# Anisotropy in constraint 4D Gauss-Bonnet gravity

Shahab Shahidiα<EMAIL_ADDRESS>Nima Khosravi${}^{\beta},^{\gamma}$
<EMAIL_ADDRESS>αSchool of Physics, Damghan University, Damghan,
41167-36716, Iran.  βDepartment of Physics, Shahid Beheshti University,
1983969411, Tehran, Iran,
γSchool of Physics, Institute for Research in Fundamental Sciences (IPM), P.
O. Box 19395-5531, Tehran, Iran

###### Abstract

Recently a new 4D Einstein-Gauss-Bonnet theory has been introduced [Phys. Rev.
Lett. 124 (2020) 081301] with a serious debate that it does not possess a
covariant equation of motion in $4D$. This feature, makes impossible to
consider non-symetric space-times in this model, such as anisotropic
cosmology. In this note, we will present a new proposal to make this happen,
by introducing a Lagrange multiplier to the action which eliminates the higher
dimensional term from the equation of motion. The theory has then a covariant
$4D$ equation of motion which is useful to study the less symmetric metrics.
On top of FRW universe, the constraint theory is equivalent to the original
$4D$ Einstein-Gauss-Bonnet gravity. We will then consider the anisotropic
cosmology of the model and compare the theory with observational data. We will
see that the theory becomes non-conservative and the matter density abundance
falls more rapidly at larger redshifts compared to the conservative matter
sources.

###### pacs:

04.20.Cv; 04.50.Gh; 04.50.-h; 04.60.Bc

##  I Introduction

The Einstein’s theory of general relativity has witnessed serious
observational and theoretical challenges since its birth in 1915 as a way to
describe the gravitational interaction. At the beginning, the theory has been
challenged from reconciling the common sense with the predictions of the
theory and afterward from solar system/local tests to cosmological/large scale
experiments. However, one of the most controversial issued about the theory is
to correctly describing the late time accelerated expansion of the universe.
The first proposal was given by Einstein himself, considering the cosmological
constant to the original Einstein-Hilbert action cosmologicalconstantreview .
This idea together with the inclusion of cold dark matter to elucidate local
observations, is now well-known as the $\Lambda$CDM model

$\displaystyle S=\int d^{4}x\sqrt{-g}(R-2\Lambda)+S_{m},$ (1)

where $\Lambda$ is the cosmological constant, and $S_{m}$ is the action for
all baryonic matter fields plus dark matter.

Nowadays, we believe that the cosmological constant, suffers from
theoretical/phenomenological problems cosmologicalconstantreview , tempting
cosmologists to consider other options, generally known as dark energy models.
The simplest possibility of adding a dark energy to the Einstein’s theory is
to promote the cosmological constant to a dynamical field. This field could be
a scalar, vector or in general some higher spin field scalarvectortheories .
However, one can consider some modifications of the Einstein-Hilbert action
itself which could involve higher dimensions higherdimension , granting mass
to the mass-less graviton massivegravity , considering richer geometries
weylcartan , or changing the gravitational interactions by modifying the Ricci
scalar in the Einstein-Hilbert action fRreview . All of these ideas are
generally known as modified theories of gravity.

Changing Ricci scalar in the action could be done in numerous ways. The
simplest way is to substitute the Ricci scalar with an arbitrary function of
$R$, resulting in a so-called $f(R)$ theory of gravity fRreview . The theory
can be easily proven to be equivalent to a specific class of Brans-Dicke
theory and hence it is free of Ostrogradski ghost. As a result it can be
safely considered as a geometrical candidate for dark energy. Another
possibility to generalize the Ricci scalar in the Einstein-Hilbert action is
to add some higher order terms constructed from Ricci and Riemann tensors. In
general these higher order terms possess extra unhealthy degrees of freedom
which will eradicate the validity of the theory. It is well-known that the
only terms which cancels the ghost degrees of freedom would be obtained from
the Lovelock prescription lovelock . However, Lovelock terms are all vanished
in four dimensional space-times and as a result only the Ricci scalar remains
as a healthy candidate to describe gravitational interactions. The first non-
trivial Lovelock term is the Gauss-Bonnet invariant which is non-zero for
$D>4$, where $D$ is the space-time dimension. An interesting idea to put the
effects of this term in the theory is to assume that the universe is
fundamentally five dimensional. Then we will project the extra dimension to
our ordinary four dimensional space-time and the effect of Gauss-Bonnet term
will appear in our $4D$ universe brane .

Recently, an idea appears in the literature, presenting a new way to make some
non-vanishing effects of the Gauss-Bonnet Lagrangian in $4D$ 4Dfirst . The
idea comes from the fact that the vanishing of the Gauss-Bonnet invariant in
$4D$ is due to the presence of an overal factor $D-4$. The author then claimed
that if one considers a $D$-dimensional abstract space-time and re-scales the
Gauss-Bonnet parameter $\alpha$ to $\alpha/(D-4)$, one then obtain some non-
vanishing contributions of the Gauss-Bonnet invariant in $4D$ by taking the
limit $D\rightarrow 4$. The consequences of the theory is widely investigated
in the literature, exploring its effects in black-holes blackholes , wormholes
wormholes , compact stars compactstars  and also in cosmology othercosmology .
Also, many works has been done to generalize the idea to higher/lower
dimensions, relations to other theories and also its quantum aspects of the
theory other works . For example, in cao , the authors showed that in order to
have a well-defined linearized theory the metric should be locally conformally
flat. Also, many works have been done to constraint the new re-scaled Gauss-
Bonnet parameter from observational data constraintonparameter . In mota  the
authors obtained the so far strongest constraint on the theory parameter,
namely, $(\alpha=2.69\pm 11.67)\times 10^{48}eV^{-2}$.

However, the most important issue related to the $4D$ Einstein-Gauss-Bonnet
theory is that despite the fact that the Lagrangian has an overall factor
$D-4$, this is not the case for the equation of motion. As have been shown in
the literature comment  and we will also review in the next section, the
equation of motion can be decomposed to two separate parts, one has the $D-4$
factor and the other which is proportional to the Weyl tensor, does not has
the factor. As a result, for a generic non-symmetric space-time, the second
part will make the equation of motion divergent. Also, it has been pointed out
that for non-symmetric space-times, there is no canonical way to define a
higher dimensional metric, and the theory seems ambiguous for generic space-
times and the limiting process is not well-defined. As a result, the theory
does not have a covariant $4D$ equation of motion.

There are some attempts in the literature to resolve these problems, for
example to couple the theory to a scalar field comment . In this paper, we put
forward a new idea to deal with the above issues. As we have seen above, the
main difficulty for having a covariant equation of motion is that the Gauss-
Bonnet term has a term without the factor $D-4$. As a result in the limiting
process this term causes problems for generic space-times. We will covariantly
set this part to zero from the action by imposing a Lagrange multiplier. This
will allow us to have a covariant $4D$ equation of motion and the limiting
process would be unambiguous.

In this paper, we will show that the isotropic cosmology of the new model
would be identical to the original theory. We will then consider the
anisotropic universe in this model. It should be noted that the same process
could not be done in the original theory since, as we discussed above, there
is an ambiguity in defining the higher dimensional metric. In our new
framework, however, there is no need for higher dimensional metric since we
have a covariant 4D equation of motion.

##  II The model

Let us start with the 4D Gauss-Bonnet gravity theory introduced in 4Dfirst .
The action in $D$ dimensions can be written as

$\displaystyle S=\int$ $\displaystyle
d^{D}x\sqrt{-g}\bigg{[}\kappa^{2}(R-2\Lambda)+\mathcal{L}_{m}+\beta\,\mathcal{G}\bigg{]},\
$ (2)

where $\Lambda$ is the cosmological constant, $\mathcal{L}_{m}$ is the matter
Lagrangian and $\mathcal{G}$ is the Gauss-Bonnet term defined as

$\displaystyle\mathcal{G}=R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}-4R^{\mu\nu}R_{\mu\nu}+R^{2}.$
(3)

The 4D Gauss-Bonnet recipe 4Dfirst  is to rescale the Gauss-Bonnet coupling
constant $\beta$ to $\alpha/(D-4)$, so that the Gauss-Bonnet term acquire a
non-vanishing part in the limit $D\rightarrow 4$.

