chapter 3 kinematics analysis for continuum robotic ...docinsa.insa-lyon.fr › these › 2005 ›...

Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II

Gang CHEN Thèse INSA de Lyon, LAI 2005 83

Chapter 3

Kinematics Analysis for Continuum Robotic

Manipulator: EDORA II



3

CHAPTER 3 ........................................................................................................................................................ 83

KINEMATICS ANALYSIS FOR CONTINUUM ROBOTIC MANIPULATOR: EDORA II.................... 83

3.1 Three essential parameters characterize the deflected shape of EDORA II ...................................... 85

3.2 Kinematics analysis using basic geometry ........................................................................................ 88 3.2.1 Basic geometry for kinematic analysis ...................................................................................................... 88

3.2.2 Derivation of orientation angle of the bending plan .................................................................................. 89

3.2.3 Derivation of bending angle α .............................................................................................................. 90

3.2.4 Summary ................................................................................................................................................... 91

3.3 Derivation of kinematics relating to internal pressure of each chamber ........................................... 91 3.3.1 The experiment setting and results ............................................................................................................ 92

3.3.2 Relationship between deflected shape with relation to the applied pressure of each chamber .................. 93

3.4 Velocity Kinematics.......................................................................................................................... 94 3.4.1 Non-redundant case ................................................................................................................................... 95

3.4.2 Redundant case.......................................................................................................................................... 95

3.5 Inverse velocity kinematics ............................................................................................................... 96

3.6 Validation of kinematic model .......................................................................................................... 98 3.6.1 The sensor choice and experimental setup................................................................................................. 99

3.6.1.1. The miniBIRD.......................................................................................................................... 99 3.6.1.2. Experimental setup................................................................................................................. 100

3.6.2 Validation of bending angle..................................................................................................................... 101

3.6.3 Validation of orientation angle ................................................................................................................ 103

3.6.4 Verification of correlation among each chamber..................................................................................... 105

3.6.5 Estimation of a correction parameter....................................................................................................... 106

3.7 Conclusions..................................................................................................................................... 109



Since continuum robotic manipulators do not have link joints, which makes them

completely different from the conventional robot, new problems are thus generated on how to

build the kinematics of these manipulators along with the corresponding control problems.

Some modeling of hyper-redundant robotic manipulators inspired the works of the continuum

robot manipulator. [CHIRIKJIAN 92] [CHIRIKJIAN 93] [CHIRIKJIAN 94] [CHIRIKJIAN 95]

proposed a great deal of theory that had laid the foundation for the kinematics of hyper-

redundant robots. Most of their research used modal analysis to describe the robot’s kinematics.

Their sophisticated analysis was based on treating the robot as a “string”. However, their modal

approach does not take into account the physical constraints of real continuum robots, and the

resulting algorithms are complex, non-intuitive, and hard to integrate with conventional robot

algorithms. [MOCHIYAMA 98], [MOCHIYAMA 99] and [MOCHIYAMA 01] presented

research in the area of kinematics and the shape correspondence between a hyper-redundant

robot and a desired spatial curve. The idea was to define the kinematics of the robot by

associating it with a predetermined curve. [GRAVAGNE 00a] [GRAVAGNE 00b]

[GRAVAGNE 00c] applied several different approaches for analyzing the kinematics of

continuum robots in his work. Most recently, by using the concept of Denavit-Hartenberg for

the modeling conventional robotic manipulator, [HANNAN 01] proposed an innovate method

for the continuum style robotic manipulator- the Elephant trunk. His model utilized the concept

of constant curvature sections, and incorporated them through the use of differential geometry

into a modified Denavit-Hartenberg procedure to determine the kinematics. The importance of

this is that the Denavit-Hartenberg procedure is the most commonly used approach for

determining the kinematics of conventional robots. Thus, the theory and analysis method of the

conventional robot can be easily used for modeling and controller design. Then, [BYRAN 05]

improved the virtual Denavit-Hartenberg-based approach by optimizing the virtual joint

configurations for the modeling of a Multi-Section Continuum Robot: Air-Octor [BYRAN 05].

In this chapter, a different kinematic model will be developed for EDORA II in detail.

3.1 Three essential parameters characterize the deflected shape of

EDORA II

As was presented in several textbooks [SCIAVICCO 00], joint angles and link lengths

provide an easy and physically realizable description of a conventional robotic manipulator

when embedded in its kinematic model, but for continuum robotic manipulators this no longer

holds true due to the continuous nature of their design. In a continuum robot, most often, there

are no clearly identifiable places where joints and links can be defined. Therefore, a kinematic

model must use new parameters that more appropriately describe the continuous and deflected

shape of continuum robots. The kinematic model introduced here strongly uses the concept of



curvature to describe the shape of the manipulator. This concept is very natural for curved

structures, and is exploited in the work [HANANN 02]. In this dissertation, the concept of

curvature has been extended to three dimensions, but the kinematics of EDORA II will be

developed directly from the direct geometry between the actuator inputs and the chosen

parameters without using D-H transformation. Thus three parameters (figure 3.2) have been

chosen to characterize the position and the orientation of the tip with respect to the bottom of

the manipulator as done in our previous prototype EDORA [THOMANN 03] [CHEN 03]

[CHEN 04]. They are described as the following:

• L is the virtual length of the center line of the robotic manipulator;

• α is the bending angle in the bending plane;

• φ is the orientation of the bending plane;

It is worth to note that [SUZUMORI 92], [LANE 99],[BAILLY 04b], [HANNAN 03],

[GRAVAGNE 02] also used almost the same set of parameters for modeling of proper

continuum robots.

