- Email: [email protected]
Deformable link segment analysis for prosthetic foot-ankle components: Kinematics
Journal Pre-proofs Deformable Link Segment Analysis for Prosthetic Foot-Ankle Components: Kinematics Stacey R. Zhao, J. Timothy Bryant, Qingguo Li PII: DOI: Reference:
S0021-9290(19)30805-X https://doi.org/10.1016/j.jbiomech.2019.109548 BM 109548
To appear in:
Journal of Biomechanics
Accepted Date:
23 November 2019
Please cite this article as: S.R. Zhao, J. Timothy Bryant, Q. Li, Deformable Link Segment Analysis for Prosthetic Foot-Ankle Components: Kinematics, Journal of Biomechanics (2019), doi: https://doi.org/10.1016/j.jbiomech. 2019.109548
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
Deformable Link Segment Analysis for Prosthetic Foot-Ankle Components: Kinematics Stacey R. Zhaoa,b,∗, J. Timothy Bryanta,b, Qingguo Lia a Mechanical b Human
and Materials Engineering, Queen’s University, Kingston, Canada Mobility Research Centre, Queen’s University and Kingston Health Sciences Centre, Kingston, Canada
Abstract Approaches in the literature for estimating prosthetic foot-ankle power typically require calculating the segment deformation velocity. This, in turn, necessitates approximating the segment angular velocity. Methods can be distinguished by the way in which a segment is defined and the assumptions used for estimating the segment angular velocity. However, isolating foot-ankle performance from overall prosthetic system performance is limited by uncertainties in the definition of angular velocity of a deformable segment. A deformable link segment (DLS) analysis is proposed that provides a means for estimating deformation velocity of a deformable segment without first approximating the angular velocity: the deformation velocity and angular velocity are solved simultaneously at each instant during the stance phase of gait. DLS analysis was compared to two approaches in the literature: the distal foot (DF) model and the unified deformable (UD) segment model during over-ground walking for three trans-tibial prosthesis users. DLS and UD segment estimates of deformation velocity were comparable when applied to the UD segment. Furthermore, DLS analysis enables modelling of deformable prosthetic foot-ankle components separately from other prosthetic componentry. The method is proposed as a rigorous approach to estimating angular velocity and deformation velocity of passive prosthetic foot-ankle components for subsequent calculation of deformation power and en∗ Corresponding
author Email address: [email protected] (Stacey R. Zhao)
Preprint submitted to Journal of Biomechanics
November 29, 2019
ergy performance of these devices. Keywords: prosthetic feet, mechanical power, deformation velocity, angular velocity, gait
2
1
Introduction
2
Modern high-function prosthetic foot-ankle components are designed to store
3
energy during loading through deformation of elastic components and to re-
4
turn energy in late stance to aid in forward propulsion (Versluys et al., 2009).
5
These devices have allowed patients to regain mobility and provide indepen-
6
dence to perform activities of daily living (Raschke et al., 2015; Romo, 1999;
7
Nielson et al., 1988; Torburn et al., 1990). However, feet considered to be higher
8
function devices have not been shown to consistently and significantly improve
9
user performance (Alaranta et al., 1991; Gitter et al., 1991; Hafner et al., 2002;
10
Low et al., 2017; Macfarlane et al., 1991). To address this lack of coherence,
11
recent studies have developed methods to evaluate the structural behaviour of
12
specific prosthetic components during gait (Takahashi et al., 2012).
13
Key to understanding energy performance of prosthetic foot components
14
during gait is to recognize the relationship between energy storage and return
15
and structural deformation of the device (American Orthotics and Prosthetics Association,
16
2010). In early power analyses, the foot was modeled as a single rigid body
17
(Robertson & Winter, 1980). More recent models incorporate deformable re-
18
gions (Geil, 1997; Siegel et al., 1996; Takahashi et al., 2012, 2017; Zelik & Honert,
19
2018).
20
Two approaches from the literature, a distal foot model introduced by Siegel et al.
21
(1996) and Geil (1997), and a unified deformable segment model described by
22
Takahashi et al. (2012), are similar methods that first estimate the deformation
23
velocity of a segment, and then compute deformation power. To estimate de-
24
formation velocity, it is necessary to describe the kinematics of the deforming
25
region. In practice, there are conceptual and technical challenges when defin-
26
ing the angular velocity of a non-rigid segment. These have been mitigated
27
by relying on rigid body assumptions to quantify angular velocity (Geil, 1997;
28
Prince et al., 1994; Siegel et al., 1996; Takahashi et al., 2012), which may not
29
be valid for some devices (Zelik & Honert, 2018).
30
It is proposed that the kinematic assumptions underlying analyses of en-
3
31
ergy storing prosthetic components be re-evaluated. The objective of this study
32
is to develop a Deformable Link Segment (DLS) analysis by which an equiv-
33
alent linkage model is used to determine the deformation velocity without an
34
approximation of the angular velocity from a rigid region of the device. The
35
deformation velocity can subsequently be used to estimate deformation power
36
associated with a deformable segment. The current paper describes the theo-
37
retical development of the technique and compares its kinematic estimates to
38
those computed using existing methods in the literature.
39
Methods
40
Equivalent Linkage Models
41
To establish a consistent terminology the distal foot (DF) model (Table 1A)
42
(Geil, 1997; Siegel et al., 1996) and the unified deformable (UD) segment model
43
(Table 1B)(Takahashi et al., 2012) can be described using the concept of an
44
equivalent linkage. An instantaneous link is defined between proximal and distal
45
reference points, and contains a deformable element. The proximal reference
46
point defines the proximal boundary of the deformable region that is captured
47
by a specific model. The distal reference point is the ground contact point,
48
approximated by the centre of pressure (COP). Since the COP progresses along
49
the plantar surface of the foot during stance, there is a new distal reference point,
50
hence a new linkage, defined at each instant of time. The foot is considered to
51
roll without slip such that the distal end point is assumed stationary and the
52
linkage is considered to instantaneously rotate about this point in the sagittal
53
plane. Of note is that the equivalent linkage is modeled with a prismatic joint
54
between the proximal and distal reference points. This introduces a constraint
55
that the magnitude of the deformation velocity, vdef , is equivalent to the rate
56
of change of the length of the linkage.
57
Deformation velocity of the linkage is computed using the reference point
58
positions and velocities, and an estimate of the segment angular velocity. For
59
a rigid body, the segment angular velocity can be calculated from the relative
4
60
velocity of known points on the body by imposing the rigid assumption. How-
61
ever, for a deformable segment, the estimation of segment angular velocity is
62
less clear.
63
In the distal foot model (Table 1A), a deformable segment is defined distal to
64
the foot centre of mass (COM). The foot angular velocity, ω ¯ F , is first calculated
65
using a rigid body analysis and a foot marker set (Geil, 1997). This estimate
66
is then applied to the distal deformable segment to determine the deformation
67
velocity, v¯def = r¯˙DF (Geil, 1997; Siegel et al., 1996). Applying the DF model to
68
modern prosthetic foot-ankle components involves uncertainty in the definition
69
of the COM and angular velocity of a deformable segment. In addition, only
70
structures distal to the COM are modeled, neglecting energy storage and return
71
related to deformable structures proximal to the foot COM.
72
In the UD segment model (Table 1B), a proximal rigid region of the segment
73
is used to estimate the angular velocity of the entire body. When applied to a
74
lower-limb prosthetic system, the angular velocity of the proximal shank, ω ¯ S , is
75
used to represent both the rigid and deformable portions of the segment. The
76
deformation velocity, v¯def = r¯˙UD , is then estimated (Takahashi et al., 2012).
77
While this model has been shown to produce useful estimates of overall pros-
78
thetic system power, it is limited in its ability to isolate the performance of
79
foot-ankle devices from other deformable components (Takahashi, 2012). The
80
aim of the current work is to establish a method to evaluate a deformable region,
81
such as a foot-ankle device, without the requirement for a proximal rigid region.
82
Theoretical Development
83
Deformable Link Segment (DLS) analysis is based on an equivalent linkage
84
model with the deformable region of interest bounded by the proximal and distal
85
reference points. The proximal reference point can be located at any trackable
86
point on a prosthetic system and an estimate of segment angular velocity using
87
a proximal rigid region is not required. As such, a fully deformable prosthetic
88
foot-ankle device, shown in Table 1C, or a specific region of the device can
89
be evaluated separately from other structures. The segment angular velocity 5
90
and deformation velocity are considered unknown and are determined simulta-
91
neously. The deformation velocity may then be used to calculate deformation
92
power of a prosthesis.
93
Consider an equivalent linkage model applied to a fully deformable pros-
94
thetic foot-ankle device (Figure 1). The proximal reference point, P , is located
95
conveniently between the device and the pylon to define a foot-pylon interface
96
(FPI). The instantaneous position, Pj , j = 1, 2, n, is assumed to be known
97
from tracking using motion capture. The distal reference point, D, is the in-
98
stantaneous COP. Note that D does not move along the plantar surface of the
99
foot, rather there is a new contact point, Dj , j = 1, 2, n, at each instant. It
100
is assumed that the instantaneous position of the COP is obtained from force
101
platform measurements gathered during motion analysis. At each instant, a
102
new link is defined with a unique structural stiffness. The foot is thus modelled
103
as a series of equivalent linkages, P Dj , j = 1, 2, n.
104
Only one linkage is considered for each instant of stance. The foot is assumed
105
to roll without slip such that Dj has zero velocity and link P Dj is assumed to
106
rotate about Dj with instantaneous angular velocity ωP D,j . Unlike the DF
107
and UD segment models, the angular velocity of the deformable segment is
108
considered unknown and the value of ωP D,j is computed at each instant based
109
on the motion of points P and D.
110
Motion of the system can first be approximated with a sagittal plane analysis
111
corresponding to the YZ plane of the global ground-fixed reference frame (XYZ),
112
as shown in Figure 1. The angle of link P D is represented by γ, measured with
113
respect to the global Y axis.
114
The position vector r¯P D is determined from the difference between the known
115
positions of P and D. The instantaneous angular orientation, γ, of link P D
116
ˆ components of r¯P D . Thus, r¯P D is is calculated from the Y (Jˆ) and Z (K)
117
represented in the sagittal plane as: ˆ r¯P D = rP D cos γ Jˆ + rP D sin γ K
6
(1)
118
119
The relative velocity of P with respect to D is obtained by differentiating Equation 1 with respect to time: ˆ v¯P D = (r˙P D cos γ − rP D ωP D sin γ)Jˆ + (r˙P D sin γ + rP D ωP D cos γ)K
(2)
120
where the magnitude of the sagittal segment angular velocity ωP D = γ˙ and is
121
considered unknown.1
122
ˆ The velocity of P is measured and known such that v¯P D = vP y Jˆ + vP z K.
123
Since the foot is assumed to roll on the ground without slip, the velocity of D
124
ˆ Grouping like terms gives: is zero, v¯P = v¯P D = vP y Jˆ + vP z K. Jˆ : vP y = r˙P D cos γ − rP D ωP D sin γ ˆ : vP z = r˙P D sin γ + rP D ωP D cos γ K
125
which can be written in matrix form as: vP y cos γ = vP z sin γ
126
(3)
−rP D sin γ r˙P D rP D cos γ ωP D
(4)
Rearranging Equation 4 and computing the matrix inverse gives:
r˙P D ωP D
=
cos γ
v P Y cos γ vP Z
sin γ
− rP1D sin γ
1 rP D
(5)
127
where the magnitude of the deformation velocity, r˙P D , and angular velocity,
128
ωP D , are unknowns.
129
This approach is distinct from those presented in the literature because
130
Equation 5 allows an approximation of deformation velocity from direct mea1 In this derivation, it should be noted that γ is instantaneously known, while γ ˙ = ωP D , representing the instantaneous angular velocity of the link, is unknown. While it is conceivable dγ to calculate the angular velocity of the linkage as dt , this represents the rate of change in angular orientation between sequential linkages, calculated from the change in γ over time, and does not represent the instantaneous angular velocity of the linkage.
