A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait
The presence of compliant elements in biped running mechanisms generates a smoother motion and decreases impact forces. Biological creatures that have a complicated actuation system with parallel and series elastic elements in their muscles demonstrate very efficient and robust bipedal gaits. The ma...
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irk-123456789-1882702023-02-19T01:27:35Z A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait Dadashzadeh, B. Allahverdizadeh, A. Esmaeili, M. Fekrmandi, H. The presence of compliant elements in biped running mechanisms generates a smoother motion and decreases impact forces. Biological creatures that have a complicated actuation system with parallel and series elastic elements in their muscles demonstrate very efficient and robust bipedal gaits. The main difficulty of implementing these systems is duplicating their complicated dynamics and control. This paper studies the effects of an actuation system, including Hill-type muscles on the running efficiency of a kneed biped robot model with point feet. In this research, we implement arbitrary trajectories compatible with the initial condition of the robot, and we calculate the necessary muscle forces using an analytical inverse dynamics model. To verify the results, we execute the direct dynamics of the robot with the calculated control inputs to generate the robot’s trajectory. Finally, we calculate the contractile element force of the muscles and its cost of transport, and we investigate the effects of the muscles’ elements on reducing or increasing the cost of transport of the gait and maximum actuating forces. Наявність податливих елементів у механізмах виконання двоногої ходи породжує плавність руху і зменшує сили удару. Біологічні істоти, які мають складну систему спрацьовування з паралельними і серіями еластичних елементів в м'язах, демонструють дуже ефективну і надійну двоногу ходу. Основними труднощами реалізації цих систем є дублювання їх складної динаміки і контролю. Вивчаються наслідки спрацьовування системи, в тому числі м'язів типу Хілла на ефективність роботи коліна у моделі двоногої ходи. У цьому дослідженні введено довільні траєкторії, сумісні з початковим станом робота, і розраховано необхідні сили м'язів за допомогою аналітичної зворотної моделі динаміки. Для перевірки результатів виконано пряму динаміку робота з обчисленням імпульсів управління для генерації траєкторії робота. Розраховано скорочувальну силу елемента м'язів і вартість його передачі для робота, і досліджено вплив елементів м'язів на зменшення або збільшення витрат на ходу і максимальні виконавчі сили. 2020 Article A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait / B. Dadashzadeh, A. Allahverdizadeh, M. Esmaeili, H. Fekrmandi // Прикладная механика. — 2020. — Т. 56, № 4. — С. 133-144. — Бібліогр.: 18 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/188270 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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| description |
The presence of compliant elements in biped running mechanisms generates a smoother motion and decreases impact forces. Biological creatures that have a complicated actuation system with parallel and series elastic elements in their muscles demonstrate very efficient and robust bipedal gaits. The main difficulty of implementing these systems is duplicating their complicated dynamics and control. This paper studies the effects of an actuation system, including Hill-type muscles on the running efficiency of a kneed biped robot model with point feet. In this research, we implement arbitrary trajectories compatible with the initial condition of the robot, and we calculate the necessary muscle forces using an analytical inverse dynamics model. To verify the results, we execute the direct dynamics of the robot with the calculated control inputs to generate the robot’s trajectory. Finally, we calculate the contractile element force of the muscles and its cost of transport, and we investigate the effects of the muscles’ elements on reducing or increasing the cost of transport of the gait and maximum actuating forces. |
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| author |
Dadashzadeh, B. Allahverdizadeh, A. Esmaeili, M. Fekrmandi, H. |
| spellingShingle |
Dadashzadeh, B. Allahverdizadeh, A. Esmaeili, M. Fekrmandi, H. A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait Прикладная механика |
| author_facet |
Dadashzadeh, B. Allahverdizadeh, A. Esmaeili, M. Fekrmandi, H. |
| author_sort |
Dadashzadeh, B. |
