Dynamic Simulation of Vertebral Column and Control of the Posture

using a Parallel Mechanism

Mouna Souissi

, Walid Amokrane

, Zefeng Wang

and Gerard Poisson

PRISME HEI-Campus centre, 2 Allee Jean Vaillee 36000 Ch

ateauroux, France

FMM HEI-Lille, 13 Rue de Toul 59000 Lille, France

PRISME Laboratory, University of Orl

eans - INSA CVL, France

Keywords:

Simulation, Dynamic Brace, Human Spine, Vertebral Column, Mechanical Structure, Kinematics.

Abstract:

Adolescent Idiopathic Scoliosis (AIS) is a deformity of spine which occurs during growth. This paper presents

a novel method for simulation of a 2D and 3D trunk model and the adaptation of an existing parallel mechanism

to design parallel joints that can be used to correct abnormal postures of the human spine affected by scoliosis.

A 3D model of the system has been elaborated for simulation and design. Simulations results show that this

mechanism is able to permit some bending motions of human torso, taking into account the speciﬁcations of

forward, backward and left/right sideways bending amplitudes.

1 INTRODUCTION

1.1 Human Spine Description

The human vertebral column extends from the skull

to the pelvis and is made up of 33 individual verte-

brae (Abbott et al., 2007). These later are stacked on

top of each other. They are grouped into four regions

(Fig. 1): cervical region, thoracic region, lumbar re-

gion, sacrum and coccyx.

(a)

(b)

Most titlted

vertebra

above apex

Apex

Most tilted

vertebra

below apex

Cobb angle

CERVICAL

THORACIC

LUMBAR

SACRUM

COCCYX

Figure 1: (a) Human Spine. (b) Cobb’s angle.

The cervical and lumbar regions enjoy the great-

est freedom of movement. In the thoracic region, mo-

tion is limited. Motions of a human spine are char-

acterized by the actuation planes that are the sagittal

plane, coronal plane, and axial plane (Fig. 2). The ba-

sic movements of the spine can be classiﬁed by using

planes, as follows:

• In Sagittal plane: Flexion and Extension

• In Coronal plane: Right and Left Lateralization

• In Longitudinal plane: Rotation

Deformity of spine is any abnormality of the for-

mation, alignment, or shape of the vertebral column.

Among the existing deformities, the deformation of

the spine is due to the scoliosis.

(a)

(b)

Figure 2: (a) Sagittal plane, coronal plane, and axial plane.

(b) Force applied to person to move in sagittal plane.

Souissi, M., Amokrane, W., Wang, Z. and Poisson, G.

Dynamic Simulation of Vertebral Column and Control of the Posture using a Parallel Mechanism.

DOI: 10.5220/0006397700170023

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 2, pages 17-23

ISBN: Not Available

1.2 Spine of the Adolescent Idiopathic

Scoliosis (AIS)

A scoliosis curve usually looks a bit like a backward

C shape and involves the spine bending sideways to

the right, which is also called dextroscoliosis. Side-

ways spinal curvature on the left side of the back is

more like a regular C shape and called levoscoliosis.

Scoliosis is categorized into several types depending

on the cause and age of the curve development, 2%

to 3% of the population suffer from scoliosis (Ander-

son, 2007). There are four common types of scolio-

sis curves: right thoracic curve, right thoracolumbar

curve, right lumbar curve, double major curve. The

two parts affected by scoliosis are the thorax and lum-

bar regions.

To measure coronal plane deformity on antero

posterior plain radiographs in the classiﬁcation of sco-

liosis Cobb angle is used. Cobb angle is deﬁned as the

angles formed between a line drawn parallel to the su-

perior endplate of one vertebra above the fracture and

a line dawn parallel to the inferior endplate of the ver-

tebra one level below the fracture (Fig. 1).

A diagnosis of scoliosis is conﬁrmed when the

Cobb angle, (J. Zhang, 2009) (Behairy, 2000) (Cobb,

1984) (Delorme, 2000), is 10 degrees or greater,

which is measured through a standard radiographic

examination. x-rays of the entire spine are taken to

evaluate the front and side curvature. The Cobb an-

gle is used on x-rays to measure the angle between

the most angulated vertebrae that make up the cur-

vature , (Longstein, 1994) (A. H. W. V. Eeuwijk,

2006) (Lafage, 2004). Lines are drawn on the x-rays

or a computer program assists to calculate the angle.

