1 
 
A General Optimal Control Model of Human Movement Patterns 
I:  Walking Gait 
 
Stuart Hagler 
Oregon Health & Science University 
Portland, OR, USA 
haglers@ohsu.edu 
 
Abstract:  The biomechanics of the human body gives subjects a high degree of freedom in how they can 
execute movement.  Nevertheless, subjects exhibit regularity in their movement patterns.  One way to 
account for this regularity is to suppose that subjects select movement trajectories that are optimal in some 
sense.  We adopt the principle that human movements are optimal and develop a general model for human 
movement patters that uses variational methods in the form of optimal control theory to calculate 
trajectories of movement trajectories of the body.  We find that in this approach a constant of the motion 
that arises from the model and which plays a role in the optimal control model that is analogous to the role 
that the mechanical energy plays in classical physics.  We illustrate how this approach works in practice by 
using it to develop a model of walking gait, making all the derivations and calculations in detail.  We finally 
show that this optimal control model of walking gait recovers in an appropriate limit an existing model of 
walking gait which has been shown to provide good estimates of many observed characteristics of walking 
gait. 
1 Introduction 
The biomechanics of the human body give a subject a high degree of freedom in how a movement may 
be executed.  Nevertheless, subjects exhibit great regularity in their movement patterns.  One way to 
account for this observed regularity is to suppose that subjects regularly choose movements that are optimal 
in some rigorous sense.  Variational methods can generate movement trajectories by finding the trajectory 
that is optimal in the sense of minimizing some measure of cost.  These methods are a standard part of the 
toolkit physicists use calculate physical trajectories in classical mechanics.  In this paper, we adopt a 
variational approach to creating models of human movement that generate movement trajectories body.  
We do this by generalizing the kinds of variational methods used by physicists by using the mathematics 
of optimal control theory to calculate trajectories for movements of the body.  The aim of the optimal 
control approach to modeling human movements is to produce a model that can generate trajectories for 
arbitrary movements of the human body.  We suppose that all human movements are variations on a 
common motor control process only differing in the part of the body being used and the goals being met.  
We therefore believe that we should study human movements beginning with a modeling approach which 
can be used to construct the trajectories for any human movement and then look at how the model may 
need to be modified to correctly describe specific movements.  However, as it makes things clearer to 
illustrate the optimal control modeling approach using a specific movement, we illustrate the approach in 
this paper using walking gait. 
We can think of the problem of the optimal control of the human body in the following terms.  The 
human body is a skeleton that is moved by the actions of muscles generating forces at points on the skeleton.  
 
 
2 
 
A movement takes the skeleton from one pose into another over some interval of time.  We can identify a 
small number of points on the body as being the points whose trajectories are controlled during the 
movement and constrain the rest of the body so that it must move in some way that is consistent with the 
motion of the controlled points as in [1-5].  The points themselves are controlled by setting the n-th time 
derivative of their position to a sequence of arbitrary values throughout the time interval in which the 
movement occurs; these n-th time derivatives of the positions are the controls of the body.  We choose the 
sequences of values for the controls by finding those that generate a movement of the skeleton that 
minimizes some measure of cost associated with the movement. 
One choice of control is the jerk (the third time-derivative of the position).  It has been invoked as a 
general principle for understanding human movements and has been used as the basis for calculating 
movement trajectories. [6-11]  We have previously used optimal control models built on jerk-control to 
describe vertical movements of the torso in the course of a simple squatting exercise [12] and movements of 
the hand to control a computer mouse. [13]  In constructing the optimal jerk-control model of the computer 
mouse in [13] we developed the mathematical apparatus of optimal jerk-control models in detail.  The choice 
of the jerk as the control means that while the jerk can be set to an arbitrary sequence of values over the 
interval in which the movement occurs, the position, velocity, and acceleration are continuous functions of 
time.  Looking to Newton’s second law, we see that having the acceleration be continuous in time means 
that the forces acting on the controlled points of the body must be continuous in time.  Therefore, choosing 
jerk as the control implicitly means constructing a model in which the forces generated by the muscles are 
continuous in time.  Again, looking to Newton’s second law, we see that controlling the jerk corresponds to 
controlling the yank (the first time-derivative of the force) of the muscles.  Thus, a model in which the jerk 
is controlled looks like a more conventional optimal control model in which the yank is controlled when the 
model is reformulated in terms of the muscle forces. 
The key component of an optimal control model is the cost functional which assigns a cost to every 
possible movement trajectory.  The cost functional contains the body of physical theory that describes the 
system while finding the solution that minimizes the cost functional is simply a matter of established 
mathematics.  We must therefore be clear about how the cost functional is constructed in the model we 
develop.  Our approach is to begin with what we take to be the most unobjectionable supposition about 
human movement – namely, that, all other things being equal, subjects will move in a way that minimizes 
the metabolic energy expended in executing the movement.  In [14-16], we constructed and validated a 
model that estimates the metabolic energy expended in executing a walking gait.  This model estimates the 
metabolic energy using the values of the muscle forces required to execute the walking gait.  However, the 
model in [14-16] only uses the forces and not the yanks, and we have argued that the optimal control model 
should be formulated in terms of control of the yanks.  We therefore associate a cost with the yanks of a 
movement and introduce terms in the yank that are mathematically convenient for the calculation of the 
movement trajectories. 
In this paper, we illustrate the general optimal control approach to modeling human movement patterns 
using walking gait.  In [14-16], we constructed a simple biomechanical model of walking gait using the 
principle of movement optimality that could account for several characteristics of walking gait, namely:  (i) 
the metabolic energy expenditure per step for the range of avg. walking speeds and avg. step lengths 
observed by Atzler & Herbst, [17] (ii)  the regular pattern of combinations of avg. walking speed and avg. 
step length selected by subjects as observed by Grieve, [18] (iii) the lower limit of avg. walking speeds for 
very slow walking gaits selected by pre-rehabilitation Parkinson’s disease subjects observed by Frazzitta et 
 
 
3 
 
al. [19] and for very slow walking gaits by older adults by Hagler et al., [20] and (iv) the functional forms 
of measures of the energy efficiency across a range of walking gaits observed by Donovan & Brooks. [21]  
Since this model moves the body using fixed trajectories, it only describes walking gait down to scales of 
about a step.  As the optimal control model of walking gait that we construct in this paper generates 
movement trajectories, we expect that it provides a means of extending the model in [14-16] to a model 
able to describe walking gait at scales below that of a step.  We therefore take an interest in wedding the 
model in [14-16] to that constructed here both to show that we can extend that model and to show that we 
can recover established results for walking gait from [14-16] using this model.  We do this by showing that 
we can recover the model in [14-16] as a limiting case of the optimal control model of walking gait. 
This paper is structured as follows.  We first provide some definitions and summarize the modeling of 
movement and metabolic energy we developed for walking gait in [14-16], show how we, in general, propose 
to construct cost functionals to describe human movements using optimal control theory, and outline the 
mathematics used to derive movement trajectories from these cost functionals (Sec. 2).  We construct the 
optimal control model for walking gait by generalizing the model that we used in [14-16] from specified to 
arbitrary trajectories of movement for skeleton, plugging the resulting model into a cost functional, and 
finding the optimal trajectories (Sec. 3).  We finally show that the trajectories of the torso and swing foot 
in the optimal control model can be made to approximate those used in the model in [14-16] in the case of 
symmetric and uniform walking gait in the limit of low yanks (Sec. 4).  We therefore find that we can 
generalize the model developed in [14-16] to scales below that of a step using an optimal control model. 
2 Optimal Control Model 
We begin by outlining the optimal control approach to modeling human movements that we adopt in 
this paper.  We first define some convenient terminology and mathematical notation.  We then review the 
model that we developed in [14] to describe movements of the body; it is composed of three parts:  (i) the 
segment model that gives the skeleton that we use to describe the poses of the body, (ii) the model we use 
to describe the metabolic energy of a movement, and (iii) the model we use to describe the perceived muscle 
forces.  Working from the relationship between the muscle forces and the metabolic energy of the movement, 
we develop the cost functional that we use to generate movement trajectories.  We finally provide an outline 
of the mathematics that we use to calculate the optimal trajectories given a cost functional for a movement. 
2.1 Terminology & Notation 
For a position variable 𝑥𝑥(𝑡𝑡), the first- and second-order time-derivatives are the velocity 𝑣𝑣(𝑡𝑡) = 𝑥𝑥̇(𝑡𝑡) and 
the acceleration 𝑎𝑎(𝑡𝑡) = 𝑥𝑥̈(𝑡𝑡).  We follow [13] and denote the higher order time-derivatives as the jerk 𝑥𝑥⃛(𝑡𝑡), 
the snap 𝑥𝑥⃛̇(𝑡𝑡), the crackle 𝑥𝑥⃛̈(𝑡𝑡), and the pop 𝑥𝑥⃛⃛(𝑡𝑡).  For a force variable 𝐹𝐹(𝑡𝑡) (e.g. example representing the 
net force of the muscles acting on a point on the body as a function of time), we denote the first-order time-
derivative as the yank 𝐹𝐹̇(𝑡𝑡).  We denote an ordinary function as 𝜉𝜉(∙) using parentheses, and a functional (a 
function of a function) as 𝜉𝜉[∙] using brackets.  We denote the set of all values 𝜉𝜉𝑛𝑛 as {𝜉𝜉𝑛𝑛}.  For a movement 
taking a time 𝑇𝑇, and beginning at time −𝑇𝑇/2 and ending at time 𝑇𝑇/2, the we denote the time average of a 
variable 𝜉𝜉 associated with the movement by 〈𝜉𝜉〉 = (1/𝑇𝑇) ∫
𝜉𝜉𝜉𝜉𝜉𝜉
𝑇𝑇/2
−𝑇𝑇/2
.  A function 𝜉𝜉(𝑡𝑡) is odd if 𝜉𝜉(𝑡𝑡) = −𝜉𝜉(−𝑡𝑡), 
and it is even if 𝜉𝜉(𝑡𝑡) = 𝜉𝜉(−𝑡𝑡).  We find it contributes to the clarity to distinguish the metabolic and 
mechanical energies of a movement of how the metabolic energy (i.e. the stored energy the body consumes 
when activating muscles) and mechanical energy (i.e. the kinetic and potential energies that describe the 
trajectories of the segments of the body as they move through space) by expressing them in different units.  
 
