Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors.
ARTICLE
International Journal of Advanced Robotic Systems
Planar Task Space Control of a Biarticular
Manipulator Driven by Spiral Motors
Regular Paper
Ahmad Zaki bin Hj Shukor1 and Yasutaka Fujimoto1,*
1 Department of Electrical and Computer Engineering, Yokohama National University
* Corresponding author E-mail: fujimoto@ynu.ac.jp
Received 30 May 2012; Accepted 20 Jul 2012
DOI: 10.5772/51742
© 2012 Shukor and Fujimoto; licensee InTech. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract This paper elaborates upon a musculoskeletal‐
inspired robot manipulator using a prototype of the spiral
motor developed in our laboratory. The spiral motors
represent the antagonistic muscles due to the high
forward/backward drivability without any gears or
mechanisms. Modelling of the biarticular structure with
spiral motor dynamics was presented and simulations
were carried out to compare two control methods,
Inverse Kinematics (IK) and direct‐Cartesian control,
between monoarticular only structures and biarticular
structures using the spiral motor. The results show the
feasibility of the control, especially in maintaining air
gaps within the spiral motor.
Keywords biarticular manipulator, inverse kinematics
control, work space control, spiral motor, musculoskeletal
1. Introduction
Over recent decades robots have been inspired by the
motion of living things, i.e., humans and animals [1] [2].
Various structures of robots have been designed, from
industrial robotic manipulators to humanoids and mobile
legged robots for different purposes. Success stories
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include ASIMO, DLR, Big Dog and many others.
However, most designs are implemented by using serial
joint servos at the end of each limb, resulting in large joint
actuators at the base to support the desired large
forces/torques (i.e., end‐effector output, friction). In the
end, the weight and size of the actuators contribute
largely to the overall specication of the robot. In
addition, difficulties in actuation, for example, backlash
and low drivability/low compliance, add to problems in
controlling such structures.
On the other hand, the anatomy of the musculoskeletal
human body provides new insights into the design of
robots. By looking closely at the structure of the
human/animal musculoskeleton, the joint/hinges are not
directly actuated, but affected by the antagonistic
contraction or extension of the muscles connected to the
bones. The muscles can be clustered into monoarticulars
(single joint articulated) and biarticulars (two joint
actuated). The dynamic domain of the human arm
muscles (from shoulder to elbow) have been studied in
[3] [4] [5]. The study in [3] applies the Hill’s muscle model
and attempted to analyse different damping and elastic
properties, and later applied task position feedback
control to monitor position, velocity and accelerations in
IntZaki
J Adv
Sy, and
2012,
Vol. 9, 156:2012
Ahmad
binRobotic
Hj Shukor
Yasutaka
Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
1
joint and ttaskspace. On
O
the variaation of mu
uscle
positioning, the potential elds producced by the inteernal
forces are afffected and alsso the speed of convergencce of
motion. By means of sim
mulation, the human reacching
w
achieved
d without inv
verse dynamiccs or
movements were
optimal trajectory derivation.
Some design
ns that includ
de biarticular muscle actuaation
can be referred to in [6] an
nd [7]. These d
designs use ro
otary
b
mu
uscle
motors, beltss and pulleyss to realize biarticular
torques. In
n [6], stiffn
ness and in
nverse dynaamics
feedforward control was applied. The lancelet (anceestor
milar
of the verrtebrae) has inspired [77] using sim
mechanisms to realizze a virtual triarticcular
mming motio
on with the properties off AC
sigmoid/swim
the
motor
con
ntrol.
Thesse
designs
include
biarticular/triarticular actu
uation, but tthe structuress are
a serial or open‐link mechanisms. Ano
other
maintained as
different dessign was show
wn in [8], wh
hereby the ro
otary
motors are placed
p
at the base and the joint angless are
pulled by wiires. Feedforw
ward control w
was also applieed to
gged
resolve forcce redundanccy. Other deesigns of leg
musculoskeleetal mechanisms can be fou
und in [9] [10] [11].
These structu
ures utilize th
he pneumatic rrubber musclees as
actuators, po
ossibly the cllosest candidaate to the hu
uman
muscle. Thesse muscles exttend or contraact, dependin
ng on
the amount o
of air pressuree supplied. Th
he small size o
of the
actuators enaables the desig
gn of the legs to be space saaving,
as the actuators are placed
d closely or attached to the link.
Jumping an
nd hopping motions were
w
successsfully
implemented
d, but becausee of the use off air supply, tthese
structures arre less mobille for autono
omous operattions.
Also, pneum
matic muscless are difficullt to control and
inaccurate du
ue to their non
nlinear elasticiity properties [12].
A combinatio
on of pneumatics/cables is aalso shown in [13].
or that we propose forr musculoskeeletal
The actuato
actuation is tthe spiral mottor [14]. It is aa novel high th
hrust
force actuato
or with high forward/back
kdrivability. From
F
the rst prottotype, the in
nternal permaanent version was
developed and
a
currently the helical surface permaanent
otype is availaable [15].
magnet proto
2. Biarticular Manipulator K
Kinematics
An example of the biartiicular muscle structure can
n be
found in the human arm, consisting of antagonistic p
pairs
of monoartiicular (musclles affecting one joint) and
biarticular muscles
m
(musscles affecting
g two joints). By
using the spiiral motor, thee antagonistic pair of exorr and
extensor can be combined by using only
y one spiral m
motor,
thus reducin
ng the numberr of actuators from the shou
ulder
to the elbow
w to only three actuators, in
nstead of six. The
exor/extenso
combined
or muscles of the pllanar
manipulator of the biartticular structture is shown
n in
Figure 1.
2
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
Figu
ure 1. Simplified
d diagram and m
manipulator dessign
ulder angle (qq1) and elbow
w
The main angles are the shou
m in Figure 1
1
anglle (q2). Based on the simplied diagram
and using trigon
nometric iden
ntities, the reelationship off
m with joint an
ngles q1 and q2 is derived (a1
musscle lengths, lmi
and a2 are link len
ngths).
��� � ����� � ���� � ���� ��� ���� ��
��� � ����� � ���� � ���� ��� ��� ��
��� � �
�
�
����
� ��
� ��
� ��� � ��
���� �� c�� �� �
����� �� c��
c �� � ����� �
���� ������ � �� �
(1))
ween connecction length
hs of thee
Disttances betw
mon
noarticulars frrom the joint angles are lab
belled ��� , ��� ,
,
��� and ��� whilee connection lengths of biarticular
b
aree
���� and ���� . Thee extension/co
ontraction of m
monoarticularr
musscles affect th
he respective jjoint angles to which they
y
attach, i.e., ��� afffects the join
nt angle �� (sshoulder) and
d
bow). Howeveer, the biarticu
ular muscle lm33
��� affects �� (elb
is reedundant becaause it affects b
both �� and �� .
ween muscle velocities, ���
Baseed on (1), thee relation betw
and joint velocitiees, �� is given as;
�
���
�
���
�
(2)
� ���
� � � ���� � � � �
�
�
���
�
ace Jacobian,,
Wheere ���� is the musclee‐to‐joint spa
obta
ained from paartial derivatiives of (1) wiith respect to
o
shou
ulder and elbo
ow angles, sho
own in (3).
���
��
����
� ����
����
��
� � ���
�
����
��
� ����
����
��� �
���� �
(3))
��� �
���� �
���
�
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where
��� ��� �� ���� �
����
����
��
;
� �;
���
���
���
����
��� ��� �� ���� �
����
� �;
��
;
���
���
���
���� ����� �� ������ � � ���� ���� ������ � �� �
�
;
���
���
���� ����� �� ������ � � ���� ���� ������ � �� �
�
���
���
It needs to be highlighted that this Jacobian is different
from that of [6], because in the mentioned reference it is
not structure‐dependent. In our case, this Jacobian varies
with the connection points (i.e., ��� , ��� , ��� , ��� ,
���� ��� ���� ). Next, the equation relating joint torques, �
(vector of shoulder and elbow torques) and muscle forces,
��� (vector of muscle 1, muscle 2 and muscle 3 forces) can
be described in (4) as;
� ���
��
�� � � ���� � �� ��� �
(4)
�
� ���
Also, the equation relating the joint torques, � and the
end‐effector forces, �� for a 2R planar manipulator is;
��
���
�� � � � � � � �
(5)
�
��
where � is the 2R planar Jacobian matrix from task space
to joint space which contains the following elements;
�� ������ � � �� ��� ��� � �� � ��� ��� ��� � �� �
�
� �
�� ��� ��� � �� �
�� ������ � � �� ��� ��� � �� �
(6)
The equation relating to the end‐effector forces, �� (vector
of end‐effector forces) and the muscle forces, ��� can be
determined as follows;
� ���
���
��
� � � � �� ���� � �� ��� �
(7)
��
� ���
From Equation (6), the end‐effector forces plot of the
biarticular structure can be obtained. Figure 2 shows the
static end‐effector forces of structures with biarticular
forces and the structures without. The forces applied for
��� [N], ��� [N] and ��� [N] are;
���� ��� � ���� � ��� ��� (8)
���� ��� � ���� � ��� ���
���� ��� � ���� � ��� ���
The end‐effector output force produced by biarticular
forces shows a hexagonal shape which is more
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homogenous than the tetragonal shape of the end‐effector
forces without biarticular actuation. This is one of the
many advantages of biarticular manipulators compared
to monoarticular actuation only.
