d})^{s^{\prime\prime}},$

we will identify $(\mathcal{P},\mathcal{Q})$ with the point in
$\mathrm{R}^{M(s^{\prime},d,n)}\times\mathrm{R}^{M(s^{\prime\prime},d,n+p)}$
whose coordinates give the coefficients of the polynomials in
$\mathcal{P},\mathcal{Q}$. We will denote the vector of coefficients of
$\mathcal{P}$ by $A$ and those of $\mathcal{Q}$ by $B$, for
$A=\mathbf{a},B=\mathbf{b}$, we will denote by
$\mathcal{P}_{\mathbf{a}},\mathcal{Q}_{\mathbf{b}}$ the corresponding tuples
of polynomials having these coefficients.

For a pair. $(\widetilde{\Phi},\widetilde{\Psi})$, of Boolean formulas,
$(\mathcal{P},\mathcal{Q})\in\mathrm{R}^{M(s^{\prime},d,n)}\times\mathrm{R}^{M(s^{\prime\prime},d,n+p)}$,
we denote by

$S_{\widetilde{\Phi}}(\mathcal{P})={\mathcal{R}}(\widetilde{\Phi}(\mathcal{P})),$

and by $\mathbf{f}_{\widetilde{\Psi}}(\mathcal{Q})$, the semi-algebraic map
$\mathrm{R}^{n}\rightarrow\mathrm{R}^{p}$, such that

$\mathrm{graph}(\mathbf{f})={\mathcal{R}}(\widetilde{\Psi}(\mathcal{Q})).$

Treating the coefficient vectors $A,B$ of $\mathcal{P},\mathcal{Q}$ in the
input of Algorithm 5 as parameters, and using a parametrized version of
Algorithm 5 (see discussion in Subsection 3.1.2), we obtain a finite set of
polynomials $\mathcal{H}\subset\mathrm{R}[A,B]$, and for each
$H\in\mathrm{Cc}(\mathcal{H})$, finite sets
$\mathcal{C}_{H}\subset\mathrm{R}[A,B,Y_{1},\ldots,Y_{p}]$,
$\mathcal{D}_{H}\subset\mathrm{R}[A,B,Y_{1},\ldots,Y_{p},Y_{1}^{\prime},\ldots,Y_{p}^{\prime}]$,
simplicial complexes $\Delta_{H,C,i}$, and homomorphisms
$\widetilde{\phi}_{H,D,\ell}$, such that

$\mathcal{C}_{H}(\mathbf{a},\mathbf{b})\subset\mathrm{R}[Y_{1},\ldots,Y_{p}],\mathcal{D}_{H}(\mathbf{a},\mathbf{b})\subset\mathrm{R}[Y_{1},\ldots,Y_{p},Y_{1}^{\prime},\ldots,Y_{p}^{\prime}],$

and $\Delta_{H,C,i}$, $\widetilde{\phi}_{H,D,\ell}$, is the output of
Algorithm 5 with input

$\mathcal{P}_{\mathbf{a}},\mathcal{Q}_{\mathbf{b}},\widetilde{\Phi}(\mathcal{P}_{\mathbf{a}}),\widetilde{\Psi}(\mathcal{Q}_{\mathbf{b}}),\ell$

for all $(\mathbf{a},\mathbf{b})\in H$.

It now follows using the same argument and notation as in the proof of Theorem
2, that the number of strong equivalence classes amongst the poset modules

$\left(\mathbf{P}_{S_{\widetilde{\Phi}(\mathcal{P}_{\mathbf{a}})},\mathbf{f}_{\widetilde{\Psi}(\mathcal{Q}_{\mathbf{b}})},T,\ell}\right)_{(\mathbf{a},\mathbf{b})\in\mathrm{R}^{M(s^{\prime},d,n)}\times\mathrm{R}^{M(s^{\prime\prime},d,n+p)},T\in(\mathrm{R}^{p})^{N}}$

is bounded by

$\mathrm{card}(\mathrm{Cc}(\mathcal{H}\cup\bigcup_{H\in\mathrm{Cc}(\mathcal{H})}\widetilde{\mathcal{D}_{H}})),$

where

$\widetilde{\mathcal{D}_{H}}:=\bigcup_{1\leq i,j\leq
N}\\{P(Y_{i,1},\ldots,Y_{i,p},Y_{j,1},\ldots,Y_{j,p})\mid
P\in\mathcal{D}_{H}\\}.$

Note that

$\displaystyle C$ $\displaystyle:=$
$\displaystyle\mathrm{card}(\bigcup_{H\in\mathrm{Cc}(\mathcal{H})}\mathcal{D}_{H}),$
$\displaystyle C^{\prime}$ $\displaystyle:=$
$\displaystyle\mathrm{card}(\mathcal{H}),$ $\displaystyle K$ $\displaystyle:=$
$\displaystyle\deg(\mathcal{H}\cup\bigcup_{H\in\mathrm{Cc}(\mathcal{H})}\mathcal{D}_{H}),$

are all bounded in terms of $s,d,n,\ell,p$ but independent of $N$.

Also note that the number of variables in the polynomials $\mathcal{H}$ equals
$M$, and that in the polynomials in

$\bigcup_{H\in\mathrm{Cc}(\mathcal{H})}\widetilde{\mathcal{D}_{H}}$

equals $pN+M$, where $M=M(s^{\prime},d,n)+M(s^{\prime\prime},d,n+p)$.

