Mathematical Model of a Human Leg

The Switched Linear System Approach

Artur Babiarz, Adam Czornik, Michał Niezabitowski and Radosław Zawiski

Institute of Automatic Control, Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland

Keywords:

Human Leg, Switched Linear System, State-dependent Switching.

Abstract:

This article presents a novel approach to modelling of the human leg with the use of linear switched systems.

Second order differential equations forming a two-segments leg model moving in a vertical plane are shown.

State space linear equations describing given model are derived. A linear switched system for such model

is presented, where the switching function is modelled as state-dependent. Based on this approach a linear

system is presented, which is composed of four subsystems between which switching occurs depending on two

state variables. These variables represent angular displacements. As a consequence, a state space division is

shown together with a linear system describing human leg in this setting. Finally, a set of simulations presents

differences between standard linear modelling approach and a switched linear system approach.

1 INTRODUCTION

Hybrid systems have become very popular during the

last decade (Sun, 2006), (Sun and Ge, 2011). There

are numerous adaptations of these systems into prac-

tical solutions. They own their popularity to possi-

bilities of applying them to both, continuous and dis-

crete dynamical systems. Switched systems are sub-

group of hybrid systems, where the switching signal

may depend on the logic-based switching function.

Switched systems can be categorized into two groups:

autonomous and controlled (Liberzon, 2003). In ad-

dition, each of these types of systems can be classiﬁed

into state-dependent and time-dependent. In this pa-

per, we consider a switched linear systems:

˙x = A

σ(·)

x+ B

σ(·)

u,

y = C

σ(·)

x+ D

σ(·)

u

(1)

deﬁned for all t ≥ 0 where x ∈ R

n

is a state, u ∈ R

m

is control, y ∈ R

q

is output, σ(·) : P → {1, 2, . . ., N}

is a switching rule and A

i

, B

i

,C

i

, D

i

, i = 1, 2, . . . , N are

constant matrices.

The authors are not aware of any traces in the

literature of modelling of human leg in the above

framework. In the paper (Bai et al., 2001), authors

present dynamics of a human arm, as a second or-

der continuous object with delay. Such mathemat-

ical model is convenient from the point of view of

generating a trajectory and executed computer simu-

lation. The approach presented in (Zhao et al., 2008)

is based on classical mechanics, which approximates

the limb as a combination of rigid links. Using the

Euler-Lagrange formalism a matrix second order dif-

ferential equation can be obtained. Matrices in this

equation describe moments of inertia, Coriolis forces

and gravity forces. Depending on the individual case

(Zhao et al., 2008) it is possible to take into account

various additional forces from external factors. In

the same article, the authors also take into account

a ground reaction force during the movement of the

foot.

Recently, researchers devote an increasing atten-

tion to mechanical systems such as exoskeletons.

Their main purpose is to aid and strengthen move-

ment attributes of a human body (Pons et al., 2007),

(Kong and Tomizuka, 2009) and to support motion

of people with reduced mobility (Sekine et al., 2013).

The authors of (Csercsik, 2005) describe the attempt

to create a control system of the lower limb exoskele-

ton with a PID controller. They also perform a stabil-

ity analysis by means of Nyquist criterion. The paper

contains a full description of the experiments for one

degree of freedom only. Obviously, such approach at

the design of control systems is appropriate (Zawiski

and Błachuta, 2012), (Błachuta et al., 2014), but it

does not fully reﬂect the dynamics of the movement

having seven degrees of freedom, and in a further per-

spective is a signiﬁcant limitation. Another group of

articles tackles the research focusing on impedance

control (Burdet et al., 2006), (Chang et al., 2013). In

90

Babiarz A., Czornik A., Niezabitowski M. and Zawiski R..

Mathematical Model of a Human Leg - The Switched Linear System Approach.