Figure 2. Cartesian view of end‐effector forces for structures
with (red) and without (blue) biarticular muscle forces
3. The Spiral Motor
Being specically developed for musculoskeletal robotics,
the third prototype spiral motor’s main feature is its
direct drive high thrust force actuator with high
backdrivability. This three phase permanent magnet
motor consists of a helical structure mover with
permanent magnet and stator. The linear motion is
derived from the spiral motion of the mover to drive the
load. In addition, the motor has high thrust force
characteristics because the ux is effectively utilized in its
three dimensional structure [14]. This spiral motor does not
include ball screw mechanisms, thus friction is negligible
under proper air gap (magnetic levitation) control.
Figure 3. Spiral motor parts (a) Neodymium magnet (b) Silicon
steel stator yoke (c) Mover attached with magnet and Teflon
sheet (d) Wound and assembled half stator
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
3
Figure 4. Spiral motor illustration
As seen in Figure 4, there is a small air gap between the
mover and stator. This small distance is ideally 700 um
between the Teon sheet of the mover (at centre position)
and the stator yoke. This air gap displacement, �� , is
related to the linear displacement, x, and angular
displacement, �� in (11). If the gap displacement is
maintained, i.e., the surface of mover and stator are
untouched,
magnetic
levitation
is
realized.
Simultaneously, by a vector control strategy, the angular
control provided by q‐axis currents will give forward or
backward thrust while d‐axis currents control linear
displacements and magnetic levitation, as shown in the
plant dynamics of the spiral motor in (9) to (12);
(9)
�� �� � �� �� � �� �� � ���
�� �� � �� �� � ���� �� � �� �� � � ��
(10)
�� � � � ��
(11)
��
��
��
(12)
where �� �� and �� �� denote spiral motor force and torque,
�� �� represents the force generated during magnetic
levitation, �� is force constant and �� is torque constant.
��� and �� are force and angular disturbances. Gap
displacements, �� , are related to linear, x, and angular
displacements, � , by the lead length, �� (12). One
revolution of � would result in a displacement of length
�� if the gap displacement is zero.
Figure 5. Assembled short‐length spiral motor
3.1 Direct Drive Control of the Spiral Motor
Assume ��� and �� are control inputs for gap
displacement and angular displacement [15], which will
match its respective acceleration terms;
4
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
� � ��� �
(13)
��� � ��� ���� � �� � � ��� ����
�
�
�
�� � ���� � ��� ����� � �� � ��� ����� � � � (14)
� is the gap velocity
where ��� is the gap reference and ���
reference. ��� , ��� , ��� and ��� are PD gains for gap and
angular positions and velocities. Also, the relation
between linear and angular acceleration is known as;
(15)
�� � ��� � ���
By replacing the linear acceleration term in (9) with (15) ,
force dynamics of (9) can also be expressed as;
(16)
� ���� � ���� � �� �� � �� �� � ���
Replacing the acceleration terms with control inputs
(subscript n denotes nominal value);
(17)
�� ���� � ��� � � ��� ����� � ��� �� � ����
Thus, the d‐axis current reference (that will be tracked by
a PI current controller) will have the following form;
�
����� � � ��� ���� � ��� � � ��� �� � ���� � (18)
��
By applying the same technique to the torque dynamics
in (9) for the q‐axis current reference;
�� �� � �� ����� � � ��� ���� � ��� � � ���� � � ��� (19)
�
��� �� � ���� ���� � ��� � � ���� � � ��� � (20)
����� �
��
and disturbance for linear displacement and angular
displacement (in Laplace domain) are estimated as;
�
(21)
���� � ���� ��� ����� � �� �� � ��� �� �
�
�
��� � ���� ��� ����� � � ��� ����� � �� �� � � ��� �� �
�
(22)
where �� and �� are the gain of the disturbance observer
for linear and angular displacements. Until this point,
direct drive control can be achieved at gap values of 0
mm. But due to disturbances (i.e., manufacturing
accuracy of mover and stator), the current at 0 mm is not
zero. Therefore, a neutral point (zero power) that induces
zero currents for d‐axis current is desired and low power
control is realized as follows;
(23)
�� ���� � ���� � �� �0� � �� �� � ����
�
�
�� ����� � ��� � 0 � �� ���� � � ���
��� �
�� ��� �0 �������� ��
��
As acceleration terms are usually affected by noise (due
to differentiation), for initial testing, we simplify (24) to
become;
��� �
����
��
(24)
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Figure 6. Direcct drive experim
mental results w
with zero d‐axiss current control (a) Forward d
drive (b) Gap diisplacements (reed denotes zero
o
d‐axis gap refeerence) (c) Curreents (d) Reversee drive (e) Gap d
displacement (f) Currents
nd reverse mo
otion
Experiments for direct driive forward an
were carried
d out using th
he monoarticcular spiral motor
m
with a strok
ke of 0.06 m.
m TMS320C67713‐225MHz were
w
used for dataa collection an
nd control. Tw
wo DC suppliees for
the forward and backward inverters were
w
used. At first,
the direct driive control waas applied to 00 mm (x posittion).
We regard th
his as the initiial gap stabilizzation phase. This
is required because
b
once magnetic leviitation is realized,
then forward
d and backwaard motions can
c
be perforrmed
smoothly. IIf magnetic levitation an
nd forward//back
performed sim
multaneously, the gap migh
ht not
motions are p
be maintaineed and damaage could bee inflicted on
n the
stator and mover. Afterr d‐axis currrent stabilized at
around 0 A, the forward rreference wass given. Then after
A again, the reeverse reference of
the current stabilize at 0 A
ure 6 shows the experimeental
0 mm was applied. Figu
results for diirect drive forrward/reverse motion with zero
d‐axis curren
nt control (after gap initialiization). Note that
in [15], direcct drive contro
ol was perform
med without zero
d‐axis contro
ol.
4. Singularity
y/manipulability of Biarticu
ular Structure
This kinemattic parameterss can be used
d to determinee the
work space of the manip
pulator. The parameters
p
off the
three spiral m
motors are sho
own in Table 11.
� � �� ��
����� � � �� cos��� � �� �
� � �� ���� ��� � � �� ���
� ��� � �� �
(25))
(26))
�� � ��� � ��� � ��
�� � ��� � ��
(27))
(28))
For the biarticulaar manipulato
or, kinematic limits
l
are duee
to th
he maximum (�� � �� ) and m
minimum length (�� � �� ) off
the actuators, sho
own as;
By referring
r
to E
Equation (1), jo
oint angle iis bounded to
o
the connection leengths ��� and
d ��� . The sam
me applies forr
noarticular ���
d
mon
� . For us, we chose ���� � ��� and
��� � ��� (same arrrangement fo
or both monoa
articulars).
Then, the ranges o
nd �� are;
of allowable aangles for �� an
��� ��
���� ���
�
� �� � �� � ��� ��
���� ���
�
(29))
Spiral Motor Type
Parameters
Monoarticullar
Biarticullar
Weight (kg)
1.5
2.3
Stroke (mm)
60
70
ngth (mm)
Extended len
256.5
380.5
Contracted leength (mm)
196.5
310.5
100
150
Estimated m
max. force [N]
The forward kin
nematics (end‐‐effector position) of a 2R
R
man
nipulator can be calculated by using the
t
following
g
equation (x and y
y are Cartesian
n coordinates);;
Table 1. Spirall motor specificaations for musclles
Figu
ure 7. Closed‐kin
nematics of biarrticular manipulator
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Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
5
Figure 8. Man
nipulability plo
ots for (a) al1=0.14m,
=
(b) al1=0.16m,
=
(c) al1=0.18m
=
and (d) al1=0.20m for similar elbow monoarticularr
connection (���� � ��� ). Biarticular connection
ns (al31 and al32) set at 0.06 m
)
where
� � ���� � ���� � ���������� � ����� ���
mum lengths of monoarticu
ulars
This implies that the minim
incur maxim
mum angles an
nd vice versa. Thus, consideering
biarticular llength limits,, manipulabiility [16] can
n be
ure 8 shows the manipu
ulability plotss for
plotted. Figu
different con
nnecting pointss of monoarticcular muscles.. The
he angle limiits is due to
o biarticular and
shape of th
monoarticulaar length limitts. It can be seen that the furrther
the connectiion points off monoarticullars are from
m the
gle, the larger the work spacce.
shoulder ang
ned by finding
g the determin
nants
Manipulabiliity was obtain
of the mu
uscle‐joint Jaccobian and end‐effector‐‐joint
Jacobian term
ms. Dark bluee represents lo
ow manipulab
bility
regions, and
d at zero manipulability
m
y, singularity
y is
induced. Red
d represents th
he high manip
pulability regio
on.
