Passivity-based Control for 2DOF Robot Manipulatorswith Antagonistic Bi-articular Muscles
Hiroyuki Kawai, Toshiyuki Murao, Ryuichi Sato and Masayuki Fujita
Abstract— This paper investigates a passivity-based control
for two degree of freedom(2DOF) robot manipulators withantagonistic bi-articular muscles which are passing over adjacent two joints and acting the both joints simultaneously.
The manipulator dynamics of three muscle torques, we callthe bi-articular manipulator dynamics, is constructed in orderto design the control input. Stability analysis with respect toour proposed control law is discussed by using the important
property which is concerned with the passivity, although thepassivity of the bi-articular manipulator dynamics can not beshown on account of antagonistic bi-articular muscles explicitly.
Finally, simulation results are shown in order to confirm the
proposed method.

(i-a1) pectoralis major(i-b1) teres major(i-a2) anterior deltoid(i-b2) posterior deltoid

I. I NTRODUCTION
Modern robots are expected to safely and dependably cohabitat with humans in homes and workplaces, providing
support in services, healthcare, assistance and so on [1].
When mechanical systems are working under dynamical
environments, sensory information is needed to behave autonomously. The authors have proposed a vision based
control [2] and vision and force based control [3] in order
to control the motion of the rigid robot manipulators inan efficient manner. Although rigid robot manipulators canmove with high torque and high speed, these would not
be suitable as modern robots which interact human motion,i.e. rehabilitation, human support, surgery and so on.
On the other hand, human motion involves neurons,
muscles, chemical reactions, bones, joints, and ligaments.
Recently, analysis of human motion and robot motion control
by using the mechanism of the human body increasingly
gains attention. For example, the configuration of the affected
human limb(s) can be controlled at each joint by using
rehabilitation robots, so that missing motor synergies can
now be compensated for severely disabled patients [4]. M.
Kuschel et al. [5] have proposed a mathematical model
for visual-haptic perception of compliant objects based on
psychophysical experiments. Wang et al. [6] dealt with aneural network based inverse optimal neuromuscular electrical simulation controller to enable the lower limb to tracka desired trajectory. Antagonistic bi-articular muscles, which
are passing over adjacent two joints and acting the both joints
simultaneously as shown in Fig. 1, are known as one of
H. Kawai is with Department of Robotics, Kanazawa Institute of Technology, Ishikawa 921-8501, Japan hiroyuki@neptune.kanazawa-it.ac.jp
T. Murao is with Master Program of Innovation for Design and Engineering Advanced Institute of Industrial Technology, Tokyo 140-0011, Japan
R. Sato is with Department of Robotics, Kanazawa Institute of Technology, Ishikawa 921-8501, Japan
M. Fujita is with Department of Mechanical and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8550, Japan

(ii-a) brachioradialis(ii-b) triceps branchii
lateral head(right part)(iii-b) triceps branchii(iii-a) biceps brachii
long head(left part)long head(left part)

Copyright 2003-2004 University of Washington. All rights reserved
including all photographs and images. No re-use, re-distribution orcommercial use without prior written permission of the authors andthe University of Washington.

Fig. 1. Muscles of Arm. (i)Antagonistic mono-articular muscles attachedto the shoulder joint consist of two flexor muscles, i.e., pectoralis major andanterior deltoid, and two extensor muscles, i.e., teres major and posteriordeltoid. (ii)Antagonistic mono-articular muscles attached to the elbow joint
consist of brachioradialis and triceps branchii lateral head. (iii)Antagonisticbi-articular muscles attached to both the shoulder and the elbow joint consistof biceps brachii long head and triceps branchii long head.

most important mechanisms of the human body associatedwith motion. Kumamoto et al give us the effects of theexistence of antagonistic bi-articular muscles [7]–[9]. Oh
and Hori [10] have proposed two-degree-of-freedom control
for robot manipulator with antagonistic bi-articular muscles.
However, stability analysis is not discussed in these works
explicitly.
This paper deals with a passivity-based control for two degree of freedom(2DOF) robot manipulators with antagonisticbi-articular muscles. Control objectives are both a regulation
and a trajectory tracking like a standard robot motion control.
Stability analysis with respect to our proposed control law is
discussed by using the important property which is concernedwith the passivity. This is one of main contributions of thisresearch. The simulation results show the validity of the
proposed method.

kj

a
b

f

Fj

bj

xj

uj
c

J2

d

q2

f3
J2

f1

q1
J1

x

(a) 2DOF robot arm(j = 1,2,3)

e

e1

f2

T1

e2

e3

T2

Ff2

Ff1

q1

J1

q2
r

r
Fe2

Fe1

(b) Human arm modelFig. 2. (a)2DOF robot arm. (b)Human arm model. Two couples of theantagonistic mono-articular muscles of f1 and e1 , and of f2 and e2 are
attached to the joints of J1 and J2 , respectively. A couple of the antagonisticbi-articular muscles f3 and e3 are attached to both joints of J1 and J2 .