Figure 3.1 Schema for the complete assembly of the experimental platform



Figure 3.2 Frame of reference for EDORA II

Consider an EDORA II shown in Figure 3.1. It is supported at the bottom end in a way such that

it stands vertically and the top end can move freely with the pressure variation in three

chambers. Figure 3.2 shows the frame of reference O-XYZ which is fixed at the base of the

manipulator. The X-axis is the one which passes by the center of the bottom end and the center

of the chamber 1. The XY-plane defines the plane of the bottom of the actuator, and the z-axis

is orthogonal to this plane. The frame of reference UVW is attached to the top end of the

manipulator. So the bending angle α is defined as the angle between the O-Z axis and O-W

axis. The orientation angle φ is defined as the angle between the O-X axis and O-T axis, where

O-T axis is the project of O-W axis on the plan X-O-Y. The notation is explained as the

following:

i: chamber index, i = 1, 2, 3

R: radius of curvature of the center line of the robotic tip

iL : arc length of the ith chamber

L0 : initial length of the chamber

iP : pressure in the chamber i

S : effective surface of the chamber

iR : radius of curvature of the ith chamber

ε : stretch length of the virtual center line

iL∆ : the stretch length of the ith chamber



3.2 Kinematics analysis using basic geometry

Controlling the deflected shape of the manipulator requires a kinematic model relating

deflected shape in terms of extension and bending to actuator inputs. This section will focus on

the deflected shape to the length of three chambers. Given three known chamber length, 1L , 2L ,

3L and the constant distance r from the center of EDORA II to the center of each pressurized

chamber, the following equations allow computation of the resulting length L , the bending

angle α and the orientation angle φ .

Two assumptions have been done for simple analysis to obtain the kinematics of EDORA II.

• There is no axial displacement;

• The load effects are ignored;

• Even though EDORA II can bend till 120°, the bending angle α is now constrained between

0 / 2< α ≤ π because the deflected shape will be more complicated when the bending angle

is more than 90°.

With three presumptions, the deflected shape of EDORA II at the bending moment is

assumed to be an arc of a circle, just as most researchers did for their continuum robots

[SUZUMORI 92][LANE 99][BAILLY 04b][HANNAN 03][GRAVAGNE 02]. In addition to

this, three chambers have the same bending angles except the different arc lengths.

3.2.1 Basic geometry for kinematic analysis

Given the bending angle at the bending moment, then the key relation for the kinematics

is given as:

i iL R= α (3.1)

or given in the form of the stretch length of virtual central line:

i i 0 i i 0L / R (L L ) / R (L ) / Rα = = + ∆ = + ε (3.2)

and

i i 0L L L∆ = − Angle iφ is defined as the angle of bending plan relative to the chamber i, shown in Figure 3.3

1

2

3

- 120

120

φ = φ⎧⎪ φ = − φ⎨⎪ φ = − − φ⎩

(3.3)

By using these angles, the radius of curvature R of the chamber i can be represented as:

i iR = R - rcos( )φ (3.4)

and :



i iL = L - rcos( ) α φ (3.5)

where r is the distance between the center of the manipulator and the center of the chamber .

Figure 3.3 Definition of angle iφ

As 3

ii 1

cos 0=

φ =∑ , one can be deduced from (3.4) and (3.5) that:

31

i3i 1

R R=

= ∑ (3.6)

31

i3i 1

L L=

= ∑ (3.7)

These two equations explain that the deformation of the manipulator on the whole is the average

of three chambers.

3.2.2 Derivation of orientation angle of the bending plan

By using equation (3.5) in the first two chambers, then

1 1 2 2L r cos( ) L r cos( )+ α φ = + α φ (3.8)

replacing (3.3) and expanding the cosines, then α and L can be obtained

2 1L L2r 3cos 3 sin

−α =

φ − φ (3.9)

2 11

2(L L )cosL L

3cos 3 sin− φ

= +φ − φ

(3.10)

continuing using (3.5) on chamber 3 and replacing α and L,

2 13 1

2(L L )(3cos 3 sin )L L

3cos 3 sin− φ − φ

= +φ + φ

(3.11)

ϕ

r2

3

1r1

r2

r3

Plan ),,( ztOdefined by φ

x



finally the orientation angle tan φ is found as:

2 3 1 2 3a tan 2( 3(L L ),2L L L )φ = − − − (3.12)

the function atan2(y,x) is a function used in MATLAB which extends the function a tan(y / x) in

the quadrant for the point (x,y). It has the following form:

a tan(y / x) if x 0 and y 0a tan(y / x) - if x 0 and y 0

at an 2(y,x) a tan(y / x) if x 0 /2 if x = 0 and y>0

- /2 if x = 0 and y<0

+ π < ≥⎧⎪ π < <⎪⎪= >⎨⎪π⎪

π⎪⎩

(3.13)

It is worth noting that the equation (3.12) is undetermined when 2 33(L L ) 0− = and

1 2 32L L L 0− − = , i.e. 1 2 3L L L= = . In this case, the manipulator demonstrates the pure

elongation, which will not be discussed in this chapter.