 
4 
 
We express the metabolic energy using calories and mechanical energy using joules; these two measures of 
energy are related to each other by 1.0 cal = 4.2 J. 
2.2 Segment, Metabolic Energy, & Perceived Muscle Force Models 
We model the skeleton of the human body as in [14-16] using a segment model consisting of a system of 
 (“nu”) segments attached at N  joints.  An example of such a model can be found in [22] using segments 
for the feet, lower legs, upper legs, hands, lower arms, upper arms, torso, and head.  A movement is described 
by a set of trajectories 𝑥𝑥⃗𝑛𝑛(𝑡𝑡) of points on the body and the rest of the body is constrained to move sensibly 
given the motion of this specified set of points.  Associated with each controlled point on the body is an 
effective mass 𝑚𝑚𝑛𝑛, a trajectory 𝑥𝑥⃗𝑛𝑛(𝑡𝑡), and a net muscle force 𝐹𝐹⃗𝑛𝑛(𝑡𝑡) that determines the acceleration of that 
point according to Newton’s second law using the effective mass. 
The metabolic rate 𝑊𝑊̇ (𝑡𝑡) of expending metabolic energy is the sum of metabolic rates 𝑊𝑊̇𝑛𝑛(𝑡𝑡) associated 
with each joint 𝑊𝑊̇ (𝑡𝑡) ≈ ∑
𝑊𝑊̇𝑛𝑛(𝑡𝑡)
𝑁𝑁
𝑛𝑛=1
.  Following [14], the metabolic rate associated with a joint is given by a 
function:  
 








,
.
F
E
n
n
n
n
n
n
W
t
W
F
t
W
F
t
v
t








  
(1) 
Further continuing the argument in [14], the metabolic rates 𝑊𝑊̇𝑛𝑛
𝐹𝐹(𝑡𝑡) and 𝑊𝑊̇𝑛𝑛
𝐸𝐸(𝑡𝑡) take on approximate 
mathematical forms given by the lowest order of a Taylor series expansion given some physical constraints 
that we do not give here; the mathematical forms are: 
 










2
.
,
,
F
n
n
n
n
E
n
n
n
n
n
n
W
F
t
F
t
W
F
t
v
t
F
t
v
t












  
(2) 
The metabolic rates 𝑊𝑊̇ (𝑡𝑡), 𝑊𝑊̇𝑛𝑛(𝑡𝑡), 𝑊𝑊̇𝑛𝑛
𝐹𝐹(𝑡𝑡), and 𝑊𝑊̇𝑛𝑛
𝐸𝐸(𝑡𝑡) have dimensions of metabolic power [𝑐𝑐𝑐𝑐𝑐𝑐∙𝑠𝑠−1]. The 
quantities ε𝑛𝑛 and η𝑛𝑛 are constant parameter values characterizing the associated metabolic rates.  The 
parameters η𝑛𝑛 may take on different values when the muscles add or remove mechanical energy to or from 
a segment, though we require they be constant in each case.  The metabolic energies 𝑊𝑊𝑛𝑛
𝐹𝐹ൣ𝐹𝐹⃗𝑛𝑛൧  associated 
with generating the muscle forces in a segment are: 
 
/2
2
2
/2
.
T
F
n
n
n
n
n
n
T
W
F
F dt
F
T












  
(3) 
The metabolic energies 𝑊𝑊𝑛𝑛
𝐹𝐹 have dimensions of metabolic energy [𝑐𝑐𝑐𝑐𝑐𝑐].   
In [14], we gave a simple model for the perceived muscle force ψ𝐹𝐹,𝑛𝑛ൣ𝐹𝐹⃗𝑛𝑛൧ for the subject over the course of 
a movement in which it is approximately related to the muscle force 𝐹𝐹⃗𝑛𝑛(𝑡𝑡) as: 
 


2
2
,
0,
1 /
.
F n
n
n
n
F
F
F








  
(4) 
Here 𝐹𝐹0,𝑛𝑛 is a constant with dimensions of force [𝑁𝑁] so ψ𝐹𝐹,𝑛𝑛 is dimensionless.  In the static case where there 
is a constant muscle force 𝐹𝐹⃗𝑛𝑛 supporting a weight without moving the body, the perceived muscle force 
ψ𝐹𝐹,𝑛𝑛ൣ𝐹𝐹⃗𝑛𝑛൧ approximates to Stevens’ power law for this case (ψ𝐹𝐹,𝑛𝑛 ≈ ൫ห𝐹𝐹⃗𝑛𝑛ห /𝐹𝐹0,𝑛𝑛൯
1.7). [23, 24] 
 
 
 
5 
 
2.3 Cost Functional 
The most unobjectionable choice for the cost functional 𝐽𝐽[∙] of a movement is the metabolic energy 
expended in executing the movement.  In this case, the model would select the movement trajectory that 
minimizes the metabolic energy.  However, if we assume that subjects possess a mechanism by which they 
can perceive the cost of a movement, then we find that we cannot use the total metabolic energy expended.  
The model that we have given in Sec. 2.2 provides a mechanism by which subjects can perceive the 
magnitudes of forces but does not provide a similar mechanism to perceive the total metabolic energy.  
Instead, we can use the model for the perception of muscle forces in (4) to construct an estimator for the 
portion of the metabolic energy expended generating the muscle forces.  This would allow us to construct 
the cost functional would be  𝐽𝐽ൣ൛𝐹𝐹⃗𝑛𝑛ൟ൧ = 𝑊𝑊𝐹𝐹ൣ൛𝐹𝐹⃗𝑛𝑛ൟ൧ ≈ ൫∑൫ε𝑛𝑛𝐹𝐹0,𝑛𝑛
2 ൯ψ𝐹𝐹,𝑛𝑛ൣ𝐹𝐹⃗𝑛𝑛൧ 
𝑛𝑛
൯𝑇𝑇.  However, we have argued that 
the optimal control model should involve control of the yanks of the muscles and the yank appears nowhere 
in this cost functional.  We must include terms in the yank in the cost functional. 
We need to construct terms of the form 𝑗𝑗𝑛𝑛ቀ𝐹𝐹⃗𝑛𝑛̇ ቁ associating a cost with a yank 𝐹𝐹⃗𝑛𝑛̇ .  We assume that the 
cost is zero when the yank is zero, and the cost is independent of the direction of the yank so that the term 
takes the form 𝑗𝑗𝑛𝑛൫𝐹𝐹̇𝑛𝑛
2൯.  Taking a Taylor series expansion of this and truncating to the lowest order, we find 
𝑗𝑗𝑛𝑛൫𝐹𝐹̇𝑛𝑛
2൯ ≈ 𝛼𝛼𝑛𝑛𝐹𝐹̇𝑛𝑛
2 for some constant parameter value 𝛼𝛼𝑛𝑛.  Thus, the cost associated with the yanks for a 
movement takes the form ∫
∑𝑗𝑗𝑛𝑛൫𝐹𝐹̇𝑛𝑛
2൯
𝑛𝑛
𝑑𝑑𝑑𝑑
𝑇𝑇/2
−𝑇𝑇/2
 ≈ ∫
∑𝛼𝛼𝑛𝑛
𝑛𝑛
𝐹𝐹̇𝑛𝑛
2𝑑𝑑𝑑𝑑
𝑇𝑇/2
−𝑇𝑇/2
.  To arrive at a cost functional in which the 
jerk is controlled for a movement taking a beginning at time −𝑇𝑇/2 and ending at time 𝑇𝑇/2, use a cost 
functional 𝐽𝐽ቂቄ𝐹𝐹⃗𝑛𝑛, 𝐹𝐹⃗̇
𝑛𝑛ቅቃ given by: 
 




/2
2
2
0,
,
/2
,
.
T
n
n
n
n
F n
n
n
n
T
n
n
J
F F
F
F
T
F dt




































  
(5) 
We note that the cost 𝐽𝐽 has dimensions of metabolic energy [𝑐𝑐𝑐𝑐𝑐𝑐]. 
We can rewrite the yank-control cost functional 𝐽𝐽ቂቄ𝐹𝐹⃗𝑛𝑛, 𝐹𝐹⃗̇
𝑛𝑛ቅቃ in (5) as: 
 




/2
2
2
/2
,
.
T
n
n
n
n
n
n
T
n
J
F F
F
F
dt















  
(6) 
As we want to use (6) to derive the trajectory that minimizes the cost, we must further rewrite (6) in terms 
of the trajectory.  We expect that each of the forces 𝐹𝐹⃗𝑛𝑛(𝑡𝑡) can be expressed as a function of the set of 
functions ൛𝑥𝑥⃗𝑛𝑛, 𝑥𝑥⃗̇𝑛𝑛, 𝑥𝑥⃗̈𝑛𝑛, 𝑥𝑥⃗⃛𝑛𝑛ൟ associated with the trajectories of the various points on the body that the subject 
controls and that we can rewrite the yank-control cost functional 𝐽𝐽ቂቄ𝐹𝐹⃗𝑛𝑛, 𝐹𝐹⃗̇
𝑛𝑛ቅቃ in terms of the set of functions 
൛𝑥𝑥⃗𝑛𝑛, 𝑥𝑥⃗̇𝑛𝑛, 𝑥𝑥⃗̈𝑛𝑛, 𝑥𝑥⃗⃛𝑛𝑛ൟ as the jerk-control cost functional: 
 






/2
/2
,
,
,
,
,
,
.
T
n
n
n
n
n
n
n
n
T
J
x
x
x
x
L
x
x
x
x
dt




















  
(7) 
2.4 Lagrangian Method of Solving the Optimal Control Problem 
One way to calculate the trajectory that minimizes the cost functional in (7) using the Lagrangian method 
of solving the optimal control problem.  This method begins by identifying the Lagrangian of the system; 
it is the integrand 𝐿𝐿൫൛𝑥𝑥⃗𝑛𝑛, 𝑥𝑥⃗̇𝑛𝑛, 𝑥𝑥⃗̈𝑛𝑛, 𝑥𝑥⃗⃛𝑛𝑛ൟ൯ in (7) and has dimensions of metabolic rate [𝑐𝑐𝑐𝑐𝑐𝑐∙𝑠𝑠−1].  We must first 
 
 
6 
 
identify all the independent components that make up the Lagrangian by specifying the scalar components 
that make up the vector 𝑥𝑥⃗𝑛𝑛(𝑡𝑡) as 𝑥𝑥𝑚𝑚,𝑛𝑛(𝑡𝑡).  Once we have done this, as we have shown in [13], we can then 
find the trajectories 𝑥𝑥𝑚𝑚,𝑛𝑛(𝑡𝑡) that optimize the value of the cost functional in (7) by solving the system of 
differential equations given by: 
 
2
3
2
3
,
,
,
,
0 .
m n
m n
m n
m n
L
d
L
d
L
d
L
x
dt
x
x
x
dt
dt





























  
(8) 
Equation (8) gives a critical point for the system and it remains a further step to prove that the critical 
point is a minimum.  For most cost functionals that we would apply to human movements we expect a 
critical point would also be a minimum. 
2.5 Hamiltonian Methods of Solving the Optimal Control Problem 
A second way to the trajectory that minimizes the value of the cost functional in (7) is by using the 
Hamiltonian method of solving the optimal control problem. This method begins by defining the generalized 
coordinate vectors 𝑄𝑄𝑚𝑚,𝑛𝑛, controls 𝑢𝑢𝑚𝑚,𝑛𝑛, and generalized momentum vectors 𝑃𝑃𝑚𝑚,𝑛𝑛 as: 
 
