#  Study of semi-inclusive charged-current electron and muon neutrino
scattering from 40Ar in the energy range of the MicroBooNE experiment

J. M. Franco-Patino  Departamento de Física Atómica, Molecular y Nuclear,
Universidad de Sevilla, 41080 Sevilla, Spain  Dipartimento di Fisica,
Università di Torino, Via P. Giuria 1, 10125 Turin, Italy  INFN, Sezione di
Torino, Via Pietro Giuria 1,10125, Turin, Italy  S. Dolan  CERN, European
Organization for Nuclear Research, CH-1211 Geneva, Switzerland  R. González-
Jiménez  Grupo de Física Nuclear, Departamento de Estructura de la Materia,
Física Térmica y Electrónica and IPARCOS, Facultad de Ciencias Físicas,
Universidad Complutense de Madrid, CEI Moncloa, Madrid 28040, Spain  M. B.
Barbaro  Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, 10125
Turin, Italy  INFN, Sezione di Torino, Via Pietro Giuria 1,10125, Turin, Italy
J. A. Caballero  Departamento de Física Atómica, Molecular y Nuclear,
Universidad de Sevilla, 41080 Sevilla, Spain  Instituto de Física Teórica y
Computacional Carlos I, Granada 18071, Spain  G. D. Megias  Departamento de
Física Atómica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla,
Spain

###### Abstract

In this work we present a full comparison of all semi-inclusive muon and
electron neutrino cross sections with 40Ar target measured by the MicroBooNE
Collaboration with the predictions of an unfactorized model based on the
relativistic distorted wave impulse approximation (RDWIA) and the SuSAv2-MEC
model implemented in the neutrino event generator GENIE. The predictions based
on the RDWIA approach with a realistic description of the initial state and a
phenomenological relativistic complex optical potential for the description of
final state interactions describe better the measured cross sections than
GENIE-SuSAv2 and RDWIA with a purely real potential.

##  I Introduction

Neutrino-nucleus interactions are one of the key components in neutrino
oscillation experiments [1]. Uncertainties associated with nuclear modeling
are an important source of systematic error in both current neutrino
oscillation experiments, NOvA [2] and T2K [3], and future programs and
experiments such as SBN [4], DUNE [5], and Hyper-Kamiokande [6]. Many future
neutrino experiments, including DUNE [7, 8, 9] and the SBN [4] program, will
employ liquid argon time projection chambers (LArTPCs) as detectors. As a
consequence, neutrino-argon cross-section measurements have great importance,
even more considering that the main focus of neutrino-nucleus measurements in
the past has been lighter nuclei like 12C and 16O. The aim of accelerator-
based neutrino oscillation experiments is generally the measurement of charged
current meson-less (CC0$\pi$) neutrino interactions with complex nuclei. In
low-energy experiments like T2K or MiniBooNE charged-current quasielastic
(CCQE) scattering of the neutrino by a single nucleon at rest with some fixed
binding energy is assumed to reconstruct the neutrino energy from the
kinematics of the lepton detected in the final state. Then corrections due to
nuclear effects and non-CCQE contributions to the CC0$\pi$ signal are made to
estimate the energy reconstruction biases. These nuclear effects include
initial-state physics and final-state interactions (FSI), while the non-CCQE
contributions correspond to two-particle-two-hole (2p2h) interactions, where
the neutrino interacts with a pair of bound nucleons that are highly
correlated, and interactions that produce a pion that is absorbed inside the
nuclear medium. By detecting more particles in the final state, for instance a
lepton and an ejected proton, we can obtain measurements that are highly
sensitive to nuclear effects relative to inclusive measurements where only the
final state lepton is detected. This kind of reactions, denominated semi-
inclusive reactions [10], has been studied within the plane-wave impulse
approximation (PWIA) [11, 12, 13] and including FSI using the relativistic
distorted wave impulse approximation (RDWIA) [14, 15]. From the experimental
point of view, T2K [16] and MINER$\nu$A [17, 18] collaborations have published
$\nu_{\mu}$-CC0$\pi$ cross section measurements on 12C with one muon and at
least one proton in the final state (denoted CC0$\pi$Np).

In Ref. [15] we analyzed the T2K and MINER$\nu$A CC0$\pi$Np measurements on
12C within the unfactorized RDWIA approach.In this work we extend the analysis
to the latest semi-inclusive MicroBooNE measurements on 40Ar with two
different topologies: zero pions, one lepton and at least one proton
(CC0$\pi$Np) [19, 20] and zero pions, one lepton and exactly one proton
(CC0$\pi$1p) [21] in the final state. As we did in [15] for 12C, in this paper
we will also compare the semi-inclusive 40Ar measurements with the predictions
from the inclusive model SuSAv2 [22, 23, 24], based on superscaling [25], that
has been implemented in the neutrino event generator GENIE [26, 27]. Strictly
speaking, this model is only capable to predict inclusive cross sections as
function of the leptonic variables. However, assuming a factorization of the
leptonic vertex and the initial nuclear state, it is possible for an inclusive
model implemented in a neutrino event generator to make predictions about the
ejected proton kinematics [28, 29]. To the quasielastic cross sections
evaluated in the above mentioned approaches we will add the contribution of
meson exchange current (MEC) in the 2p2h channel (following the approach of
Refs. [30, 31, 32]) and the pion absorption contribution (following the
approach of [33, 34]). These non-QE contributions will be calculated with
GENIE for a full comparison with the available cross-section measurements.

The paper is organized as follows: In Section II we summarize the basic
formalism for semi-inclusive neutrino-nucleus processes emphasizing the
ingredients of the RDWIA approach. We discuss the model for the initial state
and the description of the final state interactions using different
approaches. Section III contains a detailed analysis of the results obtained
comparing the theoretical predictions and data for CC0$\pi$Np and CC0$\pi$1p
topologies. We consider both muon and electron neutrino scattering processes
on 40Ar. Finally, in Section IV we draw our main conclusions.

##  II Semi-inclusive Neutrino-nucleus quasielastic scattering within RDWIA

Following previous publications [14, 15, 35], in this section we briefly
summarize the formalism to describe the one proton knockout channel where an
incoming neutrino with momentum $\mathbf{k}$ interacts with a nucleus $A$, and
a lepton and a proton, with momenta $\mathbf{k^{\prime}}$ and
$\mathbf{p_{N}}$, respectively, are detected in coincidence. In the laboratory
frame, the flux-averaged six-differential semi-inclusive cross section is
given by:

$\displaystyle\left<\frac{d\sigma}{dk^{\prime}d\Omega_{k^{\prime}}dp_{N}d\Omega^{L}_{N}}\right>$
$\displaystyle=\frac{G_{F}^{2}\cos^{2}{\theta_{c}}k^{\prime
2}p_{N}^{2}}{64\pi^{5}}\int
dk\,\Phi(k)\frac{W_{B}}{E_{B}f_{\text{rec}}}L_{\mu\nu}H^{\mu\nu},$ (1)

where $\Omega_{k^{\prime}}$ and $\Omega_{N}^{L}$ are, respectively, the solid
angles of the final lepton and the ejected proton, $\Phi(k)$ is the neutrino
energy distribution (flux), the residual nucleus $B$ can be left in an excited
state with invariant mass $W_{B}$ and total energy $E_{B}$ , $L_{\mu\nu}$ and
$H^{\mu\nu}$ are the leptonic and hadronic tensors, and $f_{\text{rec}}$ is
the recoil factor. All the information about initial-state dynamics and FSI is
contained inside the hadronic tensor $H^{\mu\nu}$, which is built as the
bilinear product of the matrix elements of the nuclear current operator
between the initial and final nuclear states [14, 15, 35]. Assuming that the
impulse approximation is valid, the initial neutrino interacts with only one
neutron of the target that is knocked out and turned into a proton. Then the
proton travels through the residual nucleus suffering FSI until it exits the
nucleus.

In RDWIA the initial nucleons are described by a relativistic bound-state wave
function obtained within the relativistic mean field (RMF) approach [36]. We
follow a procedure similar to the one described in [14, 15] and use a
realistic missing energy profile $\rho\left(E_{m}\right)$ for 40Ar where each
of the seven independent-particle shell model (IPSM) states $\alpha$ are
modeled as Gaussian distributions, with a central value $\mu_{\alpha}$,
standard deviation $\sigma_{\alpha}$ and occupancy $S_{\alpha}$. An additional
$s$-wave shell, called background, is included to account for the correlated
nucleons that are not in the IPSM states.

