Open-loop Control for 2DOF Robot Manipulatorswith Antagonistic Bi-articular Muscles
Keisuke Sano, Hiroyuki Kawai, Toshiyuki Murao and Masayuki FujitaaAbstract— This paper investigates open-loop control, which
does not need the joint angles and velocities, for two degreeof freedom(2DOF) robot manipulators with antagonistic biarticular muscles which are passing over adjacent two joints
and acting the both joints simultaneously. The manipulator
dynamics of three muscle torques, we call the bi-articular
manipulator dynamics, is constructed in order to design thecontrol input. Stability analysis with respect to our proposed
control law is discussed based on the Lyapunov method. Our
approach is inspired by the fact that humans do not measure thejoint angles and velocities explicitly, Finally, simulation results
are shown in order to confirm the proposed method and given
a design procedure in order to assign the tuning parameter.

I. I NTRODUCTION
Recently, activity of robots in homes, workspaces, providing support in services, healthcare and assistance are
expected. These robots are not expected to have high torqueand high speed, but have safely and dependably. Robot
manipulators which have actuators in each joint can movewith high torque and high speed, so that we can consider that
there might be an inflicting injury on human. These robots
are not suitable as modern robots which interact human, i.e.,rehabilitation, care, surgery, and so on [1].
On the other hand, neurons, muscles, bones, joints andligaments are related to human motion. Recently, analysis
of human motion and robot control using the mechanism of
human body attract attention. For example, the configurationof the affected human limb(s) can be controlled at each joint
with rehabilitation robots, so that missing motor synergies
can now be compensated for severely disabled patients [2].
M. Kuschel et al. [3] have proposed a mathematical model
for visual-haptic perception of compliant objects based on
psychophysical experiments. Wang et al. [4] dealt with aneural network based inverse optimal neuromuscular electrical simulation controller to enable the lower limb to tracka desired trajectory. One of the most important mechanismsof human motion is antagonistic bi-articular muscles, which
are passing over adjacent two joints and acting the bothjoints simultaneously. Since robot manipulators with antagonistic bi-articular muscles do not need high torque, theinflicting injury on human decreases. Kumamoto et al. giveus the effects of the existence of antagonistic bi-articular
K. Sano is with Graduate Program in Mechanical Engineering, Kanazawa
Institute of Technology, Ishikawa 921-8501, Japan
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
M. Fujita is with Department of Mechanical and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8550, Japan

b

f

c

y

e
J2

d

q2

f3
J2

f1

q1
J1

e2

e3

J1

x

(a) 2DOF robot arm

e1f2(b) Human arm modelFig. 1. (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 .

muscles [5]–[7]. Oh et al. have proposed two-degree-offreedom control for robot manipulators with antagonisticbi-articular muscles [8][9]. While stability analysis has notbeen discussed in these works, we have proposed passivitybased control and discussed the stability analysis explicitly
in [10]. However, the proposed control law utilize the joint
angles and velocities, although humans do not measure theseinformations explicitly.
This paper deals with open-loop control, which does notneed the joint angles and velocities, for 2DOF robot manipulators by using antagonistic bi-articular muscles. Firstly,
the brief summary of passivity-based control for 2DOF robot
manipulators with antagonistic bi-articular muscles is given.
Secondly, we propose an open-loop control law and discuss
the stability analysis of the error dynamics based on the
Lyapunov method. Finally, the simulation results show thevalidity of the proposed control law and give a design
procedure in order to assign the tuning parameter.
II. P REVIOUS W ORKS
A. Bi-articular Manipulator Dynamics
The dynamics of n-link rigid robot manipulators can be
written as
M (q)q̈ + C(q, q̇)q̇ + g(q) = T(1)

where q, q̇ and q̈ are the joint angle, velocity and acceleration,
respectively. T is the vector of the input torque. M (q) ∈kj

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 manipulators as shownin Fig. 1(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)