| title |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait |
| title_short |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait |
| title_full |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait |
| title_fullStr |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait |
| title_full_unstemmed |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait |
| title_sort |
case study on influence of utilizing hill-type muscles on mechanical efficiency of biped running gait |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188270 |
| citation_txt |
A Case Study on Influence of Utilizing Hill-type Muscles on Mechanical Efficiency of Biped Running Gait / B. Dadashzadeh, A. Allahverdizadeh, M. Esmaeili, H. Fekrmandi // Прикладная механика. — 2020. — Т. 56, № 4. — С. 133-144. — Бібліогр.: 18 назв. — англ. |
| series |
Прикладная механика |
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2025-07-16T10:15:42Z |
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| fulltext |
2020 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 56, № 4
ISSN0032–8243. Прикл. механика, 2020, 56, № 4 133
B . D a d a s h z a d e h 1 * , A . A l l a h v e r d i z a d e h 1 ,
M . E s m a e i l i 1 , H . F e k r m a n d i 2
A CASE STUDY ON INFLUENCE OF UTILIZING HILL-TYPE MUSCLES
ON MECHANICAL EFFICIENCY OF BIPED RUNNING GAIT
1Department of Mechatronics Engineering, School of Engineering Emerging Technologies,
University of Tabriz, Tabriz, Iran;
2Department of Mechanical Engineering, South Dakota School of Mines & Technology,
Rapid City, USA, *e-mail: b.dadashzadeh@tabrizu.ac.ir
Abstract. The presence of compliant elements in biped running mechanisms generates a
smoother motion and decreases impact forces. Biological creatures that have a complicated
actuation system with parallel and series elastic elements in their muscles demonstrate very
efficient and robust bipedal gaits. The main difficulty of implementing these systems is du-
plicating their complicated dynamics and control. This paper studies the effects of an actua-
tion system, including Hill-type muscles on the running efficiency of a kneed biped robot
model with point feet. In this research, we implement arbitrary trajectories compatible with
the initial condition of the robot, and we calculate the necessary muscle forces using an ana-
lytical inverse dynamics model. To verify the results, we execute the direct dynamics of the
robot with the calculated control inputs to generate the robot’s trajectory. Finally, we calcu-
late the contractile element force of the muscles and its cost of transport, and we investigate
the effects of the muscles’ elements on reducing or increasing the cost of transport of the
gait and maximum actuating forces.
Key words: biped running, gait planning, hill-type muscle, cost of transport.
1. Introduction.
Despite numerous research on planning and controlling bipedal running, there is still no
biped robot capable of running as efficiently as animals. In this area, the main difference
between robots and animals is that, in robots, muscles are replaced with electric or pneumat-
ic motors. Developing actuators that operate with a dependency on muscle structure can be a
solution for future robotic efficiency. In this research, we consider Hill-type muscles instead
of rotational motors as actuators of a biped robot model. Additionally, we plan periodic run-
ning gaits including a stance phase, take-off event, flight phase, and touch-down event.
By investigating muscle structure, Hill [1] described its mechanical model elements as
follows: one contractile element; a series elastic element; and a parallel elastic element. This
model has been used by biomechanics researchers as a base model for muscles. Lichtwark
[2] et al. used ultrasonic sensors to detect length variations of muscle elements during walk-
ing and running. Iida et al. [3] proposed bi-articular springs on the thigh and shin, instead of
muscles for a robot with minimal hip actuation, which could generate walking and a special
running gait in experiments.
Pneumatic artificial muscles have been developed as a mechanical realization of mus-
cles. However, they have some drawbacks; pneumatic muscles have small movement
strides, and it is difficult to supply compressed air for a biped robot. Hosoda et al. [4] used
antagonistic pneumatic actuators to generate biped walking and running gaits and showed
that pneumatic actuators have good characteristics to generate human-like motion. Niiyama
134
[5] developed a biped robot, named Athlete, with pneumatic artificial muscles that could run
eight steps with a forward speed of 3 m/s.