The values of the parameters are extracted from $\left(e,e^{\prime}p\right)$
experimental data using titanium as target [37] and are summarized in Table 1.
Since the parameterization extracted from exclusive electron data is for
protons and we require the neutrons missing energy profile, the positions of
the Gaussian distributions $\mu_{\alpha}$ shown in Table 1 except the
$1f_{\frac{7}{2}}$ shell are shifted by an amount equal to the difference
between the neutrons and protons RMF eigenvalues for 40Ca (the closest
symmetric nucleus), namely
$E^{n}_{\alpha}\left({{}^{40}}\textnormal{Ca}\right)-E^{p}_{\alpha}\left({{}^{40}}\textnormal{Ca}\right)$.
Lastly for the $1f_{\frac{7}{2}}$ shell we set it to the experimental neutron
separation energy and shift the rest of the shells by 1 MeV to conserve the
difference between the shells positions. This value, 1 MeV, corresponds to the
difference between the 1$f_{7/2}$ shell energy given by the RMF model and the
experimental neutron separation energy.

Within the RDWIA approach FSI are included by solving the coupled differential
equations derived from the Dirac equation with scalar and vector potentials
that include the short-range strong interaction and the Coulomb potential.
Regarding the potential that describes the strong interaction we consider two
possibilities: a phenomenological complex relativistic optical potential (ROP)
fitted to reproduce elastic proton-nucleus scattering data and the same RMF
potential used to describe the initial state but multiplied by a
phenomenological function that weakens the potential for increasing nucleon
momenta [38, 39] (ED-RMF). The parameterization of the ROP used in this work
is the energy-dependent A-independent calcium parameterization (EDAI-Ca) [40].

$\alpha$ |  |  $\mu_{\alpha}$ (MeV) |  |  $\sigma_{\alpha}$ (MeV) |  | $S_{\alpha}$
---|---|---|---|---|---|---
$1s_{1/2}$ |  | 63.602 |  | 11.63 |  | 1.07
$1p_{3/2}$ |  | 48.899 |  | 5.52 |  | 0.725
$1p_{1/2}$ |  | 42.568 |  | 3.17 |  | 1.0
$1d_{5/2}$ |  | 24.554 |  | 2.17 |  | 0.375
$1d_{3/2}$ |  | 20.918 |  | 7.0 |  | 0.6975
$2s_{1/2}$ |  | 21.383 |  | 7.0 |  | 1.0
$1f_{7/2}$ |  | 9.869 |  | 8.0 |  | 0.765
Table 1: Parameterization of the missing energy distribution for the 22
neutrons in 40Ar. The missing energy distributions are modeled as Gaussian
distributions with $\mu_{\alpha}$ the mean value and $\sigma_{\alpha}$ the
standard deviation. The spectroscopic factors $S_{\alpha}$ give the relative
occupancy of the shell respect to the pure shell model occupancy. The
parameterization for 40Ar was taken from the ${}^{48}_{22}$Ti distribution
obtained from the analysis $\left(e,e^{\prime}p\right)$ experimental data
[37]. A correction based on RMF results was applied to obtain the
$\mu_{\alpha}$ values for neutrons (see the text). The parameterization of the
background is the same as in [14].

##  III Results and discussion

In this section we show MicroBooNE CC0$\pi$Np [19, 20] and CC0$\pi$1p [21]
measurements compared with the quasielastic predictions using the unfactorized
RDWIA approach with different treatments of FSI and the implementation of the
SuSAv2 model in GENIE [28]. To the quasielastic results we add the 2p2h-MEC
calculated with the model [30] and implemented with GENIE [31, 32] and the
pion absorption [33] contribution calculated with GENIE. The processing of
GENIE output and its comparison to experimental data was made using the
NUISANCE framework [41]. Phase space restrictions applied for the comparison
with the different experimental measurements are summarized in Table 2. The
RDWIA results will be presented with colored bands that show the uncertainty
in the theoretical calculation due to the uncertainty on the modeling of the
40Ar missing energy profile. The parameters that model this profile, shown in
Table 1, have been assigned the uncertainties summarized in Table 3. The bands
are constructed by randomly sampling the values of the missing energy profile
parameters within their uncertainties with a uniform probability distribution.
The number of neutrons in the background is such that the total number is
fixed to 22, and the calculation is done only if the background contains
between 15-25% of the 22 neutrons, which is consistent with previous studies
[42, 43, 44]. The modeled missing energy profile is shown in Fig. 1 (red band)
compared with the RMF predictions (blue lines). A $\chi^{2}$-based analysis is
presented in next section when discussing the results obtained for the cross
sections.

1$\mu$CC0$\pi$Np1$\mu$CC0$\pi$Np |  | $k^{\prime}$ |  |  | $\cos{\theta_{l}}$ |  |  | $p_{N}$ |  |  | $\cos{\theta_{N}^{L}}$ |  |  | $\phi_{N}^{L}$ |  |  | $\theta_{\mu p}$ |  |  | $\delta p_{T}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
All analyses |  |  $>0.1$ GeV |  |  | - |  |  | 0.3-1.2 GeV |  |  | - |  |  | - |  |  | - |  |  | - |
1eCC0$\pi$Np1eCC0$\pi$Np |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
All analyses |  |  $>30.5$ MeV |  |  | - |  |  |  $>$ 0.3105 GeV |  |  | - |  |  | - |  |  | - |  |  | - |
1$\mu$CC0$\pi$1p1$\mu$CC0$\pi$1p |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
All analyses |  | 0.1-1.5 GeV |  |  | $-0.65<\cos{\theta_{l}}<0.95$ |  |  | 0.3-1.0 GeV |  |  | $>0.15$ |  |  | 145-215∘ |  |  | 35-145∘ |  |  |  $\delta p_{T}<0.35$ GeV |

Table 2: Phase-space restrictions applied to $\nu_{\mu}$-40Ar CC0$\pi$Np [19] and CC0$\pi$1p [21] and $\nu_{e}$-40Ar CC0$\pi$Np [20] cross section measurements performed by MicroBooNE collaboration. The opening angle $\theta_{\mu p}$ is defined as the angle between the muon and the ejected proton and $\delta p_{T}=\left|{\bf k^{\prime}_{T}}+{\bf p_{N,T}}\right|$ is the transverse momentum imbalance [45] defined as the sum of the projections in the plane perpendicular to the neutrino direction of the muon and proton momenta. The index ”L” over the proton angles means they are defined in the laboratory frame (neutrino direction fixed in the $\hat{z}$ axis). $\alpha$ |  |  $\Delta\mu_{\alpha}$ ($\%$) |  |  $\Delta\sigma_{\alpha}$ ($\%$) |  |  $\Delta S_{\alpha}$ ($\%$)
---|---|---|---|---|---|---
$1s_{1/2}$ |  | 7 |  | 20 |  | 70-90
$1p_{3/2}$ |  | 10 |  | 20 |  | 70-90
$1p_{1/2}$ |  | 12 |  | 20 |  | 70-90
$1d_{5/2}$ |  | 10 |  | 20 |  | 50-80
$1d_{3/2}$ |  | 10 |  | 50 |  | 50-80
$2s_{1/2}$ |  | 10 |  | 33 |  | 50-80
$1f_{7/2}$ |  | - |  | 50 |  | 50-80
Table 3:  Uncertainties associated to the parameterization of the missing
energy profile for 40Ar shown in Table 1. The position of the
$1f_{\frac{7}{2}}$ shell was set to the experimental neutron separation energy
with a negligible uncertainty. The number of nucleons present in the
background is set to be between 15-25$\%$ of the 22 neutrons in 40Ar.
$\;\Delta S_{\alpha}$ is the uncertainty in $\%$ associated to a shell
$\alpha$. Therefore, $\Delta S_{\alpha}=100\%$ means that the occupancy of the
shell can be up to twice the central value presented in Table. 1.

(a)  (b)

Figure 1: Missing energy profile of 40Ar in linear scale (top) described by
the parameterization given in Tables 1 and 3. The colored band correspond to
the uncertainties summarized in Table 3, the vertical blue lines show the
positions of the seven RMF shells and the black dashed line shows the mean
value of the distribution. The same profile but in logarithmic scale (bottom)
is also presented to show better the differences in the peaks position between
the RMF model and the profile used in this work.

###  III.1 CC0$\pi$Np

In Fig. 2 we compare the two RDWIA models previously described (ROP and ED-
RMF) and the GENIE-SuSAv2 predictions with MicroBooNE 1$\mu$CC0$\pi$Np data
for 40Ar [19]. The cross sections are shown as function of the final muon and
proton kinematics and also the opening angle $\theta_{\mu p}$, which is
defined as the angle between the muon and the ejected proton momenta. The
experimental cross sections are given as function of reconstructed variables,
while our models predict the results as function of true variables. Therefore,
we have applied the smearing matrix [19] to all the theoretical results shown
in Fig. 2. The cross sections as function of $k^{\prime\textnormal{reco}}$ and
$p_{N}^{\textnormal{reco}}$ are the ones more affected by the smearing, the
matrices for which are shown in Fig. 3. In particular, it should be noted that
the kinematic bin associated to largest muon and proton momenta have a large
contribution coming from the non-diagonal terms of the smearing matrix which
produces an increase of the theoretical cross sections in the last bin of both
$k^{\prime\textnormal{reco}}$ and $p_{N}^{\textnormal{reco}}$ distributions
and an overestimation of the experimental cross sections, specially in the
$k^{\prime\textnormal{reco}}$ distribution. All the models presented in the
$k^{\prime\textnormal{reco}}$ distribution in Fig. 2 exhibit large $\chi^{2}$
values compared with the degrees of freedom (d.o.f), which is mainly caused by
the last bin. Taking as reference the ROP model, the $\chi^{2}/\textit{d.o.f}$
ratio drops from 177/6 to 6.5/5 if we eliminate the last bin from the
$\chi^{2}$ calculation.

