Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of

Humanoid Balance

Vittorio Lippi

and Fabio Molinari

Technische Universit

at Berlin, Fachgebiet Regelungssysteme, Einsteinufer 17 D-10587, Berlin, Germany

Keywords:

Posture Control, Humanoid, Stability.

Abstract:

Human posture control models are used to analyse neurological experiments and control of humanoid robots.

This work focuses on a well-known nonlinear posture control model, the DEC (Disturbance estimate and

Compensation). In order to compensate disturbances, unlike other models, DEC feedbacks signals coming from

sensor fusion rather than raw sensory signals. In previous works, the DEC model is shown to predict human

behavior and to provide a control system for humanoids. In this work, the stability of the system in the sense of

Lyapunov is formally analysed. The theoretical ﬁndings are combined with simulation results, in which an

external perturbation of the support surface reproduces a typical scenario in posture control experiments.

1 INTRODUCTION

Mathematical models of human balance are used for

the analysis of neurological experiments (van der

Kooij et al., 2007; van der Kooij et al., 2005; van

Asseldonk et al., 2006; Goodworth and Peterka, 2018;

Mergner, 2010; Engelhart et al., 2014; Pasma et al.,

2014; Jeka et al., 2010; Boonstra et al., 2014), and

for the control of humanoid robots. Most of human

posture control studies exploit linear models such as

the independent channel model (Peterka, 2002), that

assumes a linear and time invariant behaviour (Engel-

hart et al., 2016). Linear models have the advantage of

being simple to analyse and relatively easy to be ﬁt on

data. However, experiments reveal that human posture

control exhibits important non-linearities.

In this work, we study the stability of a non-linear

bio-inspired posture control system, the DEC, Distur-

bance estimate and Compensation (Mergner, 2010).

The DEC model consists of a servo control loop and

a compensation of external disturbances estimated

on the basis of sensory inputs. The control princi-

ple can be addressed as “feed forward disturbance

correction” (Luecke and McGuire, 1968; Roffel and

Betlem, 2007; Zhong et al., 2012) or, in German,

“St

orgr

oßenaufschaltung” (Bleisteiner and Mangoldt,

2013). Throughout this paper, the DEC is used to

model a scenario where the subject stands on a tilting

https://orcid.org/0000-0001-5520-8974

https://orcid.org/0000-0003-2617-962X

Gravitational

Vertical

Vestibular

ሶ

෕

𝜶

𝑩𝑺

signal

∫

()

Leaky integrator

Proprioception

ෝ

𝜶

𝑩𝑺

Signal, used

for control

ෝ

𝜶

𝑭𝑺

-a

Figure 1: Posture control model. On the left: illustration of

the scenario and deﬁnition of angles used in text. On the

right: schema of the bio-inspired sensor fusion.

surface. We analyse the effect of a dead-band nonlin-

earity that affects the sensory-based estimate of the

support surface tilt. Such nonlinearity, common in

literature, is assumed on the basis of the behaviour ob-

served in humans. The formal conditions for Lyapunov

stability are investigated.

The paper is organized as follows: in Section 2, the

control problem is introduced and the body mechanics

is described; Section 3 provides details about human-

inspired sensor fusion and actuation; the conditions

726

Lippi, V. and Molinari, F.

Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of Humanoid Balance.

DOI: 10.5220/0009970307260733

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 726-733

ISBN: 978-989-758-442-8

𝑷𝑫

Body

Dynamics

𝑷𝑫

Body sway

estimator

Passive

Desired Position

Neural

controller

Proprioception

Vestibular

𝜶

𝑩𝑭

𝑲

𝑮

Gravity compensation

෕

𝜶

𝑩𝑺

ෝ

𝜶

𝑩𝑺

𝜶

𝒓𝒆𝒇

Figure 2: A scheme of a controller based on the DEC con-

cept. The sensory inputs are used to reconstruct the physical

disturbances acting on the body (in this scenario gravity and

support surface tilt). The system is controlled by a servo

controller, consisting of a PD regulator setting the desired

body position, and a direct compensation of the estimated

gravity disturbance.

for stability are obtained in Section 4, where evidence

is also provided; Section 5 presents simulations and a

qualitative discussion of the system behaviour; conclu-

sions and future work are presented in Section 6.

