A congruence theorem for compact embedded hypersurfaces in
$\mbox{${\mathbb{S}}$}^{n+1}_{+}$

Allan Freitas and Felippe Guimarães

> Abstract We prove a codimension reduction and congruence theorem for compact
> $n$-dimensional submanifolds of $\mbox{${\mathbb{S}}$}^{n+p}$ that admit a
> mean convex isometric embedding into $\mbox{${\mathbb{S}}$}^{n+1}_{+}$ using
> a Reilly type formula for space forms.
>
>
> Mathematics Subject Classification: 53A07, 53C42.
> Keywords: Isometric embedding, Reilly type formula, Isometric rigidity.

## 1\. Introduction

In the realm of submanifold theory, a relevant research topic is the
uniqueness of isometric immersions $f\colon M^{m}\to N^{n}$ between Riemannian
manifolds, where the superscript denotes the dimension. Since any isometric
immersion composed with an isometry $\tau\colon N^{n}\to N^{n}$, $g=\tau\circ
f$, remains an isometric immersion, uniqueness is considered in terms of
congruences by isometries of the larger space. In this context, an isometric
immersion is considered isometrically rigid if all other isometric immersions
are congruent to it through an isometry of the ambient space. When the ambient
space is a simply connected space form, that is, a simply connected Riemannian
manifold with constant sectional curvature, some results in the literature
ensure local isometric rigidity under conditions on the non-degeneracy of the
second fundamental form, namely [9, 1, 5, 2] (see [3] for a more detailed
discussion). While globally the results are more scarce, here we mention the
main result in [17] which ensures that, given two isometric immersions
$f_{1},f_{2}:\Sigma^{n}\rightarrow\mbox{${\mathbb{R}}$}^{n+1}$, $i=1,2$,
$n\geq 3$, from a compact hypersurface, pointwise they have the same second
fundamental form, up to a sign (cf. [3, Theorem 13.2] for a more general
statement). Global results that guarantee the congruence of the isometric
immersion of a compact hypersurface in a space form, if it has constant scalar
curvature, were obtained in [8], and results for when it is convex, in the
sense that the second fundamental form is positive definite in a suitable
normal direction, can be found in [6] (see [4] and references therein for a
good discussion). In [16], the author showed that the sphere is the only
compact hypersurface with constant scalar curvature embedded in Euclidean
space. Additionally, a congruence result was presented, making ingenious use
of Reilly’s formula presented in [15].

######  Theorem 1.1 (Theorem 2 in [16]).

Let $\Psi\colon M^{n}\rightarrow\mbox{${\mathbb{R}}$}^{n+1}$, $n\geq 2$, be a
compact hypersurface embedded hypersurface. Suppose that the mean curvature of
$\Psi$, with respect to the interior normal $H$, is nonnegative. If
$\Psi^{\prime}\colon M^{n}\rightarrow\mbox{${\mathbb{R}}$}^{n+p}$, $p\geq 1$
is an isometric immersion with mean curvature vector $\mathcal{H}$ satisfying
$\|\mathcal{H}\|\leq H$ everywhere, then $\Psi^{\prime}=i\circ\Psi$ where $i$
is a totally geodesic inclusion
$i\colon\mbox{${\mathbb{R}}$}^{n+1}\rightarrow\mbox{${\mathbb{R}}$}^{n+p}$. In
particular, $\Psi^{\prime}$ and $\Psi$ are congruent.

In the last few decades, with the aim of obtaining Alexandrov-type theorems in
ambient spaces beyond Euclidean space, generalizations of Reilly’s formula
have emerged, particularly noteworthy are those for space forms presented in
[14] and more recently for sub-static manifolds as discussed in [12]. By
employing such formulations, we are able to derive a congruence theorem
similar to Theorem 1.1 for hypersurfaces in a hemisphere on the sphere. In the
last few decades, aiming to obtain Alexandrov-type theorems in ambient spaces
beyond Euclidean space, generalizations of Reilly’s formula have emerged.
Notably, generalizations for space forms presented in [14] and, more recently,
for sub-static manifolds as discussed in [12] have proven useful for deriving
interesting integral formulas. Specifically by employing the formulation
presented in [14], we are able to derive a congruence theorem for
hypersurfaces in a hemisphere on the sphere, which is analogous to Theorem
1.1.

######  Theorem A.

Let $\Psi\colon M^{n}\rightarrow\mbox{${\mathbb{S}}$}^{n+1}_{+}$, $n\geq 2$,
be a compact hypersurface embedded into $\mbox{${\mathbb{S}}$}^{n+1}_{+}$.
Suppose that the mean curvature of $\Psi$, with respect to the interior normal
$H$, is nonnegative. If $\Psi^{\prime}\colon
M^{n}\rightarrow\mbox{${\mathbb{S}}$}^{n+p}$, $p\geq 1$ is an isometric
immersion with mean curvature vector $\mathcal{H}$ satisfying
$\|\mathcal{H}\|\leq H$ everywhere, then $\Psi^{\prime}=i\circ\Psi$ where $i$
is a totally geodesic inclusion
$i\colon\mbox{${\mathbb{S}}$}^{n+1}\rightarrow\mbox{${\mathbb{S}}$}^{n+p}$. In
particular, $\Psi^{\prime}$ and $\Psi$ are congruent.

######  Remark 1.2.

Observe that in Theorem A we do not assume embedding and that the image for
$\Psi^{\prime}$ is contained within a hemisphere. It is worth mentioning that
the hypersurface must have some points with strictly positive mean curvature,
as we know that a minimal hypersurface cannot be contained within a
hemisphere.

This type of result is naturally connected to the question:

> “Given two isometric immersions $f_{1},f_{2}:\Sigma^{n}\rightarrow M^{n+1}$,
> $i=1,2$, with the same mean curvature, do they differ only by a rigid motion
> of the ambient space?”

A good discussion about this question can be found in [11] and references
therein, as well as a related result for hypersurfaces in a spatial
Schwarzschild or AdS-Schwarzschild manifold. Integral formulas have been
important for achieving results on intrinsic rigidity, specifically regarding
the extension of geometric properties from a manifold which is the boundary of
a region to the region itself. This investigation is distinct from the main
question of this article, yet the techniques show similarities, as showed in
[7]. In [7], the approaches introduced in [16] are used to prove the following
result.

######  Theorem 1.3.

(Proposition 2 in [7]) Let $M^{n}$ be a smooth compact connected Riemannian
manifold with connected boundary $\Sigma=\partial M$ and $Rc\geq 0$. If
$\phi:\Sigma\rightarrow\mbox{${\mathbb{R}}$}^{n}$ is an isometric immersion
with $\|\mathbf{H}_{\phi}\|\leq H$ on $\Sigma$, where $\mathbf{H}_{\phi}$ is
the mean curvature vector of the immersion $\phi$, then $M$ is flat. If $\phi$
is also an embedding, $M$ is isometric to a domain in
$\mbox{${\mathbb{R}}$}^{n}$.

This is a generalization of Theorem 1.1 by requiring that the ambient space
has non-negative Ricci curvature, which leads to the conclusion that the
region must be flat. Further research related to this result was conducted in
[7, 10, 13], placing a lower bound on Ricci curvature compared to space forms.
However, unlike Theorem 1.3, which required the boundary to be, in a sense,
mean convex, the results in [13] were achieved under a stronger hypothesis of
convexity, similar to those in [6]. The authors also propose the following
question:

> “Let $(\Omega,g)$ be an $n$-dimensional, compact Riemannian manifold with
> boundary $\Sigma$. Let $D\subset S^{n}_{+}$ be a bounded domain with smooth
> boundary $\partial D$, where $S^{n}_{+}$ is the standard n-dimensional
> hemisphere. Suppose the following
>
>   * •
>
> Ric $\geq(n-1)g$.
>
>   * •
>
> There exists an isometry $X:\Sigma\rightarrow\partial D$.
>
>   * •
>
> $H\geq H_{S}\circ X$, where $H,H_{S}$ are the mean curvature of
> $\Sigma,\partial D$ in $(\Omega,g),S^{n}_{+}$ respectively.
>
>

>
> Is $(\Omega,g)$ isometric to $D$ in $S^{n}_{+}$?”

A positive answer to this question would imply that our Theorem could be
improved in a similar way to Theorem 1.3.

