A Computational Study of the Swing Phase of the Gait with Standard

and Spring-Loaded Crutches

Marko Ackermann and Bruno Augusto Taissun

Abstract— Crutches have suffered few functional modifica-tions over their long history, with improvements largely limitedto aesthetics and weight reduction aspects. The large energeticcost of the gait with crutches and problems associated to theirlong-term use impose a heavy burden to the users. In order tomitigate some of the mentioned problems, alternative designshave been proposed over the past few decades. Among them,the idea of incorporating an elastic element to the crutches toreduce impact forces transmitted to the upper extremities andto promote energy storage and release has been indicated inthe specialized literature as a potential solution, in particularfor the crutch gait styles more similar to the normative humangait such as the “two-point” and the “swing-through” . In fact,tendon elasticity has been shown to reduce energy consumptionduring animal and human locomotion by means of energystorage in the initial and mid stance-phase and release inthe push-off phase of the gait cycle. In spite of the greatpotential of this idea, appropriate stiffness curves for theelastic element are poorly studied in the literature. This studyaims at investigating appropriate stiffness values for the elasticelement of spring-loaded crutches by means of computationalsimulations using a model of the swing phase of the “swing-through” gait style. The findings show that the stiffness shouldbe tuned carefully to ensure improved gait quality. Spring-loaded crutches undoubtedly reduce impact forces transmittedto upper limbs and shoulder at touch down but they candeteriorate performance with respect to foot clearance andeffort at the shoulder when compared to stiff crutches if stiffnessis not carefully selected.

I. INTRODUCTION

Locomotion disorders affect a substantial percentage of the

world population. Among the various ambulatory assistive

devices such as wheelchairs, walkers, canes and lower limb

orthoses, crutches figure as the most frequently used. It

is, therefore, surprising that crutches have undergone few

functional modifications over their long history [1], with

the improvement effort focused on aesthetics and weight

reduction [2].

Crutches allow patients who would otherwise require

wheelchair to stand and walk. The benefits of walking and

standing rather than sitting include improved blood circula-

tion, reduced bladder infections, prevention of contractures

and social inclusion [3]. Nonetheless, walking with crutches

is costly with some gait styles requiring as much as 80%

more energy expenditure per unit of distance traveled than

This work was not supported by any organizationM. Ackermann is with the Department of Mechanical Engineering, Centro

Universitario da FEI, Av. Humberto de A. C. Branco, 3972, Sao Bernardodo Campo, Brazil mackermann@fei.edu.br

B. A. Taissun is with the Department of Mechanical Engineering, CentroUniversitario da FEI, Av. Humberto de A. C. Branco, 3972, Sao Bernardodo Campo, Brazil bruno taissun@hotmail.com

normal gait [4]. Furthermore, the large forces transmitted to

the upper limbs during crutch ambulation can cause pain and

discomfort, and on the long-run conditions such as crutch

palsy, aneurysms, and thromboses [5], [6].

Alternative crutch designs have been proposed in the

literature to reduce the magnitude of the forces transmitted

to the upper limbs and the energy requirements during

locomotion [3], including the suspension crutch, the rocker

crutch, the prosthetic foot crutch and the spring-loaded

crutch. Among these options, the latter, which features an

added spring element, appears to be the most promising. In

fact, Segura and Piazza [5] and LeBlanc and colleagues [3]

stress the potential of spring-loaded crutches and Shoup [7]

estimates the reduction in metabolic energy requirements in

about 25%, in particular for crutch gait styles more similar to

normal walking such as the “swing-through” and the “two-

point”. The present study will focus on the “swing-through”

gait, a relatively fast gait style characterized by advancing

both crutches together and then swinging both feet together

to beyond the line of the crutches [6], refer to Fig. 1. This

style is indicated to patients with inability to fully bear

weight on both legs and is frequently adopted by paraplegic

patients using bilateral lower limb orthoses.

Fig. 1. Phases of the “swing-through” gait style. Temporal informationextracted from [8].

The energy storage capability of spring-loaded crutches is

expected to reduce energy requirements during locomotion

with crutches. In fact, energy storage in elastic elements

is extensively exploited in human and animal locomotion

to reduce energy consumption [9]. For instance, it is well-

documented that the energy storage in the Achilles tendon

at the beginning of the stance phase and its release for

propulsion at push off is a valuable mechanism to reduce

energy consumption in running [9], [10]. This observation

have inspired the desing of energy-saving lower limb pros-

theses and robots, e.g. [11], [9].

