BMS algebra from residual gauge invariance

in light-cone gravity

Sudarshan Ananth†, Lars Brink∗ and Sucheta Majumdar‡

$\dagger$ Indian Institute of Science Education and Research

Pune 411008, India

${}^{*}\,$Department of Physics, Chalmers University of Technology

S-41296 Göteborg, Sweden

$\ddagger$ Université Libre de Bruxelles and International Solvay Institutes,

Campus Plaine CP231, B-1050 Brussels, Belgium

Abstract

We analyze the residual gauge freedom in gravity, in four dimensions, in the
light-cone gauge, in a formulation where unphysical fields are integrated out.
By checking the invariance of the light-cone Hamiltonian, we obtain a set of
residual gauge transformations, which satisfy the BMS algebra realized on the
two physical fields in the theory. Hence, the BMS algebra appears as a
consequence of residual gauge invariance in the bulk and not just at the
asymptotic boundary. We highlight the key features of the light-cone BMS
algebra and discuss its connection with the quadratic form structure of the
Hamiltonian.

###### Contents

  1. 1 Introduction
  2. 2 Gravity in the light-cone gauge
    1. 2.1 Perturbative expansion
    2. 2.2 Residual gauge transformations
  3. 3 BMS symmetry in light-cone gravity
    1. 3.1 Invariance of the light-cone Hamiltonian
    2. 3.2 Light-cone realization of the BMS algebra
    3. 3.3 Supertranslations and the quadratic form Hamiltonian
  4. 4 Concluding remarks
  5. A Light-cone Poincaré algebra in $d=4$
  6. B Allowed residual gauge transformations
  7. C Connections to the Carroll group
  8. D Supertranslation generator at order $\kappa$

##  1 Introduction

The Bondi-van der Burg-Metzner-Sachs (BMS) group is an infinite-dimensional
enhancement of the Poincaré group, that arises as the asymptotic symmetry
group at null infinity for asymptotically flat spacetimes [1, 2]. In [3], the
BMS group was extended to include superrotations. These asymptotic symmetries
have been related to soft theorems for gauge theories in the recent years [4],
following which there has been a renewed interest in the study of these
symmetries.

The study of asymptotic symmetries is very sensitive to the boundary
conditions and gauge choices imposed on the fields. At spatial infinity, for
instance, only the Poincaré algebra (and not the BMS algebra) can be
canonically realized under the standard boundary conditions in the Hamiltonian
formulation of general relativity [5]. Recent work however showed that, by
relaxing those boundary conditions, the BMS group can be recovered at spatial
infinity, thereby, resolving the puzzle between the asymptotic structure at
spatial and null infinity [6, 7]. This raises an important question: How much
of the gauge freedom can be fixed in the theory without losing the residual
reparameterizations that are associated with the BMS symmetry?

In the light-cone formulation of gravity in four dimensions, one can gauge
away the unphysical degrees of freedom and describe the dynamics of the theory
in terms of the two physical states of the graviton. In the usual Bondi gauge,
the BMS group appears as the asymptotic symmetry of gravity at infinity. In
light-cone gravity, we will show instead that we obtain the BMS algebra from
the invariance of the Hamiltonian under residual gauge transformations in the
bulk.

In a previous paper [8] we have studied the problem by constructing non-linear
representations of the Poincaré algebra on two field degrees of freedom with
helicity $+2$ and $-2$ respectively. We performed the study in the light-cone
frame where one of the light-cone coordinates is the evolution parameter
“time” and the conjugate momentum, the Hamiltonian. The representation is
unique up to possible counterterms [9]. When we ask if the Hamiltonian is
invariant under an extended symmetry we find indeed that it is invariant under
the BMS symmetry, which shows that this symmetry is also present in the bulk.
It should be mentioned though that our formulation is a perturbative one in
$\kappa$, and we only demonstrate this to the lowest order, but experience
from earlier studies suggests that the symmetries will hold at higher orders.
Furthermore since it is an infinite series with an ever-increasing number of
graviton fields we have to restrict our studies to weak fields.

The question here is then to ask if the remaining residual gauge invariance
corresponds to just the BMS algebra on the two helicity fields $h$ and
$\bar{h}$ or if it can be further extended. This formulation is often thought
to be one where the gauge symmetry is completely fixed apart from some freedom
in the definition of the inverse $\partial_{-}$ (the light-cone momentum that
is taken to be a space derivative). But, in this paper, we show that there is
still some residual gauge freedom in the theory, which leads to the light-cone
realization of the BMS algebra and only that. The paper is organized as
follows. We start with a brief review of the light-cone gauge-fixing of the
Einstein-Hilbert action followed by the perturbative expansion in terms of the
helicity fields $h$ and $\bar{h}$. In section 2.2, we discuss the remaining
reparameterization freedom in the theory. In section 3, we focus on a special
class of reparameterizations, dubbed “helicity-preserving”, and derive how the
fields transform under them by demanding invariance of the light-cone
Hamiltonian. We then present the symmetry algebra underlying these
reparameterizations, which yields the BMS algebra in light-cone gravity. In
section 3.3, we define a canonical generator for the supertranslations, from
which we obtain the quadratic form structure for the Hamiltonian previously
found in [9, 10]. We conclude with some remarks on the structure of the BMS
algebra in the light-cone gauge and possible extensions to Yang-Mills theory,
higher-spin and supersymmetric theories.

##  2 Gravity in the light-cone gauge

With the metric $(-,+,+,+)$, the light-cone coordinates are defined as

$\displaystyle x^{\pm}=\frac{1}{\sqrt{2}}(x^{0}\pm x^{3})\,,$ (2.1)

with the corresponding derivatives $\partial_{\pm}$. The transverse
coordinates and derivatives are

$\displaystyle x=\frac{1}{\sqrt{2}}\,(\,{x_{1}}\,+\,i\,{x_{2}}\,)\ ;$
$\displaystyle{\bar{\partial}}=\frac{1}{\sqrt{2}}\,(\,{\partial_{1}}\,-\,i\,{\partial_{2}}\,)\,,$
$\displaystyle\bar{x}=\frac{1}{\sqrt{2}}\,(\,{x_{1}}\,-\,i\,{x_{2}}\,)\ ;$
$\displaystyle{\partial}=\frac{1}{\sqrt{2}}\,(\,{\partial_{1}}\,+\,i\,{\partial_{2}}\,)\,.$
(2.2)

The Einstein-Hilbert action on a Minkowski background reads

$\displaystyle
S_{EH}=\int\,{d^{4}}x\;\mathcal{L}\,=\,\frac{1}{2\,\kappa^{2}}\,\int\,{d^{4}}x\;{\sqrt{-g}}\,\,{\mathcal{R}}\,,$
(2.3)

where $g=det\,(\,{g_{\mu\nu}}\,)$ is the determinant of the metric.
$\mathcal{R}$ is the curvature scalar and $\kappa^{2}=8\pi G$ is the coupling
constant derived from the gravitational constant. The corresponding field
equations are

$\displaystyle\mathcal{R}_{\mu\nu}\,-\,\frac{1}{2}\,g_{\mu\nu}\mathcal{R}\,=\,0\,.$
(2.4)

We impose the following three gauge choices [11, 12] on the dynamical variable
$g_{\mu\nu}$

$\displaystyle g_{--}\,=\,g_{-i}\,=\,0\,,\quad i=1,2\,.$ (2.5)

These choices are motivated by the fact that in Minkowski space, we have
$\eta_{--}\\!=\eta_{-i}\\!=0$. We also choose

$\displaystyle\begin{split}g_{+-}\,&=\,-\,e^{\phi}\,,\\\
g_{i\,j}\,&=\,e^{\psi}\,\gamma_{ij}\,.\end{split}$ (2.6)

where $\phi,\psi$ are real parameters and $\gamma^{ij}$ is a real, symmetric
matrix with unit determinant. Field equations that do not involve time
derivatives $(\partial_{+})$ are constraint relations as opposed to true
equations of motion, which have explicit time derivatives. The
$\mu\\!=\\!\nu\\!=\\!-\;$ constraint from (2.4) yields

$\displaystyle
2\,\partial_{-}\phi\,\partial_{-}\psi\,-\,2\,\partial^{2}_{-}\psi\,-\,(\partial_{-}\psi)^{2}\,+\,\frac{1}{2}\partial_{-}\gamma^{ij}\,\partial_{-}\gamma_{ij}\,=\,0\,,$
(2.7)

which may be solved by making the fourth and final gauge choice

$\displaystyle\phi\,=\,\frac{\psi}{2}\,.$ (2.8)

Note that this gauge choice relates $g_{+-}$ and $g_{ij}$. This choice
implies, from (2.7), that

$\displaystyle\psi\,=\,\frac{1}{4}\,\frac{1}{\partial^{2}_{-}}\,(\partial_{-}\gamma^{ij}\,\partial_{-}\gamma_{ij})\,.$
(2.9)

Other constraint relations eliminate $g_{++}$ and $g_{+i}$ resulting in the
following action

$\displaystyle S\,$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{4}x\;e^{\psi}\left(2\,\partial_{+}\partial_{-}\phi\,+\,\partial_{+}\partial_{-}\psi-\frac{1}{2}\,\partial_{+}\gamma^{ij}\partial_{-}\gamma_{ij}\right)$
(2.10)
$\displaystyle-e^{\phi}\gamma^{ij}\left(\partial_{i}\partial_{j}\phi+\frac{1}{2}\partial_{i}\phi\partial_{j}\phi-\partial_{i}\phi\partial_{j}\psi-\frac{1}{4}\partial_{i}\gamma^{kl}\partial_{j}\gamma_{kl}+\frac{1}{2}\partial_{i}\gamma^{kl}\partial_{k}\gamma_{jl}\right)$
$\displaystyle-\frac{1}{2}e^{\phi-2\psi}\gamma^{ij}\frac{1}{\partial_{-}}R_{i}\frac{1}{\partial_{-}}R_{j}\,,$

where

$\displaystyle
R_{i}\,\equiv\,e^{\psi}\left(\frac{1}{2}\partial_{-}\gamma^{jk}\partial_{i}\gamma_{jk}-\partial_{-}\partial_{i}\phi-\partial_{-}\partial_{i}\psi+\partial_{i}\phi\partial_{-}\psi\right)+\partial_{k}(e^{\psi}\,\gamma^{jk}\partial_{-}\gamma_{ij})\,.$

This is the closed form expression for the light-cone gravity action [11, 12]
\- purely in terms of the physical degrees of freedom in the theory.

###  2.1 Perturbative expansion

We now examine the perturbative expansion of the closed form expression
obtained above. The order $\kappa^{2}$ result was first presented in [12, 13]
while the $\kappa^{3}$ result was derived in [14]. We parameterize the matrix
$\gamma_{ij}$ as

$\displaystyle\gamma_{ij}\,=\,(e^{\kappa H})_{ij}\,,$ (2.11)

where $H$ is a traceless matrix since $det\,(\,\gamma_{ij})=1$. We choose

$\displaystyle H\,=\,\begin{pmatrix}h_{11}&h_{12}\\\
h_{12}&-h_{11}\end{pmatrix}\,;\quad
h\,=\,\frac{(h_{11}+i\,h_{12})}{\sqrt{2}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar{h}\,=\,\frac{(h_{11}-i\,h_{12})}{\sqrt{2}}\
,$ (2.12)

The Lagrangian (density) in terms of $h$ and $\bar{h}$ to order $\kappa$ now
reads

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\bar{h}\,\Box\,h\,+\,2\,\kappa\,\bar{h}\,\partial_{-}^{2}\left[-\,h\,\frac{\bar{\partial}^{2}}{\partial_{-}^{2}}h\,+\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right]\,+\,\text{c.c.}\
,$ (2.13)

with the d’Alembertian
$\Box\,=\,2\,(\,\partial\,{\bar{\partial}}\,-\,\partial_{+}\,\partial_{-}\,)$.
At the next order, time derivatives need to be removed using field
redefinitions and the resulting quartic Lagrangian was computed in [12, 13].
The corresponding Hamiltonian density for gravity reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\partial\bar{h}\,\bar{\partial}h\ +\
2\,\kappa\,\partial_{-}^{2}\bar{h}\,\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\
-\
\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)+c.c.\
+\ \mathcal{O}(\kappa^{2})\ .$ (2.14)

The light-cone action for gravity is invariant under Poincaré transformations
in four dimensions [15]. As $x^{+}$ is treated as the evolution parameter, the
conjugate momentum $P^{-}$ is the Hamiltonian operator. The light-cone
Poincaré generators split into two kinds - the kinematical ones ($\mathbb{K}$)
which do not involve time derivatives $\partial_{+}$ and the dynamical ones
($\mathbb{D}$) that do and hence receive non-linear contributions in the
interacting theory. The dynamical generators or “Hamiltonians” in Dirac’s
language [16] take the field forward in light-cone time.