Now using using Theorem 5 we obtain that

$\displaystyle\mathrm{card}(\mathrm{Cc}(\mathcal{H}\cup\bigcup_{H\in\mathrm{Cc}(\mathcal{H})}\widetilde{\mathcal{D}_{H}}))$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{pN+M}\binom{CN^{2}+C^{\prime}}{pN+M}4^{j}K(2K-1)^{pN+M-1}$
$\displaystyle=$ $\displaystyle N^{(1+o(1))N}.$

Finally the theorem follows from the fact there at most
$\sum_{s^{\prime}+s^{\prime\prime}=s}2^{2^{2s^{\prime}}}\times
2^{2^{2s^{\prime\prime}}}$ pairs of Boolean formulas
$(\widetilde{\Phi},\widetilde{\Psi})$ to consider. ∎

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# RUSOpt: Robotic UltraSound Probe Normalization with Bayesian Optimization
for In-plane and Out-plane Scanning

Deepak Raina12∗, Abhishek Mathur3, Richard M. Voyles2, Juan Wachs2, SH
Chandrashekhara4,
Subir Kumar Saha1 This work was supported in part by SERB (India) - OVDF Award
No. SB/S9/Z-03/2017-VIII; PMRF - IIT Delhi under Ref.
F.No.35-5/2017-TS.I:PMRF; National Science Foundation (NSF) USA under Grant
#2140612; Daniel C. Lewis Professorship, Amazon Research Award in Robotics and
PU-IUPUI Seed Grant.1Indian Institute of Technology (IIT), Delhi, India
({deepak.raina, saha}@mech.iitd.ac.in); 2Purdue University (PU), Indiana, USA
({draina, rvoyles, jpwachs}@purdue.edu); 3IIT, Goa, India
(abhishek.mathur.20063@iitgoa.ac.in); 4All India Institute of Medical Sciences
(AIIMS), Delhi, India (drchandruradioaiims@gmail.com).∗Corresponding author is
Deepak Raina

###### Abstract

The one of the significant challenges faced by autonomous robotic ultrasound
systems is acquiring high-quality images across different patients. The proper
orientation of the robotized probe plays a crucial role in governing the
quality of ultrasound images. To address this challenge, we propose a sample-
efficient method to automatically adjust the orientation of the ultrasound
probe normal to the point of contact on the scanning surface, thereby
improving the acoustic coupling of the probe and resulting image quality. Our
method utilizes Bayesian Optimization (BO) based search on the scanning
surface to efficiently search for the normalized probe orientation. We
formulate a novel objective function for BO that leverages the contact force
measurements and underlying mechanics to identify the normal. We further
incorporate a regularization scheme in BO to handle the noisy objective
function. The performance of the proposed strategy has been assessed through
experiments on urinary bladder phantoms. These phantoms included planar,
tilted, and rough surfaces, and were examined using both linear and convex
probes with varying search space limits. Further, simulation-based studies
have been carried out using 3D human mesh models. The results demonstrate that
the mean ($\pm$SD) absolute angular error averaged over all phantoms and 3D
models is $\boldsymbol{2.4\pm 0.7^{\circ}}$ and $\boldsymbol{2.1\pm
1.3^{\circ}}$, respectively.

##  I Introduction

Ultrasound imaging, also known as sonography, is a widely used medical imaging
technique that can provide valuable information about the structure and
function of organs and tissues. It is used in a wide range of diagnostic and
therapeutic procedures [1]. Ultrasound imaging has several advantages over
other imaging techniques such as X-rays or CT scans. For example, it does not
use ionizing radiation, making it safer for patients and clinicians. It is
also non-invasive and relatively inexpensive compared to other imaging
techniques. While ultrasound imaging is a valuable tool in modern medicine,
there are also some drawbacks to this technique. The quality of the ultrasound
images obtained is highly dependent on the skill and experience of the
operator [2]. This dependence on the operator’s skill and expertise can lead
to variability in image quality, which can affect the accuracy of the
diagnosis. Additionally, access to expert sonographers is limited in rural
areas, leading to long wait times for patients needing ultrasound imaging.
Patients who need these services may have to travel to urban areas, which can
result in increased travel costs and time [3, 4].

To address the limited availability of sonographers in rural areas, several
telerobotic or fully autonomous Robotic Ultrasound System (RUS) have been
introduced in recent years [5, 6, 7, 8, 9]. The objective of RUS is to acquire
the image with best quality, so that it is easy to diagnose and won’t require
a trained/skilled radiologist. By using RUS, it is possible to maneuver
ultrasound probe with a consistent, predetermined and precise contact force,
position and orientation. These are critical parameters that greatly impact
the quality of acquired ultrasound imaging [10].

Figure 1: The two rotations carried out to acquire urinary bladder ultrasound
images. (a) describe the ideal probe orientation normal to the point of
contact on phantom; (b) and (c) describe the non-ideal probe orientation
during out-plane an in-plane rotation of probe. The ultrasound images acquired
in these cases result in artifacts like edge or structure shadowing due to
inappropriate acoustic coupling.