The Fourth IEEE RAS/EMBS International Conferenceon Biomedical Robotics and BiomechatronicsRoma, Italy. June 24-27, 2012

978-1-4577-1200-5/12/$26.00 ©2012 IEEE 1476

In spite of the promising benefits of spring-loaded

crutches, there is scarce literature on its appropriate design.

For instance, the effects of different stiffness ranges and

profiles on gait quality is poorly understood. Perhaps the

best attempt was done by Shortell et al. [2]. The manuscript

describes experiments with subjects (the number is not spec-

ified) with spring-loaded crutches of adjustable stiffness. Af-

ter subjective evaluation by subjects, Shortell and colleagues

recommend an average stiffness of approximately 22 kN/m

for each crutch and set a lower bound of 16 kN/m for which

crutches are considered to be “too compliant”, and an upper

bound of 30 kN/m for which the users considered the crutch

“too rigid” with little difference to a rigid crutch. Segura

and Piazza [5] adopt this average value in experiments to

investigate the influence of spring-loaded crutches on ground

reaction forces and obtain inconclusive results.

Computational studies are even more scarce. The authors

of the present manuscript found only one paper [12] using

computational simulations to determine appropriate stiffness

values, which found very compliant crutches with stiffness

of less than 4 kN/m to be “optimal”. However, such low

values are much lower than the minimum bound suggested

in [2] and it is likely they would severally compromise foot

clearance. Moreover, the model utilized in [12] is composed

of a single lumped mass and does not consider upper and

lower limb motion or crutch mass. A better understanding

of the possible benefits of spring-loaded crutches as well as

appropriate stiffness ranges, if any, is clearly necessary. The

present work aims at contributing to this effort by proposing

a more realistic model of the “swing-phase” of the “swing-

through” gait with crutches and discussing the potential of

spring-loaded crutches on the light of simulation results.

II. METHODS

A. Model

A model of the swing phase of the “swing-through” gait

style, Fig. 1, was adapted from a standard crutch model

by Rovick et al. [8]. The model, which is illustrated on

the left-hand side of Fig. 2, is composed of three rigid

bodies representing: 1) the rigid crutches and the arms; 2) the

trunk and the had; and 3) the lower limbs and the orthoses.

The model is contained in the sagittal plane and has three

degrees of freedom. The knee joints remain extended, as

usual in paraplegic locomotion with lower limb orthoses, so

that thighs and shanks can be represented by a single rigid

body. The crutch pivots about the contact point between its

tip and the floor. In order to simulate gait with spring-loaded

crutches this model was modified by the addition of an elastic

element to the crutch resulting in a variable length crutch and

a model with a total of four degrees of freedom, right-hand

side of Fig. 2. It should be noted that standard crutches have

usually a rubber tip whose stiffness is much greater than

the spring stiffness range considered in this study. The cor-

responding equations of motion were derived symbolically

using Matlab’s Symbolic Toolbox from the Newton-Euler

equations using the d’Alembert’s principle [13]. The adopted

generalized coordinates are depicted in Fig. 2.

Fig. 2. Model of the swing phase of the “swing-through” gait with rigidcrutches on the left, and with spring-loaded crutches on the right. The arrowsindicate positive angle directions.

The anthropometric parameters, including mass, moment

of inertia, segment length and position of the center of mass

for each segment are extracted from [8] and [14], refer

to Tab. IV in Appendix. Most spinal cord injuries (SCI)

in paraplegic patients affect, at least partially, the ability

to control hip flexors and extensors. For these reason, the

simulations in the present study assume no muscle control

at the hip resulting in no moment applied about this joint.

Thus, the only way subjects have to control the motion along

the swing phase is by applying a moment at the shoulder.

B. Simulations

The models introduced in the previous section were

used to determine the optimal motion between fixed, pre-

determined initial and final states of the model, resulting in a

two-point boundary value problem. The problem of finding

the optimal motion was formulated as an optimal control

problem and solved by transforming it into a nonlinear

parameter optimization problem [15], [16].