The inversee kinematic singularity of planar 2R
manipulator [24] can be seen at the elbow angle of 0
f the biarticcular manipullator.
radians. Thiss is also true for
In addition, due to musclle Jacobian terrms, singulariity is
also evident at the shoullder angle off 0 radians. Some
S
examples of singularity an
nd manipulab
bility poses for the
manipulator aree illustrated in
n Figure 9.
biarticular m
In the case that either
e
shoulder or elbow monoarticular
m
r
omes fully extended, th
he shoulder//elbow anglee
beco
beco
omes zero, and
d singularity iis observed. H
However, both
h
anglles could nott become zerro simultaneo
ously, due to
o
biarrticular length
h limit (refer Fiigure 7).
Hig
gher manipulability poses arre induced wh
hen lengths off
ort, and at n
near minimum
m lengths off
musscles are sho
musscles (or maxiimum angles),, maximum manipulability
m
y
regiion is observed
d.
If the connectiion points o
of shoulder and elbow
w
noarticulars are
a
different, the followin
ng results off
mon
recip
procal condition of the Jaccobian (anoth
her method to
o
deteermine ill‐con
ndition [25]) iss shown in Figure
F
10 and
d
Figu
ure 11;
Figure 9. Exaamples of singu
ular and manip
pulability posees for
biarticular stru
ucture.
6
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
(a)
(b)
Figu
ure 10. Recipro
ocal condition p
plot for ��� � ��
� � , ��� � 0.� m
m
��� � 0.14 m; (a) ����
6 m (b)����� � ����� � 0.10 m
� � ���� � 0.06
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nts are virtuaally cut open
n and LE form
mulation wass
poin
used
d to obtain mass,
m
gravity and centrifu
ugal matrices..
Equ
uation (29) dep
picts the generralized equatio
ons of motion
n
of a closed‐chain manipulator;
�� �� ��� � ���� � � � ��� � � � �� �����
(29))
�� ����� � ���
(a)
(a)
(b)
(b)
on plot; ��� � ��
� � , ��� � 0.�� m
m and
Figure 11. Recciprocal conditio
��� � 0.� m; (a) ���� � ���� � 0.06 m (b)���
0
m
�� � ���� � 0.�0 m
F
10 and 11, if the vallue of ��� or �
��� is
As seen in Figure
small, the co
orresponding shoulder/elbo
ow angle beco
omes
narrow and vice versa. Also,
A
the furth
her the biarticcular
muscle conn
nection point is
i from the hinges,
h
the sm
maller
the work sp
pace becomes. By increasin
ng the biarticcular
connection distance,
d
ill‐co
ondition is furrther avoided,, but
with less worrk space.
s
the reeciprocal con
ndition plot with
Figure 12 shows
different con
nnection pointts of the biartiicular muscle.. It is
obvious thatt by connectin
ng the end off each side off the
biarticular muscle
m
differen
ntly, significaant changes to
o the
work space aand reciprocal condition is sseen.
5. Biarticular Plant Dynam
mics Modelling
g
hain manipula
Most planar 2R robots are serial/open‐ch
ators.
There is much
m
researcch on contrrol of these 2R
manipulatorss. Modelling of unconstrained open‐cchain
manipulatorss uses Ordinaary Differentiial Equations (via
Lagrange (LE
E) or Newton Euler (NE) fo
ormulation), w
while
closed‐chain manipulatorss require Diffferential Algeb
braic
Equations du
ue to constraaint equationss [17]. Consttraint
www.intechopen.com
(c)
Figu
ure 12. Recipro
ocal condition plot; ��� � ��
� � � 0.� m; (a))
���� � 0.06 m, ���� � 0.�0 m (b)������ � 0.0� m, ����
� � 0.06 m; (c))
��� � ��� � 0.�� m �
���� � 0.06 m, �
���� � 0.�0 m
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
7
8
where � denotes joint angle/muscle length vector, � is the
vector of constraints, �� ���is the mass matrix, ���, �� � are
centrifugal/Coriolis terms, ���� are gravity terms, � are
generalized forces/torques, ��� � are the constraint forces
(from the closed‐loop kinematics). The term �� ���
represents disturbance forces applied. �� is transposed
from (6) and ��� is a vector of x and y Cartesian
disturbance force. We will show the constraint‐related
terms. As mentioned, others can be obtained via LE, NE
or other dynamics formulation.
Constraint equations (from trigonometry) are derived as;
� 0
�� ������ � ���� � ��� ������ �
�� ������ � � ��� ������ �
� 0
�� ������ � � ��� � ��� ������ �
� 0
� 0
�� ������ � � ��� ������ �
�� ������ � � ���� � �� ������ � � ���� ��� ��� � �� � � 0
� 0
�� ������ � � �� ������ � � ���� ������ � �� �
Also, the relative acceleration at the constraints are zero
( �� , ��� are the constraint Jacobian and Jacobian derivative),
(30)
�� �� � ��� �� � 0
The constraint Jacobian (unmentioned terms are zero) is
described as;
J��� … J���
�
� � (31)
J� � � �
J��� … J���
���� � ��� ������ � ;
���� � ��� ������ �;
���� � ������ �;
���� � ���� ������ � ;
���� � ������ � ;
���� � �� ������ �;
���� � ��� ������ �;
���� � ��� ������ � ;
���� � ������ �;
���� � ���� ������ � ;
���� � ��� ��� �;
���� � �� ������ �;
���� � ���� ������ � �� � � �� �� ���� � ;
���� � ���� ������ � �� � ;
���� � ��� ��� �;
���� � ��� ������ �;
���� � ����� ������ � �� � � �� ��� ��� �;
���� � � ���� ������ � �� � ;
���� � ��� ��� �;
���� � �� ������ �;
The constraint Jacobian derivative is derived as;
J��� … J���
�
� � (32)
J�� � � �
J��� … J���
���� � ��� ������ ����
���� � ��� ������ ���� � ������ � �� � ;
���� � ������� � ��� ;
���� � ��� ������ � ��� ;
���� � ��� ������ ���� � ������ ���� ;
���� � ��� ������ � ��� ;
���� � ������ � ��� ;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������� � ��� ;
���� � ��� ������ � ��� ;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������ ���� ;
���� � �� ������ � ��� � ���� ���� � ��� ���� ��� � �� �;
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
���� � ���� ���� � ��� ���� ��� � �� �;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������� � ��� ;
���� � �� ������ � ��� � ���� ���� � ��� ���� ��� � �� �;
���� � ���� ���� � ��� ���� ��� � �� �;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������ ���� ;
By combining these two equations, the following
generalized closed‐chain model can be derived;
� � ��� � �� ���
�
���� ��
� (33)
�� � � �
� �
���
0 �
��� ��
To include the spiral motor plant dynamics in the
generalized closed‐chain model, (33) has to be modied
to the following;
� � ��� � �� ���
�� 0 ���� ��
��
�
(34)
� 0 ��
0 � ���� � �
0
��� 0
��� ��
�
The inertia terms from the angular rotation now emerge
(�� , �� , �� ). With this modification, the d and q‐axis currents
could also be seen in simulation (refer Equations (9) to
(12)). To visualize the plant model, the dimensions of the
left‐hand terms would become;
���� … ���� 0 0 0 ����� … ����� �q�� �
� �
�
�
� � �
�
�
� �� � �
��
… ���� 0 0 0 ����� … ����� � �q�� �
� ���
�
…
0
�� 0 0
0
…
0 � �� �
� 0
…
0
0 �� 0
0
…
0 � �� �
� 0
� �
0
…
0
0
0
�
0
…
0
�
�
� �� �
0
…
0 � λ
������ … ����� 0 0 0
� ��
�
�
� � �
�
�
� �� � �
� �
������ … ����� 0 0 0
0
…
0 � �λ �
�
(35)
The linear acceleration of the spiral motors (i.e., �� � , �� �, �� �)
are equivalent to �� � , �� � , �� � and rotational acceleration of
the motors are ��� , ��� , ��� . After integrating twice, a gap can
then be obtained via (11). �� are spiral motor inertias.