## 2\. The proof

In the proof, we utilize a generalization of Reilly’s formula by Qiu and Xia
(cf. [14, Theorem 1.1]), which particularly yields an intriguing identity in
the spherical case:

(2.1) $\displaystyle\int_{\Omega}\cos
r(x)((\bar{\Delta}f+(n+1)f)^{2}-|\bar{\nabla}^{2}f+fg|^{2})d\Omega$
$\displaystyle=\int_{M}\cos r(x)(2u\Delta z+nHu^{2}+h(\nabla z,\nabla z))$
$\displaystyle\quad+2nuz)dA$
$\displaystyle\quad+\int_{M}\overline{\nabla}_{\nu}[\cos r(x)](|\nabla
z|^{2}-nz^{2})dA.$

Here $\Omega\subset\mbox{${\mathbb{S}}$}^{n+1}_{+}$ is a domain with boundary
$M$, $r(x)=d_{g}(x,p)$ for some fixed point $p\in\Omega$ and
$f\colon\Omega\rightarrow\mbox{${\mathbb{R}}$}$ is a smooth function.
Furthermore, $\bar{\nabla}$ and $\bar{\Delta}$ indicate the gradient and the
Laplacian operators in $\mbox{${\mathbb{S}}$}^{n+1}_{+}$, $z=f|_{M}$ and
$u=\bar{\nabla}_{\nu}f$, where $\nu$ is the unit outward normal of $M$.
Furthermore, $\Delta$ and $\nabla$ denote the correspondent intrinsic
operators in $M$, $h(X,Y)=g(\bar{\nabla}_{X}\nu,Y)$ and $nH=tr_{g}h$ are the
second fundamental form and the mean curvature (with respect to $\nu$) of $M$,
respectively.

###### Proof of Theorem A.

Let $\tilde{\Psi}=\mathcal{I}\circ\Psi^{\prime}$ where
$\mathcal{I}:\mbox{${\mathbb{S}}$}^{n+p}\rightarrow\mbox{${\mathbb{R}}$}^{n+p+1}$
is the umbilical inclusion. Choose a global parallel basis
$\\{e_{1},e_{2},\ldots,e_{n+p+1}\\}$ of $\mbox{${\mathbb{R}}$}^{n+p+1}$ and
define $\tilde{\Psi}_{i}:M^{n}\rightarrow\mbox{${\mathbb{R}}$}$ given by
$\tilde{g}(\tilde{\Psi},e_{i})$ where $\tilde{g}$ is the standard metric of
$\mbox{${\mathbb{R}}$}^{n+p+1}$. Also, consider
$\Omega\subset\mbox{${\mathbb{S}}$}^{n+1}_{+}$ the bounded domain such that
$\partial\Omega=M$. Before using (2.1) for a suitable function $f$, let’s
simplify some terms. Let $\\{v_{1},...,v_{n}\\}$ be an orthonormal basis in
$M$, and using that $\Psi$ is a isometric immersion, we have that

$g(\nabla\tilde{\Psi}_{i},v_{s})=v_{s}\tilde{g}(\tilde{\Psi},e_{i})=\tilde{g}(\tilde{\Psi}_{*}v_{s},e_{i}),$

which also implies that
$\tilde{\Psi}_{*}\nabla\tilde{\Psi}_{i}=e_{i}-\sum^{p+1}_{k=1}\tilde{g}(e_{i},\xi_{k})\xi_{k}$,
where $\\{\xi_{1},\xi_{2},\ldots,\xi_{p+1}\\}$ is a normal basis of
$\tilde{\Psi}$(??). In particular

$\sum^{n+p+1}_{i=1}|\nabla\tilde{\Psi}_{i}|^{2}=\sum^{n+p+1}_{i=1}\left(1-\sum^{p+1}_{k=1}\tilde{g}(e_{i},\xi_{k})^{2}\right)=n,$

and

(2.2)
$\begin{split}\sum_{i=1}^{n+p+1}h(\nabla\tilde{\Psi}_{i},\nabla\tilde{\Psi}_{i})&=\sum_{i=1}^{n+p+1}\sum_{s,k=1}^{n}\tilde{g}(\tilde{\Psi}_{*}v_{s},e_{i})\tilde{g}(\tilde{\Psi}_{*}v_{k},e_{i})h(v_{s},v_{k})\\\
&=\sum_{s,k=1}^{n}g_{sk}h(v_{s},v_{k})=nH.\end{split}$

As
$\tilde{\Psi}(M^{n})\subset\mbox{${\mathbb{S}}$}^{n+p}\subset\mbox{${\mathbb{R}}$}^{n+p+1}$
it follows that $\|\tilde{\Psi}(x)\|^{2}=1$ for all $x\in M^{n}$. This implies

(2.3)
$\sum^{n+p+1}_{i=1}\left[|\nabla\tilde{\Psi}_{i}|^{2}-n(\tilde{\Psi}_{i})^{2}\right]=0.$

Now, for $1\leq i\leq n+p+1$, let
$f_{i}\colon\overline{\Omega}\rightarrow\mbox{${\mathbb{R}}$}$ be the solution
of the problem

$\begin{cases}\Delta f_{i}+(n+1)f_{i}=0\ &\quad\text{in}\quad\Omega\\\
f_{i}=\tilde{\Psi}_{i}&\quad\text{on}\quad M\end{cases}$

By considering $u_{i}=\frac{\partial f_{i}}{\partial\nu}$ and
$U=(u_{1},\ldots,u_{n+p+1})$, using (2.2) and
$\Delta\tilde{\Psi}=n\mathcal{H}-n\tilde{\Psi}$, we have:

(2.4) $\displaystyle\sum^{n+p+1}_{i=1}$
$\displaystyle\left[2u_{i}\Delta\tilde{\Psi}_{i}+nHu_{i}^{2}+h(\nabla\tilde{\Psi}_{i},\nabla\tilde{\Psi}_{i})+2nu_{i}\tilde{\Psi}_{i})\right]$
$\displaystyle=\left[2\tilde{g}(U,\Delta\tilde{\Psi})+nH\|U\|^{2}+nH+2n\tilde{g}(U,\tilde{\Psi})\right]$
$\displaystyle=\left[2\tilde{g}(U,n\mathcal{H}-n\tilde{\Psi})+nH\|U\|^{2}+nH+2n\tilde{g}(U,\tilde{\Psi})\right]$
$\displaystyle=n\left[H\|U\|^{2}+2\tilde{g}(U,\mathcal{H})+H\right].$

Applying the Reilly’s identity (2.1) for each $f_{i}$, summing up from $1$ to
$n+p+1$ and using (2.3) and (2.4) we get the following expression

$-\int_{\Omega}\cos
r(x)\sum^{n+p+1}_{i=1}(|\nabla^{2}f_{i}-\frac{1}{n+1}(\Delta
f_{i})g|^{2})d\Omega=\int_{M}n\cos
r(x)\left[H\|U\|^{2}+2\tilde{g}(U,\mathcal{H})+H\right].$

Moreover, using that
$2\tilde{g}(U,\mathcal{H})\geq-2\|U\|\|\mathcal{H}\|\geq-2\|U\|H$ it follows

$0\geq-\int_{\Omega}\cos
r(x)\sum^{n+p+1}_{i=1}(|\nabla^{2}f_{i}-\frac{1}{n+1}(\Delta
f_{i})g|^{2})d\Omega\geq\int_{M}H\cos r(x)\left[\|U\|-1\right]^{2}.$

Hence, $\|U\|=1$ in every point where $H\neq 0$, and
$|\nabla^{2}f-\frac{1}{n+1}(\Delta f_{i})g|=0$ for all $1\leq i\leq n+p+1$.
This tells us that

(2.5) $\begin{split}\|U\|=1\ \text{at }x\in
M&\Rightarrow\|\tilde{\Psi}_{*}X\|=|X|\ \text{for all }X\in
TM\oplus\text{span}\\{\nu\\};\\\ |\nabla^{2}f_{i}-\frac{1}{n+1}(\Delta
f_{i})g|=0\ \text{at
}x\in\Omega&\Rightarrow\nabla^{2}f_{i}=-f_{i}g.\end{split}$

The rest of the proof can be addressed in a more general manner, but here we
will present it for the specific setting of this work, and it can be seen as
an argument similar to Obata’s rigidity. Let
$f=(f_{1},f_{2},\ldots,f_{n+p+1})\colon\Omega\to\mbox{${\mathbb{R}}$}^{n+p+1}$
and define $A=\\{x\in\Omega\mid\|f(x)\|^{2}=1\text{ and
}\|f_{*}(x)(w)\|=|w|\text{ for all }w\in T_{x}\Omega\\}$. Due to the existence
of a point $x\in M$ with $H(x)\neq 0$ (which exists in view of Remark 1.2) and
from (2.5), we deduce that $A\neq\emptyset$. Given the smoothness of $f$, $A$
is closed. In order to prove that $A=\Omega$ remaining to show that $A$ is
open. First, the solutions of the equations $\nabla^{2}f_{i}=-f_{i}g$ for all
$1\leq i\leq n+p+1$ can be extended to an open neighborhood $\Omega^{a}$ of
$\Omega$, thus avoiding dealing with boundary points. In order to solve the
equation in (2.5) explicitly, let $y\in A$ and $U\subset\Omega^{a}$ be a
totally normal neighborhood of $y$. It follows from $\nabla^{2}f_{i}=-f_{i}g$
for all $1\leq i\leq n+p+1$ that