Most scoliosis curves are between 10 to 40 degrees in

magnitude. Although radiographic measurements are

used to decide treatment, a small degree of error ex-

ists when comparing radiographs. A change of 5 de-

grees is usually needed to document an actual change

in curve progression.

Treatment options for idiopathic scoliosis could

include: observation, bracing (Schiller and Eberson,

2010) (Aulisa and Aulisa, 2014) and surgery. Bracing

is the application of external corrective forces onto the

spine and trunk. It can be rigid or ﬂexible. But con-

ventional rigid braces present some limitations: dif-

ﬁcult to move, lower self-esteem, more fatigue and

lower compliance, heavy and non-breathable and un-

comfortable.

These braces are designed to be worn 16 to 23

hours a day. And as the child grows, the required ex-

ternal forces to correct the abnormal posture change

along the length of the curve and over the course of

treatment. Our objective is to achieve a dynamic bra-

ce. It is actuated by 2 motors placed on adjacent

rings to control the force or position applied on the

human body. In this paper we propose to investigate

the kinematics design of a vertebral column for hu-

man. Three kinds of experiments are carried out, ﬁrst

experiment of spine in the sagittal plane, second ex-

periment in coronal plane and third experiment in ax-

ial plane. Simulations of experiments let to choose

the motors of the dynamic brace.

The paper is organized as follows. Section II is

dedicated to trajectory planning and inverse kinemat-

ics of human model in sagittal plane. Section III pro-

poses a dynamics model for the human spine. Section

IV describes the kinematics of the parallel mechanism

proposed to correct the posture of spine affected by

(AIS). Section V is devoted to conclusion and per-

spectives.

2 TRAJECTORY PLANNING AND

INVERSE KINEMATICS OF

HUMAN MOTION

2.1 Model of Human

A spine is a complex remarkable mechanical struc-

ture optimized by thousands generations of humans.

It transmits the weight of the upper body to the pelvis

and is subjected to internal forces whose magnitudes

are many times the entire body weight depicts the hu-

man model used for simulation in the sagittal plane

(y,z). The model is constituted by rectangular seg-

ments, namely head, trunk, pelvis, femurs, tibias and

arms. Masses and heights of the different body parts

are derived from a human kid-size model. Total

height is 1.40 [m]. Total mass is M

=40[Kg] without

pitch joints in the vertebral column.

This mass increases by 0.01 M

for each pitch

joint. The mass of the thorax is equal to the mass

of the trunk when the robot has no articulated spine.

Table 1: Masses and heights of body parts.

Part Masses [Kg] Height [m]

Head 0.08M

0.07H

Arms(2) 0.1M

0.47H

Tibias(2) 0.12M

0.27H

Thighs(2) 0.18M

0.22H

Pelvis 0.02M

0.04H

Trunk (0.5+0.01δ

) M

0.4H

Pitch joint 0.01 M

Trunk

Lumbar part N

0.01 M

Trunk

− H

T horax

Thorax M

Trunk

(1- N

Trunk

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

Table 1 gives masses and heights of the different

parts. The trunk is composed of a thorax and a lumbar

part. Foot mass was taken into account in the tibias.

Arms are connected to the shoulders at the top of the

thorax.

In order to investigate the inﬂuence of the number

of vertebrae on pick up an object from the ﬂoor, a dy-

namics study of the model in the sagittal plane was

conducted in (M. Souissi, 2011). The motion algo-

rithm uses the pseudo-inverse technique.

The input trajectory to be tracked is deﬁned as

F = [x

pelvis

,θ

pelvis

] (1)

Given the input above, the algorithm computes all

joint angles (β

• At t=0, calculate initial β

t=0

, F

t=0

, and Jacobian

t=0

• For each next position F

of input trajectory,

1. ∆F ← F

- F(t).