In particular, maintaining an appropriate probe orientation is crucial for
ensuring the optimal acoustic coupling between the probe and physiologically
different human bodies. Fig. 1 demonstrates the impact of ideal and non-ideal
probe orientation on ultrasound image quality. When the angle between the
ultrasound beam and phantom surface normal is zero, more echoes are reflected
back and received, resulting in good image quality. However, when the angle is
non-zero, resulting ultrasound images present artifacts. Several studies have
analyzed the effect of probe orientation on image quality [11, 12], indicating
that higher quality results can be achieved when the center-line of the probe
is aligned with the normal direction of the scanning surface. In orthopedic
applications, this phenomenon is even more significant because the intensity
reflection coefficient for most soft tissue-to-bone interfaces is
approximately $25\%$, compared to less than $0.01\%$ for most soft tissue-to-
soft tissue interfaces [12]. Therefore, it seems reasonable to orient the
ultrasound probe along the normal direction of the scanning surface for
several ultrasound procedures [13, 14, 15]. However, acquiring the
normalization of robotized probes remains a challenging task and continues to
be of interest to researchers.

###  I-A Related Work on Robotized Probe Normalization

Several earlier studies have used 3D patient point cloud data to achieve probe
normalization in robotic ultrasound systems [16, 14, 15, 17, 18, 19, 20].
Huang et al. [14] approximated the normal direction of the triangle formed by
three neighboring points around the intended scanning path using a depth
camera image prior to ultrasound scanning. However, the surface acquired
before the probe makes contact with the tissue may not adjust to deformations
caused by contact. Further, the method’s precision is restricted by hand-eye
calibration and potential obstructions between the camera and the scanned
tissue. Chatelain et al. [16] proposed a visual servoing method for in-plane
orientation adjustment of the probe for 3D ultrasound imaging. However, the
method fails to optimize the out-plane orientation, which is controlled using
the teleportation of a robotic arm. Jiang et al. [21] presented a method for
optimizing the in-plane orientation of robotic ultrasound probe based on
confidence map-based [22, 23] assessment of ultrasound image quality. For out-
plane orientation, they further used the force measurements to align central
axis of probe with the normal to anatomical surface. However the method
requires exploring the full fan motion of the probe at each contact point,
which would compromise the scanning speed. Another work by Jiang et al. [18]
proposes a mechanical model, which used only force sensor data to estimate the
surface normal direction. This method has been tested on phantom and in-vivo
tissues. However, the results show that the model fails to accurately identify
the normal when rotation limits are close to goal orientation or asymmetrical,
which violates the underlying assumptions of their mechanical model. The
recent work by Jiang et al. [19] extended their work to probe normalization in
the presence of patient movements. They developed a motion monitoring module
to recompute the motion trajectory in case of patient movement. For probe
orientation correction, they used the confidence-maps of ultrasound images to
differentiate between the well-connected part and the non-contacting part.

There have also been studies that developed hardware systems to achieve probe
normalization in robotic ultrasound systems. Ma et al. [20] proposed a robotic
ultrasound imaging system that uses laser distance sensors on the end-effector
of robotic arm to achieve normal positioning of the probe. However, the end-
effector may fail to identify the normal on highly complex surfaces due to
larger outer-case for laser sensor mounting. Moreover, the laser sensor
performance might get affected due to different lighting conditions [24].
Tsumura et al. [25, 26] introduced a gantry-style robot with a passive 2-DOF
end-effector that ensures the correct positioning of the ultrasound probe in
real-time for both obstetric and lung examinations. This end-effector permits
swift angle adjustments, as there are no delays in mechanical rotational
motion. Nonetheless, the passive mechanism of this device necessitates a broad
tissue contact area, limiting its flexibility.

###  I-B Contributions

Despite several previous attempts to solve the challenges in robotized probe
normalization, the previous methods have the following two major drawbacks:
(i) they rely on computational-intensive algorithms to extract the skin
surface from the depth camera, and (ii) they require time-consuming
exploration phase with a large number of steps for identifying the normal
direction. In this work, we propose an autonomous RUS by addressing the
important challenge of identifying the normal direction with respect to the
patient body. We propose a sample-efficient machine-learning-based
optimization algorithm, Bayesian Optimization (BO), to search for normal
direction by just using the force sensor data without requiring the 3D patient
information and thoroughly exploring the rotations. Earlier, BO has been
employed in various safety-critical robotic medical procedures, including but
not limited to autonomous robotic palpation [27], semi-autonomous surgical
robot [28], tuning of hip exoskeletons’ controller [29], and Autonomous-RUS
[30]. Goel et al. [30] used BO to search for high image quality region, where
only 2D probe position is considered with compliant control. However,
orientation adjustment of probe is not considered, which is essential to
conduct the clinical ultrasound procedure for patients with different
physiological conditions. The key contribution of our work are as follows:

  1. 1.

We propose a sample-efficient Bayesian Optimization (BO)-based formalism that
will identify the necessary orientation adjustment required for the robotized
ultrasound probe to align the probe in a normal direction to the scanning
surface.

  2. 2.

We propose a objective function for BO that leverages the force sensor data
and underlying mechanics to guide the normal identification process. Further,
it handles the noise in objective function measurements by incorporating a
regularization term.

  3. 3.

We validated the proposed method on urinary bladder phantom with planar,
tilted and rough scanning surfaces for normalizing both linear and convex
probe. Further, we evaluated its robustness by applying it for 3D mesh human
models, which simulate the actual physiology of humans.