(2.6) $f(x)=\cos(r_{y}(x))f(y)+\sin(r_{y}(x))f(y)_{*}(\nabla r_{y}(x)),$

where $r_{y}(x)=d(x,y)$ for $x\in\Omega$. From $y\in A$, we know that
$\|f(y)\|^{2}=1$ holds for every $y\in U$. Now considering $x\in
U\backslash\\{y\\}$, $w\in T_{x}\Omega^{a}$ with $\|w\|=1$, and
$\gamma:[0,d(x,y)]\rightarrow U$ as the geodesic connecting $\gamma(0)=x$ to
$\gamma(d(x,y))=y$, let $\alpha:[0,T)\rightarrow U$ be a unit geodesic such
that $\alpha(0)=x$ and $\alpha^{\prime}(0)=w$. First, assuming that
$\alpha=\gamma$, we observe that

$f_{*}(x)w=\frac{d}{ds}f\circ\alpha(s)\biggr{\rvert}_{s=0}=\sin(r_{y}(x))f(y)-\cos(r_{y}(x))f_{*}(y)w$

as in this case $\nabla r_{y}(\alpha(s))=-\alpha^{\prime}(d(x,y))$, and since
$y\in A$ it implies that $\|f_{*}(x)(w)\|^{2}=1=|w|^{2}$. We can now assume
that $g(w,\gamma^{\prime}(0))=0$. By using the exponential map of
$\mbox{${\mathbb{S}}$}^{n+1}_{+}\subset\mbox{${\mathbb{R}}$}^{n+2}$ at a point
$y$, we have that

$\alpha(s)=\cos r_{y}(\alpha(s))y+\sin{r_{y}(\alpha(s))}\nabla
r_{y}(\alpha(s)).$

Hence

$\frac{d}{ds}\nabla
r_{y}(\alpha(s))\biggr{\rvert}_{s=0}=\frac{w}{\sin{r_{y}(x)}}.$

This leads to the conclusion that

$f_{*}(x)w=\frac{d}{ds}f\circ\alpha(s)\biggr{\rvert}_{s=0}=\sin(r_{y}(x))f_{*}(y)\frac{d}{ds}\nabla
r_{y}(\alpha(s))\biggr{\rvert}_{s=0}=f_{*}(y)w,$

and given that $y\in A$ it follows that $U\subset A$, concluding that
$A=\Omega$. Here, by using the inclusion of the sphere in Euclidean space, we
are clearly assuming some identifications, since
$f_{*}(x)w=\sin(r_{y}(x))f_{*}(y)\left[{exp_{y}}_{*}(\frac{d}{ds}\nabla
r_{y}(\alpha(s))\rvert_{s=0})\right]^{-1}w$. Consequently, $f$ is an isometric
immersion, and as $\Omega\subset\mbox{${\mathbb{S}}$}^{n+1}_{+}$, $f$ can be
described in the entire hemisphere using an exponential chart at a point
$x\in\Omega$ as indicated in (2.6). This expression ensures that
$f(\Omega)\subset\mbox{${\mathbb{S}}$}^{n+1}\subset\mbox{${\mathbb{S}}$}^{n+p}$,
where the totally geodesic submanifold $\mbox{${\mathbb{S}}$}^{n+1}$ is
determined as the sphere centered at the origin with its tangent space given
by $f_{*}(x)(T_{x}\Omega)$. Given the boundary condition that
$f\circ\Psi=\mathcal{I}\circ\Psi^{\prime}$, this concludes the proof of the
theorem. ∎

In the proof presented, with regard to the hypothesis about the mean curvature
of the isometric immersions, we are actually using the following inequality

$\int_{\Omega}\cos(r_{y}(x))Hd\Omega\geq\int_{\Omega}\cos(r_{y}(x))\|\mathcal{H}\|d\Omega,$

where $r_{y}(x)=d(y,x)$ for some point $y\in\Omega$. Thus, we can rewrite our
result as follows.

######  Theorem B.

Let $\Psi:M^{n}\rightarrow\mbox{${\mathbb{S}}$}^{n+1}_{+}$, $n\geq 2$, be a
compact hypersurface embedded into $\mbox{${\mathbb{S}}$}^{n+1}_{+}$. Suppose
that the mean curvature $H$ of $\Psi$, with respect to the interior domain
$\Omega$, is nonnegative. Let $y\in\Omega$ and
$\Psi^{\prime}:M^{n}\rightarrow\mbox{${\mathbb{S}}$}^{n+p}$, $p\geq 1$ be an
isometric immersion then we have that the following inequality is satisfied

$\int_{\Omega}\cos(r_{y}(x))Hd\Omega\leq\int_{\Omega}\cos(r_{y}(x))\|\mathcal{H}\|d\Omega,$

where $\mathcal{H}$ is the mean curvature vector of $\Psi^{\prime}$ and
$r_{y}(x)=d(y,x)$. Moreover, if equality holds then $\Psi^{\prime}=i\circ\Psi$
where $i$ is a totally geodesic inclusion
$i:\mbox{${\mathbb{S}}$}^{n+1}\rightarrow\mbox{${\mathbb{S}}$}^{n+p}$. In
particular, $\Psi^{\prime}$ and $\Psi$ are congruent.

## Acknowledgements

Allan Freitas has been partially supported by CNPq/Brazil (Grant
316080/2021-7) and Programa Primeiros Projetos - FAPESQ/PB and MCTIC/CNPq
(Grant 2021/3175). Felippe Guimarães was supported by FAPESQ/PB.

## References

  * [1] C. B. Allendoerfer.  Rigidity for spaces of class greater than one.  Am. J. Math., 61:633–644, 1939.
  * [2] M. Dajczer and L. A. Florit.  Genuine deformations and submanifolds.  Commun. Anal. Geom., 12(5):1105–1129, 2004.
  * [3] M. Dajczer and R. Tojeiro.  Submanifold theory. Beyond an introduction.  Universitext. Springer, New York, 2019.
  * [4] R. de Lima.  A survey on convex hypersurfaces of riemannian manifolds.  Matemática Contemporânea, 50(8), 2022.
  * [5] M. do Carmo and M. Dajczer.  A rigidity theorem for higher codimensions.  Math. Ann., 274(4):577–583, 1986.
  * [6] M. P. do Carmo and F. W. Warner.  Rigidity and convexity of hypersurfaces in spheres.  Journal of Differential Geometry, 4(2):133 – 144, 1970.
  * [7] F. Hang and X. Wang.  Vanishing sectional curvature on the boundary and a conjecture of Schroeder and Strake.  Pac. J. Math., 232(2):283–287, 2007.
  * [8] C. E. Harle.  Rigidity of hypersurfaces of constant scalar curvature.  J. Differ. Geom., 5:85–111, 1971.
  * [9] W. Killing.  Analytical treatment of non-euclidean spaces.  Leipzig. Teubner. XII u. 264 Seiten (1885)., 1885.
  * [10] M. Lai.  A note on Hang-Wang’s hemisphere rigidity theorem.  Math. Z., 296(3-4):901–909, 2020.
  * [11] C. Li, P. Miao, and Z. Wang.  Uniqueness of isometric immersions with the same mean curvature.  J. Funct. Anal., 276(9):2831–2855, 2019.
  * [12] J. Li and C. Xia.  An integral formula and its applications on sub-static manifolds.  J. Differ. Geom., 113(3):493–518, 2019.
  * [13] P. Miao and X. Wang.  Boundary effect of Ricci curvature.  J. Differ. Geom., 103(1):59–82, 2016.
  * [14] G. Qiu and C. Xia.  A generalization of Reilly’s formula and its applications to a new Heintze-Karcher type inequality.  Int. Math. Res. Not., 2015(17):7608–7619, 2015.
  * [15] R. C. Reilly.  Applications of the hessian operator in a riemannian manifold.  Indiana University Mathematics Journal, 26(3):459–472, 1977.
  * [16] A. Ros.  Compact hypersurfaces with constant scalar curvature and a congruence theorem.  J. Differ. Geom., 27(2):215–220, 1988.
  * [17] R. Sacksteder.  The rigidity of hypersurfaces.  J. Math. Mech., 11:929–939, 1962.

Allan Freitas

Departamento de Matematica,
Universidade Federal da Paraíba, João Pessoa (Brazil).

e-mail<EMAIL_ADDRESS>
Felippe Guimarães

Departamento de Matematica,
Universidade Federal da Paraíba, João Pessoa (Brazil).

e-mail<EMAIL_ADDRESS>


# Robot Adaptation for Generating Consistent Navigational
Behaviors over Unstructured Off-Road Terrain

Sriram Siva1, Maggie Wigness2, John G. Rogers2, and Hao Zhang1 1Sriram Siva
and Hao Zhang are with the Human-Centered Robotics Lab in the Department of
Computer Science, Colorado School of Mines, Golden, CO 80401, USA. Email:
{sivasriram<EMAIL_ADDRESS>Wigness and John Rogers are with the US
Army Research Laboratory (ARL), Adelphi, MD 20783, USA. Email:
{maggie.b.wigness.civ<EMAIL_ADDRESS>This work was supported
by NSF CAREER Award IIS-1942056 and ARL SARA Program W911NF-20-2-0107. The
authors also would like to thank Mr. Eric Spero at ARL for the discussion of
this work and for facilitating the experimentation.