$\displaystyle\mathbb{K}:\quad\\{P,\bar{P},P^{+},J,J^{+},\bar{J}^{+},J^{+-}\\}$
$\displaystyle\mathbb{D}:\quad\\{P^{-}\equiv H,J^{-},\bar{J}^{-}\\}$ (2.15)

All the relevant commutators of the Poincaré algebra in four dimensions are
listed in appendix A. The commutators fall broadly into three varieties

$\displaystyle[\,\mathbb{K},\,\mathbb{K}\,]~{}=~{}\mathbb{K}\,,$
$\displaystyle[\,\mathbb{K},\,\mathbb{D}\,]~{}=~{}\mathbb{D}\,,$
$\displaystyle[\,\mathbb{D},\,\mathbb{D}\,]~{}=~{}0\,.$ (2.16)

The fields $h$ and $\bar{h}$ transform with helicity, $\lambda=2$ and
$\lambda=-2$, respectively under the little group in four dimensions and thus,
represent the two physical states of graviton.

### The Hamiltonian

The Hamiltonian $H\equiv P^{-}$ in (2.14) follows from the action in (2.13) -
which was obtained by gauge-fixing the covariant Einstein-Hilbert Lagrangian.
There is an alternate approach that also leads to the same result - this is to
recognize that the Hamiltonian is also an element of the Poincaré algebra and
hence can be determined entirely simply closing all the commutators in the
symmetry algebra [15]. This is a necessary step because Lorentz invariance is
not manifest on the light-cone and must be explicitly checked. It is
interesting to note, however, that the Hamiltonian is explicitly helicity-
covariant when expressed in terms of $h$ and $\bar{h}$. Having reached this
point, we note now that there are still some reparameterizations allowed -
more specifically, transformations that (i) leave the Hamiltonian invariant
and (ii) preserve all gauge choices made so far. These residual
reparameterizations will be the focus of the next subsection.

###  2.2 Residual gauge transformations

Having obtained a perturbative expansion of the gauge-fixed action, we now
turn to examining the issue of residual reparameterization invariace. To this
end, we note that under

$x^{\mu}~{}\rightarrow~{}x^{\mu}\ +\ \xi^{\mu}\,,$

the first gauge choice $g_{--}=0$ in (2.5) holds as long as

$\xi^{+}~{}=~{}f(x^{+},x^{j})\quad\text{so}\quad\partial_{-}\xi^{+}~{}=~{}0\,.$
(2.17)

The second gauge condition $g_{-i}=0$ in (2.5) then requires that

$\displaystyle\partial_{-}\xi^{j}\,g_{ij}\ +\
\partial_{i}\xi^{+}\,g_{+-}=0\,.$ (2.18)

This relates $\xi^{j}$ to $\xi^{+}=f(x^{+},x^{j})$

$\xi^{k}~{}=~{}-\partial_{i}f\,\frac{1}{\partial_{-}}(g_{+-}g^{ik})\
+Y^{k}\,,$ (2.19)

where $Y^{k}$ does not depend on $x^{-}$. The fourth gauge condition (2.8)
further restricts the form of different components of $\xi^{\mu}$.

Confining ourselves to residual gauge transformations on the fields $h$,
$\bar{h}$ we look for transformations that leave the light-cone Hamiltonian
invariant. The light-cone gauge conditions and the subsequent perturbative
expansion of $\gamma_{ij}$ constrain the form of allowed residual
reparameterizations, the details of which can be found in appendix B. This
choice of residual gauge transformations, when expressed in the $(x,\bar{x})$
coordinates, corresponds to the following “helicity-preserving”
reparameterizations

$\displaystyle x$ $\displaystyle\rightarrow$ $\displaystyle x\ +\ Y(x)\,,$
(2.20) $\displaystyle\bar{x}$ $\displaystyle\rightarrow$
$\displaystyle\bar{x}\ +\ \overline{Y}(\bar{x})\,,$ (2.21) $\displaystyle
x^{+}$ $\displaystyle\rightarrow$ $\displaystyle x^{+}\,+\
f(x,\bar{x},x^{+})\,,$ (2.22) $\displaystyle x^{-}$ $\displaystyle\rightarrow$
$\displaystyle x^{-}\ +\ \xi^{-}\,,$ (2.23)

such that the parameters satisfy

$\displaystyle\partial
Y=\partial_{-}Y=0\,,\quad\bar{\partial}\overline{Y}=\partial_{-}\overline{Y}=0\,,\quad\partial_{-}f=0\,,$
(2.24)

and the parameter $\xi^{-}$ is completely determined in terms of
$f,Y,\overline{Y}$ as in (B.13).

##  3 BMS symmetry in light-cone gravity

###  3.1 Invariance of the light-cone Hamiltonian

Before we compute the symmetry algebra underlying these reparameterizations,
we must ensure invariance of the light-cone Hamiltonian expressed in terms of
the helicity fields (2.14). We must do this since the elimination of the
unphysical degrees of freedom might further reduce the residual symmetry.
Thus, invariance of the Hamiltonian under the new reparameterizations
guarantees that Lorentz covariance is not violated. This requirement will
indeed put some additional constraints on the parameters $Y$ and
$\overline{Y}$ as we show below.

This will, in turn, ensure that we can define a canonical generator, $G_{\xi}$
for these residual gauge transformations in the reduced phase space of
$(h,\bar{h})$ such that it commutes with the Hamiltonian

$\delta_{\xi}H~{}=~{}0\quad\Rightarrow\quad[G_{\xi},H]\,~{}=~{}0.$ (3.1)

We can then study the algebra of these residual gauge generators with the
canonical Poincaré generators defined in the $(h,\bar{h})$ phase space, which
are presented in Appendix A. One can check the invariance of the light-cone
Hamiltonian under a given transformation order by order in $\kappa$ as follows

$\displaystyle\delta_{\xi}H$ $\displaystyle=$
$\displaystyle\delta_{\xi}^{(0)}\,H^{(0)}\,+\,\delta_{\xi}^{(\kappa)}\,H^{(0)}\,+\,\delta_{\xi}^{(0)}\,H^{(\kappa)}\,+\,\mathcal{O}(\kappa^{2})~{}=~{}0\,.$
(3.2)

Under the reparameterizations (2.20) and (2.21) with the parameters satisfying
(2.24), the fields $h$ and $\bar{h}$ transform as

$\displaystyle\delta_{Y,\overline{Y}}\,h$ $\displaystyle=$ $\displaystyle
Y(x)\,\bar{\partial}h\,+\,\overline{Y}(\bar{x})\,\partial
h\,+\,(\partial\overline{Y}-\bar{\partial}Y)\,h\,,\ $
$\displaystyle\delta_{Y,\overline{Y}}\,\bar{h}$ $\displaystyle=$
$\displaystyle
Y(x)\,\bar{\partial}\bar{h}\,+\,\overline{Y}(\bar{x})\,\partial\bar{h}\,-\,(\partial\overline{Y}-\bar{\partial}Y)\,\bar{h}\,.$
(3.3)

These transformations leave the Hamiltonian invariant if one assumes

${\bar{\partial}}^{2}Y=\partial^{2}\overline{Y}=0\,,$ (3.4)

which restricts the parameters $Y$ and $\overline{Y}$ to be at most linear in
$x$ and $\bar{x}$ respectively. Thus, the above condition reduces the
$Y,\overline{Y}$ reparameterizations to a part of the Poincaré
transformations. With these assumptions, it is easy to show that the light-
cone Hamiltonian is invariant under (3.1) up to the cubic order. More
importantly, no new corrections to $\delta_{Y,\overline{Y}}h$ of order
$\kappa$ or higher are required, thus, classifying these under the kinematical
part of the light-cone Poincaré transformations (2.1).

We now examine the time reparameterizations labeled by the parameter $f$,
which transforms the fields as

$\delta_{f}h=f\partial_{+}h~{}=~{}f\frac{\partial\bar{\partial}}{\partial_{-}}h\,.$
(3.5)

This transformation leaves the free light-cone Hamiltonian invariant. However,
in order to establish the invariance at order $\kappa$

$\displaystyle\delta_{f}^{(\kappa)}\,H^{(0)}\,+\,\delta_{f}^{(0)}\,H^{(\kappa)}~{}=~{}0\,,$
(3.6)

we must add corrections to $\delta_{f}h$ at $\mathcal{O}(\kappa)$. Thus, the
field $h$ transforms non-linearly under $f$ as follows

$\displaystyle\delta_{f}h$ $\displaystyle=$ $\displaystyle f(x,\bar{x},x^{+})\
\Bigg{\\{}\,\frac{\partial\bar{\partial}}{\partial_{-}}h\,+\,2\,\kappa\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,$
(3.7)
$\displaystyle+\,2\,\kappa\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right)+\,\mathcal{O}(\kappa^{2})\Bigg{\\}}\,,$

and $\delta_{f}\bar{h}$ is simply the complex conjugate of the above
expression. Therefore, the above time reparameterizations are a symmetry of
the light-cone Hamiltonian, provided one adds corrections to $\delta_{f}h$ at
every order in $\kappa$, making these transformations dynamical.
Interestingly, in order to prove the invariance of the Hamiltonian, the
explicit form of $f$ is not required. It is important to note though that the
invariance of the Hamiltonian under these transformations is strictly proven
only to first order in the coupling constant, $\kappa$. But, extensions to
higher orders should follow without any formal difficulties, although the
explicit calculations could prove cumbersome. A key difference from the
previous analysis in [8] is that the condition $\partial_{-}f=0$ here follows
from (2.17), which is a consequence of the light-cone gauge fixing of the
Einstein-Hilbert action. In [8], the same condition is obtained by demanding
the invariance of the light-cone Hamiltonian under local extensions of the
Poincaré transformations. This reflects how the focus of this paper is
primarily on the residual gauge symmetry of the graviton fields $h$ and
$\bar{h}$, in contrary to [8], where the goal was to obtain possible
extensions of the light-cone Poincaré algebra with local parameters.

###  3.2 Light-cone realization of the BMS algebra

The fourth gauge condition (2.8) precisely fixes the $x^{+}$ dependence of $f$
as in (B.10)

$\displaystyle f(x^{+},x,\bar{x})$ $\displaystyle=$ $\displaystyle
T(x,\bar{x})\ +\ \frac{1}{2}\,x^{+}\,(\partial\overline{Y}\ +\
\bar{\partial}Y)\,,$ (3.8)

which reproduces the light-cone BMS symmetry in [8]. Here, the parameters $Y$
and $\overline{Y}$ obey (2.24) and (3.4), indicating that the only independent
parameter in $f$ is $T(x,\bar{x})$. Besides, this choice of $f$ also coincides
with the conformal Carroll case as we have discussed in appendix C. As opposed
to the geometric aspects of the conformal Carroll group in [17], our focus
lies on the field-theoretic aspects of gravity and the Lie algebra
representation associated with the symmetries of the theory.