The equation of motion of the metric field in $D$ dimensions can be obtained
from the action (2) as

$\displaystyle\kappa^{2}$ $\displaystyle\left(G_{\mu\nu}+\Lambda
g_{\mu\nu}\right)+\frac{\alpha}{D-4}\mathcal{G}_{\mu\nu}=\frac{1}{2}T_{\mu\nu},$
(4)

where we have defined the Gauss-Bonnet tensor

$\displaystyle\mathcal{G}_{\mu\nu}$
$\displaystyle=2RR_{\mu\nu}-4R_{\mu\alpha}R^{\alpha}_{\nu}-4R_{\mu\alpha\nu\beta}R^{\alpha\beta}$
$\displaystyle+2R_{\mu\alpha\beta\sigma}R_{\nu}^{~{}\alpha\beta\sigma}-\frac{1}{2}g_{\mu\nu}\mathcal{G}.$
(5)

Here, $T_{\mu\nu}$ is the energy-momentum tensor defined as

$\displaystyle
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}}.$ (6)

In order to have a covariant $4D$ equation of motion, it is needed that the
tensor $\mathcal{G}_{\mu\nu}$ has an overall factor $D-4$. In fact it can be
easily verified that the trace of the Gauss-Bonnet tensor has the above
property and as a result the equation of motion (4) in $D$ dimensions is

$\displaystyle\kappa^{2}(D-2)R-2D\kappa^{2}\Lambda=-T-\alpha\mathcal{G}.$ (7)

However, in general, this is not the case for the tensor
$\mathcal{G}_{\mu\nu}$. Specifically, one can decompose the Gauss-Bonnet part
$\mathcal{G}_{\mu\nu}$ comment  as

$\displaystyle\mathcal{G}_{\mu\nu}=C_{\mu\nu}+(D-4)S_{\mu\nu},$ (8)

where

$\displaystyle
C_{\mu\nu}=2W_{\mu\alpha\beta\gamma}W_{\nu}^{~{}\alpha\beta\gamma}-\frac{1}{2}W_{\alpha\beta\gamma\delta}W^{\alpha\beta\gamma\delta}g_{\mu\nu},$
(9)

is the Lanczos-Bach tensor where $W_{\alpha\beta\gamma\delta}$ is the Weyl
tensor, and

$\displaystyle S_{\mu\nu}$
$\displaystyle=\frac{2}{(D-2)^{2}}\Big{[}DR\,R_{\mu\nu}-2(D-2)R_{\mu\alpha\nu\beta}R^{\alpha\beta}$
$\displaystyle+(D-1)g_{\mu\nu}\Big{(}R_{\alpha\beta}R^{\alpha\beta}+\frac{D^{2}-D+2}{4(D-1)^{2}}R^{2}\Big{)}$
$\displaystyle-2(D-1)R_{\mu\alpha}R_{\nu}^{~{}\alpha}\Big{]}.$ (10)

It is then obvious that in the limit $D\rightarrow 4$ of the equation of
motion (4), the term $S_{\mu\nu}$ remains finite and gives us a non-trivial
contribution of the Gauss-Bonnet Lagrangian in $4D$. However, the term
$C_{\mu\nu}$ in the field equation (4) which is proportional to the Weyl
tensor does not have a $(D-4)$ factor and is in general non-vanising. As a
result the new factor $1/(D-4)$ could not be canceled in this term and the
equation of motion becomes infinite. So, there is no well-defined $4D$
covariant equation of motion of the Gauss-Bonnet theory in this prescription.

It should be noted that due to the Bach-Lanczos theorem, the very expression
in (9) identically vanishes in $4D$ space-times. So, the vanishing of the
Gauss-Bonnet tensor in $4D$ has two steps. One part becomes identically zero
and the other part has an overall factor $D-4$ rendering its vanishing in
$4D$.

In the new approach presented above, we start with a higher dimensional
metric, where the tensor $C_{\mu\nu}$ is non-vanishing in general, and then
tend $D\rightarrow 4$. This causes a two-fold problem. Firstly, we are only
interested in the $4D$ space-times with $4D$ metric field. But there is no
canonical way to define a higher dimensional metric and as a result, taking
the limit $D\rightarrow 4$ is not unique. This problem gets worth when the
$4D$ metric is asymmetric, like Bianchi space-times. Here, there is no
preferred way to define higher dimensional metric.

Secondly, the limit $D\rightarrow 4$ is not continuous. This can be seen by
considering the fact that the tensor $C_{\mu\nu}$ is identically zero in $4D$
but it is not in higher dimensions. So we got $\frac{0}{0}$ when we naively
take the limit $D\rightarrow 4$ and as a result we can not obtain a generally
convariant $4D$ equation of motion for Gauss-Bonnet theory.

It should be noted that the above problems seems unimportant for symmetric
spacetimes, i.e. when one considers FRW cosmology, blackholes, compact stars,
even scalar perturbations around FRW space times, since there is a canonical
way to define a higher dimensional metric and for that metric the tensor
$C_{\mu\nu}$ vanishes. So, there is no degeneracy in taking the limit
$D\rightarrow 4$ of the field equation (4). However, this does not imply that
the theory has a well-defined covariant equation of motion in $4D$.

It should be mentioned that there is another possibility that we set
$C_{\mu\nu}=0$ by hand and only consider the $S_{\mu\nu}$ part of the Gauss-
Bonnet term. This solves the problems introduced above, but results in the
non-conservation of the energy-momentum tensor. More precisely, in the case of
vanishing $C_{\mu\nu}$, the covariant divergence of the field equation (4)
gives

$\displaystyle\nabla_{\alpha}T^{\alpha}_{~{}\nu}=\alpha\bigg{[}$
$\displaystyle\nabla_{\nu}\left(R_{\alpha\beta}R^{\alpha\beta}-\frac{1}{6}R^{2}\right)+R_{\nu\alpha}\nabla^{\alpha}R$
$\displaystyle-2R^{\alpha\beta}\nabla_{\beta}R_{\nu\alpha}+4R_{\nu\alpha\beta\gamma}\nabla^{\gamma}R^{\alpha\beta}\bigg{]},$
(11)

which is non-zero. Here, a note about this equation is in order. As is well-
known, the covariant divergence of any covariant theory is conserved due to
the Noether theorem. The covariant divergence of all terms in the LHS of
equation (4) (in $D\neq 4$) are identically zero, which implies that the
matter energy-momentum tensor becomes conserved. As we have seen in equation
(8), the Guass-Bonnet term could be decomposed into two terms where the
covariant divergence of the tensors $S_{\mu\nu}$ and $C_{\mu\nu}$ are not
independently zero. In the $4D$ Gauss-Bonnet theory, the tensor $C_{\mu\nu}$
set to be zero by the limiting process. However, since the covariant
derivative of the tensor $C_{\mu\nu}$ is not vanishing, the non-conservative
nature of the tensor $S_{\mu\nu}$ remains in the theory which is the RHS of
equation (II). In fact, we have used the conservation equation of the metric
field equation to obtain equation (II) which is in agreement with Noether’s
theorem. We should note that if we insist that the matter is conserved in this
theory, we have to constrain the tensor $S_{\mu\nu}$ to be independently
conserved (which is equation (11) without the LHS).

In this paper, we put forward the above argument by imposing the vanishing of
the tensor $C_{\mu\nu}$ to the action (2) through a Lagrange multiplier. The
new constrained action becomes

$\displaystyle S=\int d^{D}x\sqrt{-g}\bigg{(}$
$\displaystyle\kappa^{2}(R-2\Lambda)+\mathcal{L}_{m}$
$\displaystyle+\frac{\alpha}{D-4}\mathcal{G}+\frac{1}{2}\lambda^{\mu\nu}C_{\mu\nu}\bigg{)},$
(12)

where the symmetric tensor $\lambda_{\mu\nu}$ is a Lagrange multiplier. The
procedure here is the same as 4Dfirst ; after obtaining the field equations,
we perform the limit $D\rightarrow 4$ to obtain a $4D$ covariant equation of
motion.

The variation should be performed with respect to the metric and the Lagrange
multiplier $\lambda_{\mu\nu}$. As a result the theory would have two equations
of motions for $g_{\mu\nu}$ and $\lambda_{\mu\nu}$ which can be used to
determine all the variables. In our case, the variation of the action (II)
with respect to $\lambda_{\mu\nu}$ gives $C_{\mu\nu}=0$. We then consider the
limit $D\rightarrow 4$ of this equation. However, $C_{\mu\nu}$ identically
vanishes in $4D$ and there is no equation of motion for the Lagrange
multiplier in $4D$. As a result in this model, the Lagrange multiplier would
become an arbitrary tensor which could be determined by physical
considerations.