7
131
surements of the trajectories of P and D without defining a proximal rigid region
132
or estimating the COM position and angular velocity of the segment.
133
Note that the direction of r¯P D changes throughout stance. In some applica-
134
tions, such as the computation of power, it is desirable to represent deformation
135
velocity in a global ground based reference frame consistent with the measured
136
ground reaction forces. The deformation velocity vector, v¯def = r¯˙P D , is given
137
ˆ components: by Equation 6 in ground-based Y (Jˆ) and Z (K) ˆ v¯def = r¯˙P D = r˙P D cos γ Jˆ + r˙P D sin γ K
138
139
(6)
Experimental Protocol Three volunteer trans-tibial prosthesis users participated in a research ethics
140
board approved gait study at the Human Mobility Research Laboratory (Kingston,
141
Canada) after giving informed consent. Participants were active individuals
142
(K3-K4) with a traumatic amputation etiology and no lower-limb co-morbidities
143
(Table 2). Complete details of the methodology can be found in (Zhao, 2018).
144
Lower limb kinematics, ground reaction forces and centre of pressure (COP)
145
were recorded using 14 optical motion tracking cameras (Oqus 400, Qualisys,
146
Gothenburg, Sweden) and six embedded force plates (Custom BP model, AMTI,
147
Watertown, MA, USA). Subjects performed level over-ground walking trials at
148
a self-selected walking speed with their prescribed prosthetic foot-ankle device
149
until a minimum of five clean force platform strikes were recorded for the pros-
150
thetic side. Details of the prosthetic componentry can be found in the Sup-
151
plementary Material. Footwear was standardized by using the same model of
152
flexible athletic shoe for all subjects.
153
Marker trajectories, ground reaction forces and COP data were recorded us-
154
ing Qualisys Track Manager (Qualisys, Gothenburg, Sweden) and subsequently
155
R R analyzed using purpose-scripted code in MATLAB (R2012a, MathWorks ,
156
Natick, United States). Angular velocity and deformation velocity estimates
157
were calculated for three regions of the prosthetic system. Data correspond-
158
ing to the stance phase were resampled to 101 data points representing 100% 8
159
stance time. Results were then ensemble averaged across steps to provide a
160
mean profile and 95% confidence interval for each subject.
161
Interestingly, while Siegel et al. (1996) reported distal foot velocity for the
162
natural foot, to the authors’ knowledge there is no report of deformation ve-
163
locities of prosthetic foot-ankle devices in the literature. As such, the reported
164
deformation velocity results focus on the Z components. Preliminary analysis
165
indicated that the Z component of deformation velocity was found to be the
166
dominant contributor to the overall power during walking, due in part to the
167
dominance of the vertical ground reaction force. While the X and Y compo-
168
nents of UD and DF velocities were not negligible, a pilot study determined
169
their contribution to the overall power was less than approximately 10%.
170
DLS Analysis
171
Proximal Reference Point Definition. For application to the foot-ankle compo-
172
nent, the proximal reference point (P ) was defined as the proximal boundary
173
of the prosthetic foot-ankle structure. The trajectory was tracked using 1-3
174
reflective markers placed on the foot-pylon interface (FPI); these were the only
175
markers necessary for DLS analysis of the foot-ankle component. Additional
176
details are provided in the Supplementary Material.
177
For comparison to UD segment analysis, DLS analysis was applied to the
178
UD segment, with P located at the UD segment centre of mass and tracked
179
using reflective markers placed on the proximal rigid region of the prosthetic
180
system. For comparison to the DF model, DLS analysis was applied to the DF
181
segment, with P located at the foot centre of mass and tracked using reflective
182
markers placed on the foot.
183
Analysis. DLS deformation velocity was calculated in three stages: 1) The po-
184
sition vector r¯P D was calculated from the difference between the position of P
185
and the COP; the instantaneous length, rP D , was then determined from the
186
magnitude of r¯P D . 2) The instantaneous angle of link P D with respect to the
187
global horizontal axis, γ, was calculated from the Y and Z components of r¯P D .
9
188
3) The magnitude of the DLS deformation velocity, r˙P D , and angular velocity,
189
ωP D , were calculated according to Equation 5. The magnitude of the DLS defor-
190
mation velocity, r˙P D , was expressed in global YZ components using Equation 6.
191
This analysis was applied to: the UD segment providing r˙P D−UD and ωP D−UD ;
192
the DF segment giving r˙P D−DF and ωP D−DF ; and the foot-ankle component
193
(Table 1C) providing r˙P D−F and ωP D−F . Note that r¯P D−F , r¯P D−DF , and
194
r¯P D−UD differ by the location of the proximal reference point P .
195
DF and UD Analysis
196
For comparison to methods in the literature, 11-14 additional markers were
197
required to facilitate estimations for the foot angular velocity, UD segment
198
(shank) angular velocity, and distal foot deformation velocity. Markers placed
199
on the proximal rigid region of the prosthesis were used to estimate the shank
200
angular velocity and track the COM of the UD segment using previously de-
201
scribed procedures (Crimin et al., 2014; Geil et al., 1999; Smith et al., 2014).
202
The angular velocity of the prosthetic foot-ankle device was estimated using
203
markers on the shoe located in positions corresponding to the calcaneus, first
204
and fifth metatarsal heads, and FPI. Angular velocities were computed using
205
the right hand rule convention.
206
Results
207
Comparison Between UD and DLS Methods
208
Deformation velocity estimates for the UD segment computed using DLS
209
analysis (r¯˙P D−UD ) and UD segment analysis (r¯˙UD ) are compared in Figure 2a-
210
c. The mean (line) and 95% confidence interval (shaded) are shown for the Z-
211
component of each velocity measure. A negative velocity indicates the length of
212
the instantaneous link is decreasing, thus representing deformation. A positive
213
velocity indicates the link length is increasing from a deformed state, returning
214
towards the non-deformed state.
215
Z-component deformation velocity estimates for DLS analysis and UD seg-
216
ment analysis are comparable when applied to the same UD segment, with root 10
217
mean square (RMS) differences for the three subjects ranging from 33.5 to 37.8
218
mm/s. For prosthetic systems without deformable componentry proximal to
219
the FPI (e.g. Subject 1, Figure 2a) deformation velocity estimates for the UD
220
segment computed using DLS analysis (r¯˙P D−UD ) are also similar to estimates
221
for the foot-ankle segment (r¯˙P D−F ), with RMS differences of 49.1 mm/s. How-
222
ever, for Subject 3 (Figure 2c) whose prosthetic system included a mechanical
223
pump proximal to the FPI, larger RMS differences of 106.2 mm/s were observed
224
between r¯˙P D−UD and r¯˙P D−F .
225
Angular velocity estimates for the UD segment computed using DLS anal-
226
ysis and UD segment analysis are compared in Figure 2d-f. For UD segment
227
analysis, angular velocity, (ωS ), was estimated from markers on the proximal
228
rigid region of the prosthetic system using a rigid body model of the shank.
229
This is compared to the angular velocity of the UD segment computed using
230
Equation 5 (ωP D−UD ). Absolute differences between ωS and ωP D−UD are up to
231
1.3 rad/s in early stance, 0.5 rad/s in mid stance and 2.7 rad/s in late stance for
232
Subject 1, with similar trends for Subject 2 and 3. For prosthetic systems with-
233
out deformable componentry proximal to the FPI (e.g. Subject 1, Figure 2d)
234
angular velocity estimates for the UD segment computed using DLS analysis
235
(ωP D−UD ) are comparable to estimates for the foot-ankle segment (ωP D−F ),
236
particularly from 10-85% stance, with absolute differences less than 0.3 rad/s.
237
However, for Subject 3 (Figure 2f) whose prosthetic system included a mechani-
238
cal pump proximal to the FPI, larger differences were observed between angular
239
velocity estimates ωP D−UD and ωP D−F , up to 2.3 rad/s in late stance.
240
Differences between deformation velocities r¯˙P D−UD and r¯˙UD are likely due to
241
the angular velocity approximations and specifically, rotation of the foot-ankle
242
component relative to the shank component. Larger deformation velocities were
243
consistent with differences in angular velocity; ωP D−UD was larger than ωS in
244
magnitude during early and late stance for all subjects.
11
245
Comparison Between DF and DLS methods
246
Deformation velocity estimates for the DF segment computed using DLS
247
analysis (r¯˙P D−DF ) and the DF model (r¯˙DF ) are compared in Figure 3a-c. The
248
mean (line) and 95% confidence interval (shaded) are shown for the Z-component
249
of each velocity measure. Z-component deformation velocity estimates for DLS
250
analysis and the DF model are distinct, particularly in late stance. Subject 1
251
displayed a RMS difference of 77.3 mm/s overall, with a difference of 141.0 mm/s
252
in late stance; these trends were similar for Subjects 2 and 3. The DF model
253
estimates trend towards positive values in late stance, representing return to the
254
undeformed shape. Deformation velocity estimates for DLS analysis applied to
255
the DF segment are closer to zero and are at times negative in late stance,
256
this may indicate that the distal foot structures further deform or maintain a
257
deformed state in late stance.
258
Angular velocity estimates for the DF segment computed using DLS analysis
259
and the DF model are compared in Figure 3d-f. For the DF model, angular
260
velocity, (ωF ), was estimated using markers on the foot and a rigid body model.
261
This is compared to the angular velocity of the DF segment computed using
262
Equation 5 (ωP D−DF ). The angular velocity of the foot-ankle segment (ωP D−F )
263
is also shown for comparison. For all subjects, DLS angular velocity estimates
264
for the distal foot are distinct from DLS angular velocity estimates for the
265
foot-ankle segment, with differences up to 2.1 rad/s for Subject 1. From 10-
266
85% stance, DLS angular velocity estimates for the distal foot (ωP D−DF ) are
267
comparable to the estimates derived from a rigid body model of the foot (ωF ),
268
with absolute differences less than 0.5 rad/s for all subjects. In the rigid body
269
model, the main axis of the foot was defined between markers on the heel and
270
metatarsal heads (Robertson et al., 2013). While this estimate may represent
271
the motion of the distal region of the foot, these results suggest that ωP D−F may
272
provide a more representative estimate for the angular velocity of the overall
273
foot-ankle component, including the effects of the deformable ankle region.
12
274
Angular Velocity Approximations
275
DLS angular velocity of the foot-ankle component (ωP D−F ) is plotted in
276
Figure 4 in comparison to values for the sagittal foot (ωF ) and shank (ωS )
277
computed using rigid body models of the foot and shank, respectively. If the
278
prosthetic foot-ankle device were rigid, the foot and shank angular velocities
279
would be equivalent. In a deformable system, these measures are expected to
280
be distinct, as observed in this study. Absolute differences between ωF and ωS
281
are up to 1.9 rad/s in early stance, 1.5 rad/s in mid stance and 1.7 rad/s in late
282
stance for Subject 1. These trends were similar for Subjects 2 and 3.
283
For all subjects, DLS angular velocity was similar to foot angular velocity in
284
early and late stance, but distinct from the shank throughout all regions. The
285
“foot flat” region for Subject 1 between 25-50% stance shows a foot angular
286
velocity that approaches zero. In contrast, angular velocity of the shank ranged
287
from -1.1 to -1.7 rad/s and DLS angular velocities ranged from -0.6 to -0.9
288
rad/s. DLS analysis appears to capture an appropriate value that represents
289
both motion of the foot and deformation of the proximal foot-ankle region during
290
foot flat that is distinct from motion of the shank.
291
Discussion
292
The DF model and UD segment analysis require a proximal rigid region or
293
rigid body assumption for approximation of the segment angular velocity. In
294
contrast, DLS analysis can be applied to various regions of the prosthetic system,
295
without requiring an angular velocity approximation for the desired segment.
