Human Balance - Anatomy & ZMP

DESIGNING MECHATRONIC SYSTEMS FOR REHABILITATION
FINAL PROJECT
PARIS, FRANCE, JANUARY 11TH, 2018
Human Balance
LUÍS RITA
3702256, luis.domingues_rita@etu.upmc.fr
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2
Introduction
With the progressively ageing of the population, the proportion of elders is
strongly increasing (Fig. 1). Linked to this stage of life are the many physical
impairments that arise due to an increased frailty caused by disease or simply by
the wear of body parts. In the following pages, we will study some of the most
important organs and systems associated with balance maintenance. And, when
not working properly, they may lead to injury or premature deaths (Fig. 2).
It may be also interesting to observe the normalized number of fatal falls
comparing with other accidents that are common in our societies.
Highly developed countries are the ones that present higher rate of
deaths caused by fall. Being the aged population in higher risk.
Fig. 3 – Age-adjusted death rates, by cause of death among adults aged 65 and over: US, 2000-2013. (3)
Fig. 1 – Median age by country, CIA World Factbook 2016. (1)
Traffic accidents are one of the
main causes of death that
governments are more actively trying
to solve. But, in fact, fall deaths in
elders are more recurrent (Fig. 3).
Plus, comparing the death rate in car
accidents in young people and the
corresponding caused by falls, the
number is similar. (12)
Fig. 2 – Deaths due to falls per million persons in 2012. (2)

3
Anatomy
In the human body, the main responsible for balance control are: vision,
vestibular system and somatosensory system. (4)
Vision (5)
In the most posterior part of the eye, there is
a structure called retina, which contains rods
and cones responsible for converting
luminous signals into electric impulses, later
transmitted through the optic nerve to the
Central Nervous System (Fig. 4):
® Rods are cylindrical receptors which
are able to see in very low light
conditions (a strike of a single photon
can activate them). They are
responsible for the black and white
vision, and without them, humans
could not see in dark conditions;
® There are three different types of
cones, the ones responding to long, medium and short wavelengths are
called, respectively: L-cones, M-cones and S-cones. Differently than the
previous ones, they need stronger light conditions to become activated.
A clear vision allows people to locate themselves in the
surrounding environment, to access their relative distances to
different objects (e.g. – when someone is walking along a
street and a storefront passed first moves into and then
beyond the range of peripheral vision).
Vestibular System (6)
Constituted by the vestibular apparatus present in each ear
is responsible for motion, equilibrium and spatial orientation.
Each apparatus, in turn, contains an utricle and saccule
responsible for detecting gravity and linear movement. As well as, three
semicircular canals filled with a liquid called endolymphatic fluid which through
Fig. 4 – Anatomy of a human eye. (5)
Fig. 5 – Vestibular apparatus
present in human ears. (6)

4
lagging mechanism caused by inertia, (induced by the rotational movements)
exerts a pressure on the sensors present in the canal, leading to a sequence of
impulses sent to the brain. Depending on the source of these electrical signals, it
will capture the type of movement in their genesis. A scheme of these structure
is depicted in Fig. 5.
Unbalanced situations arise when contradictory information is sent by
different vestibular apparatus present in each ear.
Somatosensory System (7)
Proprioceptive and exteroceptive information from skin, muscles and joints
involves the detection of stretch or pressure signals through sensory receptors
present in these locations.
An intuitive example is during tiptoeing gate, when the body is able to feel
the increased pressure on the fingers comparing to the plantar surface, and
realize its position regardless of using its visual capabilities. Or during a walk in
a stable/unstable surface, the gate is well adapted by the brain to the referred
situation.

5
Zero Moment Point
Zero Moment Point or ZMP is a concept that was introduced for the 1st
time in
January 1968 by Miomir Vukobratović at The Third All-Union Congress of
Theoretical and Applied Mechanics in Moscow. Although, it was only properly
named in the papers that arose between the years of 1970 and 1972. Becoming
a worldwide phenomenon by that time. (8)
Since many decades ago, humanoid (biped) robots are part of a field that
many interest has been causing among the scientific community. Issues related
to gait planning and control have contributed to an increased attention of ZMP.
The importance of this concept is that it allows researchers to objectively say in
which circumstances the robot will be find in a state of dynamic equilibrium or not.
(9)
Human gait is a complex process. Although, it is not the point here to
thoroughly characterize it, it is well worth it to remember that 2 distinct phases
are present: stance and swing. Due to this fact, it would be very complicated to
replicate every detail. Thus, researchers have been focusing in several points:
(9)
® Possibility of rotation of the body (caused by strong disturbances) around
the edges of one foot;
® Gait repeatability which is inseparable of a normal gait feature;
® Interchangeability between 1-foot/2-feet support.
Although, before explaining the concepts linked to this recent approach of
biped robot movement, it is essential to understand the limitations and methods
used in the past. After, other than specifically characterize what is the ZMP, it will
be clarified how to locate it. An extension of the last concept will be derived -
FZMP. A clear distinction between CoP (Centre of Pressure) and ZMP will be
also provided. All of this, focusing in a walking gate where forces are being
applied in the foot which completely rests in the floor.
Static Stability: Centre of Gravity (CoG) (10)
The gate of the 1st
biped robots was based exclusively in the concept of
CoG. Thus, the goal was to assure that the projection of this point in the ground