The equation of motion of the metric tensor after taking the limit becomes

$\displaystyle\kappa^{2}(G_{\mu\nu}+\Lambda g_{\mu\nu})+\alpha
s_{\mu\nu}+L_{\mu\nu}=\frac{1}{2}T_{\mu\nu},$ (13)

where we have defined

$\displaystyle s_{\mu\nu}$ $\displaystyle=\lim_{D\rightarrow 4}S_{\mu\nu}$
$\displaystyle=\frac{1}{2}\Big{[}4R\,R_{\mu\nu}-4R_{\mu\alpha\nu\beta}R^{\alpha\beta}$
$\displaystyle+3g_{\mu\nu}\Big{(}R_{\alpha\beta}R^{\alpha\beta}+\frac{7}{18}R^{2}\Big{)}-6R_{\mu\alpha}R_{\nu}^{~{}\alpha}\Big{]},$
(14)

and

$\displaystyle L_{\mu\nu}$
$\displaystyle=\frac{1}{4}W^{2}\lambda_{\mu\nu}+\lambda^{\alpha\beta}W_{\mu\alpha}^{~{}~{}\gamma\delta}W_{\nu\beta\gamma\delta}+3\lambda^{\beta\gamma}R^{\alpha}_{~{}(\mu}W_{\nu)\beta\alpha\gamma}-\frac{1}{2}\lambda
R^{\alpha\beta}W_{\mu\alpha\nu\beta}-R^{\alpha\beta}\lambda^{\gamma}_{(\mu}W_{\nu)\alpha\beta\gamma}+W_{\mu\beta\nu\gamma}R^{\alpha(\beta}\lambda^{\gamma)}_{~{}\alpha}$
$\displaystyle+2\lambda^{\alpha\beta}W_{\alpha\gamma\beta\delta}W_{\mu~{}~{}\nu}^{~{}\gamma\delta}-2\nabla^{\beta}\nabla^{\gamma}W_{\alpha\beta\gamma(\mu}\lambda^{\alpha}_{~{}\nu)}-\lambda\nabla^{\alpha}\nabla^{\beta}W_{\mu\beta\nu\alpha}-\lambda^{\alpha\beta}\big{(}\Box
W_{\mu\alpha\nu\beta}+2\nabla^{\gamma}\nabla_{\beta}W_{\alpha(\mu\nu)\gamma}+2\nabla_{(\mu}\nabla^{\gamma}W_{\nu)\alpha\beta\gamma}\big{)}$
$\displaystyle-\frac{1}{3}W_{\mu\alpha\nu\beta}(2R+3\Box)\lambda^{\alpha\beta}+W_{\mu\alpha\nu\beta}\big{(}\nabla^{\alpha}\nabla_{\gamma}\lambda^{\beta\gamma}+\nabla^{\beta}\nabla_{\gamma}\lambda^{\gamma\alpha}-\nabla^{\beta}\nabla^{\alpha}\lambda\big{)}+2W_{\alpha\beta\gamma(\mu}\big{(}\nabla_{\nu)}\nabla^{\beta}\lambda^{\gamma\alpha}-\nabla^{\gamma}\nabla^{\beta}\lambda^{\alpha}_{~{}\nu)}\big{)}$
$\displaystyle-\frac{1}{8}\Big{[}\lambda
W^{2}-8R^{\alpha\beta}\lambda^{\gamma\delta}W_{\alpha\gamma\beta\delta}+8\lambda^{\alpha\beta}\nabla^{\delta}\nabla^{\gamma}W_{\alpha\gamma\beta\delta}+8W_{\alpha\gamma\beta\delta}\nabla^{\delta}\nabla^{\gamma}\lambda^{\alpha\beta}\Big{]}g_{\mu\nu},$
(15)

is the contribution from the new Lagrange multiplier term in the action. In
the above expression we have defined

$\displaystyle
W^{2}=W_{\alpha\beta\gamma\delta}W^{\alpha\beta\gamma\delta},\qquad\lambda=\lambda^{\mu}_{~{}\mu}.$
(16)

Some points should be clarified in this step. First, as we have discussed
above, we do not have an equation to determine the Lagrange multiplier
$\lambda_{\mu\nu}$. Secondly, one can easily verify that the tensor
$L_{\mu\nu}$ vanishes in 4D. This is because this tensor is obtained from
variation of the Lanczos-Bach tensor in $D$ dimensions and then taking the
$D\rightarrow 4$. Since this limiting process continues for this term, the
tensor $L_{\mu\nu}$ could be seen as a variation of the Lanczos-Bach tensor in
$4D$. Since in $4D$ the Lanczos-Bach tensor vanishes identically, the
$L_{\mu\nu}$ does not contribute to the above equations of motion. Thirdly,
the energy-momentum tensor is not conserved in this model which can be easily
proved by taking the covariant divergence of the equation of motion (13). The
result is exactly equation (II). Here, the non-conservative nature of the
energy-momentum tensor is due to the non-minimal coupling between the Lagrange
multiplier term and the metric. It should be noted that the implication of
matter conservation from Noether’s theorem, works if one have a pure (or
minimally coupled) gravity theory. However, due to the Lagrange multiplier
term, our model is not purely (or minimally coupled) gravity theory. In
summary, the non-conservation of energy-momentum tensor is inherited from the
tensor $s_{\mu\nu}$ in (II).

The resulting equation of motion in for $4D$ Gauss-Bonnet theory in four
dimensions can be written as

$\displaystyle\kappa^{2}(G_{\mu\nu}$ $\displaystyle+\Lambda
g_{\mu\nu})+\frac{\alpha}{2}\Big{[}4R\,R_{\mu\nu}-4R_{\mu\alpha\nu\beta}R^{\alpha\beta}$
$\displaystyle+3g_{\mu\nu}\Big{(}R_{\alpha\beta}R^{\alpha\beta}-\frac{7}{18}R^{2}\Big{)}-6R_{\mu\alpha}R_{\nu}^{~{}\alpha}\Big{]}=\frac{1}{2}T_{\mu\nu},$
(17)

Using the definition of the Weyl tensor, one can also write

$\displaystyle\kappa^{2}$ $\displaystyle(G_{\mu\nu}+\Lambda
g_{\mu\nu})+\alpha\Bigg{[}\frac{2}{3}R\,R_{\mu\nu}-R_{\mu\alpha}R_{\nu}^{~{}\alpha}$
$\displaystyle+\frac{1}{2}\left(R_{\alpha\beta}R^{\alpha\beta}-\frac{1}{2}R^{2}\right)g_{\mu\nu}-2R^{\alpha\beta}W_{\alpha\mu\beta\nu}\Bigg{]}=\frac{1}{2}T_{\mu\nu},$
(18)

Actually, if one assumed that the Lanczos-Bach tensor where vanishing from the
first place, the same result would appear. In our procedure, this by hand
cancellation is replaced by a dynamical cancellation through the Lagrangian.

It should be noted that the new term proportional to $\alpha$ in equation (II)
was obtained from quantum gravity point of view in qggb  where the authors
wrote all the terms which are first order in Weyl tensor with the property
that its covariant divergence being zero in conformally flat spacetimes. In
this paper, our procedure is classical but we have also kept the first
Lovelock invariant which corresponds to the first order Weyl tensor. However,
since we have considered the most general spacetimes, the energy-momentum
tensor is not covariantly conserved. Rewriting the conservation equation (II)
in terms of the Weyl tensor, one obtains

$\displaystyle\nabla^{\nu}T_{\mu\nu}=2\alpha
C_{\mu\alpha\beta\nu}\nabla^{\nu}R^{\alpha\beta}.$ (19)

The above equation shows that the matter sector is covariantly conserved in
any conformally flat spacetimes which is compatible with qggb .