Figure 2: Overview of the BO framework for identifying the normal of the
robotized ultrasound probe to scanning region.

##  II Methodology

Figure 2 provides an overview of the proposed methodology, which aims to
identify the normal direction with respect to the contact surface during
robotic ultrasound procedures. To achieve this, the proposed methodology
employs Bayesian Optimization (BO), which uses only force sensor measurements
from a wrist-mounted force sensor. An objective function for BO has been
formulated using underlying contact mechanics to guide the identification
process. BO computes the probabilistic estimate of the unknown objective
function in the scanning region of the human body and uses an acquisition
function to select the next best pose to evaluate based on the current belief
about the function. Algorithm $1$ provides a detailed flowchart of the
proposed methodology.

###  II-A Bayesian Optimization formulation

If $A$ represents the region of scanning on human body, then the objective
function of BO can be formulated as:

$\max_{\boldsymbol{p}\in A}{\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}))$ (1)

where $\boldsymbol{f}(\boldsymbol{p})=[f_{x},f_{y},f_{z}]$ represents the
probe forces at pose $\boldsymbol{p}$ along the three axis $x$, $y$ and $z$.
${\mathcal{F}}$ represents the unknown objective function to be maximized
during the BO search. Note that the probe pose represents a vector consisting
of probe position ($\boldsymbol{x}$) and orientation ($\boldsymbol{o}$) as
$\boldsymbol{p}=(\boldsymbol{x},\boldsymbol{o})$. The estimator used in BO is
Gaussian Process (GP) model, which defines an unknown function ${\mathcal{F}}$
by the mean function $\boldsymbol{\mu}(\cdot)$ and covariance or kernel
function $\boldsymbol{\kappa}(\cdot,\cdot)$. If a set of unknown function
value estimates is given by
$\bar{\boldsymbol{\mathcal{F}}}=[{\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}_{1}),\cdots,{\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}_{n})]$
at set of probe poses, given by
$\boldsymbol{\bar{p}}=[\boldsymbol{p}_{1},\cdots,\boldsymbol{p}_{n}]$, then
the GP regression can predict the unknown function value estimate at new probe
pose $\boldsymbol{p}^{\dagger}$ and is given by

$\mathcal{P}({\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}^{\dagger})|\boldsymbol{p}^{\dagger},\bar{\boldsymbol{p}},\bar{\boldsymbol{\mathcal{F}}})=\mathcal{N}(\boldsymbol{k}\boldsymbol{K}^{-1}\bar{\boldsymbol{\mathcal{F}}},\boldsymbol{\kappa}(\boldsymbol{p}^{\dagger},\boldsymbol{p}^{\dagger})-\boldsymbol{k}\boldsymbol{K}^{-1}\boldsymbol{k}^{T})$
(2)

where,

$\boldsymbol{k}=\begin{bmatrix}\boldsymbol{\kappa}(\boldsymbol{p}_{*},\boldsymbol{p}_{1})&\cdots&\boldsymbol{\kappa}(\boldsymbol{p}_{*},\boldsymbol{p}_{n})\end{bmatrix}$
$\boldsymbol{K}=\begin{bmatrix}\boldsymbol{\kappa}(\boldsymbol{p}_{1},\boldsymbol{p}_{1})&\cdots&\boldsymbol{\kappa}(\boldsymbol{p}_{1},\boldsymbol{p}_{n})\\\
\vdots&\ddots&\vdots\\\
\boldsymbol{\kappa}(\boldsymbol{p}_{n},\boldsymbol{p}_{1})&\cdots&\boldsymbol{\kappa}(\boldsymbol{p}_{n},\boldsymbol{p}_{n})\end{bmatrix}$

In the above formulation, the kernel matrix $\boldsymbol{\kappa}$ has been
formulated using the sum of radial basis function and white kernel function
as:

${\kappa}(\boldsymbol{p}_{i},\boldsymbol{p}_{j})=\sigma_{r}\exp\bigg{(}\frac{-||\boldsymbol{p}_{i}-\boldsymbol{p}_{j}||^{2}}{2l^{2}}\bigg{)}+\sigma_{w}\boldsymbol{{I}_{n}}$
(3)

where $\sigma_{r}$, $\sigma_{w}$ and $l$ is the overall variance, noise
variance and length-scale, respectively, representing the BO hyperparameters
$\boldsymbol{\theta}$. To optimize the performance of a GP model with the
kernel in eq. (3), the hyper-parameters represented by $\boldsymbol{\theta}$
need to be estimated. This is done by maximizing the log marginal likelihood
using a Limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) solver, which
is an algorithm used for numerical optimization.

