Minimising the User’s Effort during Wheelchair Propulsion using an Optimal Control Problem

Abstract

This paper proposes a study of the optimal control problem with state constraint, using two types of a power-assist wheelchair propulsion. The cost function is given by the metabolic function, which represented by acompromise between the work exerted by the joints muscles (mechanical effect) and an efficiency functionthat converts chemical into mechanical energy (biomechanical effect). The dynamic wheelchair is given by asimple model, which connects the push force to the wheelchair speed. An upper bound constraint is consideredin order to limit the energy consumed by the motor. This study used an approach that calls the Pontryagin’smaximum principle, the optimal solution varies with the parameters of the problem. Finally, a numericalcomparison is enabled using two types of assistance: constant and proportional. This numerical comparisonis based on the framework of the optimal control theory with two different costs. The first cost is given by theintegral of the squared user’s force and the second by the integral of the metabolic function. This Numericalresults show that the user provides less effort with metabolic cost than with the energy user’s force.

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Full text

Minimising the user’s effort during wheelchair propulsion using an
optimal control problem
Ouazna OUKACHA1 a, Chouki SENTOUH1 b and Philippe PUDLO1 c
1University Polytechnique Hauts-de-France, CNRS, UMR 8201-LAMIH, F-59313 Valenciennes, France
{Ouazna.Oukacha, Chouki.Sentouh, Philippe.Pudlo}@uphf.fr
Keywords: Power-assist wheelchair propulsion, optimal control, biomechanical, metabolic energy cost, efficiency,
optimal strategy.
Abstract: This paper proposes a study of the optimal control problem with state constraint, using two types of a power-
assist wheelchair propulsion. The cost function is given by the metabolic function, which represented by a
compromise between the work exerted by the joints muscles (mechanical effect) and an efficiency function
that converts chemical into mechanical energy (biomechanical effect). The dynamic wheelchair is given by a
simple model, which connects the push force to the wheelchair speed. An upper bound constraint is considered
in order to limit the energy consumed by the motor. This study used an approach that calls the Pontryagin’s
maximum principle, the optimal solution varies with the parameters of the problem. Finally, a numerical
comparison is enabled using two types of assistance: constant and proportional. This numerical comparison
is based on the framework of the optimal control theory with two different costs. The first cost is given by the
integral of the squared user’s force and the second by the integral of the metabolic function. This Numerical
results show that the user provides less effort with metabolic cost than with the energy user’s force.
1 INTRODUCTION
In the long term, the manual wheelchair user can
cause upper limb pain and degeneration. Most of the
users have difficulty to avoid an obstacle or change
direction. In order to decrease the human suffer-
ing caused by this activity, many working tasks are
performed manual wheelchair level (motor, frame
structure, etc.) (Pezzuti et al., 2006). In addition,
many researches are made in biomechanical (Hori-
uchi et al., 2014) and (Luhtanen et al., 1987). This pa-
per proposes a human-machine interaction and within
this framework, a power assist manual wheelchair
propulsion is achieved using an optimal control prob-
lem (Cooper et al., 2002) and (Cuerva et al., 2016).
Usually, the model of optimal control problem in-
spired from (Oukacha and Boizot, 2020), altering the
energy of the motor vehicle by the effort required for
the user in a manual wheelchair propulsion. The mo-
tor energy takes into account as an upper bound con-
straint. It is then a optimal control problem with state
constraint.
ahttps://orcid.org/00000-003-3669-8124
bhttps://orcid.org/0000-0003-1548-9665
chttps://orcid.org/0000-0002-0339-7336
In order to reduce these efforts, a metabolic cost
function, also called ”biomechanical metabolic” is
used. This metabolic function is the quantity of
energy consumed by a person during a muscular
activity (Ardigo et al., 2005) and (Horiuchi et al.,
2014). Generally, this function measured the amount
of oxygen used by the muscles. During the muscle
contraction, the cells use the ATP as an energy source,
which is produced by hydrolysis of ATP to ADP
(chimical energy). In the literature, the metabolic
cost function can be expressed mathematically,
thanks to the tests made by different scientists and
it has several forms. In reference to (Ardigo et al.,
2005) and (Horiuchi et al., 2014), this function could
be formulated as a second degree equation, which
depend only of the linear speed of the wheelchair.
As stated above, this metabolic function could be
designated as the energy expended during a manual
wheelchair propulsion. This function measures the
mean of the Oxygen uptake and the carbon dioxide
output (Yang et al., 2009). Since the effort provides
by a person in a manual wheelchair propulsion,
depends on his/her upper limb ability, this function
may also be presented by all the joint moments of the
shoulder, elbow and wrist (Rozendaal et al., 2003).
Finally, the metabolic function can be generated