###### Abstract

Terrain adaptation is an essential capability for a ground robot to
effectively traverse unstructured off-road terrain in real-world field
environments such as forests. However, the expected robot behaviors generated
by terrain adaptation methods cannot always be executed accurately due to
setbacks such as wheel slip and reduced tire pressure. To address this
problem, we propose a novel approach for consistent behavior generation that
enables the ground robot’s actual behaviors to more accurately match expected
behaviors while adapting to a variety of unstructured off-road terrain. Our
approach learns offset behaviors that are used to compensate for the
inconsistency between the actual and expected behaviors without requiring the
explicit modeling of various setbacks. Our approach is also able to estimate
the importance of the multi-modal features to improve terrain representations
for better adaptation. In addition, we develop an algorithmic solver for our
formulated regularized optimization problem, which is guaranteed to converge
to the global optimal solution. To evaluate the method, we perform extensive
experiments using various unstructured off-road terrain in real-world field
environments. Experimental results have validated that our approach enables
robots to traverse complex unstructured off-road terrain with more
navigational behavior consistency, and it outperforms previous methods,
particularly so on challenging terrain.

##  I Introduction

Over the past several years, autonomous ground robots have been increasingly
deployed in off-road field environments to address real-world applications,
including disaster response, homeland defense, and planetary exploration [1,
2]. Field environments are challenging for ground robots to navigate over
because the terrain is unstructured and cannot be fully modeled beforehand
[3], and also the environment typically exhibits a wide variety of
characteristics (e.g, changing terrain types, and varying slope and
roughness), as demonstrated in Fig. 1. While operating in a field environment,
robots need to generate effective navigational behaviors for traversing over
the off-road terrain in order to successfully complete navigation tasks. Robot
terrain adaptation is then an essential capability for robots to adapt their
navigational behaviors according to unstructured off-road terrains [4].

Given its importance, the research problem of robot terrain adaptation has
been widely investigated over the past several years. Previous learning-based
methods can be divided into two broad categories: terrain classification and
terrain adaptation. The first category of approaches apply a robot’s
exteroceptive and proprioceptive sensory data to classify terrain types and
estimate traversability for a robot to navigate over the terrain [5, 6, 7, 8,
9]. This category also includes methods that model terrain complexity for
planning robot navigation tasks [3, 10]. The second category of methods focus
on directly generating adaptive navigational behaviors according to terrain in
order to successfully complete navigation tasks [3, 11, 12, 13]. Specifically,
learning from demonstration (LfD) is widely used to transfer human expertise
to robots in order to achieve human-level robot navigational control [4, 13,
14].

Figure 1: This motivating scenario illustrates the necessity for ground robots
to generate consistent navigational behaviors while operating in real-world
field environments. The off-road environment is unstructured and exhibits a
variety of characteristics, including changing terrain types and slopes. When
ground robots are deployed in this environment, their actual behaviors often
do not match the expected behaviors, e.g., due to wheel slip. Thus, the
capability of consistent behavior generation is essential for ground robots
while they navigate over unstructured off-road terrain.

Despite their promising performance in complex off-road environments, the
expected navigational behaviors generated by these approaches cannot always be
executed accurately, i.e., the actual robot navigational behaviors are not
consistent with the expected behaviors. This inconsistency is mainly caused by
setbacks [15, 16] that are defined as factors that increase the difficulty for
a robot to achieve its expected navigational behaviors. Example setbacks
include wheel slip, heavy robot payload, and reduced tire pressure. Existing
learning-based approaches (e.g., LfD) for robot navigation generally ignore
these setbacks, which often leads to not being able to consistently complete
the learned navigational behaviors when deployed over unstructured off-road
terrain. The challenge of how to generate consistent navigational behaviors
for robot terrain adaptation has not been well addressed.

In this paper, we develop a novel approach for consistent behavior generation
that enables a robot’s actual navigational behaviors to match the expected
behaviors while adapting to a variety of unstructured off-road terrain during
navigation. Our approach learns offset behaviors for the robot to compensate
for the inconsistency between the actual and expected navigational behaviors
without explicitly modeling the setbacks, while also adaptively navigating
over various terrain. In addition, our approach is able to integrate multi-
modal features to characterize terrain and automatically estimate the
importance of these features. This is all implemented in a unified
mathematical regularized optimization framework with a theoretical convergence
guarantee.

The novelty of this paper is twofold:

  * •

We propose a novel formulation to generate consistent navigational behaviors
via generating offset behaviors. We also introduce two regularization terms to
learn important terrain features and historic behavior differences to enable
robot adaptation to unstructured terrain.

  * •

We propose a new optimization algorithm to address the formulated convex
regularized optimization problem with dependent variables, which holds a
theoretical guarantee to effectively converge to the global optimal solution.

As an experimental contribution, we provide a comprehensive performance
evaluation of learning-based terrain adaptation methods by designing a set of
live robot navigation scenarios over a wide variety of individual and complex
unstructured off-road terrain.

The remainder of the paper is organized as follows. A review related work is
provided in Section II. The proposed approach is discussed in Section III.
After describing the experimental results in Section IV, we conclude the paper
in Section V.

##  II Related Work

In this section, we provide a review of existing methods of terrain
classification and robot adaptation that have been used to enhance autonomous
navigation.

###  II-A Terrain Classification

Terrain classification methods use sensory data from a robot to classify the
terrain. Many earlier methods were developed to address the specific needs of
larger vehicles [17]. For example, in the DARPA Grand Challenge, Urmson et al.
[18] designed a pre-selected set of optimal speeds for each terrain, which
uses sensory data cues collected over many months to categorize individual
types of terrain. Some methods used a pre-existing terrain map to achieve
high-speed terrain navigation [19, 20]. Techniques were also designed to use
terrain ruggedness data to evaluate navigational behaviors in an online
fashion for ground vehicles [6, 21].

Recently, learning-based methods have been used to classify terrains. For
example, an SVM classifier was used on terrain features learned from Hidden
Markov Models to identify the terrain [22]. Color-based terrain classification
was performed to generate robot navigational behaviors by labeling obstacles
[23]. However, these methods rely on a discrete categorization of terrain
types to perform robot navigation. In real-world field environments,
unstructured off-road terrains have a wide variety of characteristics and
cannot be easily categorized into distinctive types. Thus, these methods
typically do not work well to characterize complex unstructured off-road
terrain and cannot directly enable robot adaptation to unstructured terrain in
real-world field environments.

###  II-B Robot Adaptation

The second category of methods focus on enabling robots to intelligently adapt
to unstructured terrain. The general robot adaptation problem is commonly
investigated in robotics [24, 25, 26, 27, 28] using high-level behavior
models. Case-based reasoning was used to adapt a robot in rapidly changing
environments [29]. Earlier works considered ground speed as the optimization
variable and formulated a method for trading progress and velocity with
changing terrain characteristics [30]. Recently, an array of inertial and
ultrasonic sensors were used to recognize soil properties that allow robots to
perform terrain adaptation effectively [31].

Learning based methods for terrain adaptation have gained significant
attention because of their effectiveness and flexibility [32]. Early work
addressed terrain adaptation from the perspective of online learning that
updates model parameters in the execution phase [33]. Although successful,
this approach lacks the ability to fast adapt on the fly to deal with sudden
terrain changes. To overcome this challenge, methods were developed that
generate navigational behaviors according to the predicted terrain
characteristics [34]. Similarly, reinforcement learning based navigation was
developed to generate stable locomotion patterns for terrain adaptation [35].
Methods based on inverse reinforcement learning were used to mimic expert
navigation controls to achieve human-level maneuverability [13]. More
recently, a unified method for simultaneous terrain classification and
apprenticeship learning showed promising results on robot adaptation to
unstructured terrain [4].

Given the promise of the previously mentioned learning-based terrain
adaptation techniques, almost all methods focus solely on learning the
expected behaviors, without addressing the problem of generating consistent
navigation behaviors. Our approach is introduced to address this challenge
that has not been well addressed by the existing learning-based robot terrain
adaptation methods.

##  III Approach

In this section, we discuss the proposed method for consistent navigational
behavior generation and our algorithm to solve the formulated regularized
optimization problem.