On the initial surface $x^{+}=0$ such that $f=T$, the BMS transformations in
light-cone gravity to order $\kappa$ read

$\displaystyle\delta_{Y,\overline{Y},T}\,h$ $\displaystyle=$ $\displaystyle
Y(x)\,\bar{\partial}h+\overline{Y}(\bar{x})\,\partial h\
+(\partial\overline{Y}-\bar{\partial}Y)\,h+T\,\frac{\partial\bar{\partial}}{\partial_{-}}\,h$
(3.9)
$\displaystyle-\,2\,\kappa\,T\,\partial_{-}\,\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\
-\
2\,\kappa\,T\,\frac{1}{\partial_{-}}\left(\frac{{\partial}^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\right)\,$
$\displaystyle-\,2\,\kappa\,T\,\frac{{\partial}^{2}}{\partial_{-}^{3}}\,(\bar{h}\,\partial_{-}^{2}h)\,+\
4\,\kappa\,T\,\frac{\partial}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{2}h\right)\,,$

where the parameters $Y$, $\overline{Y}$ and $T$ satisfy

$\partial
Y=\partial_{-}Y=0\,,\quad\bar{\partial}\overline{Y}=\partial_{-}\overline{Y}=0\,,\quad\partial_{-}T=0\,,$
(3.10)

and

$\bar{\partial}^{2}Y~{}=~{}\partial^{2}\overline{Y}~{}=~{}0\,.$ (3.11)

Two such transformations close on another reparameterization

$\displaystyle\left[\,\delta(Y_{1},\overline{Y}_{1},T_{1})\,,\
\delta(Y_{2},\overline{Y}_{2},T_{2})\,\right]\,h$ $\displaystyle=$
$\displaystyle\ \delta(Y_{12},\overline{Y}_{12},T_{12})\,h\,,$ (3.12)

with the new parameters defined as

$\displaystyle Y_{12}$ $\displaystyle\equiv$ $\displaystyle
Y_{2}\,\bar{\partial}\,Y_{1}\ -\ Y_{1}\,\bar{\partial}\,Y_{2}\,,$ (3.13)
$\displaystyle\overline{Y}_{12}$ $\displaystyle\equiv$
$\displaystyle\overline{Y}_{2}\,\partial\,\overline{Y}_{1}\ -\
\overline{Y}_{1}\,\partial\,\overline{Y}_{2}\,,$ (3.14) $\displaystyle T_{12}$
$\displaystyle\equiv$ $\displaystyle[Y_{2}\,\bar{\partial}\,T_{1}\ +\
\overline{Y_{2}}\,\partial\,T_{1}\ +\frac{1}{2}\,T_{2}(\bar{\partial}Y_{1}\
+\,\partial\overline{Y}_{1})]\ -\ (1\leftrightarrow 2)\,.$ (3.15)

This is the light-cone realization of the BMS algebra in four dimensions [8].
The “supertranslations” labeled by the function $T(x,\bar{x})$ enhance the
dynamical part of the Poincaré algebra into an infinite-dimensional set, while
the kinematical part of the algebra is the same as in (2.1)

$\displaystyle\mathbb{K}~{}\rightarrow~{}\mathbb{K}\,,$
$\displaystyle\mathbb{D}~{}\rightarrow~{}\mathbb{D}(T)\,,$ (3.16)

such that the light-cone BMS algebra in (3.12) takes the form

$\displaystyle[\,\mathbb{K},\,\mathbb{K}\,]~{}=~{}\mathbb{K}\,,$
$\displaystyle[\,\mathbb{K},\,\mathbb{D}(T)\,]~{}=~{}\mathbb{D}(T)\,,$
$\displaystyle[\,\mathbb{D}(T),\,\mathbb{D}(T)\,]~{}=~{}0\,.$ (3.17)

Thus, the key feature of the light-cone BMS algebra is that the dynamical part
of the Poincaré algebra is enlarged to accommodate the supertranslations. This
is different from the BMS algebra in covariant formulations, where the four
spacetime translations of the Poincaré algebra are enhanced by the
supertranslations. However, Lorentz invariance of the theory dictates that the
enhancement involves only one single local parameter $T(x,\bar{x})$, which is
in keeping with the original BMS group found in [1, 2]. The Poincaré subgroup
of the BMS algebra can be obtained by imposing the condition
$\partial^{2}T=\bar{\partial}^{2}T=0$. This restricts $T$ to be at most linear
in $x$ or $\bar{x}$, thereby, reducing the dynamical part to $\mathbb{D}$
given in (2.1). In appendix B, we discuss in more details the Poincaré
subgroup of the light-cone BMS algebra (B).

###  3.3 Supertranslations and the quadratic form Hamiltonian

As we have established that $\delta_{T}h$ are canonical transformations in the
$(h,\bar{h})$ phase space, we can, now, define a canonical generator $G_{T}$
for supertranslations through (3.1)

$\displaystyle G_{T}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\partial_{-}\bar{h}\,(\delta_{T}h)~{}=~{}\int\,d^{3}x\,\partial_{-}\bar{h}\,T\,\frac{\partial\bar{\partial}}{\partial_{-}}h\,+\,\mathcal{O}(\kappa)\,,$
(3.18)

where we have used the equation of motion for $h$ obtained from (2.13). One
can derive the transformation law for the fields $h$ and $\bar{h}$ order by
order in $\kappa$ from the above generator using the brackets111The fields $h$
and $\bar{h}$ satisfy
$[h(x),\bar{h}(y)]\,=\,\frac{1}{\partial_{-}}\delta^{(3)}(x-y)\,,\quad[\,h(x),h(y)\,]\,=\,[\,\bar{h}(x),\bar{h}(y)\,]\,=\,0\,.$
(3.19)

$\displaystyle\delta_{T}h~{}=~{}[G_{T},h]\,,$
$\displaystyle\delta_{T}\bar{h}~{}=~{}[G_{T},\bar{h}]\,.$ (3.20)

At the lowest order, the generator can be brought to the form

$\displaystyle G_{T}^{(0)}$ $\displaystyle=$
$\displaystyle\int\,d^{3}x\,\left(T\,\partial\bar{h}\,\bar{\partial}h\,+\,\partial
T\,\bar{h}\,\bar{\partial}h\right)\ $ (3.21) $\displaystyle=$
$\displaystyle\int
d^{3}x\,\left(T\,\partial\bar{h}\,\bar{\partial}h+\frac{1}{2}\partial
T\,\bar{h}\,\bar{\partial}h+\frac{1}{2}\,\bar{\partial}T\,h\,\partial\bar{h}\right)\,.$

At the cubic order, $G_{T}$ reads [15]

$\displaystyle G_{T}^{(\kappa)}$ $\displaystyle=$ $\displaystyle\kappa\int
d^{3}x\,T\,\partial_{-}\bar{h}\Bigg{\\{}2\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,$
(3.22)
$\displaystyle+\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right)\Bigg{\\}}\,.$

After some partial integrations, the details of which can be found in Appendix
D, the supertranslation generator up to order $\kappa$ reads

$\displaystyle G_{T}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,T\,\mathcal{D}\bar{h}\,\overline{\mathcal{D}}h\,+\,\,\int
d^{3}x\,\Bigg{\\{}\frac{1}{2}\,\partial
T\,\bar{h}\,\bar{\partial}h\,-\,\kappa\,\partial
T\,\bar{h}\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}h-\bar{h}\partial_{-}^{2}\partial
h\right)$ (3.23) $\displaystyle-$ $\displaystyle\,2\,\kappa\,\partial
T\,\frac{1}{\partial_{-}^{2}}(\bar{h}\partial_{-}^{2}h)\,{\partial}\bar{h}\,+\,\frac{1}{2}\,\kappa\,\partial^{2}T\,\bar{h}\,\frac{1}{\partial_{-}^{2}}(\bar{h}\partial_{-}^{2}h)\,\Bigg{\\}}+\
\text{c.c.}\,,$

where the “covariant” derivative $\overline{\mathcal{D}}h$ reads [9, 10]

$\displaystyle\overline{\mathcal{D}}h$ $\displaystyle=$
$\displaystyle\bar{\partial}h\,+\,2\,\kappa\,\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}h\,-\,\bar{h}\partial_{-}^{2}\partial
h\right)\,.$ (3.24)

For constant time translations, $T=a$, the corresponding generator becomes the
Hamiltonian of the theory given by

$\displaystyle G_{T=a}~{}=~{}H~{}=~{}\int
d^{3}x\,a\,\mathcal{D}\bar{h}\,\overline{\mathcal{D}}h\,.$ (3.25)

Thus, the light-cone Hamiltonian for gravity in four dimensions can be recast
into a positive semi-definite structure, which we call quadratic form,
indicating that the energy of the system is manifestly positive. In an earlier
work [10], we had derived the order $\kappa^{2}$ terms in $\bar{\mathcal{D}}h$
proving that this feature extends to the second order in coupling constant as
well. However, in this analysis, we learn that the quadratic form structure of
the Hamiltonian is actually fixed by the BMS symmetry. The terms involving
derivatives of $T$ in (3.23) are not arbitrary as these are related to the
spin corrections to the boost generators $J^{-}$ and $\bar{J}^{-}$ required
for the closure of the light-cone Poincaré algebra. Any further partial
integrations in $\partial$ and $\bar{\partial}$ will spoil this structure and
hence, lead to ill-defined boost generators222This can be compared to the
asymptotic symmetry analysis at spatial infinity, where the boost generators
require special treatment in order to have well-defined Poincaré generators in
the Hamiltonian formulation [6, 7].. In fact, we can read off corrections to
the boost generators order by order in $\kappa$ from the general expression
for supertranslation generator to higher orders. The quadratic form expression
(3.25) resembles the generator of supertranslations at null infinity proposed
by Bondi, van der Burg, Metzner, and Sachs [1, 2], which shows that the flux
of the gravitational energy at null infinity is positive. Proof of the
positive energy theorem in gravity [18, 19, 20] typically involves some
spinor-like variables. In the light-cone formulation, however, one can derive
the quadratic form Hamiltonian without resorting to any spinor variables. It
is important to note though that we are concerned with the energy density in
the bulk, not the flux of gravitational energy at null infinity.

Interestingly, the covariant derivative $\overline{\mathcal{D}}h$ in the
quadratic form Hamiltonian plays a role similar to the “News tensor” in the
Bondi frame, which is related to vacuum configurations. Therefore, we can
define the vacuum configurations of gravity in the light-cone formulation as
the states which satisfy the condition

$\displaystyle\overline{\mathcal{D}}h$ $\displaystyle=$
$\displaystyle\bar{\partial}h\,+\,2\,\kappa\,\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}h\,-\,\bar{h}\partial_{-}^{2}\partial
h\right)\,+\mathcal{O}(\kappa^{2})~{}=~{}0\,.$ (3.26)

It would be instructive to use this equation to classify different vacuum
configurations and explore its possible implications for the quantum theory,
as the fields $h$ and $\bar{h}$ indeed correspond to the two physical states
of the graviton.

##  4 Concluding remarks

In this paper, we discussed the BMS symmetry in the light-cone gauge from the
perspective of residual gauge freedom in the theory. The invariance of the
Hamiltonian under these reparameterizations leads to the correct
transformation laws for the fields, which then realize the BMS algebra in four
dimensions. The most notable characteristics of the light-cone BMS algebra is
the enhancement of the dynamical part of the Poincaré algebra. In the light-
cone formalism, the dynamical transformations, which are non-linearly realized
on the fields, provide us with a powerful framework to construct interacting
actions from the closure of the symmetry algebra [15]. This is particularly
interesting for higher-spin theories as constructing interacting actions with
higher-spin fields is a very complex issue. It would, therefore, be worthwhile
to explore the implications of these non-linear transformations for the
interacting theory.

Although ours is a perturbative analysis, this light-cone approach offers a
unique insight into the structure of the BMS symmetry, that also shares some
interesting features with the BMS literature for the full Einstein theory. One
important similarity with the spatial infinity analysis is that the invariance
of the Hamiltonian reduces the parameters, $Y$ and $\overline{Y}$, to Lorentz
rotations, which eliminates superrotations from the theory. In light-cone
gravity, one can study the BMS symmetry in a physical gauge, which might help
us better understand its connection with on-shell amplitudes [21, 22] and soft
theorems [23, 24].

The fact that the light-cone Hamiltonian for gravity in four dimensions can be
expressed as a positive semi-definite quadratic form puts this theory in a
special class of theories that admit such Hamiltonians. The occurrence of such
simple structures in field theories is often indicative of hidden symmetries
[25, 26]. In this paper, we have attributed the quadratic form structure in
light-cone gravity to the invariance of the Hamiltonian under BMS
supertranslations. Similar analyses for residual gauge symmetry in Yang-Mills
and higher-spin theories, which also admit such quadratic form Hamiltonians
[27, 28], could bring forth some deeper links between these theories and
gravity. Similarly, extensions to supersymmetric theories [29] could be
instrumental in explaining the quadratic form Hamiltonians found in maximally
supersymmetric Yang-Mills and supergravity in four dimensions.

## Acknowledgments

We thank Glenn Barnich and Marc Henneaux for many discussions. The work of SM
is partially supported by the ERC Advanced Grant “High-Spin-Grav”, by FNRS-
Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by
funds from the Solvay Family.