The shape of the $p_{N}^{\textnormal{reco}}$ distribution is correctly
reproduced by the ED-RMF or ROP models once the 2p2h and other contributions
are taken in account, although the ED-RMF model overestimates the measurements
in the $0.65<p_{N}^{\textnormal{reco}}<0.9$ GeV range. It is interesting to
note that the GENIE-SuSAv2 model overestimates the experimental measurement in
almost the entire range of $p_{N}^{\textnormal{reco}}$.

The shape and magnitude of the $\cos{\theta_{l}}^{\textnormal{reco}}$ and
$\cos{\theta_{N}^{L}}^{\textnormal{reco}}$ angular distributions (Fig. 2) are
well described by all the models except at very forward muon scattering angles
and $\cos{\theta_{N}^{L}}^{\textnormal{reco}}<0.0$ where all the models
overestimate the cross-section measurements. Regarding the $\theta_{\mu
p}^{\textnormal{reco}}$ distribution, all models overestimate the data for
$\theta_{\mu p}^{\textnormal{reco}}<1.5$ rad. Moreover, the peak position is
shifted to the left with respect to the data. Both occurrences are mainly due
to the large contribution associated to pion production.

Figure 2: MicroBooNE CC$0\pi$Np $\nu_{\mu}$-40Ar cross sections as function of
the reconstructed muon and proton momenta and scattering angles and the
opening angle $\theta_{\mu p}^{\textnormal{reco}}$. All curves include the
2p2h and pion absorption contributions (shown separately) evaluated using
GENIE. Experimental results are from [19]. The bands, drawn for the ED-RMF and
ROP models, represent the uncertainties associated with the modeling of the
initial nuclear state (see the text). The $\chi^{2}/\textit{d.o.f}$ ratio is
given in brackets in the legend of each distribution.

(a)  (b)

Figure 3: Smearing matrix of the reconstructed variables
$k^{\prime\textnormal{reco}}$ (top) and $p_{N}^{\textnormal{reco}}$ (bottom)
taken from [19]. In both cases, higher bin number means higher value of the
momentum.

In Fig. 4 we compare the different theoretical models with MicroBooNE
1eCC0$\pi$Np data on 40Ar [20] as function of the electron energy and
scattering angle and the final proton kinetic energy and scattering angle.
Additionally, for the $T_{N}$ distribution presented in Fig. 4, MicroBooNE
collaboration provides one extra data point (0 $<T_{N}<$ 50 MeV) that
corresponds to events with one electron, no protons above $T_{N}$ = 50 MeV
threshold and at least one proton below the threshold. Although the
experimental measurements are statistically limited and the error bars are
large, the ROP model seems to describe better all the measurements presented
in Fig. 4. It is interesting to point out the similar predictions by GENIE-
SuSAv2 and ED-RMF of all the distributions except $T_{N}$ below 0.2 GeV where
GENIE-SuSAv2 overestimates the measurements. Notice that $T_{N}=0.2$ GeV is
close to $p_{N}=0.6$ GeV, which matches the point where GENIE-SuSAv2 and ED-
RMF models begin to differentiate in the $p_{N}^{\textnormal{reco}}$
distribution shown in Fig. 2.

Figure 4: MicroBooNE CC$0\pi$Np $\nu_{e}$-40Ar cross sections as function of
the electron scattering angle and energy ($E_{l}$) and proton kinetic energy
($T_{N}$) and scattering angle. All curves include the 2p2h and pion
absorption contributions (shown separately) evaluated using GENIE.
Experimental results are from [20]. The bands, drawn for the ED-RMF and ROP
models, represent the uncertainties associated with the modeling of the
initial nuclear state (see the text). The single black point in the $T_{N}$
distribution between $0<T_{N}<50$ MeV corresponds to an extra 1e0p0$\pi$ (one
electron, no protons above 50 MeV threshold and at least one proton with
momentum below 50 MeV threshold) measurement performed by MicroBooNE [20]. For
this single point, additional phase space restrictions on the electron energy
($E_{l}>$ 0.5 GeV) and the electron scattering angle ($\cos{\theta_{l}}>$ 0.6)
are applied. The $\chi^{2}/\textit{d.o.f}$ ratio is given in brackets in the
legend of each distribution. The $\chi^{2}/\textit{d.o.f}$ analysis of the
$T_{N}$ distribution is performed with the 1eCC$0\pi$Np data.

###  III.2 CC0$\pi$1p

The MicroBooNE CC0$\pi$1p $\nu_{\mu}-^{40}$Ar measurements are shown in Figs.
5 and 6 as function of the muon and proton kinematics, together with the RDWIA
and GENIE-SuSAv2 predictions.

A noticeable difference respect to the CC0$\pi$Np topology is the negligible
contribution of the 2p2h channel in all the distributions. The reason is the
kinematic cuts applied to the CC0$\pi$1p signal which are summarized in Table
1. From a theoretical point of view, the CC0$\pi$1p topology, leaving aside
the non-quasielastic contributions, is closer to the picture drawn by the ROP
model, in which the imaginary part of the optical potential subtracts all the
inelastic nuclear FSI, leaving only the elastic channel (i.e. the outgoing
proton interacting only elastically with the residual system). The inclusion
of the background part of the spectral function introduces, in an effective
way, states with multiple nucleons being knocked out. Note that this
contribution is very minor in the cross sections shown in Figs. 5 and 6, in
fact, the result obtained after its subtraction is contained within the
colored band shown for both RDWIA predictions. The results presented as
function of the proton kinematics show a good agreement between the ROP and
data, while the ED-RMF and GENIE-SuSAv2 predictions overestimate the measured
cross sections for most of the bins. Regarding the lepton kinematics, the bins
around the peak of the $k^{\prime}$ distributions are slightly underestimated
by the ROP but the ED-RMF and GENIE-SuSAv2 overestimate the data in the rest
of the bins.

In Fig. 5, we also present the predictions as function of the reconstructed
neutrino energy and $Q^{2}_{\text{CCQE}}$, which are defined as follows

$\displaystyle E_{\nu}^{\textnormal{cal}}=E_{l}+T_{N}+40\;\textnormal{MeV}\,,$
$\displaystyle
Q^{2}_{\textnormal{CCQE}}=\left(E_{\nu}^{\text{cal}}-E_{l}\right)^{2}-\left(\mathbf{k}-\mathbf{k^{\prime}}\right)^{2}$
(2)

with $E_{l}$ the muon energy and $T_{N}$ the kinetic energy of the ejected
proton. Both RDWIA calculations tend to underestimate (ROP) or overestimate
(ED-RMF) the measurements as function of $Q^{2}_{\text{CCQE}}$, being the ED-
RMF prediction closer to data in the bin that excludes forward muon angles,
i.e. $-0.65<\cos{\theta_{l}}<0.8$. In the case of the cross section as
function of $E_{\nu}^{\text{cal}}$, all the models overpredict the data in the
tail of the distribution (large $E_{\nu}^{\text{cal}}$-values).

Finally, the $\cos{\theta_{l}}$ distribution is shown in Fig. 6. All the
predictions are within the experimental uncertainty except for the forward
angle bin that is overestimated. However, a recent work by the MicroBooNE
collaboration [46] shows a $\cos{\theta_{l}}$ distribution that is reproduced
correctly by different neutrino generators. This suggests that the discrepancy
observed in Fig. 6 at small muon scattering angles might be due to the use of
an old version of the GENIE configuration that accounts for efficiency
corrections and beam-induced backgrounds.

