# Generating robotic elliptical excisions with human-like tool-tissue
interactions

Artūras Straižys1, Michael Burke1,2 and Subramanian Ramamoorthy1,3 1School of
Informatics, University of Edinburgh2Department of Electrical and Computer
Systems Engineering, Monash University3Work supported by a grant from the UKRI
Strategic Priorities Fund to the UKRI Research Node on Trustworthy Autonomous
Systems Governance and Regulation (EP/V026607/1, 2020-2024).For the purpose of
open access, the author(s) has applied a Creative Commons Attribution (CC BY)
license to any Accepted Manuscript version arising.

###### Abstract

In surgery, the application of appropriate force levels is critical for the
success and safety of a given procedure. While many studies are focused on
measuring in situ forces, little attention has been devoted to relating these
observed forces to surgical techniques. Answering questions like “Can certain
changes to a surgical technique result in lower forces and increased safety
margins?” could lead to improved surgical practice, and importantly, patient
outcomes. However, such studies would require a large number of trials and
professional surgeons, which is generally impractical to arrange. Instead, we
show how robots can learn several variations of a surgical technique from a
smaller number of surgical demonstrations and interpolate learnt behaviour via
a parameterised skill model. This enables a large number of trials to be
performed by a robotic system and the analysis of surgical techniques and
their downstream effects on tissue. Here, we introduce a parameterised model
of the elliptical excision skill and apply a Bayesian optimisation scheme to
optimise the excision behaviour with respect to expert ratings, as well as
individual characteristics of excision forces. Results show that the proposed
framework can successfully align the generated robot behaviour with subjects
across varying levels of proficiency in terms of excision forces.

##  I Introduction

Surgical excision implies the application of physical forces necessary for
tissue separation [1]. A successful procedure requires appropriate levels of
excision forces - sufficient for cutting, yet conservative to avoid damaging
tissue (excessive force can account for more than half of the medical errors
committed by surgical trainees [2]). Cutting forces, on the other hand,
strongly depend on the configuration of the blade [3, 4, 5], and therefore on
the excision technique. In order to derive optimal surgical techniques, in the
context of surgical training or autonomous surgery, tool-tissue interaction
forces and their downstream effect on the tissues must be studied in a
controlled manner. A comprehensive analysis of a wide range of behaviours
across different levels of expertise is needed to identify good and bad
practices and to measure their benefit or harm.

Unfortunately, such studies require a large number of participants at
different stages of their professional development, which is extremely time-
consuming and challenging in terms of logistics. As an alternative, we can use
a considerably smaller number of subjects to teach a robot performing the
excisions from various levels of proficiency and apply machine learning
techniques to interpolate the behaviour between the demonstrations. A suitably
parameterised behaviour model would let us generate a large number of trials
with tight control over the process parameters, such as excision velocity,
blade insertion angle, etc. Such a robotic setup would facilitate the
exploration of the downstream effects of the cutting technique on the tissues,
and therefore provide a deeper insight into its efficacy and safety.
Importantly, this gives us the ability to align robotic cutting behaviour with
the desired characteristics of excision forces to analyse various techniques
and their influence on tissue outcome.

A standard approach to accomplish this relies on imitation learning techniques
[6], such as the Dynamic Movement Primitive (DMP) [7], to learn excision
behaviours directly from demonstrations. In the case of the classical DMP
formulation, the encoded policy can be generalised via hyper-parameters, such
as the goal state and temporal scaling, as well as the coupling terms [8].
Unfortunately, although this allows exploration around the demonstrated
trajectories, the parameterised policies are restricted to individual
demonstrations with no relation to one another. Alternatively, one could apply
DMP formulations that leverage multiple demonstrations to capture behaviour
variability as a separate task parameter [9, 10], which can be interpolated to
synthesise unseen behaviours [11]. In the context of learning surgical
excisions, these methods allow explicit capture of different “styles” of
cutting tissues, which is particularly relevant for our task. However, these
methods are unsuitable to encode the end-effector’s pose trajectory in
Cartesian space, as the orientation component requires special treatment of
the SO(3) structure [12]. This presents challenges to apply the above methods
to learn cutting skills, as the position and orientation trajectories of a
blade are inherently coupled due to the nonholonomic nature of the cutting
motion [13, 14].