2 PROBLEM DESCRIPTION

Human posture dynamics in the sagittal plane is usu-

ally modelled as an inverted pendulum. Depending

on the scenario, the model can be a single inverted

pendulum (SIP), see, e.g., (Mergner et al., 2003; Jafari

et al., 2019), or a multiple inverted pendulum, see, e.g.,

(Alexandrov et al., 2017; Hettich et al., 2013; Lippi

et al., 2013; Lippi and Mergner, 2017; Abedi and

Shoushtari, 2012). The number of degrees of freedom

(DoF) representing the body dynamics is in general

linked to the intensity of external stimuli, see (Atkeson

and Stephens, 2007) for further details. In this work,

we consider a SIP model that is used to represent the

upright stance in presence of small disturbances. In

literature, models with 2 DoF have been used for mod-

elling balance in the frontal plane (Goodworth and

Peterka, 2010; Lippi et al., 2016).

In the remainder of this paper, we consider the fol-

lowing scenario: the subject is balancing on a tilting

platform whose inclination is controlled by an exter-

nal input. In order to keep the equilibrium despite

the tilting movement, the orientation in space of the

inverted pendulum is actively controlled by an ankle

movement.

Formally,

∈ R

denotes the angle between the

body (pendulum axis) and the vertical (gravity axis).

∈ R

denotes the angle between the body and the

axis normal to the tilting platform. The third consid-

ered angle is

∈ R

that represents the angle of the

tilting platform with regards to the vertical axis. From

Figure 1, this three angles are linearly dependent, i.e.,

= α

− α

. (1)

The torque provided by the ankle is

∈ R

, whereas

the one produced by the gravitational force is T

∈ R.

The pendulum dynamics is described by

+ T

, (2)

where

∈ R

is the moment of inertia of the body

around the ankle joint and

∈ R

is the torque describ-

ing passive stiffness and damping, i.e.,

= K

+ K

(3)

The passive stiffness, i..e,

, and damping, i.e.,

causes additional destabilisation. However, as in (Ott

et al., 2016), they have a role in stabilising the system

dynamics in presence of delays.

3 HUMAN-INSPIRED SENSORS

AND ACTUATION

3.1 Sensors’ Information

Signal

∈ R

provides the estimate of

(body

sway) obtained by the vestibular system (or, for hu-

manoids, by the IMU). On the other hand, signal

∈ R

is the proprioceptive input measured at the an-

kle (for humanoids, given by an encoder). Both signals

are estimates of the corresponding physical quantities.

Their derivatives with respect to time are

∈ R

sensed by the vestibular system, and

∈ R

, sensed

by the proprioceptive system.

In what follows, we put a check symbol above all

measured variables (i.e.,

ˇx

). On the other hand, all

estimated variables have the ”hat” symbol above them

(i.e., ˆx).

3.2 Gravity Compensation

The gravity force is the largest effect acting on the

body, see (Zebenay et al., 2015). Formally, the torque

produced by this force is

= m

· g ·h

· sin (α

where

is the gravity acceleration,

∈ R

the body

mass, and

∈ R

the height of the centre of mass.

Under the assumption of a small angle,

' m

· g · h

· α

. (4)

The ankle torque, actively produced by the subject in

order to keep the body standing despite the tilting plat-

form, i.e.,

in (2), also compensates for this gravity

Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of Humanoid Balance

727

Table 1: List of variables and their deﬁnition. The second column contains equation numbers, in parentheses, or section number

depending on where the variable is deﬁned.

Variable Deﬁned in Deﬁnition

(1) Ankle joint angle

(2) Body sway respect to the vertical

§2 Support surface rotation

(8) Support surface rotation estimate based on sensor fusion (vestibular+proprioceptive)

(10) Body sway estimate based on sensory input (vestibular+proprioceptive)

§3.1 Body sway estimate based on vestibular input

disturbance. In fact, let

be the component of

compensating gravity, such that

= −T

+ T

, (5)

where

is the ankle torque’s component not due

to gravity compensation. Gravity is slightly under-

compensated in humans, see, e.g., (Mergner et al.,

2009; Hettich et al., 2014), thus, as in (Ott et al., 2016),

we assume an arbitrary gain, i.e.,

∈ R

, for gravity

compensation. Thus,

= K

. (6)

3.3 Support Surface Tilt Compensation

In order to reproduce the behaviour observed in hu-

mans, see, e.g., (Mergner et al., 2009; Mergner et al.,

2003; Hettich et al., 2015; Hettich et al., 2014), the

control input

is not computed by directly using the

measured quantity

, but an estimate of

, say

. To this end, ﬁrst, the inspection of human be-

haviour suggests to use signal

∈ R

, obtained by

using both vestibular and proprioceptive sensed val-

ues, to estimate the tilting platform’s angle. Denote

this estimate by

∈ R

. This value is then used for

computing

, which closes the control loop.