The right‐hand terms of (34) would become;
��� ��� 0 … 0 ��
� �
� ��
��
��� … ��� ���
�
�
0 … 0� ��
� �� ��
� � ��
� �
�
�
�
�
�� � � � 0
���
0 0 … 0� � 0 ��
��
�
��� … ��� ���
� �
�
� … 0� � � ��
� �
�0
0 0 … 0� � 0 ��
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
�
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
���� … ���� ���
�
�
�
� �� � �
� �
�
�
���� … ���� ���
�
�
(36)
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The unactu
uated variablles are �� tto �� . Thus,, its
correspondin
ng �� to �� are alway
ys 0 (no input
torque/force)). �� is the vecctor of spiral motor
m
torquess. ��
to �� (monoaarticular 1, monoarticular
m
2 and biarticcular
forces) are giiven by the folllowing equatiion;
���
��
� ��� � ��� ���
�
��� � � ����
� ��� � ��� ���
� � (37)
��
���
�
�
�
�
� ��
�� ��
�
In simplified terms, the plaant can be desscribed by (38));
��
��
���� � � 0
���
�
0
��
0
��
����
0 �
0
���
��� � �� ���
��
� (38)
�
��� ��
ontrol
6. Work Spacce Position Co
vely;
nipulators hav
ve been researrched extensiv
Robotic man
ntrol
some of the m
many examples are torque sensorless con
[18], SCAR
RA manipulaator trackin
ng control [19],
feedforward and compu
uted torque control [20] and
position con
ntrol of consstrained robo
otic system [21].
Previously d
described in [222] are some aapproaches (th
he IK
d the Direct C
Cartesian (DC)) approach) fo
or the
approach and
general struccture of a biarticular man
nipulator, i.e.,, not
actuator speccific. In [23] otther methods w
were investigaated.
In this paperr, the generaliized work space control [222] is
extended to tthe specific sp
piral motor case for the purrpose
of viewing the effects on gap and possition control with
on of Carteesian load disturbance. By
the additio
considering aa circular task
k space trajecto
ory, the refereences
for x and y‐axis positions aare shown as;
x�t� � � cos�ω�t�� � x�
(39)
y�t� � � sin�ω�t�� � y�
(40)
ω�t� � 2���
2
� cos�t��
(41)
where A is th
he radius of th
he circle and xx� and y� are x
x and
y coordinatess of the centree of the circle.
From 0 to 0.5
0 seconds, gap
g
stabilizattion is perforrmed
without giviing any posiition referencce (either in joint
space or worrk space). Thiis phase is to
o allow the gaap to
stabilize at ceentre (0mm) p
position. From
m 0.5 to 6 seco
onds,
the position rreference of th
he circle trajecctory (5 cm rad
dius)
is given. In addition, Carrtesian disturb
bance on the end‐
ven between 11 to 3 seconds.
effector is giv
Here we preesent comparrisons of a monoarticular
m
only
actuated stru
ucture with th
hat of a biarticular structurre by
using the IK approacch and thee DC apprroach
ns, i.e., PD, dissturbance obseerver
(monoarticular control gain
ntical). Note th
hat for work sspace
cut‐off frequeencies are iden
control, zero d‐axis currentt control is not applied.
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5.1 IInverse Kinemaatics Approach
By using the IK
K approach [224], joint spacce trajectoriess
(elbow/shoulder angles) are ob
btained from the Cartesian
n
reference trajectorries (x and y);
��
�� ��� �
���� ����
��� ��
q� � t�n��� �
�
�
q� � t�n��
� t�n�� ��
�
��D�
D
(42))
�� ������ �
� ��� ������ �
(43))
�
(44))
From
m joint referen
nces, muscle kinematics (1) can be used
d
to obtain musccle referencees (i.e., leng
gth, velocity,,
ulder, elbow monoarticular
m
r
acceeleration). x� � , x� �, x� � are shou
and biarticular sp
piral motor lin
near acceleratiions.
�
�� ���� � ���
� �� ����
(45))
��
��
� �
�� ���� � ���
� �� ����
��
��
�
� �� ����
�� ���� � ���
��
��
(46))
(47))
alculated as in
n
The rotational accceleration refeerences are ca
(48) to (50) with
h ����� � 0 (gap referencess are zero forr
non
n‐zero d‐axis cu
urrent controll);
�� ��� ��� ����
������ � �
(48))
��
�� ���� ��� ����
�
(49))
����� �
�
�
�� ���� ��� ����
�
(50))
����� �
�
�
m these refereences, dq‐axiss control (refeer (13) to (20)))
From
can be constructeed for each mu
uscle to generrate the forcess
��� , �� �� � �n� to�q
q��s ���� , ���, ��� � �s in ���� �n� ����.
Figu
ure 13. IK contro
ol for manipulattor driven by sp
piral motors
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
9
In this part, only Y‐axis disturbance of 10 [N] is given. The
initial circle trajectory response in Figure 14 shows that the
biarticular structure exhibits better accuracy. Initial errors
due to the gap stabilization can be seen at the near start
position (0.15, 0.53). Then the effect disturbance of 10 [N]
can be seen as the response is deviating from the
reference. Cartesian errors are shown in Figure 15; ‘mono’
denotes monoarticular structure responses while ‘biart’
denotes biarticular structure responses.
Force responses from muscles are shown in Figure 17.
Both forces are bounded. Next, the torque responses
(rotational part of spiral motor) are shown in Figure 18.
Figure 17. Spiral motor forces for the IK approach
Figure 14. Close‐up view of circle trajectory responses from
monoarticular structure and biarticular structure
Figure 18. Spiral motor torque for the IK approach
Figure 15. Cartesian errors for the IK approach
Figure 19. D‐axis current for the IK approach
Figure 16. Gap responses for the IK approach
For Figure 16, ‘mono gap1’ and ‘mono gap2’ refer to the
monoarticular muscle gaps in the monoarticular structure,
while ‘biart gap1’, ‘biart gap2’, ‘biart gap3’ refer to the
shoulder monoarticular gap, elbow monoarticular gap and
biarticular gap in the biarticular structure. Referring to gap
displacement responses in Figure 16, in the initial gap
stabilization phase (0 to 0.5 s), more interaction is obvious
in the biarticular case, but the biarticular structure shows
better gap control during position tracking.
10 Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
Figure 20. Q‐axis current for the IK approach
Figures 19 and 20 show the d‐ and q‐axis currents generated.
As in the spiral motor simplified plant model, the d‐axis is
related to muscle (spiral motor) forces while the q‐axis is
related to spiral motor torques. It can be said that the
biarticular forces reduce the effort of the elbow monoarticular
muscle, but increase shoulder muscle actuation.
www.intechopen.com
For this app
proach, muscle force interaactions are cleearly
visible (seen
n in oscillatio
ons) as expeccted, becausee the
muscles are independentlly controlled without
w
any force
f
distribution.
5.2 Direct Carttesian Approachh
For the DC approach, the forces from
m work space//task
space domaiin are controlled in the sho
oulder and ellbow
joint space (virtually),
(
th
hen transferred to the actu
uator
(muscle) spacce by the jointt‐muscle Jacob
bian (3).
hows the con
ntrol diagram
m and the con
Figure 21 sh
ntrol
equation is shown
s
in Equ
uation (51) to
o (55). ���� � ����� are
Cartesian end
d‐effector position responsees.
0 � � �����
��
��
��� � � �����
��
��
��
��
�� � � �
���� �� � � � � � � � � � �
��
0
��
�
�
��
� �
0
��
where
�̂
���
� �� � � �
�̂ �
0
����
0 �� � �� ���
�̂
��
��� � � � �
��� �� � �����
�̂ �
0
��
� �
���
� � � ��� �� �
�� �
� (52)
�� � � ��� �� �
�
����� � � ����
����
����
��
The inverse Jacob
bian to obtain
n the muscle fo
orces, ���� aree
domatrix. �� iss
calculated by using Moore’s inverse pseud
Carttesian accelerration, ��� is tthe inverse of
o (6), � � is thee
Jaco
obian time derrivative, � is 22R planar mass matrix and
d
gi arre disturbance observer gaains. Now tha
at the musclee
forcces’ terms are o
obtained from
m the virtual jo
oint torques, itt
musst be distributted to the gap
p and angularr terms of thee
spirral motor. From
m Equation (116), the linear forces for thee
spirral motor is ag
gain shown;
�� ���� � ���� � �� ����� � ��� � � ���
(54))
��
v
(uxg) is the same as
a in (13), butt
The gap control variable
g
the angular contrrol variable is obtained by manipulating
uation (54) (ssubscript i reefers to respeective musclee
Equ
num
mber);
� �� �
���
�
��
�
International Journal of Advanced Robotic Systems
Planar Task Space Control of a Biarticular
Manipulator Driven by Spiral Motors
Regular Paper
Ahmad Zaki bin Hj Shukor1 and Yasutaka Fujimoto1,*
1 Department of Electrical and Computer Engineering, Yokohama National University
* Corresponding author E-mail: fujimoto@ynu.ac.jp
Received 30 May 2012; Accepted 20 Jul 2012
DOI: 10.5772/51742
© 2012 Shukor and Fujimoto; licensee InTech. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract This paper elaborates upon a musculoskeletal‐
inspired robot manipulator using a prototype of the spiral
motor developed in our laboratory. The spiral motors
represent the antagonistic muscles due to the high
forward/backward drivability without any gears or
mechanisms. Modelling of the biarticular structure with
spiral motor dynamics was presented and simulations
were carried out to compare two control methods,
Inverse Kinematics (IK) and direct‐Cartesian control,
between monoarticular only structures and biarticular
structures using the spiral motor. The results show the
feasibility of the control, especially in maintaining air
gaps within the spiral motor.