under the efficiency function.
In the works cited above, the metabolic function is
achieved through the analysis of experimental data. In
this article, the metabolic function value is calculated
by solving an optimal control problem. Moreover,
we interested more specifically in this latter function.
Thus, the metabolic function is given by a compro-
mise between the work of the push force and effi-
ciency function, which is only related to the linear
speed of the wheelchair (Cooper, 1990b). There are
several different types of the efficiency: gross, net,
true, etc. as described in (Hintzy et al., 2002) and
(Luhtanen et al., 1987). In this work, we will fo-
cus on the gross efficiency, contrary of what is done
in (Cooper, 1990a), where it studied an optimal con-
trol problem using the net efficiency that appears in
the dynamical model. The advantage of using the
gross efficiency is that it allows to take into account
the energy expended at rest.
In this paper, we tackle an optimal control prob-
lem with two approaches: an analytical study and a
numerical resolution. First, a general study is per-
formed with the help of Pontryagin’s Maximum Prin-
ciple (PMP) (Pontryagin et al., 1962). The PMP is
a very effective approach, where these theoretical re-
sults obtained that fits perfectly with the experimental
tests. For example, the article (Berret et al., 2008)
deals an optimal control problem in biomechanical,
where the authors have shown that the optimal solu-
tion obtained by the PMP similar to the experimen-
tal results. In the second step, a numerical method
is used to provide a solution to a complex optimal
control problem. The numerical simulations uses real
data, for example, the efficiency function model is re-
trieved in (Cooper et al., 2002). This function for-
mulated through experimental tests and the curve ef-
ficiency profile according to (Ardigo et al., 2005).
In addition, a comparison is made between the op-
timal control problem given in (Cuerva et al., 2016)
and a new problem which is will be presented. In the
previous paper, a comparison of three different types
of power-assist wheelchair propulsion is made using
an optimal control problem.
The section 2 is devoted to a more detailed de-
scription of the optimal control problem, the model
dynamic wheelchair and the running cost. Section 3
is dedicated to the presentation of the approach solv-
ing the optimal control problem. In the section 4,
we present the numerical results of each power-assist
wheelchair propulsion. The sections 5 discusses the
obtained results. Finally, the section 6 presents the
conclusion and perspectives of this study.
2 MODEL OF THE OPTIMAL
CONTROL PROBLEM
In this optimal control problem with state con-
straint, we will focus particularly on the metabolic
cost function which is generalized by an efficiency
function. The dynamic wheelchair-user system is
given by the Netwon’s second law of motion.
2.1 Dynamic wheelchair propulsion
The equations of the motion wheelchair-user model
are determined by a first order dynamic system. As-
sume that air resistance is negligible and the rolling
resistance of the wheelchair-user system propelled at
linear speed in a straight line. Following the Fig-
ure 1, the movement of a person assisted by a manual
wheelchair is given:
¨x=1
MFp+Fm−sign(˙x)Fr−C˙x(1)
where, x, ˙x, ¨xare the longitudinal position, the lin-
ear velocity, the linear acceleration of wheelchair re-
spectively, Fpis the user’s force, Fmis the motor
force, Fris the rolling resistance force, Cis the vis-
cous damping coefficient and Mis the total mass
(user + wheelchair). The motor force is proportional
to the user’s force, since it acts of a power-assist
wheelchair. The control strategy is given by the Fp
and Fm=Fm(Fp).
Fp
Fm
Fr
C
x
Figure 1: Dynamic Model of the motion wheelchair-user
2.2 Metabolic cost function
The energy consumption of the body can be de-
fined by a metabolic function during the propulsion
wheelchair. The metabolic function to minimize, rep-
resenting as the ratio between the power of push
and biomechanical efficiency (Oukacha and Boizot,
2020):
J=P
ρ(2)