##  III Isotropic cosmology

As an example of a symmetric space-time, let us try the new constrained 4D
Gauss-Bonnet theory on the FRW space-time. Assume that the line element is

$\displaystyle ds^{2}=a(t)^{2}\eta_{\mu\nu}dx^{\mu}dx^{\nu},$ (20)

where $a(t)$ is the scale factor and $\eta_{\mu\nu}$ is the Minkowski metric.
We consider a perfect fluid which is characterized by energy density $\rho$
and thermodynamic pressure $p$, with the energy-momentum tensor given by

$\displaystyle T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+pg^{\mu\nu}.$ (21)

The Friedmann and Raychaudhuri equations can be written as

$\displaystyle 3\kappa^{2}H^{2}$
$\displaystyle=\frac{1}{2}a^{2}\rho+\kappa^{2}\Lambda
a^{2}-\frac{3\alpha}{a^{2}}H^{4},$ (22)

and

$\displaystyle 2\kappa^{2}\dot{H}$ $\displaystyle=\frac{2}{3}\kappa^{2}\Lambda
a^{2}-\frac{1}{6}a^{2}(\rho+3p)+\frac{2\alpha H^{2}}{a^{2}}(H^{2}-2\dot{H}).$
(23)

The conservation of the energy-momentum tensor can be written as

$\displaystyle\dot{\rho}$ $\displaystyle+3H(\rho+p)=0.$ (24)

##  IV Anisotropic cosmology

In this section, we will consider the cosmological implications of the
anisotropic universe in the constraint 4D Gauss-Bonnet theory. The metric is
taken as

$\displaystyle ds^{2}=-dt^{2}+a_{1}^{2}dx^{2}+a_{2}^{2}(dy^{2}+dz^{2}),$ (25)

where $a_{1}$, $a_{2}$ are directional time dependent scale factors. We assume
that the universe is filled with an isotropic matter content with energy-
momentum tensor of the form

$\displaystyle T^{\mu}_{~{}\nu}=\textmd{diag}(-\rho,p,p,p),$ (26)

where $\rho$ and $p$ are energy density and thermodynamic pressures of the
fluid respectively.

Let us define the mean Hubble and the anisotropy parameter as

$\displaystyle H_{i}$ $\displaystyle=\frac{\dot{a}_{i}}{a_{i}},\quad\Delta
H_{i}=H-H_{i}\quad i=1,2,$ $\displaystyle 3H$ $\displaystyle=H_{1}+2H_{2},$
$\displaystyle 3A$ $\displaystyle=\left(\frac{\Delta
H_{1}}{H}\right)^{2}+2\left(\frac{\Delta H_{2}}{H}\right)^{2}.$ (27)

Also we define the deceleration parameter as

$\displaystyle q=-1+\frac{d}{dt}\left(\frac{1}{H}\right).$ (28)

It should be noted that in the case of isotropic universe with $a_{1}=a_{2}$,
the anisotropy parameter vanishes and the mean Hubble and deceleration
parameters become the standard Hubble and deceleration parameters of isotropic
universe. Since the universe becomes isotropic at late times, we will compare
the mean Hubble and deceleration parameters of the anisotropic 4D Gauss-Bonnet
universe with late time observational data.

Figure 1: The evolution of the Hubble and deceleration parameters as a
function of redshift $z$ for $\beta=0.001$ (dashed), $0.05$ (dotted), $0.1$
(dot-dashed) and $0.5$ (long dashed). The red solid line represents the
evolution in the $\Lambda$CDM model. Also, error bars correspond to the
observational data obshubble .

The Friedman and Raychaudhuri equations can be written as

$\displaystyle 3\kappa^{2}(1-B^{2})H^{2}$
$\displaystyle=\kappa^{2}\Lambda+\frac{1}{2}\rho-3\alpha\Bigg{[}(1-15B^{2}+36B^{3}+3B^{4})H^{4}+2B(1+2B)\dot{B}H^{3}$
$\displaystyle+2B\dot{B}H\dot{H}+B^{2}\dot{H}^{2}+\big{(}\dot{B}^{2}+B^{2}(1+2B)\dot{H}\big{)}H^{2}\Bigg{]},$
(29)
$\displaystyle\kappa^{2}\bigg{[}2(1-B)\dot{H}+3(1-B)^{2}H^{2}-2H\dot{B}\bigg{]}$
$\displaystyle=-\frac{1}{2}p+\kappa^{2}\Lambda-\alpha\Bigg{[}3(5B^{4}+7B^{2}-4B+1)H^{4}-2(2+B)(1-4B)\dot{B}H^{3}$
$\displaystyle+6B\dot{B}H\dot{H}+3B^{2}\dot{H}^{2}+\big{(}3\dot{B}^{2}+2(4B^{3}+9B^{2}-6B+2)\dot{H}\big{)}H^{2}\Bigg{]},$
(30)
$\displaystyle\kappa^{2}\bigg{[}3(1+B+B^{2})H^{2}+H\dot{B}+(2+B)\dot{H}\bigg{]}$
$\displaystyle=-\frac{1}{2}p+\kappa^{2}\Lambda+\alpha\Bigg{[}3(7B^{4}+6B^{3}+8B^{2}-2B-1)H^{4}+12B\dot{B}H\dot{H}+6B^{2}\dot{H}^{2}$
$\displaystyle+2(5B^{2}+14B-1)\dot{B}H^{3}+\big{(}6\dot{B}^{2}+2(5B^{3}+18B^{2}-3B-2)\dot{H}\big{)}H^{2}\Bigg{]}.$
(31)

Also, the conservation equation of the matter content can be written as

$\displaystyle\dot{\rho}+3H(\rho+p)=$
$\displaystyle-12\alpha\Bigg{(}B(1+2B)H^{2}+\frac{d}{dt}(BH)\Bigg{)}$
$\displaystyle\times\Bigg{(}3B(1-B)(1+2B)H^{3}+(4+B)\dot{B}H^{2}+2\dot{B}\dot{H}+B\ddot{H}+\big{(}\ddot{B}+B(5+B)\dot{H}\big{)}H\Bigg{)},$
(32)

where we have defined $B\equiv\sqrt{A/2}$. It should be explicitly checked
that the tensor $L_{\mu\nu}$ in equation (II) vanishes in the case of Bianchi
universe. Also, the above conservation equation shows that contrary to the
isotropic FRW universe, the matter sector is no longer conserved in this case.

In order to solve the above set of dynamical equations, we assume that the
universe is filled with radiation, with energy density $\rho_{r}$ and pressure
$p_{r}=\rho_{r}/3$ and also pressure-less dust with energy density $\rho_{m}$.
We then define the following set of dimensionless parameters

$\displaystyle\rho$ $\displaystyle=\rho_{r}+\rho_{m},\quad
H=H_{0}h,\quad\beta=\alpha\kappa^{2}H_{0}^{2},$
$\displaystyle\Omega_{\Lambda}$
$\displaystyle=\frac{\Lambda}{3H_{0}^{2}},\quad\bar{\rho}_{i}=\frac{\rho_{i}}{6\kappa^{2}H_{0}^{2}},\quad
i=r,m,$ (33)

where $H_{0}$ is the current value of the Hubble parameter.

In order to compare the model with observations it is customary to transform
the field equations to the redshift coordinates, defined as

$\displaystyle 1+z=\frac{1}{a}.$ (34)

We will assume that all the non-conservative sources is handled by dust. As a
result the radiation component of the cosmic fluid would be conserved. One can
then obtain

$\displaystyle\bar{\rho}_{r}=(1+z)^{4}\Omega_{r0},$ (35)

where $\Omega_{r0}$ is the current radiation density abundance.

Considering the Friedman and Raychaudhuri equations at $z=0$ and denoting that
by definition $h(z=0)=1$, one can obtain the cosmological constant as

$\displaystyle\Omega_{\Lambda}=1+\beta-\Omega_{r0}-\Omega_{m0},$ (36)

where $\Omega_{m0}$ is the current value of the dust density abundance. Also,
one can obtain the following constraint on the derivatives of the function $B$
and $h$ as

$\displaystyle B^{\prime}(0)=0,\qquad
h^{\prime}(0)=\frac{3\Omega_{m0}+4\Omega_{r0}}{2+4\beta}.$ (37)

In figure (1), we have plotted the mean Hubble paramter and also the
deceleration parameter as a function of the redshift for different values of
$\beta=0.001$ (dashed), $0.05$ (dotted), $0.1$ (dot-dashed) and $0.5$ (long
dashed). The red solid line represents the evolution in the $\Lambda$CDM
model. The error bars are associated with the observational data on the Hubble
parameter obshubble . One can see from the figures that the Hubble parameter
increases as the parameter $\beta$ decreases. As a result the model predicts
smaller universe for smaller values of $\beta$. This can also be seen from the
deceleration parameter where smaller values of $\beta$ imply he the universe
has more deceleration at larger redshift. In figure (2), we have plotted the
anisotropy parameter $B$ as a function of redshift.

Figure 2: The evolution of the anisotropy parameter as a function of redshift
$z$ for $\beta=0.001$ (dashed), $0.05$ (dotted), $0.1$ (dot-dashed) and $0.5$
(long dashed).