Figure 1: The proposed framework to generate human-like excision behaviours
operates as follows. At a high level, an objective function takes in the
sawing parameter $\rho$ and outputs an observation similarity score $y$. The
goal of the optimiser is to find the value for $\rho$ that maximises the
observation variable $y$. The inner structure of the proposed objective
function consists of the proposed trajectory generator ($\mathcal{T}$) that
given the scalar $\rho\in[1,10]$ and a nominal trajectory $\tau_{\text{nom}}$,
generates the pose trajectory of the blade $\tau$. Next, the robot executes
the trajectory $\tau$, and collected forces $\psi$ are converted to a set of
performance features by model $\mathcal{F}$. The obtained features are then
used to define the final objective function, e.g. the similarity score between
the robot-executed task and the performance of an actual surgeon, which
updates the optimiser and generates a new excision behaviour.

As an alternative, in this paper we introduce a simple parametric model of
elliptical excision that decomposes the skill into nominal and behaviour-
driven components, each of which can be learned from demonstrations. Here, we
focus on modelling a sawing movement as the most dominant characteristic of
the elliptical excision technique [15] observed in human trials [16]. Our
model encodes the behaviour with a single real-valued parameter $\rho$ that
determines the amount of sawing movement applied during excision. We then show
how this model can generate a variety of human-like excision trajectories, and
apply Bayesian optimisation to align the generated behaviour with expert
ratings.

Finally, as the core contribution of this paper, we propose a framework for
aligning human-like robotic elliptical excisions with the desired
characteristics of excision forces and demonstrate its applicability for
analysing excision skills. In this framework (Fig. 1), we generate an
elliptical excision behaviour using the parametric model described above,
which generates the pose trajectory of the blade, specified by a behaviour
parameter $\rho$ and a nominal cutting trajectory. The robot executes the
generated trajectory in a real-world experiment. A suitable model
characterising the excision behaviour from the excision force measurements
[16, 17, 18] allows us to 1) calibrate the robotic behaviours to match the
desired characteristics of tool-tissue interaction, and 2) use the aligned
robotic behaviour to analyse the excision techniques in a well-controlled and
repeatable manner. Here, we propose a Bayesian Optimisation scheme to minimise
the number of phantom excisions required for behaviour tuning.

Figure 2:  Measured individual position (top row) and orientation (middle row)
trajectories of the blade for each of the demonstrated behaviour (the darker
green lines correspond to smoother excisions). (Bottom row) Differenced
measurements of $x$, $y$ and pitch versus differenced measurements of $z$.
Note: the original measured trajectories are shown as black dashed lines in
the above plots.

##  II Parametric generation of human-like excision trajectories

The primary objective of our excision trajectory generator is to produce
realistic pose trajectories of the blade, mirroring those typically observed
during an elliptical excision procedure. In particular, we are interested in
learning the auxiliary sawing movement of the blade, which can play a role in
assisting the excision by lowering the cutting forces required for the task
[3]. In this section, we introduce a generative model of blade trajectories
with a single parameter $\rho$ that specifies the amount of sawing movement
applied to any desired elliptical excision motion.