3.2.3 Derivation of bending angle α

Then by combining equation (3.2) and equation (3.4), R is obtained:

3

ii 1

13

i 1i 1

LR r cos

L 3L

=

=

= φ−

∑

∑ (3.14)

Since there exists the following trigonometric relation:

2 2

xcos(a tan 2(y, x)) , (x,y) (0,0)x y

= ∀ ≠+

(3.15)

Then the radius of bending shape is finally described as:

3

ii 1

L

r LR

2==

ξ

∑ (3.16)

and 2 2 2L 1 2 3 1 2 1 3 2 3L L L L L L L L Lξ = + + − − − (3.17)

After R is obtained, by using (3.2) and (3.7), the bending angle α can be easily gotten:

L23rξ

α = (3.18)



3.2.4 Summary

In this section, the kinematics model based on basic geometry has been dealt with. The

concept of kinematics is just used for easy understanding with an analogy to the conventional

robot. Knowing the lengths of three chambers, analytical expressions for three system

parameters which characterize the deflected shape at the bending moment are the following:

2 3 1 2 3a tan 2( 3(L L ),2L L L )φ = − − − (3.12)

L23rξ

α = (3.18)

11 2 33L (L L L )= + + (3.7)

Written in matrix form, it can be expressed as:

X f (q)= (3.19)

where TX ( , ,L)= α ϕ , T1 2 3q (L ,L ,L )=

with the assumption that the deflected shape is an arc of a circle, so the expression in the

Cartesian coordinate system can be easily calculated from figure 3.4:

Lx= (1- cos ) cos

Ly = (1- cos ) sin

Lz = sin

⎧ α φ⎪ α⎪⎪ α φ⎨ α⎪⎪ α⎪ α⎩

(3.20)

3.3 Derivation of kinematics relating to internal pressure of each

chamber

Since the deflected shape of the manipulator is controlled by the pressure differential of

three chambers by using servovalves, the kinematic model obtained in the above section should

be developed relating to the three input pressure of chambers. So the relationship between the

length variation of each chamber and the applied pressure in the chambers is determined, then

the deflected shape is determined relating to the applied pressure.



3.3.1 The experiment setting and results

Considering that the exterior forces to the top end of EDORA are negligible and that the

mass is negligible, the stretch length of each chamber is assumed to be proportional to the

pressure variation in each chamber as is done by many researchers on continuums robots. In our

case, however, the strong nonlinearity relating the stretch length of chamber to the pressure

variation has been shown in preliminary experiments. This relationship is described as:

i iL f (P )∆ = (3.21)

where if (P ) is the nonlinear function of iP and this section will deal with this function through

experiments.

The stretch length of a single chamber is measured when the pressure is applied from 0

bar to the pressure maximum which can make the actuator bend 90°, while the pressures of the

other two chambers are kept at zero bar. For precise measurements, the pressure in the chamber

is kept constant by using a closed-loop controller. Since the deformation of the each chamber is

an arc of a circle, it’s difficult to find a suitable sensor to measure the arc length. A simple

method is then used to measure the length of the chamber. When the chamber is stretching

under pressure, a fine string is placed right outside the stretched chamber, so the length of the

string can be considered approximately as the length of the chamber.

Results obtained from experiments proved that there is a non-linear behavior (hysteresis)

between the pressure and the stretch length of each chamber (figure 3.4). A polynomial model is

thus used to approximate the measurements. A criteria based on the norm of the mean error is

used to select the order of each polynome fitting the data. This analysis shows that a second

order polynomial approximation allows to fit significantly the actual data as is shown on figure

3.4. The corresponding results can be written as:

21 1 1min 1 1max

1 1 11 1min

22 2 2min 2 2max

2 2 2

3.8P + 24.8P if P P PL f (P )0 if P <P

4.4P +15.7P if P P PL f (P )0

⎧ < <⎪∆ = = ⎨⎪⎩

< <∆ = =2 2min

23 3 3min 3 3max

3 3 33 min

if P <P

7.9P 33.9P if P P PL f (P )0 if P <P

⎧⎪⎪⎪

⎧⎪ ⎪⎨ ⎨

⎪⎪ ⎩⎪ ⎧− + < <⎪⎪∆ = = ⎨⎪ ⎪⎩⎩

(3.22)

Where iminP (i = 1, 2, 3) is the threshold of the working point of each chamber and their values

equal 1min

2min

3min

P = 0.7 barP = 0.8 barP = 0.8 bar

⎧⎪⎨⎪⎩

and imaxP (i = 1, 2, 3) is the maximum pressure that can be applied to each chamber.



0.6 0.8 1 1.2 1.4 1.6 1.8 2 -5

0

5

10

15

20

25

30

35

40

Pressure of the chamber (bar)

Dis

plac

emen

t of t

he c

ham

ber (

mm

)

The relationship between the input pressure and the stretch l h

Measurement of Chamber 1Measurement of Chamber 2Measurement of Chamber 3Elongation/pressure model of chamber 1Elongation/pressure model of chamber 2Elongation/pressure model of chamber 3

0.7 Bar

0.8 Bar

Figure 3.4 The model between the stretch length and the applied pressure of each chamber

3.3.2 Relationship between deflected shape with relation to the applied

pressure of each chamber

After the relationships have been determined between the stretch length of the each

chamber and the applied pressure, then the corresponding length of each chamber under the

pressure variation is expressed as the following:

1 0 1 0 1

2 0 2 0 2

3 0 3 0 3

L L L L f (P )L L L L f (P )L L L L f (P )

= + ∆ = +⎧⎪ = + ∆ = +⎨⎪ = + ∆ = +⎩

(3.23)

By inserting equation (3.23) into (3.7) and (3.12), then new equations are obtained as followed: 1

0 1 2 33L L (f (P ) f (P ) f (P ))= + + + (3.24)