Human arm model can be simplified as three pairs ofantagonistic muscles as depicted in Fig. 1. Generally, thejoint torque T will be designed as control input directly inrobot motion control. A couple of bi-articular muscles areattached to both joints as shown in Fig. 2. The joint torques
are described as
Ti = (Ff i − Fei )rp + (Ff 3 − Fe3 )rp= (uf i − uei )rp − (uf i + uei )ki rp2 qi − (uf i + uei )bi rp2 q̇i+(uf 3 − ue3 )rp − (uf 3 + ue3 )k3 rp2 (q1 + q2 )−(uf 3 + ue3 )b3 rp2 (q̇1 + q̇2 ). (i = 1, 2)(3)
In Fig.2, Ff j and Fej (j = 1, 2, 3) are output forces by
flexor muscle and by extensor muscle. uf j and uej represent
contractile forces of flexor muscle and of extensor muscle.
rp , kj and bj are the radius of the joint, coefficients w.r.t.
elastic and coefficients w.r.t. viscosity[5]. The contractile
forces of flexor muscle and of extensor muscle have following relationship[6].
uf j + uej = 1 (j = 1, 2, 3).

(4)

Because the contractile force of flexor muscle uf j can bedecided by an actuator, muscle torques are defined as τi :=(2uf i − 1)rp . Then the joint torques (3) can be transformed
into
Ti = τi + τ3 − ki rp2 qi − k3 rp2 (q1 + q2 )−bi rp2 q̇i − b3 rp2 (q̇1 + q̇2 ).

(i = 1, 2)(5)

In order to design the bi-articular muscle torque τ ∈ R3
as the control input, we define the antagonistic bi-articular
muscles torque as1
2(m2 lg2+ I˜2 )(q̈1 + q̈2 ) + g(m2 lg2 )C122
+k3 rp2 (q1 + q2 ) + b3 rp2 (q̇1 + q̇2 )= M2 (q̈1 + q̈2 ) + g(m2 lg2 )C12

τ3 =+k3 rp2 (q1 + q2 ) + b3 rp2 (q̇1 + q̇2 ).

(6)

From Eqs. (2), (5) and (6), the manipulator dynamics of

bj

Fj
xj

uj
(j=1, 2, 3)

Ff3
T1

T2

Ff2

Ff1

q1

q2
rp

rp
Fe2

Fe1
Fe3

Fig. 2. Visco-elastic muscle model [6]. Fj : output force, uj : contractile
force, kj : coefficient w.r.t. elastic, bj : coefficient w.r.t. viscosity, xj :contracting length, rp : radius of the joint.

three antagonistic muscle 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 l1 )C1
0+
g(m2 lg2 )C12



 

q1
q̇1
τ1
 + Br  =  τ2 
q2
q̇2
+Kr 
(7)
q1 + q2
q̇1 + q̇2
τ3
2
2
+ m2 l12 + I˜1 , M2 = 12 (m2 lg2
where M1 = m1 lg1+ I˜2 )
and R = m2 l1 lg2 . Si , Ci , Sij and Cij means sin qi , cos qi ,sin(qi + qj ) and cos(qi + qj ) (i, j = 1, 2), respectively.
mi , li , lgi and I˜i are the weight of the link i, the lengthof the link i, the distance from the center of a joint i tothe center of the gravity point of the link i and the momentof inertia about an axis through the center of mass of thelink i. Kr := diag {k1 , k2 , k3 } rp2 ∈ R3×3 and Br :=diag {b1 , b2 , b3 } rp2 ∈ R3×3 mean the matrices w.r.t. elastic
and viscosity. Moreover, we define the extended joint angle
vector for the manipulator dynamics of three antagonisticmuscle torques as


q1
.
q2
θ=
(8)
q1 + q2

The manipulator dynamics with antagonistic bi-articular
muscles, we call the bi-articular manipulator dynamics[10],can be described asMb (θ)θ̈ + Cb (θ, θ̇)θ̇ + gb (θ) + Kr θ + Br θ̇ = τ