##  Appendix A Light-cone Poincaré algebra in $d=4$

In this section, we present the light-cone Poincaré algebra in four
dimensions. A general Poincaré trasnformation in four dimensions

$\delta x^{\mu}~{}=~{}\omega^{\mu}_{\nu}x^{\nu}+a^{\mu}\,,$ (A.1)

in light-cone coordinates, takes the unusual form

$\delta\begin{pmatrix}x^{+}\\\ x^{-}\\\ x\\\
\bar{x}\end{pmatrix}~{}=~{}\begin{pmatrix}\omega_{+-}&0&-\bar{\omega}_{-}&-\omega_{-}\\\
0&-\omega_{+-}&\bar{\omega}_{+}&\omega_{+}\\\
\omega_{+}&\bar{\omega_{-}}&-i\omega_{12}&0\\\
\bar{\omega_{+}}&\omega_{-}&0&i\omega_{12}\end{pmatrix}\begin{pmatrix}x^{+}\\\
x^{-}\\\ x\\\ \bar{x}\end{pmatrix}\ +\ \begin{pmatrix}a^{+}\\\ a^{-}\\\ a\\\
\bar{a}\end{pmatrix}$ (A.2)

where the two real $(\omega_{+-},\omega_{12})$ and two complex
$(\omega_{+},\omega_{-})$ parameters label the Lorentz group. We define

$\displaystyle J^{+}\ =\ \frac{J^{+1}+iJ^{+2}}{\sqrt{2}}\ ,\quad\bar{J}^{+}\
=\ \frac{J^{+1}-iJ^{+2}}{\sqrt{2}}\ ,\quad J\ =\ J^{12}\ ,\quad H\ =\ P^{-}\
.$ (A.3)

All the non-vanishing commutators of the Poincaré algebra are listed below

$\displaystyle[H,J^{+-}]\ =\ -iH\ ,\quad$ $\displaystyle[H,J^{+}]=-iP\
,\qquad\qquad\quad[H,\bar{J}^{+}]\ =\ -i\bar{P}$ (A.4)
$\displaystyle[P^{+},J^{+-}]\ =\ iP^{+}\ ,\quad$ $\displaystyle[P^{+},J^{-}]\
=\ -iP\ ,\qquad\qquad[P^{+},\bar{J}^{-}]\ =\ -i\bar{P}$
$\displaystyle[P,\bar{J}^{-}]\ =\ -iH\ ,\quad$ $\displaystyle[P,\bar{J}^{+}]\
=\ -iP^{+}\ ,\qquad\qquad[P,J]\ =\ P$ $\displaystyle[\bar{P},J^{-}]\ =\ -iH\
,\quad$ $\displaystyle[\bar{P},J^{+}]\ =\ -iP^{+}\ ,\qquad\qquad[\bar{P},J]\
=\ -\bar{P}$ $\displaystyle[J^{-},J^{+-}]\ =\ -iJ^{-}\ ,\quad$
$\displaystyle[J^{-},\bar{J}^{+}]\ =\ iJ^{+-}+J\ ,\qquad[J^{-},J]\ =\ J^{-}$
$\displaystyle[\bar{J}^{-},J^{+-}]\ =\ -i\bar{J}^{-}\ ,\quad$
$\displaystyle[\bar{J}^{-},J^{+}]\ =\ iJ^{+-}-J\ ,\qquad[\bar{J}^{-},J]\ =\
-\bar{J}^{-}$ $\displaystyle[J^{+-},J^{+}]\ =\ -iJ^{+}\ ,\quad$
$\displaystyle[J^{+-},\bar{J}^{+}]\ =\ -i\bar{J}^{+}\ ,$
$\displaystyle[J^{+},J]\ =\ J^{+}\ ,\quad$ $\displaystyle[\bar{J}^{+},J]\ =\
-\bar{J}^{+}\ .$

In case of light-cone gravity, the Poincaré generators are canonically
realised on the phase space of $h$ and $\bar{h}$ as follows

$\displaystyle P^{-}\,=\,\int d^{3}x\,\mathcal{H}\ ,\quad P\,=\,\int
d^{3}x\partial_{-}\bar{h}\,\partial h\,,\quad\overline{P}\,=\,\int
d^{3}x\partial_{-}\bar{h}\,\bar{\partial}h\,,\quad P^{+}\,=\,\int
d^{3}x\partial_{-}\bar{h}\,\partial_{-}h\,,$ $\displaystyle J\,=\,i\,\int
d^{3}x\,\partial_{-}\bar{h}\,(x\bar{\partial}-\bar{x}\partial-\lambda)\,h\
,\quad J^{+-}\,=\,\int
d^{3}x\,(\partial_{-}\bar{h}\,x^{-}\partial_{-}h-x^{+}\mathcal{H})\,,$
$\displaystyle J^{+}\,=\,\int
d^{3}x\,\partial_{-}\bar{h}\,(x^{+}\partial+x\partial_{-})\,h\ ,\quad
J^{-}\,=\,\int
d^{3}x\,\left\\{x\mathcal{H}+\partial_{-}\bar{h}\left(x^{-}\partial-\lambda\frac{\partial}{\partial_{-}}\right)h+\mathcal{S}^{-}\right\\}\,,$
$\displaystyle\bar{J}^{+}\,=\,\int
d^{3}x\,\partial_{-}\bar{h}\,(x^{+}\bar{\partial}+\bar{x}\partial_{-})\,h\
,\quad\bar{J}^{-}\,=\,\int
d^{3}x\,\left\\{\bar{x}\mathcal{H}+\partial_{-}\bar{h}\,\left(x^{-}\bar{\partial}-\lambda\frac{\bar{\partial}}{\partial_{-}}\right)\,h+\bar{\mathcal{S}}^{-}\right\\}\,,$

where the spin corrections read

$\displaystyle\delta_{s^{-}}h$ $\displaystyle=$
$\displaystyle-2\,\frac{\partial}{\partial_{-}}h\,-4\,\kappa\,\partial_{-}\left(h\frac{\bar{\partial}}{\partial_{-}^{2}}h-\frac{1}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,+\,\mathcal{O}(\kappa^{2})\,,$
$\displaystyle\delta_{\bar{s}^{-}}h$ $\displaystyle=$ $\displaystyle
2\,\frac{\bar{\partial}}{\partial_{-}}h\,-4\,\kappa\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial}{\partial_{-}^{2}}\bar{h}\partial_{-}^{4}h-\frac{1}{\partial_{-}}\bar{h}\partial_{-}^{3}\partial
h+3\frac{\partial}{\partial_{-}}\bar{h}\partial_{-}^{3}h-3\bar{h}\partial_{-}^{2}\partial
h\right)\,+\,\mathcal{O}(\kappa^{2})\,.$

##  Appendix B Allowed residual gauge transformations

Any residual gauge transformation must leave the light-cone gauge choices
invariant. These put some constraints on the form of the allowed gauge
parameters $\xi^{\mu}$.

The first gauge condition $g_{--}=0$ leads to the constraint

$\displaystyle\delta g_{--}=0$ $\displaystyle\Rightarrow$
$\displaystyle\partial_{-}\xi^{+}\,g_{+-}~{}=~{}0$ (B.1)

This condition is easily satisfied if we chose the parameter $\xi^{+}$ as

$\xi^{+}~{}=~{}f(x^{+},x^{j})\quad\text{such that}\quad\partial_{-}f~{}=~{}0$

The condition $g_{-i}=0$ leads to

$\displaystyle\delta g_{-i}=0$ $\displaystyle\Rightarrow$
$\displaystyle\partial_{-}\xi^{j}\,g_{ij}\ +\ \partial_{i}\xi^{+}\,g_{+-}=0$
(B.2)

For the second condition to hold, we can solve for $\xi^{j}$ in terms of
$\xi^{+}=f$

$\xi^{k}~{}=~{}-\partial_{i}f\,\frac{1}{\partial_{-}}(g_{+-}g^{ik})\ +Y^{k}\ \
,$ (B.3)

where the integration constant $Y^{k}$ does not depend on $x^{-}$. The gauge
condition (2.8) relates the $g_{-+}$ component to the determinant of the
metric $g_{ij}$. Thus, we first consider $\delta g_{-+}$ to obtain

$\delta\phi~{}=~{}f\,\partial_{+}\phi+\xi^{-}\partial_{-}\phi+\xi^{k}\,\partial_{k}\phi+\partial_{+}f+\partial_{-}\xi^{-}-\partial_{k}f\,g_{+i}\,g^{ik}$
(B.4)

As evident from the expression of $\phi$ in (2.9), the above equation is
complicated involving the non-local $\frac{1}{\partial_{-}}$ operators. But in
the perturbative expansion, one can simplify the equation in orders of
$\kappa$ and obtain non-trivial relations between the gauge parameters. At the
lowest order, one finds

$\partial_{+}f+\partial_{-}\xi^{-}~{}=~{}0$ (B.5)

We can, alternatively, obtain $\delta\phi$ from the variation of the
determinant of $g_{ij}$

$\delta g=g\,g^{ij}\delta g_{ij}\,,$ (B.6)

with $g$ given by

$g=det(g_{ij})=2\psi=\phi$ (B.7)

where the last equality follows from the fourth gauge choice (2.8). When
compared with $\delta\phi$ in (B.4) to the lowest order, we get

$\partial_{+}f=\frac{1}{2}\partial_{i}Y^{i}\,.$ (B.8)

This constraint fixes the time dependence of the parameter $f$

$f(x^{+},x^{i})=\frac{1}{2}\partial_{i}Y^{i}\,x^{+}+T(x^{i})\,,$ (B.9)

which in the $(x,\bar{x})$ coordinate reads

$f(x^{+},x,\bar{x})=\frac{1}{2}x^{+}(\partial\overline{Y}+\bar{\partial}Y)+T(x,\bar{x})\,.$
(B.10)

Further consistency checks may be performed on the remaining components of the
metric. Since both $g_{++}$ and $g_{+i}$ are at least of order $\kappa$, we
have the following conditions at the zeroth order

$\displaystyle\delta g_{++}=0$ $\displaystyle\Rightarrow$
$\displaystyle\partial_{+}\xi^{-}=0\,,$ (B.11) $\displaystyle\delta g_{+i}=0$
$\displaystyle\Rightarrow$
$\displaystyle\partial_{i}\xi^{-}=\partial_{+}\xi_{i}\,,$ (B.12)

which along with (B.5) completely determines the form of $\xi^{-}$ in terms of
$f$ and $\xi^{i}$

$\xi^{-}=-(\partial_{+}f)\,x^{-}+(\partial_{+}\xi_{i})\,x^{i}\,.$ (B.13)

### Poincaré subgroup within the BMS

From the above conditions, one can write a general BMS transformation on the
light-cone coordinates as

$\displaystyle\delta x$ $\displaystyle=$ $\displaystyle\alpha\,+\,\beta
x\,+\,\sigma x^{+}\,,$ (B.14) $\displaystyle\delta\bar{x}$ $\displaystyle=$
$\displaystyle\bar{\alpha}\,+\,\bar{\beta}\bar{x}\,+\,\bar{\sigma}x^{+}\,,$
(B.15) $\displaystyle\delta x^{+}$ $\displaystyle=$ $\displaystyle
Re(\beta)x^{+}\,+\,T(x,\bar{x})$ (B.16) $\displaystyle=$ $\displaystyle
Re(\beta)x^{+}\,+\,[\,t_{0}+t_{1}x+\bar{t_{1}}\bar{x}+\ldots]\,,$
$\displaystyle\delta x^{-}$ $\displaystyle=$
$\displaystyle\gamma\,-\,Re(\beta)x^{-}\,+\,\sigma
x\,+\,\bar{\sigma}\bar{x}\,,$ (B.17)

where the parameters $t_{0},\gamma$ are real and $\alpha,\beta,\sigma,t_{1}$
are all complex. We can then read off the Poincaré subgroup inside the BMS
group by identifying the parameters in the above set of equations to those in
(A.2) as follows

$\displaystyle(a^{+},a^{-},a,\bar{a})~{}=~{}(t_{0},\gamma,\alpha,\bar{\alpha})$
$\displaystyle\omega_{+-}=Re(\beta)\,,\quad\omega_{12}=Im(\beta)\,,\quad(\omega_{+},\omega_{-})~{}=~{}(\sigma,t_{1})\,.$
(B.18)

The extension of the Poincaré group to the BMS group is, thus, parameterized
only by the higher order expansions of $T(x,\bar{x})$ in (B.16).

##  Appendix C Connections to the Carroll group

For null hypersurfaces like the light-cone surfaces $x^{\pm}=constant$, the
BMS algebra in four dimensions is isomorphic to the conformal Carroll group
[17]. The conformal Carroll group consists of two-dimensional conformal
algebra on the spatial coordinates augmented by supertranslations in the time
direction, $u$.

$\displaystyle x\ \rightarrow\phi(x)\ ,\ \quad u\
\rightarrow\Omega(x)^{\frac{1}{2}}\,[u\ +\ \alpha(x)]\,,$ (C.1)

where $\Omega(x)$ is the scaling factor associated with the conformal
transformations on $x$.