2 3 1 2 3a tan 2( 3(f (P ) f (P )), 2f (P ) f (P ) f (P ))φ = − − − (3.25)

In the same way, the equation (3.18) can be transformed as: 2 2 2

L 1 2 3 1 2 1 3 2 3f (P ) f (P ) f (P ) f (P )f (P ) f (P )f (P ) f (P )f (P )ξ = + + − − − (3.26)

But for the sake of easier distinction, it is named as: 2 2 2

P 1 2 3 1 2 1 3 2 3f (P ) f (P ) f (P ) f (P )f (P ) f (P )f (P ) f (P )f (P )ξ = + + − − − (3.27)

so the bending angle can be expressed in applied pressure:

P23rξ

α = (3.28)



And the matrix form of this model is given by :

Pf ( )=X q (3.29)

where T( , ,L)= α φX , Tp 1 2 3(P ,P ,P )=q .

3.4 Velocity Kinematics

In the conventional joint/link robotic manipulator, differential kinematics are presented

to explain the relationship between the joint velocities and the corresponding manipulator linear

and angular velocity. This is used to coordinate the motion of the individual joints in order to

move the manipulator in a specified direction at a specified speed. Analogous to this concept in

conventional kinematics analysis, the velocity kinematics for continuum robots can be written

as:

J =X q (3.30)

where ∈X is the task space vector, i.e. position and/or orientation, q is the joint space vector,

J is the Jacobian matrix and is a function of q , and the dot implies differentiation with respect

to time, i.e. ddt

. For the manipulator EDORA II, the task space is represented by the position

and orientation of the end-tip of EDORA II:

( )T L= α φX

and the joint space vector is chosen as the applied pressure in each chamber because of the final

control implementation is needed to calculate the applied pressure in three chambers.

( )Tp 1 2 3P P P=q

So there are two methods to calculate the Jacobian matrix using the analytical technique. The

first one is to directly use the differentiation of the direct kinematics function with respect to the

joint variables expressed in pressure, i.e. equation (3.23) (3.24) and (3.27); the other option is

through an indirect method. Firstly, the differentiation of the direct kinematics function with

respect to the joint variables expressed in length of each chamber, i.e. (3.7) (3.12) and (3.18),

then partial derivative relating the length to the applied pressure of each chamber will be

calculated. The Jacobian is shown as:

X LJL P

∂ ∂=∂ ∂

(3.31)

In this thesis, the second method is used to calculate the Jabobian matrix for calculation

considerations.



3.4.1 Non-redundant case

By choosing the task space vector X , it’s natural to compute the Jacobian matrix via

differentiation of the direct kinematics function with respect to the joint variables. This method

is called analytical technique and the Jacobian matrix can be written as:

31 2

1 1 2 2 3 3

31 2

1 1 2 2 3 3

31 2

1 1 2 2 3 3

LL L

L P L P L PLL L

J L P L P L P

LL LL L L L P L P L P

⎛ ⎞∂∂ ∂∂α ∂α ∂α⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟∂∂ ∂∂φ ∂φ ∂φ= ⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟∂∂ ∂∂ ∂ ∂⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(3.32)

according to equations (3.24) (3.25)and (3.28), and knowing that (x,y) (0,0)∀ ≠ :

2 2

g(x, y) f (x, y) f (x, y) g(x, y)x xa tan 2(f (x, y),g(x, y))

x g(x, y) f (x, y)

∂ ∂−∂ ∂ ∂=∂ +

(3.33)

and from equation(3.23), the velocity kinematics is obtained:

' ' '1 2 3 2 1 3 3 1 21 2 3

L L L

' ' '2 3 3 1 1 21 2 3

L L L

' '1 2

2L L L 2L L L 2L L Lf (P ) f (P ) f (P )

3r 3r 3rd

3(L L ) 3(L L ) 3(L L )d f (P ) f (P ) f (P )

2 2 2dL

1 1f (P ) f (P ) 3 3

− − − − − −ξ ξ ξ

α⎛ ⎞− − −⎜ ⎟φ =⎜ ⎟ ξ ξ ξ⎜ ⎟

⎝ ⎠

1

2

3'

3

dPdPdP

1 f (P )3

⎛ ⎞⎜ ⎟⎜ ⎟

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

(3.34)

where 2 2 2L 1 2 3 1 2 1 3 2 3L L L L L L L L Lξ = + + − − − , is defined by the equation (3.17) and '

if (P ) is the

derivative of if(P ) (i = 1, 2, 3) concerning to the pressure iP (i = 1, 2, 3) .

3.4.2 Redundant case

Although three parameters ( )T Lα φ can uniquely determine the position and the

orientation of end-tip of EDORA II in the task space, it’s difficult to place a sensor to measure

the length of virtual center line. However, if only orientation vector ( )T α φ is considered, there

is not much effect on the application of tubular exploration because the orientation is enough

for the guidance. In this case, the robotic manipulator is functionally redundant because the

number of components of task space is less than the number of degrees of freedom. Thus the

velocity kinematics are given as:



r r pJ =X q (3.35)

where Tr ( )= α φX

then Jacobian matrix with relation to the three pressure in the chamber is given as following:

31 2

1 1 2 2 3 3r

31 2

1 1 2 2 3 3

LL L

L P L P L PJ

LL L

L P L P L P

∂∂ ∂∂α ∂α ∂α⎛ ⎞⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟=⎜ ⎟∂∂ ∂∂φ ∂φ ∂φ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

where rJ is the submatrix of J constructed by the first two rows and three columns of J , this is

