# Walking for short distances and turning in lower-limb amputees: a study in
low-cost prosthesis users

Nidhi Seethapathi1,3, Anil Kumar Jain2 and Manoj Srinivasan1
1 Mechanical and Aerospace Engineering,
The Ohio State University, Columbus OH 43210, USA
2Santokba Durlabhji Memorial Hospital, Jaipur, Rajasthan 302015, India
3 Department of Bioengineering, University of Pennsylvania,
Philadelphia PA 19104, USA

###### Abstract

Preferred walking speed is a widely-used performance measure for people with
mobility issues. Often these speeds are measured walking in a straight line
and walking over short distances. However, daily walking involves walking for
bouts of different distances and walking with turning. Here, we observe
walking for short distances and walking in circles in subjects with unilateral
lower-limb amputation using a Jaipur Foot prosthetic leg. We find that the
preferred walking speeds are lower for unilateral amputees for short
distances, but the distance-dependence of preferred walking speed is less
steep than for non-amputees. In circle walking, unilateral amputees slow down
when walking in circles of smaller radii and above-knee amputees in particular
walk faster when the prosthetic leg is outside the circle. Using simple
mathematical models, we show that the observed preferred walking speeds are
predicted by minimal energy cost with increased costs for walking, changing
speeds, and turning for amputees compared to non-amputees. These results
predict that the cost of changing speeds and turning may be substantial in
amputees but still not as high as the constant-speed cost. These findings will
inform prosthesis design and rehabilitation therapy to better assist changing
speeds and turning tasks in amputee walking.

##  1 Introduction

Preferred overground walking speed is commonly used to quantify a human
subject’s mobility improvement after being fit with a new prosthetic leg or
after undergoing physical therapy or rehabilitation from stroke, other injury,
or movement disorder [1, 2]. Such walking speeds are estimated using a variety
of tests in the lab, most commonly using the 6 minute walk test [3] or the 10
m walk test [4], but also by having subjects walk other short distances such
as 3 m [5], 4 m [6], 5 m [7], and 15 m [8, 9]. In healthy adults with no
movement disorders, the preferred walking speed for walking in a straight line
is distance-dependent [10]: the speed is systematically lower for shorter
distances and this distance-dependence can be explained by the larger
energetic cost of speeding up and slowing down for shorter distances [10].
Here, we propose this distance dependence of walking speed as a more complete
measure of preferred walking speed (than measuring the speed at just one
distance) and characterize it in unilateral amputees. This distance dependence
of walking speed can also help shed light on daily walking behavior in humans,
in which a considerable percentage of walking occurs in short bouts [11],
especially in amputees [12].

Another important aspect of daily walking behavior is that people often need
to walk along curves or with some turning. Indeed, for subjects in one
previous study, between 8% and 50% of all walking steps in daily life involved
turning [13]. So effective mobility requires ability to walk with turning. As
a way of quantifying turning ability and as an additional measure of the
effectiveness of a prosthetic leg or a physio-therapeutic intervention, here
we propose the measurement of preferred speeds while walking in circles of
different radii and characterize such walking speeds in unilateral amputees.
Although such curved walking interventions have been used for other
populations [14, 15, 16], there is less literature on the subject for amputee-
walking. In healthy non-amputee adults, the tangential speed of walking
depends on the curvature of the circle walked: slower walking for smaller
circles or higher curvature, explained by the increased energetic cost of
walking with turning [17, 18]. Here, we test the hypothesis that walking
speeds depend on circle radius in unilateral amputees and whether the
prosthesis foot being the outer or inner foot affects circle-walking speeds.

Metabolic energy optimality has been used to make predictions for a number or
aspects of overground and treadmill walking behaviors in healthy non-amputee
adults [19, 20, 21]. Indeed, as noted above, the human preference slower
walking speeds for shorter distances and slower walking speeds for smaller
radii and higher curvatures have been attributed to optimization of the
corresponding energy costs [10, 17, 18]. However, there is less work making
such energy optimality based predictions for other population with amputations
or other movement disorders [22, 23, 24]. Here, we examine this energy
minimization hypothesis in the context of amputee walking for short distances
and in curves.

Amputees walking with passive prosthetic legs usually have a higher steady
state metabolic cost [25] compared to non-amputees (but see recent work that
find no significant cost difference in some amputee sub-populations [26]).
However, it is not known whether amputees also have a higher cost of speeding
up and slowing down when walking. The interactions between the walking cost
and the changing speed costs may decide how amputees walk over short
distances. The higher the cost of speeding up and slowing down, the greater it
will affect the optimal walking speed at short distances. Similarly, while it
is known that walking along curves costs more energy than walking in a
straight line in non-amputee adults [17, 18], the energy cost of turning has
not been characterized in amputees. Thus, by viewing preferred walking
behavior in amputees through the lens of energy minimization, these short
distance and circle walking tasks provide insight into how their cost of
changing speeds and the cost of turning may compare to that of non-amputee
individuals.

Here, we studied a small population of unilateral amputees, both above- and
below- knee, using the Jaipur foot prosthesis, a low cost prosthesis used
widely in the developing world [27]. This prosthesis, developed by P. K. Sethi
and co-workers [27], was designed for facilitating practices common in India,
such as bare foot or shod walking over unpaved uneven terrain and squatting or
cross-legged sitting on the floor. The Jaipur foot is used in over 22
countries and by hundreds of thousands of amputees, most widely used second
only to the SACH foot [28]. Thus, this study adds to the small number of
biomechanical studies (e.g., [29, 30]) on the Jaipur foot, which is under-
studied despite being so widely used.