We, therefore, consider conformal Carroll transformations in the light-cone
coordinates on a constant $x^{-}$ surface

$\displaystyle x~{}\rightarrow~{}\omega(x)\
,\quad\bar{x}~{}\rightarrow~{}\overline{\omega}(\bar{x})\,,$ (C.2)
$\displaystyle
x^{+}~{}\rightarrow~{}\Omega(x,\bar{x})^{\frac{1}{2}}\,[\,x^{+}\ +\
\alpha(x,\bar{x})\,]\,.$ (C.3)

An infinitesimal conformal transformation on $x,\bar{x}$ is given by

$\displaystyle x$ $\displaystyle\rightarrow$ $\displaystyle\omega(x)\ =\ x\ +\
Y(x)\ ,$ (C.4) $\displaystyle\bar{x}$ $\displaystyle\rightarrow$
$\displaystyle\overline{\omega}(\bar{x})\ =\ \bar{x}\ +\
\overline{Y}(\bar{x})\ .$ (C.5)

Thus, from (C.3), we find that the time-coordinate $x^{+}$ transforms
infinitesimally as

$\displaystyle x^{+}$ $\displaystyle\rightarrow$
$\displaystyle\Omega(x,\bar{x})^{\frac{1}{2}}\,\left[x^{+}\,+\
\alpha(x,\bar{x})\right]~{}=~{}[1+(\bar{\partial}Y+\partial\overline{Y})]^{\frac{1}{2}}\,\left(x^{+}\,+\
\alpha(x,\bar{x})\right)$ (C.6) $\displaystyle\sim$
$\displaystyle\left\\{1+\frac{1}{2}(\bar{\partial}Y+\partial\overline{Y})\right\\}\left(x^{+}\,+\
\alpha(x,\bar{x})\right)~{}\sim~{}x^{+}\ +\ f(x^{+},x,\bar{x})\,,$

where

$\displaystyle f(x^{+},x,\bar{x})$ $\displaystyle=$ $\displaystyle
T(x,\bar{x})\ +\ \frac{1}{2}\,x^{+}\,(\bar{\partial}Y\
+\,\partial\overline{Y})\,.$ (C.7)

Here we have ignored the terms of higher orders in $Y$ and $\overline{Y}$.
Thus, the residual reparameterizations considered in (3.8) can indeed be
interpreted as infinitesimal conformal Carroll transformation on the constant
$x^{-}$ null hypersurface.

##  Appendix D Supertranslation generator at order $\kappa$

We consider the supertranslation generator at order $\kappa$

$\displaystyle G_{T}^{(\kappa)}$ $\displaystyle=$ $\displaystyle\kappa\int
d^{3}x\,T\,\partial_{-}\bar{h}\Bigg{\\{}2\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,$
(D.1)
$\displaystyle\quad+\,\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right)\Bigg{\\}}$
$\displaystyle=$ $\displaystyle\mathcal{X}+\mathcal{Y}\,,$

where $\mathcal{X}$ contains the terms involving $\bar{h}hh$

$\displaystyle\mathcal{X}$ $\displaystyle=$ $\displaystyle 2\,\kappa\int
d^{3}x\,T\,\partial_{-}\bar{h}\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\,-\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)\,,$
(D.2)

and $\mathcal{Y}$ contains terms involving $\bar{h}\bar{h}h$

$\displaystyle\mathcal{Y}$ $\displaystyle=$ $\displaystyle\kappa\int
d^{3}x\,T\,\partial_{-}\bar{h}\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\,\partial_{-}^{2}h\,-\,2\,\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}\partial
h\,+\,\bar{h}\,\partial_{-}^{2}\partial^{2}h\right)\,.$ (D.3)

Let us focus on the $\mathcal{X}$ terms, which upon partial integrations, can
be expressed as

$\displaystyle\mathcal{X}$ $\displaystyle=$ $\displaystyle
2\kappa\int\,d^{3}x\,T\partial_{-}\bar{h}\,\partial_{-}\left(h\,\frac{{\bar{\partial}}^{2}}{\partial_{-}^{2}}h\right)\,-\,2\kappa\int\,d^{3}x\,T\partial_{-}\bar{h}\,\partial_{-}\left(\,\frac{\bar{\partial}}{\partial_{-}}h\,\frac{\bar{\partial}}{\partial_{-}}h\right)$
(D.4) $\displaystyle=$
$\displaystyle-\,2\kappa\int\,d^{3}x\,\frac{\bar{\partial}}{\partial_{-}^{2}}(T\,\partial_{-}^{2}\bar{h}h)\,\bar{\partial}h+2\kappa\int\,d^{3}x\,T\,\frac{1}{\partial_{-}}\left(\frac{\bar{\partial}}{\partial_{-}}h\,\partial_{-}^{2}\bar{h}\right)$
(D.6) $\displaystyle=$
$\displaystyle-\,2\kappa\int\,d^{3}x\,\bar{\partial}T\,\frac{1}{\partial_{-}^{2}}(\partial_{-}^{2}\bar{h}h)\,\bar{\partial}h-\,2\kappa\int\,d^{3}x\,T\,\frac{\bar{\partial}}{\partial_{-}^{2}}(\partial_{-}^{2}\bar{h}h)\,\bar{\partial}h$
$\displaystyle+2\kappa\int\,d^{3}x\,T\,\frac{1}{\partial_{-}}\left(\bar{\partial}h\,\partial_{-}^{2}\bar{h}\,+\,\frac{\bar{\partial}}{\partial_{-}}h\,\partial_{-}^{3}\bar{h}\right)$
$\displaystyle=$ $\displaystyle-2\,\kappa\int
d^{3}x\,\bar{\partial}T\,\frac{1}{\partial_{-}^{2}}(h\partial_{-}^{2}\bar{h})\,\bar{\partial}h\,+\,2\,\kappa\int
d^{3}x\,T\,\frac{1}{\partial_{-}^{2}}\left(\frac{\bar{\partial}}{\partial_{-}}h\,\partial_{-}^{3}\bar{h}\,-\,h\,\partial_{-}^{2}\bar{\partial}\bar{h}\right)\bar{\partial}h\,.\qquad\quad$
(D.7)

Similarly, the $\mathcal{Y}$ terms can be simplified as follows

$\displaystyle\mathcal{Y}$ $\displaystyle=$ $\displaystyle\kappa\int
d^{3}x\,T\,\partial_{-}\bar{h}\frac{1}{\partial_{-}^{3}}\left(\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\partial_{-}^{4}h-2\frac{\partial}{\partial_{-}}\bar{h}\partial_{-}^{3}\partial
h+\bar{h}\partial_{-}^{2}\partial^{2}h\right)$ (D.10) $\displaystyle=$
$\displaystyle-\kappa\int
d^{3}x\,T\,\partial_{-}^{2}\left(\frac{1}{\partial_{-}^{2}}\bar{h}\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}\right)\partial_{-}^{2}h+2\kappa\int
d^{3}x\,\partial_{-}\partial\left(T\frac{1}{\partial_{-}^{2}}\bar{h}\frac{\partial}{\partial_{-}}\bar{h}\right)\partial_{-}^{2}h\qquad$
$\displaystyle-\kappa\int
d^{3}x\,\partial^{2}\left(T\,\bar{h}\frac{1}{\partial_{-}^{2}}\bar{h}\right)\partial_{-}^{2}h$
$\displaystyle=$ $\displaystyle-2\kappa\int d^{3}x\,\partial
T\,\bar{h}\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}h-\bar{h}\partial_{-}^{2}\partial
h\right)+\kappa\int
d^{3}x\,\partial^{2}T\,\bar{h}\,\frac{1}{\partial_{-}^{2}}(\bar{h}\partial_{-}^{2}h)\qquad$
$\displaystyle-2\kappa\int
d^{3}x\,T\,\partial_{-}^{2}h\left(\bar{h}\frac{\partial^{2}}{\partial_{-}^{2}}\bar{h}-\frac{\partial}{\partial_{-}}\bar{h}\,\frac{\partial}{\partial_{-}}\bar{h}\right).$

Note that the second line in the above equation is the complex conjugate of
$\mathcal{X}$ in (D.2). Thus, we obtain

$\displaystyle\mathcal{Y}$ $\displaystyle=$ $\displaystyle-2\kappa\int
d^{3}x\,\partial
T\,\bar{h}\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}h-\bar{h}\partial_{-}^{2}\partial
h\right)+\kappa\int
d^{3}x\,\partial^{2}T\,\bar{h}\,\frac{1}{\partial_{-}^{2}}(\bar{h}\partial_{-}^{2}h)$
(D.11) $\displaystyle-2\,\kappa\int
d^{3}x\,{\partial}T\,\frac{1}{\partial_{-}^{2}}(\bar{h}\partial_{-}^{2}h)\,{\partial}\bar{h}\,+\,2\,\kappa\int
d^{3}x\,T\,\frac{1}{\partial_{-}^{2}}\left(\frac{\partial}{\partial_{-}}\bar{h}\,\partial_{-}^{3}{h}\,-\,\bar{h}\,\partial_{-}^{2}\partial
h\right){\partial}\bar{h}\,.\qquad\quad$

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# Shared Autonomy via Variable Impedance Control and Virtual Potential Fields
for Encoding Human Demonstrations*

Shail Jadav†,1,2, Johannes Heidersberger†,2, Christian Ott3,4 and Dongheui
Lee2,4 *This work was supported by funding from the DFG Priority Programme
(SPP 2134) Active Self and IIT Gandhinagar Overseas Research
Fellowship.$\dagger$These authors contributed to this paper equally. 1 Shail
Jadav is Indian Institute of Technology Gandhinagar, India
<EMAIL_ADDRESS>Shail Jadav, Johannes Heidersberger, & Dongheui Lee
are with Autonomous Systems, Technische Universität Wien (TU Wien), Austria,
<EMAIL_ADDRESS><EMAIL_ADDRESS>Christian Ott
is with Automation and Control Institute, Technische Universität Wien (TU
Wien), Austria<EMAIL_ADDRESS>Christian Ott and Dongheui Lee are
with the Institute of Robotics and Mechatronics, German Aerospace Center
(DLR), Germany.

###### Abstract

This article introduces a framework for complex human-robot collaboration
tasks, such as the co-manufacturing of furniture. For these tasks, it is
essential to encode tasks from human demonstration and reproduce these skills
in a compliant and safe manner. Therefore, two key components are addressed in
this work: motion generation and shared autonomy. We propose a motion
generator based on a time-invariant potential field, capable of encoding
wrench profiles, complex and closed-loop trajectories, and additionally
incorporates obstacle avoidance. Additionally, the paper addresses shared
autonomy (SA) which enables synergetic collaboration between human operators
and robots by dynamically allocating authority. Variable impedance control
(VIC) and force control are employed, where impedance and wrench are adapted
based on the human-robot autonomy factor derived from interaction forces.
System passivity is ensured by an energy-tank based task passivation strategy.
The framework’s efficacy is validated through simulations and an experimental
study employing a Franka Emika Research 3 robot.

##  I INTRODUCTION

In recent years, robotic systems have transitioned from conventional
applications in structured industrial environments to more dynamic human-robot
collaborative scenarios, such as co-manufacturing of furniture [1]. Such
collaborative frameworks have the possibility of leveraging the advanced
cognitive capabilities of humans in conjunction with the superior attributes
of robotic manipulators, like high accuracy of motion repetition without
fatigue, thereby enhancing overall system performance [2, 3, 4, 5, 6]. In this
study, we investigate complex tasks involving human-robot collaboration (Fig.
1), such as furniture co-manufacturing, which necessitates the integration of
multiple components, including motion generation algorithms informed by
learning from demonstration (LfD), shared autonomy (SA) protocols to balance
control between human and robot, as well as variable impedance and force
control mechanisms to adapt to changing task requirements and to ensure
compliant behavior.

Figure 1: Experimental setups illustrating shared autonomy in human-robot
collaboration for furniture co-manufacturing tasks (left) and button-pressing
tasks (right).

A prominent example of LfD-based motion planning for SA are dynamical systems
(DS), which are known to provide stability guarantees concerning global
attractors [7]. The domain of machine learning-based DS has witnessed numerous
extensions, including, but not limited to, incremental learning [8, 9], shared
control [10], reactivity to moving targets [11], obstacle avoidance [12]. The
Stable Estimator of Dynamical System (SEDS) is an algorithm based on Gaussian
mixture models, offering stability guarantees by attracting towards a global
point [7]. However, there is a trade-off between stability and accuracy, often
failing to precisely encode highly nonlinear demonstrations [13, 14].
Additionally, SEDS is limited to modeling trajectories that monotonically
decrease in distance to the target over time [15]. To address this limitation,
authors of [16] introduced a method based on SEDS through a Linear Parameter
Varying (LPV) re-formulation of the model (LPV-DS). This approach can learn
more intricate trajectories compared to the original SEDS, but it still falls
short in encoding highly complex motions, especially evident in the leaf shape
(see Fig. 4) within the LASA dataset of handwriting [13]. The authors in [17]
proposed an alternative approach that learns potential functions from
demonstrations, addressing the limitation in precisely encoding complex shapes
with the previously mentioned motion generators. While this method is capable
of generating complex shapes, it is not applicable to closed-loop
trajectories. Additionally, this approach employs a predefined impedance model
based on the trajectory, thereby complicating the integration of dynamic
authority shifts between human and robot through modulation of impedance
characteristics. Furthermore, this method does not encode the requisite wrench
profile for task execution, a critical component for successful human-robot
collaboration that requires precise wrench modulation.