296
Of note, methods in the literature are unable to isolate the foot-ankle compo-
297
nent from other deformable structures due to the uncertainty regarding how
298
to estimate the angular velocity of a fully deformable segment. Isolation of the
299
foot-ankle component from other deformable structures of the prosthetic system
300
(e.g. shock-absorbing/telescoping pylon or mechanical pump) was demonstrated
301
using DLS analysis with the proximal reference point at the FPI. Applying DLS
302
analysis to various regions of the foot-ankle component would provide additional
13
303
insights regarding the contribution of individual design features to the overall
304
function of the device; however, this was outside the scope of the current study.
305
The largest differences between methods are typically observed in the 0-20%
306
and 80-100% stance regions of deformation velocity estimates. These early and
307
late stance regions of deformation velocity correspond to collision and push-off
308
regions of power.
309
Application to Other Deformable Segments
310
DLS analysis can be extended to evaluate the deformation velocity of other
311
componentry. It is noted that Equation 5 describes a special case when the
312
distal reference point velocity is zero. This reflects an application to a foot
313
rolling without slip or any deformable region for which the distal reference point
314
velocity is instantaneously defined by the COP. For a more general case, the
315
ˆ is added to the derivation. distal reference point velocity, v¯D = vDY Jˆ + vDZ K,
316
The generic form of Equation 5 thus becomes:
r˙P D
cos γ
= 1 ωP D − rP D sin γ
v − v DY PY cos γ vP Z − vDZ
sin γ 1 rP D
(7)
317
The general form of the DLS analysis can be applied to an intermediate
318
segment, such as a sub-region of the foot-ankle device, a shock-absorbing pylon,
319
or other deformable segment, for which the positions of the proximal and distal
320
ends of the segment can be tracked. If the reaction force at one of the reference
321
points is also known, the deformation velocity can subsequently be used to
322
calculate deformation power.
323
Deformation and Geometric Effects
324
Consider an equivalent linkage model that defines a position vector, r¯P D ,
325
between the COP and a proximal reference point. The time derivative of this
326
d vector ( dt r¯P D ) captures the rate of change in length throughout stance phase
327
due to both foot geometry and deformation. To estimate the instantaneous
328
deformation velocity (and subsequently the rate of change of elastic potential 14
329
energy), care must be taken to ensure that deformation effects are isolated
330
from the geometric effects associated with rigid body motion. While the final
331
deformed state of a foot-ankle component can be determined by the rollover
332
shape (Hansen et al., 2000; Major et al., 2012), estimating deformation directly
333
requires a reference to the undeformed shape (the change in r¯P D due to device
334
geometry). Isolating deformation velocity (r¯˙P D ) from the overall rate of change
335
d in link length ( dt r¯P D ) provides a means of characterizing deformation effects
336
separately from geometric effects, without requiring the undeformed shape.
337
The influence of foot geometry is shown in Figure 5 using the principle of
338
superposition. First consider a hypothetical rigid foot component for which
339
the foot sole is an non-circular arc. The foot rotates from its initial position
340
(Figure 5a) to a new position, which defines an intermediate state (Figure 5b).
341
The length of the link at the intermediate state, r¯P D (j 0 ) is greater than at
342
the initial position, r¯P D (j − 1), due to the shape of the foot and its change
343
in orientation. The new link r¯P D (j 0 ) is undeformed in the intermediate state.
344
Deformation is observed by the superposition of the contact force, F , on the link
345
at the intermediate state. During deformation, the length of the link reduces
346
from the intermediate state in Figure 5c, r¯P D (j 0 ), to the final state in Figure 5d,
347
r¯P D (j).
348
The effect of geometry due to a change in angular orientation of the foot
349
and the effect of deformation due to the applied force occur simultaneously
350
throughout stance, and are thus superimposed. DLS analysis isolates the rate of
351
change in link length due to deformation (r¯˙P D ) from the overall rate of change in
352
d link length ( dt r¯P D ) by considering the segment as a series of sequential linkages
353
and solving for r¯˙P D and the angular velocity of each linkage instantaneously.
354
An incremental change in angular orientation is associated with progression of
355
the contact point, and the definition of a new linkage. This results in no change
356
in elastic potential energy of the system (Figures 5a and 5b). However, when
357
deformation takes place in the presence of a force, there is a change in the elastic
358
potential energy state (Figures 5c and 5d). As the system progresses, this elastic
359
potential energy of link at j is then transferred to the subsequent link, j + 1. 15
360
Limitations and Recommendations
361
Future work includes extension of the technique to estimate deformation
362
power and stance work. However, caution is required whenever applying this
363
method beyond the analyses presented.
364
Prostheses with Articulations. For interpretation of r¯˙P D as deformation veloc-
365
ity, care must be taken to ensure that deformation effects are isolated from the
366
geometric effects. For passive non-articulating prostheses, using DLS analysis to
367
model the segment as a series of equivalent linkages where the angular velocity
368
represents instantaneous linkage rotation facilitates representation of the defor-
369
mation velocity as the instantaneous change in link length due to deformation
370
excluding geometric effects.
371
Passive articulating prostheses may include joints with rotational stiffness.
372
In this case, there are two deformation effects influencing energy storage and
373
return: structural deformation of sub-segments and angular deflection through
374
joint rotation. There are also two geometric effects influencing the link length:
375
kinematic motion of the overall segment and rotation of the sub-segments about
376
the joint. For articulating segments it is recommended to apply DLS analysis
377
separately to each sub-segment. Each sub-segment would be an independent
378
DLS and the common point would act as a joint. As such, the joint would
379
transmit power between segments as in a conventional link segment analysis.
380
While joint linear and relative angular velocities and joint force could be deter-
381
mined using the application of current methods, determining the joint moment
382
could present challenges. In particular, elastic elements at the joint that resist
383
angulation would affect the net joint moment due to the relative angle between
384
segments. This, in turn, complicates the equilibrium analysis. However, the
385
prospect of determining the effect of angular deformation affords an interesting
386
avenue of future study that would provide further insight into energy manage-
387
ment in these systems.
388
Three Dimensional Analysis. This study assumes that prosthetic foot-ankle de-
389
formation during walking is dominated by motion in the sagittal plane. DLS 16
390
analysis described in this work is limited to a two-dimensional analysis, in which
391
the sagittal plane of walking is aligned with the YZ plane in the global reference
392
frame. However, power estimates are the product of force and velocity terms
393
that may result in significant off-plane contributions, particularly during more
394
complex activities involving turning and impact. Future work will consider ex-
395
tension of DLS analysis to provide three-dimensional estimates of deformation
396
velocity.
397
Uncertainty in COP Measurement. Low vertical ground reaction force values
398
in early and late stance are known to be associated with inaccuracies in COP
399
recordings (Robertson et al., 2013). In addition, the orientation of the foot dur-
400
ing early and late stance may be considered to be relatively unstable due to
401
a smaller base of support, leading to additional deviations in the COP trajec-
402
tory. As such, DLS deformation velocity estimates for segments where the link
403
length is relatively small (e.g. the DF segment) may be more sensitive to COP
404
inaccuracies.
405
Conflict of Interest Statement
406
The authors have no financial or personal relationships with other people or
407
organizations that could inappropriately influence this work.
408
Acknowledgements
409
Financial support for this study was received from the Natural Sciences and
410
Engineering Research Council of Canada, the Ontario Graduate Scholarship
411
Program, and the Donald and Joan McGeachy Chair in Biomedical Engineering.
412
The authors wish to thank Ms. Christina Ippolito, MASc, EIT for assistance
413
with manuscript preparation, Dr. Ron Anderson, Ph.D, P.Eng for critically
414
reviewing the study and the final manuscript, and Mr. Martin Robertson, BSc,
415
CP(c) for recruiting gait analysis participants and providing clinical support.
17
416
Alaranta, H., Kinnunen, A., Karkkainen, M., Pohjolainen, T., & Heliovaara, M.
417
(1991). Practical benefits of Flex-Foot in below-knee amputees. Journal of
418
Prosthetics and Orthotics, 3 , 179–181.
419
420
American Orthotics and Prosthetics Association (2010). AOPA Prosthetic Foot Project Report. AOPA, .
421
Crimin, A., Mcgarry, A., Harris, E. J., & Solomonidis, S. E. (2014). The effect
422
that energy storage and return feet have on the propulsion of the body: A
423
pilot study. Proceedings of the Institution of Mechanical Engineers. Part H,
424
Jounral of Engineering in Medicine, 228 , 908–915.
425
Geil, M. D. (1997). Effectiveness evaluation and functional theoretical modeling
426
of dynamic elastic response lower limb prosthetics. Ph.D. thesis The Ohio
427
State University.
428
Geil, M. D., Parnianpour, M., & Berme, N. (1999). Significance of nonsagittal
429
power terms in analysis of dynamic elastic response prosthetic feet. Journal
430
of Biomechanical Engineering, 121 , 521–524.
431
Gitter, A., Czerniecki, J. M., & DeGroot, D. M. (1991). Biomechanical analysis
432
of the influence of prosthetic feet on below-knee amputee walking. American
433
Journal of Physical Medicine Rehabilitation, 70 , 142–148.
434
Hafner, B. J., Sanders, J. E., Czerniecki, J., & Fergason, J. (2002). Energy
435
storage and return prostheses: does patient perception correlate with biome-
436
chanical analysis? Clinical Biomechanics, 17 , 325–344.
437
Hansen, A. H., Childress, D. S., & Knox, E. H. (2000). Prosthetic foot roll-over
438
shapes with implications for alignment of trans-tibial prostheses. Prosthetics
439
and Orthotics International , 24 , 205–215.
440
Low, S. E., Tsimiklis, A., Zhao, S. R., Davies, T. C., & Bryant, J. T. (2017). Ef-
441
fect of Passive Dynamic Prosthetic Feet on the Improvement of Gait in Adult
442
Unilateral Transtibial Amputees as Compared to SACH Feet: A Systematic
18
443
Review. In Proceedings of the 16th International Society for Prosthetics and
444
Orthotics World Congress. Capetown, South Africa.
445
Macfarlane, P., Nielsen, D., Shurr, D., & Meier, K. (1991). Perception of walking
446
difficulty by below-knee amputees using a conventional foot versus the flex-
447
foot. Journal of Prosthetics and Orthotics, 3 , 114–119.
448
Major, M. J., Kenney, L. P., Twiste, M., & Howard, D. (2012). Stance phase
449
mechanical characterization of transtibial prostheses distal to the socket: A
450
review. Journal of Rehabilitation Research and Development , 49 , 815–830.
451
Nielson, D. H., Shurr, D. G., Golden, J. C., & Meier, K. (1988). Comparison
452
of energy cost and gait efficiency during ambulation in below-knee amputees
453
using different prosthetic feet - A preliminary report. Journal of Prosthetics
454
and Orthotics, 1 , 24–31.
455
Prince, F., Winter, D. A., Sjonnesen, G., & Wheeldon, R. K. (1994). New
456
technique for the calculation of the energy stored, dissipated, and recovered
457
in different ankle-foot prostheses. IEEE Transactions on Rehabilitation En-
458
gineering, 2 , 247–255.
459
Raschke, S. U., Orendurff, M. S., Mattie, J. L., Kenyon, D. E., Jones, O. Y.,
460
Moe, D., Winder, L., Wong, A. S., Moreno-Hern´ andez, A., Highsmith, M. J.,
461
Sanderson, D. J., & Kobayashi, T. (2015). Biomechanical characteristics,
462
patient preference and activity level with different prosthetic feet: A random-
463
ized double blind trial with laboratory and community testing. Journal of
464
Biomechanics, 48 , 146–152.
465
466
Robertson, D. G., Caldwell, G. E., Hamill, J., Kamen, G., & Whittlesey, S. N. (2013). Research Methods in Biomechanics. (2nd ed.). Human Kinetics.
467
Robertson, D. G., & Winter, D. A. (1980). Mechanical energy generation, ab-
468
sorption and transfer amongst segments during walking. Journal of Biome-
469
chanics, 13 , 845–854.