$\boldsymbol{\theta}^{*}=\arg\max_{\boldsymbol{\theta}}\log\prod\mathcal{N}({\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}_{i}))|\boldsymbol{\mu}_{\boldsymbol{\theta}}(\boldsymbol{p}_{i}),\boldsymbol{K})$
(4)

Input: Scanning Region $A$, max. iterations $N_{max}$;

Initialize $\bar{\mathbf{p}}=\\{\\}$, $\bar{\boldsymbol{\mathcal{F}}}=\\{\\}$;

for  _$i=1,...,N_{max}$ _ do

$\boldsymbol{p}_{i}\leftarrow\text{argmax}_{\boldsymbol{p}\in
A}EI(\boldsymbol{p})$;

if  _termination criteria reached_ then

stop;

else

Probe at $\boldsymbol{p}_{i}$, compute objective function ${\mathcal{F}}_{i}$;

Set $\bar{\mathbf{p}}\leftarrow\bar{\mathbf{p}}\cup\\{p_{i}\\}$,
$\bar{\boldsymbol{\mathcal{F}}}\leftarrow\bar{\boldsymbol{\mathcal{F}}}\cup\\{\mathcal{F}_{i}\\}$;

$\boldsymbol{\theta}\leftarrow\text{argmax}\log\prod\mathcal{N}({\mathcal{F}}(\boldsymbol{f}(\boldsymbol{p}_{i}))|\boldsymbol{\mu}_{\boldsymbol{\theta}}(\boldsymbol{p}_{i}),\boldsymbol{K})$;
Re-estimate GP;

return Top probe pose with normal direction to $A$;

Algorithm 1 Bayesian Optimization for Robotized Probe Normalization

###  II-B Objective function of BO

When a constant force ${f}^{d}$ is exerted along the probe axis, it generates
a reaction force ${f}_{z}$ along the normal of the scanning region to balance
the applied force ${f}^{d}$. Consequently, the reaction forces in two mutually
perpendicular directions are generated in response to the change in the
relative orientation of probe and unknown normal. The concept of reaction
force is well-defined in theoretical mechanics, which suggests that if an
external force (${f}^{d}$) is exerted in the direction perpendicular to normal
of contact surface, then the corresponding reaction forces in the remaining
two orthogonal directions must be zero. Based on this principle, we formulated
the objective function of BO (${\mathcal{F}}$) to maximize the reaction force
${f}_{z}$ and minimize the reaction forces along the other two axis. The
mathematical form of ${\mathcal{F}}$ is given as:

${\mathcal{F}}=\frac{f_{z}}{f_{x}f_{y}+\lambda||\boldsymbol{f}||_{2}+\epsilon}$
(5)

Figure 3: Ultrasound probe generates reactive forces $[f_{x},f_{y},f_{z}]$
when a desired force $f_{d}$ is applied along its axis at a specific
orientation to the scanning surface

where $\epsilon$ is a constant term added in the denominator to avoid the
division by zero. $\lambda||\boldsymbol{f}||_{2}$ represents the
regularization term, which penalizes the noise in the force sensor by adding a
positive value to the denominator of the objective function. The objective
function with regularization term can help to stabilize the objective function
values and reduce its noisy nature. The regularization term penalizes the
reactive force vector, which will prevent the function from becoming too
sensitive to small fluctuations in the input, usually because of noisy sensor
data. This term would be particularly important in applications where the
sensor data in objective function is susceptible to noise or other sources of
variability. The $\lambda$ in regularization term is termed as penalty and is
usually a positive constant value. The $||\boldsymbol{f}||_{2}$ denotes the
$L2$-norm (also known as euclidean norm) of the reactive force vector
$\boldsymbol{f}$, which is given by:

$||\boldsymbol{f}||_{2}=\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}$ (6)

The choice of $\lambda$ value is critical in determining the strength of the
regularization term. A small value of $\lambda$ would result in a weak
regularization and high noise in objective function measurements, while a
large value of $\lambda$ will result in low noise, but a large number of
samples will be required for converging to small errors. In the current
approach, we have chosen $\lambda$ using the iterative selection method, which
involves solving the optimization problem for a range of values of $\lambda$
and selecting the value as per performance requirement.

###  II-C Acquisition function of BO

To determine the next pose of a probe to sample from during BO search, an
acquisition function is used. Given the stochastic nature of the unknown
function estimates, we used Expected Improvement (EI) acquisition function for
optimizing the next probe pose. The EI function expresses the expected gain in
performance if the next probe pose is selected based on the highest predicted
value. The improvement is evaluated based on the difference between the
highest predicted value and the current best value obtained so far. This
allows the acquisition function to balance the exploration of new regions with
the exploitation of the current best results.

The EI function takes into account the posterior mean
$\boldsymbol{\mu}_{\bar{\boldsymbol{\mathcal{F}}}}(\boldsymbol{p})$ and
variance $\boldsymbol{\sigma}_{\bar{\boldsymbol{\mathcal{F}}}}^{2}(p)$ of the
GP model to evaluate the potential improvement of exploring with a given pose.
The EI function can be written as a combination of the cumulative distribution
function (CDF) and probability density function (PDF) of the Gaussian
distribution. Specifically, it is given by the formula:

$EI(\boldsymbol{p})=\begin{cases}(\boldsymbol{\mu}_{\bar{\boldsymbol{\mathcal{F}}}}(\boldsymbol{p})-\boldsymbol{\mathcal{F}}^{+}-\xi)\boldsymbol{\Phi}(\boldsymbol{Z})+\boldsymbol{\sigma}_{\bar{\boldsymbol{\mathcal{F}}}}^{2}(\boldsymbol{p})\boldsymbol{\phi}(\boldsymbol{Z})&\text{if
}\boldsymbol{\sigma}_{\bar{\boldsymbol{\mathcal{F}}}}^{2}(\boldsymbol{p})>0\\\
0&\text{if
}\boldsymbol{\sigma}_{\bar{\boldsymbol{\mathcal{F}}}}^{2}(\boldsymbol{p})=0\end{cases}$

(7)

where $\boldsymbol{\Phi}$ and $\boldsymbol{\phi}$ are the CDF and PDF of the
standard normal distribution, respectively. $Z$ is a standard normal variable
that is calculated based on the posterior mean and variance of the GP model.
$\boldsymbol{\mathcal{F}}^{+}$ is the best function value found so far, and
$\xi$ is a parameter that controls the trade-off between exploration and
exploitation.