,
,
,
,
,
,
,
1,
,
2,
,
3,
,
,
,
.
m n
m n
m n
m n
m n
m n
m n
m n
m n
m n
Q
x
x
x
u
x
P
p
p
p


















  
(9) 
We note that the generalized coordinate vectors 𝑄𝑄𝑚𝑚,𝑛𝑛 has been defined so that the various elements have 
different dimensions; it can be made to have the same dimensions for all its elements through the 
introduction of appropriately dimensioned constants.  We have discussed the interpretation of the elements 
of the generalized momentum vectors 𝑃𝑃𝑚𝑚,𝑛𝑛 in [13].  The Hamiltonian 𝐻𝐻 is obtained by taking the Legendre 
transform of the Lagrangian.  In our case, this amounts to combining the Lagrangian in (7) and the 
generalized coordinate vectors 𝑄𝑄𝑚𝑚,𝑛𝑛 and generalized momentum vectors 𝑃𝑃𝑚𝑚,𝑛𝑛 in (9) as shown in [13], namely: 
 
,
,
,
.
m n
m n
m n
H
P
Q
L





  
(10) 
The Hamiltonian 𝐻𝐻 has dimensions of metabolic rate [𝑐𝑐𝑐𝑐𝑐𝑐∙𝑠𝑠−1].  Since we must require the 𝑃𝑃𝑚𝑚,𝑛𝑛
𝛵𝛵𝑄𝑄̇𝑚𝑚,𝑛𝑛 to 
have dimensions the same dimensions as the Lagrangian and the Hamiltonian, we require the components 
of the generalized momentum vector 𝑃𝑃𝑚𝑚,𝑛𝑛 to have the necessary dimensions to make that true.  As we have 
shown in [13], we can find the trajectories 𝑥𝑥𝑚𝑚,𝑛𝑛(𝑡𝑡) that minimize the value of the cost functional in (7) by 
solving the system of differential equations given by: 
 











,
,
,
,
,
,
,
/
0 .
m n
m n
m n
m n
m n
Q
H
P
A
P
H
Q
B
H
u
C









  
(11) 
Pontryagin’s minimum principle (see e.g., [25] Chapter 5) tells us that the Hamiltonian takes a constant 
value when calculated along the optimal trajectory (i.e. the trajectory that solves the systems in (8) and 
(11)).  We call this constant value the generalized energy Ψ, so along the optimal trajectory we find: 
 
 
7 
 
 

.
H t
  
(12) 
The generalized energy Ψ has dimensions of metabolic rate [𝑐𝑐𝑐𝑐𝑐𝑐∙𝑠𝑠−1].  The generalized energy Ψ plays a 
role in our optimal control model which is analogous to the role the mechanical energy plays in classical 
mechanics due to the fact that it is conserved over the course of an optimal movement as indicated in (12)
.  Along an optimal trajectory it gives a constant of the motion that provides information about the entire 
trajectory. 
3 Optimal Control Model of Walking Gait 
We now use the general optimal control approach to modeling human movements outlined in Sec. 2 to 
construct a model of walking gait.  We first provide some useful anthropometric values.  We then proceed 
using the same segment model for walking gait that we used in [14-16]; namely a simple two segment model 
consisting of two legs attached at single point representing the “torso.”  However, where we used specified 
trajectories of the body in [14-16], in the present treatment, we generalize the model to use arbitrary 
trajectories.  We construct a yank-control cost functional in terms of these muscle force trajectories, which 
we then rewrite as a jerk-control cost functional in terms of the body segment trajectories.  We then find 
the optimal trajectories of the body using the Lagrangian method of the optimal control problem and 
calculate the generalized energies for the trajectories using the Hamiltonian method.  We conclude by 
looking at the mechanical energy of walking gait. 
3.1 Some Anthropometric Values 
A subject with mass 𝑀𝑀 and height 𝐻𝐻 has a mass in each leg (i.e. thigh, shank, and foot) of about μ = 𝑟𝑟𝑟𝑟, 
and the length of the leg of about 𝐿𝐿 = ρH where 𝑟𝑟 = 0.16 and ρ = 0.53. [22]  The mass of the torso carried 
by the stance leg during a step is 𝑚𝑚 = (1 −2𝑟𝑟)𝑀𝑀. 
3.2 Walking Gait Model 
We model the body using a two-segment model with one segment for each leg − the legs are straight, do 
not bend at the knee, but can change length.  The mass of the torso is placed in the torso which is the 
point where the two segments meet; the mass of each leg is placed in the foot at the far end of the leg 
segments from the torso.  During a step, the torso maintains a constant height while the swing foot rises 
only a negligible height above the ground.  We include a discontinuity at heel-strike.  During one step, one 
leg is the stance leg which supports the torso as the torso moves over it, and the other is the swing leg 
which swings under the torso; the feet of the two legs are the stance foot and swing foot, respectively.  The 
stance foot remains fixed on the ground while the swing moves to the next position; the legs lengthen or 
shorten as needed by the movement − we intend the amount of lengthening and shortening to be consistent 
with reasonable knee-bending during walking.  We define the unit vector 𝑣𝑣ො to be the direction of motion of 
the torso.  We describe steady state walking gaits already in progress (i.e. ignoring starting and stopping) 
using two gait parameters: (i) the avg. walking speed 𝑣𝑣 of the torso, and (ii) the avg. step length 𝑠𝑠 that 
corresponds to the distance the torso travels each step.  The gait parameter 𝑣𝑣 giving the walking speed 
should not be confused with the velocities 𝑣𝑣⃗𝑛𝑛 appearing in the metabolic energy model or the unit vector 𝑣𝑣ො; 
the diacritical mark or the absence of one suffices to distinguish them.   
We assume that the metabolic energy associated with holding the body up against gravity is 
approximately constant over all reasonable trajectories of body for a walking gait and does not contribute 
 
 
8 
 
to the cost of a walking gait.  We allow that some amount of mechanical energy may be conserved during 
walking although we do not specify the mechanism.  We allow the subject to apply an external force 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) 
to an external object – in the case analyzed in this paper, by pulling a cart.  We may treat the external 
force as an independent variable or constraining it in some way to be a function of the motion of the torso 
as 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒({𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡, 𝑥𝑥̇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡, 𝑥𝑥̈𝑡𝑡𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜, 𝑥𝑥⃛𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡}).  For example, in the case of walking while pulling a cart, we may want 
to constrain the cart to maintain a constant position relative to the torso.  In practice, since the cart is 
pulled by the arm, its position relative to the torso may vary somewhat due to motion of the arm.  We 
choose, therefore, to solve for the case where the external force 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) is not a function of the trajectory 
and allow the position of the cart to vary relative to the torso.  We denote the muscle force applied by the 
stance leg to the torso by 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) and the force applied by the swing leg to the swing foot by 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡); we 
assume the stance foot is fixed on the ground during the step.  When the subject applies an external force 
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) to pull some object, there is a force opposing the horizontal motion of the torso so that it can be 
written −𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡)𝑣𝑣ො.  The force the body must apply to compensate for the external force is 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡)𝑣𝑣ො. 
We look first at the muscle forces 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) and 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) applied by the stance leg to the torso as illustrated 
in Fig 1.  During the first half of the step, gravity pulls the torso in the direction −𝑣𝑣ො, while, during the 
second half, gravity pulls the torso in the direction 𝑣𝑣ො.  The torso already has the mechanical energy needed 
to carry it forward with the required speed, so the muscle forces counteract the force of gravity and do 
external mechanical work. 
 
 
 
 
 
 
Figure 1.  The motion of the torso over the stance leg during one step of 
walking gait.  The torso moves over the leg at a speed 𝒙𝒙̇ 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 while producing 
and external force 𝒇𝒇𝒆𝒆𝒆𝒆𝒆𝒆.  The muscles must provide the necessary force to 
compensate for the effect of gravity on the torso’s speed. 
 
 
9 
 
The angle 𝜃𝜃(𝑡𝑡) of the stance leg with respect to the vertical determines the effect of gravity on the torso.  
We define 𝜃𝜃(𝑡𝑡) so that it is negative during the first half of the step, and positive during the second half.  
The horizontal force related to the torque on the inverted pendulum by gravity is 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 
(1/2)𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2𝜃𝜃 where 𝑔𝑔 denotes the acceleration of gravity; we find that 𝜃𝜃(𝑡𝑡) and 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) satisfy: 
 




sin
,
ˆ
1 / 2
sin2
.
torso
st
torso
ext
L
x
F
mx
mg
f
v








  
(13) 
For small angles 𝜃𝜃 corresponding to small avg. step lengths 𝑠𝑠, we can take (1/2)𝑠𝑠𝑠𝑠𝑠𝑠2𝜃𝜃 ≈ sin𝜃𝜃: 
 


sin
,
ˆ
sin
.
torso
st
torso
ext
L
x
F
mx
mg
f
v








  
(14) 
Thus, in the case of small avg. step lengths 𝑠𝑠, the force 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) of the stance leg on the torso is approximately 
given by: 
 
 



ˆ
/
.
st
torso
torso
ext
F
mx
mg
L x
f
v





  
(15) 
We look next at the muscle forces 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) applied by the torso to the swing leg as illustrated in Fig. 2.  
We require the swing foot to move horizontally with acceleration  𝑥𝑥⃗̈𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡).  During one step, the body 
moves one avg. step length, and the swing foot travels one stride length horizontally in direction 𝑣𝑣ො.  The 
swing leg begins and ends the swing at rest, and so must accelerate and decelerate as required over the 
course of the swing.  During the first half of the step, gravity pulls the swing leg in the direction 𝑣𝑣ො, while, 
during the second half, gravity pulls the swing leg in the direction −𝑣𝑣ො. 
 
The angle 𝜑𝜑(𝑡𝑡) of the swing leg with respect to the vertical determines the effect of gravity on the swing 
leg.  We define 𝜑𝜑(𝑡𝑡)  so that it is negative during the first half of the step, and positive during the second 
Figure 2.  The motion of the swing leg under the torso during one step of 
walking gait.  The swing leg moves symmetrically under the torso, accelerating 
with an acceleration 𝒙𝒙̈ 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇.  The muscles must provide the force that generates 
the acceleration and compensate for the effect of gravity on the swing leg. 
 