19
470
471
Romo, H. D. (1999). Specialized prostheses for activities. An update. Clinical Orthopaedics, 361 , 63–70.
472
Siegel, K. L., Kepple, T. M., & Caldwell, G. E. (1996). Improved agreement of
473
foot segmental power and rate of energy change during gait: inclusion of distal
474
power terms and use of three-dimensional models. Journal of Biomechanics,
475
29 , 823–827.
476
Smith, J. D., Ferris, A. E., Heise, G. D., Hinrichs, R. N., & Martin, P. E. (2014).
477
Oscillation and reaction board techniques for estimating inertial properties of
478
a below-knee prosthesis. Journal of visualized experiments, 87 , 1–16.
479
480
Takahashi, K. Z. (2012). Total power profiles of anatomical and prosthetic belowknee structures. Ph.D. thesis University of Delaware.
481
Takahashi, K. Z., Kepple, T. M., & Stanhope, S. J. (2012). A unified deformable
482
(UD) segment model for quantifying total power of anatomical and prosthetic
483
below-knee structures during stance in gait. Journal of Biomechanics, 45 ,
484
2662–2667.
485
Takahashi, K. Z., Worster, K., & Bruening, D. A. (2017). Energy neutral: the
486
human foot and ankle subsections combine to produce near zero net mechan-
487
ical work during walking. Scientific Reports, 7 .
488
Torburn, L., Perry, J., Ayyappa, E., & Shanfield, S. L. (1990). Below-knee
489
amputee gait with dynamic elastic response prosthetic feet: A pilot study.
490
Journal of Rehabilitation Research and Development , 27 , 369–384.
491
Versluys, R., Beyl, P., van Damme, M., Desomer, A., van Ham, R., & Lefeber, D.
492
(2009). Prosthetic feet: state-of-the-art review and the importance of mimick-
493
ing human ankle-foot biomechanics. Disability and Rehabilitation: Assistive
494
Technology, 4 , 65–75.
495
Zelik, K. E., & Honert, E. C. (2018). Ankle and foot power in gait analy-
496
sis: Implications for science, technology and clinical assessment. Journal of
497
Biomechanics, 75 . 20
498
499
Zhao, S. R. (2018). Deformable link segment model for analysis of prosthetic foot-ankle components. Ph.D. thesis Queen’s University.
21
500
(Siegel et al., 1996)
Deformable Region Modeled. Footankle structures distal to foot COM. Angular Velocity. Approximated using a foot rigid body model. Limitations. Application to prosthetic foot-ankle device requires defining kinematics of a deformable segment (¯ vCOM,F and ω ¯ F ). By definition, the model captures deformation of the distal region only.
v¯def = r¯˙DF = v¯COM,F − ω ¯ F × r¯DF
Deformable Region Modeled. Structures distal to UD COM.
(Takahashi et al., 2012)
(B) UD Segment Model
(A) Distal Foot Model
Table 1: Summary of approaches for estimating deformation velocity associated with deformable foot-ankle devices. An equivalent linkage model is used to describe each approach: (A) The distal foot model applied to a prosthetic foot-ankle device provides estimates of velocity, v¯def = r¯˙DF , associated with deformable structures of the distal region of the foot. (B) The unified deformable (UD) segment model provides estimates of velocity, v¯def = r¯˙U D , associated with deformable structures below the knee. (C) Deformable link segment analysis applied to a prosthetic foot-ankle device provides estimates of deformation velocity, v¯def = r¯˙P D , associated with the foot-ankle device.
Angular Velocity. Approximated using a shank rigid body model. Limitations. It is not possible to isolate the foot-ankle device from other deformable prosthetic componentry, such as a deformable pylon, mechanical pump or connector located distal to the UD COM (Takahashi, 2012). For foot-ankle devices with functional ankle regions, the foot angular velocity may differ from that of the shank.
Segment Analysis
(C) Deformable Link
v¯def = r¯˙U D = v¯COM,U D − ω ¯ S × r¯U D
Deformable Region Modeled. Footankle structures distal to the foot-pylon interface (FPI). Ã
r˙P D ωP D
!
"
cos γ = − 1 sin γ r PD
sin γ 1 cos γ r PD
#Ã
vP Y vP Z
ˆ v¯def = r¯˙P D = r˙P D cos γ Jˆ + r˙P D sin γ K
!
Angular Velocity. Determined simultaneously along with deformation velocity. Limitations. Application to 3D analysis in development.
Notation: r¯˙ , Deformation velocity; r¯, Position vector; v¯, Velocity of a reference point; ω ¯ , Angular velocity; r, Instantaneous length of r¯; ω, Sagittal angular velocity; γ, Instantaneous angular orientation of link P D.
22
Table 2: Subject demographics for pilot gait study
Subject
Age (yrs)
Mass (kg)
K-Level
Prescribed Device
Subject 1 Subject 2 Subject 3
61 31 36
53 64 67
K3 K4 K4
TruLife Seattle Catalyst9 ¨ Ossur Re-Flex ShockTM R ¨ Ossur Vari-Flex XC
23
501
Figure Captions Figure 1: Foot-ankle device modeled as prismatic link P D. Point P is a point fixed on the device located at the foot-pylon interface. Point D is the instantaneous COP position representing the contact point between the foot and ground. Instantaneously, link P D can be considered to rotate about D with angular velocity ωP D . (Note that P is not a pin joint and link P D does not rotate with respect to the foot, rather the foot and link together rotate instantaneously about D). The angular orientation of link P D is represented by γ with respect to the global (YZ) reference frame.
Figure 2: Deformation velocity and angular velocity estimates for the UD segment over the stance phase of gait. Mean (line) and 95% confidence interval (shaded region) are shown for: Subject 1 (n = 9), Subject 2 (n = 7), and Subject 3 (n = 11). (a)-(c): The Z component of the UD segment deformation velocity computed using DLS analysis, r¯˙P D−U D , is compared to that computed using UD segment analysis, r¯˙U D . Results for DLS analysis applied to the foot-ankle segment, r¯˙P D−F , are indicated by a dashed line for reference. (d)-(f): Angular velocity of the UD segment estimated from DLS analysis, ωP D−U D , compared to angular velocity approximated from the proximal rigid region of prosthetic system and a rigid body model of the shank, ωS . Results for DLS analysis applied to the foot-ankle segment, ωP D−F , are indicated by a dashed line for reference.
Figure 3: Deformation velocity and angular velocity estimates for the DF segment over the stance phase of gait. Mean (line) and 95% confidence interval (shaded region) are shown for: Subject 1 (n = 9), Subject 2 (n = 7), and Subject 3 (n = 11). (a)-(c): The Z component of the DF deformation velocity computed using DLS analysis, r¯˙P D−DF , is compared to that computed using the DF model, r¯˙DF . Results for DLS analysis applied to the foot-ankle segment, r¯˙P D−F , are indicated by a dashed line for reference. (d)-(f): Angular velocity of the DF segment estimated from DLS analysis, ωP D−DF , compared to angular velocity approximated using a rigid body model of the foot, ωF . Results for DLS analysis applied to the foot-ankle segment, ωP D−F , are indicated by a dashed line for reference.
24
Figure 4: DLS sagittal angular velocity, ωP D−F , of the foot-ankle device (black) compared to angular velocities approximated using rigid body models of the foot, ωF , and shank, ωS for each subject. Mean (line) and 95% confidence interval (shaded) across trials are shown for: (a) Subject 1 (n = 9), (b) Subject 2 (n = 7), and (c) Subject 3 (n = 11).
Figure 5: Superposition of geometric and deformation effects in the DLS analysis. First, a rigid foot rotates from its initial position (a) to a new position (b), which defines an intermediate state. The length of the link at the intermediate state, r¯P D (j 0 ), is greater than at the initial position, r¯P D (j − 1), due to the shape of the foot and its change in orientation. Next, the deformation of the segment is isolated by the instantaneous change in length of the link due to a contact force. During deformation, the length of the link reduces from the intermediate state (c), r¯P D (j 0 ), to the final state (d), r¯P D (j).
25
502
Supplementary Material
503
Tracking the Foot-Pylon Interface
504
For application to the foot-ankle component, the proximal reference point
505
(P ) was defined as the proximal boundary of the prosthetic foot-ankle structure.
506
The trajectory was tracked using 1-3 reflective markers placed on the foot-
507
pylon interface (FPI); these were the only markers necessary for DLS analysis
508
of the foot-ankle component. For Subjects 1 and 3, the FPI was defined as the
509
centre of the proximal rigid adapter on the prosthetic foot where it connected
510
to the pylon, as shown in Figure 6. For Subject 2, the FPI was defined at the
511
connection between the composite keel and shock absorbing pylon to isolate the
512
response of the keel. In this case, the FPI position was tracked using a marker
513
cluster on a nearby rigid region of the prosthetic structure.
514
Rollover Shape
515
Equivalent linkage models of the foot-ankle component are related to the
516
concept of a rollover shape. Major et al. (2012) defined the rollover shape as “a
517
spatial mapping of the center of pressure location along the plantar surface of the
518
foot relative to a shank-based coordinate frame”. The rollover shape provides
519
an approximation of the overall deformed state of a foot throughout stance and
520
represents the equivalent rigid body shape of the foot for a specific loading
521
scenario (Hansen et al., 2000). The rollover shape of a foot can be determined
522
by transforming link r¯P D to a local reference frame fixed to the FPI. This metric
523
is influenced by geometry of the device design and deformation due to applied
524
load (Major et al., 2012). Thus, both geometric factors and deformation affect
525
the change in length of link r¯P D .
526
Regions of the Prosthetic System
527
DLS deformation velocity of the prosthetic foot-ankle component (r¯˙P D−F )
528
is plotted in comparison to the distal foot model (r¯˙DF ) and UD segment model
529
(r¯˙UD ) in Figure 7 for each subject. A negative velocity indicates the length of
26
530
the instantaneous link is decreasing, thus representing deformation. A positive
531
velocity indicates the link length is increasing from a deformed state, returning
532
towards the non-deformed state.
533
As shown in Figure 7, DLS deformation velocity, r¯˙P D−F , is greater in mag-
534
nitude compared to the DF model, r¯˙DF , and lower in magnitude compared to
535
the UD model, r¯˙UD . For all subjects, values for r¯˙DF are near zero from 15-75%
536
stance. The early and late stance regions of deformation velocity estimates cor-
537
respond to collision and push-off regions of power. The largest differences in
538
magnitude between methods are typically observed in the 0-20% and 80-100%
539
stance regions.
540
Deformation velocity estimates based on the three approaches are expected
541
to differ when the approaches were used to model different regions of the pros-
542
thetic device. For example, the DF model includes only structures distal to the
543
foot COM. The low magnitude of r¯˙DF compared to the other approaches reflects
544
the smaller deformable region of the device that is captured by the model. The
545
UD segment represents the largest region, including the foot-ankle device and
546
other componentry such as a mechanical pump (Subject 3).
27
547
Supplementary Material - Figure Captions Figure 6: Prescribed prosthetic foot-ankle components: (a) Subject 1’s prescribed foot-ankle device; (b) Subject 2’s prescribed device with integrated shock absorbing pylon; and (c) Subject 3’s prescribed foot-ankle device. Note that Subject 3’s prosthetic system included a mechanical pump proximal to the foot-ankle component. For all conditions, the location of the FPI is shown for reference. Foot shells and standardized athletic shoes were also worn. ¨ (a) Photography courtesy of Trulife; (b) and (c) Photography courtesy of Ossur.
Figure 7: Deformation velocity for each subject over the stance phase of gait. The Z components of the distal foot deformation velocity, r¯˙DF , the unified deformable segment deformation velocity, r¯˙U D , and DLS deformation velocity, r¯˙P D−F , are compared. Mean (line) and 95% confidence interval (shaded region) are shown for: (a) Subject 1 (n = 9), (b) Subject 2 (n = 7), and (c) Subject 3 (n = 11). Distal foot deformation velocity, r¯˙DF , typically exhibited lower values than the DLS model whereas the the UD segment deformation velocity, r¯˙U D , typically displayed larger values than the DLS model.
