Planar Task Space Control of a Biarticular Manipulator Driven by Spiral Motors.

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International Journal of Advanced Robotic Systems

Planar Task Space Control of a Biarticular

Manipulator Driven by Spiral Motors

Regular Paper

Ahmad Zaki bin Hj Shukor

1

and Yasutaka Fujimoto

1,*

1 Department of Electrical and Computer Engineering, Yokohama National University * Corresponding author E-mail: [email protected]

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

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 specication 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

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joint and t positioning, forces are aff motion. By movements w optimal traje

Some design can be referr motors, belts torques. In feedforward of the ver mechanisms sigmoid/swim motor con biarticular/tr maintained a different des motors are p pulled by wi resolve forc musculoskele These structu actuators, po muscle. Thes the amount o actuators ena as the actuat Jumping an implemented structures ar Also, pneum inaccurate du A combinatio

The actuato actuation is t force actuato the rst prot developed a magnet proto

2. Biarticular

An example found in the of monoarti biarticular m using the spi extensor can thus reducin to the elbow combined  manipulator Figure 1.

taskspace. O the potential fected and als

means of sim were achieved ctory derivati

ns that includ ed to in [6] an s and pulleys n [6], stiffn

control was rtebrae) has to realiz mming motio ntrol. Thes iarticular actu as serial or o sign was show placed at the ires. Feedforw ce redundanc etal mechanis ures utilize th ossibly the cl se muscles ext of air pressure ables the desig ors are placed nd hopping d, but because

re less mobil matic muscles ue to their non on of pneuma

or that we the spiral mot or with high totype, the in and currently

otype is availa

Manipulator K

of the biarti human arm, icular (muscl muscles (mus iral motor, the

be combined ng the number w to only thre exor/extenso

of the biart

On the varia elds produc so the speed mulation, the

d without inv on.

de biarticular nd [7]. These d

s to realize b ness and in applied. The

inspired [7 ze a vir on with the se designs

uation, but t pen‐link mec wn in [8], wh

base and th ward control w

cy. Other de ms can be fou he pneumatic r losest candida tend or contra e supplied. Th gn of the legs d closely or at motions w e of the use of le for autono s are difficul nlinear elastici tics/cables is a

propose for tor [14]. It is a forward/back nternal perma the helical s able [15].

Kinematics

icular muscle consisting of les affecting scles affecting e antagonistic by using only r of actuators

e actuators, in or muscles

ticular struct

ation of mu ced by the inte of convergenc human reac verse dynamic

muscle actua designs use ro biarticular mu nverse dyna

lancelet (ance 7] using sim

tual triartic properties of

include the structures

hanisms. Ano hereby the ro e joint angles was also applie esigns of leg und in [9] [10]

rubber muscle ate to the hu

act, dependin he small size o to be space sa tached to the were success f air supply, t omous operat lt to control ity properties also shown in

r musculoske a novel high th kdrivability. F anent version

urface perma

structure can antagonistic p one joint) g two joints)

pair of exor y one spiral m from the shou nstead of six.

of the pl ture is shown

uscle ernal ce of ching cs or

ation otary uscle amics estor milar cular f AC the s are other otary s are ed to gged [11]. es as uman ng on of the aving, link. sfully these tions. and [12]. [13]. eletal hrust From was anent

n be pairs and . By r and motor, ulder The lanar n in

Figu The angl and mus and �� Dist mon �� mus atta �� is re Base and Whe obta shou

ure 1. Simplified main angles le (q2). Based

using trigon scle lengths, lm

a2 are link len

� ��

� � �� c

tances betw noarticulars fr and �� while and ��_��. The scles affect th

ch, i.e., �� af affects �� (elb edundant beca

ed on (1), the joint velocitie

� ��

ere �� is ained from pa ulder and elbo

�

d diagram and m are the shou

on the simp nometric iden mi with joint an ngths).