One can see from the figure that the universe becomes isotropic at late times.
Also, in figure (3), we have depicted the dust matter density abundance as a
function of $z$. The red solid curve corresponds to the conservative case.

Figure 3: The evolution of the dust matter density abundance as a function of
redshift $z$ for $\beta=0.001$ (dashed), $0.05$ (dotted), $0.1$ (dot-dashed)
and $0.5$ (long dashed). The red solid line represents the evolution of the
conservative matter source.

It should be noted that in the 4D-Gauss-Bonnet theory, the matter sector is
not conserved. This can be seen from the figure, where the curves fall more
rapidly at larger redshifts compare to the conservative case. In summary, we
note that the present theory can in principle explain late time observational
data. However, more analysis would be needed to determine the viability of the
theory as an alternative theory of general relativity.

##  V Conclusions

In this paper, we have considered a new way to resolve the problem of having
an equation of motion in $4D$ Einstein-Gauss-Bonnet gravity by introducing a
Lagrange multiplier term to the action. This new term will remove the unwanted
part of the Gauss-Bonnet equation of motion which is not proportional to the
factor $D-4$ and as a result enabled us to write a covariant equation of
motion for the theory. This new term however, make the $4D$ matter field to be
non-conservative. In four space-time dimensions, the equation of motion of the
Lagrange multiplier satisfied identically and as a result we have an arbitrary
tensor field which is not determined in the theory. This signaling a new
symmetry which could be unraveled in the future. For symmetric space-times,
the new constraint theory is equivalent to the original theory. So, all the
works done in the original context could be trivially applied to the new
constrained version. The richness of this new constrained 4D Gauss-Bonnet
theory is that the field equation is now four dimensional and as a result we
could consider less symmetric space-times like Bianchi types or rotating black
hole solutions in this context. Importantly, the analysis of perturbations,
which needs broken homogeneity and isotropicity, should be done in our
framework. The same analysis could not be done in the original version of the
theory because of the ambiguity of defining higher dimensional metrics for
these space-times. We have then considered an anisotropic universe describing
by the Bianchi type I universe as an example. We have solved the equations
numerically and see that the mean Hubble parameter tends to the current Hubble
parameter at late times. More precisely, at redshifts smaller than unity,
there is no significant deviations between the mean Hubble parameter and the
observational data. The anisotropy parameter is also an increasing function of
the redshift, predicting that the universe becomes isotropic at late times.
Overall, we have shown that the constraint 4D Gauss-Bonnet theory can explain
observational data on the Hubble parameter.

It should be noted that in the Bianchi space-time, the matter energy-momentum
tensor is no longer conserved. In figure (3), we have plotted the evolution of
the matter density abundance as a function of redshift. It can be seen from
the figure that the matter density fall more rapidly at larger redshifts
signaling that more matter is transformed to geometry at higher redshifts.

We should note that the non-conservative nature of the matter sector in this
model would results in a classical creation of particles. The same property is
also considered in other gravitational theories with non-conservative matter
energy-momentum tensor fRT . This would make a constraint on the parameter
$\alpha$. However, we can easily check that the present theory has a
Schwarzschild solution with the property that the LHS of equation (II)
vanishes. As a result, one concludes that the creation of particles will not
happen in this special case. Of course for more general non-Schwarzschild
solutions, the creation rate of particles in this theory would be non-zero.

At last, we have to say that more analysis would be needed to fully understand
the nature of theory together with is observational constraints.

## acknowledgments

We would like to thank Zahra Haghani for very useful discussions and comments.

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# Reactive Base Control for On-The-Move Mobile Manipulation in Dynamic
Environments

Ben Burgess-Limerick1,2, Jesse Haviland1,2, Chris Lehnert1, Peter Corke1 This
research was supported by the QUT Centre for Robotics and CSIRO Data61.1
Queensland University of Technology Centre for Robotics (QCR), Brisbane,
Australia 2 CSIRO Data61, Brisbane, Australia

###### Abstract

We present a reactive base control method that enables high performance mobile
manipulation on-the-move in environments with static and dynamic obstacles.
Performing manipulation tasks while the mobile base remains in motion can
significantly decrease the time required to perform multi-step tasks, as well
as improve the gracefulness of the robot’s motion. Existing approaches to
manipulation on-the-move either ignore the obstacle avoidance problem or rely
on the execution of planned trajectories, which is not suitable in
environments with dynamic objects and obstacles. The presented controller
addresses both of these deficiencies and demonstrates robust performance of
pick-and-place tasks in dynamic environments. The performance is evaluated on
several simulated and real-world tasks. On a real-world task with static
obstacles, we outperform an existing method by 48% in terms of total task
time. Further, we present real-world examples of our robot performing
manipulation tasks on-the-move while avoiding a second autonomous robot in the
workspace. See benburgesslimerick.github.io/MotM-BaseControl for supplementary
materials.

##  I Introduction

Performing mobile manipulation tasks while a robot’s base remains in motion
can significantly reduce execution time compared with approaches in which the
robot pauses to perform manipulations. This is particularly valuable in multi-
step tasks where, for example, the robot can be driving towards a second
location while picking up an object. Recent works have explored control
methods for achieving such Manipulation on-the-Move (MotM) [1, 2].

Systems that rely on planning are able to claim optimality of generated
trajectories and avoid apriori-known obstacles in the scene. However, planning
approaches suffer in real-world environments where they cannot react to
perception errors, localisation error, or inaccurate control and are unable to
perform in environments with unpredictably moving objects and obstacles.

In our previous work [1], we introduced an architecture for achieving reactive
MotM and demonstrated grasping of unpredictable, dynamic objects while on the
move. However, the implementation presented previously uses a mobile base
controller that does not consider obstacle avoidance. In this work, we develop
a mobile base controller for integration into the reactive MotM architecture
that minimises task time while avoiding static and dynamic obstacles. In
addition, we augment the quadratic program solved in the redundancy resolution
module with a constraint that provides obstacle avoidance for the arm.

Our reactive approach enables robust performance in complex environments with
dynamic obstacles. Fig. 1(a) shows a frame from a real-world trial where our
system is performing a grasp while on-the-move and avoiding a second robot
moving in the scene. The digital twin illustrated in Fig. 1(b) shows the
system’s understanding of the space, including the real-time detections of the
obstacles in the environment.

(a) Real-world manipulation on-the-move with dynamic obstacles.

(b) Digital Twin rendering of perceived state for Fig. 1(a). The system’s
local plan is illustrated with transparent robots.

Figure 1: Our method can perform robust, reactive manipulation on-the-move
while avoiding static and dynamic obstacles.

The principal contributions of this work are:

  1. 1.

a reactive base control system that minimises total task time for multi-step
tasks by performing manipulation tasks on-the-move

  2. 2.

a redundancy resolution module that enables obstacle avoidance for the arm and
base while performing manipulation on-the-move tasks

  3. 3.

the first demonstration of reactive manipulation on-the-move in real-world
environments with static and dynamic obstacles

These capabilities are demonstrated in numerous simulated and real-world
experiments.

##  II Related Works

Base control methods for mobile manipulation guide the mobile base to a pose
from which a manipulation target can be reached without violating kinematic
constraints or colliding with obstacles. This is often achieved through
explicit base control where an optimal base pose is determined and the robot
is navigated to that pose with a conventional mobile robot controller [3].
Alternatively, implicit or holistic base control methods typically start with
a desired end effector motion and use the combined kinematics of the base and
manipulator to achieve the motion.

We review common methods for base control in mobile manipulation and the
additional challenges for base control while performing manipulation on-the-
move. Recent surveys of control strategies for mobile manipulation include
[4].

###  II-A Explicit Base Control

####  II-A1 Optimal Base Placement

Numerous planners have been proposed to compute the base pose from which to
best perform a manipulation task. In general, the goal is to find a pose for
which the target is reachable with a collision-free configuration of the robot
[5]. Approaches typically aim to generate solutions that are optimal against
some other metric such as manipulability [6, 7], or stiffness of the robot
[8]. Other approaches aim to minimise task time by calculating base poses from
which multiple targets can be reached without repositioning [9, 10]. Time
efficiency can be optimised on multi-step tasks by choosing a pose based on
where the robot must go after the immediate target [3, 11]. Reactivity can be
achieved by frequently recomputing the optimal base pose [3].

####  II-A2 Mobile Base Control

Once an optimal base pose has been computed, the robot is driven to the pose
with a mobile base controller. Typically, a hierarchical planner is used that
combines a global and local planner to enable reactive obstacle avoidance
while driving to the goal pose [12].