###  III-A Problem Formulation

As a robot traverses over terrain, at each time step, we extract multi-modal
features from observations acquired from multiple sensors installed on the
robot (e.g., visual camera, LiDAR, and IMU). We concatenate all features
extracted at time point $t$ into a vector and denote it as
$\mathbf{x}^{(t)}\in\mathbb{R}^{q}$, where $q=\sum_{j=1}^{m}q_{j}$, $q_{j}$ is
the dimensionality of the $j$-th feature modality, and $m$ is the number of
modalities. We represent features extracted from a sequence of consecutive $c$
time points as a data instance
$\mathbf{x}=[\mathbf{x}^{(t)};\dots;\mathbf{x}^{(t-c)}]\in\mathbb{R}^{d}$,
where $d=c\times q$. We further denote the set of $n$ data instances for
training our approach as
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}]\in\mathbb{R}^{d\times n}$.

We use $\mathbf{Y}=[\mathbf{y}_{1},\dots,\mathbf{y}_{n}]\in\mathbb{R}^{r\times
n}$ to denote the expected navigational behaviors of the robot associated with
$\mathbf{X}$, where $\mathbf{y}_{i}\in\mathbb{R}^{r}$ is a vector of $r$
behavior control variables corresponding to $\mathbf{x}_{i}$. These variables
represent the behaviors that the robot is expected to execute (e.g., velocity,
motor torque, and steering angle) when traversing over the terrain. Due to
momentum, we know the robot’s behaviors executed at the present time are
dependent on previous time steps, so we estimate the robot’s behaviors
$\mathbf{y}_{i}$ using
$\mathbf{x}_{i}=[\mathbf{x}_{i}^{(t)};\dots;\mathbf{x}_{i}^{(t-c)}]$, taking
into account the history of $c$ observations. Then, the problem of
navigational behavior estimation can be formulated as:

$\min_{\mathbf{W}}\|\mathbf{Y}-\mathbf{W}^{\top}\mathbf{X}\|_{F}^{2}+\lambda_{1}\|\mathbf{W}\|_{M}$
(1)

where $\mathbf{W}\in\mathbb{R}^{d\times r}$ is a weight matrix that is
illustrated in Fig. 2. Each element $\mathbf{w}^{j(k)}_{i}\in\mathbf{W}$
indicates the importance of the $i$-th feature modality in $\mathbf{x}^{(k)}$
for generating the $j$-th navigational behavior.

Figure 2: Illustration of the weight matrix $\mathbf{W}$, which can be
decomposed into $[\mathbf{W}_{1},\dots,\mathbf{W}_{m}]$. Each $\mathbf{W}_{i}$
(e.g., the green and blue matrices on the right) represents the importance of
the $i$-th feature modality.

The first term in Eq. (1) is a loss function that encodes the error of using a
linear model parameterized by $\mathbf{W}$ to project $\mathbf{X}$ to
$\mathbf{Y}$. The second term is a regularization term named the feature
modality norm and is mathematically defined as:

$\|\mathbf{W}\|_{M}=\sum_{i=1}^{m}\|\mathbf{W}_{i}\|_{F}=\sum_{i=1}^{m}\sqrt{\sum_{j=1}^{r}\sum_{k=t}^{t-c}(\mathbf{w}^{j(k)}_{i})^{\top}(\mathbf{w}^{j(k)}_{i})}$
(2)

where $\|.\|_{F}$ is the Frobenius norm. The feature modality norm groups
together weights within a feature modality and enforces sparsity among
different modalities, thus, identifying the most descriptive features for
behavior generation. This is a critical capability for ground robot navigation
since different features typically capture different characteristics of the
unstructured terrains (e.g., color, slope, and roughness), and have different
effects toward generating navigational behaviors. The trade-off hyperparameter
$\lambda_{1}$ in Eq. (1) is used to balance the loss and the regularization
term.

The problem formulation in Eq. (1) allows ground robots to adapt their
navigational behaviors according to different terrain features. However, due
to setbacks that reduce the effectiveness of robot navigation, such as wheel
slip, heavy payload, and reduced tire pressure [36], the robot’s actual
behaviors do not match the expected behaviors.

###  III-B Consistent Navigational Behavior Learning

The key novelty of this paper is that we propose a principled method for
ground robots to generate consistent navigational behaviors that adapt to
unstructured terrain. Our approach monitors the difference between the actual
and the expected navigational behaviors caused by the setbacks, and computes
an offset to reduce the difference. This allows our approach to achieve
consistent robot behaviors without the requirement of explicitly modeling all
the setbacks.

Formally, we denote the actual behaviors executed by the robot as
$\hat{\mathbf{Y}}=[\hat{\mathbf{y}_{1}},\dots,\hat{\mathbf{y}_{n}}]\in\mathbb{R}^{r\times
n}$, where $\hat{\mathbf{y}}_{i}\in\mathbb{R}^{r}$ denotes the actual
behaviors executed by the robot when observing $\mathbf{x}_{i}$. We define
that the actual behaviors $\hat{\mathbf{y}_{i}}$ is composed by the expected
behaviors $\mathbf{y}_{i}$ and the offset behaviors
$\mathbf{v}_{i}\in\mathbb{R}^{r}$, i.e.,
$\hat{\mathbf{y}_{i}}=\mathbf{y}_{i}+\mathbf{v}_{i}$. The offset behaviors
$\mathbf{v}_{i}$ is computed as
$\mathbf{v}_{i}=\mathbf{U}^{\top}\mathbf{e}_{i}$, where
$\mathbf{e}_{i}=[(\hat{\mathbf{y}}_{i}^{(t)}-\mathbf{y}^{(t)}_{i});\dots;(\hat{\mathbf{y}}^{(t-c)}_{i}-\mathbf{y}^{(t-c)}_{i})]\in\mathbb{R}^{rc}$
denotes a vector of differences between actual and expected behaviors in the
previous $c$ time steps.
$\mathbf{U}=[\mathbf{u}^{1},\dots,\mathbf{u}^{r}]\in\mathbb{R}^{rc\times r}$
is the weight matrix, and $\mathbf{u}^{j}\in\mathbb{R}^{rc}$ indicates the
importance of $\mathbf{e}_{i}$ towards generating the $j$-th element in
$\mathbf{v}_{i}$. Using data from a history of $c$ time steps allows our
method to consider inertia (i.e., resistance to change in behaviors) during
navigation.

Then, generating consistent navigational behaviors can be formulated as:

$\min_{\mathbf{U},\mathbf{W}}\|\hat{\mathbf{Y}}-\mathbf{W}^{\top}\mathbf{X}-\mathbf{U}^{\top}\mathbf{E}\|_{F}^{2}+\lambda_{1}\|\mathbf{W}\|_{M}$
(3)

where $\mathbf{E}=[\mathbf{e_{1}},\dots,\mathbf{e}_{n}]\in\mathbb{R}^{rc\times
n}$. The loss function models the actual behavior by considering both
$\mathbf{X}$ and $\mathbf{E}$ to achieve consistent navigational behaviors.
Because of inertia, historical data from different past time steps may
contribute differently towards generating offset behaviors (e.g., a heavier
robot with bigger inertia often needs to consider a longer history). Thus, we
propose a new regularization term to explore which time steps in the
historical data are more important for generating the offset behaviors. We
name this regularization term the temporal norm, which is expressed by:

$\|\mathbf{U}\|_{T}=\sum_{k=t}^{t-c}\|\mathbf{U}^{(k)}\|_{F}=\sum_{k=t}^{t-c}\sqrt{\sum_{j=1}^{r}\|{\mathbf{u}}^{j(k)}\|_{2}^{2}}$
(4)

where
$\mathbf{U}^{(k)}=[\mathbf{u}^{1(k)},\dots,\mathbf{u}^{r(k)}]\in\mathbb{R}^{r\times
r}$ is the weight matrix, and $\mathbf{u}^{j(k)}$ indicates the importance of
the behavior difference $\mathbf{e}^{(k)}$ for generating the offset
$\mathbf{v}^{(k)}$ at the $k$-th time step. This norm groups together weights
for the vector of behavior differences at each time step, and enforces
sparsity between weights at different time steps to identify the most
important time steps.

Using both norms to generate consistent robot navigational behaviors while
identifying important feature modalities and historical time steps, the final
objective function becomes:

$\min_{\mathbf{U},\mathbf{W}}\|\hat{\mathbf{Y}}-\mathbf{W}^{\top}\mathbf{X}-\mathbf{U}^{\top}{\mathbf{E}}\|_{F}^{2}+\lambda_{1}\|\mathbf{W}\|_{M}+\lambda_{2}\|\mathbf{U}\|_{T}$
(5)

where $\lambda_{1}\geq 0$ and $\lambda_{2}\geq 0$ are the trade-off hyper-
parameters to balance the loss function and regularization terms.