Figure 5: MicroBooNE CC$0\pi$1p $\nu_{\mu}$-40Ar cross sections as function of
the muon and proton momenta, proton polar scattering angle,
$Q^{2}_{\textnormal{CCQE}}$ and $E_{\nu}^{\textnormal{cal}}$ as defined in Eq.
III.2. All curves include the 2p2h and pion absorption contributions (shown
separately), evaluated using GENIE. Experimental results were taken from [21].
The bands drawn for the ED-RMF and ROP models are related with the
uncertainties associated with the modeling of the initial nuclear state (see
the text). The $\chi^{2}/\textit{d.o.f}$ ratio is given in brackets in the
legend of each distribution except for $Q^{2}_{\textnormal{CCQE}}$ and
$E_{\nu}^{\textnormal{cal}}$ for which the covariance matrices are not
available. Figure 6: MicroBooNE CC$0\pi$1p $\nu_{\mu}$-40Ar cross sections as
function of the muon scattering angle. All curves include the 2p2h and pion
absorption contributions (shown separately), evaluated using GENIE.
Experimental results were taken from [21]. The bands drawn for the ED-RMF and
ROP models are related with the uncertainties associated with the modeling of
the initial nuclear state (see the text). The $\chi^{2}/\textit{d.o.f}$ ratio
is given in brackets in the legend of each distribution.

##  IV Conclusions

This paper presents a comparison of all semi-inclusive $\nu_{\mu}-^{40}$Ar and
$\nu_{e}-^{40}$Ar cross section measurements with two different theoretical
approaches: RDWIA calculations using ED-RMF and ROP FSI models, and the SuSAv2
model implemented in GENIE. For the RDWIA models the calculation is performed
taking into account conservative uncertainties associated to the modeling of
the spectral function used for the description of the initial state. Despite
using large uncertainties for the spectral function parameterization,
especially for the neutron occupation numbers as can be seen in Table 3, the
cross section calculations deviate by at most 6-7$\%$ from the mean values for
all the variables and in the region where the QE cross section is large.

Among the two RDWIA approaches considered in this work, the ROP model provides
the best overall agreement with data for both CC0$\pi$Np and CC0$\pi$1p
topologies, although the former present large uncertainties. It is worth
mentioning in the CC0$\pi$Np case the accordance between the ROP predictions
and data as function of the muon and proton kinematics, except for the forward
muon scattering angles (see Fig. 6).

With regard to the SuSAv2 model implemented with GENIE, the present study
shows that the semi-inclusive RDWIA models are in better agreement with data
for most of the kinematics explored, particularly at the kinematics explored
by the CC0$\pi$Np topology for protons with momentum below 0.6 GeV for both
electron and muon neutrinos. This comparison provides useful information on
the kinematical regions where the SuSAv2-GENIE results are solid and those
where they are not reliable.

To conclude, the analysis performed in this work also points to a possible
overestimation of the pion absorption contribution implemented in GENIE in
some particular kinematical situations, for instance the distribution as
function of $\theta_{\mu p}^{\textnormal{reco}}$. This suggests implementing
alternative models for the description of this process, as well as a semi-
inclusive calculation of the 2p2h contributions.

###### Acknowledgements.

This work is part of the I+D+i projects PID2020-114687GB-100, and
RTI2018-098868-B-I00 funded by MCIN (AED), and by the Junta de Andalucia
(grants FQM160, SOMM17/61015/UGR and P20-01247); it is supported in part by
the University of Tokyo ICRR’s Inter-University Resarch Program FY2021 &
FY2022 (JAC, MBB, JMFP, GDM, RGJ, JMU), by the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No. 839481 (GDM), by the Project University of Turin (project BARM-
RILO-21) and by INFN (national project NUCSYS) (MBB and JMFP), and by the
government of Madrid and Complutense University under project PR65/19-22430
(R.G.-J). J.M.F.P. acknowledges support from a fellowship from the Ministerio
de Ciencia, Innovación y Universidades. Program FPI (Spain).

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# DeliGrasp: Inferring Object Properties with LLMs for
Adaptive Grasp Policies

William Xie1, Jensen Lavering1, Nikolaus Correll1 1All authors are with the
University of Colorado at Boulder, Boulder, CO. Corresponding email:
<EMAIL_ADDRESS>
###### Abstract

Large language models (LLMs) can provide rich physical descriptions of most
worldly objects, allowing robots to achieve more informed and capable
grasping. We leverage LLMs’ common sense physical reasoning and code-writing
abilities to infer an object’s physical characteristics—mass $m$, friction
coefficient $\mu$, and spring constant $k$—from a semantic description, and
then translate those characteristics into an executable adaptive grasp policy.
Using a current-controllable, two-finger gripper with a built-in depth camera,
we demonstrate that LLM-generated, physically-grounded grasp policies
outperform traditional grasp policies on a custom benchmark of 12 delicate and
deformable items including food, produce, toys, and other everyday items,
spanning two orders of magnitude in mass and required pick-up force. We also
demonstrate how compliance feedback from DeliGrasp policies can aid in
downstream tasks such as measuring produce ripeness. Our code and videos are
available at: https://deligrasp.github.io

##  I Introduction

Large language models (LLMs) are able to supervise robot control and learning
in manipulation from high-level step-by-step task planning [1, 2, 3] to low-
level motion planning [4, 5]. LLMs additionally aid in robot manipulation via
understanding a given object’s semantic properties and delineating appropriate
grasp positions conditioned on those semantic affordances [6, 7, 8].

These works usually assume that the acts of “picking” and “placing” are
straightforward tasks. This is not the case for contact-rich manipulation.
Existing methods for LLM-supervised robot control cannot account for tasks
like grasping a paper airplane or origami, deformable plastic bags containing
rice or dry noodles, or particularly ripe fruits and vegetables. Current
methods for grasping such delicate objects require custom sensing solutions
combined with adaptive force-control models [9, 10, 11] and are tested on a
limited set of items. LLMs thus provide an opportunity to leverage their
common-sense physical reasoning [12] to produce grasp skills which are both
force-adaptive and for the open-world. This is particularly important for
semi-structured environments like supermarkets that are subject to a
constantly rotating stock or dealing with loose food items such as fruits,
vegetables, and pastries that come in a large variety of changing shapes.

We propose DeliGrasp, an extension of LLM-supervised robot learning to
contact-rich manipulation, and posit that LLMs can infer the physical
characteristics of any arbitrary gripper-object interaction, including mass,
spring constant, and friction. We formulate an adaptive grasp controller with
slip detection derived from the inferred characteristics, endowing LLMs
embodied with any current-controllable gripper with adaptive, open-world grasp
skills for objects spanning a range of weight, size, fragility, and
compliance.

We pair DeliGrasp with an open-world localization pipeline which, given an
“object description,” identifies the object and an initial grasp position. The
same “object description” and associated “grasp verb” are the inputs to
DeliGrasp, which generates a grasp policy: executable Python code controlling
a gripper’s compliance, force, and aperture as a complete grasp skill for the
given description.

Experimentally, we observe that DeliGrasp performs successful, minimally-
damaging grasps on a custom benchmark of delicate objects which traditional
grasping methods are not capable of. We also demonstrate how LLMs can leverage
compliance feedback from DeliGrasp policies in downstream tasks such as
picking ripe produce. These generalizable, open-world grasping skills for
minimal deformation as well as real sensor data can be used to address
difficult manipulation tasks across environments and modes of HRI.

##  II Related Work

LLMs, with an internet-scale amount of common sense information, are able to
combine high-level task knowledge, physical context, and robot affordances.
For task and motion planning, LLMs can generate navigation and pick-and-place
instructions to complete complex and novel tasks [1]. LLM code-writing further
augments robot capabilities with closed-loop control [2], new skill generation
[3], and translating language to robot parameters for low-level dynamic
control [4]. In conjunction with learning methods [13, 8], LLMs have also
shown semantic understanding of grasping such as identifying appropriate grasp
locations [14, 6, 7], but do not address the grasping dynamics themselves.

Grasping delicate objects can be achieved by relying on the compliance of soft
grippers, which still requires semantic information for control if these
grippers must grasp both lightweight and heavy objects [15]. In this paper, we
adopt a control strategy that relies on interaction force and slip velocity
measurements from a rigid gripper similar to [16, 9, 10, 17] to perform
minimally deforming grasps and to measure spring constants [18]. We use
semantic object descriptions together with an LLM to (1) select appropriate
parameters for robust and damage-free grasping of delicate objects, and (2)
use tactile inspection to classify relative fruit ripeness and connect them
back to common-sense policies on how they can be consumed.

##  III Methods

As there does not exist a dataset focusing on delicate objects, we construct
our own preliminary delicate objects dataset. We source dataset objects
primarily from supermarkets, kitchens, and food pantries. Tabulated object
mass and object-specific thresholds for unsuccessful, “deforming” grasps are
shown in Appendix A-A.

Our pipeline takes as input a “grasp verb” and an “object description”. For
the delicate objects benchmark, we use “pick” as the “grasp verb,” and the
specific “object descriptions” used are provided in A-A. Our perception
method, adapted from [6], takes the “object description” and semantically
segments the object from an RGB-D image with OWL-ViT [19] and “Segment
Anything” [20]. We crop the corresponding depth image with the generated
segmentation mask, produce a point cloud object representing the segmented
object, and perform Principal Component Analysis to compute a grasp pose that
is aligned with the first three principal axes as well as a minimum object
grasp width[21].

