The Self-energy of Nucleon for the Photoproduction of Pion

in the Low-energy Region

Susumu Kinpara

Institute for Quantum Medical Science
Chiba 263-8555, Japan

###### Abstract

The scattering amplitude of the photoproduction of $\pi^{+}$ is calculated
using the helicity formalism. The pseudovector coupling $\pi$-N interaction
with the non-perturbative term is applied to the perturbative expansion. The
parameters for the self-energy are taken from the results of the $\pi$-N
scattering in our previous study.

## 1 Introduction

Nucleon is a fundamental object and plays a central role in a lot of nuclear
phenomena. The assumption is made true by elucidating the interaction between
particles and the mechanism of the reaction processes. To proceed the study
the field theoretical method is essential and it is main purpose to examine
the validity of the techniques of the calculation by applying to phenomena
known as the experimental facts. For the nucleon is under the nuclear force
and it is described by the meson-exchange model the elements of the system are
mainly nucleon and pion which is endowed the lightest mass that is the longest
range of the force.
Once the interaction is prepared the calculation of the observable is made by
the perturbative expansion of the T-product. The procedure of the quantum
electrodynamics is an example for the investigation of the pion-nucleon system
to some extent. There are differences between them. In the latter case which
is the subject of the present study the interaction is the derivative type and
in usual the renormalization is not applicable. Therefore the non-perturbative
relation on the basis of the equation of the motion is valuable to draw some
conclusions about the vertex part. Since the axial-vector current is not
conserved the non-perturbative term arises apart from the cancellation between
the non-covariant terms in the perturbative calculation. It is interesting
that the non-perturbative term left includes the self-energy and the
divergence is removed along with the counter terms.
The renormalized propagator and the vertex with the non-perturbative term
makes possible to calculate any processes related to nucleon. While the
results are free from the divergences the coupling constant remains to be
uncertain. The strength is an adjustable parameter associated with the degree
of the off-shell behavior although the on-shell condition relates the
pseudovector interaction to the pseudoscalar one. It should be allowed to
shift the pseudovector coupling constant from the standard value $f\sim$ 1 due
to the effect of the non-perturbative term. As long as the knowledge of the
numerical results of us the magnitude of $f$ is required to make smaller so as
to understand the experimental results.
In spite of the modification of the $\pi$-N interaction for preparing the
renormalized propagator it is not useful to consider the property of the
$\pi$-N coupling constant since the self-energy does not depend on it in the
present lowest-order calculation. Then the scattering process with the
external line of pion is appropriate to examine the magnitude and the
dependence on the variables such as the energy. The pion-nucleon scattering
has two legs of pion on the diagram and it is a basic process capable of the
approach by the systematic formulation as has been shown in our previous study
[1].
The model of the non-perturbative method is suitable for the description of
the intermediate energy region. Meanwhile the low-energy region below the
$\it\Delta$-resonance needs another approach to avoid the inclusion of the
higher-order terms which could give the drastic change of the form of the
self-energy. The matrix inversion is a practical mean giving the low energy
parameters of the $\pi$-N scattering in terms of the self-energy. Our purpose
of this study is to investigate the role on the photoproduction of pion.

## 2 The self-energy for the helicity amplitude

The pion-nucleon system is formulated by the lagrangian of the fields and the
type of the $\pi$-N interaction is the pseudovector coupling. It is known that
the system is described by the pseudoscalar coupling too. The former
interaction is still a controversial issue because the renormalization is not
likely to give the finite result of the correction of the higher-order graphs
for the nucleon propagator. For the vertex part of the interaction these two
types are connected by the non-perturbative term containing the part of the
self-energy. In general the form of the self-energy is given as

$\displaystyle\Sigma(p)=Mc_{1}(p^{2})-\gamma\cdot p\,c_{2}(p^{2})$ (1)

in terms of the coefficients $c_{i}(p^{2})$ ($i$=1,2) as a function of the
four-momentum $p$ of nucleon. The $M$ is the nucleon mass and the
$\gamma_{\mu}$ is the gamma matrix.
Assuming the expansion of $c_{i}(p^{2})$ in the vicinity of $p^{2}=M^{2}$ the
form of $\Sigma(p)$ is determined by preparing coefficients of the series
expansion. In our previous study the scattering parameters of the $S$-wave for
the pion-nucleon elastic scattering has been used to construct $\Sigma(p)$ by
solving the equation of the matrix inversion on four unknown coefficients up
to the order of $(p^{2}-M^{2})^{2}$. The remaining two variables are dependent
on the solutions and the relations between them are settled by the condition
of the renormalization. For only the $S$-wave survives in the limit of the
pion momentum $\vec{p}^{\;2}\rightarrow 0$ of the $\pi$-N scattering the
choice of them as the input is thought to be suitable for the description of
the low-energy region.
The set of the output values ($c\equiv
c_{1}^{(0)}=c_{2}^{(0)}\,$,$\,c_{2}^{(1)}\,$,$\,c_{1}^{(2)}\,$,$\,c_{2}^{(2)}\,$)
is responsible for the construction of the propagator $G(p)$ through
$\alpha(p^{2})$ and $\beta(p^{2})$ in $G(p)=(\alpha(p^{2})\gamma\cdot
p+\beta(p^{2})M)/(p^{2}-M^{2})$. It is verified that the condition of the
renormalization $\alpha(M^{2})=\beta(M^{2})=1$ is satisfied automatically. The
$\alpha(p^{2})$ and $\beta(p^{2})$ are expanded in powers of $p^{2}-M^{2}$
similar to $c_{i}(p^{2})$. Applying the renormalized propagator to the
calculation of the amplitude the exact one is replaced with the approximate
one $\alpha(p^{2})\approx 1+\alpha^{(1)}(p^{2}-M^{2})$ eliminating the terms
of the order $O((p^{2}-M^{2})^{2})$. The procedure is also used to obtain
$\beta(p^{2})$.
These are expressed by the set of the coefficients and given as follows

$\displaystyle\qquad\qquad\quad\alpha^{(1)}=\frac{1}{4M^{2}}\cdot\frac{c^{2}}{1+c}-c^{(1)}_{2}+M^{2}(c_{1}^{(2)}-c_{2}^{(2)})\qquad\qquad\qquad\qquad$
(2)
$\displaystyle\beta^{(1)}=\alpha^{(1)}+\frac{1}{2M^{2}}\cdot\frac{c}{1+c}\qquad\qquad\qquad\qquad\qquad$
(3)

In the previous study the term of the $c_{i}^{(2)}$ has not been added [1].
Inclusion of the term makes the numerical results worse and then the terms of
the higher-order would be required to improve the results. Turning the scalar
meson $\sigma$ off the coefficient $c$ results in $c\sim-1$ and the propagator
$G(p)$ becomes too large to obtain the reasonable values as seen in the
$\pi$-N elastic scattering. The process mediated by the $\sigma$-meson is
essential to calculate the parameters of the $\pi$-N elastic scattering.
The self-energy has an effect on the vertex part to correct the scattering
amplitudes for various phenomena. In the quantum electrodynamics (QED) the
vertex part is connected to the propagator and the relation is known as the
Ward-Takahashi (W-T) identity. This relation is satisfied also in the case of
the nucleon interacting with the photon and the pion. Particularly the
anomalous interaction of the photon-nucleon-nucleon vertex is ascribed to the
identity. Using the value of the coupling parameter $f\sim 0.8$ and choose one
of the models of the self-energy the simultaneous fit to the strength of the
anomalous part of proton and neutron is attained.
Similar to the prescription of the non-perturbative relation in QED the
$\pi$-N-N vertex $\Gamma(p,q)$ is found out to accompany the term as