##  III Results

TABLE I: Absolute angular error between ground truth and predicted values of
normal for out-plane and in-plane rotation, respectively, of linear probe for
different search limits [min, max]. Each value represents the Mean$\pm$S.D.
for 5 runs of BO

Phantom surface | Out-plane rotation (deg.) | In-plane rotation (deg.) |  Avg.
---|---|---|---
$[-5,5]$ | $[-10,5]$ | $[-5,10]$ | $[-10,-10]$ | $[-15,5]$ | $[-5,15]$ | $[-15,15]$ | $[-5,5]$ | $[-10,5]$ | $[-5,10]$ | $[-10,-10]$ | $[-15,5]$ | $[-5,15]$ | $[-15,15]$
Planar | $1.25$ | $1.57$ | $2.77$ | $3.21$ | $3.36$ | $2.93$ | $3.37$ | $1.11$ | $1.65$ | $2.32$ | $1.93$ | $2.13$ | $2.29$ | $3.15$ | $2.36$
$\pm 0.4$ | $\pm 0.8$ | $\pm 1.1$ | $\pm 0.3$ | $\pm 1.3$ | $\pm 0.8$ | $\pm 1.1$ | $\pm 0.4$ | $\pm 0.2$ | $\pm 0.5$ | $\pm 1.2$ | $\pm 0.8$ | $\pm 1.2$ | $\pm 0.5$ | $\pm 0.4$
Tilted | $1.45$ | $1.72$ | $2.19$ | $2.20$ | $2.75$ | $3.5$ | $3.80$ | $1.23$ | $1.82$ | $2.21$ | $2.76$ | $3.23$ | $3.90$ | $4.61$ | $2.67$
$\pm 0.2$ | $\pm 0.3$ | $\pm 0.7$ | $\pm 0.4$ | $\pm 0.6$ | $\pm 0.9$ | $\pm 1.7$ | $\pm 0.4$ | $\pm 1.1$ | $\pm 0.7$ | $\pm 0.3$ | $\pm 0.9$ | $\pm 0.7$ | $\pm 0.1$ | $\pm 0.6$
Rough | $1.37$ | $2.21$ | $2.38$ | $2.56$ | $2.22$ | $2.71$ | $3.32$ | $1.61$ | $1.72$ | $1.13$ | $2.11$ | $2.05$ | $2.72$ | $3.12$ | $2.23$
$\pm 0.3$ | $\pm 1.1$ | $\pm 1.4$ | $\pm 0.5$ | $\pm 1.8$ | $\pm 0.9$ | $\pm 1.7$ | $\pm 0.4$ | $\pm 1.5$ | $\pm 0.9$ | $\pm 1.3$ | $\pm 0.9$ | $\pm 1.2$ | $\pm 0.7$ | $\pm 1.0$
Avg. | $1.35$ | $1.83$ | $2.44$ | $2.65$ | $2.77$ | $3.04$ | $3.49$ | $1.31$ | $1.73$ | $1.88$ | $2.26$ | $2.47$ | $2.97$ | $3.63$ | $\boldsymbol{2.44}$
$\pm 0.3$ | $\pm 0.7$ | $\pm 1.1$ | $\pm 0.4$ | $\pm 1.2$ | $\pm 0.86$ | $\pm 1.5$ | $\pm 0.4$ | $\pm 1.2$ | $\pm 0.7$ | $\pm 0.93$ | $\pm 0.9$ | $\pm 0.9$ | $\pm 0.7$ | $\boldsymbol{\pm 0.7}$

###  III-A Implementation details

We validated the proposed framework in both real and simulated environment.
The real environment consists of a Sawyer 7-DOF robotic arm (Rethink Robotics,
Germany) with an ultrasound probe attached to its end-effector using a
customized gripper, as shown in Fig. 4.

Figure 4: Experimental setup of RUS with urinary bladder phantom having three
different scanning surfaces.

This arm has a wrist-mounted 6-axis force sensor, which is used for real-time
contact force measurement during the scanning. The two ultrasound probes are
used, one is Linear L12-5N40-M3 and the other is a Convex MC10-5R10S-3 probe,
which is connected to a Telemed ultrasound machine (Telemed Medical Systems,
Italy). The scanning is conducted on urinary bladder phantom
(YourDesignMedical, USA). The phantom surface is flat, however it has been
modified to represent tilted and rough surface to evaluate the robustness of
the framework for different human physiological conditions. The rough surface
has been made by using a Ballistics gel (Clear Ballistics, USA) layer.

Figure 5: Simulation environment of robotic ultrasound system with 3D human
mesh models denoted as H0, H1 and H2.