Keywords biarticular manipulator, inverse kinematics
control, work space control, spiral motor, musculoskeletal
1. Introduction
Over recent decades robots have been inspired by the
motion of living things, i.e., humans and animals [1] [2].
Various structures of robots have been designed, from
industrial robotic manipulators to humanoids and mobile
legged robots for different purposes. Success stories
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include ASIMO, DLR, Big Dog and many others.
However, most designs are implemented by using serial
joint servos at the end of each limb, resulting in large joint
actuators at the base to support the desired large
forces/torques (i.e., end‐effector output, friction). In the
end, the weight and size of the actuators contribute
largely to the overall specication of the robot. In
addition, difficulties in actuation, for example, backlash
and low drivability/low compliance, add to problems in
controlling such structures.
On the other hand, the anatomy of the musculoskeletal
human body provides new insights into the design of
robots. By looking closely at the structure of the
human/animal musculoskeleton, the joint/hinges are not
directly actuated, but affected by the antagonistic
contraction or extension of the muscles connected to the
bones. The muscles can be clustered into monoarticulars
(single joint articulated) and biarticulars (two joint
actuated). The dynamic domain of the human arm
muscles (from shoulder to elbow) have been studied in
[3] [4] [5]. The study in [3] applies the Hill’s muscle model
and attempted to analyse different damping and elastic
properties, and later applied task position feedback
control to monitor position, velocity and accelerations in
IntZaki
J Adv
Sy, and
2012,
Vol. 9, 156:2012
Ahmad
binRobotic
Hj Shukor
Yasutaka
Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
1
joint and ttaskspace. On
O
the variaation of mu
uscle
positioning, the potential elds producced by the inteernal
forces are afffected and alsso the speed of convergencce of
motion. By means of sim
mulation, the human reacching
w
achieved
d without inv
verse dynamiccs or
movements were
optimal trajectory derivation.
Some design
ns that includ
de biarticular muscle actuaation
can be referred to in [6] an
nd [7]. These d
designs use ro
otary
b
mu
uscle
motors, beltss and pulleyss to realize biarticular
torques. In
n [6], stiffn
ness and in
nverse dynaamics
feedforward control was applied. The lancelet (anceestor
milar
of the verrtebrae) has inspired [77] using sim
mechanisms to realizze a virtual triarticcular
mming motio
on with the properties off AC
sigmoid/swim
the
motor
con
ntrol.
Thesse
designs
include
biarticular/triarticular actu
uation, but tthe structuress are
a serial or open‐link mechanisms. Ano
other
maintained as
different dessign was show
wn in [8], wh
hereby the ro
otary
motors are placed
p
at the base and the joint angless are
pulled by wiires. Feedforw
ward control w
was also applieed to
gged
resolve forcce redundanccy. Other deesigns of leg
musculoskeleetal mechanisms can be fou
und in [9] [10] [11].
These structu
ures utilize th
he pneumatic rrubber musclees as
actuators, po
ossibly the cllosest candidaate to the hu
uman
muscle. Thesse muscles exttend or contraact, dependin
ng on
the amount o
of air pressuree supplied. Th
he small size o
of the
actuators enaables the desig
gn of the legs to be space saaving,
as the actuators are placed
d closely or attached to the link.
Jumping an
nd hopping motions were
w
successsfully
implemented
d, but becausee of the use off air supply, tthese
structures arre less mobille for autono
omous operattions.
Also, pneum
matic muscless are difficullt to control and
inaccurate du
ue to their non
nlinear elasticiity properties [12].
A combinatio
on of pneumatics/cables is aalso shown in [13].
or that we propose forr musculoskeeletal
The actuato
actuation is tthe spiral mottor [14]. It is aa novel high th
hrust
force actuato
or with high forward/back
kdrivability. From
F
the rst prottotype, the in
nternal permaanent version was
developed and
a
currently the helical surface permaanent
otype is availaable [15].
magnet proto
2. Biarticular Manipulator K
Kinematics
An example of the biartiicular muscle structure can
n be
found in the human arm, consisting of antagonistic p
pairs
of monoartiicular (musclles affecting one joint) and
biarticular muscles
m
(musscles affecting
g two joints). By
using the spiiral motor, thee antagonistic pair of exorr and
extensor can be combined by using only
y one spiral m
motor,
thus reducin
ng the numberr of actuators from the shou
ulder
to the elbow
w to only three actuators, in
nstead of six. The
exor/extenso
combined
or muscles of the pllanar
manipulator of the biartticular structture is shown
n in
Figure 1.
2
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
Figu
ure 1. Simplified
d diagram and m
manipulator dessign
ulder angle (qq1) and elbow
w
The main angles are the shou
m in Figure 1
1
anglle (q2). Based on the simplied diagram
and using trigon
nometric iden
ntities, the reelationship off
m with joint an
ngles q1 and q2 is derived (a1
musscle lengths, lmi
and a2 are link len
ngths).
��� � ����� � ���� � ���� ��� ���� ��
��� � ����� � ���� � ���� ��� ��� ��
��� � �
�
�
����
� ��
� ��
� ��� � ��
���� �� c�� �� �
����� �� c��
c �� � ����� �
���� ������ � �� �
(1))
ween connecction length
hs of thee
Disttances betw
mon
noarticulars frrom the joint angles are lab
belled ��� , ��� ,
,
��� and ��� whilee connection lengths of biarticular
b
aree
���� and ���� . Thee extension/co
ontraction of m
monoarticularr
musscles affect th
he respective jjoint angles to which they
y
attach, i.e., ��� afffects the join
nt angle �� (sshoulder) and
d
bow). Howeveer, the biarticu
ular muscle lm33
��� affects �� (elb
is reedundant becaause it affects b
both �� and �� .
ween muscle velocities, ���
Baseed on (1), thee relation betw
and joint velocitiees, �� is given as;
�
���
�
���
�
(2)
� ���
� � � ���� � � � �
�
�
���
�
ace Jacobian,,
Wheere ���� is the musclee‐to‐joint spa
obta
ained from paartial derivatiives of (1) wiith respect to
o
shou
ulder and elbo
ow angles, sho
own in (3).
���
��
����
� ����
����
��
� � ���
�
����
��
� ����
����
��� �
���� �
(3))
��� �
���� �
���
�
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where
��� ��� �� ���� �
����
����
��
;
� �;
���
���
���
����
��� ��� �� ���� �
����
� �;
��
;
���
���
���
���� ����� �� ������ � � ���� ���� ������ � �� �
�
;
���
���
���� ����� �� ������ � � ���� ���� ������ � �� �
�
���
���
It needs to be highlighted that this Jacobian is different
from that of [6], because in the mentioned reference it is
not structure‐dependent. In our case, this Jacobian varies
with the connection points (i.e., ��� , ��� , ��� , ��� ,
���� ��� ���� ). Next, the equation relating joint torques, �
(vector of shoulder and elbow torques) and muscle forces,
��� (vector of muscle 1, muscle 2 and muscle 3 forces) can
be described in (4) as;
� ���
��
�� � � ���� � �� ��� �
(4)
�
� ���
Also, the equation relating the joint torques, � and the
end‐effector forces, �� for a 2R planar manipulator is;
��
���
�� � � � � � � �
(5)
�
��
where � is the 2R planar Jacobian matrix from task space
to joint space which contains the following elements;
�� ������ � � �� ��� ��� � �� � ��� ��� ��� � �� �
�
� �
�� ��� ��� � �� �
�� ������ � � �� ��� ��� � �� �
(6)
The equation relating to the end‐effector forces, �� (vector
of end‐effector forces) and the muscle forces, ��� can be
determined as follows;
� ���
���
��
� � � � �� ���� � �� ��� �
(7)
��
� ���
From Equation (6), the end‐effector forces plot of the
biarticular structure can be obtained. Figure 2 shows the
static end‐effector forces of structures with biarticular
forces and the structures without. The forces applied for
��� [N], ��� [N] and ��� [N] are;
���� ��� � ���� � ��� ��� (8)
���� ��� � ���� � ��� ���
���� ��� � ���� � ��� ���
The end‐effector output force produced by biarticular
forces shows a hexagonal shape which is more
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homogenous than the tetragonal shape of the end‐effector
forces without biarticular actuation. This is one of the
many advantages of biarticular manipulators compared
to monoarticular actuation only.