(9)

Kr
DesiredTrajectory

qd
q_d

q__d

Mb(q)v_ +Cb(q,q_)v+gb(q)nonlinear compensation

-

Kr

+
+

t

+ -

Manipulator
Dynamics+

DesiredTrajectory

q_

qd
q_d

q__d

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

Br

+
Br

Kr
q

-

+

Kr

+
+

q
t

+

+ -

Manipulator
Dynamics

BrBr

Bi-articular Manipulator Dynamics

Controllerq_Bi-articular Manipulator Dynamics

Controller

Fig. 4.

Fig. 3. Block diagram of the closed-loop system and the previous control
law

Block diagram of the proposed control law.

III. O PEN - LOOP C ONTROL FOR B I - ARTICULAR
M ANIPULATOR DYNAMICS

where the elements of Mb (θ) ∈ R3×3 , Cb (θ, θ̇) ∈ R3×3 andgb (θ) ∈ R3 are correspond to Eq. (7). Then the bi-articular
manipulator dynamics has following important properties.
Property 1: Under the conditions M1 + M2 > 2R andM1 M2 > R2 , the inertia matrix Mb (θ) preserves the positive
definiteness 1 .
Property 2: Ṁb (θ) − 2Cb (θ, θ̇) is skew-symmetric.
Because Property 1 and 2 are very important factors forstability analysis, we construct the bi-articular manipulator
dynamics (9) which satisfies them.

In this section, we present the analysis of the open-loop
model of 2DOF robot manipulators with antagonistic biarticular muscles.

B. Passivity-based Control

This is noted that θ and θ̇ are replaced with θd and θ̇d in
Eq. (10). The block diagram of the proposed control law is
depicted in Fig. 4 . Substituting Eq. (9) into Eq. (13), theerror dynamics can be obtained as

The control objective of the manipulators with antagonisticbi-articular muscles is that both joint angle and the joint
velocity coincide with the desired ones, respectively. For the
bi-articular manipulator dynamics, we have considered thecontrol law asτ = Mb (θ)v̇ + Cb (θ, θ̇)v + gb (θ) − ė + Kr θd + Br θ̇
(10)where v, e and a desired joint angle is defined as v = θ̇d −Kr e, e = θ −θd and θd = [qd1 qd2 qd1 +qd2 ]T . Substituting
Eq. (10) into Eq. (9), the closed-loop system can be obtainedasMb (θ)ṡ + Cb (θ, θ̇)s + s = 0(11)where s := ė + Kr e. The block diagram of the closed-loopsystem is depicted in Fig.3.
Theorem 1: [10] The equilibrium point (e, ė) = 0 for theclosed-loop system (11) is asymptotic stable.
Theorem 1 can be proved by using the following Lyapunov
function1
V = sT Mb (θ)s + eT Kr e(12)2
via Property 1 and 2 which are concerned with the passivity, although the passivity of the bi-articular manipulator
dynamics can not be shown on account of antagonistic biarticular muscles explicitly. Indeed, this control law is similar
to one of passivity-based control laws for robot manipulators
which is well-known as the Slotine and Li scheme [12]. Moredetails are available in [10].
1 In the ideal case with l = l = 2l12g1 = 2lg2 , the conditions can be
simple as m1 > 3/2m2 .

A. Open-loop Control Law
Because it is inferred that humans do not measure the joint
angles and velocities explicitly, we propose the following
open-loop control law
τ = Mb (θd )θ̈d + Cb (θd , θ̇d )θ̇d + gb (θd )+Kr θd + Br θ̇d .