28
S0021-9290(19)30805-X https://doi.org/10.1016/j.jbiomech.2019.109548 BM 109548
To appear in:
Journal of Biomechanics
Accepted Date:
23 November 2019
Please cite this article as: S.R. Zhao, J. Timothy Bryant, Q. Li, Deformable Link Segment Analysis for Prosthetic Foot-Ankle Components: Kinematics, Journal of Biomechanics (2019), doi: https://doi.org/10.1016/j.jbiomech. 2019.109548
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
Deformable Link Segment Analysis for Prosthetic Foot-Ankle Components: Kinematics Stacey R. Zhaoa,b,∗, J. Timothy Bryanta,b, Qingguo Lia a Mechanical b Human
and Materials Engineering, Queen’s University, Kingston, Canada Mobility Research Centre, Queen’s University and Kingston Health Sciences Centre, Kingston, Canada
Abstract Approaches in the literature for estimating prosthetic foot-ankle power typically require calculating the segment deformation velocity. This, in turn, necessitates approximating the segment angular velocity. Methods can be distinguished by the way in which a segment is defined and the assumptions used for estimating the segment angular velocity. However, isolating foot-ankle performance from overall prosthetic system performance is limited by uncertainties in the definition of angular velocity of a deformable segment. A deformable link segment (DLS) analysis is proposed that provides a means for estimating deformation velocity of a deformable segment without first approximating the angular velocity: the deformation velocity and angular velocity are solved simultaneously at each instant during the stance phase of gait. DLS analysis was compared to two approaches in the literature: the distal foot (DF) model and the unified deformable (UD) segment model during over-ground walking for three trans-tibial prosthesis users. DLS and UD segment estimates of deformation velocity were comparable when applied to the UD segment. Furthermore, DLS analysis enables modelling of deformable prosthetic foot-ankle components separately from other prosthetic componentry. The method is proposed as a rigorous approach to estimating angular velocity and deformation velocity of passive prosthetic foot-ankle components for subsequent calculation of deformation power and en∗ Corresponding
author Email address: [email protected] (Stacey R. Zhao)
Preprint submitted to Journal of Biomechanics
November 29, 2019
ergy performance of these devices. Keywords: prosthetic feet, mechanical power, deformation velocity, angular velocity, gait
2
1
Introduction
2
Modern high-function prosthetic foot-ankle components are designed to store
3
energy during loading through deformation of elastic components and to re-
4
turn energy in late stance to aid in forward propulsion (Versluys et al., 2009).
5
These devices have allowed patients to regain mobility and provide indepen-
6
dence to perform activities of daily living (Raschke et al., 2015; Romo, 1999;
7
Nielson et al., 1988; Torburn et al., 1990). However, feet considered to be higher
8
function devices have not been shown to consistently and significantly improve
9
user performance (Alaranta et al., 1991; Gitter et al., 1991; Hafner et al., 2002;
10
Low et al., 2017; Macfarlane et al., 1991). To address this lack of coherence,
11
recent studies have developed methods to evaluate the structural behaviour of
12
specific prosthetic components during gait (Takahashi et al., 2012).
13
Key to understanding energy performance of prosthetic foot components
14
during gait is to recognize the relationship between energy storage and return
15
and structural deformation of the device (American Orthotics and Prosthetics Association,
16
2010). In early power analyses, the foot was modeled as a single rigid body
17
(Robertson & Winter, 1980). More recent models incorporate deformable re-
18
gions (Geil, 1997; Siegel et al., 1996; Takahashi et al., 2012, 2017; Zelik & Honert,
19
2018).
20
Two approaches from the literature, a distal foot model introduced by Siegel et al.
21
(1996) and Geil (1997), and a unified deformable segment model described by
22
Takahashi et al. (2012), are similar methods that first estimate the deformation
23
velocity of a segment, and then compute deformation power. To estimate de-
24
formation velocity, it is necessary to describe the kinematics of the deforming
25
region. In practice, there are conceptual and technical challenges when defin-
26
ing the angular velocity of a non-rigid segment. These have been mitigated
27
by relying on rigid body assumptions to quantify angular velocity (Geil, 1997;
28
Prince et al., 1994; Siegel et al., 1996; Takahashi et al., 2012), which may not
29
be valid for some devices (Zelik & Honert, 2018).
30
It is proposed that the kinematic assumptions underlying analyses of en-
3
31
ergy storing prosthetic components be re-evaluated. The objective of this study
32
is to develop a Deformable Link Segment (DLS) analysis by which an equiv-
33
alent linkage model is used to determine the deformation velocity without an
34
approximation of the angular velocity from a rigid region of the device. The
35
deformation velocity can subsequently be used to estimate deformation power
36
associated with a deformable segment. The current paper describes the theo-
37
retical development of the technique and compares its kinematic estimates to
38
those computed using existing methods in the literature.
39
Methods
40
Equivalent Linkage Models
41
To establish a consistent terminology the distal foot (DF) model (Table 1A)
42
(Geil, 1997; Siegel et al., 1996) and the unified deformable (UD) segment model
43
(Table 1B)(Takahashi et al., 2012) can be described using the concept of an
44
equivalent linkage. An instantaneous link is defined between proximal and distal
45
reference points, and contains a deformable element. The proximal reference
46
point defines the proximal boundary of the deformable region that is captured
47
by a specific model. The distal reference point is the ground contact point,
48
approximated by the centre of pressure (COP). Since the COP progresses along
49
the plantar surface of the foot during stance, there is a new distal reference point,
50
hence a new linkage, defined at each instant of time. The foot is considered to
51
roll without slip such that the distal end point is assumed stationary and the
52
linkage is considered to instantaneously rotate about this point in the sagittal
53
plane. Of note is that the equivalent linkage is modeled with a prismatic joint
54
between the proximal and distal reference points. This introduces a constraint
55
that the magnitude of the deformation velocity, vdef , is equivalent to the rate
56
of change of the length of the linkage.
57
Deformation velocity of the linkage is computed using the reference point
58
positions and velocities, and an estimate of the segment angular velocity. For
59
a rigid body, the segment angular velocity can be calculated from the relative
4
60
velocity of known points on the body by imposing the rigid assumption. How-
61
ever, for a deformable segment, the estimation of segment angular velocity is
62
less clear.
63
In the distal foot model (Table 1A), a deformable segment is defined distal to
64
the foot centre of mass (COM). The foot angular velocity, ω ¯ F , is first calculated
65
using a rigid body analysis and a foot marker set (Geil, 1997). This estimate
66
is then applied to the distal deformable segment to determine the deformation
67
velocity, v¯def = r¯˙DF (Geil, 1997; Siegel et al., 1996). Applying the DF model to
68
modern prosthetic foot-ankle components involves uncertainty in the definition
69
of the COM and angular velocity of a deformable segment. In addition, only
70
structures distal to the COM are modeled, neglecting energy storage and return
71
related to deformable structures proximal to the foot COM.
72
In the UD segment model (Table 1B), a proximal rigid region of the segment
73
is used to estimate the angular velocity of the entire body. When applied to a
74
lower-limb prosthetic system, the angular velocity of the proximal shank, ω ¯ S , is
75
used to represent both the rigid and deformable portions of the segment. The
76
deformation velocity, v¯def = r¯˙UD , is then estimated (Takahashi et al., 2012).
77
While this model has been shown to produce useful estimates of overall pros-
78
thetic system power, it is limited in its ability to isolate the performance of
79
foot-ankle devices from other deformable components (Takahashi, 2012). The
80
aim of the current work is to establish a method to evaluate a deformable region,
81
such as a foot-ankle device, without the requirement for a proximal rigid region.
82
Theoretical Development
83
Deformable Link Segment (DLS) analysis is based on an equivalent linkage
84
model with the deformable region of interest bounded by the proximal and distal
85
reference points. The proximal reference point can be located at any trackable
86
point on a prosthetic system and an estimate of segment angular velocity using
87
a proximal rigid region is not required. As such, a fully deformable prosthetic
88
foot-ankle device, shown in Table 1C, or a specific region of the device can
89
be evaluated separately from other structures. The segment angular velocity 5
90
and deformation velocity are considered unknown and are determined simulta-
91
neously. The deformation velocity may then be used to calculate deformation
92
power of a prosthesis.
93
Consider an equivalent linkage model applied to a fully deformable pros-
94
thetic foot-ankle device (Figure 1). The proximal reference point, P , is located
95
conveniently between the device and the pylon to define a foot-pylon interface
96
(FPI). The instantaneous position, Pj , j = 1, 2, n, is assumed to be known
97
from tracking using motion capture. The distal reference point, D, is the in-
98
stantaneous COP. Note that D does not move along the plantar surface of the
99
foot, rather there is a new contact point, Dj , j = 1, 2, n, at each instant. It
100
is assumed that the instantaneous position of the COP is obtained from force
101
platform measurements gathered during motion analysis. At each instant, a
102
new link is defined with a unique structural stiffness. The foot is thus modelled
103
as a series of equivalent linkages, P Dj , j = 1, 2, n.
104
Only one linkage is considered for each instant of stance. The foot is assumed
105
to roll without slip such that Dj has zero velocity and link P Dj is assumed to
106
rotate about Dj with instantaneous angular velocity ωP D,j . Unlike the DF
107
and UD segment models, the angular velocity of the deformable segment is
108
considered unknown and the value of ωP D,j is computed at each instant based
109
on the motion of points P and D.
110
Motion of the system can first be approximated with a sagittal plane analysis
111
corresponding to the YZ plane of the global ground-fixed reference frame (XYZ),
112
as shown in Figure 1. The angle of link P D is represented by γ, measured with
113
respect to the global Y axis.
114
The position vector r¯P D is determined from the difference between the known
115
positions of P and D. The instantaneous angular orientation, γ, of link P D
116
ˆ components of r¯P D . Thus, r¯P D is is calculated from the Y (Jˆ) and Z (K)
117
represented in the sagittal plane as: ˆ r¯P D = rP D cos γ Jˆ + rP D sin γ K
6
(1)
118
119
The relative velocity of P with respect to D is obtained by differentiating Equation 1 with respect to time: ˆ v¯P D = (r˙P D cos γ − rP D ωP D sin γ)Jˆ + (r˙P D sin γ + rP D ωP D cos γ)K
(2)
120
where the magnitude of the sagittal segment angular velocity ωP D = γ˙ and is
121
considered unknown.1
122
ˆ The velocity of P is measured and known such that v¯P D = vP y Jˆ + vP z K.
123
Since the foot is assumed to roll on the ground without slip, the velocity of D
124
ˆ Grouping like terms gives: is zero, v¯P = v¯P D = vP y Jˆ + vP z K. Jˆ : vP y = r˙P D cos γ − rP D ωP D sin γ ˆ : vP z = r˙P D sin γ + rP D ωP D cos γ K
125
which can be written in matrix form as: vP y cos γ = vP z sin γ
126
(3)
−rP D sin γ r˙P D rP D cos γ ωP D
(4)
Rearranging Equation 4 and computing the matrix inverse gives:
r˙P D ωP D
=
cos γ
v P Y cos γ vP Z
sin γ
− rP1D sin γ
1 rP D
(5)
127
where the magnitude of the deformation velocity, r˙P D , and angular velocity,
128
ωP D , are unknowns.
129
This approach is distinct from those presented in the literature because
130
Equation 5 allows an approximation of deformation velocity from direct mea1 In this derivation, it should be noted that γ is instantaneously known, while γ ˙ = ωP D , representing the instantaneous angular velocity of the link, is unknown. While it is conceivable dγ to calculate the angular velocity of the linkage as dt , this represents the rate of change in angular orientation between sequential linkages, calculated from the change in γ over time, and does not represent the instantaneous angular velocity of the linkage.
7
131
surements of the trajectories of P and D without defining a proximal rigid region
132
or estimating the COM position and angular velocity of the segment.