 
10 
 
half.  As the acceleration of the swing leg is 𝑥𝑥̈𝑓𝑓𝑓𝑓𝑜𝑜𝑜𝑜(𝑡𝑡).  The horizontal muscle forces that must be applied to 
the leg to generate the acceleration is μ𝑥𝑥̈𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + μgsinφcosφ; we find that 𝜑𝜑(𝑡𝑡) and  𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) for the first half 
of the step satisfy: 
 




sin
,
/ 2
/ 2,
ˆ
/ 2 sin2
.
foot
torso
sw
foot
L
x
x
T
t
T
F
x
g
v













  
(16) 
For small angles 𝜃𝜃 corresponding to small avg. step lengths 𝑠𝑠, we can take (1/2)𝑠𝑠𝑠𝑠𝑠𝑠2𝜃𝜃 ≈ sin𝜃𝜃: 
 


sin
,
ˆ
sin
.
foot
torso
sw
foot
L
x
x
F
x
g
v










  
(17) 
Thus, in the case of small avg. step lengths 𝑠𝑠, the force 𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡) of the swing leg on the swing foot is 
approximately given by:  
 
 




ˆ
/
.
sw
foot
foot
torso
F
x
g
L
x
x
v







  
(18) 
3.3 Cost Functional 
The cost functional for the two-segment model of walking gait is (cf. (6)): 
 


/2
2
2
2
2
/2
.
T
st
st
sw
sw
st
st
sw
sw
T
J
F
F
F
F
dt












  
(19) 
To keep the expression of the cost functional and the resulting expressions derived from it relatively 
uncluttered, we define three frequency parameter values 𝜔𝜔𝑠𝑠𝑠𝑠, 𝜔𝜔𝑠𝑠𝑠𝑠, and 𝜔𝜔𝑠𝑠𝑠𝑠 given by: 
 
2
2
2
,
,
/
.
st
st
st
sw
sw
sw
sp
g
L










  
(20) 
Using these frequency parameters, the cost functional 𝐽𝐽 may be rewritten in terms of the trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) 
of the torso, the trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing foot, and the trajectory  𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) of the external force as: 
 
 












/2
2
2
2
/2
/2
2
2
2
/2
/2
2
2
2
/2
/2
2
2
2
/2
/
/
.
T
st
torso
sp torso
ext
T
T
sw
foot
sp
foot
torso
T
T
st
torso
sp torso
ext
T
T
sw
foot
sp
foot
torso
T
J
m
x
x
f
m
dt
x
x
x
dt
m
x
x
f
m
dt
x
x
x
dt





































  
(21) 
To calculate the general optimal trajectory for walking gait using the cost functional in (21) would require 
that we solve a single system of differential equations combining the trajectories of 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡), 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡), and 
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡).  If we restrict our attention to normal, healthy walking gaits, we can simplify the problem of finding 
the optimal trajectory somewhat.  We expect that for normal, healthy walking gaits the torso will move in 
 
 
11 
 
a way that is reasonably approximated by a constant horizontal velocity 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) ≈ vt.  We can rewrite 
this as: 
 
.
foot
torso
foot
x
x
x
vt



  
(22) 
Substituting (22) into (21), we obtain an approximate cost functional that describes normal, healthy walking 
gaits: 
 












/2
2
2
2
/2
/2
2
2
2
/2
/2
2
2
2
/2
/2
2
2
2
/2
/
/
.
T
st
torso
sp torso
ext
T
T
sw
foot
sp
foot
T
T
st
torso
sp torso
ext
T
T
sw
foot
sp
foot
T
J
m
x
x
f
m
dt
x
x
v
dt
m
x
x
f
m
dt
x
x
vt
dt




































  
(23) 
The approximate cost functional in (23) consists of a linear combination of terms either involving the 
trajectory of the torso and the external force, or the trajectory of the swing foot.  We may therefore separate 
the cost functional in (23) into two cost functionals that can be solved independently.  The first cost 
functional is 𝐽𝐽𝑠𝑠𝑠𝑠 measures the cost of moving the torso over the stance leg while applying the external force; 
it is: 
 




/2
2
2
2
/2
/2
2
2
2
/2
/
/
.
T
st
st
torso
sp torso
ext
T
T
st
torso
sp torso
ext
T
J
m
x
x
f
m
dt
m
x
x
f
m
dt


















  
(24) 
The second cost functional is 𝐽𝐽𝑠𝑠𝑠𝑠 measures the cost of moving the swing foot under the torso; it is: 
 








/2
2
2
2
/2
/2
2
2
2
/2
.
T
sw
sw
foot
sp
foot
T
T
sw
foot
sp
foot
T
J
x
x
v
dt
x
x
vt
dt


















  
(25) 
3.4 Optimal Trajectories of the Torso & External Force 
The optimal trajectories of the torso and external force minimize the cost functional 𝐽𝐽𝑠𝑠𝑠𝑠 in (24).  We first 
calculate the optimal trajectories using the Lagrangian method of solving the optimal control problem.  This 
gives a system of differential equations that separates into two problems:  (i) calculating the optimal 
trajectory of the torso over the stance leg, and (ii) calculating the optimal trajectory of the external force.  
We then calculate constants of the motion in the form of the generalized energies using the Hamiltonian 
method of solving the optimal control problem. 
3.4.1 Lagrangian Method 
The Lagrangian 𝐿𝐿𝑠𝑠𝑠𝑠 governing the motion of the torso and external force is the integrand of the cost 
functional 𝐽𝐽𝑠𝑠𝑠𝑠 in (24); that is: 
 
 
12 
 
 




2
2
2
2
2
2
/
/
.
st
st
torso
sp torso
ext
st
torso
sp torso
ext
L
m
x
x
f
m
m
x
x
f
m














  
(26) 
Since we are treating the external force 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) independently of the trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the torso, a 
system of two differential equations is obtained from the Lagrangian 𝐿𝐿𝑠𝑠𝑠𝑠 using (8).  The first differential 
equation of the system is obtained taking the partial derivatives of 𝐿𝐿𝑠𝑠𝑠𝑠 using the various time-derivatives of 
𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡); it is: 
 
















2
2
2
2
2
2
2
1
2
2
1
2
0.
torso
sp torso
st
torso
sp torso
sp
torso
sp torso
st
torso
sp torso
ext
st ext
sp
ext
st ext
x
x
x
x
x
x
x
x
m
f
f
m
f
f





































  
(27) 
The second differential equation of the system is obtained taking the partial derivatives of 𝐿𝐿 using the 
various time-derivatives of 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡); it is: 
 








2
2
2
1
2
0.
torso
sp torso
st
torso
sp torso
ext
st ext
x
x
x
x
m
f
f















  
(28) 
The system of two differential equations given in (27) and (28) is solved by trajectories  𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) and 
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡).  The trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) that solves the system is: 
 













1
2
3
4
5
6
sinh
cosh
sinh
cosh
sinh
cosh
.
torso
sp
sp
sp
sp
st
st
x
t
c
t
c
t
c t
t
c t
t
c
t
c
t












  
(29) 
The trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the torso consists of four terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 related to the motion of the stance leg 
and torso as an inverted pendulum acting under the force of gravity and two terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 related to the 
force of muscle acting on the torso.  The four terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 taken together are the solution to a fourth-
order differential equation corresponding to the solution of an optimal control problem in which the 
acceleration 𝑥𝑥̈𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) is controlled.  The additional two terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 provide a description of how the 
acceleration 𝑥𝑥̈𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) is in turn controlled by the forces generated by muscles.  The trajectory 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) of the 
external force that solves the system is: 
 





1
2
sinh
cosh
.
ext
st
st
f
t
k
t
k
t




  
(30) 
The trajectories of the torso and the external force are parameterized by the eight parameters 𝑐𝑐1, …, 𝑐𝑐6, and 
𝑘𝑘1 and 𝑘𝑘2 in (29) and (30).  We determine these parameter values by specifying the initial and final 
conditions the trajectories must satisfy. 
The torso must traverse a step length 𝑠𝑠 in the avg. step time 𝑇𝑇.  It must begin each step with some 
velocity 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
 ≥ 0 and acceleration 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, and end each step with some velocity 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 ≥ 0 and acceleration 
𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
; that is: 
 
 
13 
 
 












/ 2
/ 2,
/ 2
/ 2,
/ 2
,
/ 2
,
/ 2
,
/ 2
.
torso
torso
torso
torso
torso
torso
torso
torso
torso
torso
x
T
s
x
T
s
x
T
V
x
T
V
x
T
A
x
T
A

















  
(31) 
Combining (29) and (31) givens a linear system of six equations that allows us to solve the six parameters 
𝑐𝑐1, …, 𝑐𝑐6 in (29) in terms of the five parameters 𝑠𝑠, 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, and 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 in (31) using conventional 
methods of linear algebra. Thus, we can describe the optimal trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the torso entirely by 
specifying the step-length, and the velocities and accelerations immediately before and after heel-strike 
together with the step-time.  We discuss the physical meaning of 𝑉𝑉𝑡𝑡𝑜𝑜𝑟𝑟𝑟𝑟𝑟𝑟
−
, 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, and 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 in (31) 
further in Sec. 3.7. 
The external force must begin each step with some force 𝐹𝐹−, and end each step with some force 𝐹𝐹+; that 
is: 
 




/ 2
,
/ 2
.
ext
ext
f
T
F
f
T
F





  
(32) 
Since we have allowed for a discontinuity due to heel-strike we allow 𝐹𝐹− ≠ 𝐹𝐹+.  The two parameters 𝑘𝑘1 and 
𝑘𝑘2 in (30) are determined entirely by the two parameters 𝐹𝐹− and 𝐹𝐹+ in (32) together with the step-time 𝑇𝑇.  
Thus, we can describe the optimal trajectory 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) of the external force entirely by specifying the external 
forces immediately before and after heel-strike together with the step-time.  We also need to average the 
required external force 𝐹𝐹𝑒𝑒𝑒𝑒𝑒𝑒 
 

.
ext
ext
f
t
F

  
(33) 
Combining (15), (29), and (30), we find that the magnitude ห𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡)ห of the force generated on the torso 
by the stance leg is: 
 



















3
4
2
2
5
1
2
2
6
2
cosh
sinh
sinh
cosh
.
st
sp
sp
sp
st
sp
st
st
sp
st
F
t
m
c
t
c
t
mc
k
t
mc
k
t



















  
(34) 
Of the eight parameters 𝑐𝑐1, …, 𝑐𝑐6, and 𝑘𝑘1 and 𝑘𝑘2 that determine the optimal trajectories 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the 
torso (29) and 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) of the external force (30), only six, 𝑐𝑐3,…, 𝑐𝑐6, 𝑘𝑘1, and 𝑘𝑘2 affect the magnitude ห𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡)ห 
of the force generated on the torso by the stance leg.  The remaining two, 𝑐𝑐1, 𝑐𝑐2, describe the proper state 
of the torso so that the walking gait can be carried out correctly using the muscle activations given in (34)
.  
3.4.2 Hamiltonian Method 
The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 governing the trajectories of the torso and external force is given by: 
 
.
st
x
x
f
f
st
H
P Q
P Q
L







  
(35) 
The generalized coordinates are 𝑄𝑄𝑥𝑥 and 𝑄𝑄𝑓𝑓 are: 
 