##  2 Methods

### Subject population.

The experimental protocol was approved by the Ohio State University
Institution Review Board and all subjects participated with informed verbal
consent. All subjects ($N=12$ with 11 males, 1 female, $65.75\pm 12.6$ kg with
prosthesis and shoes, height $1.67\pm 0.09$ meters and age $39.67\pm 15.31$
years, mean $\pm$ s.d.) were unilateral amputees, out of which 7 subjects were
above-knee amputees and 5 were below-knee amputees. All subjects had a Jaipur
Foot prosthesis [27], manufactured and fit in the SDMH hospital in Jaipur; all
walking trials were also conducted at this location. Subjects walked
independently without using canes, crutches, hand rails, or other assistive
devices. The subjects did not carry any additional instrumentation. The
subjects performed two kinds of walking trials: (i) walking for short
distances and (ii) walking in circles, as described below (Figure 1).

### Walking for short distances.

Subjects were instructed to walk in a straight-line for five different short
distances: 4 m, 6 m, 8 m, 10 m and 23 m (Figure 1a). There were four trials
for each distance and trial order was randomized. Subjects were asked to “walk
the way they usually walk” and they had to start and end each trial standing
still, so they had to speed up from and slow down to rest. Average walking
speeds were estimated by measuring the time duration for each trial.

Figure 1: Overground walking experiment setup. We measured the preferred
walking speed of walking for unilateral amputees wearing a passive prosthetic
leg in two conditions: a) walking a range of short distances, starting and
stopping each bout at rest and b) walking in circles of different radii, both
clockwise and anti-clockwise.

### Walking in circles.

Subjects were asked to walk in circles of three different radii: 1 m, 2 m and
3 m (Figure 1b), completing 5, 4 and 3 laps, respectively, for these radii.
For each radius, subjects performed two trials, once with the prosthetic leg
inside the perimeter and once with the prosthetic leg outside the perimeter of
the circle. Trial order was randomized over the circle radii and walking
directions. The average speed was obtained by measuring the total walking
duration and averaging over all laps for each trial. Subjects walked with the
circle between their two feet, maintaining a non-zero step width, rather than
step on the circle with both feet.

### Mathematical model: walking for short distances.

For short distance walking, we compare the experimentally observed preferred
walking speed results to the walking speed predicted by minimizing the total
metabolic cost of the walking bout. For simplicity, we assume that people
walking a distance $D$ start from rest (specified in experiment), then
instantaneously speed up to some speed $v$, continue at that speed for the
whole distance and then instantaneously come to rest again. Thus, the total
cost of walking the distance includes the cost of accelerating from rest to
speed $v$ at the start, walking at constant speed $v$ and then decelerating to
rest at the end of the walking bout, given by the following equation
(analogous to the approach in [10]):

$E_{\mathrm{met}}=(a_{0}+a_{1}v+a_{2}v^{2})\frac{D}{v}+\lambda\left(\frac{1}{\eta_{\mathrm{pos}}}+\frac{1}{\eta_{\mathrm{neg}}}\right)\left(\frac{1}{2}mv^{2}\right).$
(1)

Here, $\dot{E}=a_{0}+a_{1}v+a_{2}v^{2}$ in Wkg-1, with $v$ in ms-1 models the
metabolic rate of walking $\dot{E}$ at a constant speed $v$ for both amputees
and non-amputees, with $a_{0}=4.97$, $a_{1}=-5.78$, and $a_{2}=5.62$ for
above-knee amputees [31], $a_{0}=3.24$, $a_{1}=-2.19$, and $a_{2}=2.89$ for
below-knee amputees [32], and $a_{0}=2.22$, $a_{1}=0$, and $a_{2}=1.155$ for
non-amputees [31]. All of these relations for $\dot{E}$ result in a classical
U-shaped relationship between energy cost per unit distance and speed of
walking. The quantity $\lambda$ provides a scaling factor between the kinetic
energy $mv^{2}/2$ and the energy cost required to achieve it, with
$\eta_{\mathrm{pos}}=0.25$ and $\eta_{\mathrm{neg}}=1.2$ are traditional
muscle efficiencies for performing positive and negative mechanical work,
respectively [33]. Because we did not directly estimate the cost of changing
speeds in amputees here (as in [10]), we tested different $\lambda$ values to
best explain the observed walking behavior.

The speed that minimizes the short-distance cost of walking $E_{\mathrm{met}}$
is given by the implicit equation: $\lambda
v^{3}\left({\eta_{\mathrm{pos}}}^{-1}+{\eta_{\mathrm{neg}}}^{-1}\right)/(a_{0}-a_{2}v^{2})=D$.
This relation implies that shorter distance bouts should have lower speeds.

### Mathematical model: Walking in circles.

Following [17, 18], who directly measured the cost for walking in circles for
non-amputee subjects, we analogously propose that the metabolic rate of
walking in a circle for amputees is
$\dot{E}=a_{0}+a_{1}v+a_{2}v^{2}+b_{2}(v/R)^{2}$, where $R$ is the circle
radius, with $a_{0,1,2}$ values as above. The cost per distance is given by
$\dot{E}/v=a_{0}/v+a_{1}+a_{2}v+b_{2}v/R^{2}$, which is minimized by the speed
$v=\sqrt{a_{0}/(a_{2}+b_{2}/R^{2})}$. Thus, the prediction is that the optimal
speed is smaller for smaller radius $R$. In the following, we chose $b_{2}$ to
best explain the speed reduction exhibited by our amputee subjects.

For both short-distance walking and circle walking, we also determine the
speeds that are within 1% of the minimum energy cost, because the energy
landscapes are usually flat and a small change in speed near the minimum
energy usually results in a much smaller energy change.