� ��

�� c��

ween connec rom the joint e connection e extension/co he respective j ffects the join bow). Howeve

ause it affects b

e relation betw es, �� is given

��

the muscle artial derivati ow angles, sho

��

�

��

manipulator des

ulder angle (q

lied diagram ntities, the re ngles q1 and q2

� ��

��

��c��

ction length angles are lab lengths of b ontraction of m

joint angles t nt angle �� (s er, the biarticu both �_� and �_�

ween muscle as;

� �_��

e‐to‐joint spa ives of (1) wi own in (3).

��

�

sign

q1) and elbow m in Figure 1 elationship of 2 is derived (a1

� �� (1)

hs of the belled ��, ��, biarticular are monoarticular o which they shoulder) and ular muscle lm3

�.

velocities, ��

(2

ace Jacobian, ith respect to

(3)

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

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 specically 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.

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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 Teon 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;

�� (13) �� (14)

where �� is the gap reference and �� is the gap velocity 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;

�� 0� � �� (23) �� 0 � ��

�

��

�

��0��_��

As acceleration terms are usually affected by noise (due to differentiation), for initial testing, we simplify (24) to become;

��_��

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Figure 6. Direc d‐axis gap refe

Experiments were carried with a strok used for data the forward the direct dri We regard th is required b then forward smoothly. I motions are p be maintaine stator and around 0 A, the current s 0 mm was results for di d‐axis curren in [15], direc d‐axis contro

4. Singularity

This kinemat work space three spiral m

Parameters

Weight (kg) Stroke (mm) Extended len Contracted le

Estimated m

Table 1. Spiral

ct drive experim erence) (c) Curre

for direct dri d out using th

ke of 0.06 m a collection an and backwar ive control wa his as the initi because once d and backwa If magnetic

performed sim ed and dama mover. After the forward r tabilize at 0 A applied. Figu irect drive for nt control (aft ct drive contro ol.

y/manipulabil

tic parameters of the manip motors are sho

ngth (mm) ength (mm)

max. force [N]

l motor specifica

mental results w ents (d) Reverse

ive forward an he monoartic m. TMS320C67 nd control. Tw d inverters w as applied to 0 ial gap stabiliz magnetic levi ard motions c levitation an multaneously,

age could be r d‐axis curr reference was A again, the re ure 6 shows rward/reverse

er gap initiali ol was perform

ity of Biarticu

s can be used pulator. The p own in Table 1

Spiral

Monoarticul

1.5 60 256.5 196.5

100

ations for muscl

with zero d‐axis e drive (e) Gap d

nd reverse mo cular spiral m 713‐225MHz w wo DC supplie were used. At 0 mm (x posit zation phase. itation is real can be perfor nd forward/ the gap migh e inflicted on rent stabilize s given. Then everse referen the experime motion with ization). Note med without

ular Structure

d to determine parameters of 1.

Motor Type

lar Biarticul

2.3 70 380.5 310.5

150

les

s current contro displacement (f

otion motor were es for first, tion). This ized, rmed /back ht not n the d at after ce of ental zero that zero

e the f the

lar

The man equ

For to th the

By r the mon �� The

Figu

l (a) Forward d ) Currents

forward kin nipulator can ation (x and y

� � �� the biarticula he maximum actuators, sho

referring to E connection le noarticular �� (same ar n, the ranges o

��

ure 7. Closed‐kin

drive (b) Gap di

nematics (end‐ be calculate y are Cartesian

�� co �� ar manipulato

(�� ) and m own as;

�� Equation (1), jo engths ��_� and ��. For us,

rrangement fo

of allowable a

� �_��

� � ��

nematics of biar

isplacements (re

‐effector posi d by using t n coordinates);

s�� or, kinematic l minimum len

� �� oint angle i d ��_�. The sam

we chose �� or both monoa

angles for �� an

� ��

rticular manipu

ed denotes zero

tion) of a 2R the following ;

