n-1},r^{\prime}(q_{n})\right]D^{-1}_{s}(q_{s})B_{s}$ and
$B_{\omega}=[I^{n-1},0^{n-1}]^{\top}$. As a result, the closed-loop system has
to possess strong disturbance rejection properties.

The roles of this disturbance term will be beneficial for us though as it will
adjust the equilibria of $h_{s}(x_{s})$ and $L_{f_{s}}h_{s}(x_{s})$ under
stabilising controllers with adjustable (and measurable) basins of attraction
to successfully satisfy state, control and GRF constraints. To do this, we
need to design an update law for the primer variable vector $\omega(t)$ such
that the finite-time convergence of the solutions to $\Gamma_{\omega}$ in the
closed-loop system is unaffected.

As far as the design of $u$ is of concern, any nonlinear controller (or linear
controllers if the nonlinear terms are bounded and the bounds are known) can
satisfy the VCs. We will limit ourselves to the following modest feedback law
$u=-L_{g}L_{f}h_{s}(x_{s})(L_{f}^{2}h_{s}(x_{s})+K_{p}y+K_{d}\dot{y})$ where
$K_{i}\in\mathbb{R}^{(n-1)\times(n-1)}$ are constant matrices and instead will
remain focused on deforming $\Gamma_{\omega}$ in order to satisfy our
constraints. Hence, we will assume stabilising supervisory controllers that
guarantee the enforcement of the virtual constraints, however, their
disturbance rejection properties has to be carefully considered.

The control law given above can generate GAS at the equilibrium point of the
system given by (6) when the primer variable vector is time-invariant, i.e.,
$\omega(t)=\omega$. Subsequently, $\Gamma_{\omega}$ takes the following form

$\Gamma_{\omega}=\left\\{x_{s}\in X\mid
h_{s}\left(x_{s}\right)=-K_{P}^{-1}K_{D}\omega(t),L_{f_{s}}h_{s}\left(x_{s}\right)=-\omega(t)\right\\}$
(7)

where the equilibrium points for $h_{s}(x_{s})$ and $L_{f_{s}}h_{s}(x_{s})$
under $\omega(t)$ are obtained by solving

$\left[\begin{array}[]{l}y_{\omega}\\\
\dot{y}_{\omega}\end{array}\right]=\left[\begin{array}[]{cc}-K_{P}^{-1}K_{D}&K_{P}^{-1}\\\
-I^{n-1}&0\end{array}\right]\left[\begin{array}[]{l}\omega(t)\\\
0\end{array}\right]$ (8)

Of course, realizing GAS property under $\omega(t)$, i.e., when the primer
variable is time-varying will not be achieved in a trivial fashion and
requires an extra condition to be satisfied. We will discuss this in the proof
of GAS property of the closed-loop system later.

## 4 Constraints Derivation

Consider the configuration variable vector
$q_{s}=\left[q^{\top}_{Act},p^{\top}_{1},q_{n}\right]^{\top}$ where
$q_{Act}\in\mathbb{R}^{m}$ is the actuated joint angle, $m$ is the number of
actuated joints and $p_{1}\in\mathbb{R}^{2}$ is the stance leg contact point.
The control matrix $B_{s}(q_{s})$ for
$u=[u^{\top}_{Act},u^{\top}_{Thrust},F^{\top}]^{\top}$ in the Euler-Lagrange
equations can take the following form

$B^{\top}_{s}(q_{s})=\left[\begin{array}[]{ccc}I^{m}&0&0\\\ \frac{\partial
p_{Thrust}}{\partial q_{Act}}&\frac{\partial p_{Thrust}}{\partial
p_{1}}&\frac{\partial p_{Thrust}}{{\partial q_{n}}}\\\
0&I^{2}&0\end{array}\right]$ (9)

where $p_{Thrust}$ is the physical location of the thruster action
$u_{Thrust}$. Notice that based on how the thruster actions look like the
underactuated coordinate $q_{n}$ can be actuated. The following restriction
dynamics can be obtained at every point on $\Gamma_{\omega}$

$\ddot{q}_{n}=-\alpha^{-1}\left(\beta_{1}\left(\dot{q}_{n}\right)^{2}+\beta_{2}\right)$
(10)

In this equation $\beta_{2}=B^{*}_{s}G_{s}(q_{n})$, where $G_{s}(q_{n})$
contains terms affected by gravity, and $\beta_{1}$ is given by

$\beta_{1}=B^{*}_{s}\left(D_{s}(q_{n})\left[r^{\prime\prime\top},0\right]^{\top}+\Sigma^{n}_{i=1}\left[r^{\prime\top},1\right]^{\top}Q_{i}(q_{n})\left[r^{\prime\top},1\right]\right)$
(11)

where $Q_{i}(q_{n})$ are the Christoffel Symbols. A similar algebraic
relationship for the constraints $[u^{\top}_{Act},F^{\top}]^{\top}$ can be
obtained which is skipped here.