2. Calculate pseudo-inverse J

from Jacobian J.

3. ∆β ← J

∆F.

4. β(t + ∆t) ← β + ∆β.

5. Calculate F(t + ∆t) using β(t + ∆t) from for-

ward geometric model.

6. Calculate Jacobian J using β(t + ∆t) for next

step.

Where F is the force applied by segment i to i− 1, and

(β

) is the joint angle, between segment i − 1 and i.

2.2 Simulation and Results

-0.2 0 0.2 0.4

0.2

0.4

0.6

0.8

1.2

1.4

0 0.2 0.4

-0.2 0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1.2

1.4

0 0.2 0.4

0.6

-0.2 0 0.2 0.4 0.6

-0.2 0 0.2 0.4 0.6-0.2 0 0.2 0.4 0.6

-0.2 0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1.2

1.4

0 0.2 0.4

0.6

-0.2 0 0.2 0.4 0.6

0.6

0 vertebra

1 vertebra

12 vertebrae

(a)

(b)

(c)

X(m)

Z(m)

Figure 3: Knee ﬂexion to pick up object on ﬂoor. This mo-

tion lasts 3 [sec]. x

pelvis

= 0.19[m] and z

pelvis

= 0.2[m]. (a)

0 vertebra, (b) 1 vertebra and (c) 12 vertebrae.

The experiment consists of pick an object from the

ﬂoor (Fig. 3). This simulation aims to determine the

inﬂuence of the number of vertebra pitch joints in

matter of sum of work in vertebrae and the inclination

of thorax after motion. The cervical region, sacrum

and coccyx are ﬁxed. The vertebrae are placed in

the lumbar and thoracic region between the ﬁrst lum-

bar vertebra and the 7th vertebra of thorax, the region

most affected by the (AIS). The experiment was car-

ried out with 0 vertebra, with 1 vertebra and with 12

vertebrae in the spine.

Time(s)

Inclination (deg)

Figure 4: Inclination (degrees) of thorax in ﬂexion of knees

to pick up an object from the ﬂoor.

Fig. 4 shows the thorax inclination for differ-

ent number of pitch joints. The inclination of trunk

increases signiﬁcantly when the number of vertebra

joints increases. From 12 joints, there is an increase

of nearly 50% of thorax inclination compared with the

conﬁguration without articulated spine.

Time(s)

Work (j)

Figure 5: Spine work in the knee ﬂexion.

Fig. 5 shows the work in spine for 0, 1, 2, 3, 4 and

12 vertebrae. The work is less important for a col-

umn composed of a single joint. The more vertebrae,

the more the work increases. For 12 vertebrae work

reaches 19[ j].

Simulation experiments carried out in the sagittal

plane have shown that pitch joints in the vertebral col-

umn is so important. Rigid brace can limit the mo-

Dynamic Simulation of Vertebral Column and Control of the Posture using a Parallel Mechanism

tions of kid. According to experiments we propose to

work with 12 joints placed in the trunk.

3 DYNAMIC SIMULATION OF

SPINE

3.1 Dynamics

Fig. 2 shows the actuation planes. In this section the

conterclockwise torque T

about center of mass pro-

duced by the applied force F

is calculated:

= F

(2)

Where L

is the length of child from center of

mass to head and equal to 0.50[m].

The opposite restoring torque T

due to the upper

body’s weight is:

= P.D (3)

Where D is the diameter of waist and assuming that

the mass of upper body of the child is 20[kg], his

weight P is:

P = 20.9.8 = 196[N] (4)

for D = 0.101[m] and T

= 19.8[N.m]. The restoring

torque produced by the weight of upper body is there-

fore [N.m]. Trunk is on the verge of toppling when

the magnitudes of these two torques are equal; that is:

= T

(5)

So the force required to move trunk of child in sagittal

plane is :

= T

= 19.8/0.30 = 39.6[N] (6)

Regarding equation (6) force F

depends on the dis-

tance between the center of mass and the point of ap-

plication of force. If F

is near of the center of gravity,

torque will be so important. And since the center of

mass is close to the lumbar part, the force at the level

of the lumbar vertebrae is the most important, which

explains the dimensions of the vertebrae at this level

comparing with the cervical and thoracic vertebrae.