The Spring Loaded Inverted Pendulum (SLIP) model with a spring and point mass has
been proven as a basic and efficient model for biped walking and running [6]. Also, the
Point Mass Biped (PMB) model consisting of a point mass and prismatic force actuator can
generate biped walking and running gaits by minimizing mechanical work costs [7], as well
as smoother force profiles by minimizing a hybrid cost function including mechanical work,
leg force, and its derivatives [8]. PMB model has proven to be a more general model than a
SLIP model, capable of generating arbitrary trajectories for bipedal running with single-
hump and double-hump ground reaction force (GRF) profiles [9]. Minimal robots have been
fabricated based on these models. For example, monopod II [13] has an energy expenditure
closer to these basic models than real robots like HRP-2L [14]. However, it has very limited
motion capabilities. To generate efficient biped running gaits, the gait model of the main
biped robot can be optimized, or a minimal optimal template can be tracked by the main
robot. Guo et al. [10] proposed a nonlinear optimization algorithm to generate optimal biped
running gaits with different velocities on level ground and stairs. Also, the SLIP model has
been used as a template to control the running of real biped robots using HZD [11] and force
control [12].
Since biological actuation systems perform better than mechanical robots, this work
simulates and investigates the benefits or drawbacks of the Hill-type muscle in bipedal run-
ning by studying the effects of muscle parameters on running efficiency. The existence of a
spring series with the motor isolates external impulsive forces from the motor but makes it
difficult to control [12, 18]. Where compliance exists in parallel to the motor, spring torque
can reduce the required torque of the motor [15,16]. With a Hill-type muscle, the benefits or
drawbacks of both cases may influence the system. Also, a PMB model instead of a SLIP
model is used as a template for the multibody biped model.
2. Muscled Biped Robot Model.
2.1. Multibody Robot Model. This paper considers a 5-link planar biped model with
point feet for dynamic modeling and gait planning, as shown in Fig. 1. Thigh and shin links
have mass and moment of inertia that are actuated by Hill-type muscles. The torso angle is
assumed to be locked at a vertical orientation
5( 0). All of the links have the length of
0,5 m. To simplify the system and build an
efficient robotic architecture similar to mus-
culoskeletal systems, a single, tensional-
compressive Hill-type muscle is considered
for each joint instead of biarticular tensional
muscles. This is because a robotic actuator
can exert both tensional and compressive
forces whereas a biological muscle can exert
only tensional force. To enable the muscles
to exert torque to links, small and massless
bone extensions with equal lengths,
6 7 8, , ,exL L L L are attached rigidly to thigh
1, the torso, and thigh 2, respectively, with
fixed angles. The fixed angles are assumed
to be kneeex,86 and ..,7 hipex
Lengths and angles of these extensions will
be considered as parameters in gait design.
The mass of the shin is 2 kg, the thigh is 2
kg, and the torso is 58 kg.
The running gait consists of a stance
Fig 1. The 5-link biped model with
muscle actuation
135
phase, take-off event, flight phase, and touch-down event, and their dynamic models are
needed for gait planning and simulations.
In the stance phase, point A is pivoted to the ground according to Fig. 1. The robot has 4
DOF1 with the generalized coordinates of 1 2 3 4[ , , , ] .T
s q Input vector 1 2 3[ , , ,u F F F
4 ]TF consists of muscle output forces. The Lagrange equation is used to obtain the dynamic
model of the system
,
d L L
dt
Q
q q
(1)
in which L is the Lagrangian function that is the subtraction of kinetic and potential ener-
gies, and Q is the generalized force vector. The positions of each joint and link’s COM2 are
written as a function of generalized coordinates, and their velocities are obtained by symbol-
ic differentiation. The kinetic and potential energies of the system are written as a function
of generalized coordinates and their time derivatives. Kinetic energy is the sum of transla-
tional and rotational kinetic energies of all the links, and potential energy consists of their
gravitational potential energies. The muscles and their bone extensions are assumed to be
massless, and, because their output forces are considered in the robot model, they do not
contribute to kinetic and potential energy terms. Muscle lengths ,,, DFCDAB and PS are
written as a function of generalized coordinates, and the generalized forces are calculated
using virtual works as
.4,3,2,1,4321
i
q
PS
F
q
DF
F
q
CD
F
q
AB
FQ
iiii
i (2)
The dynamic equations of the stance phase are summarized as
( ) ( , ) ( ) ,s s s s s s s sD C B u q q q q q (3)
in which sD is 44 inertia matrix, sC is 14 Coriolis and gravity matrix, and sB is 44
input matrix.