The power associated to the work Wis:
P=dW
dt =Fp˙x
The work performed during a muscular activity
is not constant, because the pace and the capacity
of the person change with time. The wrist motion
in handrim wheelchair propulsion can generate in
two stages: the concentric and eccentric muscle con-
tractions, which produced by the positive and nega-
tive work respectively (Williams, 1985). Therefore,
the work of the push can be described as a non-
differentiable function that is similar to the absolute
work. The absolute work of the propulsion force done
by a muscle during a period t0to Tfis:
W=
Tf
Z
t0
|Fp˙x|dt
The cost function which allows to measure the quan-
tity consumed to push the wheelchair from a starting
position at time zero to some final position at final
time Tf, is given by:
J=
Tf
Z
0
|Fp˙x|
ρ(˙x)dt (3)
According to (Cooper, 1990b), the biomechanical
efficiency can be formulated by the second equation,
which depends of the linear wheelchair velocity. The
curve profile obtained from this equation is illustrated
in Figure 2:
ρ(˙x) = 1
100 −0.55 ˙x2+7.02 ˙x+3.15
2 4 6 8
⋁
0.05
0.10
0.15
0.20
0.25
ρ(⋁)
Figure 2: Efficiency function model.
This Efficiency is calculated from the ratio be-
tween the power output and the power of the mea-
sured oxygen consumption.
2.3 Problem statement
An optimal control problem is formulated by the dy-
namic system similar to the one presented in (Cuerva
et al., 2016). The cost function is given by the
metabolic cost function presented above. Moreover,
a control bound and a fixed final time are considered
in order to satisfy the upper and lower bounds of the
user’s force and the duration of the motion. The prob-
lem is given by the following equations:



















































Minimise J=
Tf
R
0
|Fpv|
ρ(v)dt
Subject to
˙x=v;
˙v=1
MFp+Fm−Frsign(v)−C v
˙
E=|Fmv|
E≤W1
|˙v| ≤ 3
x(0) = 0
v(0) = 0!, x(Tf) = 10
v(Tf) = 0!
Lb ≤Fp≤Ub
Tf>0 : f ixed
(4)
Where W1=3
4Wis the energy consumed by the mo-
tor, when this maximum value (W) is calculated by
solving the optimisation problem (4), without user
interaction (Fm=0). Thus W=219.5J. A bound-
ary constraint (|˙v≤3|) is made regarding to the ac-
celeration ant the deceleration to ensure user’s com-
fort (Karmarkar et al., 2008).
Following (Berret et al., 2008), the problem (4)
admits at least one solution. According to (Oukacha
and Boizot, 2020), the optimal control problem (4)
is independent of the position x, the trajectories in-
cluding an arc with v<0 is not an optimal trajec-
tory. Thus, the speed is positive or null (v≥0) during
wheelchair propulsion, as suggested in (Cuerva et al.,
2016).
The optimal control problem (4) could be defined in
free final time. In this case, the time is an unknown
variable of this problem, we then have an additional
constraint.
3 Pontryagin’s maximum principle
A study is based on the Pontryagin’s Maximum
Principle (PMP) (Pontryagin et al., 1962), which
gives a necessary optimality condition of the optimal
control problem. Let us introduce λ= (λ1,λ2,λ3)the
adjoint vector of the state vector X= (x,v,E)and the
Hamiltonian function is defined by:

H(X,λ,Fp) = −|Fp|v
ρ(v)+λ1v+λ3|Fm(Fp)|v+µ(E−W1)
+λ2
1
MFp+Fm(Fp)−Fr−C v
Let (X∗,F∗
p)be an optimal solution, then the PMP as-
serts the existence of an absolutely continuous func-
tion λ:[0,Tf]→ℜ3and µis the Lagrange multiplier
of the constraint. The necessary optimality conditions
are as follows:
1. There exists t→µ(t)≥0, such that the adjoint
vectors satisfies:











˙
λ1=−∂H(X,λ,Fp)
∂x=0
˙
λ2=−∂H(X,λ,Fp)
∂v
=|Fp|
ρ(v)−|Fp|ρ0(v)v
ρ(v)2−λ1+λ2C
M−λ3|Fm(Fp)|
˙
λ3=−∂H(X,λ,Fp)
∂E=−µ
where ρ0(v) = ∂ρ(v)
∂v
2. The mapping t→µ(t)is continuous along the
boundary arc and verifies:
µ(t)(E(t)−W1) = 0,∀t∈[0,Tf]
3. H(X,λ,Fp)is constant, since the optimal control
problem (4) is autonomous. The optimal control
maximizes almost everywhere the Hamiltonian:
H=max
Fp
H(X,λ,Fp)
=λ1v−λ2
M(Fr+Cv) + µ(E−W1) + max
Fp
{φ(t)}
4. If the final time is free, then the Hamiltonian
H(X,λ,Fp) = 0 along the trajectory.
5. The candidate control strategy is given by:
Fp=argmax
Fp
{φ(t)}(5)
Where φ(t) = −|Fp|v
ρ(v)+λ3|Fm|v+λ2
MFp+Fm, is
called the switching function.
The optimal control strategy vanishes in terms the
Fm=Fm(Fp). In the section that follows, both assis-
tance will be present, where her solution varies be-
tween: bang-bang, inactivated and singular arc. An
extremal is a solution λof the above equations. A
portion of the trajectory is a bang type, when the con-
trol variable is equal to its maximum, or its minimum.
Atrajectory with inactivation is defined when the
control variable is null over a time interval. How-
ever, a singular arc is a portion of the trajectory along
which the control does not achieved its upper, as in
Figure 5.
The problem solving is based on the trape-
zoidal direct collocation method developing in Mat-
lab (Betts, 2010).
2
4
6
8
t
-1.5
-1.0
-0.5
0.5
1.0
1.5
FpHtL
Bang
Singular arc
Inactivation
Bang
Figure 3: Control strategy example with |Fp| ≤ 1
4 Numerical results
This part is devoted to present the results of nu-
merical simulation studies, using the constant and
proportional assistance. For all simulations, the
data are retrieved from the article (Cuerva et al.,
2016), their values are: M=110kg,Fr=8.9N,C=
4.6N s/m.
4.1 Constant assistance
The constant assistance is defined by a simple gain
(K1), which represents the assistance force provided
by the motor. The value of this gain is connected to
the user’s force (Fp), with respect to a threshold. An
approximation of the push force is given by a numer-
ical approach of the threshold. This push force (Fp) is
expressed as a hyperbolic tangent function:
Fm(Fp) = K1 tanh(Fp)(6)
The energy used by the motor is:
E(Fm,v) =
Tf
Z
0
|Fmv|dt =
Tf
Z
0
K1|tanh(Fp)|v dt (7)
We have |tanh(Fp)| ≤ 1, the equation (7) become:
E(Fm,v)≤
Tf
R
0
K1v dt =K1
Tf
R
0
dx
dt dt =K1
Tf
R
0
dx
Therefore, E(Fm,v)≤K1x(Tf)and we have
E(Fm,v)≤W1. In extreme case, the mo-
tor consumes E(Fm,v) = W1. Consequently:
E(Fm,v) = W1≤K1x(Tf) =⇒K1≥W1
x(Tf)=16.46
In this case, the switching function φ(t)is:
φ(t) = −|Fp|v
ρ(v)+K1λ3|tanh(Fp)|v+λ2
MFp+K1tanh(Fp)
They noted that the switching function has a quite
complex expression. Thus, an analytical study the
sign of this function is not easy. A numerical method
is used to solve the optimal control problem (4). Fig-
ures 4 shows some optimal solutions, which varied
according to the upper bound controls included.