For simulation experiments, we built a physics-based simulation framework
using the open-source Pybullet physics engine [31], as shown in Fig. 5.
Further, we imported 3D mesh models of the human body using SMPL-X [32], which
represent the actual physiology of real humans. These models were developed
for capturing human posture using RGB and depth images [32]. Our simulation
platform will help us to do thousands of experiments safely, which would be
difficult to conduct with real-world robotic system and human subjects. For
both simulation and real experiments, we used the value of $\epsilon=1.0$,
$\xi=0.45$ and $\lambda=0.3$ for BO. The robot is under hybrid-position
control [5], where z-axis of robot is under force control.

###  III-B Validation metrics

To verify the accuracy of the obtained normal direction vector in the
simulated environment, we utilized a mesh of the body being scanned. We
compared the predicted normal to the true normal of the scanning region
obtained from the mesh geometry. The mesh of the body returns the triangles
(faces) and the coordinates of the connecting points that connect the
triangles. We used a ray-mesh intersection method to determine the
intersection between the probe point and mesh. This method utilizes a Bounding
Volume Hierarchy (BVH) tree to quickly eliminate parts of the mesh that the
ray does not intersect with. After identifying potential intersecting
triangles, the method performs more precise intersection tests using
Gilbert–Johnson–Keerthi (GJK) algorithm. This is implemented using Trimesh
[33] python library. For experiments with real robots, point cloud data has
been utilized to estimate the normal vector. This was accomplished using the
PCL Python library and subsequently corrected based on known geometry. The
true normal vector for planar and rough phantoms lies along the vertical
z-axis in the robot base frame. Similarly, the normal vector for tilted
phantoms was estimated using a geometric method, as the tilt angle was known
beforehand.

###  III-C Validation on urinary-bladder phantom

To evaluate the algorithm’s robustness for clinical ultrasound applications,
we conducted a set of BO experiments on phantoms with flat, tilted, and rough
surfaces. We performed these experiments for seven different limits ([min.,
max.]) of the search space. Each BO run was initiated with a probe positioned
at the center of the phantom and random initial orientation. The resulting
angular differences between the estimated normal and the actual normal are
presented in Table I. Each value represents the mean and standard deviation
across five BO runs. In total, we conducted $210$ BO runs for results
presented in Table I.

From Table I, it can be observed that the average angular error across all
runs is $2.44\pm 0.7^{\circ}$, which is lower than the error reported for
human sonographers in [18] as $3.21\pm 1.7^{\circ}$. Additionally, in $165$
out of $210$ runs (i.e., $78.6\%$ of the runs), the error was found to be less
than $3.0^{\circ}$. It is noteworthy that the error values remained consistent
across different search space limits for both in-plane and out-plane
rotations. When grouped by rotation direction, the max. average error for out-
plane and in-plane rotation is found to be close to each other (difference
less than $0.5^{\circ}$). These results indicate that the rotation direction
does not have a significant impact on the normal identification. However, the
maximum error of in-plane rotation ($4.71^{\circ}$) is less than that of out-
plane rotation ($5.5^{\circ}$). The variation in the optimization performances
for in-plane and out-plane rotations can be attributed to the linear probe’s
structure, for which the length is significantly greater than the width. Thus,
it can be concluded that the in-plane optimization results are more accurate
in comparison to out-plane optimization. There are a few sub-optimal cases
(error $>3^{\circ}$) that occurred in fewer than $22.4\%$ of the runs. This
typically occurs when the search space limit is high, which can be attributed
to the decrease in the performance of BO for larger search spaces. The effect
of this error in the normal identification can be illustrated through Fig. 1.
As the rotation angle moves away from normal ($>3^{\circ}$) orientation, the
image shows artifacts like edge or structure shadowing.

We conducted additional experiments using a convex probe for scanning. For
curved probe, the significant difference in the ultrasound image is only
observed after $15^{\circ}$. Therefore, we tested it for larger search space
limits, which are presented in Table II. We performed 30 BO runs, and the
results showed an average angular difference of $2.31\pm 0.7^{\circ}$ across
all runs. In 23 out of 30 runs (i.e., 76.7% of runs), the error was less than
$3.0^{\circ}$. Notably, the convex probe’s error was even lower than that of
the linear probe,

TABLE II: Absolute angular error for out-plane and in-plane rotation of convex
probe. Each value represents the Mean$\pm$S.D. for five runs of BO

Phantom surface | Out-plane rotation (deg.) | In-plane rotation (deg.) |  Avg.
---|---|---|---
$[-15,-15]$ | $[-20,20]$ | $[-15,-15]$ | $[-20,20]$
Planar | $2.11\pm 0.8$ | $2.45\pm 0.3$ | $2.01\pm 0.6$ | $1.85\pm 1.1$ | $2.11\pm 0.7$
Tilted | $1.99\pm 1.0$ | $2.23\pm 0.5$ | $2.12\pm 0.9$ | $2.43\pm 0.5$ | $2.19\pm 0.7$
Rough | $2.33\pm 1.1$ | $2.89\pm 0.8$ | $2.45\pm 0.3$ | $2.80\pm 0.9$ | $2.62\pm 0.8$
Avg. | $2.14\pm 1.0$ | $2.52\pm 0.5$ | $2.19\pm 0.6$ | $2.36\pm 0.9$ | $\boldsymbol{2.31\pm 0.7}$

which is attributed to the former’s smaller tip, resulting in smoother force
variations.