In the flight phase, the robot undergoes a ballistic trajectory and has 6 DOF. General-
ized coordinates of the flight phase are chosen as 1 2 3 4[ , , , , , ]T
f E Ex y q including link
angles and hip position. Similarly, the dynamic model of the flight phase is written as
( ) ( , ) ( ) ,f f f f f f f fD C B u q q q q q (4)
in which fD is 66 matrix, fC is 16 , and sB is .46
The take-off event is an instantaneous transition from the stance to flight phase and takes
place when the vertical component of GRF reaches zero. In this instance, the initial state of
the flight phase is captured as
[ ; ( ); ( )],f s E s E sx y q q q q (5)
in which superscript + indicates the moment just after the event and – indicates just before
the event.
The touch-down event takes place at the end of the flight phase when the toe-tip position
reaches ground. The contact is assumed to be fully plastic with no rebound, after which
point S makes an ideal pivot to the ground. The toe of the robot should have a sufficient
coefficient of friction with the ground to prevent sliding effects. The touch-down map,
which is used to convert pre-touch-down velocities to post-touch-down velocities, is derived
using Lagrange’s impact equation,
ˆ( ) ( ) ,f f f fD q q q Q (6)
1 Degree of Freedom
2 Center of Mass
136
and two constraint equations of post-contact foot pivoting to the ground. More details can be
found in[16].
2.2. Muscle Model as the Robot Actuating System. Having the necessary forces of
the muscle ends (F) known for the running gait, we use a Hill-type muscle model to calcu-
late its contractile element force ),( hF as shown in Fig. 2. The only difference between the
considered muscle model in this paper and a traditional Hill-type muscle model is that the
control input hF generated by a force actuator can be tensile or compressive. 1k and 1c
represent a stiffness and damping parallel to the force actuator, corresponding to perimysi-
um, endomysium, epimysium, and sarcolemma mechanical properties in a muscle. 2k is a
series elastic component corresponding to a tendon.
To calculate the muscle actuating force ,hF known as the contractile element force, in
terms of muscle external force ,F dynamic equations of the muscle components (Fig. 2) are
used.
2 ,0( )m a bF k x x x ; (7)
1 ,0 1( )h a a aF k x x c x F . (8)
The time derivative of Eq. (7) is written as
2 ( )m aF k x x . (9)
From kinematic analysis of the desired gait, muscles lengths mx and their time deriva-
tives mx are known numerically as a function of time. Also, muscles forces F and their
time derivatives F are known from inverse dynamics, as mentioned in Sections 3.1 and 3.2.
Eqs. (7) to (9) are 3 equations with 3 unknowns, ah xF , and ,ax which yield muscle con-
tractile element force hF as a function of time.
Fig 2. The Hill-type muscle model.
3. Running Gait Planning.
There are two approaches of using forward dynamics or inverse dynamics for gait plan-
ning. In the first approach, an optimization problem with respect to initial condition of the
robot and actuator force profiles is solved to obtain a periodic gait satisfying the constraints.
In this research, the second approach is used because the system is fully actuated. This ap-
proach first considers a desired trajectory for the robot and then utilizes inverse dynamics to
calculate the amount of muscle force needed to generate the desired gait. Then, a muscle
model is used to calculate the actuation force of the muscle’s contractile element.