 
14 
 
 
,
,
,
.
x
torso
torso
torso
f
ext
Q
x
x
x
Q
f










  
(36) 
The controls are the jerk 𝑥𝑥⃛𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) and the yank 𝑓𝑓̇𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡): 
 




,
.
x
torso
f
ext
u
t
x
t
u
t
f
t




  
(37) 
We calculate the generalized momentum vectors 𝑃𝑃𝑥𝑥 and 𝑃𝑃𝑓𝑓  and the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 along the optimal 
trajectory of the torso in Appendix 1; the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 and generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 of the torso satisfy: 
 








2
2
2
2
2
2
st
st
st
st
st
st
st
st
st
torso
st
st
st
torso
st
H
F
F
F
F
x
F
F
x




















  
(38) 
When moving in an optimal trajectory between heel-strikes, the generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is an approximate 
constant of the motion of the torso and external force insofar as the approximation in (22) holds.  Although 
we refrain from working out its exact form here, the constant generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is expressed in terms 
of the eight parameters 𝑐𝑐1, …, 𝑐𝑐6, and 𝑘𝑘1 and 𝑘𝑘2 in (29) and (30) using (38). 
3.5 Optimal Trajectory of the Swing Foot 
We calculate the optimal trajectory for the motion of the swing foot under the torso by finding the 
trajectory that minimizes the cost functional 𝐽𝐽𝑠𝑠𝑠𝑠 in (25) following the same procedure that we have used in 
Sec. 3.4. 
3.5.1 Lagrangian Method 
The Lagrangian 𝐿𝐿𝑠𝑠𝑠𝑠 governing the motion of the swing foot is the integrand of the cost functional 𝐽𝐽𝑠𝑠𝑠𝑠 in 
(25): 
 








2
2
2
2
2
2
.
sw
sw
foot
sp
foot
sw
foot
sp
foot
L
x
x
v
x
x
vt














  
(39) 
We obtain single equation of motion from the Lagrangian by taking the partial derivatives of 𝐿𝐿 using the 
various time-derivatives of 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡); it is: 
 












2
2
2
2
2
2
2
0.
foot
sp
foot
sw
foot
sp
foot
sp
foot
sp
foot
sw
foot
sp
foot
x
x
x
x
x
x
x
x
vt


























  
(40) 
The trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) that solves (40) is: 
 













1
2
3
4
5
6
sin
cos
sin
cos
sinh
cosh
.
foot
sp
sp
sp
sp
sw
sw
x
t
vt
c
t
c
t
c t
t
c t
t
c
t
c
t













  
(41) 
 
 
15 
 
The trajectory of the swing foot is parameterized by the six parameters 𝑐𝑐1, …, 𝑐𝑐6, in (41).  As was the case 
with 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡), we observe that the trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡)  consists of four terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 related to the motion 
of the swing leg as a simple pendulum acting under the force of gravity and two terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 related to 
the force of muscle acting on the swing foot.  We may again understand the two terms in 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 as providing 
a description of how the acceleration 𝑥𝑥̈𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) is controlled by the muscles.   
The swing foot must traverse an avg. stride length 2𝑠𝑠 in the avg. step time 𝑇𝑇.  It must begin each step 
resting on the ground with no velocity or acceleration but may end each step with some velocity 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 ≥ 0 
and acceleration 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
, assuming impact on the ground bring it to a complete stop for reasonable values of 
𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 and 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
.  The initial and final conditions of the motion of the swing foot for a single step are: 
 












/ 2
,
/ 2
,
/ 2
0,
/ 2
,
/ 2
0,
/ 2
.
foot
foot
foot
foot
foot
foot
foot
foot
x
T
s
x
T
s
x
T
x
T
V
x
T
x
T
A















  
(42) 
Combining (41) and (42) gives a linear system of six equations that allows us to solve the six parameters 
𝑐𝑐1, …, 𝑐𝑐6 in (41) in terms of the three parameters 𝑠𝑠, 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
, and 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 in (42). Thus, we can describe the 
optimal trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing foot entirely by specifying the step-length, and the velocity and 
acceleration immediately before heel-strike together with the step-time.  We discuss the physical meaning 
of 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
, and 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 in (42) further in Sec. 3.7. 
Combining (18) and (41), we find that the magnitude ห𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡)ห of the force generated on the swing foot 
by the swing leg is: 
 















3
4
2
2
5
6
cos
sin
sinh
cosh
.
sw
sp
sp
sp
sp
sw
st
st
F
t
c
t
c
t
c
t
c
t
















  
(43) 
We observe that of the six parameters 𝑐𝑐1, …, 𝑐𝑐6, that determine the optimal trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing 
foot in (41), four, 𝑐𝑐3, …, 𝑐𝑐6, affect the magnitude ห𝐹𝐹⃗𝑠𝑠𝑠𝑠(𝑡𝑡)ห of the force generated on the swing foot by the 
swing leg.  The remaining two, 𝑐𝑐1 and 𝑐𝑐2, describe the proper state of the swing foot so that the walking 
gait can be carried out correctly using the muscle activations given in (43). 
3.5.2 Hamiltonian Method 
The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 governing the trajectory of the swing foot is given by: 
 
.
sw
sw
H
P Q
L




  
(44) 
The generalized coordinates vector Q is: 
 
,
,
.
foot
foot
foot
Q
x
x
x









  
(45) 
Thus, the generalized coordinates vector Q contains the position 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡), velocity 𝑥𝑥̇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡), and acceleration 
𝑥𝑥̈𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡).  The control is the jerk 𝑥𝑥⃛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡): 
 

.
foot
u t
x
t

  
(46) 
 
 
16 
 
We calculate the generalized momentum vector 𝑃𝑃 and the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 along the optimal trajectory 
of the swing foot in Appendix 2; the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 and generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 in this case satisfy: 
 








2
2
2
2
2
2
.
sw
sw
sw
sw
sw
sw
sw
sw
sw
foot
sw
sw
sw
foot
sw
H
F
F
F
F
x
F
F
x




















  
(47) 
When moving in an optimal trajectory between heel-strikes, the generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is an approximate 
constant of the motion of the swing foot insofar as the approximation in (22) holds.  Although we refrain 
from working out its exact form here, the constant generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is expressed in terms of the six 
parameters 𝑐𝑐1, …, 𝑐𝑐6, in (41) using (47). 
3.6 Mechanical Energy of Walking Gait 
The kinetic mechanical energy 𝑈𝑈𝐾𝐾(𝑡𝑡) of a walking gait can be calculated from the trajectories of the 
segments using the usual definition of kinetic energy from classical mechanics; it is: 
 




2
2
1 / 2
1 / 2
.
K
torso
foot
U
mx
x





  
(48) 
The kinetic mechanical energy 𝑈𝑈𝐾𝐾(𝑡𝑡) in (48) changes over time reflecting an ongoing process of changes in 
mechanical energy to walking gait.  We will assume that, whenever one of the torso or swing foot gains 
kinetic mechanical energy while the other loses it, the body transfers kinetic mechanical energy between the 
two.  Kinetic mechanical energy may change for two reasons:  (i) interaction with the environment, or (ii) 
action of the muscles on the segments of the body.  If we assume that the only interaction with the 
environment that happens is a change in kinetic mechanical energy at heel-strike, then any other change in 
kinetic mechanical energy must be due the action of muscles.   
The change in kinetic mechanical energy at heel-strike is given by the values of the parameters 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 and 
𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 just before heel-strike and 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
 just after heel-strike.  For the case where 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
 < 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, the torso 
has lost some amount of kinetic mechanical energy at heel-strike while for the case where 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
 > 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, 
the torso has gained some amount of kinetic mechanical energy.  For the case where  𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 > 0, the swing 
foot has lost some amount of kinetic mechanical energy at heel-strike.  As we have assumed that, whenever 
possible, the body transfers kinetic mechanical energy between the torso and the swing foot, the change in 
kinetic mechanical energy ∆𝑈𝑈𝐾𝐾(𝑡𝑡) at heel-strike is: 
 










2
2
2
1 / 2
1 / 2
.
K
torso
torso
foot
U
m
V
V
V


















  
(49) 
The muscles must act during the following step to restore kinetic mechanical energy to the body so that 
this change in mechanical energy can occur at the next heel-strike.  The optimal trajectories 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) and 
𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the torso and swing foot, respectively, describe the effect of this action of by the muscles on 
scales below a step. 
Using (48), we find the rate of change of the kinetic mechanical energy over the course of the walking 
gait cycle is: 
 
.
K
torso torso
foot
foot
U
mx
x
x
x








  
(50) 
 
 
17 
 
The action of muscles accrues a cost in metabolic energy we can associate a metabolic rate with (50).  Thus, 
we should be able to relate (50) to a metabolic rate for intervals between heel-strikes.  Rather than provide 
a detailed analysis of how the muscles act on the torso and the swing foot separately, we follow (2) and 
simply assume that we can make an approximation 𝑊𝑊̇  ≈ η+/−𝑈𝑈̇𝐾𝐾 where η+/− may take on one value when 
𝑈𝑈̇𝐾𝐾 > 0 and another when 𝑈𝑈̇𝐾𝐾 < 0 so that 𝑊𝑊̇  ≥ 0.  Thus, the metabolic rate can differ between the cases 
where mechanical energy is added to or removed from the body.  Therefore, metabolic rate associated with 
the muscles mediating the change in kinetic mechanical energy is:   
 


/
.
torso torso
foot
foot
W
mx
x
x
x











  
(51) 
Integrating over a step, (51) provides an additional metabolic energy not explicitly contained in the model 
used in [14-16], but implicitly contained in it in the form of an additive constant that was introduced to 
facilitate fitting the model to the empirical data in Atzler & Herbst. [17] 
3.7 Discussion 
When constructing the general form of a cost functional for a movement in (5), we required that there 
be no discontinuities in the force.  Nevertheless, the model of walking gait we use precisely does have such 
discontinuities at heel-strike.  In fact, there are two distinct discontinuities happening simultaneously in 
our model at heel-strike:  (i) the impact of the previous swing foot on the ground and (ii) the lifting of the 
new swing foot from the ground. Thus, we can argue that, at the scale at which we are working, while there 
might be no discontinuities in the muscle forces at heel-strike, there are discontinuities related to the 
discontinuous interaction between the body and the ground. 
Between heel-strikes, the trajectories of the torso and swing foot are completely determined by the 
parameter values 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, 𝑉𝑉𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
, 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, and 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
, and 𝐹𝐹− and 𝐹𝐹+ in (31), (32), and (42).  At 
heel-strike, these parameter values summarize any jarring of the body by the impact of the swing foot with 
the ground.  For the intervals of time between heel-strikes these parameters serve to completely determine 
the trajectories of the torso and swing foot as the body recovers from the previous heel-strike and prepares 
for the next.  In effect, we can look at the optimal control model of walking gait that we have developed as 
one in which the subject controls each heel-strike to produce the initial and final conditions required to 
generate the desired trajectories of the torso and swing foot between heel-strikes.  In this respect, the 
optimal control model is a kind of generalization of  the passive-dynamic walking model [26] in which a 
simple walking mechanism steps down a slightly inclined plane with the mechanism moving in a physically 
determined way under the influence of gravity and with the impact at heel-strike driving the mechanism 
into the next step while gravity provides for the restoration of lost kinetic mechanical energy. 
While the optimal control model of walking begins to develop a more detailed description of walking gait 
on scale below that of a step, it leaves out the details of many aspects of walking gait such the double 
support portion of walking gait where both feet are on the ground, the thrust on the torso at toe off, the 
lifting of the swing foot up from the ground, and any vertical motion of the torso.  One of the advantages 
of using a variational approach to modeling movement trajectories, both in physics and in the present study 
of the biomechanics of human movement, is the ease with which existing models can be modified to produce 
more detailed models by the introduction of additional terms in the cost functional.  Thus, the relatively 
simple optimal control model of walking gait that we construct here may be readily modified to 
accommodate more aspects of walking gait, such as the one we have listed, by expanding the cost functional 
we have given with additional terms that describe further details of walking gait.  Work in this direction 
 