Formally, by (1), one has

−

. (7)

The estimate of

simulates the human behaviour.

This is done by feeding

into function

ρ(·)

, which

is then integrated through a leaky integrator, i.e.,





− c

dτ (8)

with c

∈ R

and the threshold function deﬁned as

ρ(α) :=











α + θ if α ≤ −θ

0 if − θ < α < θ

α − θ if θ ≤ α

, (9)

for θ ∈ R

With this piece of information at hand, we compute

, the quantity used in T

, by employing (1), i.e.,

. (10)

3.4 Other Disturbances

In order to completely describe the effect of the envi-

ronment on the body, other disturbances should also

be taken into account. Field forces can be produced,

for example, by an horizontal translation of the sup-

port surface

f s

leading to a torque

trans

= ¨x

f s

·h

·m

and an external touch that can be estimated as

ext

− T

. A robotic control applying also these dis-

turbances is described in (Zebenay et al., 2015). Cur-

rently, a model of human support surface translation

compensation is still object of research, and there are

no evidences yet of a direct compensation of such

disturbances. In this work only gravity will be consid-

ered.

3.5 Servo Control

As in (Ott et al., 2016), the system is controlled

through a

controller with proportional coefﬁcient,

respectively derivative coefﬁcient, being

∈ R

, re-

spectively K

∈ R, i.e.,

= K

ε + K

dε

, (11)

where the error variable ε is deﬁned as

ε :=

− α

ref

, (12)

with the desired position being, in general,

ref

= 0

All delays involved into the processing of sensory in-

puts and the motor control are not considered in this

analysis. In neurology the concept was proposed in

(Merton, 1953) to explain the role of the muscle stretch

reﬂex for the control of posture and movements: a PD-

controller adjusts the force of the muscles so as to

produce the desired pose or movement.

Remark 1.

Gravity is compensated by directly using

the available measure

, whilst

uses the esti-

mated value

. This is due to the fact that the non-

linearity has been experimentally observed only on the

latter (Mergner et al., 2009; Mergner et al., 2003; Het-

tich et al., 2014). Qualitatively, the effect on the thresh-

old applied on

is that the slower the platform tilt-

ing is, the less it is compensated (gain nonlinearity).

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

728

For very slow platform movements (i.e.

< θ

), there

is even no compensation. A similar nonlinearity ap-

plied on the gravity compensation would produce a

paradoxical behaviour.

4 STABILITY ANALYSIS

Assumption 1.

As done in (Lippi and Mergner, 2017;

Lippi et al., 2013; Mergner, 2010), we assume the

direct measurements to be equal to the corresponding

variables, i.e.,

= α

and

= α

Let the input to the system be

u =

, (13)

i.e., the tilting speed of the platform. The state vector

is four-dimensional and equal to

x =

























. (14)

Starting from (2), we derive a model for the system

at hand, by incorporating (4), (6), (8), (10), and (11).

This yields the following system:











˙x

= x

˙x

= a

+ a

+ f (u)

˙x

= bx

+ x

− bx

+ g(u)

˙x

= u

(15)

where

+ K

+ mgh

− K

, (16)

+ K

, (17)

− c

, (18)

− c

, (19)

b = C

, (20)

g(u) = ρ(u) − u , (21)

f (u) = K

g(u) − K

u. (22)

Note that the nonlinearity brought about by

ρ(·)

affects

the system only through input

. Figure 3 illustrates

f (u) and g(u).

System (15) can be also written in matrix form

i.e.,

x(t) = Ax(t) + B(u(t)), (23)

In this case we explicitly report the dependence on time.

−θ θ

−θ

g(u)

(

)

Figure 3: f (u) and g(u).

where

A :=







0 1 0 0

b 1 −b −b

0 0 0 0







(24)

and

B(u) :=







f (u(t))

g(u(t))

u(t)







. (25)

System (23) is a linear system, with nonlinearity on

the control input. The dynamics of the fourth state is

a simple integrator of input

, thus

is an eigenvalue

of the system’s dynamics. The remaining three eigen-

values can be determined by the choice of

and

our design parameters. In the following the stability

conditions are derived in an analytical way while in

previous work the stability of the DEC was demon-

strated empirically with simulations (Lippi et al., 2013)

and robot experiments (Hettich et al., 2014; Ott et al.,

2016; Zebenay et al., 2015).