Figure 4: Optimal trajectories with Lb ≥Fp≤Ubi,i=
1,.., 4 and U b1>U b2>U b3>U b4
Figure 5: Optimal control strategies with Lb ≥Fp≤
Ubi,i=1,.., 4 and U b1>U b2>U b3>U b4
In this case, the authors of the work (Cuerva
et al., 2016) solved the optimal control problem with
tanh(Fp)≈sign(Fp), since the latter problem does
not converge. In this order, a comparison is estab-
lished with the optimal control problem (4) and with
the same condition (cf. Figure 6). Therefore, the mo-
tor force and motor energy are becoming:
Fm(Fp) = K1 sign(Fp)(8)
E(Fm,v)≈
Tf
Z
0
K1v dt (9)
The energy consumed by the motor depends only
on the wheelchair speed. The gain value K1is ob-
tained from the equation (9), as follows:
E(Fm,v)≈
Tf
Z
0
K1v dt ≤3
4W=⇒K1
Tf
Z
0
dx ≤3
4W
Thus, K1≤3W
4x(Tf)=16.46. The switching function
φ(t)become:
φ(t) = −|Fp|v
ρ(v)+λ2
MFp+K1sign(Fp)
The last expression of the φ(t)of the form:
φ(t) = −K|Fp|+MFp∓N, where K=v
ρ(v),M=λ2
M
and N=λ2
MK1. As the before case, an analytical study
of this switching function is not simple.
Figure 6: Comparison between the problem (Cuerva et al.,
2016) (Ref) and problem (4) (New) with tanh(Fp)≈
sign(Fp)for the constant assistance.
In this case, the goal is to compare the problem
posed in (4) without and with tanh(Fp)≈sign(Fp).
For the same initial conditions, the optimal solutions
are illustrated by the Figures 6.
Figure 7: Comparison between the problem (4) without and
with tanh(Fp)≈sign(Fp)for the constant assistance.
4.2 Proportional assistance
This kind of assistance is inspired from the work
of (Cooper et al., 2002), where the motor force (Fm)

is given by the use’s force (Fp) multiplied by a gain
(K2) (cf. equation (10)):
Fm(Fp) = K2Fp(10)
The energy generated by the motor during the ac-
celeration and deceleration phases is given by:
E(Fm,v) =
Tf
Z
0
|Fmv|dt
=
Tf
Z
0
K2
K2+1M|˙v|+sign(v)Fr+C vv dt
The gain value K2is calculated with the same prin-
ciple adapted to the previous assistance study, then
K2=1.87. In the case of the proportional assistance,
the switching function φ(t)is given by:
φ(t) = −v
ρ(v)−K2λ3v|Fp|+λ2
M(1+K2)Fp
The last function is of the form:
φ(t) = G(t)|Fp(t)|+L(t)Fp
Where G=v
ρ(v)−K2λ3vand L=λ2
M(1+K2).
The maximisation condition of the con-
trol (Oukacha and Boizot, 2020):
Fp=argmax
Ub≤Fp≤Lb
{φ(t)}
=








Ub if L>G(bang)
Fp∈[0,Ub]if L=G(singular)
0 if −G<L<G(inactivation)
Fp∈[Lb,0]if L=−G(singular)
Lb if L<−G(bang)
Figure 8: Comparison between the problem (Cuerva et al.,
2016) and problem (4) for the proportional assistance.
Cost function value
Table 1 summarizes the results of the cost function for
the different strategies of the control, with the two as-
sistances.
Table 1: Comparison betwen the cost functions.
Cost Constant assist. Proportional assist.
JEnergy 4201.06 2115.98
JMetabolic 1208.84 —
JSign
Metabol ic 1137.22 727.48
JUB1
Metabol ic 775.47 —
JUB2
Metabol ic 984.51 —
JUB3
Metabol ic 1019.22 —
JUB4
Metabol ic 1208.84 —
Where JEnergy and JSign
Metabol ic are the costs correspond-
ing to (Cuerva et al., 2016) and the optimal control
problem (4) with t anh(Fp)≈sign(Fp), for each one
of these two assistance (Figures 6 and 8 respectively).
JMetabolic ,JU b1
Metabol ic,JU b2
Metabol ic,JU b3
Metabol ic and JU b4
Metabol ic
are the costs to the optimal strategies of the prob-
lem (4), for the constant assistance. These costs as-
sociated to Figures 7 and 4 (or Figure 5) respectively.
One notes that the both cost functions JMetabolic and
JLb4
Metabol ic represented the same control strategy.
5 Discussion
The Figures 4, 6, 7 and 8 present the optimal tra-
jectories: position, velocities, user’s force and the mo-
tor force in this order.
The Figure 4 represents the solving of the optimal
control problem (4), with the constant assistance.
The optimal strategy (Figure 5) varies to the upper
bound of the user’s force. All control strategies
started with the value of the upper bound, then its
are gradually decreasing, except for the green curve
(c.f. Figure 5 with Ub2) which pushes to the limit
and then decreased. The positions and the velocities
have a very similar profile. The peaks of user’s force
occur almost at the same time. When the higher the
applied force a small cost function value, as observed
in Table 1, i.e. For these upper bounds of the control
arranged in this sequence: U b1>U b2>U b3>U b4,
these costs functions are classified in the inverse or-
der: JUb1
Metabol ic <JU b2
Metabol ic <JU b3
Metabol ic <JU b4
Metabol ic.
Figures 6 and 8 illustrate a comparison between
the problem described in (Cuerva et al., 2016) and the
optimal control problem posed in (4). The optimal so-
lution of each problem is indicated by the indices En-
ergy and Metabolic respectively. For each assistance