(13)

Mb (θ)ë = −Cb (θ, θ̇)ė − Kr e − Br ė − h (t, e, ė) (14)where the residual dynamics h (t, e, ė) is defined as follows:h (t, e, ė) = {Mb (θ) − Mb (θd )} θ̈d+{Cb (θ, θ̇) − Cb (θd , θ̇d )}θ̇d + gb (θ) − gb (θd ).
(15)Then, the bi-articular manipulator dynamics satisfies thefollowing properties like that of n−DOF robots manipulators [13].
Property 3: There exists a constant kM > 0 such that
∥Mb (x)z − Mb (y)z∥ ≤ kM ∥x − y∥z
for all vectors x, y, z ∈ R3 .
Property 4: There exist numbers kC1 > 0, kC2 > 0 suchthat
∥Cb (x, z)w − Cb (y, v)w∥ ≤ kC1 ∥z − v∥∥w∥+kC2 ∥x − y∥∥w∥∥z∥
for all vectors v, w, x, y, z ∈ R3 .
Property 5: The vector gb (θ) is Lipschitz, that is, thereb (θ)
exists a nonnegative constant kg ≥ ∂g∂θ
such that∥gb (x) − gb (y)∥ ≤ kg ∥x − y∥for all vectors x, y ∈ R3 .
Property 6: There exist constants kh1 , kh2 ≥ 0 such thatthe norm of the residual dynamics satisfies∥h (t, e, ė) ∥ ≤ kh1 ∥ė∥ + kh2 ∥Sin(e)∥(16)

for all e, ė ∈ R3 , where Sin(e) is following saturatedfunction [14]TSin(x) = [Sin(x1 ), Sin(x2 ), · · · , Sin(xn )](17)

1
xi ≥ π2sin (xi ) |xi | < π2
Sin(xi ) =
(18)

−1
xi ≤ − π2∂Sin(x)= Cos(x) = diag{Cos(x1 ), · · · , Cos(xn )}(19)∂xfor all vector x ∈ Rn . Saturated function Sin(x) satisfies thefollowing properties• ∥Sin(x)∥ ≤ ∥x∥√n• ∥Sin(x)∥ ≤2T• ∥Sin(x)∥ ≤ Sin(x) x• ∥Cos(x)ẋ∥ ≤ ∥ẋ∥for all x, ẋ ∈ Rn .
B. Stability Analysis with Open-loop Control Law
Here, we define the state of the error dynamics with thebi-articular manipulator dynamics and the proposed control
law as x := [eT ėT ]T . 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 the error
dynamics.
Theorem 2: Given a positive scalar γ, assume
λmin {Br } > kh1 + γb
(20)
{
2
λ
{M
(θ)}
b
λmin {Kr } > Max γ 2 Max,
λmin {Mb (θ)}}{2γa + kh2 }2+ kh2 (21)4γ[λmin {Br } − kh1 − γb]where the constant a and b are given by
{
( )}
]
1[
a=
−λmin {Br } + kC1 sup θ̇d+ kh1 ,2√b = λMax {Mb (θ)} + 3kC1 .
Then the equilibrium point x = 0 for the error dynamics is
asymptotic stable,
Proof: Consider the following Lyapunov function candidate,
V (t, e, ė) =1 T1ė Mb (θ)ė + eT Kr e + γSin(e)T Mb (θ)ė
2
2
(22)

where Sin(e) is defined as the saturated function (18). To
show that the Lyapunov function candidate (22) is positive
definite, we first observe that the third term in (22) satisfies−γSin(e)T Mb (θ)ė ≥ −γλMax {Mb (θ)} ∥e∥ ∥ė∥ (23)Therefore, the Lyapunov function candidate (22) satisfies thefollowing inequality:
V (t, e, ė) ≥
[
]T [
][
]
1 ∥e∥
λmin {Kr }−γλMax {Mb (θ)} ∥e∥(24)−γλMax {Mb (θ)} λmin {Mb (θ)}∥ė∥
2 ∥ė∥and consequently, it happens to be positive definite andradially unbounded since by assumption, Kr is positive
definite and we also supported that it is chosen so as tosatisfy (21).
Using Property 2, the time derivative of the Lyapunov
function candidate (22) yields
V̇ (t, e, ė) = −ėT Br ė + γ ėT Cos(e)T Mb (θ)ė − γSin(e)T Kr e−γSin(e)T Br ė + γSin(e)T Cb (θ, θ̇)T ė
−ėT h (t, e, ė) − γSin(e)T h (t, e, ė) .