Lemma 1. If K

and K

are chosen such that

< c

− K

, (26)

+ K

< K

− mgh

− K

− c

, (27)

< K

− mgh

− K

, (28)

then system (23) has three eigenvalue with negative

real part.

Proof.

By deﬁnition of eigenvalues, the spectrum of

A is

eig(A) = {0} ∪ eig(

A),

where

A :=





0 1 0

b 1 −b





The characteristic polynomial of

, whose solutions

are

A’s eigenvalues, is:

(λ) = λ

+(b−a

)λ

+(−a

−a

b − a

)λ− b(a

Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of Humanoid Balance

729

By the Descartes Rule of Signs, we can impose that all

eigenvalues of

A have negative real part, by holding











b − a

> 0

−a

− a

b − a

> 0

−b(a

+ a

) > 0

This latter becomes a set of inequalities in

and

by incorporating (16)-(20). This yields (26)-(28), thus

concluding the proof.

Lemma 2. The solution to system (23) is

x(t) = e

x(0) +

A(t−τ)

B(u(τ))dτ. (29)

Proof.

Let

(t) := f (u(t))

(t) := g(u(t))

, and

(t) := u(t). We have

x(t) = Ax(t) +

u(t), (30)

with

u(t) = B(u(t)) (31)

where

u(t) := [u

(t), u

(t)]

and

B :=







0 0 0

1 0 0

0 1 0

0 0 1







By (Skogestad and Postlethwaite, 2007, (4.7)),

x(t) = e

x(0) +

A(t−τ)

u(τ)dτ,

which, by incorporating (31), yields (29), thus con-

cluding the proof.

In (29), the ﬁrst addendum is the free response, and

the integral is referred to as forced response. By (29),

system’s stability is determined only by matrix

, thus

the nonlinearity acting on the input, i.e.,

B(u(t))

, does

not play any role for stability. The following two

deﬁnitions of stability are extracted from (Mellodge,

2015, Chapter 3) and (Bernstein and Bhat, 1995).

Deﬁnition 1

(Asymptotic Stability)

System (30) is

asymptotically stable if and only if all the eigenvalues

of A are in the left half of the complex plane.

Deﬁnition 2

(Lyapunov Stability)

System (30) is Lya-

punov stable if and only if no eigenvalues of A are in

the right half of the complex plane and all eigenvalues

on the imaginary axis are semisimple (i.e., they have

algebraic multiplicity equal to the geometric multiplic-

ity).

Theorem 1.

and

are chosen as in Lemma 1,

system (23) is Lyapunov stable, but not asymptotically

stable.

Proof.

Consider system (30) which is an equivalent

of (23). Clearly, system (30) is Lyapunov stable

(asymptotically stable) if and only if also (23) is Lya-

punov stable (asymptotically stable).

By Lemma 1, matrix

has three eigenvalues with neg-

ative real part and one eigenvalue (the integrator in

) which is on the imaginary axis and semisimple.

By Deﬁnition 2, system (30) is Lyapunov stable. By

Deﬁnition 2, system (30) is not asymptotically stable,

thus the proof is concluded.

The non-linear system can be stabilised by choos-

ing the appropriate pair of proportional and derivative

coefﬁcients for the PD controller.

5 SIMULATION

The parameters, deﬁned on the basis of human anthro-

pometrics (see (Winter, 2009)) and previous posture

control analysis (see (Mergner et al., 2009; Mergner

et al., 2003; Hettich et al., 2014)), are shown in Table 2.

With the speciﬁc set of parameters, and by (26)-(28),

Table 2: System parameters.

Parameter Value

71.55 Kg · m

0.8

157.31 N · m

39.32 N · m · s

0.0125 s

−1

θ 0.0028 rad

m 80 Kg

h 1.80 m

we design

= −1200 N · m

and

= −1000 N · m · s.

The behavior of the system is shown in regime of free

response with no support surface tilt velocity, and

forced response with a periodic input. Speciﬁcally the

following conditions are simulated:

Condition 1: free response with

x(0) = [π/10,0.1,π/10, 0]

The free response with no support surface tilt is the

characteristic one of a linear second-order system (see

Fig. 4). This happens because the nonlinearity affects

only the input

u(t)

. The leaky integrator used in the

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

730

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.1

-0.05

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

Experiment 1. Free response

Figure 4: Free response with horizontal support surface and

initial conditions

x(0) = [π/10, 0.1,π/10,0]

The estimate

is equivalent to α

when α

= 0.