For grasping, we use a UR5 robot arm with the open-source MAGPIE gripper [22],
which has a palm-integrated Intel RealSense D-405 camera. The gripper is
powered by two separate Dynamixel AX-12 motors, which allow maximum current
control, allowing us to implement simple torque control (default output force
is 4 N). Translating torque via a four-bar linkage, the gripper force can be
controlled and sensed in a range from 0.15 N to 16 N, and has an aperture
range from 106 mm fully open to 0 mm fully closed (default closure speed of
100 $\frac{mm}{s}$).

Figure 1: Free body diagram describing gripper interactions with an object at
rest, adapted from [9].

###  III-A Grasp Force Modeling

Fig. 1 shows the interaction between our gripper and an arbitrary object with
mass $m$ and spring constant (compliance) $k$. The gripper exerts a composite
applied gripper force $F_{a}=F_{left}+F_{right}$ that leads to a frictional
force $F_{f}=\mu F_{a}$, where $\mu$ is the Coulomb friction coefficient
between the gripper and object [21] that counteracts the force of gravity $mg$
($g$ is the gravitational constant). For compliant objects, approximated as
ideal springs, we can additionally describe the left and right gripper forces
$F_{l,r}=F_{spring_{l,r}}=kx_{l,r}$, where $x$ is the compression of the
grasped object.

Typically, object slip within a gripper can only be detected after the gripper
grasps an object at rest, the gripper begins some upward acceleration
$a_{lift}$, and an object begins slipping with some downward acceleration
$a_{slip}$. Increasing $F_{a}$ to account for $ma_{lift}$ yields an adaptive
minimum applied grasp force $F_{min}=F_{a}$ which prevents slip and is
minimally deforming [16]: $F_{min}=\frac{m(g+a_{lift})}{\mu}$ Conversely, when
an object is slipping with $a_{slip}$, the applied force
$F_{a_{slip}}=\frac{m(g-a_{slip})}{\mu}$ [9]. And when $a_{lift,slip}$ are 0
and the gripper and object are at static equilibrium,
$F_{min}=\frac{mg}{\mu}$. $F_{min}$ can then be arranged in relation to these
quantities:

$F_{a_{slip}}<\frac{mg}{\mu}\leq F_{min}$ (1)

For the delicate objects dataset, we estimate a minimum applied grasping force
$F_{min}=\frac{mg}{\mu}$ with a conservative, uniform Coulomb friction
coefficient of $\mu=0.33$ (A-A).

###  III-B Problem Statement

Here $m$ and $\mu$, which determine successful (non-slip) and minimally
deforming grasp forces, as well as spring constant $k$, are unknown values.
Suppose then that a highly capable reasoning agent can reliably determine
values of said $m,\mu$, thereby approximating the lower bound of $F_{min}$,
and $k$. These values form the basis of our force-adaptive algorithm for
minimally deforming grasps on delicate objects.

Given a gripper driven by a current-controlled servo motor, we define a
closed-loop force compensation controller: starting from an estimated target
aperture that corresponds to the estimated object’s width w from RGB-D data,
we increase the gripper output force $F_{out}$ and decreasing gripper aperture
$x$ until sensing a contact force $F_{c}$ greater than the target $F_{min}$
[17]. To determine the gain terms, $\Delta F_{out}$ and $\Delta{x}$, i.e. how
fast we close the gripper and ratchet up force, the controller uses the agent-
determined $k$ and $\Delta{x}$, we change $F_{out}$ by $c\cdot k\Delta x$,
where $c=0.1$ is a dampening constant. We describe the controller in Algorithm
1.

$F_{c}=\texttt{SetGripper}(x=w,0)$

while $F_{c}\leq F_{min}$ do

$F_{out}\mathrel{{+}{=}}c\cdot k\Delta x$

$x$ $\mathrel{{-}{=}}\Delta{x}$

$F_{c}=$ SetGripper($x$, $F_{out}$)

Algorithm 1 Adaptive Grasping with Slip Detection

We task an LLM (GPT-4) with predicting these quantities for an arbitrary
object. If such predictions are accurate, performing minimally deforming
grasps on a variety of objects becomes algorithmically simpler and removes the
need for parameter tuning [17] and additional gripper sensors [9, 11]. By
default, we instruct the LLM to compute $F_{min}=\frac{mg}{\mu}$ and we do not
account for $a_{lift}$, choosing to err closer to object slip than
deformation. However, we also provide the LLM with agency to deviate from the
default $F_{min}$ depending on the “grasp verb” provided.

We query the LLM to define and generate grasps with a dual-prompt structure
similar to that of Language to Rewards [4], with an initial grasp “descriptor”
prompt which produces a structured description, which the subsequent “coder”
prompt translates into an executable Python grasp policy that modulates
gripper compliance, force, and aperture according to Alg. 1. See Appendix A-D
and A-E for the full prompts.

##  IV Experiments

Our robot experiment platform consists of a UR5 robot arm and a MAGPIE gripper
looking top-down on a table with a palm-integrated Intel RealSense D-405
camera. We benchmark DeliGrasp (DG) against three grasp policies: closing the
gripper fully or until it is output force limited (F.L.), closing the gripper
to the visual object width determined by our perception method, and an ablated
DeliGrasp policy which directly generates $F_{min}$, $\Delta F_{out}$, and
$\Delta x$ without first reasoning an object’s physical properties (DG-D). We
employ each grasp policy 10 times, objects placed randomly within a 30 x 45 cm
bounding area on a table, and do not record attempts which receive faulty
poses from perception.

TABLE I: Successful Minimally Deforming Grasps on Delicate and Deformable Objects (10 trials per object) ID | Object | DG | DG-D | Visual | F.L.
---|---|---|---|---|---
|  |  |  | Width |
1 | Paper Airplane | 10 | 10 | 0 | 0
2 | Cup (empty) | 10 | 10 | 10 | 0
3 | Dried Yuba | 9 | 7 | 3 | 0
4 | Raspberry | 9 | 8 | 0 | 0
5 | Hard Taco | 9 | 7 | 5 | 0
6 | Mandarin | 10 | 10 | 10 | 10
7 | Stuffed Toy | 7 | 8 | 0 | 10
8 | Cup (water) | 10 | 8 | 3 | 4
9 | Bag (noodles) | 7 | 4 | 0 | 5
10 | Avocado | 9 | 7 | 4 | 0
11 | Spray Bottle | 6 | 5 | 0 | 3
12 | Bag (rice) | 0 | 0 | 0 | 2

DeliGrasp and the ablated direct DeliGrasp both perform successful, minimally-
deforming grasps that the visual-only and hardware-limited grasp policies are
not capable of on objects like the paper airplane and raspberry. DeliGrasp
dominates hardware-limited and vision-only grasps in 8 out of 12 objects, and
is better or on par in 10 out of 12 objects (full DeliGrasp details are
provided in App. A-B)

We look closely at failure modes to better understand the performance of each
policy. Given two bounds of failure—slip grasps and deforming grasps—we
observe that, categorically, force-limited grasps deform and vision-only
grasps slip. Force-limited grasps succeed only with robust, compliant objects.
Vision-only grasps succeed when the objects are relatively stiff, but slight
deviations in grasp position alter the grasp width and contact area, leading
to failures with the taco, yuba, water cup, and avocado.

TABLE II: Full vs. Ablated DeliGrasp Estimated $F_{min}$ ID | $\frac{mg}{\mu}$ | $F_{max}$ | $F_{DG}$ | $F_{DG}$ | $F_{DG\\-D}$ | $F_{DG\\-D}$
---|---|---|---|---|---|---
| (N) | (N) | (N) |  Err. (N) | (N) |  Err. (N)
1 | 0.02 | 0.03 | 0.10 | 0.07 | 0.15 | 0.12
2 | 0.11 | 0.13 | 0.25 | 0.13 | 0.5 | 0.38
3 | 0.16 | 0.20 | 0.39 | 0.21 | 0.5 | 0.32
4 | 0.18 | 0.22 | 0.06 | 0.14 | 0.2 | 0.02
5 | 0.44 | 0.55 | 0.98 | 0.48 | 1.5 | 1.00
6 | 1.65 | 2.05 | 1.88 | 0.20 | 1.25 | 0.60
7 | 2.18 | 2.71 | 0.61 | 1.84 | 1.5 | 0.95
8 | 3.12 | 3.88 | 4.08 | 0.58 | 3 | 0.50
9 | 5.62 | 6.98 | 12.3 | 6.00 | 4 | 2.30
10 | 6.00 | 7.46 | 3.92 | 2.81 | 1 | 5.73
11 | 7.36 | 9.14 | 12.2 | 3.95 | 5 | 3.25
12 | 26.5 | 32.9 | 19.6 | 10.1 | 8 | 21.7

As we conduct a small number of trials, we cannot conclusively explain the
performance differences between the full and ablated DeliGrasp policies,
however, the $F_{min}$ terms estimated by each DeliGrasp policy, shown in
Table II, potentially clarify such differences. DeliGrasp primarily slips and
performs only 3 deforming grasps, all of which are minor, but sufficiently
damaging, occurring once each time with the yuba, raspberry, taco. For each of
those items, the ablated DeliGrasp-Direct estimates a higher $F_{min}$ and
deforms the same items at a higher rate, though also with minor deformations.
Conversely, the ablated DeliGrasp estimates a lower $F_{min}$ for the cup of
water, bag of noodles, avocado, and resulting grasps slip more. Both policies
slip at high rates on the stuffed animal (grasped by the tail), spray bottle,
bag of noodles, and bag of rice. These objects are non-linearly and/or very
compliant and high volume, exerting lever forces on the gripper that likely
exceed $F_{min}$.