133
Note that the direction of r¯P D changes throughout stance. In some applica-
134
tions, such as the computation of power, it is desirable to represent deformation
135
velocity in a global ground based reference frame consistent with the measured
136
ground reaction forces. The deformation velocity vector, v¯def = r¯˙P D , is given
137
ˆ components: by Equation 6 in ground-based Y (Jˆ) and Z (K) ˆ v¯def = r¯˙P D = r˙P D cos γ Jˆ + r˙P D sin γ K
138
139
(6)
Experimental Protocol Three volunteer trans-tibial prosthesis users participated in a research ethics
140
board approved gait study at the Human Mobility Research Laboratory (Kingston,
141
Canada) after giving informed consent. Participants were active individuals
142
(K3-K4) with a traumatic amputation etiology and no lower-limb co-morbidities
143
(Table 2). Complete details of the methodology can be found in (Zhao, 2018).
144
Lower limb kinematics, ground reaction forces and centre of pressure (COP)
145
were recorded using 14 optical motion tracking cameras (Oqus 400, Qualisys,
146
Gothenburg, Sweden) and six embedded force plates (Custom BP model, AMTI,
147
Watertown, MA, USA). Subjects performed level over-ground walking trials at
148
a self-selected walking speed with their prescribed prosthetic foot-ankle device
149
until a minimum of five clean force platform strikes were recorded for the pros-
150
thetic side. Details of the prosthetic componentry can be found in the Sup-
151
plementary Material. Footwear was standardized by using the same model of
152
flexible athletic shoe for all subjects.
153
Marker trajectories, ground reaction forces and COP data were recorded us-
154
ing Qualisys Track Manager (Qualisys, Gothenburg, Sweden) and subsequently
155
R R analyzed using purpose-scripted code in MATLAB (R2012a, MathWorks ,
156
Natick, United States). Angular velocity and deformation velocity estimates
157
were calculated for three regions of the prosthetic system. Data correspond-
158
ing to the stance phase were resampled to 101 data points representing 100% 8
159
stance time. Results were then ensemble averaged across steps to provide a
160
mean profile and 95% confidence interval for each subject.
161
Interestingly, while Siegel et al. (1996) reported distal foot velocity for the
162
natural foot, to the authors’ knowledge there is no report of deformation ve-
163
locities of prosthetic foot-ankle devices in the literature. As such, the reported
164
deformation velocity results focus on the Z components. Preliminary analysis
165
indicated that the Z component of deformation velocity was found to be the
166
dominant contributor to the overall power during walking, due in part to the
167
dominance of the vertical ground reaction force. While the X and Y compo-
168
nents of UD and DF velocities were not negligible, a pilot study determined
169
their contribution to the overall power was less than approximately 10%.
170
DLS Analysis
171
Proximal Reference Point Definition. For application to the foot-ankle compo-
172
nent, the proximal reference point (P ) was defined as the proximal boundary
173
of the prosthetic foot-ankle structure. The trajectory was tracked using 1-3
174
reflective markers placed on the foot-pylon interface (FPI); these were the only
175
markers necessary for DLS analysis of the foot-ankle component. Additional
176
details are provided in the Supplementary Material.
177
For comparison to UD segment analysis, DLS analysis was applied to the
178
UD segment, with P located at the UD segment centre of mass and tracked
179
using reflective markers placed on the proximal rigid region of the prosthetic
180
system. For comparison to the DF model, DLS analysis was applied to the DF
181
segment, with P located at the foot centre of mass and tracked using reflective
182
markers placed on the foot.
183
Analysis. DLS deformation velocity was calculated in three stages: 1) The po-
184
sition vector r¯P D was calculated from the difference between the position of P
185
and the COP; the instantaneous length, rP D , was then determined from the
186
magnitude of r¯P D . 2) The instantaneous angle of link P D with respect to the
187
global horizontal axis, γ, was calculated from the Y and Z components of r¯P D .
9
188
3) The magnitude of the DLS deformation velocity, r˙P D , and angular velocity,
189
ωP D , were calculated according to Equation 5. The magnitude of the DLS defor-
190
mation velocity, r˙P D , was expressed in global YZ components using Equation 6.
191
This analysis was applied to: the UD segment providing r˙P D−UD and ωP D−UD ;
192
the DF segment giving r˙P D−DF and ωP D−DF ; and the foot-ankle component
193
(Table 1C) providing r˙P D−F and ωP D−F . Note that r¯P D−F , r¯P D−DF , and
194
r¯P D−UD differ by the location of the proximal reference point P .
195
DF and UD Analysis
196
For comparison to methods in the literature, 11-14 additional markers were
197
required to facilitate estimations for the foot angular velocity, UD segment
198
(shank) angular velocity, and distal foot deformation velocity. Markers placed
199
on the proximal rigid region of the prosthesis were used to estimate the shank
200
angular velocity and track the COM of the UD segment using previously de-
201
scribed procedures (Crimin et al., 2014; Geil et al., 1999; Smith et al., 2014).
202
The angular velocity of the prosthetic foot-ankle device was estimated using
203
markers on the shoe located in positions corresponding to the calcaneus, first
204
and fifth metatarsal heads, and FPI. Angular velocities were computed using
205
the right hand rule convention.
206
Results
207
Comparison Between UD and DLS Methods
208
Deformation velocity estimates for the UD segment computed using DLS
209
analysis (r¯˙P D−UD ) and UD segment analysis (r¯˙UD ) are compared in Figure 2a-
210
c. The mean (line) and 95% confidence interval (shaded) are shown for the Z-
211
component of each velocity measure. A negative velocity indicates the length of
212
the instantaneous link is decreasing, thus representing deformation. A positive
213
velocity indicates the link length is increasing from a deformed state, returning
214
towards the non-deformed state.
215
Z-component deformation velocity estimates for DLS analysis and UD seg-
216
ment analysis are comparable when applied to the same UD segment, with root 10
217
mean square (RMS) differences for the three subjects ranging from 33.5 to 37.8
218
mm/s. For prosthetic systems without deformable componentry proximal to
219
the FPI (e.g. Subject 1, Figure 2a) deformation velocity estimates for the UD
220
segment computed using DLS analysis (r¯˙P D−UD ) are also similar to estimates
221
for the foot-ankle segment (r¯˙P D−F ), with RMS differences of 49.1 mm/s. How-
222
ever, for Subject 3 (Figure 2c) whose prosthetic system included a mechanical
223
pump proximal to the FPI, larger RMS differences of 106.2 mm/s were observed
224
between r¯˙P D−UD and r¯˙P D−F .
225
Angular velocity estimates for the UD segment computed using DLS anal-
226
ysis and UD segment analysis are compared in Figure 2d-f. For UD segment
227
analysis, angular velocity, (ωS ), was estimated from markers on the proximal
228
rigid region of the prosthetic system using a rigid body model of the shank.
229
This is compared to the angular velocity of the UD segment computed using
230
Equation 5 (ωP D−UD ). Absolute differences between ωS and ωP D−UD are up to
231
1.3 rad/s in early stance, 0.5 rad/s in mid stance and 2.7 rad/s in late stance for
232
Subject 1, with similar trends for Subject 2 and 3. For prosthetic systems with-
233
out deformable componentry proximal to the FPI (e.g. Subject 1, Figure 2d)
234
angular velocity estimates for the UD segment computed using DLS analysis
235
(ωP D−UD ) are comparable to estimates for the foot-ankle segment (ωP D−F ),
236
particularly from 10-85% stance, with absolute differences less than 0.3 rad/s.
237
However, for Subject 3 (Figure 2f) whose prosthetic system included a mechani-
238
cal pump proximal to the FPI, larger differences were observed between angular
239
velocity estimates ωP D−UD and ωP D−F , up to 2.3 rad/s in late stance.
240
Differences between deformation velocities r¯˙P D−UD and r¯˙UD are likely due to
241
the angular velocity approximations and specifically, rotation of the foot-ankle
242
component relative to the shank component. Larger deformation velocities were
243
consistent with differences in angular velocity; ωP D−UD was larger than ωS in
244
magnitude during early and late stance for all subjects.
11
245
Comparison Between DF and DLS methods
246
Deformation velocity estimates for the DF segment computed using DLS
247
analysis (r¯˙P D−DF ) and the DF model (r¯˙DF ) are compared in Figure 3a-c. The
248
mean (line) and 95% confidence interval (shaded) are shown for the Z-component
249
of each velocity measure. Z-component deformation velocity estimates for DLS
250
analysis and the DF model are distinct, particularly in late stance. Subject 1
251
displayed a RMS difference of 77.3 mm/s overall, with a difference of 141.0 mm/s
252
in late stance; these trends were similar for Subjects 2 and 3. The DF model
253
estimates trend towards positive values in late stance, representing return to the
254
undeformed shape. Deformation velocity estimates for DLS analysis applied to
255
the DF segment are closer to zero and are at times negative in late stance,
256
this may indicate that the distal foot structures further deform or maintain a
257
deformed state in late stance.
258
Angular velocity estimates for the DF segment computed using DLS analysis
259
and the DF model are compared in Figure 3d-f. For the DF model, angular
260
velocity, (ωF ), was estimated using markers on the foot and a rigid body model.
261
This is compared to the angular velocity of the DF segment computed using
262
Equation 5 (ωP D−DF ). The angular velocity of the foot-ankle segment (ωP D−F )
263
is also shown for comparison. For all subjects, DLS angular velocity estimates
264
for the distal foot are distinct from DLS angular velocity estimates for the
265
foot-ankle segment, with differences up to 2.1 rad/s for Subject 1. From 10-
266
85% stance, DLS angular velocity estimates for the distal foot (ωP D−DF ) are
267
comparable to the estimates derived from a rigid body model of the foot (ωF ),
268
with absolute differences less than 0.5 rad/s for all subjects. In the rigid body
269
model, the main axis of the foot was defined between markers on the heel and
270
metatarsal heads (Robertson et al., 2013). While this estimate may represent
271
the motion of the distal region of the foot, these results suggest that ωP D−F may
272
provide a more representative estimate for the angular velocity of the overall
273
foot-ankle component, including the effects of the deformable ankle region.
12
274
Angular Velocity Approximations
275
DLS angular velocity of the foot-ankle component (ωP D−F ) is plotted in
276
Figure 4 in comparison to values for the sagittal foot (ωF ) and shank (ωS )
277
computed using rigid body models of the foot and shank, respectively. If the
278
prosthetic foot-ankle device were rigid, the foot and shank angular velocities
279
would be equivalent. In a deformable system, these measures are expected to
280
be distinct, as observed in this study. Absolute differences between ωF and ωS
281
are up to 1.9 rad/s in early stance, 1.5 rad/s in mid stance and 1.7 rad/s in late
282
stance for Subject 1. These trends were similar for Subjects 2 and 3.
283
For all subjects, DLS angular velocity was similar to foot angular velocity in
284
early and late stance, but distinct from the shank throughout all regions. The
285
“foot flat” region for Subject 1 between 25-50% stance shows a foot angular
286
velocity that approaches zero. In contrast, angular velocity of the shank ranged
287
from -1.1 to -1.7 rad/s and DLS angular velocities ranged from -0.6 to -0.9
288
rad/s. DLS analysis appears to capture an appropriate value that represents
289
both motion of the foot and deformation of the proximal foot-ankle region during
290
foot flat that is distinct from motion of the shank.
291
Discussion
292
The DF model and UD segment analysis require a proximal rigid region or
293
rigid body assumption for approximation of the segment angular velocity. In
294
contrast, DLS analysis can be applied to various regions of the prosthetic system,
295
without requiring an angular velocity approximation for the desired segment.