0 50 100 150 200 250 300 350

Time [s]

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

Experiment 2. Free response

0 1 2 3 4 5

0.1

0.2

0.3

-0.4

-0.2

0.2

Figure 5: Free response with tilted, but not moving, support

surface and non-zero initial body sway velocity and body

lean, i.e.

x(0) = [π/10, 0.1,π/10, π/15]

The smaller plot

shows the transient during the ﬁrst 5 seconds.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.04

-0.03

-0.02

-0.01

0.01

0.02

0.03

0.04

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

Experiment 3. Forced response

Figure 6: Forced response to sinusoidal support tilt with

initial conditions

x(0) = [0, 0,0, 0]

, and input

u(t) =

0.1cos(10t).

estimate of α

is constantly at zero.

Condition 2: free response with

x(0) = [π/10,0.1,π/10, π/15]

The free response with a constant support surface tilt

is shown in Fig. 5. The response is again the char-

acteristic of a linear system, in which

behaves as

a constant signal affecting the dynamics of

and

There is a residual lean

due to the error in body

sway estimate

Condition 3: forced response with

x(0) = [0,0,0, 0]

and

u(t) = 0.1cos(10t).

-4

-3

-2

-1

p.p. Amplitude [rad]

-1

p.p. gain

peak velocity

reaches

the threshold

Figure 7: Support surface tilt to body sway gain with si-

nusoidal support tilt at different amplitude. The gain is

computed as the ratio between peak to peak amplitude for of

the input and the one of the output. Smaller support surface

tilt are associated with larger gains because they are under-

compensated due to the nonlinearity. The plateau on the left

is the zone of linear behavior that happens when the support

surface rotation speed is always under threshold

. For larger

amplitudes the gain tends asymptotically to a constant gain,

i.e. linear behavior.

The forced response shows a partial rejection of the

external disturbance. The effect of the nonlinearity

is reﬂected in the difference between

and its es-

timated value

. The simulations is repeated with

different amplitudes for the support surface tilt pro-

ﬁle, producing the results in Fig. 7. The gain, in this

context deﬁned as the ratio between peak to peak am-

plitude for of the input and the output is plotted for

different amplitudes. Smaller support surface tilt are

associated with larger gains because they are under-

compensated due to the nonlinearity. Speciﬁcally the

plateau on the left is the zone of linear behavior that

happens when the support surface rotation speed is

always under threshold

and the disturbance is not

compensated. For larger amplitudes the gain tends

asymptotically to a constant gain because the signal is

almost always above the threshold.

6 CONCLUSIONS AND FUTURE

WORK

The formal analysis of the system has provided a condi-

tion for the stability, speciﬁcally on the gains of the PD

controller, i.e.,

(26)

(28)

. This conﬁrms the idea, sug-

gested by empirical experiments with human subjects

and robots, that the nonlinearity is benign in that it

does not endanger the stability of the system. There is

the hypothesis that such dead-band nonlinearity could

be useful in cutting out vestibular noise, especially

when the support surface is not moving (that is the

most common scenario in nature). In order to study

Lyapunov Stability of a Nonlinear Bio-inspired System for the Control of Humanoid Balance

731

the effect of the threshold on noise future work may in-

tegrate methods for the analysis of stochastic systems

(Han. et al., 2018; Bj

ornsson. et al., 2018). Another

important aspect in posture control, that was not con-

sidered here, is the effect of delay. Delay imposes a

limitation on feedback gain that can be considered the

motivation for the feed-forward compensation of ex-

ternal disturbances (in this work, gravity). The effects

of delay have been studied formally in the linear case

(Antritter et al., 2014), but not yet with the nonlinear

system. As the DEC has also been applied to multi-

ple degrees of freedom scenarios (Lippi et al., 2019b;

Lippi and Mergner, 2017) the formal study may be ex-

tended to multiple inverted pendulum models. A way

to tackle the complexity of the multiple DoF problem

may require the use of numerical methods for the study

of the stability (Giesl. et al., 2018; Bj

ornsson. and

Hafstein., 2018; Giesl. and Mohammed., 2018), this

will require a particular effort considering the number

of state variables required to represent the dynamics

of the mechanical degrees of freedom, the dynamics

of the sensory estimates (e.g. the leaky integrator in

the presented work) and the ones used to represent the

delays.

ACKNOWLEDGEMENTS

This work is supported by the project

COMTEST (Lippi et al., 2019a), a

sub-project of EUROBENCH (European

Robotic Framework for Bipedal Locomotion

Benchmarking, www.eurobench2020.eu)

funded by H2020 Topic ICT 27-2017 under

grant agreement number 779963.

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