For L

= 0.1[m], torque is equal to F

= 198[N] By

bending the torso the center of gravity will be shifted

away and as the result will the restoring torque be in-

creased.

3.2 A Cad Model for Motion Simulation

In this paper a new model is proposed for the lum-

bar part of a spine. In this model we consider that

all solicitations of the spine are vertically applied at

the center of the vertebra. The 3D model has been

elaborated in ADAMS-software with the aim to run a

dynamic simulation for computing motion properties

and reaction forces (Fig. 6). In ADAMS-software the

discus has been modeled as a body that is attached

to the assembled vertebrae. Joint between all discs is

spherical. This joint is very near to the real type of

link between vertebrae.

Figure 6: A 3D ADAMS model of all body. Masses and

heights of the different body parts are derived from a human

kid-size model.

First simulation consists of tilting the entire body

to the right and left and calculating the forces applied

to the column in the thoracic and lumbar region. Fig.

7 shows the simulation with ADAMS-software and

the result of forces applied in vertebrae for the left

bending motion. Regarding the simulations, forces

applied to the thoracic and lumbar vertebrae vary be-

tween 130[N] and 107[N] for the left bending motion.

Results correspond to the equation (6). When the dis-

tance between force and vertebra is small, the magni-

tude of force is more important.

Second simulation consists of the for-

ward/backward motion. Fig. 8 shows the simulation

with ADAMS-software and the result of forces ap-

plied in vertebrae for the forward motion. Forces vary

between 300[N] and 350[N]. And third simulation

consist of the yaw motion. Forces applied to the

vertebrae reachs 145[N].

0 1 2 3 4 5 6 7 8 9 10

Time (10

-2

135

130

125

120

115

110

105

100

Force (N)

0 1 2 3 4 5 6 7 8 9 10

Time (10

-2

135

130

125

120

115

110

105

100

Force (N)

(a)

(b)

Figure 7: Forces applied to the vertebrae for the left/right

bending motion.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

0 1 2 3 4 5 6 7 8 9 10

Time (10

-2

Force (N)

350

340

330

320

310

300

290

280

270

260

0 10 20

Time (10

-2

Force (N)

340

330

320

310

300

290

280

270

260

(a)

(b)

Figure 8: Forces applied to the vertebrae for the (b) forward

Time (10

-2

Figure 9: Forces applied to the vertebrae for the yaw mo-

tion.

In the next section we propose a parallel mech-

anism to correct the abnormal posture change along

the length of the curve, taking into account the max-

imal forces applied in vertebrae in the three planes,

pitch, roll and yaw motions.

4 PARALLEL TILTING MODULE

FOR CORRECTING POSTURE

4.1 Kinematic Model

In this section we propose a dynamic brace to treat

children with scoliosis without limiting upper body

movement. Mechanism is ﬁxed close to the verte-

brae L1, L2, L3, L4 and L5. This choice is due to

the opti- mization of mechanism in matter of torque

applied to the two motors ﬁxed in the bottom plat-

form (M. Souissi, 2016). The objective of this sys-

tem is to modulate the corrective forces on the spine

in desired directions while still allowing the users to

perform typical everyday activities. The most impor-

tant motions to control are the pitch, roll and yaw mo-

tions in the actuation planes. Mechanism proposed

was simulated with ADAMS. It is ﬁxed in the lum-

bar region. The parallel mechanism, developed in

ℓ

(a)

(b)

Figure 10: (a) Perspective view of parallel mechanism in

initial position. The central rod is ﬁxed and rigid. It is at-

tached to the top platform by a Universal joint at O. The

mechanism is actuated by two revolute joints, each of them

is located at the bottom of each arm. The two arms are ar-

ranged at 90 [deg]. The arm A

is planar and remains

in the (xz) remains in this plane. It is composed of two rev-

olute joints and one U-joint. The other arm B

is ini-

tially in the (yz) plane, and does not remain in this plane if

the top platform rolls. (b)Perspective view of parallel mech-

anism after pitch and roll rotations.