##  3 Results

### Preferred walking speed for unilateral amputees depends on distance
walked.

Both above- and below-knee amputees showed, on average, a decrease in
preferred walking speed for short distances. Pooled across all above and below
knee subjects, the preferred walking speed for each of the short distances (4
m, 6 m, 8 m and 10 m) was significantly lower than the preferred walking speed
for the long-distance 23 m trial ($p<0.0025$ for each distance, left-tailed
paired $t$-test). The percentage decreases in the amputees’ preferred walking
speeds, compared to the long distance 23 m trial, are shown in Figure 2a. The
percentage decrease is higher for shorter distances: in above knee amputees,
the percentage decreases ranged from 7% for the 10 m walk to 13% for the 4 m
walk, on average. In below knee amputees, the percentage decreases ranged from
7% for the 10 m walk to 18% for the 4 m walk.

### The rate of speed decrease with distance is lower for amputees compared to
non-amputees.

Although we find that the preferred walking speed for amputees decreases with
distance walked, the rate of decrease is flatter compared to that for non-
amputee individuals (data repeated from [10]). As seen in Figure 2b, the best-
fit slope of the distance dependence of speed for non-amputee subjects is
about three times steeper than that for the above-knee amputee subjects and
about 1.5 times that for the below-knee amputees. Thus, the above-knee
amputees slowed down by a much smaller percentage for short-distance walking
trials, compared to both below knee amputees and non-amputee subjects.

Figure 2: Decrease in preferred walking speed with distance walked for
amputees. a) Amputees showed a greater decrease in preferred walking speed for
short distances speed compared to able-bodied individuals. b) The rate of
change in preferred walking speed with distance is steeper for able-bodied
individuals than for the unilateral amputees.

### Increased cost of walking and changing speeds is consistent with flatter
speed-distance relationships.

Our simple optimization-based model of short distance walking predicts a
flatter distance-dependence of optimal walking speed for both above and below
knee amputees (Figure 2,3), when we take into account the increased constant-
speed metabolic cost of walking previously measured in experiments [31, 32]
and an increased cost of changing speeds compared to able-bodied subjects
[10]. Compared to non-amputees, the scaling factor $\lambda$ for the cost of
changing speeds was increased by a factor of 1.87 for below-knee amputees and
1.90 above-knee amputees. Nevertheless, mechanistically, our model suggests
that the amputees do not slow down as much for short distances because the
cost of moving at a reduced speed for the whole distance outweighs the cost of
changing speeds from rest. As seen in Figure 3, the mean amputee preferred
walking speeds plus one standard error is within 1% of the optimal energy
costs from this model.

Figure 3: Minimization of total metabolic cost captures slower short-distance
walking speeds. The total cost of the walking a short distance includes a term
due to constant-speed cost and a changing-speed cost. We find that minimizing
this total cost predicts the observed trends in changing preferred walking
speed with distance for both amputees and non-amputees. The error bars for
human data represent standard errors, and the filled bands represent the set
of all speeds within 1% of the energy optimal energy cost.

### Preferred walking speeds for unilateral amputees walking in circles
depends on the circle radii.

We find that, irrespective of type of amputation, all the unilateral amputees
that we observed slowed down when walking along circles of smaller radii
(Figure 5), compared to walking a longer distance in a straight line. The
slope of decrease in preferred walking speed with radius of circle walked, is
quite similar for the able-bodied (0.1 s-1) and amputee populations (0.08
s-1). Using our optimization-based model to fit this decreasing speed trend,
the best-fit scaling coefficient $b_{2}$ for the cost of turning was about six
times more than non-amputees for below-knee amputees and eight times more than
non-amputees for above-knee amputees.

Figure 4: Preferred walking speeds for circle walking. a) The preferred
walking speed for all the unilateral amputees showed a decrease with radius of
the circle walked. b) Amputees, when pooled together, did not show a
significant difference in preferred walking speed when walking with the
prosthesis-leg inside versus outside the circle. c) Above knee amputees show a
greater walking speed on average when the prosthesis leg is outside the
circle. Figure 5: Optimal walking speeds for circle walking. Minimizing the
energy cost of walking in a circle predicts slower walking for smaller
circles. Error bars shown for the data correspond to one standard error about
the mean, and these are generally within the set of all speeds within 1% of
the optimal energy costs (the shaded bands shown, as in Figure 3. The non-
amputee data and model used for comparison is from [17, 18].

### Preferred walking speeds for above-knee amputees walking in circles is
dependent on turning direction.

We had subjects walk both clockwise and anti-clockwise along circles drawn on
the ground. We did this so as to check for any effects due to having the
prosthesis-leg as the pivot, as opposed to the intact leg as the pivot.
Considering all amputees together, we did not find significant differences
between the two conditions ($p=0.096$). Just considering the above-knee
amputees, with trials for all radii pooled, we found that they walked slightly
faster when the prosthetic leg was outside the circle ($p=0.0008$, mean
difference about 0.03 m/s on average, median 0.04 m/s).

### Preferred walking speeds are proportional to degree of amputation.

For all the cases above, that is, straight-line walking for different
distances and circle-walking at different radii, we found that the preferred
walking speeds for the below-knee amputees are higher than above-knee and
lower than non-amputee preferred speeds (Figures 2-3). This result for
straight-line long-distance walking over long distances is known from other
work in the past [34]. However, the result for short-distance and circle-
walking is new, even if unsurprising. In addition, we found, as have others in
the past , that the preferred walking speed for the below-knee unilateral
amputees were higher than those for the above-knee unilateral amputees.

### Preferred gait initiation swing is usually with the affected limb.

In addition to measuring the preferred walking speeds, we noted whether the
subjects stepped forward with their affected or unaffected limb for their very
first step. Stepping forward with the affected limb corresponds to the first
swing phase being with the affected limb and the first stance phase being with
the unaffected limb. We found that 9 out of 12 subjects had over 80% of their
first steps be their prosthetic foot; the other three subjects had 69%, 36%,
and 0% of their steps start with swinging the affected limb. These leading
limb preferences are similar to those found in [35].