###  III-C Generating Consistent Behaviors during Execution

After computing the optimal values of the weight matrices $\mathbf{W}$ and
$\mathbf{U}$ according to Algorithm 1, at each time step during execution, our
approach extracts a feature vector $\mathbf{x}\in\mathbb{R}^{d}$ from the
robot’s new observation. Then, it computes its corresponding offset behavior
by $\mathbf{v}=\mathbf{U}^{\top}\mathbf{e}$, where $\mathbf{e}$ denotes the
difference between the expected behavior $\mathbf{y}$ and the actual behavior
$\hat{\mathbf{y}}$ (e.g., measured using a pose estimation technique based
upon SLAM or visual odometry).

In addition, we predict the offset needed for the next time step so our
approach can proactively generate behaviors that take into account future
behavior differences. The predicted offset behavior $\tilde{\mathbf{v}}$ is
estimated by111Derivation is presented in the supplementary material at:
http://hcr.mines.edu/publication/TerrainAdapt_Supp.pdf:

$\tilde{\mathbf{v}}=\sum_{k=t}^{t-c}\Big{(}\big{(}\mathbf{U}^{(k)\top}\big{)}^{-1}\big{(}\mathbf{y}^{(k)}-\mathbf{W}^{(k)\top}\mathbf{x}^{(k)}\big{)}\Big{)}$
(6)

Including both offset behaviors for the current and future time steps, our
approach allows the robot to generate consistent actual navigational behaviors
by:

$\mathbf{y}=\mathbf{W}^{\top}\mathbf{x}+\begin{bmatrix}\mathbf{I}_{r}\\\
\mathbf{U}\end{bmatrix}^{\top}\begin{bmatrix}\tilde{\mathbf{v}}\\\
\mathbf{e}\end{bmatrix}$ (7)

where $\mathbf{I}_{r}\in\mathbb{R}^{r\times r}$ is an identity matrix.

1

Input : $\mathbf{X}\in\mathbb{R}^{d\times n}$,
$\mathbf{Y}\in\mathbb{R}^{r\times n}$, and $\mathbf{E}\in\mathbb{R}^{rc\times
n}$

2

Output : The weight matrices $\mathbf{W}\in\mathbb{R}^{d\times r}$ and
$\mathbf{U}\in\mathbb{R}^{rc\times r}$

3

4Initialize $\mathbf{W}\in\mathbb{R}^{d\times r}$ and
$\mathbf{U}\in\mathbb{R}^{rc\times r}$;

5while  _not converge_ do

6 Calculate the block diagonal matrix $\mathbf{Q}$ with $i$-th diagonal block
given as $\frac{1}{2\|\mathbf{W}_{i}\|_{F}}\mathbf{I}_{q_{i}}$;

7 Compute the matrix $\mathbf{W}$ according to Eq. (9);

8 Calculate the block diagonal matrix $\mathbf{P}$ with the $k$-th diagonal
block as $\frac{1}{2\|\mathbf{U}^{(k)}\|_{F}}\mathbf{I}_{r}$;

9 Compute the matrix $\mathbf{U}$ according to Eq. (11);

10return:  $\mathbf{W}\in\mathbb{R}^{d\times r}$ and
$\mathbf{U}\in\mathbb{R}^{rc\times r}$

Algorithm 1 The proposed algorithm to solve the formulated regularized
optimization problem in Eq. (5).

###  III-D Optimization Algorithm

As the second contribution, we design an iterative optimization algorithm to
obtain the optimal solution to the proposed regularized optimization problem
in Eq. (5). The optimization problem is challenging to solve in general
because of the two non-smooth structured regularization terms. The algorithm
is presented in Algorithm 1.

To solve for the optimal weight matrix $\mathbf{W}$, we minimize Eq. (5) with
respect to $\mathbf{W}$, resulting in:

$2\mathbf{X}^{\top}\mathbf{X}\mathbf{W}-2\mathbf{X}\hat{\mathbf{Y}}+2\mathbf{XU}^{\top}\mathbf{E}+\lambda_{1}\mathbf{Q}\mathbf{W}=0$
(8)

where $\mathbf{Q}$ is a diagonal matrix with the $i$-th diagonal block given
as $\frac{1}{2\|\mathbf{W}_{i}\|_{F}}\mathbf{I}_{q_{i}}$ and
$\mathbf{I}_{q_{i}}$ is an identity matrix. Then, we compute $\mathbf{W}$ in a
closed-form solution by:

$\mathbf{W}=(2\mathbf{X}^{\top}\mathbf{X}+\lambda_{1}\mathbf{Q})^{-1}(2\mathbf{X}\hat{\mathbf{Y}}-2\mathbf{XU}^{\top}\mathbf{E})$
(9)

Because $\mathbf{Q}$ and $\mathbf{W}$ are interdependent, an iterative
algorithm is required to compute them.

To compute $\mathbf{U}$, we calculate the derivative of the objective function
in Eq. (5) and set the equation to zero as:

$2\mathbf{E}^{\top}\mathbf{E}\mathbf{U}-2\mathbf{E}\hat{\mathbf{Y}}+2\mathbf{EW}^{\top}\mathbf{X}+\lambda_{2}\mathbf{P}\mathbf{U}=0$
(10)

where $\mathbf{P}$ is a diagonal matrix with the $k$-th diagonal block as
$\frac{1}{2\|\mathbf{U}^{(k)}\|_{F}}\mathbf{I}_{r}$, with $\mathbf{I}_{r}$
being an identity matrix. Then, we compute $\mathbf{U}$ in a closed-form
solution as:

$\mathbf{U}=(2\mathbf{E}^{\top}\mathbf{E}+\lambda_{2}\mathbf{P})^{-1}(2\mathbf{E}\hat{\mathbf{Y}}-2\mathbf{EW}^{\top}\mathbf{X})$
(11)

This $\mathbf{U}$ is then used to calculate $\mathbf{W}$ in the next
iteration.

Convergence. Algorithm 1 is guaranteed to converge to the global optimal
solution in the formulated regularized optimization problem in Eq. (5). The
proof is provided in the supplementary material${}^{\ref{supplementary}}$.

Complexity. As the formulated optimization problem in Eq. (5) is convex,
Algorithm 1 converges fast (e.g., within tens of iterations only). In each
iteration of our algorithm, computing Steps 3 and 5 is trivial. Steps 4 and 6
can be computed by solving a system of linear equations with quadratic
complexity.

##  IV Experiments

This section presents the experiment and implementation setups, and presents
an analysis of experimental results obtained by our approach and previous
terrain adaptation methods.

###  IV-A Experimental Setups

TABLE I: Quantitative results based on ten runs for scenarios when the robot traverses over individual types of unstructured terrain shown in Fig. 4. Successful runs (with no failures) are used to calculate the metrics of traversal time, inconsistency and jerkiness. Our approach is compared with LfD [14], MM-LfD [37] and TRAL [4]. | Failure Rate (/10) | Traversal Time (s) | Inconsistency |  Jerkiness (m/s3)
---|---|---|---|---
Terrain | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours
Grass | 0 | 0 | 0 | $\mathbf{0}$ | 17.9 | 18.2 | $\mathbf{17.4}$ | 17.5 | 2.82 | 3.06 | $\mathbf{1.84}$ | 2.11 | 79.59 | 81.42 | $\mathbf{75.18}$ | 76.50
Sand | 0 | 1 | 0 | $\mathbf{0}$ | 15.3 | $\mathbf{12.7}$ | 15.9 | 13.1 | 4.72 | 4.62 | 4.67 | $\mathbf{4.55}$ | 71.14 | 73.74 | 65.32 | $\mathbf{64.22}$
Gravels | 0 | 0 | 0 | $\mathbf{0}$ | 22.9 | 24.1 | 20.4 | $\mathbf{20.2}$ | 4.12 | 4.63 | 3.81 | $\mathbf{3.04}$ | 44.95 | 48.27 | 40.26 | $\mathbf{39.48}$
M.Rock | 1 | 3 | 0 | $\mathbf{0}$ | 33.2 | 36.9 | 29.4 | $\mathbf{28.4}$ | 7.92 | 9.59 | 4.21 | $\mathbf{2.41}$ | 141.48 | 144.22 | 113.34 | $\mathbf{111.18}$
L.Rock | 6 | 6 | 2 | $\mathbf{1}$ | 57.8 | $\mathbf{55.3}$ | 63.4 | 60.9 | 24.79 | 28.50 | 9.51 | $\mathbf{7.84}$ | 52.55 | 54.30 | 49.50 | $\mathbf{48.36}$
TABLE II: Quantitative results for scenarios when the robot traverses over complex unstructured off-road terrain shown in Fig. 5. Successful runs (with no failures) are used to calculate the metrics of traversal time, inconsistency and jerkiness. | Failure Rate (/10) | Traversal Time (s) | Inconsistency |  Jerkiness (m/s3)
---|---|---|---|---
Terrain | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours | LfD | MM-LfD | TRAL | Ours
Gr-M.Rock | 5 | 7 | 2 | $\mathbf{1}$ | 22.0 | $\mathbf{19.7}$ | 27.5 | 23.1 | 15.62 | 17.28 | 14.54 | $\mathbf{12.31}$ | 65.01 | 80.56 | 58.36 | $\mathbf{51.93}$
Gr-L.Rock | 8 | 9 | 3 | $\mathbf{3}$ | $\mathbf{27.2}$ | 27.4 | 29.4 | 28.8 | 93.53 | 101.26 | 68.87 | $\mathbf{51.16}$ | 34.96 | 40.51 | 28.22 | $\mathbf{24.55}$
M.Terrain I | 0 | 1 | 0 | $\mathbf{0}$ | $\mathbf{17.9}$ | 18.2 | 19.4 | 18.9 | 3.97 | 5.38 | 4.91 | $\mathbf{3.39}$ | 72.37 | 83.17 | 70.36 | $\mathbf{68.55}$
M.Terrain II | 5 | 7 | $\mathbf{4}$ | 5 | 23.1 | $\mathbf{18.1}$ | 30.2 | 28.5 | 93.37 | 95.47 | 80.43 | $\mathbf{78.82}$ | 54.13 | 77.49 | 52.51 | $\mathbf{47.93}$