Most initial states for the swing phase at “toes-off” are

prescribed and coincide with the experimentally measured

initial positions and velocities in [8], Tab. I. The final crutch

and trunk angles, σ and Φ, are prescribed as well to adhere

to the experimental data in [8], Tab. I. The duration of the

swing phase is considered fixed at 0.7 s, a value consistent

with the same experimental data. These boundary constraints

were ensured by the addition of nonlinear constraints to

the optimization. For the simulations with the spring-loaded

crutches the initial length change rate C was set to zero, as

it is reasonable to assume crutches will be in “equilibrium”

with all the weight transferred from the legs to the crutches

before “toes off” occurs. Furthermore, to ensure a fair com-

parison between rigid and spring-loaded crutches, the initial

length of the spring-loaded crutches C(t = to) matches the

length of the standard, rigid crutches, and the initial axial

force exerted by the springs in the spring-loaded crutches

matches the initial axial force at the tips of the the standard

crutches at t = to. The force applied by both bilateral springs

is computed as

F = k(C − Cn) , (1)

where k is the spring stiffness and Cn is the neutral crutch

length for which the springs exert no force. As the initial

1477

force Fo = F (t = to) and the initial crutch length Co =C(t = to) are prescribed, it is possible to compute Cn for a

given value of k.

TABLE I

BOUNDARY CONDITIONS FOR THE SIMULATIONS. (∗ - ONLY FOR

SPRING-LOADED CRUTCHES)

to = 0.0 s tf = 0.7 s

σ (o) 1.50 -15.25Φ (o) -8.25 6.31θ (o) -32.0 freeσ (o/s) -28.0 free

Φ (o/s) free free

θ (o/s) 20.0 free∗C (m) 1.51 free∗C (m/s) 0.00 free

Shoulder angle δ(t) = Φ(t) − σ(t) was parameterized by

a 5th-order polynomial and incorporated to both models as

rheonomic constraints, Eq. 4 in Appendix. In this approach,

shoulder moment can be computed as the reaction moment at

the shoulder joint that ensures the shoulder motion constraint

is satisfied. The polynomial coefficients are included in

the vector of optimization parameters and a best fit to the

experimental data by [8] was performed to find appropriate

initial guesses for the optimization, refer to Tab. V in

Appendix. The 5th-order polynomial provided a good fit of

the experimental data. For the simulations of the gait with

spring-loaded crutches the spring stiffness k is added to the

vector of optimization parameters.

Two performance indexes were considered. One represents

a measure of the subject’s “effort” required to perform the

movement and is assumed to be related to the average of the

squared shoulder moment τ over the simulation as

Js =1

T

∫ tf

to

τ2(t) dt , (2)

where the swing-phase duration is T = tf − to = 0.7 s.

The other considered performance index is related to the

axial (aligned to the longitudinal axis of the crutch) and

the transversal components of the force transmitted to the

shoulder joints by the crutches, fa and ft, respectively, as

Jf =1

T

∫ tf

to

f2

t (t) dt +1

T

∫ tf

to

(fa(t) − fa,o)2 dt , (3)

where fa,o is the initial axial force component at t = to. The

addition of the offset given by the constant fa,o in the second

term of Jf inhibits oscillations and avoids convergence to

unconventional patterns. With this offset the optimal axial

force history would be fa(t) = fa,o rather than fa(t) close

to zero which is unrealistic. The optimization problem was

solved for three different cost functions: J = Js, J = Jf

and J = Js + wJf .

Therefore, the resulting optimization problem consists of

finding optimal polynomial coefficients for the motion of the

shoulder joint and optimal stiffness values for the spring-

loaded crutches that satisfy the equations of motion and the

constraints given by the boundary conditions in Tab. I, and

minimize the cost function given by the performance index

in Eq. 2, J = Js, or by the performance index in Eq. 3,

J = Jf , or by a combination of both, J = Js + wJf ,

where the choice of w will be discussed in the next section.

The optimization problem was solved using the interior-point

algorithm available in the Matlab’s routine fmincon.m.