(25)

(26)

limits are due gth (�� ) of

(27)

(28)

is bounded to me applies for �� and articulars).

nd �� are;

(29)

lator

o

R g

) )

e f

) )

o r d

(6)

Figure 8. Man connection (��

where

This implies incur maxim biarticular l plotted. Figu different con shape of th monoarticula the connecti shoulder ang

Manipulabili of the mu Jacobian term regions, and induced. Red

The inverse manipulator radians. This In addition, also evident examples of biarticular m

Figure 9. Exa biarticular stru

nipulability plo

�� ). Biarti

� � ��

that the minim mum angles an

length limits, ure 8 shows nnecting points he angle limi ar length limit ion points of gle, the larger

ity was obtain uscle‐joint Jac ms. Dark blue d at zero m d represents th

e kinematic [24] can be s is also true f

due to muscl at the shoul singularity an manipulator are

amples of singu ucture.

ots for (a) al1=

cular connection

�

�_{� �� }

mum lengths nd vice versa. , manipulabi the manipu s of monoartic its is due to ts. It can be se f monoarticul the work spac

ned by finding cobian and e represents lo

manipulability he high manip

singularity seen at the for the biartic le Jacobian ter lder angle of nd manipulab e illustrated in

ular and manip

=0.14m, (b) al1=

ns (al31 and al32)

��

of monoarticu Thus, conside ility [16] can ulability plots cular muscles. o biarticular

en that the fur lars are from ce.

g the determin end‐effector‐ ow manipulab

y, singularity pulability regio

of planar elbow angle cular manipul

rms, singulari f 0 radians. S bility poses fo

n Figure 9.

pulability pose

=0.16m, (c) al1=

) set at 0.06 m

ulars ering n be s for . The and rther m the

nants ‐joint bility y is on.

2R of 0 lator. ity is Some r the

es for

In t beco beco angl biar

Hig mus mus regi

If mon recip dete Figu

Figu

��

=0.18m and (d)

he case that e omes fully omes zero, and

les could not rticular length

gher manipula scles are sho scles (or maxi ion is observed

the connecti noarticulars a

procal condit ermine ill‐con ure 11;

ure 10. Recipro

� 0.14 m; (a) ��

al1=0.20m for either should extended, th d singularity i t become zer h limit (refer Fi

bility poses ar ort, and at n imum angles), d.

ion points o are different,

ion of the Jac ndition [25]) is

(a)

(b)

ocal condition p

�� 0.06

similar elbow

er or elbow m he shoulder/

is observed. H ro simultaneo igure 7).

re induced wh near minimum

, maximum m

of shoulder the followin cobian (anoth s shown in F

plot for �� 6 m (b)��

monoarticular

monoarticular /elbow angle However, both ously, due to

hen lengths of m lengths of manipulability

and elbow ng results of her method to Figure 10 and

��, �� 0.� m �� 0.10 m

r

r e h o

f f y

w f o d

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Figure 11. Rec �� 0.� m; (a

As seen in F small, the co narrow and muscle conn the work sp connection d with less wor

Figure 12 s different con obvious that biarticular m work space a

5. Biarticular

Most planar There is m manipulators manipulators Lagrange (LE closed‐chain Equations du

ciprocal conditio ) �� 0

Figure 10 and orresponding

vice versa. A nection point i pace becomes distance, ill‐co

rk space. shows the re nnection point

t by connectin muscle differen and reciprocal

Plant Dynam

2R robots are much researc s. Modelling s uses Ordina E) or Newton

manipulators ue to constra

(a)

(b)

on plot; ��

0.06 m (b)��

11, if the val shoulder/elbo Also, the furth is from the h . By increasin ondition is fur

eciprocal con ts of the biarti ng the end of ntly, significa

condition is s mics Modelling

serial/open‐ch ch on contr

of unconstr ary Differenti Euler (NE) fo s require Diff aint equations

��, �� 0.�� m

� �� 0.�0 m

lue of �� or � ow angle beco her the biartic hinges, the sm ng the biartic rther avoided,

ndition plot icular muscle. f each side of ant changes to

seen. g

hain manipula rol of these

ained open‐c ial Equations ormulation), w ferential Algeb s [17]. Const

m and m

�� is omes cular maller cular , but

with . It is f the o the

ators.