DOI: 10.5220/0005230300900097

In Proceedings of the 5th International Conference on Pervasive and Embedded Computing and Communication Systems (PECCS-2015), pages 90-97

ISBN: 978-989-758-084-0

Copyright

c

2015 SCITEPRESS (Science and Technology Publications, Lda.)

the paper (Chang et al., 2013), authors present an ap-

plication of the theoretical results of impedance con-

trol for trajectory generation and control of robot mo-

tion imitating the movement of a human arm with two

degrees of freedom.

By scrutinizing the literature describing a human leg

the problem of geometric description of the human

limbs should be distinguished, along with the prob-

lem of stability during movement, the optimal con-

trol problem and the control problem with limitations

on the value of the driving torque as a control sig-

nal (Ueyama and Miyashita, 2014). The information

from (Burdet et al., 2006), (Chen, 2011), (Lee et al.,

2010), (Neumann et al., 2013) became the inspiration

to develop the idea of application of hybrid systems.

1.1 Background and Signiﬁcance

The applications of hybrid systems are based on the

results in (Pavlovic et al., 2001), (Lee et al., 2010),

(Neumann et al., 2013). According to the informa-

tion in (Lee et al., 2010) and (Neumann et al., 2013),

shape of upper and lower limbs is changing strongly

during the execution of any movement. In addition,

the construction of a single muscle and the occurrence

of muscle synergism is precisely described. As a re-

sult, we can assume that the matrix of inertia and the

distance from the center of gravity of each link, are

changing. However, changes of these parameters de-

pend on the conﬁguration of the leg.

Furthermore, research results published in (Babi-

arz et al., 2013) justify the application of hybrid sys-

tems for modeling objects with complex biomechan-

ical structure. On the other hand, results presented in

the work by (Burdet et al., 2006) and (Chen, 2011)

indicate that the human arm is unstable in the consid-

ered range of motion. In addition, the state space of

the object can be naturally divided and one can get a

family of subsystems. Those subsystems form a basis

for the modeling of the human leg by means of the

switched systems. In this setting a switching function

depends on the state vector. Consequently, we con-

clude that utilising of switching between the subsys-

tems designated by given states of the limb’s motion

is a novelapproach modeling and analysis of dynamic

properties of human limbs.

The structure of this paper is as follows. We be-

gin by presenting mathematical model of vertical hu-

man two-link leg. Next section describes proposed

switched linear systems. In this section, we show the

partition of the state space and switching functions.

Section 3 presents a comparison of human leg linear

and switched model. Moreover, the obtained results

are shown for two simulation experiments. Finally,

we conclude our proposition of mathematical model.

2 A HUMAN LEG MODEL

2.1 A State-space Model

In general terms the motion of a rigid body can be de-

scribed by second-order differential nonlinear equa-

tion resulting from Euler-Lagrange formalism

M(q) ¨q+C(q, ˙q) ˙q+ G(q) = u, (2)

where: M(q) is an inertia matrix, C(q, ˙q) is a Cori-

olis and centrifugal forces matrix, G(q) is a gravity

forces vector, u are forces and moments acting on the

system, q is an angular displacement.

The human leg model is presented in Figure 1.

The equation describing dynamics of a two-link leg

Figure 1: The model of two–link human leg.

in nonlinear state equation form (2) is

M(q) =

c

1

c

2

cos(q

1

− q

2

)

c

2

cos(q

1

− q

2

) c

3

,

C(q, ˙q) =

0 c

2

sin(q

1

− q

2

) ˙q

2

−c

2

sin(q

1

− q

2

) ˙q

1

0

,

G(q) =

−sinq

1

−c

5

sinq

2

,

c

1

= m

1

l

2

c1

+ m

2

l

2

1

+ I

1

, c

2

= m

2

l

1

l

c2

, c

3

= m

2

l

2

c2

+ I

2

,

c

4

= (m

1

l

c1

+ m

2

l

1

)g, c

5

= m

2

l

c2

g.

where m is a mass, l is a link’s length, l

c

is a distance

from the joint to the center of mass, I is a moment of

inertia, g is a gravity acceleration.