Figure 2. Cartesian view of end‐effector forces for structures
with (red) and without (blue) biarticular muscle forces
3. The Spiral Motor
Being specically developed for musculoskeletal robotics,
the third prototype spiral motor’s main feature is its
direct drive high thrust force actuator with high
backdrivability. This three phase permanent magnet
motor consists of a helical structure mover with
permanent magnet and stator. The linear motion is
derived from the spiral motion of the mover to drive the
load. In addition, the motor has high thrust force
characteristics because the ux is effectively utilized in its
three dimensional structure [14]. This spiral motor does not
include ball screw mechanisms, thus friction is negligible
under proper air gap (magnetic levitation) control.
Figure 3. Spiral motor parts (a) Neodymium magnet (b) Silicon
steel stator yoke (c) Mover attached with magnet and Teflon
sheet (d) Wound and assembled half stator
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
3
Figure 4. Spiral motor illustration
As seen in Figure 4, there is a small air gap between the
mover and stator. This small distance is ideally 700 um
between the Teon sheet of the mover (at centre position)
and the stator yoke. This air gap displacement, �� , is
related to the linear displacement, x, and angular
displacement, �� in (11). If the gap displacement is
maintained, i.e., the surface of mover and stator are
untouched,
magnetic
levitation
is
realized.
Simultaneously, by a vector control strategy, the angular
control provided by q‐axis currents will give forward or
backward thrust while d‐axis currents control linear
displacements and magnetic levitation, as shown in the
plant dynamics of the spiral motor in (9) to (12);
(9)
�� �� � �� �� � �� �� � ���
�� �� � �� �� � ���� �� � �� �� � � ��
(10)
�� � � � ��
(11)
��
��
��
(12)
where �� �� and �� �� denote spiral motor force and torque,
�� �� represents the force generated during magnetic
levitation, �� is force constant and �� is torque constant.
��� and �� are force and angular disturbances. Gap
displacements, �� , are related to linear, x, and angular
displacements, � , by the lead length, �� (12). One
revolution of � would result in a displacement of length
�� if the gap displacement is zero.
Figure 5. Assembled short‐length spiral motor
3.1 Direct Drive Control of the Spiral Motor
Assume ��� and �� are control inputs for gap
displacement and angular displacement [15], which will
match its respective acceleration terms;
4
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
� � ��� �
(13)
��� � ��� ���� � �� � � ��� ����
�
�
�
�� � ���� � ��� ����� � �� � ��� ����� � � � (14)
� is the gap velocity
where ��� is the gap reference and ���
reference. ��� , ��� , ��� and ��� are PD gains for gap and
angular positions and velocities. Also, the relation
between linear and angular acceleration is known as;
(15)
�� � ��� � ���
By replacing the linear acceleration term in (9) with (15) ,
force dynamics of (9) can also be expressed as;
(16)
� ���� � ���� � �� �� � �� �� � ���
Replacing the acceleration terms with control inputs
(subscript n denotes nominal value);
(17)
�� ���� � ��� � � ��� ����� � ��� �� � ����
Thus, the d‐axis current reference (that will be tracked by
a PI current controller) will have the following form;
�
����� � � ��� ���� � ��� � � ��� �� � ���� � (18)
��
By applying the same technique to the torque dynamics
in (9) for the q‐axis current reference;
�� �� � �� ����� � � ��� ���� � ��� � � ���� � � ��� (19)
�
��� �� � ���� ���� � ��� � � ���� � � ��� � (20)
����� �
��
and disturbance for linear displacement and angular
displacement (in Laplace domain) are estimated as;
�
(21)
���� � ���� ��� ����� � �� �� � ��� �� �
�
�
��� � ���� ��� ����� � � ��� ����� � �� �� � � ��� �� �
�
(22)
where �� and �� are the gain of the disturbance observer
for linear and angular displacements. Until this point,
direct drive control can be achieved at gap values of 0
mm. But due to disturbances (i.e., manufacturing
accuracy of mover and stator), the current at 0 mm is not
zero. Therefore, a neutral point (zero power) that induces
zero currents for d‐axis current is desired and low power
control is realized as follows;
(23)
�� ���� � ���� � �� �0� � �� �� � ����
�
�
�� ����� � ��� � 0 � �� ���� � � ���
��� �
�� ��� �0 �������� ��
��
As acceleration terms are usually affected by noise (due
to differentiation), for initial testing, we simplify (24) to
become;
��� �
����
��
(24)
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Figure 6. Direcct drive experim
mental results w
with zero d‐axiss current control (a) Forward d
drive (b) Gap diisplacements (reed denotes zero
o
d‐axis gap refeerence) (c) Curreents (d) Reversee drive (e) Gap d
displacement (f) Currents
nd reverse mo
otion
Experiments for direct driive forward an
were carried
d out using th
he monoarticcular spiral motor
m
with a strok
ke of 0.06 m.
m TMS320C67713‐225MHz were
w
used for dataa collection an
nd control. Tw
wo DC suppliees for
the forward and backward inverters were
w
used. At first,
the direct driive control waas applied to 00 mm (x posittion).
We regard th
his as the initiial gap stabilizzation phase. This
is required because
b
once magnetic leviitation is realized,
then forward
d and backwaard motions can
c
be perforrmed
smoothly. IIf magnetic levitation an
nd forward//back
performed sim
multaneously, the gap migh
ht not
motions are p
be maintaineed and damaage could bee inflicted on
n the
stator and mover. Afterr d‐axis currrent stabilized at
around 0 A, the forward rreference wass given. Then after
A again, the reeverse reference of
the current stabilize at 0 A
ure 6 shows the experimeental
0 mm was applied. Figu
results for diirect drive forrward/reverse motion with zero
d‐axis curren
nt control (after gap initialiization). Note that
in [15], direcct drive contro
ol was perform
med without zero
d‐axis contro
ol.
4. Singularity
y/manipulability of Biarticu
ular Structure
This kinemattic parameterss can be used
d to determinee the
work space of the manip
pulator. The parameters
p
off the
three spiral m
motors are sho
own in Table 11.
� � �� ��
����� � � �� cos��� � �� �
� � �� ���� ��� � � �� ���
� ��� � �� �
(25))
(26))
�� � ��� � ��� � ��
�� � ��� � ��
(27))
(28))
For the biarticulaar manipulato
or, kinematic limits
l
are duee
to th
he maximum (�� � �� ) and m
minimum length (�� � �� ) off
the actuators, sho
own as;
By referring
r
to E
Equation (1), jo
oint angle iis bounded to
o
the connection leengths ��� and
d ��� . The sam
me applies forr
noarticular ���
d
mon
� . For us, we chose ���� � ��� and
��� � ��� (same arrrangement fo
or both monoa
articulars).
Then, the ranges o
nd �� are;
of allowable aangles for �� an
��� ��
���� ���
�
� �� � �� � ��� ��
���� ���
�
(29))
Spiral Motor Type
Parameters
Monoarticullar
Biarticullar
Weight (kg)
1.5
2.3
Stroke (mm)
60
70
ngth (mm)
Extended len
256.5
380.5
Contracted leength (mm)
196.5
310.5
100
150
Estimated m
max. force [N]
The forward kin
nematics (end‐‐effector position) of a 2R
R
man
nipulator can be calculated by using the
t
following
g
equation (x and y
y are Cartesian
n coordinates);;
Table 1. Spirall motor specificaations for musclles
Figu
ure 7. Closed‐kin
nematics of biarrticular manipulator
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Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
5
Figure 8. Man
nipulability plo
ots for (a) al1=0.14m,
=
(b) al1=0.16m,
=
(c) al1=0.18m
=
and (d) al1=0.20m for similar elbow monoarticularr
connection (���� � ��� ). Biarticular connection
ns (al31 and al32) set at 0.06 m
)
where
� � ���� � ���� � ���������� � ����� ���
mum lengths of monoarticu
ulars
This implies that the minim
incur maxim
mum angles an
nd vice versa. Thus, consideering
biarticular llength limits,, manipulabiility [16] can
n be
ure 8 shows the manipu
ulability plotss for
plotted. Figu
different con
nnecting pointss of monoarticcular muscles.. The
he angle limiits is due to
o biarticular and
shape of th
monoarticulaar length limitts. It can be seen that the furrther
the connectiion points off monoarticullars are from
m the
gle, the larger the work spacce.
shoulder ang
ned by finding
g the determin
nants
Manipulabiliity was obtain
of the mu
uscle‐joint Jaccobian and end‐effector‐‐joint
Jacobian term
ms. Dark bluee represents lo
ow manipulab
bility
regions, and
d at zero manipulability
m
y, singularity
y is
induced. Red
d represents th
he high manip
pulability regio
on.