(M. Souissi, 2012a) and compared with serial system

in (M. Souissi, 2012b), is inspired from a ﬂight simu-

lator (Sabrie, 2004) and (Alexander V. Korobeynikov,

). It consists of 2 platforms – one bottom platform

and one top platform OA

, that are linked

by a central vertical rod CO and two arms arranged at

90[deg] in the initial position (Fig. 10). The central

rod CO is ﬁxed and always remains vertical. It joins

the top platform through a Universal joint whose drive

is responsible for roll and pitch motion of the top plat-

form. The arm A

is planar and is composed of

two segments, two revolute joints at A

and A

, and

one Universal joint at A

. This is the planar arm. The

arm B

also includes two segments, one revolute

joint at B

, one Universal joint at B

and one ball joint

at the attachment locus B

with the top platform. This

arm is 3D. The bottom platform is linked to coordi-

nate frame R

, centered at C with axes i, j and k.

The top platform rotates about O and is linked to co-

ordinate frame R

, frame centered at O whose axes are

, j

and k

(Fig. 10). The top platform can be pitched

about ﬁxed axis j by angle θ

, and rolled about axis

by angle θ

The two active joints are the revolute joints at A

and B

. The associated rotation angles are denoted by

α and β.

The length d

and `

are approximately half the

lengths d

and `

respectively.

This means that the motor axes should be located

in the middle with respect to the trunk half width and

the trunk half depth respectively.

Dynamic Simulation of Vertebral Column and Control of the Posture using a Parallel Mechanism

Table 2: Normalized parameters of 3D and 2D arm.

Parameter 2D arm Parameter 3D arm

0.666

0.733

0.466

1.333

0.666

4.2 Torque Considerations

The actuators will have to support the mass M of the

bottom platform. Assuming this mass is concentrated

on a point G that is ﬁxed with respect to the bottom

platform, we can express the torques τ

and τ

that

the actuators must exert to support this mass:





= J





(7)

where J is the Jacobian matrix of forward kinematics:





= J





J =



−r



= MOGg j

= MOGgi

where g is the gravity, r

, r

and r

are expressed

thanks to the scalar triple product of three vectors.

The vectors of the type u

are unit vectors along the

direction given by the pair of points X and Y. τ

and

are the gravity torques exerted at the center O of

the bottom platform, respectively about the axis j and

the axis i

Let us give a geometric interpretation. If r

= 0,

i.e. actuators are activated separately,we have:

= r

.τ

= r

.τ

The ratios r

and r

can be interpreted respectively as

the ratio of the area of the triangle A

over the

triangle OA

(Fig. 11), and the ratio of the area of

the triangle B

over the triangle OB

To reduce the active torques in the parallel mech-

anism, it is therefore necessary to reduce these two

Figure 11: Geometric interpretation of the ratio of the active

torque at A

over the gravity torque at O. According to the

scalar triple product expression, it is equal to the surface of

the triangle A

over the surface of the triangle OA

ratios as much as possible. This can be achieved by

a careful choice of the different lengths (`

) of the

mechanism, and considering the full scope of incli-

nations required from the speciﬁcations of the tele-

echography application. Intuitively through the ge-

ometrical interpretation, the surface of the triangle

should be reduced compared to the trian-

gle OA

. This can be achieved by increasing the

length A

over A

, and increasing the length OA

from OC. However, one may pay attention not to go

through ﬂat triangle conﬁgurations that represent sin-

gularities where the mechanism would not be control-

lable any more.

The motion consist of giving the angular inputs

for the pitch/roll joint of the top platform, namely θ

and θ

. The vertebral column is bent 10[deg] for-

ward, then 10[deg] backward. The angles for the joint

motors, namely α and β are calculated thanks to the

inverse geometric model (Fig. 12).

Figure 12: Angle trajectories θ

and θ

of platform and

the related motor angles α and β for the pitch inclination

motion.