296
Of note, methods in the literature are unable to isolate the foot-ankle compo-
297
nent from other deformable structures due to the uncertainty regarding how
298
to estimate the angular velocity of a fully deformable segment. Isolation of the
299
foot-ankle component from other deformable structures of the prosthetic system
300
(e.g. shock-absorbing/telescoping pylon or mechanical pump) was demonstrated
301
using DLS analysis with the proximal reference point at the FPI. Applying DLS
302
analysis to various regions of the foot-ankle component would provide additional
13
303
insights regarding the contribution of individual design features to the overall
304
function of the device; however, this was outside the scope of the current study.
305
The largest differences between methods are typically observed in the 0-20%
306
and 80-100% stance regions of deformation velocity estimates. These early and
307
late stance regions of deformation velocity correspond to collision and push-off
308
regions of power.
309
Application to Other Deformable Segments
310
DLS analysis can be extended to evaluate the deformation velocity of other
311
componentry. It is noted that Equation 5 describes a special case when the
312
distal reference point velocity is zero. This reflects an application to a foot
313
rolling without slip or any deformable region for which the distal reference point
314
velocity is instantaneously defined by the COP. For a more general case, the
315
ˆ is added to the derivation. distal reference point velocity, v¯D = vDY Jˆ + vDZ K,
316
The generic form of Equation 5 thus becomes:
r˙P D
cos γ
= 1 ωP D − rP D sin γ
v − v DY PY cos γ vP Z − vDZ
sin γ 1 rP D
(7)
317
The general form of the DLS analysis can be applied to an intermediate
318
segment, such as a sub-region of the foot-ankle device, a shock-absorbing pylon,
319
or other deformable segment, for which the positions of the proximal and distal
320
ends of the segment can be tracked. If the reaction force at one of the reference
321
points is also known, the deformation velocity can subsequently be used to
322
calculate deformation power.
323
Deformation and Geometric Effects
324
Consider an equivalent linkage model that defines a position vector, r¯P D ,
325
between the COP and a proximal reference point. The time derivative of this
326
d vector ( dt r¯P D ) captures the rate of change in length throughout stance phase
327
due to both foot geometry and deformation. To estimate the instantaneous
328
deformation velocity (and subsequently the rate of change of elastic potential 14
329
energy), care must be taken to ensure that deformation effects are isolated
330
from the geometric effects associated with rigid body motion. While the final
331
deformed state of a foot-ankle component can be determined by the rollover
332
shape (Hansen et al., 2000; Major et al., 2012), estimating deformation directly
333
requires a reference to the undeformed shape (the change in r¯P D due to device
334
geometry). Isolating deformation velocity (r¯˙P D ) from the overall rate of change
335
d in link length ( dt r¯P D ) provides a means of characterizing deformation effects
336
separately from geometric effects, without requiring the undeformed shape.
337
The influence of foot geometry is shown in Figure 5 using the principle of
338
superposition. First consider a hypothetical rigid foot component for which
339
the foot sole is an non-circular arc. The foot rotates from its initial position
340
(Figure 5a) to a new position, which defines an intermediate state (Figure 5b).
341
The length of the link at the intermediate state, r¯P D (j 0 ) is greater than at
342
the initial position, r¯P D (j − 1), due to the shape of the foot and its change
343
in orientation. The new link r¯P D (j 0 ) is undeformed in the intermediate state.
344
Deformation is observed by the superposition of the contact force, F , on the link
345
at the intermediate state. During deformation, the length of the link reduces
346
from the intermediate state in Figure 5c, r¯P D (j 0 ), to the final state in Figure 5d,
347
r¯P D (j).
348
The effect of geometry due to a change in angular orientation of the foot
349
and the effect of deformation due to the applied force occur simultaneously
350
throughout stance, and are thus superimposed. DLS analysis isolates the rate of
351
change in link length due to deformation (r¯˙P D ) from the overall rate of change in
352
d link length ( dt r¯P D ) by considering the segment as a series of sequential linkages
353
and solving for r¯˙P D and the angular velocity of each linkage instantaneously.
354
An incremental change in angular orientation is associated with progression of
355
the contact point, and the definition of a new linkage. This results in no change
356
in elastic potential energy of the system (Figures 5a and 5b). However, when
357
deformation takes place in the presence of a force, there is a change in the elastic
358
potential energy state (Figures 5c and 5d). As the system progresses, this elastic
359
potential energy of link at j is then transferred to the subsequent link, j + 1. 15
360
Limitations and Recommendations
361
Future work includes extension of the technique to estimate deformation
362
power and stance work. However, caution is required whenever applying this
363
method beyond the analyses presented.
364
Prostheses with Articulations. For interpretation of r¯˙P D as deformation veloc-
365
ity, care must be taken to ensure that deformation effects are isolated from the
366
geometric effects. For passive non-articulating prostheses, using DLS analysis to
367
model the segment as a series of equivalent linkages where the angular velocity
368
represents instantaneous linkage rotation facilitates representation of the defor-
369
mation velocity as the instantaneous change in link length due to deformation
370
excluding geometric effects.
371
Passive articulating prostheses may include joints with rotational stiffness.
372
In this case, there are two deformation effects influencing energy storage and
373
return: structural deformation of sub-segments and angular deflection through
374
joint rotation. There are also two geometric effects influencing the link length:
375
kinematic motion of the overall segment and rotation of the sub-segments about
376
the joint. For articulating segments it is recommended to apply DLS analysis
377
separately to each sub-segment. Each sub-segment would be an independent
378
DLS and the common point would act as a joint. As such, the joint would
379
transmit power between segments as in a conventional link segment analysis.
380
While joint linear and relative angular velocities and joint force could be deter-
381
mined using the application of current methods, determining the joint moment
382
could present challenges. In particular, elastic elements at the joint that resist
383
angulation would affect the net joint moment due to the relative angle between
384
segments. This, in turn, complicates the equilibrium analysis. However, the
385
prospect of determining the effect of angular deformation affords an interesting
386
avenue of future study that would provide further insight into energy manage-
387
ment in these systems.
388
Three Dimensional Analysis. This study assumes that prosthetic foot-ankle de-
389
formation during walking is dominated by motion in the sagittal plane. DLS 16
390
analysis described in this work is limited to a two-dimensional analysis, in which
391
the sagittal plane of walking is aligned with the YZ plane in the global reference
392
frame. However, power estimates are the product of force and velocity terms
393
that may result in significant off-plane contributions, particularly during more
394
complex activities involving turning and impact. Future work will consider ex-
395
tension of DLS analysis to provide three-dimensional estimates of deformation
396
velocity.
397
Uncertainty in COP Measurement. Low vertical ground reaction force values
398
in early and late stance are known to be associated with inaccuracies in COP
399
recordings (Robertson et al., 2013). In addition, the orientation of the foot dur-
400
ing early and late stance may be considered to be relatively unstable due to
401
a smaller base of support, leading to additional deviations in the COP trajec-
402
tory. As such, DLS deformation velocity estimates for segments where the link
403
length is relatively small (e.g. the DF segment) may be more sensitive to COP
404
inaccuracies.
405
Conflict of Interest Statement
406
The authors have no financial or personal relationships with other people or
407
organizations that could inappropriately influence this work.
408
Acknowledgements
409
Financial support for this study was received from the Natural Sciences and
410
Engineering Research Council of Canada, the Ontario Graduate Scholarship
411
Program, and the Donald and Joan McGeachy Chair in Biomedical Engineering.
412
The authors wish to thank Ms. Christina Ippolito, MASc, EIT for assistance
413
with manuscript preparation, Dr. Ron Anderson, Ph.D, P.Eng for critically
414
reviewing the study and the final manuscript, and Mr. Martin Robertson, BSc,
415
CP(c) for recruiting gait analysis participants and providing clinical support.
17
416
Alaranta, H., Kinnunen, A., Karkkainen, M., Pohjolainen, T., & Heliovaara, M.
417
(1991). Practical benefits of Flex-Foot in below-knee amputees. Journal of
418
Prosthetics and Orthotics, 3 , 179–181.
419
420
American Orthotics and Prosthetics Association (2010). AOPA Prosthetic Foot Project Report. AOPA, .
421
Crimin, A., Mcgarry, A., Harris, E. J., & Solomonidis, S. E. (2014). The effect
422
that energy storage and return feet have on the propulsion of the body: A
423
pilot study. Proceedings of the Institution of Mechanical Engineers. Part H,
424
Jounral of Engineering in Medicine, 228 , 908–915.
425
Geil, M. D. (1997). Effectiveness evaluation and functional theoretical modeling
426
of dynamic elastic response lower limb prosthetics. Ph.D. thesis The Ohio
427
State University.
428
Geil, M. D., Parnianpour, M., & Berme, N. (1999). Significance of nonsagittal
429
power terms in analysis of dynamic elastic response prosthetic feet. Journal
430
of Biomechanical Engineering, 121 , 521–524.
431
Gitter, A., Czerniecki, J. M., & DeGroot, D. M. (1991). Biomechanical analysis
432
of the influence of prosthetic feet on below-knee amputee walking. American
433
Journal of Physical Medicine Rehabilitation, 70 , 142–148.
434
Hafner, B. J., Sanders, J. E., Czerniecki, J., & Fergason, J. (2002). Energy
435
storage and return prostheses: does patient perception correlate with biome-
436
chanical analysis? Clinical Biomechanics, 17 , 325–344.
437
Hansen, A. H., Childress, D. S., & Knox, E. H. (2000). Prosthetic foot roll-over
438
shapes with implications for alignment of trans-tibial prostheses. Prosthetics
439
and Orthotics International , 24 , 205–215.
440
Low, S. E., Tsimiklis, A., Zhao, S. R., Davies, T. C., & Bryant, J. T. (2017). Ef-
441
fect of Passive Dynamic Prosthetic Feet on the Improvement of Gait in Adult
442
Unilateral Transtibial Amputees as Compared to SACH Feet: A Systematic
18
443
Review. In Proceedings of the 16th International Society for Prosthetics and
444
Orthotics World Congress. Capetown, South Africa.
445
Macfarlane, P., Nielsen, D., Shurr, D., & Meier, K. (1991). Perception of walking
446
difficulty by below-knee amputees using a conventional foot versus the flex-
447
foot. Journal of Prosthetics and Orthotics, 3 , 114–119.
448
Major, M. J., Kenney, L. P., Twiste, M., & Howard, D. (2012). Stance phase
449
mechanical characterization of transtibial prostheses distal to the socket: A
450
review. Journal of Rehabilitation Research and Development , 49 , 815–830.
451
Nielson, D. H., Shurr, D. G., Golden, J. C., & Meier, K. (1988). Comparison
452
of energy cost and gait efficiency during ambulation in below-knee amputees
453
using different prosthetic feet - A preliminary report. Journal of Prosthetics
454
and Orthotics, 1 , 24–31.
455
Prince, F., Winter, D. A., Sjonnesen, G., & Wheeldon, R. K. (1994). New
456
technique for the calculation of the energy stored, dissipated, and recovered
457
in different ankle-foot prostheses. IEEE Transactions on Rehabilitation En-
458
gineering, 2 , 247–255.
459
Raschke, S. U., Orendurff, M. S., Mattie, J. L., Kenyon, D. E., Jones, O. Y.,
460
Moe, D., Winder, L., Wong, A. S., Moreno-Hern´ andez, A., Highsmith, M. J.,
461
Sanderson, D. J., & Kobayashi, T. (2015). Biomechanical characteristics,
462
patient preference and activity level with different prosthetic feet: A random-
463
ized double blind trial with laboratory and community testing. Journal of
464
Biomechanics, 48 , 146–152.
465
466
Robertson, D. G., Caldwell, G. E., Hamill, J., Kamen, G., & Whittlesey, S. N. (2013). Research Methods in Biomechanics. (2nd ed.). Human Kinetics.
467
Robertson, D. G., & Winter, D. A. (1980). Mechanical energy generation, ab-
468
sorption and transfer amongst segments during walking. Journal of Biome-
469
chanics, 13 , 845–854.
19
470
471
Romo, H. D. (1999). Specialized prostheses for activities. An update. Clinical Orthopaedics, 361 , 63–70.