 
18 
 
might begin using the simple walking gait models in [27, 28] that include descriptions of such aspects of 
walking gait. 
4 Symmetric & Uniform Walking Gait in the Low Yank Limit 
We now show that the optimal control model in Sec. 3 generalizes the model in [14-16] so that we can 
look at the optimal control model as a means of generalizing the model in [14-16] to scales below that of a 
step.  While it may not be surprising that the present optimal control model is able to approximate the 
model in [14-16], it is of interest to show the conditions under which it does so.  We find that we arrive at 
the approximation by solving the optimal control model for the case of walking gaits that are symmetric 
and uniform and looking at walking gaits in which the yanks are kept relatively low. 
4.1 Symmetric Walking Gait 
We obtain a symmetric trajectory for the torso during a step by requiring it have the same velocity at 
the beginning and end of a step, and the same magnitude of acceleration at the beginning and end of a step.  
Thus, we restrict the parameter values 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
, and 𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 in (31) so that 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
 = 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
, and 
|𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
−
| = |𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
|, and we replace (31) with: 
 












/ 2
/ 2,
/ 2
/ 2,
/ 2
,
/ 2
,
/ 2
,
/ 2
.
torso
torso
torso
torso
torso
torso
torso
torso
torso
torso
x
T
s
x
T
s
x
T
V
x
T
V
x
T
A
x
T
A













  
(52) 
Similarly, we obtain a symmetric external force during a step by requiring that the external force to be the 
same at the beginning and end of a step.  Thus, we restrict the parameter values 𝐹𝐹− and 𝐹𝐹+ in (32) so that 
𝐹𝐹− = 𝐹𝐹+, and we replace (32) with: 
 




/ 2
,
/ 2
.
ext
ext
f
T
F
f
T
F



  
(53) 
Finally, we obtain a symmetric trajectory for the swing foot during a step by requiring that it have the 
same velocity at the beginning and end of a step, and the same magnitude of acceleration at the beginning 
and end of a step.  This can only happen when we take 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
+
 = 0, and 𝐴𝐴𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
+
 = 0; thus we replace (42) 
with: 
 












/ 2
,
/ 2
,
/ 2
0,
/ 2
0,
/ 2
0,
/ 2
0.
foot
foot
foot
foot
foot
foot
x
T
s
x
T
s
x
T
x
T
x
T
x
T













  
(54) 
4.2 Uniform Walking Gait 
A walking gait is approximately uniform when the torso moves with an approximately constant speed: 
 

.
torso
x
t
vt

  
(55) 
 
4.3 Optimal Trajectories of Symmetric and Uniform Walking Gait 
 
 
19 
 
In the case of symmetric walking gait, only three of the six terms in the optimal trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of 
the torso in (29) remain (have non-zero parameter values), namely those that are odd in time on the interval 
−𝑇𝑇/2 ≤ 𝑡𝑡 ≤ 𝑇𝑇/2.  This gives an optimal trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑜𝑜(𝑡𝑡) of the torso of the form: 
 







1
2
3
sinh
cosh
sinh
.
torso
sp
sp
st
x
t
C
t
C t
t
C
t






  
(56) 
In Appendix 3, we calculate the values of the parameters 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3 that give a trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of 
the torso that approximates uniformity as given in (55); these are: 
 













3
5
2
2
2
1
3
3
4
2
2
2
3
4
3
2
2
2
/
3 / 2
,
1 / 2
/
,
.
st
sp
sp
sp
st
st
sp
sp
st
sp
st
sp
st
C
C
C
C
C
v




































  
(57) 
For symmetric walking gait, only one of the two terms in the optimal trajectory 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡)  of the external 
force in (29) remains (has non-zero parameter value), namely the one that is even in time on the interval 
−𝑇𝑇/2 ≤ 𝑡𝑡 ≤ 𝑇𝑇/2: 
 



cosh
.
ext
st
f
t
K
t


  
(58)  
In Appendix 3, we calculate the value of the parameter 𝐾𝐾 that gives a trajectory 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) of the external 
force that has symmetry as given in (53); these are: 
 


1
sinh
/ 2
1
1
.
/ 2
st
ext
st
T
K
F
T
























  
(59) 
For symmetric walking gait, only three of the six terms in the optimal trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing 
foot in (41) only three remain (have non-zero parameter values), namely those that are odd in time on the 
interval −𝑇𝑇/2 ≤ 𝑡𝑡 ≤ 𝑇𝑇/2.  This gives an optimal trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing foot of the form: 
 







1
2
3
sin
cos
sinh
.
foot
sp
sp
sw
x
t
vt
C
t
C t
t
C
t







  
(60) 
In constructing the walking gait model in Sec. 3, we assumed small 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡.  We may therefore make the 
approximation 𝑠𝑠𝑠𝑠𝑠𝑠൫𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡൯ ≈ 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 - ൫𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡൯
3/6 allowing us to approximate (60) as: 
 



3
1
2
3 sinh
.
foot
sw
x
t
vt
t
t
C
t







  
(61) 
In Appendix 4, we calculate the values of the parameters 𝜅𝜅1, 𝜅𝜅2, and 𝐶𝐶3 that give a trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) that 
has symmetry as given in (54); these are: 
 
 
20 
 
 


















2
1
3
2
2
3
3
3
2
24
2
sinh
/ 2
,
12
sinh
/ 2
,
3
12
.
12
sinh
/ 2
6
cosh
sw
sw
sw
sw
sw
sw
sw
sw
T
v
T
C
T
T
T
C
T
T
C
v
T
T
T
T































































  
(62) 
4.4 Metabolic Energy in the Low Yank Limit 
We limit the model to low yanks by requiring the model parameters 𝛼𝛼𝑠𝑠𝑠𝑠 and 𝜀𝜀𝑠𝑠𝑠𝑠 in (24), and 𝛼𝛼𝑠𝑠𝑠𝑠 and 𝜀𝜀𝑠𝑠𝑠𝑠 
in (25) to have values such that for typical walking gaits we have 𝛼𝛼𝑠𝑠𝑠𝑠〈𝐹𝐹̇𝑠𝑠𝑠𝑠
2 〉 >> 𝜀𝜀𝑠𝑠𝑠𝑠〈𝐹𝐹𝑠𝑠𝑠𝑠
2 〉 and  𝛼𝛼𝑠𝑠𝑠𝑠〈𝐹𝐹̇𝑠𝑠𝑠𝑠
2 〉 >> 
𝜀𝜀𝑠𝑠𝑠𝑠〈𝐹𝐹𝑠𝑠𝑠𝑠
2 〉.  This has the effect of making the yank terms contribute most of the cost in the cost functional 
in (19) so that optimal trajectories will tend to have lower yanks.  In practice, we take the low yank limit 
by taking the limit as 𝜔𝜔𝑠𝑠𝑠𝑠 and 𝜔𝜔𝑠𝑠𝑠𝑠 in (20) become relatively low frequencies.  As the metabolic energy 
model used in [14-16] was a model of the metabolic energy per step of walking gait, we look specifically at 
the metabolic energy per step in the low yank limit. 
Following (3), the metabolic energy per step of the torso is: 
 


/2
2
/2
.
T
torso
st
st
ext ext
torso
T
W
F
f
x
dt









  
(63) 
We can split (63) into the sum 𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑊𝑊𝑠𝑠𝑠𝑠 + 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒 where 𝑊𝑊𝑠𝑠𝑠𝑠 is metabolic energy per step associated with 
generating force on the torso by the stance leg and 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒 is the metabolic energy per step associated with 
doing external work.  The metabolic energy per step 𝑊𝑊𝑠𝑠𝑠𝑠 associated with generating force on the torso is: 
 
/2
2
/2
.
T
st
st
st
T
W
F dt




  
(64) 
The metabolic energy per step 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒 associated with doing external work is: 
 
/2
/2
.
T
ext
ext
ext
torso
T
W
f
x
dt







  
(65) 
We first look the metabolic energy per step 𝑊𝑊𝑠𝑠𝑠𝑠 associated generating force on the torso by the stance leg 
in (64).  Combining (64) with (56) we obtain: 
 









/2
2
/2
2
2
3
2
sinh
sinh
cosh
.
T
st
st
sp
sp
T
st
sp
st
st
W
mC
t
mC
t
K
t
dt













  
(66) 
The constants in (66) are given in (57) and (59).  In Appendix 5, we take low yank limit by taking 𝜔𝜔𝑠𝑠𝑠𝑠→ 
0, we find: 
 
 
21 
 
 




2
2
2
3
2
/ 12
/
/ .
st
st
st
ext
W
g m
L
s
v
F s
v




  
(67) 
In this same limit, we also find: 
 
.
ext
ext
f
F

  
(68) 
The metabolic energy 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒 associated with doing external work in (65) therefore becomes: 
 
.
ext
ext
ext
W
F s


  
(69) 
Evaluating 𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 by combining (67) and (69) using the sum 𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑊𝑊𝑠𝑠𝑠𝑠 + 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒 we obtain the model of 
walking gait doing external work used in [15, 16], as we should. 
Following (3), the metabolic energy per step of the swing foot is: 
 
/2
2
/2
.
T
foot
sw
sw
T
W
F dt




  
(70) 
Combining (70) with (66), the metabolic energy per step 𝑊𝑊𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 associated with the motion of the swing foot 
is: 
 




/2
2
2
2
2
3
/2 6
sinh
.
T
foot
sw
sw
sw
T
W
t
C
t
dt








  
(71) 
In Appendix 6, we take low yank limit by taking 𝜔𝜔𝑠𝑠𝑠𝑠→ 0, we find: 
 