##  4 Discussion

Preferred walking speed is often used as a measure of progress in walking
rehabilitation for various populations, for instance persons with
neuromuscular disorders and amputees. An implicit assumption made when relying
on such measures is that higher speeds means more improvement. However, past
theoretical and experimental work on able-bodied subjects shows that the speed
at which people choose to move depends on the constraints of the motion itself
like the distance walked [10] and the curvature of the motion [17, 18]. So,
depending on the situation, people may sometimes move at a lower speed than
physically possible to satisfy some other objective, like minimizing energy.
Here, we find that these observations extend to amputee populations as well
and may have significant implications for the usage of preferred walking
speeds as a measure of performance. These implications are detailed in the
paragraphs below.

We find that above-knee and below-knee amputees slow down when walking short
distances, as do non-amputees. This implies that the distance over which the
preferred walking speed is measured and interpreted during rehabilitation may
systematically overestimate or underestimate the progress that the patient has
made. Many studies about measure progress in people with walking disorders
after rehabilitation, measure the preferred walking speed over very short
distances. In order to circumvent these distance-effects, we suggest measuring
the speed over a few distances, not just one or two. Also, when comparing the
preferred speed values for amputees to the able-bodied values, we suggest
comparing to the values for the same distance walked. This is because the
difference in walking speed between able-bodied and above-knee amputees over
short distances is lesser than the corresponding difference over longer
distances.

An alternative to in-lab measurements of preferred speeds is to track
subjects’ speeds and movements all day using body-worn sensors such as
pedometers, IMUs, GPS, etc [36, 37, 38]. Ultimately, it is these speeds during
daily living that is of relevance to quantifying mobility. Such ambulatory
measurements provide an opportunity to independently corroborate the results
in this study, by characterizing the speeds over bouts of different lengths
and walks with turns that naturally occur during daily life.

Everyday walking consists of not just straight line walking but also turning.
Here, we find that unilateral amputees also slow down when taking sharp turns
(circles of smaller radii) similar to non-amputees [17, 18]. We also find that
the corresponding walking speeds are lower for the amputees for all radii,
compared to those for able-bodied individuals. For all these reasons, we
propose that circle-walking lends itself as an additional useful measure of
walking performance during rehabilitation. Unlike the straight line walking
trials, we were not interested in capturing the effects of speeding up from
and slowing down to rest for circle walking. We believe that the multiple laps
used makes such distance-dependence effects negligible.

In the past, people have predicted aspects of walking behavior such as walking
speed, walking step width and walking step frequency using energy minimization
as a hypothesis. However, most of these studies look at constant-speed
straight line walking on treadmills. Moreover, a majority of these studies
attempt to predict non-amputee walking behavior. Here, we provide evidence
that energy-minimization can predict aspects of non-steady overground walking
behavior in a disabled population. Specifically, we have found that minimizing
a combined cost of constant-speed walking and a cost for changing speeds
captures the short-distance walking speed trends in amputees; similarly,
adding a cost for turning captures the circle-walking speed trends.

We have studied the preferred walking speeds of unilateral amputees wearing a
passive prosthetic leg (Jaipur Foot) used in a number of developing countries.
Using preferred walking speeds as a performance measure may be more relevant
where the resources available are limited, where access to other measures of
performance, such as using a gait lab with motion capture and force plates,
may be limited. Thus, we feel that our conclusions regarding the qualification
of preferred walking speeds as a performance measure would be more relevant
where walking-speed-based mobility measures would be more exclusively used.

Fitting the short-distance walking model and circle-walking model to the
amputee walking speed behavior, we found that the scaling factors for the cost
of changing speeds and for the cost of turning ($\lambda$ and $b_{2}$) are
much higher than for non-amputees. To test whether these increased cost
estimates, one could directly measure the metabolic cost of changing speed and
the cost of turning, as in [10, 17, 18]. Stability considerations may be an
alternative to metabolic cost being the determinant of reduced speeds,
especially for turning in a circle. One could examine this alternative
hypothesis by having subjects use different speeds and estimating simple
measures of stability [39, 40].

For short-distance walking, we measured the average speed over the whole bout
of the amputees. This average speed includes the acceleration and deceleration
periods. So the reduction in average speed is partly due to a greater portion
of the bout being spent in acceleration-deceleration and partly due to reduced
walking speed. This was the case in the earlier non-amputee study as well
[10]. In that study, walking slower for shorter distances was a real choice,
as the subjects could certainly cover the distance in shorter time. While we
did not test this hypothesis here by asking the amputees to walk faster than
they preferred to do, we suggest, based on prior studies with amputee
populations, that they can typically walk much faster than their preferred
speeds (e.g., [41, 31]). Specifically, we do not know of any demonstrated
instance in which someone’s preferred speed is also their maximum possible
speed, although it may be possible.

One limitation of this study is the relatively small sample size of subjects.
The experiments were conducted in a hospital in India and not all the patients
had the time or resources to participate in the study. Small sample sizes are
common with such amputee populations. Despite the small sample size, the
qualitative results for the amputees are robust. A follow-up study will be
conducted to investigate the population with a larger sample size. Another
potential limitation is that we did not limit our amputee sample by years
since amputation and age. However, we believe that this limitation is offset
by the fact that we focus here on the change in preferred speeds of each
subject under different conditions relative to his/her own long-distance
straight-line walking speed. So, the individual differences in terms of years
of being accustomed to wearing the prosthesis will not affect our results. We
compare the non-amputee walking speeds of a diverse group of college or
graduate student-age subjects [10, 17, 18] in the USA to those of Indian
amputees. It would perhaps be more appropriate to compare with age and size-
matched Indian non-amputee adults. However, past studies of variations in
walking speed due to location or country are much lesser than the differences
between non-amputee and amputee populations that we observe here [42]. So, our
fundamental findings will not change even with the more appropriate Indian
able-bodied population.