Our central hypothesis is that elliptical excision comprises two movement
components: a nominal smooth cutting motion along the ellipse and an adaptive
sawing motion that assists the excision. To better understand the interplay
between these two components, we recorded ten elliptical excisions with
varying amounts of sawing behaviour, from a highly pronounced sawing movement
to an extremely smooth excision. Fig. 2 (top and middle rows) shows the
measured position and orientation trajectories of the blade in each trial.
Notice the back-and-forth oscillation of the blade on the $XY$ plane, a
distinct feature of the sawing movement. It should be noted that sawing is
executed at a noticeably different pitch when compared to smoother executions.
Unsurprisingly, smoother excisions result in a much lower spread of the pose
trajectories, which highlights the challenge of maintaining consistent motion
when sawing. As expected, the sawing motion is most dominant along the $z$
axis, i.e. the cutting depth. However, the sawing is also reflected in the
blade’s $x$, $y$ and pitch trajectories.

Figure 3: (Top row) Comparison of measured $x$, $y$ and pitch trajectories
(blue lines) with model predictions (orange lines). The nominal trajectories
obtained by low-pass filtering raw measurements are denoted with green.
(Bottom row) Comparison of measured $z$ trajectories for three different
sawing behaviours (black lines) and corresponding synthetic trajectories
generated by the model (coloured semi-transparent lines).

Fig. 2 (bottom row) shows the trajectories of differenced $x$, $y$ and pitch
motion components, compared to a differenced $z$ trace, all from one of the
observed sawing behaviours. The entire trace of $x$, and the first half of the
parabola in $y$ trajectory, are noticeably in-phase with the $z$. In contrast,
the whole trace of the pitch, and the second half of the parabola in $y$
trajectory, are noticeably out-of-phase with the $z$ trajectory. The
relationship between $z$ and $xy$ trajectories indicates that sawing movement
is achieved by propagating the blade forward on the ascent, with residual
backward motion on the blade’s descent. The relation between $z$ and pitch
trajectories suggests that consistent modulation of the insertion angle is
also a part of the slicing motion.

###  II-A Modeling blade trajectories for elliptical excisions

Given the above insights, we model the elliptical excision behaviour as
follows. First, we decompose the cutting motion into a nominal movement
component that follows the desired smooth cutting contour, and a behaviour
component that characterises the manner of task execution (e.g. sawing vs
smooth excisions). For the $z$ trajectory, this is written as

$z_{m}=z_{n}+z_{b}$ (1)

where $z_{m}$, $z_{n}$ and $z_{b}$ are the measured, nominal and behaviour
trajectories along the $z$ axis, respectively.

Next, we assume that a nominal movement component can be approximated by low-
pass filtering of the measurements, and therefore the behaviour component can
be computed as $z_{b}=z_{m}-z_{n}$. The obtained behaviour component $z_{b}$
can be used to model the observed $x$, $y$ and pitch trajectories of the
blade, as follows:

$\begin{array}[]{lcl}&x_{m}=x_{n}+c_{x}z_{b}\\\ &y_{m}=y_{n}+c_{y}z_{b}\\\
&\beta_{m}=\beta_{n}+c_{\beta}z_{b}\end{array}$ (2)

Here $x_{m}$, $y_{m}$ and $\beta_{m}$ are the measured $x$, $y$ and pitch
trajectories, respectively, and $x_{n}$, $y_{n}$ and $\beta_{n}$ are their
corresponding nominal trajectories. $c_{x}$, $c_{y}$ and $c_{\beta}$ are the
scaling coefficients (We invert the sign of $c_{y}$ in the second half of the
$y_{m}$ parabola to correctly represent the modulation of parabolic nominal
trajectories on the $XY$ plane).

As a result, we model the $x$, $y$ and pitch trajectories as a function of
$z_{b}$. According to this model, any modulation along the cutting depth is
reflected in the $x$, $y$ and pitch trajectories. Fig. 3 (top row) compares
the actual trajectories with those predicted by our model for one of the
observed sawing behaviours. Here, we used a first-order Butterworth filter
with a cutoff frequency of 0.6 Hz to obtain the nominal trajectories from the
raw measurements and set $c_{x}$, $c_{y}$ and $c_{\beta}$ parameters to 0.2,
0.13 and -1.2 values, respectively.