III. RESULTS AND DISCUSSION

The three simulation results for the rigid, standard crutches

are shown in Tab. II for the three considered cost functions:

J = Jf which is related to the shoulder forces transmitted

by the crutches, Eq. 3; J = Js which is related to the

“effort” quantified by the shoulder moment, Eq. 2; and

J = Js + Jf/10 which is a combination of Jf and Js

and considers both, transmitted shoulder forces and applied

shoulder moment over the swing phase. The cost function

J = Js + Jf/10 with w = 1/10 was considered a suitable

combination of both performance indices. The rationale

behind this choice will be clarified further on. Information on

Tab. II includes values for the performance indices, maximal

crutch longitudinal component of the force transmitted by

the crutch to the shoulder, maximal shoulder moment, and,

respectively, horizontal and vertical speed of the body’s

center of mass at final time, t = 0.7 s. The corresponding

time series of the simulated shoulder axial force and moment

for each simulation are shown in Fig. 3.

TABLE II

OPTIMIZATION RESULTS FOR THE GAIT WITH RIGID, STANDARD

CRUTCHES.

J = Jf Js + Jf /10 Js

J 7753 1005 27.02Jf (Eq. 3) 7753 9280 10411Js (Eq. 2) 862.6 77.53 27.02max axial force (N) 836.1 863.2 869.7max shoulder moment (N.m) 98.16 16.72 11.86final CM hor. speed (m/s) 1.289 1.289 1.287final CM vertical speed (m/s) -0.318 -0.358 -0.373

Notice that the variation of performance index is consistent

with the choice of cost function, with minimization of

J = Jf reducing the value of the index Jf and leading

to lower magnitudes of axial force, and minimization of

J = Js reducing the value of index Js and leading to lower

magnitudes of the shoulder moment, Tab. II and Fig. 3. The

combined cost function J = Js + Jf/10 leads, as expected,

to intermediate results.

Note that the rather modest decrease in maximal axial

forces obtained by minimizing Jf requires very large shoul-

der moments peaking at unrealistic 100 Nm. This observation

indicates that minimizing shoulder forces alone leads to

unrealistic requirements and J = Jf is probably not the

most appropriate cost function. On the other hand, minimiz-

ing J = Js leads to low shoulder moment values while

causing a rather modest increase in shoulder forces.This

would make J = Js a good cost function candidate, but

1478

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7550

600

650

700

750

800

850

900S

ho

uld

er

Axia

l F

orc

e (

N)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−50

0

50

100

Time (s)

Sh

ou

lde

r M

om

en

t (N

m)

Jf

Js + Jf/10

Js

Fig. 3. Simulated shoulder forces on the top and shoulder moments onthe bottom for the gait with rigid, standard crutches and the three differentcost functions.

a closer inspection of the simulation results shows that the

predicted patterns are closer to the experimentally measured

by [8] when minimizing Jf in opposition to Js. This shows

the necessity of a combined cost function that takes both

performance indices into account and leads to patterns that

adhere to experimentally observed while requiring realistic

shoulder moments. This is achieved by the combination

J = Js + Jf/10, which approximates reasonably well the

experimental results by [8] as indicated in Fig. 4, and requires

low values of shoulder moment. For these reason, for the

simulations with spring-loaded crutches emphasis will be

given to simulations results obtained with this cost function.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−40

−30

−20

−10

0

10

20

30

Time (s)

Jo

int

An

gle

(º)

σ

θ

δ

Fig. 4. Simulated kinematics of gait with standard crutches using J =Js + Jf /10 as cost function, solid lines, compared to experimental datain [8], markers, where δ = Φ − σ.

In order to study the spring-loaded crutches a series of

simulations was performed with values of the combined

stiffness k of both bilateral crutches varying from k =1 kN/m to k = 100 kN/m. This range of stiffness was

chosen to cover from very compliant to almost rigid crutches.

Fig. 5 shows the optimal cost function value as a function of

stiffness k for the three considered cost functions. The dotted,

horizontal lines show the optimal cost function value for the

rigid, standard crutch when using the same cost function.

The first observation is that stiffness values close to the

lower bound of k = 1 kN/m lead to cost function values

much lower than the one obtained for the rigid crutch,

suggesting the corresponding spring-loaded crutches would

lead to much lower shoulder moments and forces. This is

consistent with the findings by [12], who, by using a rather

simple model of crutch gait, conclude that very low stiffness

crutches with stiffness of 2 kN/m to 4 kN/m are advan-

tageous. However, the simulations here show clearly that

stiffness values lower than about 10 kN/m can dramatically

compromise ground clearance, as shown by the stick figure

on the top of Fig. 6. In fact, a few simulations performed with

additional constraints imposing foot clearance led to much

larger values of the cost functions and dramatically worse

performance as vigorous correction action are required by

the user to prevent foot-ground collision.