2R chain (via while braic traint

poin used Equ of a

Figu ��

��

nts are virtua d to obtain m uation (29) dep

closed‐chain

��

ure 12. Recipro

� 0.06 m, ��

� �� 0.�� m �

ally cut open mass, gravity picts the gener manipulator; ��

(a)

(b)

(c)

ocal condition � 0.�0 m (b)�� 0.06 m, �

n and LE form and centrifu ralized equatio

� � � ��

plot; ��

�� 0.0� m, ��

�� 0.�0 m

mulation was ugal matrices. ons of motion

�_��

�� (29)

�� 0.�m; (a) �� 0.06 m; (c) s

. n

)

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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

Also, the relative acceleration at the constraints are zero ( ��, �� are the constraint Jacobian and Jacobian derivative), �� 0 (30)

The constraint Jacobian (unmentioned terms are zero) is described as; J� � � J�� … J�� J�� … J�� (31)

�� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; The constraint Jacobian derivative is derived as; J�� J�� … J�� J�� … J�� (32)

�� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; �� ; By combining these two equations, the following generalized closed‐chain model can be derived; � �� 0 � � �� _� �� (33)

To include the spiral motor plant dynamics in the generalized closed‐chain model, (33) has to be modied to the following; �� 0 �� 0 �� 0 �� 0 0 � �� _� �� (34) 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 ��_� _� _� _�� … ��_� _�� … �� 0 0 0 �� … �� 0 … 0 �� 0 0 0 … 0 0 … 0 0 �� 0 0 … 0 0 … 0 0 0 �� 0 … 0 �� … �� 0 0 0 0 … 0 � � � � � � � � � �� … �� 0 0 0 0 … 0 � � � � � � � � � � � � � � � � � � �q_�� q�� θ�� θ�� θ�� λ� � λ�� (35)

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The unactu correspondin torque/force) to �_� (monoa forces) are gi

In simplified

��

6. Work Spac

Robotic man some of the m

[18], SCAR

feedforward position con Previously d approach and general struc actuator spec

In this paper extended to t of viewing t the additio considering a for x and y‐a

where A is th y coordinates

From 0 to 0 without givi space or wor stabilize at ce the position r is given. In effector is giv

Here we pre actuated stru using the (monoarticul cut‐off freque control, zero

uated variabl ng �� to ��

). �_� is the vec articular 1, m iven by the fol

��

terms, the pla

� ��0� �0� �� 0

ce Position Co

nipulators hav many exampl

RA manipula

and compu

ntrol of cons described in [2

d the Direct C cture of a bi cific. In [23] ot

r, the generali the specific sp he effects on on of Carte a circular task xis positions a

x�t� � � y�t� � � ω�t� � 2

he radius of th s of the centre

0.5 seconds, g ing any posi rk space). Thi entre (0mm) p reference of th addition, Car ven between 1

esent compar ucture with th

IK approac

ar control gain encies are iden d‐axis current

les are �� t � are alway

ctor of spiral m monoarticular

llowing equati

��

ant can be des

�� 0 0 � �� ontrol

ve been resear es are torque ator trackin uted torque

strained robo 22] are some a Cartesian (DC) articular man ther methods w

ized work sp piral motor ca gap and pos esian load k space trajecto

are shown as;

cos�ω�t�� x sin�ω�t�� y� 2�� cos�t��

he circle and x e of the circle.

gap stabilizat ition referenc is phase is to position. From he circle trajec rtesian disturb 1 to 3 seconds.