Further, SA is useful in co-manufacturing tasks with humans [18]. There have
been numerous advancements in dynamic authority allocation between human
operators and robotic agents, existing methodologies often employ end-effector
velocity and force, electromyography (EMG), and stability constraints for
authority arbitration [19]. Admittance control methodologies have been
investigated in the context of shared autonomy, yet they exhibit constraints
in force-production tasks involving rigid bodies, due to uncertainties induced
by physical interactions [20, 21, 22].

Consequently, this paper presents a comprehensive framework for shared
autonomy with the following contribution:

  * •

We introduce a time-invariant, state-dependent motion generator that encodes
task-specific trajectories and wrench profiles, accommodates complex and
closed-loop paths, and integrates obstacle avoidance.

  * •

We propose a shared autonomy methodology that incorporates variable impedance
and force control (VIC) for accurate trajectory and wrench tracking, adapting
to authority levels and complying with human-induced perturbations.

Furthermore, we incorporate an energy-tank-based passivation strategy, akin to
that presented in [23, 24, 25], to guarantee passivity during time-varying
impedance and control parameters. Finally, we validate the proposed approach
through extensive experiments and simulations.

##  II Methods

In the proposed framework, the VIC receives reference trajectories from a
time-invariant, state-dependent motion generator, as shown in Fig. 2,
modulated via the authority arbitrator. The authority arbitrator is modulated
by human input; when force is applied by the human or any unencoded forces are
detected, the compliance of the robot is increased.

###  II-A Motion Planning using Virtual Potential Fields

Let us assume we are provided one demonstration of $\rm{T_{d}}$ datapoints
consisting of the Cartesian positions of the end-effector $\mathbf{X}_{\rm
d}\in\mathbb{R}^{3\times\rm{T_{d}}}$, the orientation of the end-effector as
Euler angles $\boldsymbol{\theta}_{\rm d}\in\mathbb{R}^{3\times\rm{T_{d}}}$,
the wrench exerted by the end-effector $\mathbf{W}_{\rm
d}\in\mathbb{R}^{6\times\rm{T_{d}}}$, as well as the Cartesian velocities
$\dot{\mathbf{X}}_{\rm d}\in\mathbb{R}^{3\times\rm{T_{d}}}$ and angular end-
effector velocities $\dot{\boldsymbol{\theta}}_{\rm
d}\in\mathbb{R}^{3\times\rm{T_{d}}}$. In the proposed time-invariant, state-
dependent approach the reference positions, velocities, and wrenches are
determined using the current robot’s end-effector position and orientation as

$\displaystyle\begin{pmatrix}\dot{\mathbf{x}}_{\rm ref}\\\
\dot{\boldsymbol{\theta}}_{\rm ref}\\\ \mathbf{w}_{\rm
ref}\end{pmatrix}=f\left(\mathbf{x}(t),\boldsymbol{\theta}(t),\mathbf{x}_{\rm
start}(t),\mathbf{x}_{\rm goal}(t),\mathbf{x}_{\rm obs}(t)\right).$ (1)

Figure 2: Overview of proposed framework consisting of a motion generator,
variable impedance control, and an authority arbitrator.

Here, $f$ is the dynamical system derived from the demonstration, which maps
the current states $\mathbf{x}$ and $\boldsymbol{\theta}$ to the desired end-
effector states $(\cdot)_{\rm ref}$. The derivation of $f$ is described in the
following.

The encoded motion should be usable anywhere within the robot’s workspace;
therefore, we need to generalize it based on new starting and ending points.
To adapt the demonstrated motion to time-varying start $\mathbf{x}_{\rm
start}(t)\in\mathbb{R}^{3}$ and goal $\mathbf{x}_{\rm
goal}(t)\in\mathbb{R}^{3}$ points, we employ scaling and rotation operations
to transform the original demonstration. The scaling factor for translations
is given by

$\displaystyle s_{L}(t)=\frac{\|\mathbf{x}_{\rm goal}(t)-\mathbf{x}_{\rm
start}(t)\|}{\|\mathbf{x}_{\rm d}^{\rm{T_{\rm d}}}-\mathbf{x}_{\rm d}^{1}\|},$
(2)

where $\ \mathbf{x}_{\rm d}^{\rm i}\in\mathbb{R}^{3}$ is the i-th sample point
of end-effector position from demonstration $\mathbf{X}_{\rm d}$.

We compute the rotation matrix $\mathbf{R}(t)\in\mathbb{R}^{3\times 3}$ using
the Rodrigues’ rotation formula [26], which aligns the scaled 3D motion with
the desired endpoint, i.e. the vectors $\mathbf{x}_{\rm d}^{\rm{T_{\rm
d}}}-\mathbf{x}_{\rm d}^{1}$ and $\mathbf{x}_{\rm goal}(t)-\mathbf{x}_{\rm
start}(t)$ are aligned. From the scaling and rotation operation, the
transformation of the position follows as

$\displaystyle\tilde{\mathbf{x}}_{\rm d}^{\rm t_{\rm d}}(t)$
$\displaystyle=s_{L}(t)\mathbf{R}(t)\Bigl{(}{\mathbf{x}}_{\rm d}^{{\rm t}_{\rm
d}}-{\mathbf{x}}_{\rm d}^{1}\Bigr{)}+\mathbf{x}_{\rm start}(t)\quad\forall{\rm
t}_{\rm d}\in\left[1,\rm{T_{\rm d}}\right].$

The linear velocities are scaled and rotated such that they are consistent
with the transformed positions in (II-A). The end-effector orientation
$\boldsymbol{\theta}_{\rm d}(t)$ and angular velocities
$\dot{\boldsymbol{\theta}}_{\rm d}$ are not transformed. Deviations of the
end-effector from the desired trajectory due to external perturbations are
compensated by feedback velocities that attract to the nearest point on that
trajectory. The index $\mathrm{i_{\text{min}}}(t)\in\left[1,\rm{T_{\rm
d}}\right]$ of the closest point from the demonstration to the current
position is determined as

$\displaystyle{\rm i}_{\text{min}}(t)=\arg\min_{{\rm i}\in\left[1,\rm{T_{\rm
d}}\right]}\|\tilde{\mathbf{x}}_{\rm d}^{\rm i}(t)-\mathbf{x}(t)\|.$ (4)

The spatial kd-tree algorithm can be employed to minimize execution time when
finding the closest point [27]. The linear and angular feedback velocities are
given by

$\displaystyle\mathbf{v}_{\rm{fb}}(t)=\boldsymbol{\Lambda}_{\rm
L}(\tilde{\mathbf{x}}_{\rm d}^{\rm i_{\rm{min}}}(t)-\mathbf{x}(t)),$
$\displaystyle\boldsymbol{\omega}_{\rm{fb}}(t)=\boldsymbol{\Lambda}_{\rm
A}(\boldsymbol{\theta}_{\rm d}^{\rm i_{\rm{min}}}(t)-\boldsymbol{\theta}(t)).$
(5)

Here $\boldsymbol{\Lambda}_{\rm L}\in\mathbb{R}^{3\times 3}$ and
$\boldsymbol{\Lambda}_{\rm A}\in\mathbb{R}^{3\times 3}$ are positive definite
feedback gains. These feedback velocities guide the end-effector toward the
desired trajectory, but they do not guarantee tracking of the trajectory.
Therefore, feed-forward velocities are introduced to aid tracking. The linear
and angular feed-forward velocities are

$\displaystyle\mathbf{v}_{\rm{ff}}(t)=\dot{\tilde{\mathbf{x}}}_{\rm d}^{\rm
i_{\rm{min}}}(t),$ $\displaystyle\boldsymbol{\omega}_{\rm
ff}(t)=\dot{\boldsymbol{\theta}}_{\rm d}^{\rm i_{\rm{min}}}(t).$ (6)

In the absence of obstacles, feedback velocities contribute to the convergence
of the end-effector toward the desired trajectories, while the feedforward
velocities facilitate the motion to the desired goal point. However, during
trajectory generation, avoiding obstacles is crucial. To ensure this,
velocities to bypass obstacles are computed as $\mathbf{v}_{\rm
obs}(t)=\mathbf{v}_{\rm r}(t)+\mathbf{v}_{\rm g}(t),$ where $\mathbf{v}_{\rm
r}(t)$ are repulsion velocities from the obstacle, pushing the robot away from
the obstacle, and $\mathbf{v}_{\rm g}(t)$ are guidance velocities, which help
to guide the robot during the obstacle avoidance. To simplify the
calculations, obstacles in this paper are represented by bounding spheres. We
define $\mathbf{n}_{\rm o}(t)\in\mathbb{R}^{3}$ as the normal vector to the
obstacle bounding sphere

$\displaystyle\mathbf{n}_{\rm o}(t)=\frac{\mathbf{x}(t)-\mathbf{x}_{\rm
obs}(t)}{\|\mathbf{x}(t)-\mathbf{x}_{\rm obs}(t)\|},$ (7)

where $\mathbf{x}_{\rm obs}(t)\in\mathbb{R}^{3}$ is the centre of the
obstacle. The repulsion velocity is directed in the normal direction away from
the surface of the bounding sphere

$\displaystyle\mathbf{v}_{\rm r}(t)=\lambda_{\rm
obs}(\mathbf{x},\mathbf{x}_{\rm obs})\mathbf{n}_{\rm o}(t),$ (8)

where $\lambda_{\rm obs}(\mathbf{x},\mathbf{x}_{\rm obs})\in\mathbb{R}^{+}$ is
a state-dependent positive gain that increases in the vicinity of the obstacle

$\displaystyle\lambda_{\rm obs}(\mathbf{x},\mathbf{x}_{\rm obs})$
$\displaystyle=\frac{1}{\left(\|\mathbf{x}(t)-\mathbf{x}_{\rm
obs}(t)\|-r\right)}.$ (9)

Here the radius of the obstacle is denoted as $r\in\mathbb{R}$. The guidance
velocities make it possible to give a preference for the obstacle avoidance
movement by setting a desired avoidance direction

$\displaystyle\mathbf{v}_{\rm g}(t)=\lambda_{\rm
obs}(\mathbf{x},\mathbf{x}_{\rm obs})\mathbf{v}_{\rm proj}(t)k_{g}(t).$ (10)

The vector $\mathbf{v}_{\rm proj}(t)\in\mathbb{R}^{3}$ consists of
$\mathbf{v}_{\rm{ff}}(t)$ and a desired direction vector
$\mathbf{v}_{\rm{dir}}$, which are projected onto the plane orthogonal to
$\mathbf{n}_{\rm o}(t)$

$\displaystyle\mathbf{v}_{\rm proj}(t)$
$\displaystyle=\frac{\mathbf{v}_{\rm{ff}}(t)}{\|\mathbf{v}_{\rm{ff}}(t)\|}-\left\langle\frac{\mathbf{v}_{\rm{ff}}(t)}{\|\mathbf{v}_{\rm{ff}}(t)\|},\mathbf{n}_{\rm
o}(t)\right\rangle\mathbf{n}_{\rm o}(t)$ (11)
$\displaystyle+\mathbf{v}_{\rm{dir}}-\left\langle\mathbf{v}_{\rm{dir}},\mathbf{n}_{\rm
o}(t)\right\rangle\mathbf{n}_{\rm o}(t).$

The dot product of two vectors is denoted as
$\left\langle\cdot,\cdot\right\rangle$. The desired direction vector
$\mathbf{v}_{\rm{dir}}$ is selected manually, e.g. in the direction of the
z-axis of the world frame to avoid the obstacle by moving over it. The
guidance gain $k_{g}(t)\in\\{0,1\\}$ activates the guidance velocity if the
angle between the normal vector to the obstacle bounding sphere and the feed-
forward velocity is bigger than $None$

$\displaystyle k_{g}(t)=\begin{cases}1,&\text{if
}\left\langle\mathbf{v}_{\rm{ff}}(t),\mathbf{n}_{\rm o}(t)\right\rangle\leq
0,\\\ 0,&\text{otherwise}.\end{cases}$ (12)

If the angle between the vectors is less than $None$, the system effectively
navigates past obstacles and onto the desired path by combining feed-forward,
feedback, and obstacle-repulsion velocities. This is facilitated by the
positive feedback mechanism described by (7).