0 2 4 6 8 10

x 104

5000

10000

Jf

0 2 4 6 8 10

x 104

1000

2000

Jf/10

+J

s

0 2 4 6 8 10

x 104

20

40

k (N/m)

Js

Fig. 5. Optimal cost function value as function of spring-loaded crutchstiffness for the three considered cost functions. Data was obtained in aseries of optimizations with fixed spring stiffness k. The circles indicate thelocation of the best local optima for each cost function other than the onesat the extremes, for k = 1 kN/m and k = 100 kN/m. Note that shownstiffness is the combined stiffness of the two bilateral crutches.

Another important observation in Fig. 5, more markedly

for J = Js + Jf/10 and J = Js, is the existence of a

stiffness range, from about k = 5 kN/m to k = 20 kN/m

for which performance deteriorate substantially compared

to performance of rigid, standard crutches shown by dotted

lines. Refer, for example, to the solution for k = 10 kN/m

and J = Js+Jf/10 in Fig. 7 which shows modest increase in

shoulder force and substantial increase in shoulder moment.

The conclusion is that this range of stiffness should be

avoided in the design of spring-loaded crutches as well.

The results obtained for the cost function J = Js +Jf/10

1479

evidence that there is a large range of stiffness, from mod-

erately low with combined stiffness of k ≈ 25 kN/m until

almost rigid, that leads to performance similar to standard

crutches as the cost function values remain close to the one

for rigid crutches, Fig. 5, and the shoulder moments and

forces are similar to the ones observed for rigid crutches,

Fig. 7. In order to better compare the performance of spring-

loaded crutches with the one for rigid, standard crutches,

the local optimum marked by circles in Fig. 5 are selected

and the corresponding simulation results are summarized in

Tab. 5.

TABLE III

OPTIMIZATION RESULTS FOR THE GAIT WITH SPRING-LOADED

CRUTCHES. SHOWN RESULTS ARE FOR LOCAL OPTIMA INDICATED BY

CIRCLES IN FIG. 5

J = Jf Js + Jf /10 Js

J 8584 1029 26.08Jf (Eq. 3) 8584 9587 10835Js (Eq. 2) 4592 70.55 26.08max axial force (N) 846.9 870.0 873.6max shoulder moment (N.m) 269.3 12.94 11.61final CM hor. speed (m/s) 1.427 1.453 1.401final CM vertical speed (m/s) -0.374 -0.302 -0.329kopt (kN/m) 4.80 67.27 58.17

At the first sight, results suggest that spring-loaded

crutches do not offer any significant benefit over standard,

rigid crutches, at least in the swing phase of the “swing-

through” gait, but a closer inspection shows a 10% increase

in the horizontal speed of the body’s center of mass (CM)

at the end of the swing phase for t = 0.7 s. This indicates

greater momentum and kinetic energy available at the end of

the swing phase and would arguably assist body progression

and reduce push off effort in the subsequent double stance

phase. This finding is consistent with experimental observa-

tions by [17]. Furthermore, a 20% decrease in the vertical,

downward speed of the body’ CM can be observed, which

could be translated into a reduction of the vertical impact

force at touch down resulting in improved gait quality.

The results for spring-loaded crutches, which look some-

what deceptive at first sight, should be interpreted carefully.

The simulations in this study are restricted to the swing-phase

and the boundary conditions are the ones measured for the

gait with rigid, standard crutches. In reality, the whole gait

cycle, not only the swing phase, would adapt to the spring-

loaded crutches, and this would certainly alter the bound-

ary conditions of the swing-phase and could bring further

benefits. Furthermore, the energy saving potential of spring-

loaded crutches would probably only become apparent in the

adjacent double stance phases where most energy is stored in

and released of the spring-loaded crutches. Avoiding energy

dissipation in theses phases was the reason the addition of

a damper was not thoroughly investigated, although a few

simulations have shown a certain amount of damping can

reduce shoulder vertical forces in the swing phase. Finally,

and perhaps most importantly, the certain reduction in impact

forces provided by compliant crutches is not taken into

−1 0 1

0

0.5

1

1.5

k = 1 kN/m

−1 0 1

0

0.5

1

1.5

k = 10 kN/m

−1 0 1

0

0.5

1

1.5

k = 60 kN/m

Fig. 6. Stick-figures illustrating simulated optimal kinematics for costfunction J = Js+Jf /10 and three different stiffness values of the bilateralspring-loaded crutches, k = 1 kN/m, k = 10 kN/m, and k = 60 kN/m,from top to bottom.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7600