It would also be useful to repeat these experiments in other subject
populations, including amputees wearing other prostheses with different
mechanical properties. The Jaipur foot is a unique prosthesis, designed for
Indian life and has features that make it mechanically different compared to
other prostheses. Specifically, it has more mobility in the ankle and subtalar
joint to allow squatting, sitting cross legged, and walking over uneven
unpaved surfaces. It has three main pieces: a heel block and fore-foot-toe
block made out of micro-cellular rubber bonded to a wooden ankle block. Tread
rubber is used on the undersurface to provide traction and the entire assembly
is covered in a skin colored compound to provide cosmesis and water proofing.
We speculate that these construction features give more compliance, allowing
its users to perform more three dimensional tasks; future work should consider
more detailed biomechanical analyses for such 3D tasks.

Finally, given the simplicity of these tasks, we propose that these distance-
dependence of walking speeds and radius-dependence of walking speeds be used
as routine measures of mobility not just in amputees, but also other subject
populations such as the elderly and those with or recovering from other
movement disorders.

### Acknowledgments.

The authors thank Dr. Harlal Singh Mali for hosting N.S. in his lab briefly
during her visit to Jaipur and for initially facilitating interactions.

### Ethics Statement

All subjects participated with informed consent and the experimental protocol
was approved by the the Ohio State University Institution Review Board.

### Funding Statement.

For this study, N.S. received funding from Schlumberger Foundation Faculty for
the Future Fellowship, The Ohio State University Global Gateway Grant and the
Alumni Grant for Graduate Research and Scholarship. MS was supported in part
by NSF CMMI grant .

### Data Accessibility.

All data and codes form will be made available upon acceptance.

### Competing Interests.

We have no competing interests.

### Authors’ Contributions.

N.S. conceived the study, collected the data, analyzed the results, performed
the mathematical analyses, and wrote the paper. A.J. provided guidance for
data collection and edited the draft. M.S. provided ideas for designing the
study, suggested additional analyses, and edited the draft.

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Antonino Ingargiola wrote much of the high level Model code and has provided
many bug fixes and improvements. Daniel B. Allan wrote much of the original
version of the high level Model code, and many improvements to the testing and
documentation. Austen Fox fixed many of the built-in model functions and
improved the testing and documentation of these. Michal Rawlik added plotting
capabilities for Models. The method used for placing bounds on parameters was
derived from the clear description in the MINUIT documentation, and adapted
from J. J. Helmus’s Python implementation in leastsqbounds.py. E. O. Le Bigot
wrote the uncertainties package, a version of which was used by lmfit for many
years, and is now an external dependency. The original AMPGO code came from
Andrea Gavana and was adopted for lmfit. The propagation of parameter
uncertainties to uncertainties in a Model was adapted from the excellent
description at
https://www.astro.rug.nl/software/kapteyn/kmpfittutorial.html#confidence-and-
prediction-intervals, which references the original work of: J. Wolberg, Data
Analysis Using the Method of Least Squares, 2006, Springer. Additional
patches, bug fixes, and suggestions have come from Faustin Carter, Christoph
Deil, Francois Boulogne, Thomas Caswell, Colin Brosseau, nmearl, Gustavo
Pasquevich, Clemens Prescher, LiCode, Ben Gamari, Yoav Roam, Alexander Stark,
Alexandre Beelen, Andrey Aristov, Nicholas Zobrist, Ethan Welty, Julius
Zimmermann, Mark Dean, Arun Persaud, Ray Osborn, @lneuhaus, Marcel Stimberg,
Yoshiera Huang, Leon Foks, Sebastian Weigand, Florian LB, Michael Hudson-
Doyle, Ruben Verweij, @jedzill4, @spalato, Jens Hedegaard Nielsen, Martin
Majli, Kristian Meyer, @azelcer, Ivan Usov, and many others. The lmfit code
obviously depends on, and owes a very large debt to the code in
scipy.optimize. Several discussions on the SciPy-user and lmfit mailing lists
have also led to improvements in this code. Other software: numpy (Harris et
al., 2020), pandas (pandas development team, 2020; Wes McKinney, 2010),
matplotlib (Hunter, 2007), sncosmo (Barbary et al., 2021), simsurvey (Feindt
et al., 2019a)

## Data Availability

The ZTF DR2 photometry is available on GitHub. The binning program is
available at https://github.com/JTerwel/late-time_lc_binner. snap is available
at https://github.com/JTerwel/SuperNova_Animation_Program. The ePESSTO+
photometry is available on the ESO archive. The late-time spectrum of SN
2020alm will be available upon request to the author.

