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
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
Gang CHEN Thèse INSA de Lyon, LAI 2005 84
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
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
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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
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
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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
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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
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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 :
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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
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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)
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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.
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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.
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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)
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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.
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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:
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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];
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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)
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
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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
2min 2 2max 2min 2 2max
3min 3 3max 3min 3 3max
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
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
Gang CHEN Thèse INSA de Lyon, LAI 2005 99
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.
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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
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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
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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.
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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 = − − −
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
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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
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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:
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Gang CHEN Thèse INSA de Lyon, LAI 2005 106
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
k21 = mutual stiffness that determine the effect of P1 on the length of the chamber 2
k13 = mutual stiffness that determine the effect of P3 on the length of the chamber 1
k31 = mutual stiffness that determine the effect of P1 on the length of the chamber 3
k23 = mutual stiffness that determine the effect of P3 on the length of the chamber 2
k32 = mutual stiffness that determine the effect of P2 on the length of the chamber 3
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.
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
Gang CHEN Thèse INSA de Lyon, LAI 2005 107
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.
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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
Amplitude of 0.1 bar
Figure 3.15 Verification of corrected with different pressure input (dead zone)
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-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
Amplitude of 0.2 bar
Amplitude of 0.1 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
Chapter 3 Kinematics Analysis for Continuum Robotic Manipulator: EDORA II
Gang CHEN Thèse INSA de Lyon, LAI 2005 110
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.