(constant and proportional), the both Figures 6 and 8,
generated a control strategy, which starts and ends at
the same values. In other words, we presume that the
users have the same maximum force in the handrim
wheelchair propulsion. For the constant assistance,
the position trajectories are identical, but the veloc-
ity profile is just similar at the beginning and end and
not achieving the maximum value. The middle part
corresponds more or less to singular arc, which is not
constant over time (Figure 5). The optimal strategy is
composed of bang-bang and singular arc. The results
also noted that the two peaks of user’s force happen-
ing at the same time, as shown in the graph of the
motor force. Because, the switching time of the both
control are produced simultaneously. In the propor-
tional assistance, the position trajectories are almost
identical. In addition, the velocity profiles are identi-
cal at the beginning and the end. Thus, a part of the
trajectory, which corresponds to the singular arc of
the control, is composed of: bang-bang, inactivation,
singular arc. In the both assistance, the cost function
value given by the optimal control problem (4) is less
than that the one achieved by (Cuerva et al., 2016)
with three methods (c.f. Table 1). This can be ex-
plained by the presence of a period where the user’
force is zero (inactivation period in the proportional
assistance) or almost zero (singular arc in the con-
stant assistance) on the time interval. Minimising a
function of this type implies the presence of inacti-
vation period. This phenomena takes place because
the work of both agonistic and antagonistic muscles
acting on a joint during rapid motion (Berret et al.,
2008), which produced by the hydrolysis of ATP to
ADP. During the inactivation period, the user’s force
applied at each joint is null. Therefore, the cost func-
tion is also equal to zero during this inactivation pe-
riod.
Figure 7 represent a comparison for optimal
control problem (4), without and with tanh(Fp) =
sign(Fp). As we can see, all the curves overlaid
in each case, which is confirmed by the cost func-
tions, since the two strategies expend almost the same
amount of energy (Table 1).
6 Conclusion
This paper addresses an optimal control prob-
lem with minimal user effort during the propul-
sion wheelchair. The optimal control problem with
state constraint is formulated using a power assisted
wheelchair propulsion. The assistance represented
a human-machine interaction, where a cooperation
between the motor force and the user’s force is en-
abled. Both assistance are described: constant and
proportional. The optimal control problem with state
constraint processed by the Pontryagin’s Maximum
Principle and then a numerical resolution is achieved
when this problem is complex. The cost function is
given by the metabolic function, which is a compro-
mise between the absolute work of the user’s force
and his/her performance in the wheelchair. Finally,
a comparison is established between two costs func-
tions using this optimal control problem.
The numerical simulations show that the proposed
metabolic cost function reduces the physical effort in
comparison with the energy of user’s force. All strate-
gies obtained by solving the optimal control prob-
lem (4) costs less that the other three approaches pre-
sented in (Cuerva et al., 2016), for each assistance.
The non-differentiable of this function allows to pro-
duce a period when the user’s force is null.
Contrary to (Oukacha and Boizot, 2020), this
work also includes singular arcs non-constants and
the control strategy vanishes in the terms of the motor
force model, which is proportional to the user’s force
applied on the handrim wheelchair propulsion.
The work is realized in QBA framework
project (Bentaleb et al., 2019). This paper presented
initial simulation results of the biomechanical part.
The future work is to develop this model in order to
realize the experimental part in the PSCHITT-PMR
platform.
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