$\displaystyle\Gamma(p,q)=\Gamma_{0}(p,q)+G(p)^{-1}\,\gamma_{5}+\gamma_{5}\,G(q)^{-1}$
(4) $\displaystyle\Gamma_{0}(p,q)=\gamma_{5}\,\gamma\cdot(p-q)+O((f/m)^{2})$
(5)

where $\Gamma_{0}(p,q)$ is the proper vertex defined by the perturbative
expansion in terms of the ratio of $f$ and the mass of pion $m$. Hence setting
the on-shell conditions $\gamma\cdot p-M\rightarrow 0$ and $\gamma\cdot
q-M\rightarrow 0$ for the incoming ($q$) and the outgoing ($p$) momenta the
lowest-order of $\Gamma(p,q)$ reduces to the vertex of the pseudoscalar
interaction as $\Gamma(p,q)\rightarrow-2M\gamma_{5}$ with the coupling
constant $G\equiv 2Mf/m$ besides the isospin.
Our interest is the calculation of the photoproduction of single pion and to
examine whether the self-energy plays a decisive role or not. The process
$\gamma+p\rightarrow\pi^{+}+n$ is taken as an example. The other types of the
photoproduction of pion reaction are treated like the present case. They are
classified by the state of the isospin. For the $\pi^{+}$ production the
initial state is same as the compton scattering and the optical theorem is
available. The procedure is tractable when the cross section in the forward
direction is understood quantitatively by taking into account the scattering
of photon by the cloud of the virtual pions or the photon-photon-proton-proton
vertex. Alternatively the scattering amplitude is calculated by the lowest-
order perturbative expansion. The perturbative part $\Gamma_{0}(p,q)$ is
approximated by $\Gamma_{0}(p,q)\approx\gamma_{5}\,\gamma\cdot(p-q)$
neglecting the higher-order corrections.
The perturbative expansion of the $S$-matrix is used to calculate the process
of the $\pi^{+}$ production. The Feynman diagram is seen in Ref. [2]. The
central part $v_{s^{\prime}s}(p^{\prime},p)$ consists of a set of four terms
put between the Dirac spinors of the initial proton state with the four-
momentum $p$ and the spin $s$ and the final neutron state with the four-
momentum $p^{\prime}$ and the spin $s^{\prime}$

$\displaystyle
v_{s^{\prime}s}(p^{\prime},p)\equiv\bar{u}^{(s^{\prime})}(p^{\prime})\gamma_{5}\,[\,a+b\,\gamma\cdot
k\,\gamma\cdot\epsilon_{\chi}+c\,\gamma\cdot\epsilon_{\chi}+d\,\gamma\cdot
k\,]\,u^{(s)}(p)$ (6) $\displaystyle
a=a^{\prime}\,q\cdot\epsilon_{\chi}=\frac{1}{q\cdot k}\,q\cdot\epsilon_{\chi}$
(7) $\displaystyle b=-\frac{1}{2}\,(\,\frac{1}{p\cdot
k}+\alpha+\beta\,)+\frac{\kappa_{p}+\kappa_{n}}{4M^{2}}\qquad$
$\displaystyle-\frac{1}{2p\cdot k}[\,\kappa_{p}(1+\frac{p\cdot
k}{2M^{2}})-\kappa_{n}\frac{p\cdot k}{p^{\,\prime}\cdot k}(1-\frac{p\cdot
q}{2M^{2}}+\frac{m^{2}}{4M^{2}})\,]$
$\displaystyle-\frac{\kappa_{p}+\kappa_{n}}{2}(\alpha+\beta)-\frac{(\kappa_{p}-\kappa_{n})\alpha}{2M^{2}}(p\cdot
q-\frac{m^{2}}{2})-\frac{\kappa_{p}\alpha\,q\cdot k}{2M^{2}}$ (8)
$\displaystyle c=-\,\frac{p\cdot
k}{M}\,\alpha-\frac{\kappa_{p}-\kappa_{n}}{2M}-\frac{\kappa_{p}+\kappa_{n}}{2M}(\alpha+\beta)p\cdot
k+\frac{\kappa_{n}(\alpha+\beta)\,q\cdot k}{2M}$ (9) $\displaystyle
d=d^{\,\prime}\,q\cdot\epsilon_{\chi}=\frac{\kappa_{n}}{2M}\,(\,\frac{1}{p^{\,\prime}\cdot
k}-\alpha-\beta\,)\,q\cdot\epsilon_{\chi}$ (10)

in which $k$, $q$ are the four-momenta of photon and pion and
$\epsilon_{\chi}$ is the polarization vector of photon with the subscript
specifying two axes perpendicular to the direction of the photon momentum
respectively. The superscripts of $\alpha^{(1)}$ and $\beta^{(1)}$ are omitted
for the sake of the brevity. When $\alpha=\beta=0$ the
$v_{s^{\prime}s}(p^{\prime},p)$ reduces to that of the pseudoscalar coupling.
The strength of the anomalous interactions of proton and neutron are taken
from the experimental values as $\kappa_{p}$ = 1.79 and $\kappa_{n}$ = $-$1.91
respectively.
To connect $a\sim d$ to the elements of the helicity formalism [3][4] the
quantization axis of the final spin state is required to be coincident with
each other. The final helicity state is transformed to the one in the usual
$z$-axis by operating the $D$-function. The resulting scattering amplitude is
a linear combination of the helicity amplitudes on the neutron spin parallel
and anti-parallel to the direction of the momentum. Consequently the elements
$A_{+}^{J}\equiv A^{J}$, $A_{-}^{J}\equiv B^{J}$, $B_{-}^{J}\equiv C^{J}$ and
$B_{+}^{J}\equiv D^{J}$ for $J=l+\frac{1}{2}$ ($l\geq 0$) are obtained by the
integration of the integrand $\xi_{i\pm}$ and $\zeta_{i\pm}$ ($i$ = 0, 1) on
the scattering angle $z={\rm cos}\,\theta$ as

$\displaystyle A_{\pm}^{J}=\frac{A}{2}\int_{-1}^{1}dz\,(\,P_{l}(z)\pm
P_{l+1}(z)\,)(\,\xi_{0\pm}+\xi_{1\pm}\,z\,)$ (11) $\displaystyle
B_{\pm}^{J+1}=\pm\sqrt{\frac{l(l+1)}{(l+2)(l+3)}}\,B_{\pm}^{J}\qquad\qquad\qquad\qquad$
$\displaystyle\qquad+\frac{A}{2}\sqrt{\frac{l+1}{l+3}}\int_{-1}^{1}dz\,(\,P_{l}(z)-P_{l+2}(z)\,)(\,\zeta_{0\pm}+\zeta_{1\pm}\,z\,)$
(12)

in which the $P_{l}(z)$ is the Legendre polynomial of the $l$-th order. The
$\xi_{i\pm}$ and $\zeta_{i\pm}$ are given as

$\displaystyle\xi_{0\pm}=W-Y\pm(X-2Z_{0})$ (13)
$\displaystyle\xi_{1\pm}=-X\mp(2Z_{1}+W-Y)$ (14)
$\displaystyle\zeta_{0\pm}=\mp\,\zeta_{1\pm}=-X\pm(W+Y)$ (15) $\displaystyle
W=\frac{\omega\,q}{\sqrt{2}}\cdot\frac{c+b(E+M+\omega)}{(E^{\prime}+M)(E+M)}=-Z_{1}$
(16) $\displaystyle
X=\frac{q^{2}}{\sqrt{2}}\cdot\frac{a^{\,\prime}+d^{\,\prime}(E-M+\omega)}{E^{\,\prime}+M}$
(17) $\displaystyle
Y=-\frac{\omega\,q}{\sqrt{2}}\,\\{\frac{a^{\,\prime}}{E+M}-d^{\,\prime}(1+\frac{\omega}{E+M})\\}-W$
(18) $\displaystyle Z_{0}=\frac{1}{\sqrt{2}}\\{c-b(E-M+\omega)\\}$ (19)

and the overall constant is

$\displaystyle
A=\sqrt{\frac{\alpha_{e}\,G^{\,2}(E^{\,\prime}+M)\,q}{8\pi\,(E+M)\,\omega}}$
(20)

with the neutron energy $E^{\,\prime}$, the proton energy $E$ and the photon
energy $\omega$ in the center of mass system ($q\equiv|\,\vec{q}\,|$). The
$\alpha_{e}$ is the fine structure constant.