risons of a m hat of a biarti

ch and the ns, i.e., PD, dis ntical). Note th t control is not

to ��. Thus,

ys 0 (no i motor torques

2 and biartic ion;

� � �

�

scribed by (38)

�� _� ��

�

rched extensiv sensorless con ng control

control [20] otic system approaches (th

) approach) fo nipulator, i.e., were investiga

ace control [2 se for the pur sition control disturbance. ory, the refere

x�

�

x� and y� are x

tion is perfor ce (either in o allow the ga m 0.5 to 6 seco ctory (5 cm rad bance on the

monoarticular cular structur

e DC appr

sturbance obse hat for work s

applied.

, its nput s. �_� cular

(37)

);

(38)

vely; ntrol [19], and [21]. he IK or the , not ated.

22] is rpose with By ences (39) (40) (41)

x and

rmed joint ap to onds, dius) end‐

only re by roach erver space 5.1 I By (elb refe

From to acce and The (48) non From can ��, Figu Inverse Kinema

using the IK ow/shoulder rence trajector

q�� t�n�

m joint referen obtain musc eleration). x��, x�

biarticular sp

rotational acc to (50) with n‐zero d‐axis cu

m these refere be constructe

�� n� to�q

ure 13. IK contro

atics Approach

K approach [2 angles) are ob ries (x and y);

� ��_��

q�� t�n��

��

�� t�n��

nces, muscle cle reference

x��,x�� are shou

piral motor lin

��

celeration refe h ��_��_�� 0 (g urrent control

��

ences, dq‐axis ed for each mu

q��s ��, ��,�

ol for manipulat

24], joint spac btained from

��

�_��D� D

��

kinematics (1 es (i.e., leng ulder, elbow m near accelerati

��

erences are ca ap references l); ��

s control (refe uscle to gener

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tor driven by sp

ce trajectories the Cartesian

(42)

(43)

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) can be used gth, velocity, monoarticular ions.

(45)

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alculated as in s are zero for

(48)

(49)

(50)

er (13) to (20)) rate the forces

�n� ��.

piral motors

(10)

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.

Figure 14. Close‐up view of circle trajectory responses from

monoarticular structure and biarticular structure

Figure 15. Cartesian errors 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.

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 18. Spiral motor torque for the IK approach

Figure 19. D‐axis current for the IK approach

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

(11)

For this app

visible (seen

muscles are

distribution.

5.2 Direct Cart

For the DC

space domai

joint space (

(muscle) spac

Figure 21 sh

equation is s

Cartesian end

�

� �

� � �

��̂

�

�̂

� where � � ��

Figure 21. DC

proach, muscl

n in oscillatio

independentl

tesian Approach

approach, th

in are control

(virtually), th

ce by the joint

hows the con

shown in Equ

d‐effector pos

��

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0

�

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� � �

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0

_�

�

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control for man

e force intera

ons) as expec

ly controlled w

h

e forces from

led in the sho

hen transferre

t‐muscle Jacob

ntrol diagram

uation (51) to

ition response

��

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� � �

�

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0

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0

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actions are cle

cted, because

without any f

m work space/

oulder and el

d to the actu

bian (3).

m and the con

o (55). ��

es.

0

�

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� � ��̂

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�� early

e the

force

/task

lbow

uator

ntrol

�� are

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The calc Cart Jaco

gi ar

forc mus spir spir The the Equ num For add seco dep sign Figu appr Figu Nex Biar mon

inverse Jacob

ulated by us

tesian acceler

obian time der

re disturbanc

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st be distribut

ral motor. From

ral motor is ag

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gap control v

angular contr

uation (54) (s

mber);

this study, C

ded to the en

onds. The traje

icted in Figu

nificant improv

ure 22. Close‐up

roach

ure 23. Cartesian

xt, the gap

rticular mus

noarticulars, e

bian to obtain

ing Moore’s

ration, ��is t

rivative, � is 2

e observer ga

obtained from

ted to the gap

m Equation (1

gain shown;