Further, velocity modulation is crucial for harmonizing manipulator behavior
with varying levels of human autonomy, thereby facilitating a smooth
transition of authority and ensuring that the generated reference trajectory
remains within bounds. To address this, we modulate the motion generator and
consider system dynamics as

$\displaystyle\dot{\hat{\mathbf{x}}}_{\rm ref}(t)$
$\displaystyle=(1-\alpha_{\rm h}(t))^{2}\biggl{(}\mathbf{v}_{\rm
fb}(t)+\mathbf{v}_{\rm ff}(t)+\mathbf{v}_{\rm obs}(t)\biggr{)},$ (13)

here $\alpha_{\rm h}(t)\in[0,1]$ is the human authority value, where
$\alpha_{\rm h}=1$ signifies the human operator having a leadership role in
the collaborative task, see section II-B. Velocities are scaled using a
quadratic reduction factor of authority value instead of a linear one to
ensure faster decay of reference velocity. The norm of the velocity is limited
to a manually selected value $v_{\rm th}\in\mathbb{R}^{+}$ by

$\displaystyle\dot{\mathbf{x}}_{\rm
ref}(t)=\begin{cases}\dot{\hat{\mathbf{x}}}_{\rm ref}(t),&\text{if
}\|\dot{\hat{\mathbf{x}}}_{\rm ref}(t)\|\leq v_{\rm th}\\\
\dot{\hat{\mathbf{x}}}_{\rm ref}(t)\frac{v_{\rm
th}}{\|\dot{\hat{\mathbf{x}}}_{\rm ref}(t)\|},&\text{otherwise}.\end{cases}$
(14)

(13) and (14) guarantee that the generated reference velocities are bounded,
thereby enhancing system safety. The combination of feedback, feedforward, and
obstacle avoidance velocities creates the virtual potential field in order to
effectuate the desired motion with attraction to a reference trajectory,
following the reference trajectory with its velocity profile while also
avoiding detected obstacles.

Further outputs from the dynamical system in (1) are the reference angular
velocities and wrenches. The reference angular velocities are determined as
$\dot{\boldsymbol{\theta}}_{\rm ref}(t)=(1-\alpha_{\rm
h}(t))^{2}\dot{\boldsymbol{\theta}}_{\rm d}^{\rm i_{\text{min}}}(t)$. We
define the desired wrench $\mathbf{w}_{\rm ref}(t)\in\mathbb{R}^{6}$ as

$\displaystyle\mathbf{w}_{\rm ref}(t)=\beta(t)\mathbf{W}_{\text{d}}^{\rm
i_{\text{min}}}(t),$ (15)

where $\beta(t)\in\\{0,1\\}$ is given by

Here, $\beta(t)$ ensures that the robot generates force only when the position
errors are below the threshold $w_{\text{th}}\in\mathbb{R}^{+}$ and the robot
has autonomy, i.e. $\alpha_{\rm h}$ is low; otherwise, it remains zero. The
focus now transitions to the computation of authority allocation.

###  II-B Authority allocation between human and robot

The authority allocation between the human operator and the robot allows the
human to take over control of the robot by applying forces to the robot. To
facilitate a smooth transition of control between the tracking controller and
the human operator, we introduce a variable stiffness
$\mathbf{K}(t)\in\mathbb{R}^{6\times 6}$ and a variable damping
$\mathbf{D}(t)\in\mathbb{R}^{6\times 6}$, both of which are functions of the
human authority parameter $\alpha_{\rm h}(t)$. The variable stiffness is
defined as $\mathbf{K}(t)=(1-\alpha_{\rm h}(t))\mathbf{K}_{\rm max}$.
Subsequently, the variable damping is selected as
$\mathbf{D}(t)=2\sqrt{\mathbf{K}(t)}$.

The human authority parameter $\alpha_{\rm h}(t)$ is adjusted based on
external perturbations from humans or the environment

$\displaystyle\hat{\alpha}_{\rm
h}(t)=0.5\left(1+\tanh\left(\frac{3}{a}\left({\rm w}_{\rm
diff}-a-b\right)\right)\right).$ (17)

Figure 3:  Parameter influence on $\hat{\alpha}_{\text{h}}$ calculation with
$b=2$ setting the deadband for $\hat{\alpha}_{\text{h}}$, and $a$ setting the
slope of the $\hat{\alpha}_{\text{h}}$ transition to the maximum of 1.

Here, $a>0$ modulates the slope of the transition function and $b>0$
determines the width of the deadband. The perturbations are considered as the
norm of wrench difference ${\rm w}_{\rm diff}=c_{1}\|\mathbf{w}_{\rm
s}(t)-\mathbf{w}_{\rm est}(t)\|+c_{2}\|\mathbf{w}_{\rm ref}(t)-\mathbf{w}_{\rm
s}(t)\|$ between the reference wrench $\mathbf{w}_{\rm ref}(t)$, the measured
wrench $\mathbf{w}_{\rm s}(t)\in\mathbb{R}^{6}$ from a force-torque sensor,
and the wrench $\mathbf{w}_{\rm est}(t)\in\mathbb{R}^{6}$ estimated based on
joint torque measurements. The wrench difference ${\rm w}_{\rm
diff}\in\mathbb{R}^{+}$ is separated into two parts with respective scaling
factors $c_{1}\&c_{2}>0$. The first part of ${\rm w}_{\rm diff}$ allows for
the integration of wrenches applied to the robot that are not captured by the
sensor while the second part considers the error of the measured wrench
$\mathbf{w}_{\rm s}(t)$ compared to the reference wrench $\mathbf{w}_{\rm
ref}(t)$. This two-part solution not only enables the user to touch anywhere
on the robot to take over the authority but also allows for the lowering of
the robot’s authority, thereby increasing its compliance when wrenches are
applied due to accidental contact or human intervention. Finetuning the
sensitivity of $\hat{\alpha}_{\rm h}(t)$ to different kinds of external forces
is achievable through the two scaling factors, e.g. making the authority
allocation less responsive to force errors applied tool-side of the sensor by
selecting $c_{1}>c_{2}$, which can enhance the force production capabilities.
The parameters $a$, $b$, $c_{1}$, and $c_{2}$ are empirically chosen design
parameters of the authority arbitrator.

A simplified illustration of this function and its parameters is shown in Fig.
3. At every time step $\alpha_{\rm h}(t)$ is updated based on the computed
$\hat{\alpha}_{\rm h}(t)$ value using a recursive filter formulation, which
facilitates a smoother transition of $\alpha_{\rm h}$ values

$\displaystyle\alpha_{\rm h,{i}}=\alpha_{\rm
h,{i-1}}+g_{\alpha}\left(\hat{\alpha}_{\rm h}-\alpha_{\rm h,{i-1}}\right),$
(18)

where $\left(\cdot\right)_{\rm i}$ denotes the i-th time step and
$g_{\alpha}(t)\in\left[0,1\right]$ is the update gain

$\displaystyle g_{\alpha}(t)=\begin{cases}g^{\rm+},&\text{if
}\hat{\alpha}_{\rm h}>\alpha_{\rm h,{i-1}},\\\
g^{\rm-}+g^{\rm-}\left(1-\alpha_{\rm
h,{i-1}}\right)^{2},&\text{otherwise}.\end{cases}$ (19)

By setting $g^{\rm+}$ higher than $g^{\rm-}$ a faster increase rate than
decrease rate of $\alpha_{\rm h}$ can be achieved. These parameters as well as
the previously mentioned $a$, $b$, $c_{1}$, and $c_{2}$ values were manually
tuned when implementing the approach on the robot. The time-varying impedance
parameters may induce instabilities in the closed-loop system therefore we
design our closed-loop controller using the energy tank approach of passivity
theory [23, 24].

###  II-C Variable Impedance and Force Controller

We consider an n degrees of freedom manipulator’s interaction in task-space
[28], where the dynamics are characterized by

$\displaystyle\mathbf{M}_{\rm
x}(\mathbf{q})\ddot{\boldsymbol{\xi}}(t)+\mathbf{C}_{\rm
x}(\mathbf{q},\dot{\mathbf{q}},{\mathbf{v}_{n}})\dot{\boldsymbol{\xi}}(t)+\mathbf{g}_{\rm
x}(\mathbf{q})$ $\displaystyle=\mathbf{w}(t)+\mathbf{w}_{\mathrm{env}}(t).$
(20)

Here, $\mathbf{q}(t)\in\mathbb{R}^{n}$ is the vector of joint angles,
$\boldsymbol{\xi}(t)\in\mathbb{R}^{6}$ is the Cartesian pose of the
manipulator, ${\mathbf{v}_{n}}$ are null-space velocities, $\mathbf{M}_{\rm
x}(\mathbf{q})\in\mathbb{R}^{6\times 6}$ is the inertia matrix,
$\mathbf{C}_{\rm x}(\mathbf{q},\dot{\mathbf{q}})\in\mathbb{R}^{6\times 6}$
represents the Coriolis and centripetal forces, $\mathbf{g}_{\rm
x}(\mathbf{q})\in\mathbb{R}^{6}$ is the vector of conservative wrench,
$\mathbf{w}(t)\in\mathbb{R}^{6}$ is the control input to the manipulator, and
$\mathbf{w}_{\mathrm{env}}(t)\in\mathbb{R}^{6}$ represents wrench from
interactions with the environment. We define the control law as

$\displaystyle\mathbf{w}(t)$ $\displaystyle=\mathbf{M}_{\rm
x}(\mathbf{q})\ddot{\boldsymbol{\xi}}_{\rm ref}(t)+\mathbf{C}_{\rm
x}(\mathbf{q},\dot{\mathbf{q}},{\mathbf{v}_{n}})\dot{\boldsymbol{\xi}}_{\rm
ref}(t)+\mathbf{g}_{\rm x}(\mathbf{q})$ (21)
$\displaystyle-\mathbf{\bar{K}}\mathbf{e}(t)-\mathbf{\bar{D}}\dot{\mathbf{e}}(t)+\mathbf{u}(t),$

where $\mathbf{\bar{K}}\in\mathbb{R}^{6\times 6}$ is positive definite
stiffness, $\mathbf{\bar{D}}\in\mathbb{R}^{6\times 6}$ is positive definite
damping, $\mathbf{e}(t)=\boldsymbol{\xi}(t)-\boldsymbol{\xi}_{\rm ref}(t)$ is
the tracking error and $\mathbf{u}(t)\in\mathbb{R}^{6}$ is an additional
control input with time-varying impedance and wrench parameters. The motion
generator, governed by (13), generates desired reference trajectories
utilizing feedback from the manipulator’s current states. In the absence of
the control input $\mathbf{u}(t)$, the robot demonstrates high compliance to
human-induced perturbations and deviates from the reference trajectory and
wrench, a result of choosing small values for $\mathbf{\bar{K}}$ and setting
$\mathbf{\bar{D}}=2\sqrt{\mathbf{\bar{K}}}$. Conversely, $\mathbf{u}(t)$
enables the integration of the authority level for trajectory and force
tracking with variable impedance, as delineated in Section II-B. The control
input is inspired by [23, 24, 25] and defined as

$\displaystyle\mathbf{u}(t)=\zeta\Bigl{(}-\mathbf{K}(t)\mathbf{e}-\mathbf{D}(t)\dot{\mathbf{e}}+(1-\phi)\mathbf{w}_{\rm
ref}(t)$ $\displaystyle+\mathbf{K}_{\rm w}(\mathbf{w}_{\rm
env}(t)-\mathbf{w}_{\rm ref}(t))$ $\displaystyle+\mathbf{K}_{\rm
i}\int_{0}^{t}(\mathbf{w}_{\rm env}(\delta)-\mathbf{w}_{\rm
ref}(\delta))d\delta\Bigr{)}+\phi\mathbf{w}_{\rm ref}(t)$ (22)

where $\mathbf{K}(t)=(1-\alpha_{\rm h}(t))\mathbf{K}_{\rm max}$ is variable
stiffness, $\mathbf{D}(t)=2\sqrt{\mathbf{K}(t)}$ is variable damping, and
$\mathbf{K}_{\rm i}\hskip 2.84544pt\&\hskip 2.84544pt\mathbf{K}_{\rm
w}\in\mathbb{R}^{6\times 6}$ are positive-definite proportional and integral
gain for wrench error. $\zeta\in\\{0,1\\}$ and $\phi\in\\{0,1\\}$ are
associated with the energy tank, which will be explained in more detail in the
following. The tank energy is defined as
$\psi_{\mathrm{energy}}(t)=\frac{1}{2}s^{2}(t)$, where $s\in\mathbb{R}$ is the
state associated with energy-tank. For passivity analysis, we consider the
storage function as