###  II-B Learning cutting behaviour

The movement component $z_{b}$, which defines the excision behaviour, can be
learned using one of many supervised learning techniques. In this work, we
first apply the following function approximation to encode the behaviour
component $z_{b}$:

$z_{b}(t)\approx\frac{\sum_{i=1}^{N}\psi_{i}(t)\theta_{i}}{\sum_{i=1}^{N}\psi_{i}(t)},$
(3)

where $\psi_{i}(t)=\textrm{exp}(-h_{i}(t-c_{i})^{2})$ are Gaussian basis
functions, $N$ is the number of basis functions, $t$ is the timestep, $c_{i}$
and $h_{i}$ are the centres and widths of the basis functions, and
$\theta_{i}$ are the weights of the basis functions.

The vector of learned weights $\bm{\theta}=[\theta_{1},...,\theta_{N}]$ can be
viewed as a compressed representation of the $z_{b}$ time series. We fit a
second-order autoregression model to vector $\bm{\theta}$:

$\theta_{i}=c+a_{1}\theta_{i-1}+a_{2}\theta_{i-2}+\epsilon_{i}$ (4)

where $a_{k}$ and $c$ are the autoregression coefficients and bias constant,
respectively; $\epsilon_{i}$ is the white noise term.

Finally, given the excision behaviour label (1 to 10, where 1 represents the
distinct sawing cuts, and 10 denotes smooth excisions), we fit a linear model
to predict the $a_{1}$ and $a_{2}$ coefficients. Thus, given a real number
$\rho\in[1,10]$, our model generates the behaviour component $z_{b}$, which
along with a nominal cutting trajectory can be used to produce a human-like
pose trajectory of the blade with desired sawing behaviour to be executed by a
robot. Fig. 3 (bottom row) compares the $z$ trajectories sampled from our
model with the actual measurements $z_{m}$ for different sawing behaviours.

##  III Optimisation of elliptical excision technique

Optimisation of the robotic behaviour with respect to the excision force
characteristics can offer interesting prospects for studies on the effects of
surgical procedures. For example, the alignment of robot behaviour with
excision forces that match those of expert surgeons is particularly promising
for robotic surgery applications. Equally important is the ability to
faithfully recreate sub-standard excision techniques and study the common
mistakes observed in less experienced surgeons or trainees. In addition,
technique optimisation concerning force characteristics critical to the
procedure’s safety could positively contribute to the existing surgical
training and practice.

In this work, we use an excision force model described in [16], characterising
the excision performance from force measurements; however, other force-based
characterisation approaches can be readily applied in the proposed framework.
Below, we provide a brief overview of the force-based characterisation of
elliptical excision used in this study, followed by descriptions of the
objective function and the proposed method for technique optimisation.

###  III-A Performance characterisation from force measurements

A model of excision forces introduced in [16] allows characterising the
performance of the generated excision behaviours directly from the
measurements of the excision forces. Alongside descriptive statistics of
excision forces, this model parameterises task-related characteristics
correlated with elliptical excision performance, such as abruptness of task
execution flow. This approach models the elliptical excision process as a
hybrid system, with underlying continuous dynamics of viscoelastic interaction
between tissues and the blade, as well as discrete event dynamics, typically
associated with tissue re-tensioning or blade re-orientation.

First, the model approximates the process of cutting a viscoelastic object as
a continuous blade’s movement through Maxwell material using the following
constitutive law:

$\frac{\eta}{E}\dot{f}+f=\eta\dot{x}$ (5)

with $f$ is the excision force, $\dot{f}$ the time derivative of the excision
force, $\dot{x}$ is the blade’s velocity,$E$ and $\eta$ the Maxwell model’s
spring and damper coefficients, respectively.