Physical parameters of a modelled leg are pre-

sented in Table 1. Namely, the dynamics of model in

MathematicalModelofaHumanLeg-TheSwitchedLinearSystemApproach

91

Table 1: The parameters of human leg.

m[kg] l[m] l

c

[m] I [kgm

2

]

Link 1 5.7 0.32 0.14 0.061

Link 2 2.65 0.40 0.17 0.038

terms of the state vector

q

T

, ˙q

T

T

can be expressed

as (Babiarz et al., 2013), (Babiarz et al., 2014b)

d

dt

q

˙q

=

˙q

M(q)

−1

[u−C(q, ˙q) ˙q− G(q)]

. (3)

Now, a new set of variables can be assigned to each

of the state variables. In accordance with (3), the new

set of state variables and their equivalences can be ex-

pressed as

x

1

= q

1

, x

2

= q

2

, (4)

x

3

= ˙x

1

= ˙q

1

, x

4

= ˙x

2

= ˙q

2

.

We can write the general state and output equations in

the following way

˙x = Ax+ Bu, (5)

y = Cx+ Du, (6)

where

˙x =

˙q

1

˙q

2

¨q

1

¨q

2

, x =

q

1

q

2

˙q

1

˙q

2

,

y =

¨q

1

¨q

2

, u =

u

1

u

2

.

(7)

2.2 An Approach Switched Linear

System

According to section 1 and subsection 1.1, we can

design switched linear system based on (3) and (7).

Analysing the activity of the human leg, we can as-

sume that the switching times depend on state vector

and the switched linear system is controlled (Liber-

zon, 2003). According to these assumptions the math-

ematical model can be described by equations (Babi-

arz et al., 2014a)

˙x(t) = A

σ(x)

x(t) + B

σ(x)

u(t), (8)

y(t) = C

σ(x)

x(t) + D

σ(x)

u(t). (9)

We consider the switched linear systems with state-

dependent switching. For the switched system de-

scribed above we propose a state-dependent switch-

ing function

˙x =

A

1

x+ B

1

u if x

1

= 0, x

2

≥ 0

A

2

x+ B

2

u if x

1

> 0, x

2

> 0

A

3

x+ B

3

u if x

1

< 0 or x

1

> 0, x

2

= 0

A

4

x+ B

4

u if x

1

< 0, x

2

> 0

(10)

y=

C

1

x+ D

1

u if x

1

= 0, x

2

≥ 0

C

2

x+ D

2

u if x

1

> 0, x

2

> 0

C

3

x+ D

3

u if x

1

< 0 or x

1

> 0, x

2

= 0

C

4

x+ D

4

u if x

1

< 0, x

2

> 0

(11)

Under the above assumption about the division of

state space, the model’s description can be made using

four linear subsystems forming a linear switched sys-

tem (equations (10) and (11)). Switching between any

dynamics depends only on angular displacement be-

cause only this parameter inﬂuences leg’s shape and

conﬁguration during motion. The direct consequence

is that for each switching the angular velocity may be

arbitrary.

Figure 2: The ﬁrst case.

Figure 3: The second case.

Figure 4: The third case.

Figure 5: The fourth case.

PECCS2015-5thInternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

92

3 SIMULATION STUDY

This paragraph presents human leg model’s simula-

tion conditions for linear as well as for switched sys-

tem. All simulations were performed with the use of

Matlab Simulink package. A block diagram is given

in Figure 6.

Figure 6: The Simulink model.

3.1 A Linear Model

For the purpose of obtaining a linear model equation

2 was linearized about equilibrium point (x

0

;u

0

) =

([0rad, 0rad, 0

rad

s

, 0

rad

s

];[0Nm, 0Nm]). Parameters

taken for linearization are given in Table 1. The mo-

ment of inertia was calculated for truncated cone ap-

proximating the shape of a human leg.