' ' '1 2 3 2 1 3 3 1 21 2 3 1

L L L2

' ' '2 3 3 1 1 231 2 3

L L L

2L L L 2L L L 2L L Lf (P ) f (P ) f (P ) dP3r 3r 3rd

dPd 3(L L ) 3(L L ) 3(L L ) dPf (P ) f (P ) f (P )

2 2 2

− − − − − −⎛ ⎞⎛ ⎞⎜ ⎟ξ ξ ξα⎛ ⎞ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟φ − − −⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ξ ξ ξ⎝ ⎠

(3.36)

3.5 Inverse velocity kinematics

Equation (3.30) and Equation (3.35) provided the velocity of task space and angular

velocity of the robot manipulator as a linear function of joint velocities based on the non-

redundant case and redundant case. The inverse velocity kinematics is concerned with the joint

velocities and with the velocity of task space. Namely, given a desired manipulator velocity, we

find the corresponding joint velocities that cause the robot manipulator to move at the desired

velocity. In the case of non-redundant configuration, since the Jacobian matrix is square matrix

of n order and the determinant is not null, so it’s easy to calculate directly the inverse matrix of

J . So the inverse Jacobian matrix is given the following relation: 1J −=q X (3.37)

In the case of a redundant manipulator with respect to a given task, equation (3.35), the

inverse kinematic problem admits infinite solutions. This suggests that redundancy can be

conveniently exploited to meet additional constraints on the kinematic control problem in order

to obtain greater manipulability in terms of manipulator configurations and interaction with the

environment. Some typical applications using redundancy are referenced here:

• Obstacle avoidance [MACIEJAWSKI85];

• Mechanical joint limits [LIEGEOIS 77];

• Joint actuator power consumption [VUKOBRATOVIC 84];

• Avoidance of kinematic singularities [YOSHIKAWA 85a], [YOSHIKAWA 85b],

[KLEIN 87], [ANGELES 88], [CHIU 88];



A viable solution method is to formulate the problem as a constrained linear

optimization problem. [WHITNEY 69], in his pioneering work on resolved-rate control,

proposed to use the Moore-Penrose pseudoinverse of the Jacobian matrix as: T T 1

p J (J (JJ ) )+ −= =q X X (3.38)

The pseudoinverse Jacobian matrix has a least squares property that generates the minimum

norm joint velocities.

By revising the pseudoinverse minimum-norm solution, a more general solution (3.35)

and is given by:

p J [I J J]g+ += + µ −q X (3.40)

where I is the identity matrix and g is an arbitrary joint velocity vector. The homogeneous term

[I J J]g+µ − is the null space projection of the solution of (3.37). The null space solution only

generates motion in the “joint” space of the manipulator, and will produce zero movement in

task space of the robot. This null space motion is also known as the self motion of the robot.

It is worth discussing the way to specify the vector g for a convenient utilization of

redundant degrees of freedom. A typical choice is :

Ta

w(q)g k ( )q

∂=∂

where ak 0> and w(q) is a second objective function of the joint variables. Since the solution

moves along the direction of the gradient of the objective function, it attempts to locally

maximize its compatibility to the primary objective (kinematic constraint). The typical objective

functions are:

• The manipulability measurement, defined as

Tw(q) det(JJ )= (3.41)

which vanishes at a singular configuration; thus, by maximizing this measure, redundancy

is exploited to move away from singularities.

• The distance from mechanical joint limits, defined as 2n

i iave

iM imi 1

q q1w(q)2n q q=

⎛ ⎞−−= ⎜ ⎟−⎝ ⎠∑ (3.42)

where iMq and imq denotes the maximum and minimum joint limit and iq−

the middle value

of the joint range; thus, by maximizing this distance, redundancy is exploited to keep the

joint variables as close as possible to the center of their ranges.

• The distance from an obstacle, defined as

p,ow(q) min || p(q) o ||= − (3.43)



where o is the position vector of a suitable point on the obstacle (its center, for instance, if

the obstacle is modeled as a sphere) and p is the position vector of a generic point along the

structure; thus, by maximizing this distance, redundancy is exploited to avoid collision of

the manipulator with an obstacle.

For the robot manipulator EDORA II, there is a mechanical limit range for the elongation of

each chamber and the corresponding pressure applied into the chamber .

1min 1 1max 1min 1 1max



L L L , P P P

L L L , P P P

L L L , P P P

≤ ≤ ≤ ≤

≤ ≤ ≤ ≤

≤ ≤ ≤ ≤

In order to avoid this case, the objective function is constructed to be included in the

inverse Jacobian algorithm, equation (3.40), as the second criteria. This objective function

evaluate the pressure difference between the applied pressure in the chamber and the average

pressure iaveP applied in the chamber . So the cost function is expressed as the following:

23i iave

iM imi 1

P P1w(q)3 P P=

⎛ ⎞−= ⎜ ⎟−⎝ ⎠∑ (3.44)

we can then minimize w(q) by choosing:

1 1ave 2 2ave 3 3ave2 2 2

1 2 3 1M 1m 2M 2m 3M 3m

P P P P P Pw w w 2g grad w(q) = P P P 3 (P P ) (P P ) (P P )

⎛ ⎞⎛ ⎞ − − −∂ ∂ ∂= = ⎜ ⎟⎜ ⎟∂ ∂ ∂ − − −⎝ ⎠ ⎝ ⎠ (3.45)

So the solution pq to Equation (3.40) meets the minimization of two criteria simultaneously:

• Minimum norm joint velocities through J+X ;

• Secure the pressure variation of each chamber relating to the average pressure is minimal.

Now that the inverse velocity kinematics is developed, the kinematic control of can be

implemented based on (3.40) to control the position/orientation of EDORA II.

3.6 Validation of kinematic model

As described before, a theoretical kinematic model has been developed for EDORA II,

the next logical step is to provide experimental verification of the model. Since the kinematics

of EDORA II have been described as the relationship between the deflected shape and lengths

of three chambers (three pressures of each chamber), the validation of kinematics needs to have

a sensor to measure the deflected shape, i.e. the bending angle, the arc length and the

orientation angle. So this section first presents sensor choice and its experimental setup for



determining system parameters, then presents the verification of static kinematic model using

these experimental configurations.

3.6.1 The sensor choice and experimental setup

As for most continuum style robots, due to the dimension and the inability to mount

measurement device for the joint angles, the determination of the manipulator shape is a big

problem. Although there are several different technologies that could help solve this problem

with big one, such as [HIROSE 02], but they are difficult and costly to implement on a micro-

robot. Since a Cartesian frame has been analyzed with relation to the deflected shape

parameters, an indirect method is used to for the purpose of validation of kinematics with the

position measurement in 3D. With comparison and contrast of different 3D sensors, a

“miniBIRD” sensor is used for experimental validation.

Figure 3.5 MiniBIRD position and orientation measurement system

3.6.1.1. The miniBIRD

MiniBIRD is a six degree-of-freedom (position and orientation) measuring device from

Ascension Technology Corporation. A miniBIRD consists of one or more Ascension Bird

electronic units, a transmitter and one or more sensors, see figure 3.5. It offers full

functionality of our other DC magnetic trackers, with miniaturized sensors as small as 5mm

wide. Table 3.1 shows the characteristics of miniBIRD [Ascension 02]. The real-time

measurement can be easily integrated with Dspace 1005 card, described in chapter 2, through

serial communication.



Degree of Freedom 6 (position and orientation)

Range ±76.2cm

Accuracy Position: 1.8mm

Orientation : 0.5°

Resolution Position:0.5mm

Orientation: 0.1° @ 30.5cm

Measure rate Up to 120 measurements /second

Figure 3.6 Experimental setup of miniBIRD sensor

3.6.1.2. Experimental setup

The bottom of EDORA II is bounded to a fixture and the sensor is placed on the top of

EDORA II, shown in figure 3.6. The transmitter is placed at a stationary position. Thus the

position and orientation of top-end of EDORA II is read directly from the sensor –receiver- with

relation to the transmitter, and then the position of top-end of the manipulator with relation to

the bottom of the manipulator is calculated indirectly through reference transformation, shown

in figure 3.7.

Table 3.1. Characteristics of miniBIRD 500



Figure 3.7 Reference Transformation for calculating the position of top-end relative to the bottom end of EDORA II

3.6.2 Validation of bending angle

Since the bending can be expressed concerning to the length of the each chamber or

concerning to the pressure of each chamber, so two cases are used to validate bending angle.

The first one is to use the length of chamber to directly calculate the bending angle. Since the

length of each chamber under pressure has been measured as described in 3.3.1 :

1 0 1 0 1

2 0 2 0 2

3 0 3 0 3

L L L L f (P )L L L L f (P )L L L L f (P )

= + ∆ = +⎧⎪ = + ∆ = +⎨⎪ = + ∆ = +⎩

(3.23)

so the bending angle can be easily calculated by using Equation (3.18) relating the bending

angle to the chamber length.

L23rξ

α = (3.18)

where 2 2 2L 1 2 3 1 2 1 3 2 3L L L L L L L L Lξ = + + − − −

Receiver

Transmitter Reference Frame

+ X

+ Z+ Y

Measurement ofthe sensor

Position of top-end relative tothe bottom end



Then, from equation (3.28),

P23rξ

α = (3.28)

where 2 2 2P 1 2 3 1 2 1 3 2 3f (P ) f (P ) f (P ) f (P )f (P ) f (P )f (P ) f (P )f (P )ξ = + + − − −

the bending angle is calculated with relation to the three pressure input of three chambers

respectively.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 x 10 5

0

10

20

30

40

50

60

70

80

90 Verification of kinematic

Pressure in one chamber

Ben

ding

ang

le

the model in pressurethe experimental measurementsthe model in length

Figure 3.8 Comparisons of bending angle with relation to the chamber length and chamber pressure

The comparison of two results in figure 3.8 shows that the bending angle concerning the

chamber length and the chamber pressure respectively has the same characteristics. In order to

directly check the validation of the theoretical model, the miniBIRD sensor is used to measure

directly the bending angle under the corresponding pressure and the results are shown in figure

3.8. Compared with other two lines, the curve of the directly measured bending angle has some

difference with the other two curves obtained by indirect methods. This difference is explained

to be the measurement error of the chamber length brought on by the imprecise manual

measurements.

To validate the position of the end-tip in the bending moment, another experiment has been

carried out. The miniBIRD sensor is used to measure the displacement of end-tip with relation

to the reference coordinate. Theoretically, the position in the space can be easily calculated

(Equation 3.20) when the bending angle is measured.