## 3 The results of the calculation in the threshold region

When the scattering angle appears in the denominator the integral is performed
for each term of the expansion series. Each of the coefficients $a^{\prime}$,
$b$, $c$ and $d^{\,\prime}$ is a function of ${\rm cos}\,\theta$ only through
$x\equiv q_{0}-q\,{\rm cos}\,\theta$. The property is helpful to examine the
threshold regions ($q\sim 0$) where the replacement $x\approx q_{0}$ is
appropriate and the coefficients are independent of $z={\rm cos}\,\theta$.
Under the approximation the series of the elements stops at $J\,$= 3/2. The
elements remaining are given as $A_{\pm}^{3/2}=A\,\xi_{1\pm}/3$,
$A_{\pm}^{1/2}=\pm A_{\pm}^{3/2}+A\,\xi_{0\pm}$ and
$B_{\pm}^{3/2}=A\,\zeta_{0\pm}/\sqrt{3}$. The other elements ($J\geq\,$5/2)
become $0$. There exists the relation $B_{\pm}^{5/2}=0$ unexpectedly due to
the cancellation between the first and the second terms of the recursion
relation in Eq. (12). The higher components of the partial wave dropped
approximately return to their original values as the energy increases.
For the helicity elements in the low energy region the restriction $J\leq 3/2$
is allowed and the cross section $\sigma$ of the process
$\gamma+p\rightarrow\pi^{+}+n$

$\displaystyle\sigma=\pi\sum_{l=0}^{\infty}\,(l+1)\,(\,|A_{+}^{J}|^{2}+|A_{-}^{J}|^{2}+|B_{+}^{J}|^{2}+|B_{-}^{J}|^{2}\,)$
(21)

consists of a several of the terms where the summation of the $B_{\pm}^{J}$
part starts from $l=1$. At the region the elements of the $J=3/2$ part has the
approximate relation known as $A^{3/2}+B^{3/2}\approx C^{3/2}+D^{3/2}\approx
0$ making possible to reduce the number of the terms furthermore. It arises
from the form of the sum $\sim X\sim O(q^{2})$ smaller than the main terms
$W$, $Y$ and $Z_{i}$ ($i\,$=$\,0,1$) in each element.
Using the $x\approx q_{0}$ approximation the calculation of $\sigma$ is
probably applicable only to the low energy region. The lowest-order
pseudoscalar model without the anomalous interaction shows the dependence of
the energy having the broad peak around the region below the resonance energy
favorable to proceed the investigation of the correction. Regarding to the
volume it is about half as large as the experimental fact and the additional
effects are necessary to describe the mechanism of the pion production
quantitatively. The anomalous interaction is not effective against the lack of
the volume and makes the energy dependence of $\sigma$ change to increase
gradually over the resonance region.
To supply the strong positive effect the self-energy is taken into account for
the diagram containing the Dirac part of the vertex of the electromagnetic
interaction. In other words the parameters of the self-energy $\alpha$,
$\beta$ and the anomalous interactions $\kappa_{i}\,(i=p,n)$ are considered as
roughly the same order and are retained up to the first-order of $b$, $c$ and
$d^{\,\prime}$. The $a^{\prime}$ does not contain these parameters from the
outset. In the actual calculation of the cross section the $\alpha$ and
$\beta$ of the second order ($\sim\kappa_{i}\alpha$ and $\sim\kappa_{i}\beta$)
are replaced with the previous values. They are associated with the derivation
of the anomalous magnetic moment to reproduce the experimental values.
The cross section $\sigma$ is calculated as a function of the laboratory
energy. While the curve has the property of the increasing tendency it does
not change to the decrease above the resonance region. It means that the set
of $\alpha$ and $\beta$ suitable for the region is different from that of the
threshold region determined by the $\pi$-N scattering. The intermediate energy
region is actually expressed by the other set on the basis of the perturbative
calculation with the non-perturbative term. In fact the theoretical value of
the anomalous magnetic moment of nucleon is obtained by the vertex correction
with the previous set of the self-energy. The numerical values are small
($\alpha,\beta\leq 1$) in comparison with those of the low-energy $\pi$-N
scattering. To remove or replace the $\sim\alpha\kappa_{i}$ and
$\sim\beta\kappa_{i}$ ($i=$$\,p$,$\,n$) terms takes account of the anomalous
interaction arising from the joint use of the vertex correction and the self-
energy of nucleon. It is reasonable that the theoretical curve overestimates
because of the coupling constant $f$ which is reduced about 20$\%$ from the
standard value to adjust the magnetic moment of proton and neutron to the
experimental values simultaneously as shown in our previous study.
As the incident energy increases it is difficult to disregard the second term
of $x$. Particularly when $q\gg m$, $x\rightarrow 0$ in the forward direction.
The correction of $a^{\,\prime}$ is important compared with $d^{\,\prime}$ in
which the denominator is $p^{\prime}\cdot k=\omega(E+\omega-x)$ and the effect
is weaken by the other terms below the intermediate energy region. The
$a^{\,\prime}$ is expanded in powers of $z\,q/q_{0}$ as
$a^{\,\prime}\,=\,\omega^{-1}\,q_{0}^{-1}+q\,\omega^{-1}\,q_{0}^{-2}\,z+O(z^{2})$.
The correction $\delta\,a^{\,\prime}=q\,\omega^{-1}\,q_{0}^{-2}\,z$ is
important since it is comparable to $\omega^{-1}\,q_{0}^{-1}$ at $q_{0}\sim
q$.
The present calculation achieves to reach the volume of $\sigma$ at the
laboratory energy $\sim 0.3\,{\rm GeV}$ and enables us to study the connection
to the $\Delta(1232)$ resonance. The application to the helicity states is
interesting to examine the branching ratio observed by the experiment. In
favor of the time reversal invariance the amplitude $T_{\lambda}$ of the decay
from the $\Delta(1232)$ resonance to the helicity $\lambda$ state is given as
$T_{1/2}\propto A_{+}^{3/2}-A_{-}^{3/2}$ and $T_{3/2}\propto
B_{-}^{3/2}-B_{+}^{3/2}$ excluding the common factor. The overall phase is
arbitrarily determined by employing the extra phase in the polarization
vector.
Due to the relation $Z_{1}=-W$ in Eq. (16) the ratio is determined to be
$T_{3/2}\,/\,T_{1/2}=-\sqrt{3}$ irrespective of the detail of the amplitude
under the $x\approx q_{0}$. For the $\lambda=3/2$ the branching ratio is
$|T_{3/2}|^{2}/(|T_{1/2}|^{2}+|T_{3/2}|^{2})=0.75$ a little smaller than the
experimental value 0.78 – 0.79 [5]. The $\delta\,a^{\,\prime}$ correction
makes the value change from 0.75 to 0.84 by breaking the simple relation
between the elements. Recently it has been verified that the use of the
$\sim\alpha\kappa_{i}$ and $\sim\beta\kappa_{i}$ terms with the $\alpha$ and
$\beta$ obtained from the method of the matrix inversion for the $\pi$-N
scattering yields the numerical value 0.78 up to the $O(z^{3})$ order of the
$\delta\,a^{\,\prime}$ correction.

## 4 Summary and remarks

The strength of the anomalous interaction is not only a parameter and it has a
dynamical origin related to the vertex correction by the pion propagator. The
pseudovector coupling constant is adjusted because of the non-perturbative
term which creates the self-energy appropriate to the anomalous terms. In
order to construct the photoproduction of pion from the threshold to the
resonance region the amplitude needs the self-energy of nucleon determined by
the method of the matrix inversion with the $\sigma$ meson exchange process.
The dispersion theoretical method would be effective along with the previous
models of the self-energy for the $\pi$-N scattering to understand the
decrease of the cross section above the resonance region in addition to the
perturbative calculation for the intermediate region.

## References

  * [1] S. Kinpara, arXiv:nucl-th/2204.09906v1.
  * [2] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory
Harper & Row (1962).

  * [3] M. Jacob and G. C. Wick, Ann. Phys. 7, 404(1959).
  * [4] R. L. Walker, Phys. Rev. 182, 1729(1969).
  * [5] Particle Data Group, Phys. Lett. B592, p897(2004).


# ProNav: Proprioceptive Traversability Estimation for Legged Robot Navigation
in Outdoor Environments

Mohamed Elnoor, Adarsh Jagan Sathyamoorthy, Kasun Weerakoon, Dinesh Manocha

###### Abstract

We propose a novel method, ProNav, which uses proprioceptive signals for
traversability estimation in challenging outdoor terrains for autonomous
legged robot navigation. Our approach uses sensor data from a legged robot’s
joint encoders, force, and current sensors to measure the joint positions,
forces, and current consumption respectively to accurately assess a terrain’s
stability, resistance to the robot’s motion, risk of entrapment, and crash.
Based on these factors, we compute the appropriate robot trajectories and gait
to maximize stability and minimize energy consumption. Our approach can also
be used to predict imminent crashes in challenging terrains and execute
behaviors to preemptively avoid them. We integrate ProNav with a vision-based
method to navigate dense vegetation and demonstrate our method’s benefits in
real-world terrains with dense bushes, high granularity, negative obstacles,
etc. Our method shows an improvement up to 50% in terms of success rate and up
to 22.5% reduction in terms of energy consumption compared to exteroceptive
based methods.