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##  Appendix A Tables

Table 7: Spectra used to make the SN 2011fe model. All spectra were taken from WISeREP (Yaron & Gal-Yam 2012). MJD | Phase (d) | Telescope | Instrument | Wavelength coverage (Å) | Reference
---|---|---|---|---|---
55798.0 |  $-$16.0 | Lijiang-2.4m | YFOSC | 3461 – 8956 | Zhang et al. (2016)
55798.2 |  $-$15.8 | Lick-3m | KAST | 3416 – 10278 | Nugent et al. (2011)
55799.0 |  $-$15.0 | Lijiang-2.4m | YFOSC | 3502 – 8958 | Zhang et al. (2016)
55799.3 |  $-$14.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55800.2 |  $-$13.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55801.2 |  $-$12.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55802.3 |  $-$11.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55803.2 |  $-$10.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55804.2 |  $-$9.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55805.2 |  $-$8.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55806.2 |  $-$7.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55807.3 |  $-$6.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55808.2 |  $-$5.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55809.2 |  $-$4.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55811.4 |  $-$2.6 | HST | STIS | 1779 – 24965 | Mazzali et al. (2014)
55812.0 |  $-$2.0 | Gemini-N | GMOS | 3497 – 9648 | Parrent et al. (2012)
55813.2 |  $-$0.8 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55814.2 | 0.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55815.2 | 1.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55816.2 | 2.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55817.2 | 3.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55817.7 | 3.7 | HST | STIS | 1265 – 24965 | Mazzali et al. (2014)
55818.2 | 4.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55821.2 | 7.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55823.2 | 9.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55826.2 | 12.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55828.2 | 14.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55829.0 | 15.0 | Gemini-N | GMOS | 3497 – 9643 | Parrent et al. (2012)
55830.2 | 16.2 | Keck1 | LRIS | 3227 – 10242 | Stahl et al. (2020)
55831.2 | 17.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55832.0 | 18.0 | Lijiang-2.4m | YFOSC | 3577 – 8957 | Zhang et al. (2016)
55833.2 | 19.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55835.3 | 21.3 | HST | STIS | 1731 – 10221 | Mazzali et al. (2014)
55836.2 | 22.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55838.2 | 24.2 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55841.3 | 27.3 | HST | STIS | 1738 – 10221 | Mazzali et al. (2014)
55855.2 | 41.2 | HST | STIS | 1738 – 10216 | Mazzali et al. (2014)
55888.6 | 74.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55891.7 | 77.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55893.6 | 79.7 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55896.6 | 82.6 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55897.7 | 83.7 | Keck1 | LRIS | 3164 – 10126 | Stahl et al. (2020)
55901.6 | 87.6 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55903.6 | 89.6 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55911.0 | 97.0 | XLT | BFOSC | 3296 – 9693 | Zhang et al. (2016)
55911.6 | 97.6 | UH88 | SNIFS | 3296 – 9693 | Pereira et al. (2013)
55913.5 | 99.5 | Lick-3m | KAST | 3427 – 10332 | Stahl et al. (2020)
55914.0 | 100.0 | WHT-4.2m | ISIS | 3499 – 9491 | Friesen et al. (2017)
55916.0 | 102.0 | WHT-4.2m | ISIS | 3498 – 9491 | Law et al. (2009); Rau et al. (2009)
55917.0 | 103.0 | WHT-4.2m | ISIS | 3499 – 9492 | Law et al. (2009); Rau et al. (2009)
55926.0 | 112.0 | Lijiang-2.4m | YFOSC | 3366 – 9069 | Zhang et al. (2016)
55929.5 | 115.5 | Lick-3m | KAST | 3426 – 10170 | Stahl et al. (2020)
55944.5 | 130.5 | Lick-3m | KAST | 3453 – 10088 | Stahl et al. (2020)
55959.0 | 145.0 | Lick-3m | KAST | 3497 – 10000 | Law et al. (2009); Rau et al. (2009)
55980.4 | 166.4 | Lick-3m | KAST | 3441 – 10250 | Stahl et al. (2020)
55988.0 | 174.0 | WHT-4.2m | ISIS | 3495 – 9982 | Mazzali et al. (2015)
56019.4 | 205.4 | Lick-3m | KAST | 3438 – 10324 | Mazzali et al. (2015)
56040.4 | 226.4 | Lick-3m | KAST | 3437 – 10178 | Mazzali et al. (2015)
56047.0 | 233.0 | Lijiang-2.4m | YFOSC | 3392 – 9053 | Zhang et al. (2016)
56073.0 | 259.0 | WHT-4.2m | ISIS | 3495 – 9483 | Mazzali et al. (2015)
56103.0 | 289.0 | WHT-4.2m | ISIS | 3423 – 10268 | Mazzali et al. (2015)
56127.0 | 313.0 | P200 | DBSP | 3197 – 10991 | Law et al. (2009); Rau et al. (2009)
56162.2 | 348.2 | Lick-3m | KAST | 3487 – 10240 | Mazzali et al. (2015)
56194.2 | 380.2 | Keck1 | LRIS | 3232 – 10268 | Stahl et al. (2020)
56277.0 | 463.0 | Lijiang-2.4m | YFOSC | 3379 – 9337 | Zhang et al. (2016)
56778.5 | 964.5 | Keck1 | LRIS | 3074 – 10320 | Graham et al. (2015)
56831.2 | 1017.2 | LBT | MODS1 | 3098 – 10487 | Taubenberger et al. (2015)