$\displaystyle v=\frac{1}{2}\dot{\mathbf{e}}^{\top}\mathbf{M}_{\rm
x}\dot{\mathbf{e}}+\frac{1}{2}{\mathbf{e}}^{\intercal}\bar{\mathbf{K}}{\mathbf{e}}+\frac{1}{2}s^{2}.$
(23)

Utilizing the skew-symmetric property
$\frac{1}{2}\dot{\mathbf{e}}^{\intercal}(\dot{\mathbf{M}}_{x}-2\mathbf{C}_{x})\dot{\mathbf{e}}=0$,
the derivative $\dot{v}$ becomes

$\displaystyle\dot{v}=-\dot{\mathbf{e}}^{\intercal}\bar{\mathbf{D}}\dot{\mathbf{e}}+\dot{\mathbf{e}}^{\intercal}\mathbf{w}_{\mathrm{env}}+\dot{\mathbf{e}}^{\intercal}\mathbf{u}+s\dot{s}.$
(24)

Herein, the terms $\dot{\mathbf{e}}^{\intercal}\mathbf{w}_{\mathrm{env}}$ and
$\dot{\mathbf{e}}^{\intercal}\mathbf{u}$ are observed to be sign-indefinite.
By invoking Equation (II-C), $\dot{s}$ is formulated as

$\displaystyle\dot{s}$
$\displaystyle=\frac{\gamma}{s}(\dot{\mathbf{e}}^{\intercal}\bar{\mathbf{D}}\dot{\mathbf{e}}-\phi\dot{\mathbf{e}}^{\intercal}\mathbf{w}_{\rm
ref}(t))-\frac{\zeta}{s}\biggl{(}-\dot{\mathbf{e}}^{\intercal}\mathbf{K}(t)\mathbf{e}-\dot{\mathbf{e}}^{\intercal}\mathbf{D}(t)\dot{\mathbf{e}}$
$\displaystyle+(1-\phi)\dot{\mathbf{e}}^{\intercal}\mathbf{w}_{\rm
ref}(t)+\dot{\mathbf{e}}^{\intercal}\mathbf{K}_{\rm w}(\mathbf{w}_{\rm
env}(t)-\mathbf{w}_{\rm ref}(t))$
$\displaystyle+\dot{\mathbf{e}}^{\intercal}\mathbf{K}_{\rm
i}\int_{0}^{t}(\mathbf{w}_{\rm env}(\delta)-\mathbf{w}_{\rm
ref}(\delta))d\delta\biggr{)},$ (25)

where

$\displaystyle\gamma(t)=\begin{cases}1,&\text{if
}\psi_{\mathrm{energy}}(t)\leq\overline{\psi}_{\mathrm{threshold}}\\\
0,&\text{otherwise}\end{cases}$ (26)

with $\overline{\psi}_{\mathrm{threshold}}\in\mathbb{R}^{+}$ serving as an
application-dependent upper bound to preclude excessive energy accumulation in
the reservoir.

Figure 4: Comparison of the Proposed Algorithm with SEDS and LPV-DS on the
LASA handwriting dataset.

To avert singularities, it is required that $s(0)>0$. The parameter $\zeta$,
controlling the extraction of energy due to non-passive actions, is defined as

$\displaystyle\zeta(t)=\begin{cases}1,&\text{if
}\psi_{\mathrm{energy}}(t)\geq\underline{\psi}_{\mathrm{threshold}}\\\
0,&\text{otherwise},\end{cases}$ (27)

where $\underline{\psi}_{\mathrm{threshold}}\in\mathbb{R}^{+}$ constitutes a
positive lower energy boundary to ensure that the energy tank is not fully
depleted. Additionally, we use dissipative reference wrench to quickly refill
the energy tank. The parameter $\phi$, which modulates desired wrench effects,
is set as follows:

$\displaystyle\phi(t)=\begin{cases}1,&\text{if
}\dot{e}^{\intercal}\mathbf{w}_{\text{ref}}(t)<0\\\
0,&\text{otherwise},\end{cases}$ (28)

This setting allows the controller to keep applying feed-forward wrench even
if the energy tank is empty, as shown in (II-C). Substituting (II-C) in (24)
and as $\gamma,\zeta,\phi\in\\{0,1\\}$ we get
$\dot{v}\leq\dot{e}^{\intercal}\mathbf{w}_{\mathrm{env}}$, which is the
condition required for passivity for pair
($\dot{e},\mathbf{w}_{\mathrm{env}}$). The parameters of the variable
impedance controller and the energy tank are empirically chosen.

##  III Simulations and Experiments

To assess the efficacy of the proposed methodology, we conducted a comparative
evaluation against SEDS and LPV-DS, utilizing the leaf shape from the LASA
handwriting dataset, as depicted in Fig. 4. While both SEDS and LPV-DS
succeeded in reaching the endpoint, they failed to accurately replicate the
original demonstration. Although LPV-DS exhibited lower tracking error
compared to SEDS, it did not faithfully reproduce the original demonstration.

In Fig. 4., the red lines represent the original demonstrations, the black
lines with arrows indicate the streamline of the velocity field, and the solid
black lines denote the reproduced demonstrations. For our proposed approach,
we employed the mean trajectory of the demonstrations, highlighted by the
green dashed line in Fig. 4. However, it is important to note that the
proposed methodology necessitates only a single demonstration, selected from
multiple demonstrations based on optimal performance criteria. Our results
demonstrate that points originating away from the demonstration converge to
and subsequently follow this mean trajectory, ultimately reaching the goal.
This validates the capability of the proposed approach to encode complex
shapes, that do not monotonically decrease in distance to the target over
time. Similarly the proposed algorithm can be successfully deployed on the
other shapes of the LASA handwriting dataset. Additionally, we model the
obstacle avoidance scenario, as depicted in Fig. 5. In this case, the guidance
vectors align with the positive y and z axes, directing the desired
trajectories around the obstacle and ultimately converging to the goal point.

Figure 5: Demonstration of obstacle avoidance using the proposed approach,
following the desired guidance direction.

The efficacy of the proposed approach is further validated through two
distinct experimental studies. The first experiment focuses on a button-
pressing task, highlighting the approach’s proficiency in precise force
generation and responsiveness to human perturbations. The second experiment
evaluates the robot’s capability to replicate complex movements on the real
robot, akin to those depicted in Fig. 4, while ensuring safety during
collaborative task execution. Real-time obstacle avoidance is also verified in
the second experiment. A Franka Emika Research 3 robotic arm is employed in
both studies. We use a 6-axis force-torque sensor from AIDIN ROBOTICS Inc.
mounted at the robot’s end-effector to measure interactions with the
environment.

###  III-A Force production and reactivity to human perturbation

As described earlier, many tasks performed in contact with the environment
require accurate force production. In this experiment, we investigate the
ability of the proposed approach to reproduce the motion and interaction
forces required to perform a task that is taught using kinesthetic teaching.
Furthermore, it is important for the human to be able to take control of the
robot during the collaboration and that the robot’s motion is subsequently
adapted. To demonstrate these characteristics, we chose the scenario of
pressing a button where the human diverts the robot from the desired motion by
repeatedly taking over autonomy, as illustrated in Fig. 1 (right). The human
moves the robot’s end-effector to different positions, from which the robot
takes back control when the human stops applying forces. Thereafter the end-
effector moves according to the previously defined attraction and feed-forward
velocities, and finally converges to the target point (button) and presses it.
Fig. 6 shows the 3D path of the end-effector during the demonstration and the
reproduction of the motion using the proposed approach.

By applying forces the human increases the authority allocation value
according to (17) - (19), which leads to the robot assuming a more compliant
role. Fig. 7 shows the absolute desired, measured, and estimated end-effector
forces, as well as the human autonomy value $\alpha_{\rm h}(t)$. The emergency
stop button chosen for this task requires approximately 15 $\mathrm{N}$ to be
pressed, which is encoded during the demonstration.

Figure 6: 3D paths of the robot’s end-effector position from demonstration
(blue) and reproductions (red) of a button pressing task.

As the measured force in Fig. 7 shows, the force required to trigger the
button is reached during reproduction of the task. After the button is pressed
(starting at 34.3 $\mathrm{s}$ in Fig. 7), the reference force decreases
slightly and the controller starts tracking the new desired reference force.
During the force production period the maximum force error between desired and
measured force in the direction of button press of 1.313 $\mathrm{N}$ was
observed. After approximately 2.5 $\mathrm{s}$ the force error reduces to a
steady state tracking error of 0.142 $\mathrm{N}$ due to the effects of the
integral term in (II-C). Despite the control parameter not being optimized for
this specific task prior to the experiments, this highlights the force
production capabilities of the proposed approach. Furthermore Fig. 7 shows
starting at 44.2 $\mathrm{s}$ exemplary how the human applies forces to the
robot’s body (flange side of the FT sensor) is detected by the force
estimation. Due to the mismatch of FT sensor measured forces and estimated
forces, $\alpha_{\rm h}(t)$ increases with an approximate delay of
approximately 150 $\mathrm{ms}$. The human can successfully take over control
of the robot, which transitions to lower impedance.

###  III-B Encoding of complex closed-loop trajectories

In the previous experiment, we already show that with the proposed approach
the human is capable of taking over autonomy over the task by applying force
on the robot. Therefore, in this experiment, we highlight the framework’s
ability to encode complex and closed-loop motion, avoid detected obstacles,
and illustrate the workings of the energy tank. The task at hand deals with
the polishing of an object with a complex shape, see Fig. 1 (left). The shape
encoded in this experiment is similar to the leaf shape of the LASA
handwriting dataset, shown earlier in Fig. 4. The robot’s end-effector
position during demonstration, reproduction, and obstacle avoidance are shown
in Fig. 8. The encoded, counter-clockwise motion can be reproduced in a
closed-loop manner with the proposed approach.

Figure 7: Reproduction of the button pressing task where the human uses shared
autonomy to reposition the end-effector. The autonomy value $\alpha_{\rm h}$
(orange) is calculated based on the desired $\mathbf{f}_{\rm ref}$ (green),
the measured $\mathbf{f}_{\rm s}$ (blue), and the estimated $\mathbf{f}_{\rm
est}$ (red) end-effector forces. Figure 8: 3D paths of the robot’s end-
effector position from demonstration (green), reproduction (blue), and
reproduction while avoiding an obstacle (red). Figure 9: Tank energy
$\psi_{\mathrm{energy}}$ during human robot collaboration experiment. The
energy level decreases until the lower tank energy threshold
$\underline{\psi}_{\mathrm{threshold}}=10\ $\mathrm{J}$$ is reached.

Additionally, in case of a detected obstacle, the motion generator is capable
of avoiding the obstacle, passing it, and transitioning back to the
reproduction of the task. Here ArUco markers [29] placed on the object were
used for obstacle detection and the obstacle radius was selected empirically.
When the energy tank reaches its lower threshold, the robot switches to low
impedance mode and ceases to track the desired motion, a state also achieved
at the end of obstacle-avoiding motion.

As shown in (21) - (28), the implemented energy tank-based variable impedance
control is able to ensure passivity of the controller. Fig. 9 illustrates how
the tank energy depletes over time while reproducing the encoded motion of
this experiment.

At 55 $\mathrm{s}$ the energy in the tank reaches the lower limit and the
respective parts of the controller, i.e. variable stiffness and damping term
as well as the non-dissipative terms responsible for force production, are
turned off. The robot is in a state of low impedance.

##  IV Conclusion

This study introduces a framework for shared autonomy, incorporating several
key components. Initially, a time-invariant, state-dependent motion generator
based on potential fields is presented. This approach encodes both complex and
closed-loop task-specific trajectories and wrench profiles, while concurrently
facilitating obstacle avoidance. Subsequently, the framework provides a
methodology for dynamic and smooth authority transitions between human
operators and robotic systems, employing variable impedance and force control
mechanisms to ensure precise task execution and reactiveness to human-induced
perturbations. Additionally, the framework integrates an energy-tank-based
passivation strategy to maintain system passivity under time-varying
parameters. The efficacy of the proposed framework is validated through
experimental and simulation-based evaluations. Future research endeavors will
concentrate on the incremental refinement of robotic skills in dynamic
environments and the predictive modeling of human intentions to optimize
authority arbitration.

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