###### Index Terms:

Proprioception, Autonomous Navigation, legged robots

##  I Introduction

In recent years, autonomous legged robots have found applications in
surveillance/monitoring [1], exploration [2], and search and rescue [3]. A key
advantage of enabling such broad applications is their superior capabilities
in traversing complex terrains, ones that are inaccessible to wheeled and
tracked robots.

It is important to develop autonomous methods for navigation in complex
terrains, which can be broken down into three major categories: uneven/rocky
outdoor terrains, dense vegetation, and granular terrains like sand and mud.
The uneven or rocky terrains challenge the robot’s stability as they often
lack solid footholds with sudden variations in elevation [4]. Dense vegetation
introduces another layer of complexity, presenting risks of entanglement in
branches, dried grass, or bushes [5, 6], leading to unstable behaviors such as
slipping and tripping. The third category, granular terrains, often leads to
the robot’s legs sinking into surfaces like sand or mud due to their
deformability under the robot’s weight[7, 8]. Each of these terrain types
presents unique difficulties for legged robots, which can affect their
navigational capabilities.

Figure 1: Comparison of our method with other methods navigating a Spot robot
through a granular terrain: ProNav (trot: red, amble: green), Spot In built
Planner (orange), DWA (sky blue), and VERN (yellow). In this scenario, we
observe that only our method successfully reaches its goal due to its fast
adaptation and accurate proprioception-based traversability estimation.

To tackle these challenges, the robot must be able to accurately evaluate a
terrain’s traversability (a measure of the ease of motion) and then plan its
trajectories. Existing methods typically utilize exteroceptive modalities (RGB
images, lidar point clouds, and scans) [9, 10, 11, 12, 13] for traversability
estimation. However, such methods experience degradation in perception
accuracy in environments with high occlusions, poor illumination, scarce
features, etc. For example, the terrain geometry could be occluded by dense
vegetation and point clouds could be scattered by tall, thin grass reducing
the LiDAR accuracy. Moreover, certain entities (e.g. negative obstacles such
as ditches, potholes) and changes in a terrain’s properties (dry sand versus
wet sand) cannot be accurately detected by such modalities.

To overcome these limitations, several methods have fused exteroception with
proprioception to evaluate a terrain’s traversability [14, 15]. Proprioception
measures the state of the robot’s joint and body position and force feedback
[16], while exteroception sensing measures the state of the environment using
sensors such as cameras, LiDAR, etc. Although proprioception cannot provide a
look-ahead for the terrain, it more accurately represents the robot’s
stability on a terrain since unstable walking behaviors are reflected by large
changes in the positions, forces experienced at certain joints, and high
energy consumption. However, existing research works on traversability
analysis have predominantly focused on environments where the robot
experiences slippage [17, 18, 19], and have not handled regions where the
robot’s legs could get entrapped (e.g. in dense vegetation).

Main Contributions: To address these limitations, we propose ProNav
integrating exteroceptive sensing with proprioception for improved terrain
traversability estimation in a variety of environments (rocky, granular,
densely vegetated, etc). Those proprioceptive signals are measured from a
legged robot’s joint encoders,force, and current sensors. We integrate ProNav
with a vision-based method for navigating vegetated environments to reach a
goal. The novel components of our work include:

  * •

A novel terrain traversability estimation method using only proprioception
(joint positions, forces, current consumption) to characterize the stability,
and resistance to the robot’s motion on a terrain. Our method uses joint
proprioceptive signals and battery current for estimating traversability using
simple Principal Component Analysis (PCA) within $\sim 1s$ of walking on a new
terrain type using edge computing hardware.

  * •

A novel crash prediction mechanism that can foresee slipping, tripping, and
leg entrapment related crashes. This leads to an improvement of 50% in terms
of success rate in densely vegetated regions where all other methods
experienced difficulties in reaching the goal.

  * •

A navigation approach that computes velocities, and the appropriate gait mode
to optimize both stability and the consumed energy while traversing
challenging terrains. Our approach is integrated with a vision-based method
for navigating through vegetation, rocky and granular terrains.

##  II Related Works

In this section, we discuss the existing methods for estimating terrain
traversability, which forms the basis of our novel approach. Next, we analyze
the existing navigation and planning techniques for legged robots.

###  II-A Perception for Navigation

Autonomous robot navigation in challenging environments requires robots to
perceive the real world through their sensors. To this end, robots often
incorporate onboard exteroceptive, and proprioceptive sensors. We briefly
review the existing work on exteroceptive and proprioceptive perception in the
following sub-sections.

####  II-A1 Exteroceptive Sensors

A popular approach is the use of geometry-based methods which reconstruct a 3D
representation of the environment by using technologies such as LiDAR or
stereo cameras. A typical example of this can be found in [12], where an
interpolation technique estimates terrain shape, informing surface
characteristics such as height, slope, and roughness. Another approach, as
presented in [13], generates a 3D triangle mesh of the environment from a 3D
point cloud, which is then input into an online path planner for local
navigation. However, these geometry-based methods have limitations, including
difficulties with deformable surfaces such as sand, obstacles like tall grass,
and the risk of poor estimation [12, 2].

Concurrently, vision-based approaches have seen widespread application in
robot perception [9, 10, 11]. Previous work in semantic segmentation
categorizes terrain properties into traversable and non-traversable classes.
For instance, Guan et al [20] leverage a multi-head vision transformer
architecture to segregate the terrain into six distinct categories. Also,
traversability classification can be performed using anomaly detection from
multi-model images [21]. Even though such vision-based systems perform well
under perfect weather conditions, they often result in erroneous
classification due to motion blur [22] and lighting changes [23].

Several studies have also explored the potential of sensor fusion for terrain
classification [24, 25, 26, 27]. Notably, in [24], geometric and vision-based
techniques are used to deliver improved performance. In [26], reliability-
aware sensor fusion is performed to mitigate the performance degradation due
to cluttered sensing. Recently, [5] proposed VERN, which utilizes a
lightweight Siamese network to classify complex outdoor vegetation based on
traversability. The method in [28] employs IMU sensor data to learn surface
traction, bumpiness, and deformability using an online self-supervised
learning strategy. While this approach has shown promising results for a
number of terrains, others like rocks and bushes, with irregular
texture/structure, were not investigated.

####  II-A2 Proprioceptive sensors

In outdoor environments, exteroceptive sensors could receive noisy data
beacsue of factors such as degraded lighting conditions and occlusions. Also,
the environment can be extreme and challenging. For instance, the ground could
be covered by vegetation (e.g., short/tall grass, bushes) and the robot cannot
recognize the terrain type using vision or LiDAR. To overcome such issues,
there has been a continuous development in proprioceptive perception [29, 30].
Moreover, proprioception can be coupled with vision in legged robots as in
[14], where Fu et al. use the camera to create a cost map around the robot,
while the terrain traversability is mainly evaluated based on proprioceptive
feedback. That also helps in avoiding unexpected obstacles such as glass
walls.

More recently, Dey et al. [31] leveraged the proprioceptive information from a
legged robot’s joints to predict slip and fall events with high accuracy.
However, the robot was operated in a limited number of terrains such as rubble
and other uneven, underground terrains and not in densely vegetated
environments. Moreover, their proposed model primarily predicted slipping and
tripping and not for navigation. Our novel approach uses proprioceptive
feedback and current consumption from the actuators to also detect
entanglement in dense vegetation and recover the robot.

###  II-B Outdoor Navigation

Recently, many approaches have been proposed to leverage the agile mobility of
legged robots [32, 33] in unstructured outdoor environments, which is
challenging for wheeled robots [34]. Some of these works use cost maps to
represent the traversablity of the environments [35, 36]. Semantic Belief
Graph are utilized in [37] to train a policy for trajectory generation in
extreme environments. Besides that, Silver et al. [38] proposed an approach
based on learning from demonstration for autonomous navigation in natural
terrains. Proprioceptive feedback is also used in the literature [6, 14, 31].
In [6], Lee et al. utilized proprioceptive feedback to train a robot
controller using reinforcement learning. Their apparoch showed zero-shot
capabilitues when tested in outdoor settings. In general, proprioception
inherently lacks the capacity to provide information about terrains prior to
the robot’s interaction with them. This limitation motivates the integration
of ProNav with an exteroceptive-based navigation method, ensuring a more
comprehensive navigation strategy.