3.1 Stance Phase. We use a point mass biped (PMB) model that has a massless force-
actuated leg as the template for the real robot, and we consider a degree-4 polynomial trajec-
tory for it in the stance phase, as shown in Fig. 3. The leg force is denoted by rF and the leg
137
angle, with respect to vertical, by θ. The degree-4
polynomial path is similar qualitatively to the SLIP
trajectory and can generate running gaits with any
desired initial condition of the stance phase with the
correct initial and final accelerations [9]. The PMB
path in the stance phase is considered as
,0
2
1
4
2 axaxay (10)
in which coefficients ia are calculated using the
COM initial position, velocity, and acceleration.
Combining the path equation with the dynamic
equation of the PMB system leads to a second order
nonlinear differential equation [9]. By solving this
equation numerically, the trajectory is calculated as
a function of time.
Knowing the COM trajectory of the multibody robot, we need to convert it to the links’
motions. In the stance phase, the stance leg and swing leg move separately. Stance leg an-
gles 1 and 2 are calculated using the position of the fixed pivot A and hip joint E in each
instance (Fig. 4, a). Because the torso angle is assumed to be fixed and the legs’ masses are
negligible relative to the torso, we assume the robot mass is concentrated in point E. So, the
trajectory of point E is the planned degree-4 polynomial (10).
a b
Fig 4. Planning leg angles in stance phase (a) stance leg (b) swing leg.
Fig 5. A professional runner’s leg angles during the stance phase
The swing leg angles are inspired by professional runner Usain Bolt’s running gait, which is
shown in Fig. 5. He is the fastest man in the world as of 2019, holding several Olympic rec-
Fig 3. The degree-4 polynomial
trajectory of the robot COM in the
stance phase.
138
ords in running. According to the video frames, he flexes his swing knee to decrease the
moment of inertia of the swing leg as much as possible. This increases the swing leg angular
velocity without the need of muscle forces by the law of angular momentum conservation.
So, we consider the desired angle between the thigh and shin of the swing leg to be constant,
,knee during the stance phase. The swing leg angle (from hip to toe) is assumed to be sym-
metric with the stance leg, as shown in Fig. 4, b. The relative velocity relationship between
the two ends of each joint, for example
/C A AC C Av v r
, (11)
and their relative acceleration relationship
/ /( )C A AC AC C A AC C Aa a r r
, (12)
are solved together to find the angular velocity and acceleration of each link.
Muscle forces during the stance phase are calculated using inverse dynamics by substitut-
ing the angular velocity and acceleration of links into Eq. (3) in each instance.
3.2. Flight Phase. In the flight phase, the robot’s COM moves as a projectile, but its
links need to be controlled to touch down with an appropriate condition. To plan the trajec-
tory of the robot links in the flight phase, a degree-5 polynomial versus time, Eq. (13), is
considered for each link because it is sufficient to capture the initial and final velocities and
accelerations
5 4 3 2
1 2 3 4 5 6t a t a t a t a t a t a . (13)
In this equation, coefficients 1a to 6a are obtained using the desired angle, angular ve-
locity, and angular acceleration of each link at the take-off and pre-touch-down moments.
The time duration of the flight phase is derived using the PMB model. The angular velocity
and acceleration of the links as a function of time are obtained using time derivatives of Eq.
(13). Then, the muscles forces are calculated using inverse dynamics (4).
3.3. Energy Expenditure Function. To measure the energy expenditure of the robot,
we use the cost of transport (COT) function that is the consumed energy per unit weight per
unit distance traveled
1 2 3 4 5
COT
( )
W
m m m m m gL
, (14)
in which L is the step length. W is the mechanical work done during the stance and flight
phases, which is calculated for overall muscle as
4
, ,0
1
stance stance flight
stance
t t t
F i m i i m it
i
W F x dt F x dt
(15)
and for the muscle contractile element as
4
, , , ,0
1
.
stance stance flight
h
stance
t t t
F h i a i h i a it
i
W F x dt F x dt
(16)
Using these two works, we can calculate overall muscle cost of transport
hFCOT and
contractile element cost of transport
hFCOT to investigate muscle efficiency effects.