Next, we will steer $y$ and $\dot{y}$ using the primer variable $\omega(t)$ in
(6) in order to make sure the solutions of (10) stay within the constraint-
admissible space. To do this, consider the $y$-$\dot{y}$ space.

Since we assumed a pre-stabilized system – in fact all of the above
derivations only make sense if
$q_{Act}=r(q_{n})+\int_{0}^{\tau}\omega(\tau)d\tau$ and
$\dot{q}_{Act}=r^{\prime}(q_{n})\dot{q}_{n}+\omega(t)$ – it is reasonable to
evaluate the constraints $c_{l}\leq[u^{\top}_{Act},F^{\top}]^{\top}\leq c_{u}$
($c_{l}$ and $c_{u}$ are constraint lower and upper bounds) based on the
steady-state solutions, i.e., $y_{\omega}$ and $\dot{y}_{\omega}$, and ignore
the transient solutions.

Other than simplifying the nonlinear constraint satisfaction problem given in
(4), considering $[y^{\top}_{\omega},\dot{y}^{\top}_{\omega}]^{\top}$ has
another interesting result which will be explained below. Consider the set

$Y_{\omega}=\left\\{z=[y^{\top},\dot{y}^{\top}]^{\top}|[I^{n-1},-K^{-1}_{P}K_{D}]z=0\right\\}$
(12)

which is the locus of all of the steady-state solutions of the system (6). It
is possible to show that invariant sets around any point
$[y^{\top}_{\omega},\dot{y}^{\top}_{\omega}]^{\top}$ in the set defined by
$Y_{\omega}$ can be created.

## 5 Simulation Results and Concluding Remarks

Fig. 1 shows the low-order representations of our legged robot with two
thrusters attached to its torso. This section discusses the simulation setup
and results where the proposed framework is implemented on an equivalent low-
order model (e.g., variable-length inverted pendulum model) of the full
system. In this way, the low-order model is considered as a template used to
satisfy the GRF, state, etc., constraints and the full system’s joint angles
and velocities are resolved through the forward kinematics equations. The
simulation is done to show that the constraints, which are highly nonlinear
and are often resolved using costly optimizers, can be enforced in a
completely optimization-free fashion.

Figure 2: Illustration of the states constraint satisfaction. The primer
update law was applied to adjust the joints reference trajectories to satisfy
the constraints. This can be seen in the pendulum angle and virtual leg
length, i.e., the low-order representation of the Harpy system, at 0.5 s. The
algorithm prevents the pendulum angle from dropping to below 5∘ as specified
by the constraints.

The simulation has been resolved using a 4-th order Runge-Kutta scheme. The
primer algorithm explained above was used to perform the trajectory
manipulation task on the state trajectories in such a way that the constraints
are satisfied. The following states in the low-order system were considered:
the pendulum angle from the ground plane normal vector, heading angle, and the
length of the virtual leg. Then, the following four constraints were stacked
inside the nonlinear vector function $C_{ineq}$ in the inequality constraint
equation (i.e., $C_{ineq}\geq 0$). The constraints include: a minimum pendulum
angle of five degrees, the ground friction cone constraints in the $x$ and $y$
directions in the 3D model, and a minimum ground normal forces of 20 N. The
latter constraint guarantees that the stance leg-end always stays on the
ground surface.

Figure 3: Illustration of the GRF constraints at the contact point (i.e.,
stance leg-end).

The simulation results are illustrated in Figs. 2 to 4. Figure 2 displays the
state trajectories for the low-order model. As it can be seen in this figure,
a major deviation from the target reference at $t=0.5$ s is required in order
to avoid constraint violation. Fig. 3 shows the GRF profiles obtained using
the target references and compares them against the primed references. While
the target references frequently violate the no-slip constraints, the primed
trajectories operate within the permissible bounds (this can be easily
verified in Fig. 3).

Figure 4: A comparison between all of the constraints involved when primed
(top) and un-primed trajectories are applied. Almost all of the constraints
are violated when the primer is absent.

The constraints behavior under the proposed priming framework is summarized in
Fig. 4. This figure shows that the target reference lead to many violations
while the primed references do not violate the constraints. However, in a few
occasions, as it can be seen in Fig. 4, the primed references violate the
constraints. Based on our assumption outlined above in Section 4, this
temporary violations are expected. And, as it is evidently seen in, the
trajectories are attracted to the constraint admissible sets after constraint
violations occur.

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