II. M ODEL OF 2DOF ROBOT M ANIPULATORS WITH
A NTAGONISTIC B I - ARTICULAR M USCLES
A. Antagonistic Bi-articular Muscle TorqueThe dynamics of n-link rigid robot manipulators can be
written as
M (q)q̈ + C(q, q̇)q̇ + g(q) = T

Ff3

(1)

where q, q̇ and q̈ are the joint angle, velocity and acceleration,
respectively. T is the vector of the input torque. M (q) ∈Rn×n is the manipulator inertia matrix, C(q, q̇) ∈ Rn×n is
the Coriolis matrix and g(q) ∈ Rn is the gravity vector [11].
In the case of 2DOF robot manipulator as shown in Fig. 2(a),the dynamics can be concretely represented as[][]
M1 + 2M2 + 2RC2 2M2 + RC2
q̈1
2M2 + RC2
2M2
q̈2
[
][
]
−RS2 q̇2 −RS2 (q̇1 + q̇2 )q̇1+
RS2 q̇1
0
q̇2
[
] [
]
g(m1 lg1 + m2 l2 )C1 + g(m2 lg2 )C12
T1+=
g(m2 lg2 )C12
T2
(2)
(
)
2
2
where M1 = m1 lg1+ m2 l12 + I˜1 , M2 = 12 m2 lg2+ I˜2
and R = m2 l1 lg2 . mi and li are the weight and the lengthof the link i, lgi is the distance from the center of a joint
i to the center of the gravity point of the link i, I˜i is themoment of inertia about an axis through the center of massof link i (i = 1, 2). Si , Ci , Sij and Cij mean sin qi , cos qi ,sin(qi + qj ) and cos(qi + qj ), respectively.
While a real human arm has four pairs of antagonisticmuscles as shown in Fig. 1, human arm model can be
simplified as three pairs of antagonistic muscles as depicted
in Fig. 2(b) [8]. Generally, the joint torque T will be designed
as a control input directly in robot motion control. Becausea couple of bi-articular muscles are attached to both joints

Fe3
Fig. 3. Visco-elastic muscle model [8]. Fj : output force, uj : contractile
force, kj : elastic coefficient, bj : coefficient of viscosity, xj : contracting
length.

Activation Level Calculated [%]

y

ue1

100
50
0

uf1

100

uf2

50
0

ue2

100

uf3

50
ue3

0
d

e f a b cForce DirectiondFig. 4. Each muscle respond depending on the direction of the force atthe tip point [8].

as shown in Fig. 3, the joint torques are described as
Ti = (Ff i − Fei )r + (Ff 3 − Fe3 )r= (uf i − uei )r − (uf i + uei )ki r2 qi − (uf i + uei )bi r2 q̇i+(uf 3 − ue3 )r − (uf 3 + ue3 )k3 r2 (q1 + q2 )−(uf 3 + ue3 )b3 r2 (q̇1 + q̇2 ) (i = 1, 2)(3)

where Ff j and Fej are forces generated by flexor muscle andby extensor muscle, uf j and uej represent contractile forcesof flexor muscle and of extensor muscle (j = 1, 2, 3). r, kj
and bj are the radius of the joint pulley, elastic coefficientsand visco coefficients, respectively [7].
Fig. 4 shows the activation levels of each muscle responddepending on the direction of the force at the tip point. a–fmean the direction of the force at the tip point in Fig. 2(b).
From Fig. 4, the activation levels of the antagonistic pairmuscles satisfy
uf j + uej = 1 (j = 1, 2, 3).

(4)

Using this important property with respect to the antagonistic
pair muscles, the joint torques can be transformed intoKrq-

Ti = (2uf i − 1)r − ki r qi − bi r q̇i + (2uf 3 − 1)r−k3 r2 (q1 + q2 ) − b3 r2 (q̇1 + q̇2 ) (i = 1, 2) (5)22Here we assume that the contractile force of flexor muscle
uf i can be decided by an actuator. Then, the joint torques(5) are represented as
Ti = τi + τ3 − ki r2 qi − k3 r2 (q1 + q2 )−bi r2 q̇i − b3 r2 (q̇1 + q̇2 ) (i = 1, 2)(6)

where muscle torques are defined as τi := (2uf i − 1)r.
Therefore, we will design the bi-articular muscle torqueτ ∈ R3 as the control input for the 2DOF robot manipulatorswith antagonistic bi-articular muscles.
B. Dynamics of 2DOF Robot Manipulators with Antagonistic Bi-articular Muscles
In this subsection, we construct the manipulator dynamics
of three muscle torques in order to design the control input.
We now suggest that the dynamics of the antagonistic biarticular muscles torque is defined as
τ3 =