A reactive, Short Term Aborting A* (STAA*) method is presented in [13] that
demonstrates improved performance in environments with static and dynamic
obstacles compared to commonly used local planners. STAA* plans a collision-
free global path by searching through a visibility graph, and then computes
the intersection of the global path and the border of a local planning region
to develop an intermediate goal. A discretised acceleration space is used to
sample new states for exploration in a time-bounded A* search. The search uses
an obstacle-aware heuristic that ensures exploration along a collision-free
path and avoids local minima.

In most cases, mobile manipulators using explicit base controllers consider
the base and manipulator motion entirely separately and the arm does not start
moving until the base has achieved the desired pose. More recent methods have
improved task time by coordinating the arm motion with the base such that the
hand arrives at the target at the same time as the base arrives at the desired
pose [3].

###  II-B Holistic Control

Rather than considering the base and arm motion separately, some approaches
combine the subsystems with a holistic controller [14]. These methods use the
combined degrees of freedom from the mobile base and manipulator to achieve a
desired end-effector motion. Holistic control of the robot enables reduced
task time by moving both components together, as well as an improved ability
to optimise secondary objectives such as manipulability [14], obstacle
avoidance [15], or visibility [16] through exploitation of additional degrees
of freedom.

The input to the holistic control system can be from a motion planner and
executed under open-loop control [15], or reactive, where the controller uses
visual feedback for closed-loop control [14, 17]. Model Predictive Control
formulations of the problem have been used to enable collision avoidance for
both arm and base [18, 19]. A learned base controller is presented in [20]
which translates a desired end-effector velocity and local occupancy map to
base motions that ensure the omnidirectional robot avoids obstacles.

These works demonstrate holistic control of a mobile manipulator and can avoid
obstacles. However, they focus only on execution of the immediate goal and do
not consider the time efficiency for multi-step tasks achieved by performing a
manipulation task while on the move toward the next target.

###  II-C Manipulation On-The-Move

The first MotM approaches restricted base motion to constant speed, straight-
line motion [21]. Recent works plan collision-free trajectories in cluttered
environments that can complete mobile manipulation tasks in minimum time [2,
22, 23, 24, 25]. However, these approaches execute the planned trajectory
open-loop and cannot react to dynamic changes in the scene, or compensate for
perception and control errors. Consequently, these systems are often failure
prone in real-world environments.

In this work, we present a reactive base control method that avoids obstacles
while performing manipulation tasks on-the-move. Further, we enable collision
avoidance for the manipulator by implementing the approach described in [26].

##  III Base Controller

We introduce several modifications to the global and local planners described
in STAA* [13] that improve performance for manipulation on-the-move scenarios.

###  III-A Goal Orientation

The most important addition to STAA* is the inclusion of an orientation to the
goal state. Where STAA* considers driving to a point only, we include
orientation which enables poses to be achieved that smoothly connects the
immediate target with the next goal.

Figure 2: A comparison of global paths generated by our modified STAA* (path
in red) and the original (shown in yellow) for an example goal pose. The red
$x$ axis of the target frame represents the desired forward direction.

###  III-B Rotation in Global Planner Cost

The addition of orientation to the goal state requires modification to the
node cost computation used in the global A* search. STAA* uses only the
cumulative distance between nodes along a path to compute the cost. Instead,
we consider the PathRTR metric that estimates the time required to both
translate and rotate between nodes along a path. Further explanation of the
PathRTR metric is presented in [13] where it is used for the local planner.
Fig. 2 illustrates the value of including rotation costs in the global path
planner. For the scenario shown, the path in red generated by our modified
version encourages the robot to drive a smooth curve around the obstacle
connecting the start and end pose. Without rotation costs, the shortest path
passes the obstacle on the opposite side and requires significantly more
turning.

###  III-C Search Termination Conditions

The implementation of STAA* presented in [13] terminates its search when an
explored node is sufficiently close to the goal. To perform manipulation on-
the-move, we want to encourage the robot to drive through the goal at high
speed. Therefore, we also terminate when the path passes sufficiently close to
the target.

###  III-D Reduction of Proximity Grid Penalty

STAA* includes a cost on visiting nodes based on their proximity to obstacles
using an inflated occupancy grid. However, to complete mobile manipulation
tasks such as picking and placing objects from a table, the robot must
necessarily travel close to the table while interacting with objects on it.
For example, in Fig. 2, the occupancy grid is represented by the colour of the
ground around the robot, with green representing free space, and red
representing occupied space. When the target pose for the base is near an
obstacle, as is the case when grasping from a table, the penalty applied to
nodes near obstacles inhibits the exploration of states near the goal. To
limit this effect, we reduce the weight of the occupancy grid cost based on
proximity to the object pick or place location. We scale the grid penalty by
$k=\max{(0.1,\min{(t_{h}/3,1)})}$ where $t_{h}$ is the estimated time until
the goal is achieved.

###  III-E Bézier Heuristic

Figure 3: Illustration of the Bézier path evaluation. The PathRTR evaluation
between the two presented poses consists of Rotate(30°),
Translate(4$\mathrm{m}$), Rotate(90°). By comparison, the Bézier curve shown
has a total length of 4.35$\mathrm{m}$. Assuming the robot has maximum linear
and angular velocities of 0.5$\mathrm{m}\mathrm{s}^{-1}$ and
100°$\mathrm{s}^{-1}$, the estimated sequential PathRTR cost is
9.2$\mathrm{s}$, while travelling the Bézier path and combining rotation and
translation requires only 8.7$\mathrm{s}$.

The PathRTR heuristic used for the A* graph search in the local planner
assumes that translation and rotation of the robot will be performed at
maximum speed, but sequentially. This has a tendency to overestimate costs for
states where the optimal path to the goal is a smooth arc of simultaneous
rotation and translation. We introduce an additional heuristic based on a
Bézier curve that is used in place of PathRTR only when the Bézier path
results in a lower cost. This encourages the exploration of states that can be
connected to the goal through smooth curves. The Bézier curve is constructed
by adding two control points alongside the start and end points given by the
current pose and target. The first control point is positioned in front of the
robot’s current forward direction, and the second is positioned an equal
distance behind the desired end pose. The offset distance is chosen to be 25%
of the distance between the current and target pose. The optimal offset
distance is a complicated function of the relative target pose and the ratio
of the maximum linear and angular velocity capabilities of the robot. However,
we find that a value of 25% results in curves that work well for our robot in
practice. Fig. 3 illustrates an example Bézier path.

##  IV Base Placement

The optimal base placement is selected from a discretised set by evaluating
the path cost from the current robot pose to candidate base placements as well
as from the candidates to the next target (Fig. 4). Candidates are evenly
spaced around the target object in 10° increments for a total of 36 possible
base positions. Each position is assigned two possible orientations: the
robot’s forward vector can be tangential to the circle facing either clockwise
or counter-clockwise which gives a total of 72 candidates.

The radius of the circle on which the candidates are placed is dynamically
adjusted between 0.6 $\mathrm{m}$ and 0.8 $\mathrm{m}$. When no collision-free
candidates are available close to the object the radius is increased until a
solution is found. The radius limits are defined by the geometry of our robot,
0.6 $\mathrm{m}$ is calculated from the radius of the robot base plus a safety
margin, and 0.8 $\mathrm{m}$ is the largest distance at which the robot can
still perform manipulation tasks. When no viable candidates are found within
these limits, the robot will drive toward the collision-free position closest
to the target with the assumption that the system may identify a valid
candidate as the map updates with more recent lidar data.

(a) Drop Location 1

(b) Drop Location 2

Figure 4: Examples of optimal base pose selection for two different drop
locations. The red and cyan lines illustrate the path to the candidate and
path from candidate to next target respectively for the optimal base
placement.

The path cost for each candidate is evaluated using the PathRTR metric
described in [13]. The total path cost for the $i$-th candidate is given by
the weighted sum of two components

$C_{i}=C_{i,\text{C}}+1.05\cdot C_{i,\text{N}}$

where $C_{i,\text{C}}$ is the estimated cost from robot to candidate and
$C_{i,\text{N}}$ is the estimated cost from the candidate to the next target.
The weighting that biases towards minimising $C_{i,\text{N}}$ ensures that the
robot continues to drive toward the next target even when close to the object.
Decreasing this weight below 1 will tend towards a greedy solution that
optimises for the immediate task only without considering the next task.
Increasing the weight further will encourage the robot to sacrifice time
efficiency on the immediate task in favour of minimising the expected travel
time for the next task.