4. Simulation Results.
A COM trajectory as Eq. (10) is planned for the stance phase with an initial leg length
of 0,823 m,r angle of 0,360 rad, initial COM velocity of 3( ),cmx m s
0,5( ),cmy m s and initial COM acceleration of 0, .cm cmx y g Since the system (3)
is fully actuated, its inverse dynamics can be solved using the method described in Section
3.1 to make the robot’s COM undergo trajectory (10). Then, the flight inverse dynamics (4)
is solved to make the robot links undergo trajectories (13). These calculations result in re-
quired muscle forces )(),(),( 321 tFtFtF and )(4 tF for one complete step of running.
139
The constants ,exL ,,kneeex ,,hipex and kneeex, (Figs. 1 and 4) that have significant effects
on the results are considered as gait planning parameters. To study the effects of these pa-
rameters on nonlinear dynamics of the system, we consider discrete values for them within
their acceptable range as
,
,
0,1, 0,15, 0,2 ; 155 ,165 ,175 ;
155 ,165 ,175 ; 30 , 48 , 66 , 84 .
ex ex knee
ex hip knee
L m
(17)
Gait planning is executed for all combinations of these parameters’ values. The maxi-
mum force value of all muscles, ,4,3,2,1, iFi during one step in each case is shown in
Fig. 6. In this figure, the horizontal axis shows ,knee subplots of each row has a constant
,,kneeex and each column has a constant .,hipex The numbers written in each curve show its
exL value. maxF decreases with increasing ,knee decreases with ,exL decreases with
,,kneeex and increases with .,hipex Therefore, to restrict the maximum force of the muscles,
we need to choose larger ,knee greater ,exL larger ,,kneeex and smaller .,hipex However,
the bone extension length exL would be restricted by the robot geometrics.
In each simulated case, we calculate the overall muscle COT. Interestingly, FCOT is
only a function of ,knee as shown in Fig. 7, and is constant while the other three parameters
vary. This is because, although changing the bone extension length and angle varies the
muscles force and velocity profiles, the time integral of their product is constant for a specif-
ic trajectory of the robot. Therefore, to have a smaller COT, we have to choose a larger knee
of the swing leg in the stance phase.
We use a Hill-type muscle model to calculate the contractile elements’ force profiles and their
COT. The parameters of Hill-type muscles are chosen to have proper values for the robot mass
and geometrics, as shown in Table 1. With this actuating system,
hFCOT is calculated using Eq.
(16) for each running gait simulation. The plots in Fig. 8 show that
hFCOT decreases with ,knee
,exL and ,,kneeex and increases with .,hipex Theses trends are similar to the trends of .maxF
20 40 60 80 100
0
0.5
1
1.5
2
x 10
4
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.150.2
ex,hip
=155
e
x
,k
n
e
e
=
1
5
5
20 40 60 80 100
0
0.5
1
1.5
2
x 10
4
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.150.2
ex,hip
=165
20 40 60 80 100
0
0.5
1
1.5
2
x 10
4
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
ex,hip
=175
20 40 60 80 100
2000
4000
6000
8000
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
e
x
,k
n
e
e
=
1
6
5
20 40 60 80 100
2000
4000
6000
8000
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
20 40 60 80 100
2000
4000
6000
8000
10000
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
20 40 60 80 100
2000
3000
4000
5000 0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
e
x
,k
n
e
e
=
1
7
5
20 40 60 80 100
2000
4000
6000
8000
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
20 40 60 80 100
2000
4000
6000
8000
10000
0.1
knee
(deg)
F
M
,m
ax
(
N
)
0.15
0.2
Fig 6. Max muscle force variations with respect to gait parameters
140
30 40 50 60 70 80 90
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
knee
(deg)
C
O
T
F
Fig 7. Variations of FCOT with respect to the swing knee angle.