Table 8: Redshift values where 50 per cent of the simulated SNe were found to have CSM interaction. Strength is the strength of the H $\alpha$ line compared to the strength detected in SN 2015cp. Start shows how many days after the peak the interaction begins, and duration is in days as well. We fitted sigmoid functions to the results of each simulation in order to find the redshift where 50 per cent of the interactions were recovered, assuming the reference images were of the same depth as the ones used in ZTF or 0.5 or 1 mag deeper. These values are shown in $z_{50}\pm\sigma_{z_{50}}$, and $\chi^{2}_{\text{red}}$ shows the quality of the fit. |  |  |  | ZTF references |  | 0.5 mag deeper |  | 1 mag deeper
---|---|---|---|---|---|---|---|---
strength | start | duration |  | $z_{50}$ | $\sigma_{z_{50}}$ | $\chi^{2}_{\text{red}}$ |  | $z_{50}$ | $\sigma_{z_{50}}$ | $\chi^{2}_{\text{red}}$ |  | $z_{50}$ | $\sigma_{z_{50}}$ | $\chi^{2}_{\text{red}}$
0.0 | – | – |  | 0.0038 | 0.0009 | 0.72 |  | 0.0042 | 0.0019 | 1.83 |  | 0.0046 | 0.0032 | 3.47
0.1 | 100 | 100 |  | 0.0041 | 0.0008 | 0.83 |  | 0.0049 | 0.0011 | 1.82 |  | 0.0049 | 0.0033 | 3.41
0.1 | 100 | 300 |  | 0.0050 | 0.0007 | 0.79 |  | 0.0055 | 0.0014 | 1.79 |  | 0.0059 | 0.0025 | 3.42
0.1 | 100 | 500 |  | 0.0053 | 0.0006 | 0.77 |  | 0.0062 | 0.0011 | 1.76 |  | 0.0068 | 0.0018 | 3.44
0.1 | 200 | 100 |  | 0.0045 | 0.0007 | 0.78 |  | 0.0050 | 0.0016 | 1.80 |  | 0.0054 | 0.0028 | 3.38
0.1 | 200 | 300 |  | 0.0053 | 0.0006 | 0.78 |  | 0.0058 | 0.0014 | 1.79 |  | 0.0063 | 0.0022 | 3.44
0.1 | 200 | 500 |  | 0.0056 | 0.0006 | 0.77 |  | 0.0066 | 0.0010 | 1.79 |  | 0.0073 | 0.0016 | 3.47
0.1 | 300 | 100 |  | 0.0050 | 0.0006 | 0.80 |  | 0.0056 | 0.0012 | 1.80 |  | 0.0059 | 0.0023 | 3.43
0.1 | 300 | 300 |  | 0.0052 | 0.0006 | 0.80 |  | 0.0061 | 0.0044 | 1.78 |  | 0.0067 | 0.0017 | 3.46
0.1 | 300 | 500 |  | 0.0053 | 0.0005 | 0.82 |  | 0.0060 | 0.0010 | 2.03 |  | 0.0069 | 0.0016 | 3.53
0.1 | 500 | 100 |  | 0.0045 | 0.0007 | 0.82 |  | 0.0048 | 0.0014 | 2.08 |  | 0.0054 | 0.0021 | 3.52
0.1 | 500 | 300 |  | 0.0050 | 0.0005 | 0.83 |  | 0.0056 | 0.0010 | 2.09 |  | 0.0062 | 0.0017 | 3.53
0.1 | 500 | 500 |  | 0.0050 | 0.0005 | 0.82 |  | 0.0057 | 0.0009 | 2.09 |  | 0.0063 | 0.0017 | 3.52
1.0 | 100 | 100 |  | 0.0041 | 0.0007 | 0.86 |  | 0.0045 | 0.0013 | 2.10 |  | 0.0046 | 0.0027 | 3.58
1.0 | 100 | 300 |  | 0.0091 | 0.0005 | 0.85 |  | 0.0096 | 0.0012 | 1.98 |  | 0.0108 | 0.0017 | 3.40
1.0 | 100 | 500 |  | 0.0111 | 0.0004 | 0.84 |  | 0.0129 | 0.0008 | 2.08 |  | 0.0139 | 0.0013 | 3.50
1.0 | 200 | 100 |  | 0.0067 | 0.0007 | 0.80 |  | 0.0077 | 0.0012 | 2.05 |  | 0.0083 | 0.0021 | 3.51
1.0 | 200 | 300 |  | 0.0104 | 0.0005 | 0.93 |  | 0.0122 | 0.0008 | 2.06 |  | 0.0130 | 0.0014 | 3.31
1.0 | 200 | 500 |  | 0.0116 | 0.0004 | 0.84 |  | 0.0129 | 0.0008 | 2.11 |  | 0.0140 | 0.0018 | 3.63
1.0 | 300 | 100 |  | 0.0086 | 0.0004 | 0.78 |  | 0.0094 | 0.0008 | 1.95 |  | 0.0100 | 0.0013 | 3.62
1.0 | 300 | 300 |  | 0.0105 | 0.0004 | 0.75 |  | 0.0118 | 0.0007 | 2.02 |  | 0.0128 | 0.0015 | 3.61
1.0 | 300 | 500 |  | 0.0109 | 0.0004 | 0.78 |  | 0.0125 | 0.0008 | 2.11 |  | 0.0139 | 0.0013 | 3.76
1.0 | 500 | 100 |  | 0.0081 | 0.0003 | 0.83 |  | 0.0090 | 0.0008 | 1.94 |  | 0.0094 | 0.0016 | 3.69
1.0 | 500 | 300 |  | 0.0104 | 0.0003 | 0.78 |  | 0.0121 | 0.0007 | 2.10 |  | 0.0133 | 0.0012 | 3.79
1.0 | 500 | 500 |  | 0.0105 | 0.0003 | 0.80 |  | 0.0122 | 0.0007 | 2.11 |  | 0.0135 | 0.0012 | 3.80
10.0 | 100 | 100 |  | 0.0091 | 0.0014 | 0.93 |  | 0.0097 | 0.0028 | 2.13 |  | 0.0087 | 0.0058 | 3.45
10.0 | 100 | 300 |  | 0.0259 | 0.0007 | 1.36 |  | 0.0299 | 0.0009 | 1.79 |  | 0.0328 | 0.0012 | 1.83
10.0 | 100 | 500 |  | 0.0297 | 0.0005 | 1.21 |  | 0.0347 | 0.0007 | 1.83 |  | 0.0383 | 0.0010 | 1.84
10.0 | 200 | 100 |  | 0.0204 | 0.0009 | 1.69 |  | 0.0223 | 0.0012 | 2.40 |  | 0.0217 | 0.0019 | 2.54
10.0 | 200 | 300 |  | 0.0314 | 0.0004 | 1.45 |  | 0.0351 | 0.0007 | 1.98 |  | 0.0377 | 0.0010 | 2.40
10.0 | 200 | 500 |  | 0.0326 | 0.0005 | 1.50 |  | 0.0374 | 0.0007 | 2.05 |  | 0.0415 | 0.0009 | 1.95
10.0 | 300 | 100 |  | 0.0221 | 0.0023 | 1.21 |  | 0.0233 | 0.0011 | 2.01 |  | 0.0231 | 0.0019 | 2.71
10.0 | 300 | 300 |  | 0.0311 | 0.0004 | 1.30 |  | 0.0350 | 0.0007 | 1.90 |  | 0.0375 | 0.0010 | 2.27
10.0 | 300 | 500 |  | 0.0328 | 0.0004 | 1.42 |  | 0.0381 | 0.0007 | 2.22 |  | 0.0423 | 0.0009 | 2.21
10.0 | 500 | 100 |  | 0.0199 | 0.0011 | 1.80 |  | 0.0224 | 0.0014 | 2.47 |  | 0.0226 | 0.0019 | 3.04
10.0 | 500 | 300 |  | 0.0323 | 0.0004 | 1.23 |  | 0.0371 | 0.0007 | 2.12 |  | 0.0406 | 0.0009 | 2.19
10.0 | 500 | 500 |  | 0.0323 | 0.0004 | 1.22 |  | 0.0373 | 0.0007 | 2.12 |  | 0.0407 | 0.0009 | 2.12