Lx= (1- cos ) sin

Ly = (1- cos ) cos

Lz = sin

⎧ α φ⎪ α⎪⎪ α φ⎨ α⎪⎪ α⎪ α⎩

Then the comparison of theoretical and experimental results are shown in figure 3.9.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 x 10 5

0

10

20

30

40

50

60

70

Presssure in the room (Pascal)

X (m

m)

the experimental measurementsthe theoretical model in pressure

Figure 3.9 Position comparison of end-tip of EDORA II

From this figure, the results are satisfactory because the experimental data has a good

agreement at each pressure point except the first several points. Again, these errors can be

explained by the fact that the weight of the sensor has much more effect with lower pressure in

the chamber than higher pressure in the chamber. Finally, the comparison results of the two

experiments proved greatly that the kinematic model for the bending angle gives good

performances.

3.6.3 Validation of orientation angle

Another important parameter to be verified is the orientation angle, expressed in the

equation (3.25) relating to the three pressure points 1 2 3(P , P , P ) :

2 3 1 2 3a tan 2( 3(f (P ) f (P )), 2f (P ) f (P ) f (P ))φ = − − −



Since the miniBIRD can not measure the orientation angle directly, indirect methods are

required to check the orientation angle. From Equation (3.20), we can obtain the orientation

angle with relation to the XY frame coordinate.

Lx= (1- cos ) cos y arctan( )

L xy = (1- cos ) sin

⎧ α φ⎪⎪ α ⇒ φ =⎨⎪ α φ⎪ α⎩

So by using miniBIRD to measure the XY frame coordinate, the experiments are easily done to

check the orientation angle. The pressure combinations of three chambers are used are the

following:

1 2 3

2 1 3

3 1 2

1 2 3

g(P ), with P P 0 (1)g(P ), with P P 0 (2)g(P ), with P P 0 (3)g(P P ), with P 0

φ = = =φ = = =φ = = =φ = = =

2 3 1

3 1 2

(4)g(P P ), with P 0 (5)g(P P ), with P 0 (6)

⎧⎪⎪⎪⎪⎨⎪⎪φ = = =⎪

φ = = =⎪⎩

This pressure combinations, theoretically, will follow 6 principal orientation angles (0°, 60°,

120°, 180°, 240°, 300°) with the bending angle varying from 0 to the maximum in the plane (x-

o-y). Figure 3.10 shows theoretical and experimental results. From this figure, the 6 orientation

angles are close from the theoretical values.

-50 -40 -30 -20 -10 0 10 20 30 40 50 -60

-40

-20

0

20

40

60 Validation of orientation angle

X (mm)

Y (m

m)

Experimental measurements

Simulation

Figure 3.10 Comparison of orientation angle



3.6.4 Verification of correlation among each chamber

Section 3.6.2 and 3.6.3 validated the bending angle and orientation angle separately in a

static way. The special motion that EDORA II can generate results from the pressure

differentials among each chamber, this is to say, the interaction of each chamber. So it is

necessary to check this mutual interaction among each chamber. To achieve this goal, sinuous

signals of pressure with 120° delay among each servovalve with a definitive velocity are

employed to make EDORA II turn around its vertical axis (see the experimental setup figure

3.7) to see the mutual interaction of each chamber. By using miniBIRD, the coordinates of

endpoint of EDORA II can be easily obtained in XOY plane. Thus the comparison between

these coordinates and the coordinates obtained from the simulation of kinematic model

(Equation 3.20, 3.29) allows us to verify if there is any mutual interaction among each chamber

elongation.

Two comparisons are then proposed (figure 3.11). Three sinuous signals of pressure

with an amplitude of 0.4 bar and an offset of 0.9 bar are applied in the chambers of the

prototype. The path of the endpoint of EDORA II is in a form of a triangle(figure 3.11a )

because these actuators of EDORA II work in their nonlinear zone. Three sinuous signals of

pressure with an amplitude of 0.4 bar and an offset of 1.2 bar are applied in the chamber of

EDORA II. In this case, EDORA II works in the linear zone and has the approximate movement

of a circle (figure 3.11 b).

-50 -40 -30 -20 -10 0 10 20 30 40 50-40 -30 -20 -10

0 10 20 30 40 Comparison between the model without correction and the experiment

X (mm)

Y (m

m)

SimulationExperiment

-60 -40 -20 0 20 40 60 80-60

-40

-20

0

20

40

60

X (mm)

Y (m

m)

Comparison between the model without correction and experiment

SimulationExperiment

(a) (b)

Figure 3.11 Simulation et experimental results of the movement of the endpoint of EDORA II

The lines in the outer layer are the simulation results from the kinematic model relating XY

coordinates to the corresponding pressure of each chamber without mutual interactions of three

chambers. The three chamber models used are the following:



21 1 1min 1 1max

1 1 11 1min

22 2 2min 2 2max

2 2 2

3.8P + 24.8P if P P PL f (P )0 if P <P

4.4P +15.7P if P P PL f (P )0

⎧ < <⎪∆ = = ⎨⎪⎩

< <∆ = =2 2min

23 3 3min 3 3max

3 3 33 min

if P <P

7.9P 33.9P if P P PL f (P )0 if P <P

⎧⎪⎪⎪

⎧⎪ ⎪⎨ ⎨

⎪⎪ ⎩⎪ ⎧− + < <⎪⎪∆ = = ⎨⎪ ⎪⎩⎩

(3.22)

The difference between the simulation of the model and the experimental results show that there

exists an interaction among each chamber when the motion of top-end of EDORA II is

achieved. Therefore, additional parameters need to be added to reflect this behavior.