2 3
192
/ .
foot
sw
W
v
s


  
(72) 
This bears a functional resemblance to the swing foot trajectory used in [14-16].  We expect that it can 
approximate the model used in [14-16] reasonably well over the range of empirical data used in validating 
that model. 
4.5 Discussion 
In the symmetric and uniform walking gait model, the torso maintains a constant kinetic mechanical 
energy while the swing foot gains kinetic mechanical energy up to midswing and then loses mechanical 
energy from then until heel-strike.  The swing foot slows to zero velocity relative to the ground and there 
is no jarring impact with the ground.  Thus, all change in the kinetic mechanical energy of the body is 
affected by the action of the muscles.  However, it is clear from our discussion of heel-strike in Sec. 3 that 
we do not expect walking gaits to be symmetric, but instead we expect there to be a jarring due to impact 
with the ground.  We should therefore begin the process of using the optimal control model of walking gait 
to generalize the walking gait model in [14-16] to more realistic asymmetric walking gaits. 
Appendix 1 
We would like to calculate the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 of the torso as it is moved by the forces generated by the 
stance leg and generates the external force.  The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 satisfies (35); that is: 
 
 
22 
 
 
.
st
x
x
f
f
st
H
P Q
P Q
L







  
(73) 
The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 consists of a term 𝑃𝑃𝑥𝑥
𝛵𝛵𝑄𝑄̇𝑥𝑥 associated with the trajectory of the torso and 𝑃𝑃𝑓𝑓
𝛵𝛵𝑄𝑄̇𝑓𝑓 associated 
with the trajectory of the external force, and the Lagrangian 𝐿𝐿𝑠𝑠𝑠𝑠 in (26)  The generalized coordinates vectors 
𝑄𝑄𝑥𝑥 and 𝑄𝑄𝑓𝑓 given in (36). 
We first calculate the quantity 𝑃𝑃𝑥𝑥
𝛵𝛵𝑄𝑄̇𝑥𝑥 associated with the trajectory of the torso.  The generalized 
momentum vector 𝑃𝑃𝑥𝑥 and the control 𝑢𝑢𝑥𝑥 in (37) are: 
 
1
2
3
,
,
,
.
x
x
torso
P
p
p
p
u
x








  
(74) 
Using equation C in (11), we find the optimal trajectory satisfies 𝜕𝜕𝐻𝐻𝑠𝑠𝑠𝑠/𝜕𝜕𝑢𝑢𝑥𝑥 = 0; this gives: 
 


2
2
3
2
/
2
.
st
x
sp torso
ext
st
st
p
m
u
x
f
m
mF










  
(75) 
Using equation B in (11), we find that the generalized momentum vector 𝑃𝑃𝑥𝑥 for optimal trajectory satisfies 
𝑃𝑃̇𝑥𝑥= −𝜕𝜕𝐻𝐻𝑠𝑠𝑠𝑠/𝜕𝜕𝑄𝑄𝑥𝑥; this gives: 
 
2
2
2
2
0
0
0
1
0
0
2
.
0
1
0
st
sp
st
x
x
st
sp
st
st
st
F
P
P
m
F
F






































  
(76) 
Therefore, the generalized momentum vector 𝑃𝑃𝑥𝑥 is: 
 




2
2
2
2
0
2
.
1
0
st
st
st
sp
x
st
st
st
st
st
st
F
F
P
m
F
m
F
F











































  
(77) 
We finally find that 𝑃𝑃𝑥𝑥
𝛵𝛵𝑄𝑄̇𝑥𝑥 is: 
 








2
2
2
2
2
.
x
x
st
st
torso
sp
torso
st
st
st
st
torso
st
st
st
torso
P Q
F
mx
mx
F
F
x
F
F
x























  
(78) 
We next calculate the quantity 𝑃𝑃𝑓𝑓
𝛵𝛵𝑄𝑄̇𝑓𝑓 associated with the trajectory of the external force.  The generalized 
momentum vector 𝑃𝑃𝑓𝑓 and the control 𝑢𝑢𝑓𝑓 in (37) are: 
 
,
.
f
f
ext
P
p
u
f


  
(79) 
Using equation C in (11), we find optimal trajectory satisfies 𝜕𝜕𝐻𝐻𝑠𝑠𝑠𝑠/𝜕𝜕𝑢𝑢𝑓𝑓 = 0; this gives: 
 


2
2
/
2
.
st
torso
sp torso
f
st
st
p
m x
x
u
m
F










  
(80) 
 
 
23 
 
The generalized momentum vector 𝑃𝑃𝑓𝑓 (see Appendix 1) is: 
 
2
.
f
st
st
P
F



  
(81) 
We finally find that 𝑃𝑃𝑓𝑓
𝛵𝛵𝑄𝑄̇𝑓𝑓 is: 
 
2
.
f
f
st
st ext
P Q
F f





  
(82) 
Combining (26), (36), (73), (77), and (81), we find that the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 governing the motion of the 
torso is: 
 








2
2
2
2
2
2
st
st
st
st
st
st
st
st
st
torso
st
st
st
torso
st
H
F
F
F
F
x
F
F
x




















  
(83) 
Thus, the generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is an approximate constant of the motion of the torso that holds according 
to how well the approximation in (22) holds.  When moving in an optimal trajectory, the torso moves over 
the course of a step, between heel-strikes, in a way that keeps the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 approximately constant 
according to (83). 
Appendix 2 
We would like to calculate the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 of the swing foot as it is moved by the forces generated 
by the swing leg.  The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 satisfies (44); that is: 
 
.
sw
sw
H
P Q
L




  
(84) 
The Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 consists of a term 𝑃𝑃𝛵𝛵𝑄𝑄̇ associated with the trajectory of the swing foot, and the 
Lagrangian 𝐿𝐿𝑠𝑠𝑠𝑠 in (39).  The generalized coordinates vector 𝑄𝑄 given in (45). 
We calculate the quantity 𝑃𝑃𝛵𝛵𝑄𝑄̇ associated with the trajectory of the swing foot.  The generalized 
momentum vector 𝑃𝑃 and the control 𝑢𝑢 are: 
 
1
2
3
,
,
,
.
foot
P
p
p
p
u
x








  
(85) 
Using equation C in (11), we find optimal trajectory satisfies 𝜕𝜕𝐻𝐻𝑠𝑠𝑠𝑠/𝜕𝜕𝜕𝜕 = 0; this gives: 
 


2
2
3
2
2
.
sw
sp
foot
sw
sw
p
u
x
F










  
(86) 
Using equation B in (11), we find that the generalized momentum 𝑃𝑃 for optimal trajectory satisfies 𝑃𝑃̇ = 
−𝜕𝜕𝐻𝐻𝑠𝑠𝑠𝑠/𝜕𝜕𝜕𝜕; this gives: 
 
2
2
2
2
0
0
0
1
0
0
2
.
0
1
0
sw
sp
sw
sw
sp
sw
sw
sw
F
P
P
F
F






































  
(87) 
 
 
24 
 
The generalized momentum vector 𝑃𝑃 is: 
 




2
2
2
2
0
2
.
1
0
sw
sw
sw
sp
sw
sw
sw
sw
sw
sw
F
F
P
F
F
F










































 
(88) 
We finally find that 𝑃𝑃𝛵𝛵𝑄𝑄̇ is: 
 






2
2
2
2
2
.
sw
sw
sw
sw
sw
sw
foot
sw
sw
sw
foot
P Q
F
F
F
x
F
F
x


















  
(89) 
Combining (39), (45), (84), and (88), we find that the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 is: 
 








2
2
2
2
2
2
.
sw
sw
sw
sw
sw
sw
sw
sw
sw
foot
sw
sw
sw
foot
sw
H
F
F
F
F
x
F
F
x




















  
(90) 
Thus, the generalized energy 𝛹𝛹𝑠𝑠𝑠𝑠 is an approximate constant of the motion of the swing foot that holds 
according to how well the approximation in (22) holds.  When moving in an optimal trajectory, the swing 
foot moves over the course of a step, between heel-strikes, in a way that keeps the Hamiltonian 𝐻𝐻𝑠𝑠𝑠𝑠 
approximately constant according to (47). 
Appendix 3 
The optimal trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the torso for symmetric walking gait for a step taking place on the 
time interval −𝑇𝑇/2 ≤ 𝑡𝑡 ≤ 𝑇𝑇/2 has the general form given in (56); that is: 
 







1
2
3
sinh
cosh
sinh
.
torso
sp
sp
st
x
t
C
t
C t
t
C
t






  
(91) 
We would like to find parameters 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3 so that the trajectory 𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑡𝑡) of the torso is approximately 
that of uniform walking gait as given by (55); namely: 
 

.
torso
x
t
vt

  
(92) 
We expand the symmetric torso trajectory in (91) as a power series in 𝑡𝑡 using the usual series expansions 
of the sinh and cosh functions: 
 













2
1
2
1
2
0
0
2
1
3
0
2
1 !
2
!
.
2
1 !
n
n
sp
sp
torso
n
n
n
st
n
t
t
x
t
C
C t
n
n
t
C
n



















  
(93) 
As we have three parameter values 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3 to solve for, we would like to truncate the series expansion 
in (93) to three independent terms.  We do this by truncating the expansion to fifth-order in 𝑡𝑡; we find: 
 
 
25 
 
 







1
2
3
3
2
3
3
1
2
3
5
4
5
5
1
2
3
3
/ 3!
5
/ 5!.
torso
sp
sp
sp
sp
st
sp
sp
st
x
t
C
C
C
t
C
C
C
t
C
C
C
t

















  
(94) 
Combining (92) and (94), we find that we can calculate 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3 by solving a linear system of three 
equations in the three unknowns 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3: 
 
1
3
2
3
2
5
4
5
3
1
3
0 .
0
5
sp
st
sp
sp
st
sp
sp
st
C
v
C
C






































  
(95) 
The parameter values 𝐶𝐶1, 𝐶𝐶2, and 𝐶𝐶3 that satisfy (92) are: 
 













3
5
2
2
2
1
3
3
4
2
2
2
3
4
3
2
2
2
/
3 / 2
,
1 / 2
/
,
.
st
sp
sp
sp
st
st
sp
sp
st
sp
st
sp
st
C
C
C
C
C
v




































  
(96) 
Appendix 4 
The optimal trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) of the swing foot for symmetric walking gait for a step taking place on 
the time interval −𝑇𝑇/2 ≤ 𝑡𝑡 ≤ 𝑇𝑇/2 has the general form given in (61); that is: 
 



3
1
2
3 sinh
.
foot
sw
x
t
vt
t
t
C
t







  
(97) 
We would like to find a swing foot trajectory 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑡𝑡) that satisfies the initial and final conditions given in 
(54); namely: 
 












/ 2
,
/ 2
,
/ 2
0,
/ 2
0,
/ 2
0,
/ 2
0.
foot
foot
foot
foot
foot
foot
x
T
s
x
T
s
x
T
x
T
x
T
x
T