##  III Background

In this section, we explain our assumptions, define important notations used,
and our problem formulation. We also provide some background on a
exteroception-based vegetation navigation method that is used with our
proprioception-based terrain classification.

###  III-A Setup and Conventions

We assume a quadrupedal robot with 12 degrees of freedom (DOF), with 2-DOFs in
the hip, and 1-DOF in the knee of each leg. We assign numbers 1, 2, 3, and 4
to denote the front-left, front-right, rear-left, and rear-right legs
respectively, and $i$ to denote each leg. A robot coordinate frame is
established at its center of mass with positive X, Y, Z pointing forward,
left, and up respectively. Frames with similar conventions are established at
each hip and knee joint. The hip actuators are capable of moving in X and Y
directions, and the knee actuator moves along the Z direction. We measure the
positions $p^{hip-i}_{x},p^{hip-i}_{y}$, $p^{knee-i}_{z}$ velocity
$v^{hip-i}_{x},v^{hip-i}_{y}$, $v^{knee-i}_{z}$ and force
$f^{hip-i}_{x},f^{hip-i}_{y},f^{knee-i}_{z}$ exerted at a time instant $t$.
Position and velocity data at the joints are measured using encoder sensors,
and the forces experienced are measured using tactile sensors. Finally, we
assume that the current drawn ($I(t)$) from the robot’s battery can be
measured using an ammeter or a current sensor while traversing various
terrains. We define $\mathbf{X}_{t}\in\mathbb{R}^{36}$ as the set of all
positions (3), velocities (3), and forces (3) obtained from all four legs of
the robot.

###  III-B Problem Domain

The focus of our approach is to enhance the navigational capabilities of
legged robots traversing through a variety of terrains (e.g. densely
vegetated, granular, rocky) using proprioceptive feedback to predict and adapt
to changes in surface conditions. In these terrains, the robot’s legs could
slip, trip, sink, or get entrapped. We briefly explain the factors that cause
these unstable phenomena:

  * •

Poor Foothold: This causes the robot’s feet to slip in rocky or slippery
terrains because the robot’s feet do not have a firm, flat surface to support
themselves on.

  * •

Granularity: This causes the robot’s feet to sink into the terrain (e.g. sand,
mud, snow) leading to erroneous measurements of joint positions. This could
cause the robot’s controller to overcompensate to stabilize itself.

  * •

Resistance to Motion: This is typically caused by dense, pliable vegetation
that can be passed through (e.g. tall grass and bushes). Additionally, the
robot’s legs could get entangled with vegetation causing higher resistance to
motion.

To traverse various terrains, the legged robot can alternate between three
gaits: trot, crawl, and amble. Trot is the standard walking gait where the
robot walks with two legs on the ground allowing fast movements. It is stable
in hard surfaces, with moderate power consumption. On the other hand, during
crawl and amble, the robot has three legs on the ground leading to more stable
behaviors in uneven, granular, deformable surfaces. Despite their
similarities, amble has much higher velocity compared to crawl, which also
leads to a high power consumption. Amble helps traverse through environments
with high resistance to motion while also maintaining stability, which also
helps handle poor foothold terrains. Similarly, crawl maintains high stability
in granular terrains and regions with poor footholds while consuming minimal
power. Based on these definitions, our formulation can be stated as follows,

######  Formulation III.1.

To adaptively select a stable gait $g^{*}$ given collision-free, goal-directed
velocities $(v^{*},\omega^{*})$, and a set of proprioceptive signals
$\mathbf{Y}_{t}\subset\mathbf{X}_{t}$ from a legged robot to overcome
instability and prevent crashes.

##  IV ProNav: Proprioception-based Stable Navigation

In this section, we analyze and choose the relevant proprioceptive signals,
process them to characterize instabilities in terms of their
severity/magnitude and cause, and explain our gait adaptation strategy to
stabilize the robot.

###  IV-A Analysis of Proprioceptive Signals

Our goal is to choose the fewest number of proprioceptive signals that are
also excellent indicators of instability. In other words, we want to choose
the minimum subset $\mathbf{Y}_{t}\subset\mathbf{X}_{t}$, the set of all
available proprioceptive signals at time $t$, such that the terrain
traversabilty is accurately determined.

Figure 2: (a) Changes in the hip x-axis position of the robot while traversing
grass (green box), and rocks (brown box). The relatively stable readings
during grass traversal contrast the substantial fluctuations seen on the more
challenging rocky terrain due to frequent slips. (b) Force exerted by the four
actuators associated with the robot’s legs while traversing grass (green box),
and rocks (brown box). Steady readings observed on stable grass terrain
reflect ease of traversal, while the increased volatility and noticeable
spikes on the rocky terrain are indicative of increased resistance and
unevenness, causing variable load on the actuators. (c) The average current
consumption in amperes as the robot traverses concrete, grass, and sand. Lower
current consumption on concrete indicates efficient traversal. However, higher
values on sand highlight increased resistance and energy usage. Additionally,
a consistent trend in current consumption exhibited while using crawl, trot
and amble on various terrains.

Hip’s Position: Our empirical analysis revealed a strong correlation between
the amount of slip on a terrain and the change in hip position, particularly
along the X-axis and Y-axis. Figure 2 visually represents these changes as the
robot navigates three different types of terrain, each representing different
levels of traversability: grass, mulch, and rocks.

Knee’s force: Sudden peaks in the forces experienced by the robot’s knee
actuators (Fig. 2) along the Z direction indicate an absence of stable
footholds due to unevenness, causing the robot to exert more effort to
stabilize itself.

Current Consumption: The amount of current consumed while traversing various
terrains at a consistent elevation is proportional to the resistance to motion
experienced in each terrain (Fig. 2). Moreover, the robot’s gait consistently
impacts the current consumption on different terrains following the trend
$I^{Amble}>I^{Trot}>I^{Crawl}$.

###  IV-B Prepossessing Proprioceptive Signals

Our goal in preprocessing the chosen force and position data is to obtain
quantities that change drastically on various terrains, thus indicating its
properties. Vectors of the processed data are then analyzed using Principal
Component Analysis (PCA).

####  IV-B1 Preprocessing Force Data

At any time instant $t$, we consider past $n$ samples of $f^{knee-i}_{z}$ of
the $i^{th}$ leg. That is, we consider the vector
$[f^{knee-i}_{z}(t),f^{knee-i}_{z}(t-1),...,f^{knee-i}_{z}(t-n+1)]$. Next, we
obtain the mean force for the $i^{th}$ leg’s knee as
$\mu^{i}_{f}=(\sum_{j=0}^{n-1}f^{knee-i}_{z}(t-j))/n$, and then the mean force
experienced by the robot as a whole as
$\mu^{rob}_{f}=(\sum_{i=1}^{4}\mu^{i}_{f})/4$. Finally, we calculate the
difference $\Delta^{i}_{f}=\mu^{rob}_{f}-f^{knee-i}_{z}(t)$ for each leg. As
the robot walks on various terrains, $\Delta^{i}_{f}<0$ indicates that the
robot has entered a poorly traversable terrain which leads to high knee
forces, and $\Delta^{i}_{f}>0$ indicates a highly traversable terrain. To
further amplify changes in traversability, we use
$\sum_{i=1}^{4}\Delta^{i}_{f}$, and a counter that denotes the number of
spikes in the force experienced, defined as: $count=count+1$ if
$\Delta^{i}_{t}<0,i\in\\{1,2,3,4\\}$.

####  IV-B2 Preprocessing Position Data

At time instant $t$, we consider the past $m$ samples of $p^{hip-i}_{x}$. For
each leg $i$, we calculate the maximum $max_{i}$ and minimum $min_{i}$ values
of these $m$ samples and finally calculate
$\Omega^{rob}_{p-x}(t)=\sum_{i=1}^{4}\left|max_{i-x}-min_{i-x}\right|$.
$\Omega^{rob}_{p-x}$ represents the magnitude of variation in the hip
positions along the X direction. Similarly, we obtain $\Omega^{rob}_{p-y}$
along the Y direction. A high value of $\Omega^{rob}_{p-x}(t)$ or
$\Omega^{rob}_{p-y}(t)$ indicates the unavailability of stable footholds which
leads to slippage (e.g. in rocky terrains), or the presence of a granular
surface that leads to sinkage.