The inversee kinematic singularity of planar 2R
manipulator [24] can be seen at the elbow angle of 0
f the biarticcular manipullator.
radians. Thiss is also true for
In addition, due to musclle Jacobian terrms, singulariity is
also evident at the shoullder angle off 0 radians. Some
S
examples of singularity an
nd manipulab
bility poses for the
manipulator aree illustrated in
n Figure 9.
biarticular m
In the case that either
e
shoulder or elbow monoarticular
m
r
omes fully extended, th
he shoulder//elbow anglee
beco
beco
omes zero, and
d singularity iis observed. H
However, both
h
anglles could nott become zerro simultaneo
ously, due to
o
biarrticular length
h limit (refer Fiigure 7).
Hig
gher manipulability poses arre induced wh
hen lengths off
ort, and at n
near minimum
m lengths off
musscles are sho
musscles (or maxiimum angles),, maximum manipulability
m
y
regiion is observed
d.
If the connectiion points o
of shoulder and elbow
w
noarticulars are
a
different, the followin
ng results off
mon
recip
procal condition of the Jaccobian (anoth
her method to
o
deteermine ill‐con
ndition [25]) iss shown in Figure
F
10 and
d
Figu
ure 11;
Figure 9. Exaamples of singu
ular and manip
pulability posees for
biarticular stru
ucture.
6
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
(a)
(b)
Figu
ure 10. Recipro
ocal condition p
plot for ��� � ��
� � , ��� � 0.� m
m
��� � 0.14 m; (a) ����
6 m (b)����� � ����� � 0.10 m
� � ���� � 0.06
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nts are virtuaally cut open
n and LE form
mulation wass
poin
used
d to obtain mass,
m
gravity and centrifu
ugal matrices..
Equ
uation (29) dep
picts the generralized equatio
ons of motion
n
of a closed‐chain manipulator;
�� �� ��� � ���� � � � ��� � � � �� �����
(29))
�� ����� � ���
(a)
(a)
(b)
(b)
on plot; ��� � ��
� � , ��� � 0.�� m
m and
Figure 11. Recciprocal conditio
��� � 0.� m; (a) ���� � ���� � 0.06 m (b)���
0
m
�� � ���� � 0.�0 m
F
10 and 11, if the vallue of ��� or �
��� is
As seen in Figure
small, the co
orresponding shoulder/elbo
ow angle beco
omes
narrow and vice versa. Also,
A
the furth
her the biarticcular
muscle conn
nection point is
i from the hinges,
h
the sm
maller
the work sp
pace becomes. By increasin
ng the biarticcular
connection distance,
d
ill‐co
ondition is furrther avoided,, but
with less worrk space.
s
the reeciprocal con
ndition plot with
Figure 12 shows
different con
nnection pointts of the biartiicular muscle.. It is
obvious thatt by connectin
ng the end off each side off the
biarticular muscle
m
differen
ntly, significaant changes to
o the
work space aand reciprocal condition is sseen.
5. Biarticular Plant Dynam
mics Modelling
g
hain manipula
Most planar 2R robots are serial/open‐ch
ators.
There is much
m
researcch on contrrol of these 2R
manipulatorss. Modelling of unconstrained open‐cchain
manipulatorss uses Ordinaary Differentiial Equations (via
Lagrange (LE
E) or Newton Euler (NE) fo
ormulation), w
while
closed‐chain manipulatorss require Diffferential Algeb
braic
Equations du
ue to constraaint equationss [17]. Consttraint
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(c)
Figu
ure 12. Recipro
ocal condition plot; ��� � ��
� � � 0.� m; (a))
���� � 0.06 m, ���� � 0.�0 m (b)������ � 0.0� m, ����
� � 0.06 m; (c))
��� � ��� � 0.�� m �
���� � 0.06 m, �
���� � 0.�0 m
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
7
8
where � denotes joint angle/muscle length vector, � is the
vector of constraints, �� ���is the mass matrix, ���, �� � are
centrifugal/Coriolis terms, ���� are gravity terms, � are
generalized forces/torques, ��� � are the constraint forces
(from the closed‐loop kinematics). The term �� ���
represents disturbance forces applied. �� is transposed
from (6) and ��� is a vector of x and y Cartesian
disturbance force. We will show the constraint‐related
terms. As mentioned, others can be obtained via LE, NE
or other dynamics formulation.
Constraint equations (from trigonometry) are derived as;
� 0
�� ������ � ���� � ��� ������ �
�� ������ � � ��� ������ �
� 0
�� ������ � � ��� � ��� ������ �
� 0
� 0
�� ������ � � ��� ������ �
�� ������ � � ���� � �� ������ � � ���� ��� ��� � �� � � 0
� 0
�� ������ � � �� ������ � � ���� ������ � �� �
Also, the relative acceleration at the constraints are zero
( �� , ��� are the constraint Jacobian and Jacobian derivative),
(30)
�� �� � ��� �� � 0
The constraint Jacobian (unmentioned terms are zero) is
described as;
J��� … J���
�
� � (31)
J� � � �
J��� … J���
���� � ��� ������ � ;
���� � ��� ������ �;
���� � ������ �;
���� � ���� ������ � ;
���� � ������ � ;
���� � �� ������ �;
���� � ��� ������ �;
���� � ��� ������ � ;
���� � ������ �;
���� � ���� ������ � ;
���� � ��� ��� �;
���� � �� ������ �;
���� � ���� ������ � �� � � �� �� ���� � ;
���� � ���� ������ � �� � ;
���� � ��� ��� �;
���� � ��� ������ �;
���� � ����� ������ � �� � � �� ��� ��� �;
���� � � ���� ������ � �� � ;
���� � ��� ��� �;
���� � �� ������ �;
The constraint Jacobian derivative is derived as;
J��� … J���
�
� � (32)
J�� � � �
J��� … J���
���� � ��� ������ ����
���� � ��� ������ ���� � ������ � �� � ;
���� � ������� � ��� ;
���� � ��� ������ � ��� ;
���� � ��� ������ ���� � ������ ���� ;
���� � ��� ������ � ��� ;
���� � ������ � ��� ;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������� � ��� ;
���� � ��� ������ � ��� ;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������ ���� ;
���� � �� ������ � ��� � ���� ���� � ��� ���� ��� � �� �;
Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
���� � ���� ���� � ��� ���� ��� � �� �;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������� � ��� ;
���� � �� ������ � ��� � ���� ���� � ��� ���� ��� � �� �;
���� � ���� ���� � ��� ���� ��� � �� �;
���� � ��� ������ ���� � ������ � ��� ;
���� � ������ ���� ;
By combining these two equations, the following
generalized closed‐chain model can be derived;
� � ��� � �� ���
�
���� ��
� (33)
�� � � �
� �
���
0 �
��� ��
To include the spiral motor plant dynamics in the
generalized closed‐chain model, (33) has to be modied
to the following;
� � ��� � �� ���
�� 0 ���� ��
��
�
(34)
� 0 ��
0 � ���� � �
0
��� 0
��� ��
�
The inertia terms from the angular rotation now emerge
(�� , �� , �� ). With this modification, the d and q‐axis currents
could also be seen in simulation (refer Equations (9) to
(12)). To visualize the plant model, the dimensions of the
left‐hand terms would become;
���� … ���� 0 0 0 ����� … ����� �q�� �
� �
�
�
� � �
�
�
� �� � �
��
… ���� 0 0 0 ����� … ����� � �q�� �
� ���
�
…
0
�� 0 0
0
…
0 � �� �
� 0
…
0
0 �� 0
0
…
0 � �� �
� 0
� �
0
…
0
0
0
�
0
…
0
�
�
� �� �
0
…
0 � λ
������ … ����� 0 0 0
� ��
�
�
� � �
�
�
� �� � �
� �
������ … ����� 0 0 0
0
…
0 � �λ �
�
(35)
The linear acceleration of the spiral motors (i.e., �� � , �� �, �� �)
are equivalent to �� � , �� � , �� � and rotational acceleration of
the motors are ��� , ��� , ��� . After integrating twice, a gap can
then be obtained via (11). �� are spiral motor inertias.