6
was always inside the support polygon. This immediately neglects a human-like
gate and the dynamics of the body. Meaning that when the previous criteria is not
satisfied, the robot will inevitably fall. Another consequence of this approach is
the impossibility of fast movements.
Dynamic Stability: ZMP Notion (9)
Starting by identifying all the forces being applied in the foot
(considering a single-support phase and, remember, the
whole foot in contact with the ground): the weight of the foot
(𝑚" 𝑔); force applied by the upper part of the body in the
ankle/point A (𝐹%) and the reaction force of the ground (𝑅 →
(𝑅), 𝑅+, 𝑅,)). Plus, the momentum in the joint A (𝑀%) and the one produced by the
ground (𝑀 → (𝑀), 𝑀+, 𝑀,)) are also present.
Keeping in mind the foot is at rest in the floor and that the friction force acts
at the point of contact between
the horizontal plane and the
foot, the horizontal
components of 𝑅 and 𝑀 acting
on the horizontal plane are
balanced by friction.
In the case of horizontal
forces 𝑅) and 𝑅+, generated
by friction, they will cancel the
respective components of 𝐹%.
The corresponding vertical
momentum (𝑀,) generated by
the 2 ground reaction
components will cancel out the
vertical 𝑀% and the one
induced by 𝐹%. Finally, if we
consider a non-sliding
environment, the static
friction will compensate for
ZMP Definition
Point that results from the
intersection of an axis of constant
vertical momentum with the
plane of the support polygon.
Fig. 6 – Humanoid (biped) robot with the representation of forces and
momenta being applied on it. (8)

7
the horizontal components 𝑅) and 𝑅+, as well as the vertical component (ground
reaction) of the momentum – 𝑀,. On the other hand, 𝑅,, the vertical reaction
force, will balance the remaining vertical forces applied in the foot (the weight and
the one applied in the ankle).
Still not discussed was the balance of horizontal components of
momentum in the foot. It is known that every change in the momentum of A, will
be compensated by a change in position of the point where the reaction force is
being applied. Being the limit, the edge of the polygon, where no further increases
of momentum are possible to support without losing dynamic equilibrium. This is
shown in Fig. 6 (d). 𝑦 is the key distance that should be varied accordingly to the
values of the imposed momentum in the ankle. Thus, it is important to emphasize
that any time there is a variation in the ankle momentum, a change on the position
of reaction force will arise and, consequently, momentum in 𝑥 and 𝑦 directions
will not exist, again, in conditions of dynamic equilibrium. Not being included in
Fig. 6 (b).
Although, if the support polygon is too small, then the point of application
of force R will move to the foot edge and the uncompensated horizontal
components of the momentum will cause the mechanism’s rotation. Therefore, it
is a necessary and sufficient condition, to guarantee locomotion’s dynamic
equilibrium,
𝑀) = 0;
𝑀+ = 0.
Consequently, the precise explanation for the name Zero Moment Point
arises from the fact that, in dynamic equilibrium, this point has null horizontal
components.
Due to its importance in gate synthesis, a natural question arises: how do
we know the exact position of ZMP? To answer this, the equations of static
equilibrium should be introduced:
𝑅 + 𝐹% + 𝑚" 𝑔 = 0 (1)
Assuming the complete mechanism in equilibrium, then the total sum of
forces should be 0.
𝑂𝑃×𝑅 + 𝑂𝐺×𝑚" 𝑔 + 𝑀% + 𝑀 + 𝑂𝐴×𝐹% = 0 (2)

8
Where 𝑂𝑃, 𝑂𝐺 and 𝑂𝐴 are the vectors from the origin of the referential O to
the ground reaction force point (P), center of gravity (G) or ankle joint (A),
respectively.
Highlighting the fact that the definition of ZMP does not suppose that
total moment in point P to be 0. In fact, by projecting equation (2) in z-plane, 𝑀,
is given by:
𝑀, = 𝑀9: = −(𝑀%
,
+ (𝑂𝐴×𝐹%),
)
And by projecting equation (2) in the horizontal plane, the location of ZMP
is found using the following relation:
(𝑂𝑃×𝑅)<
+ 𝑂𝐺×𝑚" 𝑔 + 𝑀%
<
+ (𝑂𝐴×𝐹%)<
= 0
All the previous analysis was done supposing a single support situation, but
looking at Fig. 7 and the size of polygon in the case of double support, we
immediately understand the increased stability against disturbances in the
position on the right. (11)
In the case where the ZMP is out of the support polygon, then it is usually
called fictitious ZMP (FZMP). As bigger the distance to the edge of the foot, as
significate is the induced unbalance (Fig. 8).
Having introduced both definitions of ZMP and FZMP, it will be now
clarified the difference between CoP (Centre of Pressure) and ZMP. By definition,
CoP is the point where the pressure applied by the ground on the foot can be
replaced by a single force vector. If it balances the other forces applied in the
mechanism, then it is called ZMP as well. Being the movement considered
balanced. On the other case, where this is not possible, then CoP and ZMP do
not coincide. Once the 1st
can only be present in the contact surface, the 2nd
vanishes and gives place to another quantity already addressed – FZMP, out of
the support polygon. (12)
Fig. 7 – Single Support vs Double Support
polygon size. (9)

9
Different ZMP Definitions (9)
Since its conception, many researchers have been presenting different
definitions of ZMP. Despite they may not seem similar, in reality they do not
differ in meaning. Some examples are shown below (Fig. 9).
Fig. 9 – Two interpretations of ZMP collected from different scientific articles. (8)
Fig. 8 – 3 possible scenarios between ZMP and CoP in a non-rigid foot. (a) Dynamically
balanced movement where the CoP coincides with the ZMP. (b) Unstable gait in which there
is an overturning of the foot around the edge of the foot. There is no ZMP and the reaction
force induced by the ground is located on the tip of the foot. (c) Tiptoed movement (pointe
technique) considered dynamically balanced. (8)

10
References
Websites
1. List of countries by median age. In: Wikipedia [Internet]. 2017 [cited 2018 Jan
11]. Available from:
https://en.wikipedia.org/w/index.php?title=List_of_countries_by_median_age
&oldid=813536623
2. Falling (accident). In: Wikipedia [Internet]. 2018 [cited 2018 Jan 11]. Available
from:
https://en.wikipedia.org/w/index.php?title=Falling_(accident)&oldid=8189990
43
3. Products - Data Briefs - Number 199 - May 2015 [Internet]. [cited 2018 Jan
11]. Available from: https://www.cdc.gov/nchs/products/databriefs/db199.htm
4. The Human Balance System | Vestibular Disorders Association [Internet].
[cited 2018 Jan 9]. Available from: http://vestibular.org/understanding-
vestibular-disorder/human-balance-system
5. Human eye. In: Wikipedia [Internet]. 2018 [cited 2018 Jan 11]. Available from:
https://en.wikipedia.org/w/index.php?title=Human_eye&oldid=818786738
6. Vestibular system. In: Wikipedia [Internet]. 2017 [cited 2018 Jan 11]. Available
from:
https://en.wikipedia.org/w/index.php?title=Vestibular_system&oldid=812197
324
7. Somatosensory system. In: Wikipedia [Internet]. 2017 [cited 2018 Jan 11].
Available from:
https://en.wikipedia.org/w/index.php?title=Somatosensory_system&oldid=81
7766033
8. Zero moment point. In: Wikipedia [Internet]. 2016 [cited 2018 Jan 10].
Available from:
https://en.wikipedia.org/w/index.php?title=Zero_moment_point&oldid=73811
0493
9. Vukobratović M, Borovac B. Zero-moment point — thirty five years of its life.
Int J Humanoid Robot. 2004 Mar 1;01(01):157–73.
10. Goswami A. FRI point: A new gate planning tool to evaluate postural stability
of biped robots [Internet]. [cited 2018 Jan 10]. Available from:
https://www.cc.gatech.edu/fac/Chris.Atkeson/legs/kuff1c.pdf
11. Robot's Walking - Chapter 5 [Internet]. [cited 2018 Jan 11]. Available from:
http://www.diss.fu-
berlin.de/diss/servlets/MCRFileNodeServlet/FUDISS_derivate_0000000025

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04/05_Kapitel5.pdf;jsessionid=BB71E0C0242031B3A6B5F4E873C63233?h
osts=
12. D A Winter. Human balance and posture control during standing and walking
[Internet]. [cited 2018 Jan 10]. Available from:
https://www.cc.gatech.edu/fac/Chris.Atkeson/legs/kuff1c.pdf

Human Balance - Anatomy & ZMP

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