(25)

We now proceed to upper-bound V̇ (t, e, ė) by a negative
definite function in terms of the states e and ė. Using theproperties of the bi-articular manipulator dynamics, eachterm in Eq. (25) satisfy
2−ėT Br ė ≤ −λmin {Br } ∥ė∥(26)
2γ ėCos(e) Mb (θ)ė ≤ γλMax {Mb (θ)} ∥ė∥
T

(27)
2

−γSin(e) Kr e ≤ −γλmin {Kr } ∥Sin(e)∥T(28)−γSin(e) Br ė ≤ −γλmin {Br } ∥ė∥ ∥Sin(e)∥(29)√
2γSin(e)T Cb (θ, θ̇)T ė ≤ γ 3kC1 ∥ė∥{( )}+γkC1 sup θ̇d ∥ė∥ ∥Sin(e)∥T2−ėT h (t, e, ė) ≤ kh1 ∥ė∥ + kh2 ∥ė∥ ∥Sin(e)∥−γSin(e)T h (t, e, ė) ≤ γkh1 ∥ė∥ ∥Sin(e)∥2+γkh2 ∥Sin(e)∥ .

(30)
(31)
(32)

The bounds (26)–(32) yield that the time derivative V̇ (t, e, ė)in (25), satisfies[]T[]∥Sin(e)∥∥Sin(e)∥V̇ (t, e, ė) ≤ −γ
R(γ)
(33)
∥ė∥
∥ė∥
where the matrix R (γ) is defined as[
R (γ) =

λmin {Kr } − kh2−a − γ1 k2h2

−a − γ1 k2h2
1
γ {λmin {Br } − kh1 } − b].

According to the theorem of Sylvester, in order for the matrix
R (γ) to be positive definite it is necessary and sufficient thatthe component R11 and det{R (γ)} be strictly positive. With
respect to the first condition we stress that the matrix w.r.t.
elastic Kr must satisfy
λmin {Kr } ≥ kh2 .

(34)

On the other hand, the determinant of R (γ) is given bydet {R (γ)} =1[λmin {Kr } − kh2 ] [λmin {Br } − kh1 ]
γ
[
]2
1 kh2
− [λmin {Kr } − kh2 ] b − a +.
γ 2The latter must be strictly positive for which it is necessary
and sufficient that the matrix w.r.t. elastic Kr satisfies2

λmin {Kr } >{2γa + kh2 }+ kh2 (35)4γ [λmin {Br } − kh1 − γb]Joint Angle q1 [rad]2

Joint Angle q2 [rad]

2

0.6

1

0

1

2

3

y-axis [m]

0
4

0.4
t=2
0.2
t = 0.05

1

0

0
0

1

2
time [s]

3

-0.2

4

Joint Angle q1 [rad]Joint Angle q2 [rad]Fig. 5. Step response in the case of qd1 = 0, qd2 = π2 with a
open-loop (γ = 0.1815)

2

2

t=0
0

0.2
0.4
x-axis [m]0.6

Fig. 6. Trajectory of the arm in the case of qd1 = 0, qd2 = π2 witha open-loop (γ = 0.1815)

0.6

1

0

1

2

3

y-axis [m]

0
4

0.4
t=2
0.2
t = 0.05

1

0

0
0

1

2
time [s]