700

800

900

1000

Sh

ou

lde

r A

xia

l F

orc

e (

N)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

20

40

60

Time (s)

Sh

ou

lde

r M

om

en

t (N

m)

10kN/m

30kN/m

60kN/m

Rigid

Fig. 7. Simulated shoulder forces on the top and shoulder moments onthe bottom for gait with spring-loaded crutches and cost function J =Js +Jf /10, for four different stiffness values: k = 1 kN/m, k = 10 kN/m,k = 60 kN/m, and k → ∞ (rigid, standard crutch).

1480

account as the model does not cover phase transitions. All

these are important limitations of the model we adopt in this

study and indicate spring-loaded crutches can still prove very

advantageous. However, the authors believe that an important

contribution of this study is determining safe ranges of

spring stiffness which do not deteriorate performance in the

swing phase and could therefore be safely used to improve

performance in other phases of the gait.

IV. CONCLUSIONS

This paper proposes a realistic model of the swing phase

of the “swing-through” gait with standard and spring-loaded

crutches which considers trunk, legs and crutches motion and

allows for assessment of foot clearance. The model was used

in a series of simulations obtained by minimization either of

shoulder moments, shoulder forces or both. A combination

of both indices was proposed which led to adherence to

experimental data and realistic shoulder moments.

The simulations with spring-loaded crutches show that

combined bilateral crutches’ stiffness lower than about

25 kN/m deteriorate performance in the swing phase ei-

ther by compromising ground clearance or by significantly

increasing shoulder moments and forces. Stiffness values

greater than 25 kN/m, in turn, led to performance similar to

standard, rigid crutches, except for final CM speed patterns.

Simulations show a 10% increase in forward, horizontal

speed and a 20% decrease in downward speed, which can

improve gait quality by assisting forward progression at the

subsequent push-off and reducing impact forces at touch

down, respectively.

Apparently, spring-loaded crutches provide less striking

advantages than one would expect, but it should be stressed

that the model adopted here does not consider the adjacent

double stance phases, which are the phases where most en-

ergy storage and release in the elastic element take place and

energetic gains would probably be much more prominent.

Furthermore, the model does not include the collision at the

beginning of the first double stance phase, where a compliant

crutch would certainly reduce impact forces transmitted to

the upper limbs. Nevertheless, this study indicates a safe

range of stiffness which do not deteriorate performance in the

swing phase of gait and yet could bring substantial gains in

other phases. The development of a whole-gait cycle model

of the gait with crutches is the focus of ongoing research.

APPENDIX

A best fit to experimentally measured shoulder angle δ(t)by [8] was performed to determine optimal coefficients of

the 5th-order polynomial,

δ(t) = a5t5 + a4t

4 + a3

3+ a2t

2 + a1t + a0 , (4)

where t is time. The obtained values for the coefficients are

shown in Tab. V. These values were used as initial guess for

the coefficients in all optimizations.

TABLE IV

MODEL PARAMETERS SELECTED FROM [8] AND [14].

trunk length 0.51 mstandard crutch length 1.51 mleg length 0.97 mdistance crutch tip and crutch’s CM 1.04 mdistance shoulder joint and trunk’s CM 0.17 mdistance hip and leg’s CM 0.42 mmass of the two crutches 13.8 kgmass of the trunk 51.3 kgmass of both legs 32.5 kg

moment of inertia of both crutches about CM 2 kg.m2

moment of inertia of the trunk about CM 3.6 kg.m2

moment of inertia of both legs about CM 2.4 kg.m2

TABLE V

COEFFICIENTS FOR POLYNOMIAL OF EQ. 4 OBTAINED BY BEST FIT TO

EXPERIMENTAL DATA FROM [8], WITH δ(t) IN RADIANS.

a5 a4 a3 a2 a1 a0

-22.64 33.95 -13.82 2.04 -0.087 -0.170

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