Minimizing the 4-dimensional linear interpolation of the data leads to the minimum point
of the discrete nodes, and the spline or cubic interpolation leads to some internal points that,
due to nonlinearities of the problem, do not generate a minimal COT gait. Therefore, to find
the optimal gait parameters, we take the discrete parameters point that minimizes ,COT
hF
satisfying the geometric constraints. These optimal parameters are 84 , 0,15 ,knee exL m
, 175 ,ex knee and , 155 .ex hip
Table 1- Hill-type muscle parameters of the robot actuating system
Parameter Value
1k 1 kN m
2k 30 kN m
1c 10 Ns m
Free length of 1x 0,25 m
20 40 60 80 100
0
2
4
6
0.1
knee
(deg)
C
O
T Fh
0.15
0.2
ex,hip
=155
e
x
,k
n
e
e
=
1
5
5
20 40 60 80 100
0
2
4
6
0.1
knee
(deg)
C
O
T Fh
0.15
0.2
ex,hip
=165
20 40 60 80 100
0
2
4
6
0.1
knee
(deg)
C
O
T Fh
0.15
0.2
ex,hip
=175
20 40 60 80 100
0.5
1
1.5
2
0.1
knee (deg)
C
O
T Fh 0.15
0.2
e
x
,k
n
e
e
=1
6
5
20 40 60 80 100
0.5
1
1.5
2
0.1
knee (deg)
C
O
T Fh 0.15
0.2
20 40 60 80 100
0
1
2
3
0.1
knee (deg)
C
O
T Fh 0.15
0.2
20 40 60 80 100
0.5
1
1.5
2
0.1
knee
(deg)
C
O
T Fh 0.15
0.2
e
x
,k
n
e
e
=
1
7
5
20 40 60 80 100
0.5
1
1.5
2
0.1
knee
(deg)
C
O
T Fh
0.15
0.2
20 40 60 80 100
0
1
2
3
0.1
knee
(deg)
C
O
T Fh
0.15
0.2
Fig 8. Variations of
hFCOT ions with respect to gait parameters
141
The stick diagram of the simulated optimum gait is shown in Fig. 9. The generated COM
path of the robot in the stance phase is depicted by a curve, the planned degree-4 polynomial
(10). The swing leg angle is kept constant in the stance phase and varies to the desired value
of the next step during the flight phase.
For the optimum gait, the muscles’ external forces are calculated using inverse dynamics
in each phase; their diagrams are shown in Fig. 10. In this figure, the dashed lines depict
flight phase forces. These force profiles are the needed control inputs for the 5-link robot, in
which muscles have been replaced with force actuators. The knee force of the stance leg
)( 1F has the biggest amount )3341( N in the stance phase because it bears the robot’s
weight. The hip forces ),( 32 FF have larger peaks in the flight phase because they need to
move the legs to the desired state before touch-down with relatively large accelerations.
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x(m)
y(
m
)
Fig 9. Stick diagram of the planned running gait
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-4000
-3000
-2000
-1000
0
1000
2000
t(s)
F
o
rc
e
(N
)
F
1
F
2
F
3
F
4
Fig 10. Muscles’ external forces for the running gait.
142
Knowing the force profile of each muscle end, we calculate its contractile element force.
The resulted forces hF have profiles similar to Fig. 10 with small differences. Fig. 11 shows
the muscle ends and the muscle contractile element force profiles of muscle 1. Solid lines
are for the stance phase, and dashed lines are for the flight phase. The maximum amount of
force for this muscle is ,3478 N which is 4% greater than the muscle’s external force.
This gait has an overall muscle cost of transport of 46,0COT F and a contractile ele-
ment cost of transport of ,45,1COT
hF which has been increased considerably by using the
muscle. This result can be justified using the muscle’s external and contractile element ve-
locity profiles shown in Fig. 12. In the first half of the stance phase, the muscle elongates
(positive velocity), and it contracts in the second half (negative velocity). In both cases, the
contractile element has larger velocities that increase its consumed power and COT. There-
fore, using a single Hill-type actuation system for a biped running robot has negative effects
on both the maximum actuating force and COT. Its advantage is to isolate the external im-
pacts from the contractile element.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
t(s)
F
o
rc
e
(N
)
F
1
F
h1
Fig 11. Muscle 1’s external and contractile element
force profiles for the running gait.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-3
-2
-1
0
1
2
3
t(s)
d
x/
d
t (
m
/s
)
dx
m1
/dt
dx
a1
/dt
Fig 12. Muscle 1’s external and contractile element
velocity profiles for the running gait.