The model assumes that an excision is executed using $K$ cutting regimes,
where each regime $k$ corresponds to a constant velocity of the blade $v_{k}$.
System uncertainty is modelled using white Gaussian noise with variance
$\sigma_{k}$, $\tilde{v}_{k}\sim\mathcal{N}(v_{k},\sigma_{k}^{2})$. Cutting
regimes are switched according to $K\times K$ transition matrix $\mathbf{Q}$,
which along with $v_{k}$ and $\sigma_{k}^{2}$, can be learned by fitting a
Hidden Markov Model (HMM) to $\dot{x}$. Under the assumption of the Maxwell
model, $v_{k}$ can be obtained directly from force measurements $f$, captured
using a suitably instrumented scalpel.

The learned model parameters encode the amplitude and temporal features of the
excision forces that characterise the manner of task execution. For instance,
parameters $\\{v_{1}$…$v_{K}\\}$ describe the dominant force levels of the
excision forces, and thus the overall magnitude and spread of forces applied
during the excision. Along with transition probability matrix $\mathbf{Q}$,
which captures the temporal characteristics of the excision forces, these
parameters can encode meaningful features, such as Energy (where increased
energy reflects higher cutting forces applied for a longer duration) or
Smoothness (where increased smoothness reflects the higher probability of
sudden rise and fall of excision forces).

###  III-B Objective function

Fig. 1 shows a diagram of the proposed framework for aligning robotic cutting
behaviour with desired characteristics of excision forces, typically observed
in human trials. At every $n$-th iteration, the robot executes the excision
trajectory $\tau_{n}$ generated by the proposed trajectory model
($\mathcal{T}$) using candidate behaviour parameter $\rho_{n}$ and a fixed
nominal trajectory $\tau_{\textrm{nom}}$. After the execution, the recorded
excision force profile $\psi_{n}$ is provided to the force-based
characterisation model ($\mathcal{F}$), which encodes the characteristics of
the excision forces, as described in the previous section. The features
extracted by model $\mathcal{F}$ are then used to evaluate the objective
function $g(\rho_{n})$. Two scoring methods are used to evaluate the proposed
objective function:

####  III-B1 A single characteristic of excision forces

is used for aligning the generated robot behaviour with respect to a certain
characteristic of the excision forces, e.g. amplitude or smoothness evaluated
by model $\mathcal{F}$.

####  III-B2 Expert rating

This objective function is used to optimise the robot behaviour with respect
to a predicted expert score. [16] evaluated the magnitude-based
characteristics of the excision forces (the Amplitude and Consistency
features, captured by model $\mathcal{F}$) using 15 expert-labelled excision
trials. Here, we apply k-nearest neighbour regression (with $k=1$) to predict
the expert rating given the excision features.

Figure 4: Gaussian process model fit to six datapoints (black dots) collected
by optimising $\rho$ with respect to the Smoothness feature. (The black line
is the posterior mean, and the shaded region is 95$\%$ confidence interval).

###  III-C Bayesian optimisation of excision behaviour

We apply Bayesian Optimisation (BO)[19] to reduce the number of cutting
experiments required to generate a given human-like excision behaviour. At
each iteration, BO optimises an acquisition function $\alpha$ to choose the
next candidate sawing parameter $\rho$ for trajectory generation in order to
evaluate the objective function $g$ (as described above). We model the mapping
beween trajectory parameter $\rho$ and the objective function using a Gaussian
process (GP) [20]:

$g(\rho)\sim\mathcal{GP}\left(m(\rho),\kappa(\rho,\rho^{\prime})\right)$ (6)

where $m(\rho)$ and $\kappa(\rho,\rho^{\prime})$ are the mean and kernel
functions.

Figure 5: Results for behaviour ($\rho$) optimisation with respect to the
interpolated performance score from four experts. (Top row) Gaussian process
models fit to six observations (black dots) obtained during optimisation. The
black lines show the posterior, the shaded regions illustrate the 95$\%$
confidence intervals, and the red dashed lines highlight the optimal $\rho$
parameters for each of the experiments. (Bottom row) Contour plot of the
interpolated expert scores over the feature space of the excision force model
$\mathcal{F}$. The orange dots are the individual sample points $\rho$ used
during optimisation.