3

-0.2

4

Fig. 7. Step response in the case of qd1 = 0, qd2 = π2 with aclosed-loop[10]while it is sufficient that Br satisfies
λmin {Br } > kh1 + γb

t=0

(36)

for the light-hand side of the inequality (35) to be positive.
Observe that in this case the inequality (34) is trivially
implied by (35).
Notice that the inequalities (35) and (36) correspondprecisely to those in (20) and (21) as the tuning guidelines
for the controller. This means that R (γ) is positive definite
and therefore, V̇ (t, e, ė) is negative definite.
According to the arguments above, given a positive constant γ we may determine the matrices Kr and Br according
to Eqs. (20) and (21) in a way that the function V (t, e, ė)
given by (22) is positive definite while V̇ (t, e, ė) expressedas (33) is negative definite. For this reason, V (t, e, ė) isa strict Lyapunov function. According to Eqs. (22) and(33), the equilibrium point for the error dynamics (14) is
asymptotic stable.
Theorem 2 shows the stability for the error dynamics
combined by the bi-articular manipulator dynamics and the
proposed open-loop control law via the Lyapunov method.

0

0.2
0.4
x-axis [m]0.6

Fig. 8. Trajectory of the arm in the case of qd1 = 0, qd2 = π2 witha closed-loop[10]It is interesting to note that the joint angles θ and the joint
velocities θ̇ are exploited in the inner-loop only as depicted
in Fig. 4.
IV. S IMULATION R ESULTS
We show the simulation results in order to confirm the
proposed open-loop control law. The parameters of the biarticular manipulator dynamics used in the simulation are
m1 = 3.5[kg], m2 = 1.75[kg], l1 = 0.3[m], l2 = 0.3[m].
lg1 = 0.15[m], lg2 = 0.15[m], I˜1 = 0.026[kg·m2 ], I˜2 =0.013[kg·m2 ] and rp = 0.05[m]. The coefficients w.r.t.
viscosity are b1 = b2 = b3 = 400[Ns/m] and the coefficientsw.r.t. elastic are k1 = 3000[N/m], k2 = 2000[N/m] andk3 = 4000[N/m] for γ = 0.1815. We consider set-point
problems with the following initial values q1 (0) = 0[rad],q2 (0) = 0[rad], q̇1 = 0[rad/s] and q̇2 = 0[rad/s]. The
simulation results for the set-point problem are shown in
Figs. 5– 10. Figs. 5 and 6 depict the joint angles in thecase of qd1 = 0 and qd2 = π2 with the proposed open-loop
control law (13). For comparison, Figs. 7 and 8 depict thejoint angles in the case of qd1 = 0 and qd2 = π2 with the

0
0

1

2

3

0
1

2
time [s]

3

2

1
0

4

1

0

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

2

1

2

Joint Angle q2 [rad]Joint Angle q1 [rad]24

Fig. 9. Step response in the case of qd1 = 0, qd2 = π2 (γ = 0.1283)

previous control law[10]. In Figs. 5 and 7, solid lines are
the joint angles and dashed lines are desired ones. In Figs. 6
and 8, solid lines are the link and markers are the joints.
Although the step response with a open-loop is inferior to
the proposed closed-loop control law, these simulation results
suggest that 2DOF robot manipulators with antagonistic biarticular muscles could be controlled with the proposed openloop control law.
Here, we give a design procedure in order to assign thematrices Kr and Br in the following.
Step 1) The matrix Br satisfying the condition (20) is
chosen for a given γ.
Step 2) The matrix Kr satisfying the condition (21) isdecided.
Based on the design procedure, we design the matrices Br
and Kr for the following two given γ.
γ = 0.1283 :Kr = diag{1500, 1000, 2000}rp2 , Br = diag
{200, 200, 200}rp2 ;γ = 0.2567 :Kr = diag{6000, 4000, 8000}rp2 , Br = diag{2000, 2000, 2000}rp2 .
Figs. 9 and 10 depict the simulation results in the caseof γ = 0.1283 and γ = 0.2567, respectively. The rise timeis small for the smaller values of γ from Figs. 5, 9 and
10, while the time response is oscillatory. These simulations
show how the method can be used to adjust the rise time
and the overshoot for the proposed open-loop control system,
where the parameter γ together with the matrices Kr and Br
can be tuned for such purpose.
V. C ONCLUSIONSThis paper considers open-loop control for the 2DOF robot
manipulators with antagonistic bi-articular muscles. The stability analysis of the error dynamics has been discussed basedon the Lyapunov method. The simulation results show thevalidity of the proposed open-loop control law and the design
procedure for the matrices Kr and Br . This approach does
not deal with the uncertainties of the bi-articular manipulator
dynamics. In our future work, we will consider vision-based