We use the Clearpath Husky robot in our field experiments. The robot is
equipped with an Intel Reasense D435 color-depth camera and an Ouster OS1-64
LiDAR. The robot also has a variety of sensors to measure its internal states,
including IMU readings, wheel odometry, power consumption, motor speed and
battery status. If sensor readings have a lower frame rate, linear
interpolation is used to get a steady 30 Hz frame rate from all sensors.

Figure 3: Multimodal features used to characterized surrounding terrain.

To represent unstructured terrain, we implement multiple visual features
extracted from color images to describe different terrain characteristics,
including Histogram of Oriented Gradients (HOG) [38] to describe shape and
Local Binary Patterns (LBP) [39] to describe texture. We also compute an
elevation map from LiDAR data to represent the grid-wise elevation of the
terrain around the robot. These features are visualized in Fig. 3. During
training, expected navigational behaviors are provided by human operators who
control the robot to traverse unstructured terrains as fast as possible while
maintaining safety (e.g., no flipping or crashing). During training and
execution phases, actual robot navigational behaviors are estimated from LiDAR
based SLAM [40]. We use a sequence of fifteen frames (i.e., $c=15$), and
hyperparameters $\lambda_{1}=0.1$ and $\lambda_{2}=10$ for all experiments.

Figure 4: Individual types of unstructured terrain.  Figure 5: Complex
unstructured off-road terrain.

We compare our approach with several previous state of the art learning-based
robot navigation techniques, including Learning from Demonstration (LfD) for
robot navigation [14], multi-modal LfD (MM-LfD) [37], and Terrain
Representation and Apprenticeship Learning (TRAL) [4]. To comprehensively
evaluate the performance of robot navigation in a quantitative fashion, we use
four evaluation metrics:

  * •

_Failure Rate (FR)_ : This metric is defined as the number of times the robot
fails to complete the navigation task across a set of experimental trials. If
a robot flips or is stopped by a terrain obstacle, it is considered a failure.
Lower values of FR indicate better performance.

  * •

_Traversal Time (TT)_ : This metric is defined as the time taken to complete
the navigation task over given terrain. Smaller values of TT indicate better
performance.

  * •

_Inconsistency_ : This metric is defined as the error between the expected
behavior and the actual behavior in terms of robot poses (linear position and
angular). Lower values of inconsistency indicate better performance.

  * •

_Jerkiness_ : This metric is defined as the average sum of the acceleration
derivatives along all axes, with lower values indicating better performance.
Jerkiness indicates how smooth a robot can traverse over a terrain. Because
state estimation and SLAM methods (e.g., based on Kalman filters) may assume
smooth robot motions, jerkiness is a useful metric.

###  IV-B Navigating over Individual Types of Unstructured Terrain

In this set of experiments, the robot navigates over individual terrain using
off-road tracks. Each track is made up of one type of terrain and is
approximately ten meters long. Five types of terrain are used in our
experiments, which are illustrated in Fig. 4. Our approach is trained on data
collected while the robot is manually controlled by an expert to traverse the
terrain. Then, the learned model is deployed on the robot to autonomously
navigate over the terrain. Evaluation metrics for each method are computed
across ten trials on each type of terrain track.

The quantitative results achieved by our approach and the comparison to other
methods on the metric of failure rate and traversal time are presented in
Table I. For simple individual terrains, such as grass and gravel, all methods
allow the robot to successfully traverse over the terrain. However, for more
challenging terrain, especially large rocks, both LfD and MM-LfD have a high
failure rate, whereas, TRAL and our approach have a low failure rate. Our
approach only has one failure over the difficult large-rock terrain and
outperforms other tested methods. The presented traversal time is computed by
averaging the traversal time across all successful runs, i.e., it excludes the
failed trials captured by the FR metric. It is observed that all methods have
a similar traversal time. In successful runs, both LfD methods show less
traversal time compared to other methods over rocky terrain, although they
also have a much higher failure rate. Thus, the emphasis of high speed
traversal used by these methods produces an unreliable system when the robot
traverses unstructured off-road terrain in real-world field environments.

Table I also presents the quantitative results for the inconsistency and
jerkiness metrics. We observe that both LfD and MM-LfD methods do not perform
well and have higher values of inconsistency over individual types of terrain,
especially on the large-rock terrain. TRAL has lower inconsistency and
performs the best over the grass terrain. Our proposed method outperforms the
previous approaches and obtains the lowest averaged inconsistency value.
Finally, we also evaluate the tested methods using the jerkiness metric. An
interesting observation is that the medium-rock terrain causes the largest
jerkiness measure. This is caused by the fact that robots can navigate
smoothly over simple terrain types (e.g., grass, sand and gravel), but robots
have to move slowly over large-rock terrain and the slow motion may reduce
jerkiness. TRAL and our approach achieve relatively similar jerkiness values,
which significantly outperform LfD and MM-LfD.

((a)) Optimization hyperparameters

((b)) Sequence length

Figure 6: Analysis of our approach based on the inconsistency metric.

###  IV-C Navigating over Complex Off-road Unstructured Terrains

In the second set of experiments, we evaluate our approach when the robot
navigates over complex off-road unstructured terrain. The tracks in these
experiments either show transitions between different terrain types (i.e.,
grass to large rocks and grass to medium rocks) or a mixture of different
terrain types in real off-road environments (i.e., Mixed Terrain I and Mixed
Terrain II), as shown in Fig. 5. In the experiments, no additional training is
performed and the previously trained model from individual types of
unstructured terrains is used directly.

Table II presents the quantitative results obtained by our approach and the
comparison with other methods. In this set of experiments, it is observed that
each of the methods have a much higher failure rate in general, especially
over the Mixed Terrain II and the terrain with transitions from grass to large
rocks. Our approach and TRAL generally perform equally well and significantly
outperform LfD and MM-LfD in terms of failure rate. Similar to the experiments
over individual types of terrain, we observe that both LfD methods have much
small traversal time for successful runs, but they have a significantly higher
failure rate compared with TRAL and our approach. Moreover, LfD and MM-LfD are
outperformed with respect to inconsistency and jerkiness, with MM-LfD
performing worst among all tested methods, especially on the jerkiness metric.
In these tested scenarios of more complex off-road terrain, our approach
clearly outperforms other methods and obtains the state-of-the-art performance
in terms of consistency and jerkiness.

###  IV-D Discussion

Hyperparameter Analysis: The hyperparameters $\lambda_{1}$ and $\lambda_{2}$
in Eq. (5) are implemented to balance the loss function and regularization
terms. Fig. 6(a) depicts how the inconsistency metric changes given varying
$\lambda$ values based on cross validation during training. It is observed
that $\lambda_{1}\in(0.1,10)$ and $\lambda_{2}\in(1,10)$ result in good
performance in general. The best result is obtained when $\lambda_{1}=0.1$ and
$\lambda_{2}=10$. These values are used during our execution for all
experiments.

Figure 7:  Convergence.

Dependence on Frame Sequence: Our approach uses a sequence of historical
frames with length $c$ to generate consistent behaviors. Fig. 6(b) shows the
change of the inconsistency metric according to $c$. It is observed that our
approach generally performs well when $c\in(15,20)$, and we observe that the
inconsistency metric is worst when either a small number ($c<5$) or a big
number ($c>30$) is used under the sensing framerate of 30 Hz.

Convergence: Experimental results in Fig. 7 illustrate the fast, monotonically
decreasing convergence of Algorithm 1, which validates our theoretical proof.