The candidate with lowest $C_{i}$ is used as the goal for navigation. The
method selects base placements that are within manipulation range of the
target and efficiently connect the current robot pose with the next target.
For example, the spheres in Fig. 4 are coloured based on their path cost, with
brighter green representing lower cost. The paths for the optimal candidate
are also shown.

The path costs for each candidate are reevaluated at each controller step (20
$\mathrm{H}\mathrm{z}$) to enable reactive control and response to changes in
the environment.

##  V Arm Obstacle Avoidance

Obstacle avoidance for the manipulator is enabled by modifying the redundancy
resolution controller described in [1]. This controller solves a Quadratic
Program (QP) to calculate joint velocities for a given desired end-effector
and base velocity. Using a similar controller, reactive obstacle avoidance for
a mobile manipulator is demonstrated in [26]. The controller allows for slack
in the achieved end-effector velocity which can be exploited along with
redundant degrees of freedom to avoid obstacles.

Obstacle avoidance is implemented in the QP through the addition of an
inequality constraint which limits the velocity of points on the arm when they
are close to an obstacle. Further details on the implementation are available
in [26].

Accurately modelling the environment in sufficient detail for robust collision
avoidance in 3D is a difficult problem and outside the scope of this work. In
[26], real-world trials are performed with simulated objects whose pose can be
directly observed. In our system, we use the 2D lidar in the robot’s base to
construct an obstacle map for the arm. Any obstacles detected by the lidar are
assumed to be tall enough that the arm should avoid them. When the obstacles
are short this is a conservative assumption and the arm will unnecessarily
avoid the space above the obstacle. However, when an obstacle is larger above
the plane of the lidar (for example a table supported by a central post), the
system cannot observe the geometry at the height of the arm and there is the
risk of a collision. We mitigate this by providing the system with a pre-
generated map of most of the collision geometry in the environment. Additional
obstacles detected by the lidar are added to the map. In future work, more
detailed collision geometry could be modelled online with a 3D lidar or depth
camera, enabling improved obstacle avoidance.

We query the constructed manipulator obstacle map with a single point in the
centre of the robot’s gripper rather than computing the closest point on link
collision meshes as in [26]. This simplifies the implementation and we find
that it achieves acceptable performance in our real-world testing. The end-
effector speed toward the object is limited to

$\dot{d_{ro}}\leq\xi\frac{d-d_{s}}{d_{i}-d_{s}}$

where $\dot{d_{ro}}$ is the distance between the end-effector and the closest
obstacle, $\xi=0.6$ is a gain controlling the aggressiveness of the obstacle
avoidance, $d_{s}=0.25$ $\mathrm{m}$ is the minimum distance allowed between
end-effector and obstacle, and $d_{i}=0.6$ $\mathrm{m}$ is the threshold at
which the limit is enabled. For $d_{ro}>d_{i}$ the constraint is removed from
the QP.

##  VI Experiments

Real-world experiments are performed with our
Frankie333https://github.com/qcr/frankie_docs mobile manipulator that consists
of an Omron LD-60 differential-drive mobile base and a Franka-Emika Panda
7-DOF manipulator. Robot control is implemented using the Robotics Toolbox
[27]. The simulation and digital twin environments are implemented in Unity.

###  VI-A Baseline Comparisons

The method presented in [3] optimises task time by selecting base poses that
are conditioned on the next target location in a higher-level task. This is
similar to the method described in IV, but does not consider the challenges of
performing the tasks on-the-move. To meaningfully compare with the work
presented in [3] we recreate their experiments as closely as possible in
simulation and the real-world. [3] provides performance data for two
baselines, as well as two versions of their proposed approach. These methods
are described briefly here:

  1. 1.

Fixed Set Baseline: The robot chooses a base placement from a set of 7 fixed
candidates spaced around and facing the object table. Selection amongst the
candidates is based on maximising manipulability.

  2. 2.

Inverse Reachability Maps (IRM) Baseline: The optimal base placement is
selected based on evaluation of a manipulability metric only and does not
consider navigation costs.

  3. 3.

Greedy [3]: The base placement includes both manipulability and navigation
costs associated with a candidate but does not include navigation to the next
location.

  4. 4.

Sequential [3]: Optimal base placement is determined based on a weighted cost
combining manipulability, navigation costs to a candidate, and navigation
costs from the candidate to the next location.

Further details on these methods are available in [3]. All data relating to
these approaches has been reproduced from [3].

###  VI-B Experiment 1: Static Obstacles

A collection of 6 objects were placed on a 2.4 $\times$ 0.8 $\mathrm{m}$ table
and must be transported to two drop locations. Fig. 5 shows the experiment
layout – a dimensioned diagram is available on our project website. The table
and drop points locations are consistent with those used in [3], allowing for
meaningful comparison.

Figure 5: Layout of objects and drop locations for experiment 1. For each
trial, 6 of the 12 object positions were randomly selected and an order was
generated for them to be alternately delivered to the two drop locations. The
yellow squares indicate additional static obstacles for experiment 1a.

The 6 objects were placed randomly on the 12 possible object locations shown
in Fig. 5 and a random order was assigned. The objects must be picked up in
order and transported to alternating drop locations. This is to ensure fair
comparison with the experiments in [3], however it should be noted that
performance improvements are possible by optimally selecting the order and
drop location.

Our experiment differs from [3] in two respects. First, we use red 40
$\mathrm{m}\mathrm{m}$ cubic objects to simplify perception and grasp
synthesis which is not the focus of this work. Second, we limit object
positions to a set of 12 candidates around the perimeter of the table. Our
robot has an arm reach of 0.855 $\mathrm{m}$, significantly less than the 1.55
$\mathrm{m}$ of the ARMAR-6 robot used in [3]. Positioning the objects closer
to the table’s edge allows more room for our robot to perform reactive
grasping without colliding with the table. The reach of ARMAR-6 is sufficient
to grasp objects on the far side of the table, whereas our robot must drive
around the table, increasing the distance to be travelled. We limit the
positions to a set of 12 candidates to improve reproducibility of the
experiment.

We conducted 50 trials with random object arrangements in simulation and
performed 10 trials in the real world. The first real-world object arrangement
is hand-crafted for best comparison with the real-world trial presented in
[3], and the remaining 9 were randomly chosen. Details of the 10 real-world
scenarios are available on our project website.

###  VI-C Experiment 1a: Additional Static Obstacles

The 50 simulated experiments conducted in Experiment 1 were repeated with the
addition of 2 cuboid obstacles (shown in yellow on Fig. 5). These objects were
positioned to be consistent with those included in scenario 2 presented in
[3].

###  VI-D Experiment 2: Dynamic Obstacles

Figure 6: Simulated experiment 2 with four autonomous Temi robots providing
dynamic obstacles.

This experiment uses the same basic arrangement as experiment 1 but introduces
additional autonomous robots to the scene which function as dynamic obstacles.
In simulation, four Temi robots were tasked with autonomously driving around
the table (Fig. 6). The large number of robots relative to the size of the
environment ensures frequent interactions and therefore avoidance manoeuvres.
We also perform a real-world trial with a single Temi robot. The complex and
unpredictable interactions between the robots make the task time metric
difficult to interpret for these experiments. Instead, we demonstrate that our
system can complete the task with dynamic obstacles in the real world and
examine the data from an example simulated trial to investigate the
performance of the obstacle avoidance functionality.

##  VII Results

###  VII-A Experiment 1: Table Clearing

####  VII-A1 Simulated Results

The results presented in Fig. 7 compare the time to complete the 6 object
pick-and-place task with the baselines and approach presented in [3]. Our MotM
method reduces the total task time by 43% compared to the sequential task
optimised method presented in [3].

####  VII-A2 Real-world Results

Table I compares the real-world performance of our method with the IRM
baseline and Sequential optimised method from [3]. On a single trial (6 object
pickups) with approximately the same object arrangement and order, we
successfully cleared the table in 91 $\mathrm{s}$ where the method presented
in [3] takes 176 $\mathrm{s}$, a task time reduction of 48%. Across 10 trials
(60 object pickups) with random object arrangement, the task was completed in
an average of 100.4 $\mathrm{s}$ with a success rate of 58 out of 60. The two
failure cases resulted from grasp failures where imperfect control of the
robot while the fingers were closing caused a collision between the object and
fingers that destabilised the grasp. These failures could be recovered by
detecting the failure and allowing the robot to reopen the gripper and
reattempt the grasp.