4.3 Simulations and Result

The prototype was simulated with ADAMS-software

and trajectory planning is designed with Matlab-

SIMULINK, which is used to control the ADAMS

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

(a)

(b)

Torque (N.m)

(b)

Time(s)

(L1)

(L2)

(L3)

(L4)

(L5)

)

(L1)

(L2)

(L3)

(L4)

(L5)

)

Figure 13: Prototype used to correct the posture in the sagit-

tal(a) and coronal(b) plane.

model (M. Souissi, 2012b), minimization of torque

was done in (M. Souissi, 2016). Prototype is used to

correct the posture of the spine in two planes: sagittal

and coronal plane.

For all experiences done with ADAMS, the sagital

plane is presented by zy-plane, the axial plane is pre-

sented by xy-plane and the coronal plane is presented

by xz-plane as shown in ﬁgure (Fig. 2) .

Fig. 13 shows the torque of the two motors of roll

and pitch joint to move spine in the sagittal and coro-

nel plane. Maximal torque of the 3D arms is 10[N.m]

and torque of the 2D motor is constant and is about

6[N.m] in the ﬁrst experience. Maximal torque of the

3D arms is 10[N.m] and torque of the 2D motor is con-

stant and is about 8[N.m] in the second experience.

5 CONCLUSION AND

PERSPECTIVES

In this paper, a model for the human spine is pro-

posed for analyzing applied forces on the interverte-

bral discs through a suitable motion simulation. A

new dynamic brace has been proposed to correct the

posture of the human affected by scoliosis in two

planes, sagittal and coronal plane. The next steps of

this research will consist realization and motorization

of system according to the dynamics results obtained.

REFERENCES

A. H. W. V. Eeuwijk, S. Lobregt, F. A. G. (2006). A

novel method for digital x-ray imaging of the com-

plete spine. 344.

Abbott, J., Nagy, Z., Beyeler, F., and Nelson, B. (2007).

volume 3.

Alexander V. Korobeynikov, V. E. T. Modeling and evalu-

ating of the stewart platform.

Anderson, S. M. (2007). Spinal curves and scoliosis, radio-

logic technology. 79.

Aulisa, A.G., G. V. M. E. G. M. F. F. and Aulisa, L. (2014).

Brace treatment in juvenile idiopathic scoliosis: a

prospective study in accordance with the srs criteria

for bracing studies-sosort award 2013 winner.

Behairy, Y., H. D. H. D. M. J. M. M. (2000). Partial correc-

tion of cobb angle prior to posterior spinal instrumen-

tation. 20:398–401.

Cobb, J. (1984). Outline for the study of scoliosis. 5:261–

275.

Delorme, S., L. H. P. B. R. C. C. C. J. (2000). Pre-, intra-

, and postoperative three-dimensional evaluation of

adolescent idiopathic scoliosis. 13:93–101.

J. Zhang, E. Lou, L. H. L. D. L. H. J. V. R. Y. W. (2009). Au-

tomatic cobb measurement of scoliosis based on fuzzy

hough transform with vertebral shape prior. 22:463–

472.

Lafage, V., D. J. L. F. S. W. (2004). 3d ﬁnite element simu-

lation of cotreldubosset correction. 9:17–25.

Longstein, J. (1994). Adolescent idiopathic soliosis.

344:1407–1412.

M. Souissi, V. Hugel, P. B. (2011). Inﬂuence of the number

of humanoid vertebral column pitch joints in ﬂexion

movements.

M. Souissi, V. Hugel, P. B. (2012a). Design optimization of

parallel joint mechanism for humanoid spine. pages

997–1000.

M. Souissi, V. Hugel, P. B. (2012b). Modeling and simula-

tion of humanoid robot spine vertebra. 2:415–418.

M. Souissi, V. Hugel, P. B. (2016). Minimized-torque-

oriented design of parallel modular mechanism for hu-

manoid waist.

Sabrie, E. (2004). Analyse d’un m

ecanisme de simulation

de vol sph

erique et son contr

ole en temps reel. Facult

des sciences et de g

enie universitaire, Laval, Qu

ebec.

Schiller, J.R., T. N. and Eberson (2010). Brace management

in adolescent idiopathic scoliosis. 468:670–678.

Dynamic Simulation of Vertebral Column and Control of the Posture using a Parallel Mechanism