3.6.5 Estimation of a correction parameter

In this section, new parameters will be chosen to represent the mutual interactions

among each chamber. They will account for the coupling effect of stretching of one chamber to

that of the other two chambers. Thus 6 parameters are added to describe this effect:

k12 = mutual stiffness that determine the effect of P2 on the length of the chamber 1






Here, it is assumed that the hysteresis of each chamber actuator is negligible, and then 6 mutual

stiffnesses will be reduced to one parameter as the geometry of EDORA II is symmetric.

Thus the property of each actuator is represented as the following:

1 1 1 2 2 3 3

2 2 2 1 1 3 3

3 3 3 1 1 2 2

L f (P ) k(f (P ) f (P ))L f (P ) k(f (P ) f (P ))L f (P ) k(f (P ) f (P ))

∆ = + +⎧⎪∆ = + +⎨⎪∆ = + +⎩

then the coefficient k is obtained by minimizing the difference between the operational

coordinates (Xs, Ys) measured by miniBIRD and the operational coordinates (Xm, Ym) obtained

by simulation of the geometrical model (Equation 3.20, 3.29), figure 3.12.



System

Min J

Model XmYm

YsXs

Figure 3.12 Optimisation model

And the cost criteria is chosen as :

2 2 2 2m m s sJ(k) || X (k) Y (k) X (k) Y (k) ||= + − +

Then the coefficient k obtained is 0.3.

1 1 1 2 2 3 3

2 2 2 1 1 3 3

3 3 3 1 1 2 2

L f (P ) 0.3(f (P ) f (P ))L f (P ) 0.3(f (P ) f (P ))L f (P ) 0.3(f (P ) f (P ))

∆ = + +⎧⎪∆ = + +⎨⎪∆ = + +⎩

then figure 3.13 and figure 3.14 present the results with the correction parameter of two cases.

0 5 10 15 20 25 300.4 0.5 0.6 0.7 0.8 0.9

1 1.1 1.2 1.3 1.4

Time ( t )

Pres

sure

(bar

)

Pressures in three chambres (nonlinear)

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30Comparison between the simulation of corrected model and experiment

X (mm)

Y (m

m)

Simulation of corrected kinematic model

Experimental result

(a) (b)

e 3.13 Comparison between the result of simulation with correction parameter (continuous line) and experimental result

(a) pressure group for three chamber of EDORA II (b) the endpoint of EDORA II in the plane of XY.



0 5 10 15 20 25 300.7 0.8 0.9

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Pressures in three chambres

Time (t)

Pres

sur (

bar)

Chambre 1 Chambre 2 Chambre 3

-40 -30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50Comparison between simulation of corrected model and experiment

X (mm)

Y (m

m)

Simulation of corrected kinematic model

Experimental result

a b

e 3.14 Comparison between the result of the simulation with a correction parameter (continuous line) and experimental

result (a) pressure group for three chambers of EDORA II (b) the endpoint of EDORA II in the plane of XY.

To check the uniformity of this coefficient within its total work zone of EDORA II, three other

experiments have also been carried out to validate this coefficient for each case. Three sinuous input

pressures with an amplitude ranging from 0.1 bar to 0.3 bar are applied to three chambers of EDORA

II. By using the improved kinematic model with the corrected coefficient, the comparison shows that

this coefficient reflected the same mutual effect among each chamber during its total work zone of

EDORA II including the dead zone. Figure 3.15 and figure 3.16 present the comparison results of two

cases. Results prove right the assumption that there exists interaction between each chamber.

-40 -30 -20 -10 0 10 20 30 40 -40

-30

-20

-10

0

10

20

30 Results of different pressure ampitude (one part in the deadzone)

X

Y

Amplitude of 0.4 bar Amplitude of 0.3 bar

Amplitude of 0.2 bar


Figure 3.15 Verification of corrected with different pressure input (dead zone)



-40 -30 -20 -10 0 10 20 30 40 50 -50

-40

-30

-20

-10

0

10

20

30

40

50

X coordinnate (mm)

Y co

ordi

nnat

e (m

m)

Results of different pressure amplitude

Amplitude of 0.4 bar Amplitude of 0.3 bar



Figure 3.16 Verification of corrected with different pressure inputs (linear zone)

To summarize, the experimental results verified that there exists interaction among each

chamber when the motion of top-end of EDORA II is generated. Since three chambers are

identical, the coefficients determined from non-linear algorithm are the same for three chambers

within their work zones.

3.7 Conclusions

In this chapter, we have detailed the study of kinematics of EDORA II. Three geometric

parameters are chosen to determine the position/orientation of top-end of EDORA II. With the

assumption that the deflected shape is an arc of a circle and the effects are ignored, then we

have established the forward kinematic model of EDORA II relating these three parameters to

the length of three chambers. Unlike other works on the linearity of the actuator, the non-linear

models of each chamber were obtained through experiments. Thus, the kinematics relating to

three system parameters to three pressures were then determined. Based on the forward

kinematics analysis, the velocity of kinematics is then studied from two cases: non-redundant

and redundant. In the case of redundant manipulation with relation to the chosen variables:

bending angle and orientation angle in the task space, inverse velocity kinematics is studied.

Experiments have been done to validate the bending angle and orientation of EDORA II



respectively. To check if there is any mutual interaction among each chamber, sinuous signals

of pressure with 120° delay among each chamber with a definitive velocity were employed to

make EDORA II turn around its vertical axis. Experimental results showed that there is mutual

interaction among each chamber. Thus a new correction parameter was chosen to represent this

effect and its coefficient was determined through a non linear optimization and validation.

chapter 3 kinematics analysis for continuum robotic ...docinsa.insa-lyon.fr › these › 2005 ›...

Documents