###  III-D Validation on 3D mesh human models

In order to provide comprehensive validation, the proposed framework was also
evaluated on three 3D human mesh models in simulation. The absolute angular
differences for the out-plane and in-plane rotations are presented in Fig. 6.
The search space limit for both out-plane and in-plane experiments was set to
$[-15,15]$, and 10 runs of BO were conducted for each human model. It was
observed that the average error for H0 and H1 was less than $3.0^{\circ}$,
indicating the robustness of the method for complex human physiology. The
average error for H3 was relatively high, which can be attributed to the high
irregularity of the body surface. Across all $60$ BO runs (in-plane and out-
plane), more than $75\%$ of the runs reported an error less than
$3.0^{\circ}$. It is important to note that simulation study has been limited
to small contact point on the mesh bodies due to their rigid surfaces. Despite
this limitation, it is noteworthy that simulation studies offer a reliable
means of verifying the proposed approach for determining the normal direction
of the probe while interacting with different 3D human models.

Figure 6: Absolute angular error between ground truth normal and estimated
normal for 3D mesh models of human

###  III-E Analysis of BO objective function

The proposed framework incorporates a regularisation term in the objective
function of BO (eq. 5). Hence, we conducted an analysis to examine the effect
of this term by varying the penalty parameter, $\lambda$. The outcomes of this
analysis are presented in Table III, which are generated using linear probe
for scanning different phantom surfaces within the search space of $[-15,15]$.
It is apparent from the results that the absence of a regularization term
(i.e., $\lambda=0.0$) results in high error rates ($>3.7^{\circ}$). However,
the addition of the regularization term results in a decrease in error rates.
Although further decrease in error rates can be achieved with larger values of
$\lambda$, more samples will be required to converge to the low error rates.
This is because BO will need to sample more points to attain the global maxima
among the highly correlated function maps. Although determining the
appropriate value of $\lambda$ requires extensive experimentation, it
significantly affects the performance of the proposed framework and plays a
key role in enhancing the accuracy of normal identification.

TABLE III: Performance of BO for different regularization in the objective
function

Phantom | $\lambda$ | Out-plane rotation | In-plane rotation | Sampled points
---|---|---|---|---
Planar | $0.0$ | $3.71\pm 0.7$ | $4.85\pm 0.5$ | $50$
$0.3$ | $3.37\pm 1.1$ | $3.15\pm 0.5$ | $100$
$0.5$ | $1.95\pm 0.3$ | $2.06\pm 0.7$ | $170$
$10$ | $1.21\pm 0.2$ | $1.54\pm 0.3$ | $250$
Tilted | $0.0$ | $5.14\pm 1.1$ | $4.76\pm 0.9$ | $57$
$0.3$ | $3.81\pm 1.7$ | $4.61\pm 0.1$ | $95$
$0.5$ | $2.34\pm 0.4$ | $2.17\pm 0.7$ | $182$
$10$ | $1.43\pm 0.2$ | $1.21\pm 0.3$ | $271$
Rough | $0.0$ | $4.13\pm 0.5$ | $3.67\pm 0.6$ | $63$
$0.3$ | $3.32\pm 1.7$ | $3.12\pm 0.7$ | $121$
$0.5$ | $2.05\pm 0.2$ | $1.93\pm 0.6$ | $168$
$10$ | $1.12\pm 0.3$ | $1.32\pm 0.2$ | $263$

##  IV Conclusion

In this paper, we propose a novel Bayesian Optimization (BO) based method for
determining the normal direction of the probe to the scanning region during
robotic ultrasound procedures, considering both in-plane and out-plane
rotations of the robotized probe. Our approach introduces a novel objective
function for BO that leverages force sensor measurements and the underlying
mechanics of contact to guide the identification process. Notably, this method
offers significant advantages as it does not rely on a 3D point cloud of the
patient’s body and extensive exploration for finding normal. To evaluate its
performance, we conducted experiments using urinary bladder phantoms and 3D
mesh models of humans, employing both linear and convex probes.

However, it is important to note that this study represents a preliminary step
towards achieving an efficient estimation of the probe’s normal orientation,
and further investigation is required. One major limitation of the study is
the absence of ultrasound imaging feedback for normal identification. Our
future work will explore the joint optimization of force and ultrasound image
quality [34] using multi-objective Bayesian optimization. Another limitation
of the work is the adjustment of probe orientation at a single point. In the
future, we will explore the normalization of the probe during continuous
scanning. Furthermore, we aim to extend our framework to optimize varying
probe orientations and desired forces, thereby increasing its clinical
applicability for scanning complex physiology and different anatomical
structures under skin, bones, or muscles. To achieve this, we will explore the
integration of our method with the RUS framework presented in our previous
works [35, 36, 37], incorporating the geometrical information of the anatomy
being scanned. To further enhance the sample efficiency of the optimization
process, we will explore leveraging the domain expertise in BO. For instance,
a potential solution involves using a prior in BO as suggested in [37, 38],
which can be modeled using expert’s demonstrations. Finally, in order to
validate the proposed framework for clinical procedures, it will be necessary
to conduct assessments on humans. In future, we intend to conduct experiments
on human subjects using our robotic ultrasound system, proposed in our earlier
works [5, 39]. These experiments will provide insights into the clinical
applicability of our method, establishing its effectiveness in real-world
medical settings.

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