Fig. 10.
0.2567)

0

1

2

3

4

0

1

2
time [s]

3

4

1
0

Step response in the case of qd1 = 0, qd2 = π2 (γ =control for the bi-articular manipulator dynamics in orderto overcome this drawback and verify the validity of the
proposed open-loop control law with experiments.
R EFERENCES[1] B. Siciliano and O. Khatib (Eds), Springer Handbook of Robotics,
Springer–Verlag, 2008.
[2] E. Guglielmelli, M. J. Johnson and T. Shibata, “Guest Editorial Special
Issue on Rehabilitation Robotics,” IEEE Trans. on Robotics, Vol. 25,
No. 3, pp. 477–480, 2009.
[3] M. Kuschel, M. Di Luca, M. Buss and R. L. Klatzky, “Combination
and Integration in the Perception of Visual-Haptic Compliance Information,” IEEE Trans. on Haptics, Vol. 3, No. 4, pp. 234–244, 2010.
[4] Q. Wang, N. Sharma, M. Johnson and W. E. Dixon, “Adaptive Inverse
Optimal Neuromuscular Electrical Stimulation,” Proc. 2010 IEEE
Multi-Conference on Systems and Control, pp. 1287–1292, 2010.
[5] M. Kumamoto, T. Oshima and T. Yamamoto, “Control Properties Induced by the Existence of Antagonistic Pairs of Bi-articular Muscles,”Human Movement Science, Vol. 13, No. 5, pp. 611–634, 1994.
[6] M. Kumamoto, et al., Revolution in Humanoid Robotics, Tokyo Denki
Univ. Press, 2006.
[7] M. Kumamoto, “Engineering Review of Biological Evolution of Motion Control,” Proc. of the Int. Symp. on Application of BiomechanicalControl Systems to Precision Engineering, pp. 29–34, 2010.
[8] S. Oh and Y. Hori, “Development of Two-Degree-of-Freedom Control
for Robot Manipulator with Biarticular Muscle Torque,” Proc. of the2009 American Control Conference, pp. 325–330, 2009.
[9] V. Salvucci, Y. Kimura, S. Oh and Y. Hori, “Experimental Verificationof Infinity Norm Approach for Force Maximization of Manipulators
Driven by Bi-articular Actuators,” Proc. of the 2011 American Control
Conference, pp. 4105–4110, 2011.
[10] H. Kawai, T. Murao, R. Sato and M. Fujita, “Passivity-based Control
for 2DOF Robot Manipulators with Antagonistic Bi-articular Muscles,” Proc. of the 2011 IEEE Multi-Conference on Systems andControl, pp. 1451–1456, 2011.
[11] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control, John
Wiley & Sons, 1989.
[12] B. Brogliato, R. Lozano, B. Maschke and O Egland, Dissipative
Systems Analysis and Control: Theory and Applications (2nd Ed.),
Springer, 2007.
[13] R. Kelly, V. Santibanez and A. Loria, Control of Robot Manipulators
in Joint Space, Springer, 2005.
[14] S. Arimoto, “Fundamental Problems of Robot Control: part I, Innovations in the Realm of Robot Servo-loopes,” Robotica, Vol. 13, pp. 19–
27, 1995.