143
The COT of the biped model’s running gait with and without muscles compared to other
references are shown in Table 2. Our planned biped running gait without muscles has a good
COT compared to other multibody biped robot gaits. The COT with muscles is acceptable
compared to the real robots, which is the optimum COT value for the actuation system as-
sumed in this paper. However, it can be improved by searching other actuation system struc-
tures for Hill-type muscle actuators like bi-articular muscles.
Table 2 – Cost of Transport of different biped models
Biped System COT (J/Nm) Reference
Human 0,28 0
Monopod I 0,7 0
Monopod II 0,22 0
HRP-2L 3,57 0
Optimized gait for a biped model 1,02 0
Parallel Elastic actuating model 1,31 0
PMB model with degree 4 polynomial path 0,24 This work
The 5 link biped without muscles 0,46 This work
The 5 link biped with muscles 1,45 This work
5. Conclusions.
In this work, a 5-link biped robot model with a Hill-type muscle actuation system was
proposed to study the Hill-type muscle effects on biped running efficiency. A dynamic
model of the multibody biped model with a muscle-like actuation system was presented. A
COM trajectory with a desired initial position and velocity was planned, and the robot’s
muscle force and velocity profiles were calculated to make the robot undergo the desired
trajectory. Then, the force and velocity profiles of the muscle’s contractile elements were
found using a muscle model, and its COT was calculated. In addition, the effects of the ge-
ometric model parameters on COT and maxF was investigated.
Unexpectedly, utilizing a single Hill-type muscle for each joint caused an increase in both
the COT and maxF for the biped mechanism assumed in this work. This is because, when the
Hill-type muscle contracts or extracts actively, both the series elastic element and parallel
elastic element oppose the contractile element and increase its required force. Also, when
the contractile element contracts, the series elastic element is stretched, which increases the
contractile element contraction velocity relative to the muscle end’s velocity. Optimizing
muscle stiffness parameters to minimize
hFCOT and maxF yields to zero stiffness of the par-
allel spring and infinite stiffness of the series spring that means substitution of the Hill-type
muscle with a single force actuator. Therefore, using a single Hill-type muscle actuation
system in each joint with the proposed structure in this paper is not beneficial for robotic
energetics and benefits only impact isolation and motor protection.
In future works, the effects on overall motion energetics of the arrangement of a single
muscle, bi-articular muscles, and pre-stretched muscles will be studied. In a bi-articular
structure, muscles can generate only active tension, and the parameters of its pair muscle’s
passive tension can be investigated.
РЕЗЮМЕ. Наявність податливих елементів у механізмах виконання двоногої ходи породжує
плавність руху і зменшує сили удару. Біологічні істоти, які мають складну систему спрацьовування з
паралельними і серіями еластичних елементів в м'язах, демонструють дуже ефективну і надійну дво-
ногу ходу. Основними труднощами реалізації цих систем є дублювання їх складної динаміки і конт-
ролю. Ця стаття вивчає наслідки спрацьовування системи, в тому числі м'язів типу Хілла на ефектив-
ність роботи коліна у моделі двоногої ходи. У цьому дослідженні ми вводимо довільні траєкторії,
сумісні з початковим станом робота, і розраховуємо необхідні сили м'язів за допомогою аналітичної
зворотної моделі динаміки. Для перевірки результатів ми виконуємо пряму динаміку робота з обчис-
ленням імпульсів управління для генерації траєкторії робота. І нарешті, ми розраховуємо скорочува-
144
льну силу елемента м'язів і вартість його передачі для робота, і ми досліджуємо вплив елементів
м'язів на зменшення або збільшення витрат на ходу і максимальні виконавчі сили.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 28.02.2019 Утверждена в печать 03.03.2020
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