We do not comprehensively measure the consistency of DeliGrasp $m$ and $\mu$
predictions (App. A-B), but we anecdotally observe that re-generated $m,\mu$
predictions do not deviate significantly from the generated terms used in
experiments, resulting in a range of $\frac{2}{3}F_{min}\to\frac{4}{3}F_{min}$
for the initial $F_{min}$. While this is a large range, DeliGrasp Direct is
more volatile, producing an even larger range, $\frac{1}{3}F_{min}\to
3F_{min}$. We hypothesize that mass, a common quantity, is more reliably
predicted by an LLM, producing values more consistent than an abstract a
minimum grasping force.

###  IV-A Sensorizing DeliGrasp to Pick Ripe Produce

DeliGrasp explicitly experimentally measures a spring constant $k$ while
performing Alg 1: the adaptive grasp controller checks for contact force
$F_{c}$ with each small motion $\Delta x$, and $k=\frac{F_{c}}{\Delta x}$.
Though this measured $k$ (App. A-B) shows that DeliGrasp is not able to
accurately map object-specific knowledge to spring constants, it can be
utilized as a compliance sensor in tasks such as appraising the ripeness (a
correlate of compliance) of produce items. We perform this measurement on two
assortments of produce: avocados and tomatoes. When receiving the “grasp verb”
of “check” or “inspect for ripeness”, DeliGrasp bypasses the default minimum
grasp force value of $\frac{mg}{\mu}$ and manually sets a lower $F_{c}$ of
$0.5$ N for avocados and $0.2$ N for tomatoes.

Qualitatively, the relative ordering of the measured $k$ for each item type
corresponds with judgments from our own human hand grasps. We query an LLM
(GPT-3.5) with the item name and compliance data, and show in Table III that
it is able to reason over and answer abstract, relative questions about when
to eat them. Further qualitative questions such as “how” to eat a specific
item “right now” yield dishes appropriate for the ripeness of the produce
(App. A-C).

TABLE III: Measured Produce Compliance and LLM Reasoning

Query | Object | $k_{1}$ | $k_{2}$ | $k_{3}$ | $k_{4}$
---|---|---|---|---|---
|  | (N/m) | (N/m) | (N/m) | (N/m)
| Avocado | 65.1 | 143.5 | 335.4 | —
| Tomato | 271.6 | 331.8 | 606.2 | 1559.
At a given time, which should I eat?
Next | Avocado |  |  | ✓ |
week? | Tomato |  |  |  | ✓
Right | Avocado | ✓ |  |  |
now? | Tomato | ✓ | ✓ |  |
Three | Avocado |  | ✓ | ✓ |
Days? | Tomato | ✓ | ✓ | ✓ |
When will this go bad? (days)
| Avocado | 3–5 | 4–6 | 5–7 |
| Tomato | 1–2 | 2–4 | 3–5 | 7+

##  V Discussion

We introduce DeliGrasp, which uses LLMs to 1) infer semantic, common sense
physical information and then 2) generate adaptive grasp policies to grasp a
variety of delicate and deformable items, representative of a large class of
items—especially food and produce items, that traditional parallel jaw
grippers cannot grasp. These inferred $m,\mu,k$ characteristics enable more
consistent control than direct estimation of grasping forces, and we also
leverage DeliGrasp’s compliance feedback to measure produce ripeness.

In our future work we plan to explore other potential downstream tasks in
dense, cluttered, and unstructured settings like supermarkets, home kitchens,
and other such representative domains. There are multiple limitations, as well
as open directions for DeliGrasp. Currently, we do not investigate the rich
space of “grasp verbs” or “object descriptions” and how that affects generated
policies. Our adaptive grasping algorithm is gripper-agnostic, but we have not
validated it on other end-effectors. Furthermore, the controller is employed
only while robot is at rest, but if synchronized with robot motion, our
algorithm can be extended for in-motion slip-detection [9, 10].

In our experiments, we generate each grasp policy only once without feedback,
as typically a grasp failure leads to irreversible deformation. However,
finetuning on iterative feedback in LLM-supervised human-robot control may
improve initial policy performance, which is of high importance for delicate
objects, and generate new, complex policies [5]. Additionally, we can use
multimodal foundation models like GPT-4V for DeliGrasp, thereby generating
“grasp verbs” and “object descriptions” from images, rather than using hand-
annotated inputs, and grounding grasp descriptions in both visual and semantic
physical knowledge.

##  VI Acknowledgements

This work is supported by the NSF NRI and the USDA project COLW-2020-08988
“Autonomous Restoration and Revegetation of Degraded Ecosystems.”

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##  Appendix A Appendix

###  A-A Delicate Objects Physical Characteristics

For objects with empty entries in the “Input Phrase” column, we do not modify
or add descriptions beyond the name of the object. The squeeze bottle is a
uniquely difficult object for a parallel jaw gripper to grasp. It is extremely
sensitive to grasp position as it has high stiffness at all but the “squeeze”
point. However, if it is squeezed too hard, the bottle will spill. The bag of
rice is very heavy, though theoretically within the possible grasping force.
However due to its extreme compliance, successful grasps require as close to
full-finger contact.

TABLE IV: Delicate and Deformable Object Properties

ID | Object | Width | Mass | $\frac{mg}{\mu}$ | $F_{max}$ | “Object Description” Input | Invalidating
---|---|---|---|---|---|---|---
|  | (mm) | (g) | (N) | (N) | to LLM & VLM | Deformation
1 | Paper Airplane | 20 | 0.8 | 0.02 | 0.03 | — | crumples
2 | Cup (empty) | 75 | 3.6 | 0.11 | 0.13 | ”empty paper cup” | crumples, creased
3 | Dried Yuba | 30 | 5.5 | 0.16 | 0.20 | ”yuba (dried tofu skin)” | cracks, shatters
4 | Raspberry | 20 | 6 | 0.18 | 0.22 | — | juices, torn
5 | Hard Taco | 65 | 15 | 0.44 | 0.55 | ”hard-shell tortilla” | cracks, broken
6 | Mandarin | 50 | 56 | 1.65 | 2.05 | — | inelastic deformation
7 | Stuffed Toy | 28 | 74 | 2.18 | 2.71 | ”tail of a stuffed animal” | inelastic deformation
8 | Cup (water) | 75 | 106 | 3.12 | 3.88 | ”paper cup filled with water” | spillage
9 | Bag (noodles) | 90 | 191 | 5.62 | 6.98 | ”plastic bag containing dried noodles” | cracks
10 | Avocado | 60 | 204 | 6.00 | 7.46 | — | inelastic deformation
11 | Spray Bottle | 50 | 250 | 7.36 | 9.14 | ”squeeze bottle filled with water” | spillage
12 | Bag (rice) | 80 | 900 | 26.49 | 32.9 | ”plastic bag containing rice” | N/A

###  A-B Full Details of DeliGrasp Performance on Delicate Objects Dataset

Computed $\Delta F_{out}$ which are less than 0.01 N are set to 0.01 N.
Computed $F_{min}$ values below the force sensing threshold of 0.15 N are set
to that threshold when checking for slip or setting output force.