##  Appendix B Inputs to simsurvey simulations

The specific inputs to simsurvey used in Section 3.4.2 to determine the
detection efficiencies for SN 2015cp-like interaction are listed here.

  * •

Model: SN 2011fe + H $\alpha$ line (Section 3.4.1).

  * •

Sky distribution: RA $\in$ [0∘, 360∘], Dec. $\geq-30^{\circ}$ (The area
covered by ZTF, Bellm et al. 2019).

  * •

Volumetric rate: The SN Ia rate is $2.4\times 10^{-5}$ Mpc-3 yr-1 for $z\leq
0.1$ (Frohmaier et al. 2019). simsurvey uses this to calculate the amount of
SNe to generate at a given redshift interval.

  * •

SN peak time distribution: $58\,195\leq$ modified Julian date (MJD) $\leq
58\,487$ (between 18 March 2018 and 4 January 2019).

  * •

Galactic extinction: dust maps from Schlegel et al. (1998).

  * •

Host galaxy extinction: Cardelli et al. (1989) extinction law, with E(B – V)
drawn from an exponential distribution with exponent $\lambda=0.11$ (Stanishev
et al. 2018), the same way as host extinction was added in the original
simsurvey paper (Feindt et al. 2019b).

  * •

Telescope specifications: ZTF P48 camera, 4$\times$4 grid of CCDs with four
readout channels each, resulting in 64 separate output channels (Dekany et al.
2020).

  * •

Survey plan: ZTF observation logs between $58\,197\leq$ MJD $\leq 59\,211$
(between 20 March 2018 and 28 December 2020), ensuring all simulated SNe are
followed for a minimum of about 2 years after the peak.

##  Appendix C Late-time spectrum of SN 2020alm

After confirming the late-time detections in SN 2020alm to still be ongoing,
we obtained a spectrum using OSIRIS on the GTC on 26 July 2023, 1277 days
after the estimated peak date of the SN. As the observed spectrum was heavily
dominated by the host galaxy, we subtracted a host galaxy spectrum taken by
SDSS in 2003 to remove the host contamination. This was done after re-sampling
the host spectrum to have the same wavelength spacing as the new spectrum.
This left only the spectrum causing the late-time photometry detections, and
is shown in Fig. 14. The subtraction was confirmed to be successful by
checking that the Mg I ${\lambda 5175}$ and Na ID absorption lines were
reduced to the noise level, as these lines are not expected to be due to the
late-time signal but purely from the host galaxy. Some of the host galaxy
emission lines were not completely subtracted during this process, most
noticeably [N II] ${\lambda 6583}$ and [S II] ${\lambda\lambda 6716,6730}$,
but our resolution is inadequate to draw any conclusions from this.

The only explanation for the late-time detections that uses a second transient
at the same location is a TDE. Hammerstein et al. (2023) shows that the
intrinsic spectrum of a TDE is flat in this wavelength range, with sometimes
some narrow emission lines. Therefore, we model a TDE spectrum as a line of
constant flux density, and add Milky Way extinction (using the SFD89 dust maps
in the direction of the object; Schlegel et al. 1998) and variable host
extinction in an attempt to obtain the general shape of the observed spectrum.
We find that $0.6\leq$ $E(B-V)$${}_{\text{host}}\leq 1$ mag is adequate to
reproduce the general spectral shape and suggests that a TDE with
approximately constant colour and moderate host extinction can explain the
observed spectral excess for this event.

Figure 14: Spectrum of the late-time signal in SN 2020alm in its rest frame.
The top panel shows the late-time spectrum obtained on 26 July 2023 using
OSIRIS on the GTC, and the SDSS spectrum obtained in 2003. The bottom panel
shows the late-time excess, obtained by subtracting the SDSS host galaxy
spectrum from the observed late-time spectrum. A smoothed spectrum is shown in
blue. The smoothing was done using a rolling kernel of size 5 to average over
the values. The red lines are a simple TDE model with Milky Way and some
amount of host galaxy extinction applied (the amount is shown in the legend),
in order to get the approximate shape of the observed spectrum. Narrow
emission and absorption lines that were notable in the unsubtracted spectrum
are marked with dashed lines. The grey regions are affected by sky lines, and
should be ignored.

##  Appendix D Binned SMP light curves of the final candidates

Figure 15: Similar to Fig. 11, but showing the binned light curves generated
using scene modelling photometry instead of forced photometry. As no bins
$\geq 5\sigma$ was recovered in SN 2018grt, it is not shown here. Both SN
2019ldf and SN 2020tfc do however still have robust detections in some bands.
The best fitting alternate transients shown in dotted lines are the same as in
Fig. 11. The i-band of SN 2020tfc is not shown as the background was not
subtracted completely, resulting in a significant flux offset.