For $\alpha$, we use the Expected Improvement acquisition function with the
following closed-form expression [21]:

$\begin{array}[]{lcl}\alpha(\rho)=\left(m(\rho)-g(\rho^{+})-\epsilon\right)\Phi(Z)+\sigma(\rho)\phi(Z),\end{array}$
(7)

where

$Z=\begin{cases}\frac{m(\rho)-g(\rho^{+})-\epsilon}{\sigma(\rho)},&\text{if
}\sigma(\rho)>0\\\ 0,&\text{if }\sigma(\rho)=0\end{cases}$

Above, $m(\rho)$ and $\sigma(\rho)$ are the mean and the standard deviation of
the GP posterior, and $\Phi$ and $\phi$ are the cumulative probability
function and the probability density function, respectively. $\rho^{+}$ is the
current optimal choice of sawing parameter, and $\epsilon$ is a scalar that
sets the tradeoff between exploration and exploitation during optimisation.

We use a Matérn kernel [22] function $\kappa(\rho,\rho^{\prime})$ with the
following analytical expression:

$\kappa(\rho_{i},\rho_{j})=\frac{1}{\Gamma(\nu)2^{\nu-1}}\Big{(}\frac{\sqrt{2\nu}}{l}r\Big{)}^{\nu}K_{\nu}\Big{(}\frac{\sqrt{2\nu}}{l}r\Big{)}$
(8)

where $r=|\rho_{i}-\rho_{j}|$, $\Gamma(\cdot)$ is the gamma function,
$K_{\nu}$ the Bessel function, $\nu$ and $l$ kernel hyperparameters.

In our experiments, we set $\epsilon=2$, $\nu=2.5$ and $l=1$. These parameters
were chosen manually during setup calibration.

##  IV Experiments and Results

We performed two sets of experiments with the following objectives: 1) to find
the excision technique that maximises the smoothness feature of the applied
cutting forces, and 2) to find the excision technique whose excision forces
predict the highest expert ratings.

Fig. 4 shows the optimisation results for six iterations of smoothness feature
optimisation. The first trial was initialised with a sample
$\\{\rho_{1}=1,y_{1}\\}$, where $y_{1}$ is the force smoothness feature
evaluated by model $\mathcal{F}$. Optimisation results confirmed our
expectations that smoother excision trajectories (e.g. generated by model
$\mathcal{T}$ using larger values of $\rho$ parameter) result in smoother
excision forces.

Figure 6:  (Left) $\rho$ parameter vs evaluated smoothness feature. (Middle
and right) Contour plots of the $\rho$ parameter values obtained during
experiments. The orange dots are the individual datapoints with shown values
for $\rho$ parameter. Note: T and S denote medical trainee and professional
surgeon, respectively.

Fig. 5 shows the results for six iterations of expert score optimisation, for
each of the four experts. As before, we initialised the optimisation with
$\\{\rho_{1}=1,y_{1}\\}$ pair (this time, $y_{1}$ is the predicted expert
score, as described in section III-B2). The obtained mean of the posterior GP
predicts higher expert scores for the excision behaviours generated using
larger $\rho$ values (i.e. smoother trajectories). In other words, the results
suggest that sawing movement is more likely to be penalised by experts. In
addition, the experiment demonstrates that the sawing behaviour parameterised
by $\rho$ can achieve different modulations of excision forces. For example,
the score from Expert A highlights the rater’s preference towards excisions
with a more pronounced force modulation, as reflected by the Consistency
feature. Notice that the posterior GP successfully captures this preference
with the optimal sawing parameter $\rho\approx 6$.