1
2
+ I˜2 )(q̈1 + q̈2 ) + g(m2 lg2 )C12
(m2 lg22+k3 r2 (q1 + q2 ) + b3 r2 (q̇1 + q̇2 )= M2 (q̈1 + q̈2 ) + g(m2 lg2 )C12
+k3 r2 (q1 + q2 ) + b3 r2 (q̇1 + q̇2 ).

(7)

From Eq. (2), the manipulator dynamics of three antagonisticmuscles torques can be represented as
M1 + M2 + 2RC2 M2 + RC2
0
q̈1


M2 + RC2
M2
0 
q̈2
0
0
M2
q̈1 + q̈2−RS2 q̇2 −RS2 (q̇1 + q̇2 ) 0
q̇1

0
0 
q̇2
+  RS2 q̇1
0
00q̇1 + q̇2g(m1 lg1 + m2 l2 )C10+
g(m2 lg2 )C12



 

q1
q̇1
τ1
 + Br  =  τ2 
q̇2
q2
+Kr 
q1 + q2
q̇1 + q̇2
τ3(8)
where Kr := diag{k1 , k2 , k3 }r2 ∈ R3×3 and Br :=diag{b1 , b2 , b3 }r2 ∈ R3×3 . Moreover, we define the extended joint angle vector for the manipulator dynamics ofthree antagonistic muscles torques as


q1
.
q2
θ=
(9)
q1 + q2
Based on the well-known form of the manipulator dynamics(1), the manipulator dynamics with antagonistic bi-articulart+ -Manipulator
Dynamicsq_BrBi-articular Manipulator Dynamics

Fig. 5.

Block diagram of the bi-articular manipulator dynamics.

muscles, we call the bi-articular manipulator dynamics, canbe described asMb (θ)θ̈ + Cb (θ, θ̇)θ̇ + gb (θ) + Kr θ + Br θ̇ = τ(10)where the elements of Mb (θ) ∈ R3×3 , Cb (θ, θ̇) ∈ R3×3 andgb (θ) ∈ R3 are correspond to Eq. (8). The block diagram ofthe bi-articular manipulator dynamics is depicted in Fig. 5.
Then the bi-articular manipulator dynamics has following
important properties.
Property 1: The inertia matrix Mb (θ) preserves the positive definiteness.
Property 2: Ṁb (θ) − 2Cb (θ, θ̇) is skew-symmetric.
Remark 1: The form of the manipulator dynamics withthree muscle torques (8) is not constructed uniquely, since
the degree of freedom for the control input τ is greater thanthat for the joint q. For example, Oh and Hori [10] haveproposed diagonalized inertia matrix in order to decouple thecorrelation of joint torques, although the positive definitenessof the inertia matrix is not preserved. Because Property 1
and 2 are very important factors for stability analysis, weconstruct the bi-articular manipulator dynamics (10) which
satisfies them.
III. PASSIVITY- BASED C ONTROL L AW
The control objective of the manipulator with antagonisticbi-articular muscles is that both the joint angle and the joint
velocity coincide with the desired ones, respectively. For the
bi-articular manipulator dynamics, we propose the control
law asτ = Mb (θ)v̇ + Cb (θ, θ̇)v + gb (θ) − ė + Kr θd + Br θ̇ (11)where v, v̇, e and ė are defined asv = θ̇d − Kr e, v̇ = θ̈d − Kr ė
e = θ − θd , ė = θ̇ − θ̇dand θd := [qd1 qd2 qd1 + qd2 ]T is a desired extended joint
angle. Substituting Eq. (11) into Eq. (10), the closed-loopsystem can be obtained asMb (θ)ṡ + Cb (θ, θ̇)s + s = 0(12)where s := ė + Kr e. Here, we define the state of the closedloop system with the bi-articular manipulator dynamics andthe proposed control law as
[ ]
e
x=
.
(13)
ė

-

+
+

t

+ -

+

q_

Br

+
Br

Manipulator
Dynamics

Bi-articular Manipulator Dynamics

Controller

Fig. 6. Block diagram of the closed-loop system with the bi-articular
manipulator dynamics and the proposed control law.