A

L

=

0 0 1 0

0 0 0 1

−61.44 21.14 0 0

77.27 −65.14 0 0

, (12)

B

L

=

0 0

0 0

3.806 −4.8

−4.8 14.75

, (13)

C

L

=

−61.44 21.14 0 0

77.27 −65.14 0 0

, (14)

D

L

=

3.806 −4.8

−4.8 14.75

. (15)

3.2 A Switched Linear System

For a linear switched system equation (2) was lin-

earized about arbitrary selected working points which

are contained within particular regions Ω

i

, i =

1, 2, 3, 4 of a state space. Parameters used for calcu-

lation of matrices A

i

, B

i

,C

i

, D

i

, i = 1, 2, 3, 4 in state

equation (8) and output equation (9) are gathered in

Table 2. Values of parameter l

c

and moments of in-

ertia for each Ω

i

region are in the same table. The

moment of inertia was calculated also for a truncated

cone, the radii of which were changing depending on

the values of state vector elements x

1

and x

2

.

Table 2: The parameters of switched model.

Link 1 Link 2

m[kg] 5.7 2.65

l[m] 0.32 0.40

The case I l

c

[m] 0.14 0.17

The case II l

c

[m] 0.14 0.17

The case III l

c

[m] 0.14 0.17

The case IV l

c

[m] 0.14 0.17

The case I I[kgm

2

] 0.061 0.038

The case II I[kgm

2

] 0.061 0.038

The case III I [kgm

2

] 0.061 0.038

The case IV I [kgm

2

] 0.061 0.038

A First Example

3.2.1 Case I (Figure 2)

Matrices A

1

, B

1

,C

1

, D

1

calculated for (x

0

;u

0

) =

([0.349rad, −0.349rad, 0

rad

s

, 0

rad

s

];[0Nm, 0Nm])

working point belonging to Ω

1

region are presented

below:

A

1

=

0 0 1 0

0 0 0 1

−31 −1.8 0 0

14.52 −19.3 0 0

, (16)

B

1

=

0 0

0 0

3.05 −3.02

−3.02 13.9

, (17)

C

1

=

−31 −1.8 0 0

14.52 −19.3 0 0

, (18)

D

1

=

3.05 −3.02

−3.02 13.9

. (19)

3.2.2 Case II (Figure 3)

From the Ω

2

region a working point (x

0

;u

0

) =

([0rad, −0.523rad, 0

rad

s

, 0

rad

s

];[0Nm, 0Nm]) was

selected, for which matrices A

2

, B

2

,C

2

, D

2

were

obtained:

A

2

=

0 0 1 0

0 0 0 1

43 4.65 0 0

44.1 −35.5 0 0

, (20)

B

2

=

0 0

0 0

3.04 −3.4

−3.4 14.7

, (21)

C

2

=

43 4.65 0 0

44.1 −35.5 0 0

, (22)

MathematicalModelofaHumanLeg-TheSwitchedLinearSystemApproach

93

D

2

=

3.04 −3.4

−3.4 14.7

. (23)

3.2.3 Case III (Figure 4)

Working point (x

0

;u

0

) = ([−0.523rad, 0rad, 0

rad

s

,

0

rad

s

];[0Nm, 0Nm]) belonging to Ω

3

region was

used for linearization resulting with matrices

A

3

, B

3

,C

3

, D

3

:

A

3

=

0 0 1 0

0 0 0 1

−32.06 2.54 0 0

20.11 −26, 8 0 0

, (24)

B

3

=

0 0

0 0

3.33 −3.8

−3.8 15.5

, (25)

C

3

=

−32.06 2.54 0 0

20.11 −26, 8 0 0

, (26)

D

3

=

3.33 −3.8

−3.8 15.5

. (27)

3.2.4 Case IV (Figure 5)

Matrices A

4

, B

4

,C

4

, D

4

present in the state and

output equations of subsystem four of switched

system are obtained by linearization about (x

0

;u

0

) =

([−0.523rad, −0.523rad, 0

rad

s

, 0

rad

s

];[0Nm, 0Nm])

working point:

A

4

=

0 0 1 0

0 0 0 1

−45 12.3 0 0

58.2 −50.1 0 0

, (28)

B

4

=

0 0

0 0

3.02 −3.9

−3.9 16

, (29)

C

4

=

−45 12.3 0 0

58.2 −50.1 0 0

, (30)

D

4

=

3.02 −3.9

−3.9 16

. (31)

3.2.5 The Results

Figures 7-10 present time history of four elements

of state vector. Each ﬁgure shows comparison be-

tween consecutive elements of state vector of a

simple linear system and a switched system. Ini-

tial condition used for every simulation was x(t =

0) = [0.349rad, −0.349rad, 0

rad

s

, 0

rad

s

] and simula-

tion time was equal to 20 seconds.