472
Siegel, K. L., Kepple, T. M., & Caldwell, G. E. (1996). Improved agreement of
473
foot segmental power and rate of energy change during gait: inclusion of distal
474
power terms and use of three-dimensional models. Journal of Biomechanics,
475
29 , 823–827.
476
Smith, J. D., Ferris, A. E., Heise, G. D., Hinrichs, R. N., & Martin, P. E. (2014).
477
Oscillation and reaction board techniques for estimating inertial properties of
478
a below-knee prosthesis. Journal of visualized experiments, 87 , 1–16.
479
480
Takahashi, K. Z. (2012). Total power profiles of anatomical and prosthetic belowknee structures. Ph.D. thesis University of Delaware.
481
Takahashi, K. Z., Kepple, T. M., & Stanhope, S. J. (2012). A unified deformable
482
(UD) segment model for quantifying total power of anatomical and prosthetic
483
below-knee structures during stance in gait. Journal of Biomechanics, 45 ,
484
2662–2667.
485
Takahashi, K. Z., Worster, K., & Bruening, D. A. (2017). Energy neutral: the
486
human foot and ankle subsections combine to produce near zero net mechan-
487
ical work during walking. Scientific Reports, 7 .
488
Torburn, L., Perry, J., Ayyappa, E., & Shanfield, S. L. (1990). Below-knee
489
amputee gait with dynamic elastic response prosthetic feet: A pilot study.
490
Journal of Rehabilitation Research and Development , 27 , 369–384.
491
Versluys, R., Beyl, P., van Damme, M., Desomer, A., van Ham, R., & Lefeber, D.
492
(2009). Prosthetic feet: state-of-the-art review and the importance of mimick-
493
ing human ankle-foot biomechanics. Disability and Rehabilitation: Assistive
494
Technology, 4 , 65–75.
495
Zelik, K. E., & Honert, E. C. (2018). Ankle and foot power in gait analy-
496
sis: Implications for science, technology and clinical assessment. Journal of
497
Biomechanics, 75 . 20
498
499
Zhao, S. R. (2018). Deformable link segment model for analysis of prosthetic foot-ankle components. Ph.D. thesis Queen’s University.
21
500
(Siegel et al., 1996)
Deformable Region Modeled. Footankle structures distal to foot COM. Angular Velocity. Approximated using a foot rigid body model. Limitations. Application to prosthetic foot-ankle device requires defining kinematics of a deformable segment (¯ vCOM,F and ω ¯ F ). By definition, the model captures deformation of the distal region only.
v¯def = r¯˙DF = v¯COM,F − ω ¯ F × r¯DF
Deformable Region Modeled. Structures distal to UD COM.
(Takahashi et al., 2012)
(B) UD Segment Model
(A) Distal Foot Model
Table 1: Summary of approaches for estimating deformation velocity associated with deformable foot-ankle devices. An equivalent linkage model is used to describe each approach: (A) The distal foot model applied to a prosthetic foot-ankle device provides estimates of velocity, v¯def = r¯˙DF , associated with deformable structures of the distal region of the foot. (B) The unified deformable (UD) segment model provides estimates of velocity, v¯def = r¯˙U D , associated with deformable structures below the knee. (C) Deformable link segment analysis applied to a prosthetic foot-ankle device provides estimates of deformation velocity, v¯def = r¯˙P D , associated with the foot-ankle device.
Angular Velocity. Approximated using a shank rigid body model. Limitations. It is not possible to isolate the foot-ankle device from other deformable prosthetic componentry, such as a deformable pylon, mechanical pump or connector located distal to the UD COM (Takahashi, 2012). For foot-ankle devices with functional ankle regions, the foot angular velocity may differ from that of the shank.
Segment Analysis
(C) Deformable Link
v¯def = r¯˙U D = v¯COM,U D − ω ¯ S × r¯U D
Deformable Region Modeled. Footankle structures distal to the foot-pylon interface (FPI). Ã
r˙P D ωP D
!
"
cos γ = − 1 sin γ r PD
sin γ 1 cos γ r PD
#Ã
vP Y vP Z
ˆ v¯def = r¯˙P D = r˙P D cos γ Jˆ + r˙P D sin γ K
!
Angular Velocity. Determined simultaneously along with deformation velocity. Limitations. Application to 3D analysis in development.
Notation: r¯˙ , Deformation velocity; r¯, Position vector; v¯, Velocity of a reference point; ω ¯ , Angular velocity; r, Instantaneous length of r¯; ω, Sagittal angular velocity; γ, Instantaneous angular orientation of link P D.
22
Table 2: Subject demographics for pilot gait study
Subject
Age (yrs)
Mass (kg)
K-Level
Prescribed Device
Subject 1 Subject 2 Subject 3
61 31 36
53 64 67
K3 K4 K4
TruLife Seattle Catalyst9 ¨ Ossur Re-Flex ShockTM R ¨ Ossur Vari-Flex XC
23
501
Figure Captions Figure 1: Foot-ankle device modeled as prismatic link P D. Point P is a point fixed on the device located at the foot-pylon interface. Point D is the instantaneous COP position representing the contact point between the foot and ground. Instantaneously, link P D can be considered to rotate about D with angular velocity ωP D . (Note that P is not a pin joint and link P D does not rotate with respect to the foot, rather the foot and link together rotate instantaneously about D). The angular orientation of link P D is represented by γ with respect to the global (YZ) reference frame.
Figure 2: Deformation velocity and angular velocity estimates for the UD segment over the stance phase of gait. Mean (line) and 95% confidence interval (shaded region) are shown for: Subject 1 (n = 9), Subject 2 (n = 7), and Subject 3 (n = 11). (a)-(c): The Z component of the UD segment deformation velocity computed using DLS analysis, r¯˙P D−U D , is compared to that computed using UD segment analysis, r¯˙U D . Results for DLS analysis applied to the foot-ankle segment, r¯˙P D−F , are indicated by a dashed line for reference. (d)-(f): Angular velocity of the UD segment estimated from DLS analysis, ωP D−U D , compared to angular velocity approximated from the proximal rigid region of prosthetic system and a rigid body model of the shank, ωS . Results for DLS analysis applied to the foot-ankle segment, ωP D−F , are indicated by a dashed line for reference.
Figure 3: Deformation velocity and angular velocity estimates for the DF segment over the stance phase of gait. Mean (line) and 95% confidence interval (shaded region) are shown for: Subject 1 (n = 9), Subject 2 (n = 7), and Subject 3 (n = 11). (a)-(c): The Z component of the DF deformation velocity computed using DLS analysis, r¯˙P D−DF , is compared to that computed using the DF model, r¯˙DF . Results for DLS analysis applied to the foot-ankle segment, r¯˙P D−F , are indicated by a dashed line for reference. (d)-(f): Angular velocity of the DF segment estimated from DLS analysis, ωP D−DF , compared to angular velocity approximated using a rigid body model of the foot, ωF . Results for DLS analysis applied to the foot-ankle segment, ωP D−F , are indicated by a dashed line for reference.
24
Figure 4: DLS sagittal angular velocity, ωP D−F , of the foot-ankle device (black) compared to angular velocities approximated using rigid body models of the foot, ωF , and shank, ωS for each subject. Mean (line) and 95% confidence interval (shaded) across trials are shown for: (a) Subject 1 (n = 9), (b) Subject 2 (n = 7), and (c) Subject 3 (n = 11).
Figure 5: Superposition of geometric and deformation effects in the DLS analysis. First, a rigid foot rotates from its initial position (a) to a new position (b), which defines an intermediate state. The length of the link at the intermediate state, r¯P D (j 0 ), is greater than at the initial position, r¯P D (j − 1), due to the shape of the foot and its change in orientation. Next, the deformation of the segment is isolated by the instantaneous change in length of the link due to a contact force. During deformation, the length of the link reduces from the intermediate state (c), r¯P D (j 0 ), to the final state (d), r¯P D (j).
25
502
Supplementary Material
503
Tracking the Foot-Pylon Interface
504
For application to the foot-ankle component, the proximal reference point
505
(P ) was defined as the proximal boundary of the prosthetic foot-ankle structure.
506
The trajectory was tracked using 1-3 reflective markers placed on the foot-
507
pylon interface (FPI); these were the only markers necessary for DLS analysis
508
of the foot-ankle component. For Subjects 1 and 3, the FPI was defined as the
509
centre of the proximal rigid adapter on the prosthetic foot where it connected
510
to the pylon, as shown in Figure 6. For Subject 2, the FPI was defined at the
511
connection between the composite keel and shock absorbing pylon to isolate the
512
response of the keel. In this case, the FPI position was tracked using a marker
513
cluster on a nearby rigid region of the prosthetic structure.
514
Rollover Shape
515
Equivalent linkage models of the foot-ankle component are related to the
516
concept of a rollover shape. Major et al. (2012) defined the rollover shape as “a
517
spatial mapping of the center of pressure location along the plantar surface of the
518
foot relative to a shank-based coordinate frame”. The rollover shape provides
519
an approximation of the overall deformed state of a foot throughout stance and
520
represents the equivalent rigid body shape of the foot for a specific loading
521
scenario (Hansen et al., 2000). The rollover shape of a foot can be determined
522
by transforming link r¯P D to a local reference frame fixed to the FPI. This metric
523
is influenced by geometry of the device design and deformation due to applied
524
load (Major et al., 2012). Thus, both geometric factors and deformation affect
525
the change in length of link r¯P D .
526
Regions of the Prosthetic System
527
DLS deformation velocity of the prosthetic foot-ankle component (r¯˙P D−F )
528
is plotted in comparison to the distal foot model (r¯˙DF ) and UD segment model
529
(r¯˙UD ) in Figure 7 for each subject. A negative velocity indicates the length of
26
530
the instantaneous link is decreasing, thus representing deformation. A positive
531
velocity indicates the link length is increasing from a deformed state, returning
532
towards the non-deformed state.
533
As shown in Figure 7, DLS deformation velocity, r¯˙P D−F , is greater in mag-
534
nitude compared to the DF model, r¯˙DF , and lower in magnitude compared to
535
the UD model, r¯˙UD . For all subjects, values for r¯˙DF are near zero from 15-75%
536
stance. The early and late stance regions of deformation velocity estimates cor-
537
respond to collision and push-off regions of power. The largest differences in
538
magnitude between methods are typically observed in the 0-20% and 80-100%
539
stance regions.
540
Deformation velocity estimates based on the three approaches are expected
541
to differ when the approaches were used to model different regions of the pros-
542
thetic device. For example, the DF model includes only structures distal to the
543
foot COM. The low magnitude of r¯˙DF compared to the other approaches reflects
544
the smaller deformable region of the device that is captured by the model. The
545
UD segment represents the largest region, including the foot-ankle device and
546
other componentry such as a mechanical pump (Subject 3).
27
547
Supplementary Material - Figure Captions Figure 6: Prescribed prosthetic foot-ankle components: (a) Subject 1’s prescribed foot-ankle device; (b) Subject 2’s prescribed device with integrated shock absorbing pylon; and (c) Subject 3’s prescribed foot-ankle device. Note that Subject 3’s prosthetic system included a mechanical pump proximal to the foot-ankle component. For all conditions, the location of the FPI is shown for reference. Foot shells and standardized athletic shoes were also worn. ¨ (a) Photography courtesy of Trulife; (b) and (c) Photography courtesy of Ossur.
Figure 7: Deformation velocity for each subject over the stance phase of gait. The Z components of the distal foot deformation velocity, r¯˙DF , the unified deformable segment deformation velocity, r¯˙U D , and DLS deformation velocity, r¯˙P D−F , are compared. Mean (line) and 95% confidence interval (shaded region) are shown for: (a) Subject 1 (n = 9), (b) Subject 2 (n = 7), and (c) Subject 3 (n = 11). Distal foot deformation velocity, r¯˙DF , typically exhibited lower values than the DLS model whereas the the UD segment deformation velocity, r¯˙U D , typically displayed larger values than the DLS model.
28