  
(98) 
We can make the symmetric swing foot trajectory in (97) satisfy (98) by finding the three parameter values 
κ1, κ2, and 𝐶𝐶3 that satisfy the linear system of three equations given by: 
 






3
1
2
2
2
3
4
8 sinh
/ 2
8
4
3
4
cosh
/ 2
0 .
0
0
3
sinh
/ 2
sw
sw
sw
sw
sw
T
T
T
s
T
T
C
T
T












































  
(99) 
The parameter values κ1, κ2, and 𝐶𝐶3 that satisfy (99) are: 
 
 
26 
 
 


















2
1
3
2
2
3
3
3
2
24
2
sinh
/ 2
,
12
sinh
/ 2
,
3
12
.
12
sinh
/ 2
6
cosh
sw
sw
sw
sw
sw
sw
sw
sw
T
v
T
C
T
T
T
C
T
T
C
v
T
T
T
t































































  
(100) 
Appendix 5 
The metabolic energy per step is associated with the motion of the torso is given in (66); it is: 
 









/2
2
/2
2
2
3
2
sinh
sinh
cosh
.
T
st
st
sp
sp
T
st
sp
st
st
W
mC
t
mC
t
K
t
dt













  
(101) 
The constants are: 
 








3
4
2
2
2
3
4
3
2
2
2
1
1 / 2
/
,
,
sinh
/ 2
1
1
.
/ 2
st
sp
sp
st
sp
st
sp
st
st
ext
st
C
C
C
v
T
K
F
T




















































  
(102) 
We look at the metabolic energy per step of the torso in the limit as 𝜔𝜔𝑠𝑠𝑠𝑠→ 0.  In this limit, we find that 
the term in (101) in 𝐶𝐶2 goes to zero.  Taking the square of the integrand and noting that odd terms will 
integrate to zero, we find: 
 






/2
2
2
2
2
2
2
3
/2
/2
2
2
/2
sinh
cosh
.
T
st
st
st
sp
st
T
T
st
st
T
W
m C
t dt
K
t dt













  
(103) 
The integrals in (103) evaluate to: 
 












/2
2
/2
/2
2
/2
sinh
sinh
1
/ 2 ,
sinh
cosh
1
/ 2 .
T
st
st
T
st
T
st
st
T
st
T
t dt
T
T
T
t dt
T
T
































  
(104) 
In the limit we are looking at, we may make the approximation 𝑠𝑠𝑠𝑠𝑠𝑠ℎ(𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡) ≈ 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 + (𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡)3/6.  Therefore, 
in the limit as 𝜔𝜔𝑠𝑠𝑠𝑠→ 0, we find: 
 
 
27 
 
 




/2
2
2
3
/2
/2
2
/2
3
sinh
/ 12,
cosh
,
/
,
.
T
st
st
T
T
st
T
st
ext
t dt
T
t dt
T
C
v
K
F












  
(105) 
Thus, (101) approximates to: 
 
4
2 2
3
2
/ 12
.
st
st
sp
st
ext
W
m v T
F T




  
(106) 
Evaluating 𝜔𝜔𝑠𝑠𝑠𝑠
4  using the definition in (20) and noting that 𝑠𝑠 = 𝑣𝑣𝑣𝑣, we find: 
 




2
2
2
3
2
/ 12
/
/ .
st
st
st
ext
W
g m
L
s
v
F s
v




  
(107) 
Appendix 6 
The metabolic energy per step is associated with the motion of the swing foot is given in (71); it is: 
 




/2
2
2
2
2
3
/2 6
sinh
.
T
foot
sw
sw
sw
T
W
t
C
t
dt








  
(108) 
The constants are: 
 














2
2
3
3
3
2
sinh
/ 2
,
3
12
.
12
sinh
/ 2
6
cosh
sw
sw
sw
sw
sw
sw
T
T
C
T
T
C
v
T
T
T
T











































  
(109) 
We look at the metabolic energy per step of the torso in the limit as 𝜔𝜔𝑠𝑠𝑠𝑠→ 0.  In this limit, we find that 
the term in (108) in κ2 goes to zero.  Taking the square of the integrand, we find: 
 


/2
2
2
2
2
3
/2 sinh
.
T
foot
sw
sw
sw
T
W
C
t dt






  
(110) 
The integral in (110) evaluates to: 
 






/2
2
/2
sinh
sinh
1
/ 2 .
T
sw
sw
T
sw
T
t dt
T
T
















  
(111) 
In the limit we are looking at, we may make the approximation 𝑠𝑠𝑠𝑠𝑠𝑠ℎ(𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡) ≈ 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 + (𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡)3/6.  Therefore, 
in the limit as 𝜔𝜔𝑠𝑠𝑠𝑠→ 0, we find: 
 


/2
2
2
3
/2 sinh
/ 12.
T
sw
sw
T
t dt
T





  
(112) 
 
 
28 
 
We now approximate the constant C3 in (109), in this case we make the 𝑠𝑠𝑠𝑠𝑠𝑠ℎ(𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡) ≈ 𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡 and 𝑐𝑐𝑐𝑐𝑐𝑐ℎ(𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡) 
≈ 1 + (𝜔𝜔𝑠𝑠𝑠𝑠𝑡𝑡)2/2; we find: 
 
3
3
2
48
.
sw
C
v
T














  
(113) 
Combining (110), (112), and (113), we find: 
 
2 2
192
/
.
foot
sw
W
v
T


  
(114) 
Noting that 𝑠𝑠 = 𝑣𝑣𝑣𝑣, we find: 
 
2 3
192
/ .
foot
sw
W
v
s


  
(115) 
References 
[1] 
G. Schöner, "Recent developments and problems in human movement science and their conceptual 
implications," Ecol Psychol, vol. 7, pp. 291-314, 1995. 
[2] 
J. P. Scholz, Schöner, G., "The uncontrolled manifold concept:  identifying control variables for a 
functional task," Exp Brain Res, vol. 126, pp. 289-306, 1999. 
[3] 
J. P. Scholz, Schöner, G., Latash, M.L., "Motor control of pistol shooting:  identifying variables 
with the uncontrolled manifold," Exp Brain Res, vol. 135, pp. 382-404, 2000. 
[4] 
J. P. Scholz, Reisman, D., Schöner, G., "Effects of varying task constraints on solutions to joint 
control  in sit-to-stand," Exp Brain Res, vol. 141, pp. 485-500, 2001. 
[5] 
J. P. Scholz, Schöner, G., Hsu, W.L., Jeka, J.J., Horak, F., Martin, V., "Motor equivalent of control 
of the center of mass in response to support surface perturbations," Exp Brain Res, vol. 180, pp. 
163-179, 2007. 
[6] 
Y. Nubar, Contini, R., "A minimum principle in biomechanics," Bull Math Biol, vol. 23, pp. 377-
391, 1961. 
[7] 
W. L. Nelson, "Physical principles for economies of skilled movement," Biol Cybern, vol. 46, pp. 
135-147, 1983. 
 
 
29 
 
[8] 
N. Hogan, "An organizing principle for a class of voluntary movements," J Neurosci, vol. 4, pp. 
2745-2754, 1984. 
[9] 
T. Flash, Hogan, N., "The coordination of arm movements:  an experimentally confirmed 
mathematical model," J Neurosci, vol. 5, pp. 1688-1703, 1985. 
[10] 
Y. Uno, Kawato, M., Suzuki, R., "Formation and control of optimal trajectory in human multijoint 
arm movement," Biol Cybern, vol. 61, pp. 89-101, 1989. 
[11] 
M. G. Pandy, Garner, B.A., Anderson, F.C., "Optimal control of non-ballistic muscular movements: 
a constraint-based performance criterion for rising from a chair," J Biomech Eng, vol. 117, pp. 15-
26, 1995. 
[12] 
S. Hagler, Jimison, H., Bajcsy, R., Pavel, M., "Quantification of human movement for assessment 
in automated exercise coaching," Conf Proc IEEE Eng Med Biol Soc 2014, pp. 5836-5839, 2014. 
[13] 
S. Hagler, "On the principled description of human movements," arXiv:1509.06981 [q-bio.QM], 
2015. 
[14] 
S. Hagler, "Patterns of selection of human movements I:  Movement utility, metabolic energy, and 
normal walking gaits," arXiv:1608.03583 [q-bio.QM], 2016. 
[15] 
S. Hagler, "Patterns of selection of human movements II:  Movement limits, mechanical energy, 
and very slow walking gaits," arXiv:1610.00763 [q-bio.QM], 2016. 
[16] 
S. Hagler, "Patterns of selection of human movements III:  Energy efficiency, mechanical advantage, 
and walking gait," arXiv:1611.04193 [q-bio.QM], 2016. 
[17] 
E. Atzler, Herbst, R., "Arbeitsphysiologische studien III. teil," Pflügers Arch Gesamte Physiol 
Menschen Tiere, vol. 215, pp. 291-328, 1927. 
[18] 
D. W. Grieve, "Gait patterns and the speed of walking," Biomed Eng, vol. 3, pp. 119-122, 1968. 
 
 
30 
 
[19] 
G. Frazzitta, Maestri, R., Uccellini, D., Bertotti, G., Abelli, P., "Rehabilitation treatment of gait 
patients with Parkinson's disease and freezing:  A comparison between two physical therapy tools 
using visual and auditory cues with or without treatmill training," Movement Disorders, vol. 24, 
pp. 1139-1143, 2009. 
[20] 
S. Hagler, Austin, D., Hayes, T.L., Kaye, J., Pavel, M., "Unobtrusive and ubiquitous in-home 
monitoring:  A methodology for continuous assessment of gait velocity in elders," IEEE Trans 
Biomed Eng, vol. 57, pp. 813-820, 2010. 
[21] 
C. M. Donovan, Brooks, G.A., "Muscular efficiency during steady rate exercise II.  Effects of walking 
speed and work rate," J Appl Physiol Respir Environ Exerc Physiol, vol. 43, pp. 431-439, 1977. 
[22] 
D. A. Winter, Biomechanics and Motor Control of Human Movement, 3rd ed.: Wiley, 2005. 
[23] 
S. S. Stevens, "On the psychophysical law," Psychol Rev, vol. 64, pp. 153-181, 1957. 
[24] 
S. S. Stevens, Psychophysics:  Introduction to its perceptual, neural, and social prospects. New 
York: Wiley, 1975. 
[25] 
D. E. Kirk, Optimal control theory:  An introduction. Englewood Cliffs, NJ: Prentice-Hall, 1970. 
[26] 
T. McGeer, "Passive Dynamic Walking," Int J Robot Res, vol. 9, pp. 62-82, 1990. 
[27] 
A. D. Kuo, "A simple model of bipedal walking predicts the preferred speed-step length 
relationship," J Biomed Eng, vol. 123, pp. 264-269, 2001. 
[28] 
A. D. Kuo, "Energetics of actively powered locomotion using the simplest walking model," J 
Biomech Eng, vol. 124, pp. 113-120, 2002.