TABLE V: DeliGrasp Performance on Delicate Objects Dataset (10 trials)

|  | LLM Inferred $m,\mu,k$ and Downstream Terms | Mean Values of Experimentally Produced Terms
---|---|---|---
ID | Success | $m$ | $\mu$ | $k$ |  $F_{min}$ (N) | $\Delta F_{out}$ | $\Delta x$ | $x_{final}$ | $x_{goal}$ | $F_{out}$ | Time | $k_{max}$ | $k$
|  | (g) |  | (N/m) | $=\frac{mg}{\mu}$ | (N) | (mm) | (mm) | (mm) | (N) | (s) | (N/m) | (N/m)
1 | 10 | 5 | 0.5 | 20 | 0.098 | 0.01 | 2 | 11.9 | 7.8 | 0.54 | 4.23 | 332.46 | 184.91
2 | 10 | 10 | 0.4 | 50 | 0.25 | 0.025 | 5 | 67.83 | 59.5 | 0.5 | 2.94 | 707.93 | 389.76
3 | 9 | 20 | 0.5 | 200 | 0.39 | 0.04 | 2 | 24.74 | 19.82 | 0.6 | 4.08 | 240.27 | 157.69
4 | 9 | 5 | 0.8 | 50 | 0.063 | 0.01 | 1 | 15.4 | 5.98 | 0.28 | 7.26 | 97.46 | 60.96
5 | 9 | 50 | 0.5 | 1000 | 0.98 | 0.2 | 2 | 53.17 | 46.64 | 1.59 | 3.19 | 342.05 | 188.39
6 | 10 | 150 | 0.8 | 500 | 1.88 | 0.1 | 2 | 45.13 | 39.83 | 2.12 | 3.08 | 236.41 | 151.83
7 | 9 | 50 | 0.8 | 100 | 0.61 | 0.02 | 2 | 9.79 | 4.19 | 0.97 | 5.29 | 308.73 | 192.77
8 | 9 | 250 | 0.6 | 200 | 4.08 | 0.04 | 2 | 58.23 | 53.3 | 4.36 | 4.27 | 958.61 | 373.24
9 | 7 | 500 | 0.4 | 300 | 12.3 | 0.15 | 5 | 71.72 | 61.11 | 12.77 | 2.89 | 7484.7 | 4324.33
10 | 9 | 200 | 0.5 | 500 | 3.92 | 0.1 | 2 | 50.88 | 46.4 | 4.39 | 3.04 | 729.11 | 298.66
11 | 6 | 500 | 0.4 | 150 | 12.2 | 0.03 | 2 | 41.94 | 35.08 | 12.6 | 3.64 | 3456.4 | 1383.11
12 | 0 | 1000 | 0.5 | 200 | 19.6 | 0.1 | 5 | 35 | 27.5 | 20.8 | 5.7 | 13671. | 6192.2

We also show the corresponding values for the ablated DeliGrasp-Direct policy.
We did not record time, experimental $k$ values, or $x_{final}$ in these
experiments.

TABLE VI: Ablated DeliGrasp-Direct Performance on Delicate Objects Dataset (10
trials)

ID | Success | $F_{min}$ | $\Delta F_{out}$ | $\Delta x$ | $x_{goal}$ | $F_{out}$
---|---|---|---|---|---|---
|  | (N) | (N) | (mm) | (mm) | (N)
1 | 10 | 0.15 | 0.05 | 1 | 4.48 | 0.98
2 | 10 | 0.5 | 0.2 | 1 | 57.4 | 0.922
3 | 7 | 0.5 | 0.3 | 2 | 8.6 | 1.14
4 | 8 | 0.2 | 0.1 | 1 | 6.4 | 0.83
5 | 7 | 1.5 | 0.2 | 1 | 49.2 | 2.3
6 | 10 | 1.25 | 0.5 | 5 | 32.4 | 2.01
7 | 8 | 1.5 | 0.5 | 2 | 3.0 | 3.2
8 | 8 | 3 | 1 | 5 | 49.6 | 3.18
9 | 4 | 4 | 1 | 4 | 52.8 | 6
10 | 7 | 1 | 1 | 5 | 47.2 | 2.2
11 | 5 | 5 | 2 | 2 | 25.0 | 7.1
12 | 0 | 8 | 2 | 5 | 105. | 10

###  A-C Ripeness Reasoning with LLMs

  1. 1.

Asking ”how should I eat avocado $k_{3}$” yields: sliced avocado, grilled
avocado, salad toppings, stuffed avocado

  2. 2.

Asking ”how should I eat avocado $k_{1}$” yields: guacamole, avocado toast,
sushi rolls, smoothies

  3. 3.

Asking ”how should I eat tomato $k_{4}$” yields: sliced for burgers, grilled
or roasted, salsa, stuffed

  4. 4.

Asking ”how should I eat tomato $k_{1}$” yields: caprese salad, gazpacho,
pasta sauce

We additionally observe that multimodal models like GPT-4V or OWL-ViT alone
cannot reason ripeness over visual images. We query GPT-4V with an image of
the same items 5 times and request a ripeness ranking. For the 4 tomatoes,
GPT- 4V produces the same ranking each query, from most ripe to least ripe,
of: $[k4,k2,k3,k1]$, reasoning entirely on the “redness” of the tomatoes. For
the 3 avocados, it refuses to generate a ranking, citing a fundamental lack of
sensory capabilities.

OWL-ViT when directly queried for ”the ripest” or ”least ripe” produce
arbitrarily detects items (sometimes the same item) depending on the angle
(and thus lighting, resolution, relative object size) of the captured image.

When we pair the spring constant data with the image in a query to GPT-4V, it
ignores the visual characteristics and reasons solely on the spring data.:
Query:

⬇

I have labeled 3 avocados for you between 0, 1, 2 and provide their average
spring constants from an initial ripeness check. Spring constants are
inversely proportional to ripeness\--a low spring constant means a soft
object, and high means stiff. Combining this information with the image,
please do the following tasks:

1. Estimate the ripeness of a ripe and unripe avocado.

2. Rank the given avocados by ripeness.

E. Using the estimated ripeness range, estimate when each one will go bad
(avocados are stored in the open at room temperature).

Spring constant data:

Avocado 1: 65 N/m

Avocado 2: 143 N/m

Avocado 3: 335 N/m

Response:

⬇

To perform the tasks, we need to use the given spring constants and the visual
clues from the image to estimate the ripeness and predict when each avocado
will spoil. Typically, a ripe avocado has a spring constant at the lower end
because it is softer, while a less ripe or unripe avocado has a higher spring
constant because it is firmer.

0. Estimate the ripeness of a ripe and unripe avocado:

- A ripe avocado typically has a low spring constant, indicating softness, perhaps similar to Avocado 1, which has the lowest spring constant

- An unripe avocado has a high spring constant, as it’s stiffer, like Avocado 3, with the highest spring constant

1. Rank the given avocados by ripeness:

- Avocado 1, with the lowest spring constant range 65, would generally be the ripest.

- Avocado 2, with a spring constant range 143, would be less ripe than avo1.

- Avocado 3, having the highest spring constant range 335, would be the least ripe.

So the ripeness rank from ripest to least ripe would be:

Avocado 1 > Avocado 2 > Avocado 3

2. Estimate when each avocado will go bad:

- Avocado 1, being the ripest, based on its lower spring constant, could spoil in about 2-3 days if stored at room temperature.

- Avocado 2 is less ripe than avo1 and may take around 4-5 days to spoil at room temperature.

- Avocado 3 is the least ripe, with quite a high spring constant and could take upwards of a week to spoil, potentially around 7-10 days at room temperature.

Please note that these estimates are quite rough and actual spoilage times can
vary due to numerous factors such as specific room temperature, humidity, and
the avocado’s initial condition. The visual clues are limited, but from the
image, the physical characteristics that usually indicate ripeness (dark skin,
slightly yielding to gentle pressure) don’t provide significant additional
information beyond the spring constant data.’

###  A-D DeliGrasp Desciptor Prompt

The descriptor takes as input a “grasp verb”, “object description”, and
“object width.” It outputs a structured summary of the object, how it should
be grasped, and physical characteristics of the object.

⬇

Control a robot gripper with torque control and contact information.

This is a gripper with two independently actuated fingers.

The gripper’s parameters can be adjusted corresponding to the type of object
that it is trying to grasp.

As well as the kind of grasp it is attempting to perform.

The gripper has a measurable max force of 16N and min force of 0.15N, a
maximum aperture of 105mm and a minimum aperture of 1mm.

Some grasps may be incomplete, intended for observing force information about
a given object.

Describe the grasp strategy using the following form:

[start of description]

* This {CHOICE: [is, is not]} a new grasp.

* This grasp should be [GRASP_DESCRIPTION: <str>].

* This is a {CHOICE: [complete, incomplete]} grasp.

* This grasp {CHOICE: [does, does not]} contain multiple grasps.

* This grasp is for an object with {CHOICE: [high, medium, low]} compliance.

* This grasp is for an object with {CHOICE: [high, medium, low]} weight.

* The object has an approximate mass of [PNUM: 0.0] grams

* The object has an approximate spring constant of [PNUM: 0.0] Newtons per meter.

* The gripper and object have an approximate friction coefficient of [PNUM: 0.0]

* This grasp should set the goal aperture to [PNUM: 0.0] mm.

* If the gripper slips, this grasp should close an additional [PNUM: 0.0] mm.

* Based on object mass and friction coefficient, grasp should initially set the force to [PNUM: 0.0] Newtons.

* If the gripper slips, this grasp should increase the output force by [PNUM: 0.0] Newtons.