The block diagram of the closed-loop system is depicted
in Fig. 6. It is noted that the equilibrium point x = 0 is
equal to e = 0 and s = 0. If the equilibrium point x = 0,then the joint angle and the joint velocity coincide with thedesired ones and the control objective is achieved. We show
the following theorem concerning the stability of closed-loopsystem.
Theorem 1: The equilibrium point x = 0 for the closedloop system (12) is asymptotic stable.
Proof: Consider the following positive definite function
1
V = sT Mb (θ)s + eT Kr e.
(14)
2
The positive definiteness of the function V results from
Property 1. Differentiating (14) with respect to time, we
obtain1V̇ = sT Mb (θ)ṡ + sT Ṁb (θ)s + 2eT Kr ė
2
(
) 1
T
= s −Cb (θ, θ̇)s − s + sT Ṁb (θ)s + 2eT Kr ė
2
)
1 T(
T
= −s s + s Ṁb (θ) − 2Cb (θ, θ̇) s + 2eT Kr ė (15)2
Using Property 2, i.e., the skew-symmetry of the matrixṀb (θ) − 2Cb (θ, θ̇) yields
V̇ = −(ė + Kr e)T (ė + Kr e) + 2eT Kr ė= −ėT ė − eT Kr2 e.

(16)

From the positive definiteness of Kr , this completes theproof.
Stability analysis with respect to our proposed control law
is discussed by using Property 1 and 2 which are concernedwith the passivity, although the passivity of the bi-articular
manipulator dynamics can not be shown on account of antagonistic bi-articular muscles explicitly. Indeed, our proposed
control law is similar to one of passivity-based control laws
for robot manipulator which is well-known as the Slotineand Li scheme [12]. This is one of main contributions of
this research.
IV. S IMULATION R ESULTS
In this section, we show the simulation results in order toconfirm the proposed method. Moreover we give the possibility of a sensorless control for 2DOF robot manipulatorswith antagonistic bi-articular muscles. The parameters of the

Joint Angle q1 [rad]Mb(q)v_ +Cb(q,q_)v+gb(q)nonlinear compensation

q-

Joint Angle q2 [rad]DesiredTrajectory

Kr

+

2
1
0
0

0.5

1

1.5

2

0

0.5

1
time [s]

1.5

2

2
1
0

π
Fig. 7. Step response in the case of qd1 = , qd2 = 0 (solid:Joint angles,
2
dashed:Desired ones).

t=2

0.6

y-axis [m]

Kr
qd
q_d
q__d

0.4

0.2
t = 0.05
0

t=0
-0.2

Fig. 8.

00.2
0.4
x-axis [m]Trajectory of the arm in the case of qd1 =0.6
π, qd2 = 0.
2bi-articular manipulator dynamics used in the simulation are
m1 = 1.75[kg], m2 = 1.75[kg], l1 = 0.3[m], l2 = 0.3[m],lg1 = 0.15[m], lg2 = 0.15[m], I˜1 = 0.014[kg · m2 ], I˜2 =0.014[kg · m2 ], r = 0.05[m] and b1 = b2 = b3 = 400[Ns/m], and the effect of the gravity is ignored. Because
we guess that elastic coefficients depend on the strength ofmuscles, we select k1 = 3, 000[N/m], k2 = 2, 000[N/m] andk3 = 4, 000[N/m] based on Fig. 1.
We consider both set-point problems and trajectory tracking ones. In the cases of set-point problems, the initial
angles are q1 (0) = 0[rad] and q2 (0) = 0[rad]. In thecases of trajectory tracking problems, the initial angles are
q1 (0) = 0[rad] and q2 (0) = 1[rad].
A. Simulation Results with Proposed Control Law
The simulation results for the set-point problem are shownin Figs. 7–10. Figs. 7 and 8 depict the joint angles and the
trajectory of the arm in the case of qd1 = π2 and qd2 = 0,i.e., only shoulder joint is wanted to move. Figs. 9 and 10
describe the joint angles and the trajectory of the arm in
the case of qd1 = 0 and qd2 = π2 , i.e., only elbow joint iswanted to move. Although another joint was moved in bothcases, we consider that it is quite natural on human motion.

0.6
0
0

0.5

1

1.5

2

2

t=2
0.2
t = 0.05
0

0
0

0.5

1
time [s]

1.5

2

1
0
-1
0

2

4

6

8

2
1
0
0

2

4
time [s]68Fig. 11. Trajectory tracking problem(ω = 0.25[Hz]) (solid:Joint
angles, dashed:Desired ones).