##  V Conclusion

In this paper, we introduce a novel approach for consistent behavior
generation that enables ground robots’ actual behavior to more accurately
match expected behaviors while adapting to a variety of unstructured off-road
terrain. Our approach learns offset behaviors to compensate for the
inconsistency between the actual and expected behaviors without the need to
explicitly model various setbacks, and learns the importance of the multi-
modal features to improve the representation of terrain for better adaptation.
Additionally, we implement an optimization algorithm to solve the formulated
problem with a theoretical guarantee. Our proposed approach is extensively
evaluated in real-world field environments. Experimental results have shown
the our approach enables robots to traverse complex unstructured off-road
terrain with more consistent navigational behaviors and it outperforms
previous methods.

## References

  * [1] H.-T. L. Chiang, B. HomChaudhuri, L. Smith, and L. Tapia, “Safety, challenges, and performance of motion planners in dynamic environments,” in Robotics Research, pp. 793–808, Springer, 2020.
  * [2] L. Bruzzone and G. Quaglia, “Locomotion systems for ground mobile robots in unstructured environments,” Mechanical Sciences, vol. 3, no. 2, pp. 49–62, 2012.
  * [3] D. Silver, J. A. Bagnell, and A. Stentz, “Learning from demonstration for autonomous navigation in complex unstructured terrain,” International Journal of Robotics Research, vol. 29, no. 12, pp. 1565–1592, 2010.
  * [4] S. Siva, M. Wigness, J. Rogers, and H. Zhang, “Robot adaptation to unstructured terrains by joint representation and apprenticeship learning,” in Robotics: Science and Systems, 2019.
  * [5] F. L. G. Bermudez, R. C. Julian, D. W. Haldane, P. Abbeel, and R. S. Fearing, “Performance analysis and terrain classification for a legged robot over rough terrain,” in International Conference on Intelligent Robots and Systems, 2012.
  * [6] C. A. Brooks and K. Iagnemma, “Vibration-based terrain classification for planetary exploration rovers,” Transactions on Robotics, vol. 21, no. 6, pp. 1185–1191, 2005.
  * [7] E. Devjanin, V. Gurfinkel, E. Gurfinkel, V. Kartashev, A. Lensky, A. Y. Shneider, and L. Shtilman, “The six-legged walking robot capable of terrain adaptation,” Mechanism and Machine Theory, vol. 18, no. 4, pp. 257–260, 1983.
  * [8] E. M. DuPont, C. A. Moore, and R. G. Roberts, “Terrain classification for mobile robots traveling at various speeds: An eigenspace manifold approach,” in International Conference on Robotics and Automation, 2008\.
  * [9] R. Hudjakov and M. Tamre, “Aerial imagery terrain classification for long-range autonomous navigation,” in International Symposium on Optomechatronic Technologies, 2009.
  * [10] T. Peynot, S.-T. Lui, R. McAllister, R. Fitch, and S. Sukkarieh, “Learned stochastic mobility prediction for planning with control uncertainty on unstructured terrain,” Journal of Field Robotics, vol. 31, no. 6, pp. 969–995, 2014.
  * [11] F. Han, X. Yang, Y. Zhang, and H. Zhang, “Sequence-based multimodal apprenticeship learning for robot perception and decision making,” in International Conference on Robotics and Automation, 2017.
  * [12] P. Pastor, H. Hoffmann, T. Asfour, and S. Schaal, “Learning and generalization of motor skills by learning from demonstration,” in International Conference on Robotics and Automation, 2009.
  * [13] M. Wigness, J. G. Rogers, and L. E. Navarro-Serment, “Robot navigation from human demonstration: Learning control behaviors,” in International Conference on Robotics and Automation, 2018.
  * [14] F. Ge, W. Moore, and M. Antolovich, “Learning from demonstration using gmm, chmm and dhmm: A comparison,” in Australian Joint Conference on Artificial Intelligence, 2015.
  * [15] R. Knight, F. Fisher, T. Estlin, B. Engelhardt, and S. Chien, “Balancing deliberation and reaction, planning and execution for space robotic applications,” in International Conference on Intelligent Robots and Systems, 2001.
  * [16] C. D. B. Borges, A. M. A. Almeida, I. C. P. Júnior, and J. J. d. M. S. Junior, “A strategy and evaluation method for ground global path planning based on aerial images,” Expert Systems With Applications, vol. 137, pp. 232–252, 2019.
  * [17] A. Jumikis, “Introduction to terrain-vehicle systems,” Soil Science, vol. 110, no. 1, p. 77, 1970.
  * [18] C. Urmson, J. Anhalt, M. Clark, T. Galatali, J. P. Gonzalez, J. Gowdy, A. Gutierrez, S. Harbaugh, M. Johnson-Roberson, H. Kato, et al., “High speed navigation of unrehearsed terrain: Red team technology for grand challenge 2004,” Robotics Institute - Carnegie Mellon University, 2004\.
  * [19] S. Shimoda, Y. Kuroda, and K. Iagnemma, “Potential field navigation of high speed unmanned ground vehicles on uneven terrain,” in International Conference on Robotics and Automation, 2005.
  * [20] M. Spenko, Y. Kuroda, S. Dubowsky, and K. Iagnemma, “Hazard avoidance for high-speed mobile robots in rough terrain,” Journal of Field Robotics, vol. 23, no. 5, pp. 311–331, 2006.
  * [21] D. Sadhukhan, C. Moore, and E. Collins, “Terrain estimation using internal sensors,” in International Conference on Robotics and Automation, 2004\.
  * [22] E. Trautmann and L. Ray, “Mobility characterization for autonomous mobile robots using machine learning,” Autonomous Robots, vol. 30, no. 4, pp. 369–383, 2011.
  * [23] R. Manduchi, A. Castano, A. Talukder, and L. Matthies, “Obstacle detection and terrain classification for autonomous off-road navigation,” Autonomous Robots, vol. 18, no. 1, pp. 81–102, 2005.
  * [24] L. E. Parker, “L-ALLIANCE: Task-oriented multi-robot learning in behavior-based systems,” Advanced Robotics, vol. 11, no. 4, pp. 305–322, 1996.
  * [25] L. E. Parker, “Lifelong adaptation in heterogeneous multi-robot teams: Response to continual variation in individual robot performance,” Autonomous Robots, vol. 8, no. 3, pp. 239–267, 2000.
  * [26] B. He, S. Wang, and Y. Liu, “Underactuated robotics: a review,” International Journal of Robotics Research, vol. 16, no. 4, 2019.
  * [27] S. Nikolaidis, Y. X. Zhu, D. Hsu, and S. Srinivasa, “Human-robot mutual adaptation in shared autonomy,” in International Conference on Human-Robot Interaction, 2017.
  * [28] P. Papadakis, “Terrain traversability analysis methods for unmanned ground vehicles: A survey,” Engineering Applications of Artificial Intelligence, vol. 26, no. 4, pp. 1373–1385, 2013.
  * [29] I. Watson and F. Marir, “Case-based reasoning: A review,” The Knowledge Engineering Review, 1994.
  * [30] D. Fox, W. Burgard, and S. Thrun, “The dynamic window approach to collision avoidance,” Robotics and Automation Magazine, vol. 4, no. 1, pp. 23–33, 1997.
  * [31] S. Nabulsi, M. Armada, and H. Montes, “Multiple terrain adaptation approach using ultrasonic sensors for legged robots,” in International Conference on Climbing and Walking Robots, 2006.
  * [32] S. Thrun, “Lifelong learning algorithms,” in Learning to learn, Springer, 1998.
  * [33] A. Kleiner, M. Dietl, and B. Nebel, “Towards a life-long learning soccer agent,” in Robot Soccer World Cup, 2002.
  * [34] C. Plagemann, S. Mischke, S. Prentice, K. Kersting, N. Roy, and W. Burgard, “Learning predictive terrain models for legged robot locomotion,” in International Conference on Intelligent Robots and Systems, 2008.
  * [35] M. S. Erden and K. Leblebicioğlu, “Free gait generation with reinforcement learning for a six-legged robot,” Robotics and Autonomous Systems, vol. 56, no. 3, pp. 199–212, 2008.
  * [36] N. Sidek and N. Sarkar, “Exploiting wheel slips of mobile robots to improve navigation performance,” Advanced Robotics, vol. 27, no. 8, pp. 627–639, 2013.
  * [37] Y. Wu, R. Wang, L. F. D’Haro, R. E. Banchs, and K. P. Tee, “Multi-modal robot apprenticeship: Imitation learning using linearly decayed dmp+ in a human-robot dialogue system,” in International Conference on Intelligent Robots and Systems, 2018.
  * [38] N. Dalal and B. Triggs, “Histograms of oriented gradients for human detection,” in Conference on Computer Vision and Pattern Recognition, 2005\.
  * [39] T. Ahonen, A. Hadid, and M. Pietikainen, “Face description with local binary patterns: Application to face recognition,” Transactions on Pattern Analysis and Machine Intelligence, no. 12, pp. 2037–2041, 2006.
  * [40] T. Shan and B. Englot, “Lego-loam: Lightweight and ground-optimized lidar odometry and mapping on variable terrain,” in International Conference on Intelligent Robots and Systems, 2018.