��

variable (uxg)

rol variable is

subscript i re

�

��

�

��

Cartesian Y di

nd‐effector fro

ectory trackin

ure 22 and

vements comp

p view of circle t

n errors for the D

responses ar

cle helps r

especially duri

n the muscle fo

inverse pseud

the inverse o

2R planar ma

ains. Now tha

m the virtual jo

p and angular

16), the linear

��

is the same a

obtained by

efers to respe

� �_��

��

sturbance of

om the perio

ng and Cartes

Figure 23 w

pared to the IK

trajectory respo

DC approach

re shown in

reduce gap

ing disturbanc

orces, �� are

domatrix. �� is

of (6), �� is the

ss matrix and

at the muscle

oint torques, it

r terms of the

forces for the

�� (54)

as in (13), but

manipulating

ective muscle

(55)

10 [N] is also

od of 1 to 3

ian errors are

which shows

K approach.

nses for the DC

n Figure 24.

values for

(12)

The spiral motor force and torques are depicted in Figure 25 and Figure 26. Forces were acceptable and bounded between 60 [N] in both directions for all the muscles.

Figure 24. Gap responses for the DC approach

Figure 25. Spiral motor forces for the DC approach

Figure 26. Muscle torques for the DC approach

Figure 27. D‐axis current for the DC approach

Next, spiral motor d‐ and q‐axis currents are shown in Figure 27 and Figure 28. It can be seen that the d‐axis current is lower in the biarticular structure than in the monoarticular only structure.

Figure 28. D‐axis currents for the DC approach

7. Discussion

The two methods for position control in the task space/work space domain provide some interesting results. It is clear that the IK approach and the DC approach yield acceptable responses.

In the gap stabilization phase, IK provides a clear oscillatory response in achieving a gap at the centre (0 mm). The biarticular actuation increases the oscillations of the gaps. In the DC approach, this oscillation does not exist, in both monoarticular and biarticular structures. The DC approach provides smooth gap responses initially. At the start of position tracking (before disturbance), both methods yield good responses (i.e., variation of gap minimum, small amount of forces and currents), but during disturbance, the effects can be seen clearly. Tracking errors increase, but control was maintained. Towards the end of the disturbance, errors were acceptable.

Table 2 summarizes the comparisons of the IK and DC methods on monoarticular and biarticular structures based on Cartesian errors.

Structure/Method

Error (mm) MonoIK BiartIK MonoDC BiartDC Mean x error ‐0.070 ‐0.001 ‐0.017 ‐0.005 Mean y error ‐0.039 0.007 ‐0.013 ‐0.003 Max. x error 3.800 1.600 0.131 0.149 Max. y error 0.651 0.617 0.120 0.085

Table 2. Comparison of errors between methods/structure

In short, errors for DC method are lower than IK method and errors for biarticular structures are lower than monoarticular structures.

8. Conclusion and Future Works

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properties were initially presented. Then, the spiral motor was introduced and manipulability was discussed. Next, the modelling of the biarticular manipulator using spiral motors was briefly explained and the results of the IK approach and the DC approach for position control of the manipulator were compared between a monoarticular only structure and a biarticular structure. For work space force control schemes (with environment) of a biarticular manipulator, readers are advised to refer to [26].

The gap control is a crucial element in the spiral motor direct drive motion. It was shown that the work space position control was successful in all cases, although some better responses were shown in the biarticular manipulator. Among the advantages of the biarticular structure were the improved control of the gap of the elbow monoarticular muscle and better accuracy of trajectory tracking. In future, we plan to compare our manipulator to other biarticular manipulators available, i.e., ball‐screw mechanism, pneumatics, cables or hydraulics.

9. Acknowledgments

This work was supported by KAKENHI 24246047.

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