Specially, it is very difficult to try to move only the elbow
joint by oneself.
We give qd1 = sin(2πωt) and qd2 = 1 + sin(2πωt)
as the desired angles for the trajectory tracking problem.
Figs. 11 and 12 show the joint angles and desired ones with
ω = 0.25[Hz] and with ω = 1[Hz], respectively. Thoughboth joints have the errors until 0.5[s] in the case of thehigh frequency, they completely coincide with the desired
trajectory after 0.5[s]. From these simulation results, theasymptotic stability can be also confirmed.
B. Simulation Results with Modified Control Law
In this subsection, we modify the proposed control law(11). Because it is inferred that humans do not measure
the joint angles and velocities explicitly, we consider thefollowing sensorless control law
τ = Mb (θd )θ̈d + Cb (θd , θ̇d )θ̇d + gb (θd ) + Kr θd + Br θ̇d .
(17)
It is noted that θ and θ̇ are replaced with desired values θdand θ̇d in Eq. (11), respectively. The block diagram of themodified control law is depicted in Fig. 13. Simulation results

t=0
-0.2

0

0.2
0.4
x-axis [m]0.6

Trajectory of the arm in the case of qd1 = 0, qd2 =Fig. 10.
Joint Angle q1 [rad]Joint Angle q1 [rad]

0.4

1

π
Fig. 9. Step response in the case of qd1 = 0, qd2 = . (solid:Joint
2
angles, dashed:Desired ones).

Joint Angle q2 [rad]y-axis [m]1

Joint Angle q2 [rad]Joint Angle q1 [rad]Joint Angle q2 [rad]

2

π
.
2

1
0
-1
0

2

4

6

8

0

2

4
time [s]

6

8

2
1
0

Fig. 12. Trajectory tracking problem(ω = 1[Hz]) (solid:Joint angles,
dashed:Desired ones).

Kr
DesiredTrajectory

qd
q_d
q__d

Mb(qd)q__d+Cb(qd,q_d)q_d+gb(qd)nonlinear compensation

Kr

+
+

q
t

+

+ -

Fig. 13.

q_

Br

Br
Controller

Manipulator
Dynamics

Bi-articular Manipulator Dynamics

Block diagram of the modified control law.

by using the modified control law (17) are shown in Figs. 14–
17. Although the transient response and the tracking performance are inferior to the proposed control law (11), thesesimulation results suggest that 2DOF robot manipulators withantagonistic bi-articular muscles could be controlled with asensorless control law, i.e., the information of joint angles
and velocities would not be needed in a control law explicitly.
The stability analysis with the sensorless control law has tobe discussed in our future work.
V. C ONCLUSIONSThis paper deals with a passivity-based control for 2DOF
robot manipulators with antagonistic bi-articular muscles.
Stability analysis with respect to our proposed control law,

0.6

0
0

0.5

1

1.5

2

2

t=2
0.2
t = 0.05
0

0
0

0.5

1
time [s]

1.5

2

1
0
-1
0

2

4

6

8

2
1
0
0

2

4
time [s]68Fig. 16. Trajectory tracking problem with a sensorless(ω = 0.25[Hz])(solid:Joint angles, dashed:Desired ones).

which is one of main contributions of this paper, is discussed by using the important property which is concernedwith the passivity, although the passivity of the bi-articular
manipulator dynamics can not be shown explicitly. The
simulation results show the validity of the proposed method
and give the possibility of a sensorless control for 2DOF
robot manipulators with antagonistic bi-articular muscles. In
our future work, we will discuss the stability analysis withthe sensorless control law and integrate the vision based
control [2] or vision and force based control [3] in order
to consider human motion with five senses.
ACKNOWLEDGEMENT
Musculoskeletal Images are from the University of Washington “Musculoskeletal Atlas: A Musculoskeletal Atlas of
the Human Body” by Carol Teitz, M.D. and Dan Graney,
Ph.D.
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Systems Technology, Vol. 15, No. 1, pp. 40–52, 2007.

t=0
-0.2

0

0.2
0.4
x-axis [m]0.6πFig. 15. Trajectory of the arm in the case of qd1 = 0, qd2 =2with a sensorless.
Joint Angle q1 [rad]Joint Angle q1 [rad]

0.4

1

π
Fig. 14. Step response in the case of qd1 = 0, qd2 =with a2
sensorless (solid:Joint angles, dashed:Desired ones).

Joint Angle q2 [rad]y-axis [m]1

Joint Angle q2 [rad]Joint Angle q1 [rad]Joint Angle q2 [rad]

2

1
0
-1
0

2

4

6

8

0

2

4
time [s]

6

8

2
1
0

Fig. 17. Trajectory tracking problem with a sensorless(ω = 1[Hz])(solid:Joint angles, dashed:Desired ones).

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