####  IV-B3 Processed Input Vector

We combine the processed quantities in knee forces and hip positions with the
current drawn from the robot’s battery to construct the input vector
$A\in\mathbb{R}^{9}$ to estimate terrain traversability as,

$A(t)=[\Delta^{1}_{f},\,\Delta^{2}_{f},\,\Delta^{3}_{f},\,\Delta^{4}_{f},\,\sum_{i=1}^{4}\Delta^{i}_{f},\,count,\,\Omega^{rob}_{p-x},\Omega^{rob}_{p-y},\,I].$
(1)

All the quantities on the left in equation 1 are functions of $t$. It is
omitted for readability.

###  IV-C Terrain Traversability Estimation

To estimate a terrain’s traversability using our denoised proprioceptive
signal, we first apply Principal Component Analysis (PCA) to reduce its
9-dimensions into two principal components. This method allowed us to simplify
and effectively compare different terrains based on these components.

Figure 3: PCA applied to key proprioceptive metrics (hip actuator position,
knee actuator force, and battery current) across different terrains. (a) The
variances along the two principal components between concrete (stable) and
rocks (unstable terrain). Unstable terrains lead to high variances. During
unstable navigation, the robot’s proprioceptive signals lie outside the green
circle centered at (0, 0) with a radius of $\Gamma_{unstable}$. (b) The shift
in distribution between stable navigation (grey points), before (yellow) and
after a crash (red). If the robot’s proprioceptive signals lie outside the
circle of radius $\Gamma_{crash}$, the robot is headed towards a crash.

The resulting plots of the first two principal components in Fig. 3 reflect
that the variance of the data along the two principal components
differentiates stable (low variance) and unstable (high variance) terrains
(grass versus sand, and grass versus rocks). Using variances
$\sigma^{2}_{PC1}$ and $\sigma^{2}_{PC2}$, we define the cost of traversing a
terrain as,

$f(\sigma^{2}_{PC1},\sigma^{2}_{PC2})=\sqrt{\sigma^{2}_{PC1}+\sigma^{2}_{PC2}}$
(2)

Function $f(.)$ implicitly measures both the severity of instability and the
current/power consumed while traversing a terrain. Furthermore, we observe
that the PCA outputs are capable of predicting imminent crashes with
noticeable shifts during the moments immediately before and after crashes (see
Fig. 3).

Figure 4: ProNav System Architecture for optimal gait and velocities. We
utilize hip x-axis position, Knee force and Battery as our proprioceptive
signals. Our PCA-based classifier estimate the traversability of the terrain.
The gait adaptation selects the optimal gait for stability. The camera and
Lidar are used with the integrated exteroception-based planner for obstacle
avoidance

###  IV-D Cause of Instability

Although $f(.)$ indicates the severity of a terrain’s instability, it does not
indicate the factors (poor foothold, granularity, resistance to motion)
responsible for it. To detect the cause of instability, we utilize raw
$p^{hip-i}_{x}$, and $p^{hip-i}_{y}$ values. We found that $p^{hip-i}_{y}$ is
more indicative of poor foothold and motion resistance, like in bushes.
Conversely, granular terrain causes more changes in $p^{hip-i}_{x}$.
Therefore, we define a cost function relating the robot’s stable gaits with
the causes of instability as,

$h(gait)=\begin{cases}\frac{1}{\sum_{i=1}^{4}p^{hip-i}_{y}}\,\,&\text{if gait
= Amble}\\\
\frac{1}{\sum_{i=1}^{4}\left|p^{hip-i}_{x}-\epsilon^{hip-i}_{x}\right|}\,\,&\text{if
gait = Crawl.}\end{cases}$ (3)

where $\epsilon^{hip-i}_{x}$ is a predefined threshold for $i^{th}$ hip.
$h(.)$ is designed to return a low cost value for a gait if the predominant
cause of instability can be solved using that gait.

###  IV-E Stable Gait Adaptation

To combat instability in a terrain, our approach switches to a stable gait
based on the severity of instability $f(.)$, and cause $h(.)$. We choose an
optimal switching gait $g^{*}$ as,

$\displaystyle g^{*}=\underset{g\in\\{\text{Amble,
Crawl}\\}}{\operatorname{argmin}}h(g)\,\,\,\text{if }f(.)>\Gamma_{stable}.$
(4) $\displaystyle g^{*}=\text{None},(v^{*},\omega^{*})=(0,0)\,\,\,\text{if
}f(.)>\Gamma_{crash}.$ (5)

Here, $\Gamma_{stable},\Gamma_{crash}$ denote stability and crash thresholds
respectively. An example of $\Gamma_{stable}$ and $\Gamma_{crash}$ is shown in
Fig. 3. When the magnitude of $f(.)$ goes above $\Gamma_{crash}$, it cannot be
stabilized using any gait. Therefore, ProNav prevents a crash by halting the
robot until $f()<\Gamma_{crash}$.

###  IV-F Algorithm

The designed algorithm for our navigation system is described in algorithm 1.

Algorithm 1 ProNav: Proprioception-based Traversability Estimation

1:Initialize: $\mathbf{Y}_{t}\subset\mathbf{X}_{t}$, PCA parameters,
$\Gamma_{stable}$, $\Gamma_{crash}$, $n$, $m$

2:for each time $t$ do

3: Extract positions, velocities, and forces to $\mathbf{X}_{t}$

4: Obtain $\mathbf{Y}_{t}$ from $\mathbf{X}_{t}$

5: $\mu^{i}_{f},\mu^{rob}_{f}\leftarrow\mathbf{Y}_{t}$ // Preprocessing

6: for each leg $i$ do

7: Calculate: $\Delta^{i}_{f}$, update $count$ based on $\Delta^{i}_{f}$

8: end for

9:
$\Omega^{rob}_{p-x/y}(t)\leftarrow\sum_{i=1}^{4}\left|max_{i-x/y}-min_{i-x/y}\right|$

10: $A(t)\leftarrow$ Eq. 1

11: $\sigma^{2}_{PC1},\sigma^{2}_{PC2}$ $\leftarrow PCA(A(t))$

12: $f(.)\leftarrow f(\sigma^{2}_{PC1},\sigma^{2}_{PC2})$

13: if $f(.)>\Gamma_{stable}$ then

14: Evaluate $h(gait)$

15: $g^{*}\leftarrow\underset{g\in\\{\text{Amble,
Crawl}\\}}{\operatorname{argmin}}h(g)$

16: if $f(.)>\Gamma_{crash}$ then

17: $g^{*}\leftarrow\text{None}$

18: end if

19: end if

20:end for

###  IV-G ProNav + Exteroception

Although proprioceptive modalities help estimate the traversability of the
terrain the robot is walking on, they lack look-ahead capability (whenever
terrain is visible) that exteroceptive sensing affords. That is, the
traversability of terrains ahead of the robot cannot be assessed using past
and current proprioceptive data. Therefore, fusing exteroception and
proprioception helps bring out the best of both worlds.

ProNav can be easily combined with any navigation method that uses
exteroceptive sensing such as RGB images, 3D point clouds, etc. An overview of
such a hybrid architecture is shown in Fig. 4. The collision-free, goal-
directed velocities $(v^{*},\omega^{*})$ are extracted from an exteroception-
based planner [39, 26, 5], and the gait evaluated to be the most stable for
the current scenario by ProNav (equation 4) are used by the robot.
$(v^{*},\omega^{*})$ ensures the robot’s safety in terms of avoiding solid
obstacles, and ProNav ensures walking stability.

##  V Results and Analysis

This section outlines ProNav’s implementation, our chosen evaluation
parameters, the varied test environments, and comparisons with other methods.

###  V-A Robot Setup and Dataset

Our approach is implemented on a Spot robot, a 12-degree-of-freedom (DOF)
quadruped from Boston Dynamics. The robot provids joint feedback from its 12
actuators, its odometry, and the battery current data during its operation.
Our data collection was carried out in the diverse environments of the
University of Maryland College Park campus, on different terrains including
concrete, asphalt, grass, rocks, sand, bushes, and mulch. The resulting
dataset represents approximately 7 hours of operation, during which the joint
feedback, odometry and battery current were continuously recorded. This
collection presents a detailed account of Spot’s interaction with its
environment and its energy consumption metrics.

Figure 5: Spot navigating in different outdoor terrains using: ProNav(trot:
red, amble: dark blue), Spot In built Planner (orange), DWA (sky blue), and
VERN (yellow). The black box shows the gait switching location. We observe
that ProNav chooses the optimal gait and velocities based on the terrain
traversability, leading to better success rate, lower power consumption and
less vibration cost.