###  IV-A Excision force characteristics versus $\rho$ parameter

We analysed the $\\{\rho,y\\}$ datapoints collected in the above experiments
to explore the relationships between the sawing parameter $\rho$ and
characteristics of the excision forces. Fig. 6 (left) shows a scatter plot of
the $\rho$ parameter values vs the smoothness feature of the excision forces
evaluated by model $\mathcal{F}$. As in the first experiment, the sawing
behaviour shows a strong positive correlation with the Smoothness feature
(Pearson’s $r=0.74$, $p<0.05$). This relationship has an intuitive
interpretation that smoother excision trajectories must result in a more
uniform application of excision forces. The smoothness of the excision forces
achieved by two medical students (dotted green lines) and two practising
surgeons (dashed red lines) suggest that more experienced surgeons are likely
to apply more uniform blade trajectories.

Fig. 6 (middle) shows the contour plots of the Amplitude and Consistency
features of excision forces against the sawing parameter $\rho$. The results
show that $\rho$ has no significant correlation with the excision force
amplitude - both smooth and sawing excision trajectories can yield equally low
or high cutting forces. On the other hand, the consistency of excision forces
(a feature that reflects the inverse of the spread of force levels during
excision), highlights a strong correlation with sawing parameter $\rho$
(Pearson’s $r=0.82$, $p<0.05$). This relationship is explained by Fig. 3
(bottom row), where larger $\rho$ values correspond to the noticeably lower
variations of $z_{b}$, and as the result, to lower variations along $x$, $y$
and pitch components of the excision trajectories. This observation agrees
with a general intuition that it is more difficult to apply excision forces
consistently when sawing.

The Confidence feature (Fig. 6 right), which characterises both the uniformity
and consistency of the excision forces, shows a significant alignment with
sawing parameter $\rho$ (Pearson’s $r=0.78$, $p<0.05$). Similar to the
Amplitude feature, the experiment results show a weak relationship (Pearson’s
$r=0.58$, $p<0.05$) between $\rho$ and the Energy feature. Finally, Fig. 6
shows the model $\mathcal{F}$ characterisation of excision forces from two
professional surgeons (denoted as S) and two medical trainees with no
experience at elliptical excision task (denoted as T). Notice that model
characterisation of surgeon excisions is aligned with the higher sawing
parameter $\rho$ values, whereas the performance of trainees matches the
region of lower $\rho$ values. Again, this indicates that surgeons are likely
to exhibit smoother excision trajectories when compared to less experienced
medical students.

##  V Discussion and conclusions

In this study, we used a model of excision forces to optimise the pose
trajectories of a blade in an elliptical excision task. We proposed an
excision trajectory generator capable of producing human-like elliptical
excision behaviours parameterised by a single parameter $\rho$ that defines
the amount of sawing motion applied during excision. These models can be used
to efficiently optimise the excision technique with desired excision
characteristics using Bayesian optimisation. More specifically, we show how to
align robotic excision behaviours with features that satisfy the criteria of
surgical experts and behaviours that resemble performance characteristics of
surgeons at various experience levels. Experimental results indicate that the
proposed $\rho$-parameterisation can successfully produce cutting force
modulation that maximises the performance assessment of experts.

Our analysis suggests that professional surgeons are more likely to apply
smoother excision trajectories, whereas less experienced medical trainees
exhibit greater sawing behaviour. It is well known that sawing movement
assists the cutting process by lowering the forces required to separate the
material [3]. We hypothesise that inexperienced trainees employ this strategy
to overcome the need to apply excessive excision forces to perform a
controlled cut. Although this study is limited to scalpel trajectories only,
the cutting forces are strongly affected by the tissue tensioning controlled
by the non-dominant hand. Hence, we hypothesise that experienced surgeons,
while applying the smooth excision trajectories, actively tension the tissues
to assist the excision. This combination of excision and tissue tensioning
strategies is an interesting line of future work to better understand the
dexterous manipulation skills underlying the elliptical excision task.
Finally, we want to emphasise the promising prospect of using the proposed
framework for studying surgical techniques at various levels of expertise,
analysing common mistakes and their effects on the tissues.

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