* [optional] This grasp {CHOICE: [does, does not]} use the default minimum grasp force force.

* [optional] This grasp sets the force to [PNUM: 0.0], which is {CHOICE: [lower, higher]} than the default initial force because of [GRASP_DESCRIPTION: <str>].

[end of description]

Rules:

1. If you see phrases like {NUM: default_value}, replace the entire phrase with a numerical value. If you see {PNUM: default_value}, replace it with a positive, non-zero numerical value.

2. If you see phrases like {CHOICE: [choice1, choice2, ...]}, it means you should replace the entire phrase with one of the choices listed. Be sure to replace all of them. If you are not sure about the value, just use your best judgement.

3. If you see phrases like [GRASP_DESCRIPTION: default_value], replace the entire phrase with a brief, high level description of the grasp and the object to be grasp, including physical characteristics or important features.

4. By default the minimum grasp force can be estimated by dividing the object weight (mass * gravitational constant) by the friction coefficient: (m*g/(mu)).

5. Using knowledge of the object and the grasp description, set the initial grasp force either to this default value or an appropriate value.

6. If you deviate from the default value, explain your reasoning using the optional bullet points. It is not common to deviate from the default value.

7. Using knowledge of the object and how compliant it is, estimate the spring constant of the object. This can range broadly from 20 N/m for a very soft object to 2000 N/m for a very stiff object.

8. Using knowledge of the object and the grasp description, if the grasp slips, first estimate an appropriate increase to the aperture closure, and then the gripper output force.

9. The increase in gripper output force the maximum value of (0.01 N, or the product of the estimated aperture closure, the spring constant of the object, and a damping constant 0.1: (k*additional_closure*0.0001)).

10. I will tell you a behavior/skill/task that I want the gripper to perform in the grasp and you will provide the full description of the grasp plan, even if you may only need to change a few lines. Always start the description with [start of description] and end it with [end of description].

11. We can assume that the gripper has a good low-level controller that maintains position and force as long as it’s in a reasonable pose.

12. The goal aperture of the gripper will be supplied externally, do not calculate it.

13. Do not add additional descriptions not shown above. Only use the bullet points given in the template.

15. Use as few bullet points as possible. Be concise.

###  A-E DeliGrasp Coder Prompt

The coder takes as input the structured summary of the grasp and inferred
characteristics and generates an grasp policy which implements the adaptive
grasping algorithm 1 and compliance sensing.

⬇

We have a description of a gripper’s motion and force sensing and we want you
to turn that into the corresponding program with following class functions of
the gripper:

The gripper has a measurable max force of 16N and min force of 0.15N, a
maximum aperture of 105mm and a minimum aperture of 1mm.

‘‘‘

def get_aperture(finger=’both’)

‘‘‘

finger: which finger to get the aperture in mm, of, either ’left’, ’right’, or
’both’. If ’left’ or ’right’, returns aperture, or distance, from finger to
center. If ’both’, returns aperture between fingers.

‘‘‘

def set_goal_aperture(aperture, finger=’both’, record_load=False)

‘‘‘

aperture: the aperture to set the finger(s) to (in mm)

finger: which finger to set the aperture in mm, of, either ’left’, ’right’, or
’both’.

record_load: whether to record the load at the goal aperture. If true, will
return array of (pos, load) tuples

This function will move the finger(s) to the specified goal aperture, and is
used to close and open the gripper.

Returns a position-load data array of shape (2, n) \--> [[positions],
[loads]], average force, and max force after the motion.

‘‘‘

def set_compliance(margin, flexibility, finger=’both’)

‘‘‘

margin: the allowable error between the goal and present position (in mm)

flexibility: the compliance slope of motor torque (value 0-7, higher is more
flexible) until it reaches the compliance margin

finger: which finger to set compliance for, either ’left’, ’right’, or ’both’

‘‘‘

def set_force(force, finger=’both’)

‘‘‘

force: the maximum force the finger is allowed to apply at contact with an
object(in N), ranging from (0.1 to 16 N)

finger: which finger to set compliance for, either ’left’, ’right’, or ’both’

‘‘‘

def check_slip(load_data, force, finger=’both’)

‘‘‘

load_data: the position-load data array from set_goal_aperture

force: the force to check if the contact force is met (in N), which is set by
set_force()

finger: which finger to check the contact force for, either ’left’, ’right’,
or ’both’

Returns True if the contact force is not reached, meaning the gripper has
slipped, False otherwise (the gripper has not slipped and has a good grasp).

Also returns the average and max force experienced by the gripper in the load
data.

Example answer code:

‘‘‘

from magpie.gripper import Gripper # must import the gripper class

G = Gripper() # create a gripper object

import numpy as np # import numpy because we are using it below

new_task = {CHOICE: [True, False]} # Whether or not the task is new

# Reset parameters to default since this is a new, delicate grasp to avoid
crushing the raspberry.

if new_task:

G.reset_parameters()

# [REASONING]

goal_aperture = {PNUM: goal_aperture}

complete_grasp = {CHOICE: [True, False]}

# Initial force. The default value of object weight / friction coefficient.

initial_force = {PNUM: {CHOICE: [({PNUM: mass} * 9.81) / {PNUM: mu}, {PNUM:
different_inital_force}] }}}

# [REASONING for initial force choice]

additional_closure = {PNUM: additional_closure}

# Additional force increase. The default value is the product of the object
spring constant and the additional_closure, with a dampening constant 0.1.

additional_force = np.max([0.01, additional_closure * {PNUM: spring_constant}
* 0.0001])

# Move quickly (without recording load) to a safe aperture that is wider than
the goal aperture

G.set_goal_aperture(goal_aperture + additional_closure * 2, finger=’both’,
record_load=False)

# [REASONING]

# [PREDICTION]

G.set_compliance(1, 3, finger=’both’)

G.set_force(initial_force, ’both’)

load_data = G.set_goal_aperture(goal_aperture, finger=’both’)

# [REASONING]

# [PREDICTION]

curr_aperture = G.get_aperture(finger=’both’)

applied_force = inital_force

slip_threshold = initial_force

slippage, avg_force, max_force = G.check_slip(load_data, slip_threshold,
’both’)

# record spring constants over slip detection

prev_aperture = curr_aperture

k_avg = []

while slippage:

goal_aperture = curr_aperture - additional_closure

if np.mean(avg_force) > 0.10: # low-pass filter force readings so we don’t
increase force when there is no contact

applied_force += additional_force

G.set_force(applied_force, ’both’)

print(f"Previous aperture: {curr_aperture} mm, Goal Aperture: {goal_aperture}
mm, Applied Force: {applied_force} N.")

load_data = G.set_goal_aperture(goal_aperture, finger=’both’)

# Report data after each adjustment

curr_aperture = G.get_aperture(finger=’both’)

slippage, avg_force, max_force = G.check_slip(load_data, slip_threshold,
’both’)

# record spring constants over slip detection

distance = np.abs(curr_aperture - prev_aperture)

k_avg.append(np.mean(avg_force) * distance * 1000.0)

prev_aperture = curr_aperture

if complete_grasp:

curr_aperture = G.get_aperture(finger=’both’)

G.set_goal_aperture(curr_aperture - additional_closure, finger=’both’,
record_load=False)

print(f"Final aperture: {curr_aperture} mm, Controller Goal Aperture:
{goal_aperture} mm, Applied Force: {applied_force} N.")

else:

G.open_gripper()

‘‘‘

Remember:

1. Always format the code in code blocks. In your response all five functions above: get_aperture, set_goal_aperture, set_compliance, set_force, check_slip should be used.

2. Do not invent new functions or classes. The only allowed functions you can call are the ones listed above. Do not leave unimplemented code blocks in your response.

3. The only allowed library is numpy. Do not import or use any other library. If you use np, be sure to import numpy.

4. If you are not sure what value to use, just use your best judge. Do not use None for anything.

5. If you see phrases like [REASONING], replace the entire phrase with a code comment explaining the grasp strategy and its relation to the following gripper commands.

6. If you see phrases like [PREDICTION], replace the entire phrase with a prediction of the gripper’s state after the following gripper commands are executed.

7. If you see phrases like {PNUM: default_value}, replace the value with the corresponding value from the grasp description.

8. If you see phrases like {CHOICE: [choice1, choice2, ...]}, it means you should replace the entire phrase with one of the choices listed. Be sure to replace all of them. If you are not sure about the value, just use your best judgement.

9. Remember to import the gripper class and create a Gripper at the beginning of your code.

10. Remember to check the current aperture after setting the goal aperture and adjust the goal aperture if necessary.

11. Before checking for slip, remember to create two new variables, applied_force and slip_threshold, set equal to the initial initial_force. Slip detection continues checking the unchanged slip_threshold, but the applied_force increases.

12. Remember to reassign the goal aperture to the current aperture after completing the slip check for complete grasps.’