###  V-B Comparison Methods

We combine ProNav with VERN [5] and compare its performance with three purely
exteroceptive navigation techniques:

  * •

Spot’s in-built planner: Uses $360^{\degree}$ RGB-D cameras to detect and
avoid obstacles.

  * •

Dynamic Window Approach (DWA) [39]: Uses 2D lidar scans for obstacle
avoidance.

  * •

VERN [5]: Uses RGB images and 3D point clouds to differentiate pliable
vegetation from solid obstacles and traverse vegetated environments.

Additionally, we also perform ablation studies by removing the joint positions
and current components from our input to our PCA-based classifier.

###  V-C Evaluation Metrics

  * •

Mean power consumption: The amount of power (in Watts) consumed from the
robot’s battery averaged over all trials.

  * •

Success Rate: The ratio of successful navigation trials where the robot was
able to reach its goal without freezing or colliding with obstacles.

  * •

Mean Velocity: The robot’s average velocity along its trajectory as it
traverses various surfaces.

  * •

Vibration Cost: The cumulative sum of deviations, measured from a predefined
stable reference terrain, in the actuator positions at the robot’s hip joints
along its trajectory.

###  V-D Test Scenarios

  * •

Scenario 1: Granular terrains (small rocks and sand).

  * •

Scenario 2: Concrete, rocks, and vegetation.

  * •

Scenario 3: Sparse tall grass, fallen logs, and trees.

  * •

Scenario 4: Dense vegetation and bushes.

Metrics | Method |  Scen. 1 |  Scen. 2 |  Scen. 3 |  Scen. 4
---|---|---|---|---|---
Mean Power Consumption (watt) | In-built system |  488 |  451 |  523 |  -
DWA[39] |  496 |  402 |  392 |  -
VERN[5] |  475 |  371 |  350 |  446
w/o current+VERN |  358 |  362 |  352 |  400
w/o position+VERN |  375 |  342 |  351 |  414
ProNav+VERN |  368 |  341 |  348 |  370
Success Rate (%) | In-built system |  40 |  10 |  10 |  -
DWA[39] |  60 |  30 |  20 |  -
VERN[5] |  90 |  70 |  80 |  60
w/o current+VERN |  100 |  70 |  60 |  50
w/o position+VERN |  100 |  80 |  70 |  70
ProNav+VERN |  100 |  80 |  100 |  90
Mean Velocity (m/s) | In-built system |  0.42 |  0.33 |  0.30 |  -
DWA[39] |  0.46 |  0.35 |  0.32 |  -
VERN[5] |  0.45 |  0.49 |  0.33 |  0.30
w/o current+VERN |  0.30 |  0.43 |  0.34 |  0.41
w/o position+VERNN |  0.22 |  0.50 |  0.31 |  0.33
ProNav+VERN |  0.28 |  0.39 |  0.25 |  0.32
Vibration Cost | In-built system |  67.2 |  56.7 |  54.1 |  -
DWA[39] |  45.1 |  38.1 |  22.3 |  -
VERN[5] |  23.1 |  23.9 |  11.7 |  25.1
w/o current+VERN |  20.8 |  12.6 |  90.2 |  5.4
w/o position+VERN |  44.4 |  2.6 |  13.3 |  10.9
ProNav+VERN |  15.2 |  2.7 |  8.1 |  4.8
Table I: Navigation performance of other methods compared to other method
using four different metrics. We observe that ProNav’s performance is the
highest in success rate. Additionally, our method leads to the lowest power
consumption and vibration cost

###  V-E Analysis and Discussion

In this section, we evaluate qualitatively and quantitatively the performance
of our method and compare it with other methods. Figure 5 provides a visual
representation of the trajectories in different terrains. Our method showcases
its superiority in navigating through granular (1), rocks (5.a), unstructured
terrains (5.b), and dense vegetation (5.c). Notably, our method adapts by
choosing crawl on sand (scenario 1) and amble in other scenarios where poor
footholds and resistance to motion dominates. In comparison, the lidar-based
DWA method and Spot’s builtin system struggle in vegetation-rich scenarios (2,
3, and 4). Specifically, DWA freezes in front of vegetation and Spot’s In-
built system crashes multiple times.

Table I presents the quantitative results of our study. We observe that our
method demonstrates optimal power consumption across all tested terrains. This
efficiency is a result of its capability in detecting instability and its
optimality in gait selection. VERN, while comparable in certain scenarios, has
increased power consumption in the fourth scenario due to its default trot
gait leading to more entanglements and consequent motion resistance.
Moreover, our method consistently records the lowest vibration costs. In
contrast, Spot’s in-built system and DWA experience higher vibration costs due
to sinkage (Scenario 1), slippage (Scenario 2), and motion resistance
(Scenario 3). In the mean velocity metric, ProNav shows a reduced pace,
particularly during the gait switch to crawl in Scenario 1. The velocities of
other methods are comparatively similar.

Table II shows navigation comparisons between different gaits (crawl and
amble) as well as ProNav with its adaptive gait adjustment. We observe that
crawl has lower power consumption and vibration cost compared to amble.
However, its application in dense vegetation presents challenges; the robot
moves slowly, leading to entanglement. This challenge is particularly
experienced in Scenario 4 for the crawl gait, where we noted elevated power
consumption and vibration costs alongside a significantly diminished velocity.
In contrast, the amble gait consistently achieves superior velocities relative
to crawl. ProNav on the other hand provides the best trade off between the avg
power consumption, vibration cost and mean velocity.

Ablation Study on Proprioceptive Signals: Our ablation analysis focused on two
proprioceptive signals of ProNav: battery current and hip joints’ positions.
In our evaluations (Table I), omitting battery current resulted in notably
delayed or incorrect traversability estimations, notably impacting power
consumption and vibration costs, especially evident in Scenario 4 in dense
vegetation. Removing hip joints’ positions also hindered performance but to
lesser extent. Despite their relative performance, neither ablated
configurations could exceed the performance of the fully-integrated ProNav
system.

Metrics | Method |  Scen. 1 |  Scen. 2 |  Scen. 3 |  Scen. 4
---|---|---|---|---|---
Mean Power Consumption (watt) | Crawl |  375 |  367 |  379 |  494
Amble |  430 |  375 |  415 |  416
ProNav |  368 |  341 |  348 |  370
Mean Velocity (m/s) | Crawl |  0.28 |  0.24 |  0.14 |  0.02
Amble |  0.34 |  0.57 |  0.27 |  0.58
ProNav |  0.29 |  0.39 |  0.25 |  0.44
Vibration cost | Crawl |  17.3 |  34.1 |  44.1 |  35
Amble |  26.4 |  23.1 |  3.15 |  3.6
ProNav |  15.2 |  2.7 |  8.1 |  8.8
Table II: Navigation performance of different gaits, Crawl and amble. ProNav
selects the optimal gait based on propriception.

##  VI Conclusion, Limitations & Future work

We present ProNav, a new method that uses proprioceptive data to evaluate
terrain’s traversability in real-time for legged robots. Our method optimizes
robotic trajectories and gaits for improved stability and reduced energy
consumption. Also, the inclusion of an advanced crash prediction system
ensures safer and more efficient navigation. We also combined ProNav with a
vision-based navigation method, which improved its performance. We validate
our method in different outdoor environments and provide a detailed comparison
with other navigational methods.

However, ProNav has some limitations. Currently, it can only assess the
stability of terrain it’s directly on. It doesn’t have a way to see and avoid
bad terrains ahead of time. To solve this, we’re thinking of adding other
sensors’ modalities that can predict what lies ahead for the robot. In the
future, we plan to work on these improvements and explore other ways to make
ProNav even better.