Figure 7: Result of the ﬁrst example - x

1

signal.

Figure 8: Result of the ﬁrst example - x

2

signal.

Figure 9: Result of the ﬁrst example - x

3

signal.

Figure 10: Result of the ﬁrst example - x

4

signal.

3.3 Second Example

In the second simulation example the same sim-

ulation time of 20 seconds is set but initial

conditions are changed. The nonlinear sys-

tems described by (2) was in this case linearize

about arbitrary selected working point (x

0

, u

0

) =

([0.349rad, −0.349rad, 0

rad

s

, 0

rad

s

];[0Nm, 0Nm]).

The resulting matrices A

L

, B

L

,C

L

i D

L

are of the form

A

L

=

0 0 1 0

0 0 0 1

−31 −1.8 0 0

14.5 −19.3 0 0

, (32)

PECCS2015-5thInternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

94

B

L

=

0 0

0 0

3.05 −3.021

−3.021 13.9

, (33)

C

L

=

−31 −1.8 0 0

14.5 −19.3 0 0

, (34)

D

L

=

3.05 −3.021

−3.021 13.9

. (35)

Matrices in the state and output equation of the

switched system remained unchanged.

Another simulation was run, this time

with the initial condition x(t = 0) =

[0rad, −0.349rad, 0

rad

s

, 0

rad

s

] for both, linear

”standard” and switched system. Figures 11-14

depict, as previously, comparison of every member

of the state vector. Based on presented simulation

results for both examples we state that the modelling

with the use of switched systems gives better outcome

than standard approach with linear systems. As a re-

sult the modelled object exhibits smaller oscillations,

Figure 11: Result of the second example - x

1

signal.

Figure 12: Result of the second example - x

2

signal.

Figure 13: Result of the second example - x

3

signal.

Figure 14: Result of the second example - x

4

signal.

what translates directly into the synthesis of control

algorithms.

Although the presented model of a human leg

is largely simpliﬁed, on the current state of work

it is sufﬁcient for the analysis of dynamical prop-

erties which will be of interest to us in further re-

search (Czornik and

´

Swierniak, 2004), (Czornik and

´

Swierniak, 2005).

4 CONCLUSIONS

In future research authors aim at using linear switched

system to modelling of the human leg with seven de-

grees of freedom. Moreover, as shown in (Babiarz

et al., 2013) the subsystems of given object may be

unstable or on the stability boundary and may be un-

observable. For this reason we plan conduct the anal-

ysis of properties such as stability (Shorten, 2007) or

controllability and observability (Czornik and Niez-

abitowski, 2013), (Klamka and Niezabitowski, 2013),

(Klamka et al., 2013). In the next step we will in-

corporate systems with fractional order, as in (Hos-

seinNia et al., 2013), (Kaczorek, 2013), (Tejado et al.,

2013b), (Klamka et al., 2014) into this framework.

ACKNOWLEDGEMENTS

The work of the ﬁrst author was partially supported

by Polish Ministry of Science and Higher Edu-

cation, no. BK-265/RAu1/2014/t.2 (A.B.). The

research presented here were done by the authors

as parts of the projects funded by the National

Science Centre granted according to decisions DEC-

2012/05/B/ST7/00065 (A.C.), DEC-2012/07/N/ST7/

03236 (M.N.) and DEC-2012/07/B/ST7/01404

(R.Z.), respectively. The calculations were performed

with the use of IT infrastructure of GeCONiI Upper

Silesian Centre for Computational Science and Engi-

neering (NCBiR grant no POIG.02.03.01-24-099/13).

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