The consistency between the real-world and simulated timing data validates the
simulation as a meaningful representation of real-world performance. In
general, tasks are completed slightly faster in simulation, which we attribute
to idealisations of the perception and low-level controllers.

Figure 7: Time required to complete the simulated experiment 1 task. The box
plots for the Fixed Set, IRM, Greedy, and Sequential methods have been
recreated from the results published in [3] which do not include the time
required to complete the grasps, which they estimate at about 40$\mathrm{s}$.
Consequently, the timings from [3] have been increased by 40$\mathrm{s}$ to
compare with our results. For all experiments the objects are rectangular
prisms, with comparable grasp complexity.

###  VII-B Experiment 1a: Table Clearing with Static Obstacles

Fig. 8 presents the simulated task times for the task described in Fig. 5 with
additional static obstacles and compares our method to the baselines and
approaches presented in [3]. We demonstrate a reduction in total task time of
46% which is consistent with the performance from experiment 1. These results
demonstrate that our approach is robust to the inclusion of additional
obstacles in the environment, even when they are close to the grasp locations
and interfere with the ideal trajectory for grasping an object on the move.

Figure 8: Time required to complete the experiment 1a task with additional obstacles in simulation. Note that as in Fig. 7 results taken from [3] include grasping times. TABLE I: Comparison of task times for experiment 1 in simulation and the real world. Our real-world results are averaged across 10 different scenarios, where the results for the IRM baseline and Sequential methods are from single trials conducted in [3]. Method | Simulation | Real-World
---|---|---
| Task Time | Task Time | Success Rate
IRM Baseline [3] | 192.9 $\mathrm{s}$ | 202 $\mathrm{s}$ | 6/6
Sequential [3] | 172.3 $\mathrm{s}$ | 176 $\mathrm{s}$ | 6/6
Ours (Single Trial) | 85.6 $\mathrm{s}$ | 91 $\mathrm{s}$ | 6/6
Ours (Ten Trials) | 95.9 $\mathrm{s}$ | 100.4 $\mathrm{s}$ | 58/60

###  VII-C Experiment 2: Table Clearing with Dynamic Obstacles

We demonstrate real-world manipulation on-the-move in an environment with
unpredictably moving dynamic obstacles. The time required to complete the task
is dependent on the random interactions between our robot and the Temi acting
as a dynamic obstacle. The complexity of the interactions makes the
experiments impossible to properly replicate, and therefore it is meaningless
to compare the task-time metric over relatively few trials. However, in an
example real-world scenario, our system completed the task in 94 $\mathrm{s}$
in the presence of a dynamic obstacle. In this trial, there were 4
interactions where an evasive manoeuvre was required to avoid the other robot.
For the same object arrangement without the dynamic obstacle, the task was
completed in 91 $\mathrm{s}$. This demonstrates that our system can robustly
perform tasks in environments with dynamic obstacles while incurring minimal
increase in overall task execution time.

Fig. 9 shows the minimum distance between the end-effector and the closest
obstacle over an example simulated trial with 4 dynamic obstacles. These
results demonstrate the effectiveness of the arm obstacle avoidance
constraints described in Section V. When the distance dips into the region of
influence (shown in orange), the constraint limits the velocity toward the
obstacle. The minimum distance allowed between the end-effector and obstacles
is 0.25 $\mathrm{m}$.

Figure 9: Minimum distance between end-effector and closest detected obstacle
over a simulated trial with dynamic obstacles. Discontinuities are caused by
obstacles coming into view as the robot moves through the environment. The red
and orange regions indicate the minimum end-effector to obstacle distance and
region of influence of the constraint described in Section V.

Fig. 10 illustrates what the behaviour of the robot looks like in practice.
After reaching out to grasp the object, the system retracts the arm to avoid
the Temi while driving on towards the drop location to complete the task.

Figure 10: A series of snapshots from our system’s arm and base successfully
avoiding another robot while performing a pick-and-place task on-the-move.

##  VIII Discussion and Future Work

There are numerous avenues that can be explored in future work to improve
performance.

####  VIII-1 Obstacle Motion Prediction

Our current implementation has no predictive capability for the motion of
dynamic obstacles. Instead, dynamic obstacles are detected at each time step
and treated the same as static obstacles for path planning. We have noticed
that this results in a tendency for our system to cut in front of other robots
and people in the environment. For example, consider the case illustrated in
Fig. 11. If we assume that the Temi is stationary then the optimal path to the
object (red cube) passes in front of the obstacle. This is the path that is
currently taken by our system. However, if the Temi was modelled as a dynamic
obstacle, its time history (shown as transparent) could be used to predict
that the obstacle will keep driving forward. In that case, a better path is to
drive directly toward the object allowing the Temi to move out of the way
before our robot arrives.

The implementation of STAA* presented in [13] includes support for dynamic
obstacle motion prediction and demonstrates the desired behaviour. However,
obstacle motion prediction is not included in this work due to the difficulty
of accurately segmenting obstacles from real-world lidar data. A system that
explicitly modelled dynamic objects would allow for improved performance in
environments with other moving agents. We view this as a perception challenge
that could be addressed in future work.

Figure 11: Demonstration of our system’s tendency to cut in front of dynamic
obstacles. Transparent robots illustrate Temi’s history and Frankie’s planned
path.

####  VIII-2 3D Collision Geometry for Arm

As discussed in Section V, collision avoidance for the arm is enabled by
assuming that all obstacles detected by the base lidar are tall enough that
they should be avoided by the arm. This is both overly conservative for short
obstacles and does not prevent collision with obstacles above the plane of the
2D lidar. Including additional sensors and a perception pipeline that
accurately models 3D geometry would enable improved manipulator obstacle
avoidance and manipulation in cluttered environments such as reaching into a
bin or a cupboard.

####  VIII-3 Failure Recovery

The base control system presented in this work reactively adjusts the base
motion to keep the object in reach until the grasp is completed. This is more
thoroughly investigated in our related work presented in [28]. However, we
have not implemented a robust method for detecting grasp failures in the real
world. The robot’s gripper provides infrequent feedback on the finger position
and applied forces, which makes it difficult to detect failures through
proprioceptive methods. Instead, additional tactile or vision sensors could be
used for feedback. If grasp failures could be reliably detected, then the
presented system would control the robot such that grasps can be reattempted
until success is achieved. This would enable recovery from manipulation
failures where a second attempt can be easily executed. However, failures such
as the object falling to the floor will likely require more sophisticated
recovery behaviours including searching for the object.

####  VIII-4 Gripper Speed

The relatively slow speed of the Franka-Emika Panda gripper requires the
system to stabilise the gripper over the object for a significant period of
time (approximately 0.8 $\mathrm{s}$) while the fingers close. This limits the
speed at which the robot can be driving past while the grasp is attempted. A
gripper that closed more quickly could be used to enable faster grasping on-
the-move.

####  VIII-5 Phantom Collisions

The Franka-Emika Panda manipulator is a cobot with in-built collision
detection implemented through joint torque measurement. This is a valuable
safety feature, however, accelerations of the mobile base caused by changes in
commanded velocity as well as bumps in the terrain can impart torques on the
robot joints that are registered as collisions and result in a pause or
shutdown of the arm’s controller. Decreasing the frequency of these detections
by increasing arm compliance warrants further investigation for higher speed
performance.

####  VIII-6 Grasp Synthesis for Complex Objects

The simple perception and grasp synthesis method used in this work limits the
system to grasping objects of simple geometry and uniform colour. The addition
of a more sophisticated grasp synthesis method would enable the grasping of
more complex objects. The key challenge in this respect is developing a grasp
synthesis system that provides closed-loop feedback throughout the grasping
action, all the way until the fingers close. Although not essential, closed-
loop feedback throughout the grasping action improves performance in the
highly dynamic environment of manipulation on-the-move, and provides reliable
performance that is robust to imprecise perception and localisation, and
inaccurate robot control [29].

##  IX Conclusion

We presented a base control method that integrates with the architecture
presented in [1] to enable reactive manipulation on-the-move in complex
environments with dynamic obstacles. Further, the QP in the redundancy
resolution module is augmented to include an inequality constraint that
provides collision avoidance for the manipulator. We have explored the
system’s performance in a number of simulated and real-world experiments. We
demonstrated a reduction in task execution time of 48% compared to a state-of-
the-art method on an example real-world pick-and-place task. In addition, we
show that the mobile manipulator can perform pick-and-place tasks while its
arm and base both avoid dynamic obstacles. Several limitations of the system
are explored which highlight interesting opportunities for further research.

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