## References

  * [1] Z. Chen, T. Fan, X. Zhao, J. Liang, C. Shen, H. Chen, D. Manocha, J. Pan, and W. Zhang, “Autonomous social distancing in urban environments using a quadruped robot,” _IEEE Access_ , vol. 9, pp. 8392–8403, 2021.
  * [2] S. B. Goldberg, M. W. Maimone, and L. Matthies, “Stereo vision and rover navigation software for planetary exploration,” in _Proceedings, IEEE aerospace conference_ , vol. 5. IEEE, 2002, pp. 5–5.
  * [3] C. D. Bellicoso, M. Bjelonic, L. Wellhausen, K. Holtmann, F. Günther, M. Tranzatto, P. Fankhauser, and M. Hutter, “Advances in real-world applications for legged robots,” _Journal of Field Robotics_ , vol. 35, no. 8, pp. 1311–1326, 2018.
  * [4] E. Tennakoon, T. Peynot, J. Roberts, and N. Kottege, “Probe-before-step walking strategy for multi-legged robots on terrain with risk of collapse,” in _2020 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2020, pp. 5530–5536.
  * [5] A. J. Sathyamoorthy, K. Weerakoon, T. Guan, M. Russell, D. Conover, J. Pusey, and D. Manocha, “Vern: Vegetation-aware robot navigation in dense unstructured outdoor environments,” _arXiv preprint arXiv:2303.14502_ , 2023\.
  * [6] J. Lee, J. Hwangbo, L. Wellhausen, V. Koltun, and M. Hutter, “Learning quadrupedal locomotion over challenging terrain,” _Science robotics_ , vol. 5, no. 47, p. eabc5986, 2020.
  * [7] D. Han, R. Zhang, H. Zhang, Z. Hu, and J. Li, “Mechanical performances of typical robot feet intruding into sands,” _Energies_ , vol. 13, no. 8, p. 1867, 2020.
  * [8] H. Kolvenbach, C. Bärtschi, L. Wellhausen, R. Grandia, and M. Hutter, “Haptic inspection of planetary soils with legged robots,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 2, pp. 1626–1632, 2019.
  * [9] S. Fahmi, V. Barasuol, D. Esteban, O. Villarreal, and C. Semini, “Vital: Vision-based terrain-aware locomotion for legged robots,” _IEEE Transactions on Robotics_ , 2022.
  * [10] A. Agarwal, A. Kumar, J. Malik, and D. Pathak, “Legged locomotion in challenging terrains using egocentric vision,” in _Conference on Robot Learning_. PMLR, 2023, pp. 403–415.
  * [11] Y. Hu, B. Chen, and H. Lipson, “Egocentric visual self-modeling for legged robot locomotion,” _arXiv preprint arXiv:2207.03386_ , 2022.
  * [12] D. B. Gennery, “Traversability analysis and path planning for a planetary rover,” _Autonomous Robots_ , vol. 6, pp. 131–146, 1999.
  * [13] S. Pütz, T. Wiemann, J. Sprickerhof, and J. Hertzberg, “3d navigation mesh generation for path planning in uneven terrain,” _IFAC-PapersOnLine_ , vol. 49, no. 15, pp. 212–217, 2016.
  * [14] Z. Fu, A. Kumar, A. Agarwal, H. Qi, J. Malik, and D. Pathak, “Coupling vision and proprioception for navigation of legged robots,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 17 273–17 283.
  * [15] T. Homberger, L. Wellhausen, P. Fankhauser, and M. Hutter, “Support surface estimation for legged robots,” in _2019 International Conference on Robotics and Automation (ICRA)_. IEEE, 2019, pp. 8470–8476.
  * [16] A. H. Al-dabbagh and R. Ronsse, “A review of terrain detection systems for applications in locomotion assistance,” _Robotics and Autonomous Systems_ , vol. 133, p. 103628, 2020.
  * [17] J. Carius, R. Ranftl, V. Koltun, and M. Hutter, “Trajectory optimization for legged robots with slipping motions,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 3, pp. 3013–3020, 2019.
  * [18] S. Teng, M. W. Mueller, and K. Sreenath, “Legged robot state estimation in slippery environments using invariant extended kalman filter with velocity update,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 3104–3110.
  * [19] D. W. Haldane, P. Fankhauser, R. Siegwart, and R. S. Fearing, “Detection of slippery terrain with a heterogeneous team of legged robots,” in _2014 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2014, pp. 4576–4581.
  * [20] T. Guan, D. Kothandaraman, R. Chandra, A. J. Sathyamoorthy, K. Weerakoon, and D. Manocha, “Ga-nav: Efficient terrain segmentation for robot navigation in unstructured outdoor environments,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 3, pp. 8138–8145, 2022.
  * [21] L. Wellhausen, R. Ranftl, and M. Hutter, “Safe robot navigation via multi-modal anomaly detection,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 2, pp. 1326–1333, 2020.
  * [22] H. Seok and J. Lim, “Rovins: Robust omnidirectional visual inertial navigation system,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 4, pp. 6225–6232, 2020.
  * [23] M. Aladem, S. Baek, and S. A. Rawashdeh, “Evaluation of image enhancement techniques for vision-based navigation under low illumination,” _Journal of Robotics_ , vol. 2019, 2019.
  * [24] F. Schilling, X. Chen, J. Folkesson, and P. Jensfelt, “Geometric and visual terrain classification for autonomous mobile navigation,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 2678–2684.
  * [25] D. Wisth, M. Camurri, and M. Fallon, “Vilens: Visual, inertial, lidar, and leg odometry for all-terrain legged robots,” _IEEE Transactions on Robotics_ , 2022.
  * [26] K. Weerakoon, A. J. Sathyamoorthy, J. Liang, T. Guan, U. Patel, and D. Manocha, “Graspe: Graph based multimodal fusion for robot navigation in unstructured outdoor environments,” _arXiv preprint arXiv:2209.05722_ , 2022.
  * [27] A. E. B. Velasquez, V. A. H. Higuti, M. V. Gasparino, A. N. Sivakumar, M. Becker, and G. Chowdhary, “Multi-sensor fusion based robust row following for compact agricultural robots,” _arXiv preprint arXiv:2106.15029_ , 2021\.
  * [28] A. J. Sathyamoorthy, K. Weerakoon, T. Guan, J. Liang, and D. Manocha, “Terrapn: Unstructured terrain navigation using online self-supervised learning,” in _2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2022, pp. 7197–7204.
  * [29] T. Miki, J. Lee, J. Hwangbo, L. Wellhausen, V. Koltun, and M. Hutter, “Learning robust perceptive locomotion for quadrupedal robots in the wild,” _Science Robotics_ , vol. 7, no. 62, p. eabk2822, 2022.
  * [30] T. Haserbek, Z. Wen, X. Xie, P. Zhao, and W. An, “Model-free end-to-end learning of agile quadrupedal locomotion over challenging terrain,” in _2022 IEEE International Conference on Real-time Computing and Robotics (RCAR)_. IEEE, 2022, pp. 699–703.
  * [31] S. Dey, D. Fan, R. Schmid, A. Dixit, K. Otsu, T. Touma, A. F. Schilling, and A.-A. Agha-Mohammadi, “Prepare: Predictive proprioception for agile failure event detection in robotic exploration of extreme terrains,” in _2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2022, pp. 4338–4343.
  * [32] J. Truong, A. Zitkovich, S. Chernova, D. Batra, T. Zhang, J. Tan, and W. Yu, “Indoorsim-to-outdoorreal: Learning to navigate outdoors without any outdoor experience,” _arXiv preprint arXiv:2305.01098_ , 2023.
  * [33] S. Gangapurwala, M. Geisert, R. Orsolino, M. Fallon, and I. Havoutis, “Rloc: Terrain-aware legged locomotion using reinforcement learning and optimal control,” _IEEE Transactions on Robotics_ , vol. 38, no. 5, pp. 2908–2927, 2022.
  * [34] P. Biswal and P. K. Mohanty, “Development of quadruped walking robots: A review,” _Ain Shams Engineering Journal_ , vol. 12, no. 2, pp. 2017–2031, 2021.
  * [35] D. V. Lu, D. Hershberger, and W. D. Smart, “Layered costmaps for context-sensitive navigation,” in _2014 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2014, pp. 709–715.
  * [36] T. Overbye and S. Saripalli, “Path optimization for ground vehicles in off-road terrain,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 7708–7714.
  * [37] M. F. Ginting, S.-K. Kim, O. Peltzer, J. Ott, S. Jung, M. J. Kochenderfer, and A.-a. Agha-mohammadi, “Safe and efficient navigation in extreme environments using semantic belief graphs,” _arXiv preprint arXiv:2304.00645_ , 2023.
  * [38] D. Silver, J. A. Bagnell, and A. Stentz, “Learning from demonstration for autonomous navigation in complex unstructured terrain,” _The International Journal of Robotics Research_ , vol. 29, no. 12, pp. 1565–1592, 2010\.
  * [39] D. Fox, W. Burgard, and S. Thrun, “The dynamic window approach to collision avoidance,” _IEEE Robotics & Automation Magazine_, vol. 4, no. 1, pp. 23–33, 1997.