The right‐hand terms of (34) would become;
��� ��� 0 … 0 ��
� �
� ��
��
��� … ��� ���
�
�
0 … 0� ��
� �� ��
� � ��
� �
�
�
�
�
�� � � � 0
���
0 0 … 0� � 0 ��
��
�
��� … ��� ���
� �
�
� … 0� � � ��
� �
�0
0 0 … 0� � 0 ��
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
�
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
�� � ��� � �� ���� ��� � ��� ��� �
�
�
���� … ���� ���
�
�
�
� �� � �
� �
�
�
���� … ���� ���
�
�
(36)
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The unactu
uated variablles are �� tto �� . Thus,, its
correspondin
ng �� to �� are alway
ys 0 (no input
torque/force)). �� is the vecctor of spiral motor
m
torquess. ��
to �� (monoaarticular 1, monoarticular
m
2 and biarticcular
forces) are giiven by the folllowing equatiion;
���
��
� ��� � ��� ���
�
��� � � ����
� ��� � ��� ���
� � (37)
��
���
�
�
�
�
� ��
�� ��
�
In simplified terms, the plaant can be desscribed by (38));
��
��
���� � � 0
���
�
0
��
0
��
����
0 �
0
���
��� � �� ���
��
� (38)
�
��� ��
ontrol
6. Work Spacce Position Co
vely;
nipulators hav
ve been researrched extensiv
Robotic man
ntrol
some of the m
many examples are torque sensorless con
[18], SCAR
RA manipulaator trackin
ng control [19],
feedforward and compu
uted torque control [20] and
position con
ntrol of consstrained robo
otic system [21].
Previously d
described in [222] are some aapproaches (th
he IK
d the Direct C
Cartesian (DC)) approach) fo
or the
approach and
general struccture of a biarticular man
nipulator, i.e.,, not
actuator speccific. In [23] otther methods w
were investigaated.
In this paperr, the generaliized work space control [222] is
extended to tthe specific sp
piral motor case for the purrpose
of viewing the effects on gap and possition control with
on of Carteesian load disturbance. By
the additio
considering aa circular task
k space trajecto
ory, the refereences
for x and y‐axis positions aare shown as;
x�t� � � cos�ω�t�� � x�
(39)
y�t� � � sin�ω�t�� � y�
(40)
ω�t� � 2���
2
� cos�t��
(41)
where A is th
he radius of th
he circle and xx� and y� are x
x and
y coordinatess of the centree of the circle.
From 0 to 0.5
0 seconds, gap
g
stabilizattion is perforrmed
without giviing any posiition referencce (either in joint
space or worrk space). Thiis phase is to
o allow the gaap to
stabilize at ceentre (0mm) p
position. From
m 0.5 to 6 seco
onds,
the position rreference of th
he circle trajecctory (5 cm rad
dius)
is given. In addition, Carrtesian disturb
bance on the end‐
ven between 11 to 3 seconds.
effector is giv
Here we preesent comparrisons of a monoarticular
m
only
actuated stru
ucture with th
hat of a biarticular structurre by
using the IK approacch and thee DC apprroach
ns, i.e., PD, dissturbance obseerver
(monoarticular control gain
ntical). Note th
hat for work sspace
cut‐off frequeencies are iden
control, zero d‐axis currentt control is not applied.
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5.1 IInverse Kinemaatics Approach
By using the IK
K approach [224], joint spacce trajectoriess
(elbow/shoulder angles) are ob
btained from the Cartesian
n
reference trajectorries (x and y);
��
�� ��� �
���� ����
��� ��
q� � t�n��� �
�
�
q� � t�n��
� t�n�� ��
�
��D�
D
(42))
�� ������ �
� ��� ������ �
(43))
�
(44))
From
m joint referen
nces, muscle kinematics (1) can be used
d
to obtain musccle referencees (i.e., leng
gth, velocity,,
ulder, elbow monoarticular
m
r
acceeleration). x� � , x� �, x� � are shou
and biarticular sp
piral motor lin
near acceleratiions.
�
�� ���� � ���
� �� ����
(45))
��
��
� �
�� ���� � ���
� �� ����
��
��
�
� �� ����
�� ���� � ���
��
��
(46))
(47))
alculated as in
n
The rotational accceleration refeerences are ca
(48) to (50) with
h ����� � 0 (gap referencess are zero forr
non
n‐zero d‐axis cu
urrent controll);
�� ��� ��� ����
������ � �
(48))
��
�� ���� ��� ����
�
(49))
����� �
�
�
�� ���� ��� ����
�
(50))
����� �
�
�
m these refereences, dq‐axiss control (refeer (13) to (20)))
From
can be constructeed for each mu
uscle to generrate the forcess
��� , �� �� � �n� to�q
q��s ���� , ���, ��� � �s in ���� �n� ����.
Figu
ure 13. IK contro
ol for manipulattor driven by sp
piral motors
Ahmad Zaki bin Hj Shukor and Yasutaka Fujimoto:
Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors
9
In this part, only Y‐axis disturbance of 10 [N] is given. The
initial circle trajectory response in Figure 14 shows that the
biarticular structure exhibits better accuracy. Initial errors
due to the gap stabilization can be seen at the near start
position (0.15, 0.53). Then the effect disturbance of 10 [N]
can be seen as the response is deviating from the
reference. Cartesian errors are shown in Figure 15; ‘mono’
denotes monoarticular structure responses while ‘biart’
denotes biarticular structure responses.
Force responses from muscles are shown in Figure 17.
Both forces are bounded. Next, the torque responses
(rotational part of spiral motor) are shown in Figure 18.
Figure 17. Spiral motor forces for the IK approach
Figure 14. Close‐up view of circle trajectory responses from
monoarticular structure and biarticular structure
Figure 18. Spiral motor torque for the IK approach
Figure 15. Cartesian errors for the IK approach
Figure 19. D‐axis current for the IK approach
Figure 16. Gap responses for the IK approach
For Figure 16, ‘mono gap1’ and ‘mono gap2’ refer to the
monoarticular muscle gaps in the monoarticular structure,
while ‘biart gap1’, ‘biart gap2’, ‘biart gap3’ refer to the
shoulder monoarticular gap, elbow monoarticular gap and
biarticular gap in the biarticular structure. Referring to gap
displacement responses in Figure 16, in the initial gap
stabilization phase (0 to 0.5 s), more interaction is obvious
in the biarticular case, but the biarticular structure shows
better gap control during position tracking.
10 Int J Adv Robotic Sy, 2012, Vol. 9, 156:2012
Figure 20. Q‐axis current for the IK approach
Figures 19 and 20 show the d‐ and q‐axis currents generated.
As in the spiral motor simplified plant model, the d‐axis is
related to muscle (spiral motor) forces while the q‐axis is
related to spiral motor torques. It can be said that the
biarticular forces reduce the effort of the elbow monoarticular
muscle, but increase shoulder muscle actuation.
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For this app
proach, muscle force interaactions are cleearly
visible (seen
n in oscillatio
ons) as expeccted, becausee the
muscles are independentlly controlled without
w
any force
f
distribution.
5.2 Direct Carttesian Approachh
For the DC approach, the forces from
m work space//task
space domaiin are controlled in the sho
oulder and ellbow
joint space (virtually),
(
th
hen transferred to the actu
uator
(muscle) spacce by the jointt‐muscle Jacob
bian (3).
hows the con
ntrol diagram
m and the con
Figure 21 sh
ntrol
equation is shown
s
in Equ
uation (51) to
o (55). ���� � ����� are
Cartesian end
d‐effector position responsees.
0 � � �����
��
��
��� � � �����
��
��
��
��
�� � � �
���� �� � � � � � � � � � �
��
0
��
�
�
��
� �
0
��
where
�̂
���
� �� � � �
�̂ �
0
����
0 �� � �� ���
�̂
��
��� � � � �
��� �� � �����
�̂ �
0
��
� �
���
� � � ��� �� �
�� �
� (52)
�� � � ��� �� �
�
����� � � ����
����
����
��
The inverse Jacob
bian to obtain
n the muscle fo
orces, ���� aree
domatrix. �� iss
calculated by using Moore’s inverse pseud
Carttesian accelerration, ��� is tthe inverse of
o (6), � � is thee
Jaco
obian time derrivative, � is 22R planar mass matrix and
d
gi arre disturbance observer gaains. Now tha
at the musclee
forcces’ terms are o
obtained from
m the virtual jo
oint torques, itt
musst be distributted to the gap
p and angularr terms of thee
spirral motor. From
m Equation (116), the linear forces for thee
spirral motor is ag
gain shown;
�� ���� � ���� � �� ����� � ��� � � ���
(54))
��
v
(uxg) is the same as
a in (13), butt
The gap control variable
g
the angular contrrol variable is obtained by manipulating
uation (54) (ssubscript i reefers to respeective musclee
Equ
num
mber);
� �� �
���
�
��
