Download Modeling and Control of a Pair of Robot Fingers with Saddle Joint

2009 IEEE International Conference on Robotics and Automation
Kobe International Conference Center
Kobe, Japan, May 12-17, 2009
Modeling and Control of a Pair of Robot Fingers
with Saddle Joint under Orderless Actuations
Morio Yoshida, Suguru Arimoto, and Kenji Tahara
z
Saddle Joint
O
Abstract— A new robot hand dynamics model with rolling
constraints and with a saddle joint at one finger is proposed,
where two saddle-joint actuations are considered to be orderless. Spinning motion around the opposition axis connecting two
center points of each finger-tip contact area with an object is
faithfully treated, and a viscosity model for damping rotational
motion of the object is proposed. A class of control signals
without referring to object kinematics or using external sensing
is proposed. Finally, numerical simulation results show the
stability of motion of the overall closed-loop dynamics supplied
with the proposed control input.
y
Rolling Contacts
I. I NTRODUCTION
The whole stimuli to the human brain, which arise from
making and using tools by hands, have accelerated the
evolutionary speed of the human brain. The primatologist
Napier [1] claims that most of important movements of
the hands are based upon finger-thumb opposition shown
in Fig.1, that has contributed to the progress of humanity.
Despite such a profound proposition, there is a dearth of
robotic researches paying much attention to consideration of
control functions of multi-fingered hands.
For the sake of raveling mysteries of the human hand and
realizing its dexterous movements, many robotic researchers
devote a lot of their elaborate efforts to the research of robot
hands [2] [3] [4]. However, most of the researchers were
attracted by kinematics and planning of motions realizing
force/torque closure for stable grasp by robot fingers in a
static sense. On the other hand, rolling geometry between
two arbitrarily-shaped objects is rigorously discussed [5] [6].
However, all the researches have remained in a kinematic or
semi-dynamic meaning. In the field of multibody dynamics
[7], many models with constraints in 3-D space are presented
without modeling physically faithful interaction between a
robot finger and an object surface through 3-D rolling.
An explicit 3-D dynamical pinching model with rolling
constraints between a finger end and an object surface has
been missing.
Around the year of 2000, Arimoto et al. first showed that
2-D object pinching is realized by using a pair of robot
fingers with hemispherical ends, succeeding at embedding
This work was partially supported by Japan Society for the Promotion
of Science (JSPS), Grant-in-Aid for Scientific Research (B) (20360117).
M. Yoshida, S. Arimoto, and K. Tahara are with the RIKEN-TRICollaboration Center for Human-Interactive Robot Research, RIKEN,
Nagoya, Aichi, 463-0003, Japan [email protected]
S. Arimoto is with the Department of Robotics, Ritsumeikan University,
Kusatsu, Shiga, 525-8577, Japan [email protected]
K. Tahara is with the Organization for the Promotion of
Advanced Research, Kyushu University, Fukuoka, 819-0395, Japan
[email protected]
978-1-4244-2789-5/09/$25.00 ©2009 IEEE
x
Fig. 1.
Human pinch motion based on fingers-thumb opposition
z
z
O
O
q21z
,
x
q11
q21x
x
Saddle Joint
y
q12
Gravity
direction
r21z
O01
Oc.m.
q13
y
Z
O2 X
O1
q22
O02
Y
Fig. 2. Two robot fingers with soft tips pinching an object with parallel
flat surfaces under gravity
z
x
O'
y
Fig. 3.
Modeling of a saddle joint in human thumb [5]
rolling constraints into the overall fingers-object dynamics
[8] [9]. The degrees-of-freedom redundancy problem of the
overall fingers-object dynamics for desired tasks is overcome
by proposing a stability concept called “Stability on a
2499
manifold”. In the year of 2006 [10], modeling of motion
of pinching an object was extended from two-dimensional
to three-dimensional pinching. In the modeling of pinching
it is assumed that spinning motion of the object around the
opposition axis connecting two contact points between the
left or right finger end and the object surface has ceased
and will not arise any more due to dry friction and microdeformations near the two contact points. This assumption
leads to a five-variables model of the object dynamics. In
the year of 2008 [11] [12] [13], instead of this assumption,
the modeling problem was posed as a more faithful model
considering that spinning motion of the object is possible to
arise but viscosity friction is exerted on x axis in the frame
coordinates O − xyz (see Figure 2), which means the overall
fingers-object dynamics as a full variables model is derived.
Furthermore, in the previous paper [12], two saddle-joint
motors of the robot finger are actuated in sequence. However,
the saddle joint of the human thumb must be actuated around
the two axes respectively without making any order [14] (see
Figure 3).
In this paper, we propose a saddle joint model with
the orderless relation like the human thumb, and a robot
hand model with the saddle joint through taking account
of rolling constraints that are Pfaffian. A pair of robot
fingers with hemispherical tips made from soft material
is treated under the gravity effect. Simple control signals
based on the opposability without using object kinematics
or external sensing are shown, which are the same as the
previously-proposed control signals [11] [12] [13]. Finally,
for the sake of confirmation of the stability analysis, we carry
out numerical simulations based on the proposed model of
object-fingers dynamics and control signals.
Since the first joint J21 of the robot finger shown in Fig.4
has orderless saddle-joint actuations, the Denavit-Hartenberg
Notation is not applied to derive a geometrical relation of
robot kinematics. In this section, a kinematic geometry of
the robot finger with orderless actuations, the structure of
which is similar to a saddle-joint model of the human thumb
[14](see Figure 3), is derived. The coordinate system shown
Fig.4 is right-handed. The first joint with center O depicted
in Fig.3 is a saddle joint exerted by two actuators, one of
which rotates around x-axis and another around z- axis,
q̇21x denotes the angular velocity around x-axis and q̇21z
around z-axis respectively, where q̇ 21 = (q̇21x , 0, q̇21z )T
denotes the rotational vector of the first link with the saddle
joint in terms of frame coordinates O − xyz (see Figure
4). We set the Cartesian coordinates Oc.m.21 − X21 Y21 Z21
fixed at the first link frame of the robot hand, and denote
three orthogonal unit vectors at the first link frame in
each corresponding direction X21 ,Y21 , and Z21 by rX21 =
(rXx21 , rXy21 , rXz21 )T , rY 21 = (rY x21 , rY y21 , rY z21 )T ,
and r Z21 = (rZx21 , rZy21 , rZz21 )T as shown in Fig.4. We
derive the mass center Oc.m.21 of the right robot finger by
xm21 = (xm21 , ym21 , zm21 )T = s21 r Y 21 + (L, 0, 0)T based
s21
l 21
s22
l 22
rZ22
l22-s22
rY22
Y22
Z 22(=Z21)
Z21
y
q22
rX22
O02
z
x
.
q21x
rX21
rY21
Y21
Oc.m.22
.
q21z
Z 21
rZ21
l21-s21
X 21
Oc.m.21
J22
z
.
q22z
x
.
q21x
X 22
.
q22y
y
Fig. 4. Modeling of the right robot finger with a saddle joint which has
orderless actuations
on the frame coordinates O − xyz (see Figure 2). It is well
known that the 3 × 3 rotation matrix
R21 (t) = (rX21 , rY 21 , rZ21 ) ∈ SO(3)
(1)
which is subject to the first-order differential equation representing infinitesimal rotation as follows:
d
R21 (t) = R21 (t)Ω21 (t)
dt
where
II. G EOMETRY OF ROBOT F INGER WITH O RDERLESS
ACTUATIONS
O'
J21
⎛
0
Ω21 (t) = ⎝ q̇21z
0
−q̇21z
0
q̇21x
⎞
0
−q̇21x ⎠
0
(2)
(3)
The equations expressing the mass center Oc.m.22 of the
second link are derived by considering the relationship
between the first link’s posture and the angle q22 which
rotates around Z21 as follows:
r X22 = cos q22 rX21 + sin q22 r Y 21
rY 22 = − sin q22 rX21 + cos q22 rY 21
(4)
(5)
rZ22 = r Z21
(6)
The angular velocity q̇ 22 = (q̇22x , q̇22y , q̇22z )T of the
second link, which rotates around Z21 , can be represented by
q̇ 22 = q̇22 rZ21 +q22 ṙZ21 . Therefore the mass center Oc.m.22
defined by xm22 = (xm22 , ym22 , zm22 )T can be represented
by xm22 = l21 rY 21 + s22 rY 22 .
III. DYNAMICS
The model of pinching a rigid object with parallel flat
surfaces by two robot fingers with 3 DOFs and 3 DOFs is
schematically shown in Fig.2. The left finger (finger i = 1)
is planar with three actuators rotational around z-axis. The
2500
1
Since each contact point Oi can be expressed by the coordinates ((−1)i li , Yi , Zi ) based on the object frame Oc.m. XY Z, taking an inner product between Equation (11) and
r Y gives rise to
viscosity direction
against spinning motion
Z, rZ
l1
X
r1
O01
rX
Y
rY
Y1
rX
Yi = (x0i − x)T rY , i = 1, 2
O c.m.
Similarly, it follows that
rX
Zi = (x0i − x)T rZ , i = 1, 2
Z1
O 2 contact
point
O 1 contact
point
Fig. 5. rX , rY , and rZ vectors represent rotational movement of an object.
Viscosity works around rX vector at a contact point between a finger tip
and an object surface
vector q 11 = (q11 , q12 , q13 )T denotes the joint angles of the
left hand side finger. In the previous paper [10], it is also
assumed that spinning around the opposition axis is possible
to arise but viscosity damps rotational motion of the object
around x-axis, that is, about ωx , where ω = (ωx , ωy , ωz )T
denotes the vector of rigid body rotation in terms of frame
coordinates O-xyz (see Figure 2). This viscosity model is
effective but rough in a physical sense. Instead of it, we
introduce a more faithful viscosity model which is exerted
on rotational motion of the object around the rX vector at
two contact points between the left or right finger tip and
the object surface as shown in Fig.5.
At the same time, we introduce the cartesian coordinates
Oc.m. -XY Z fixed at the object frame and denote the three
orthogonal unit vectors at the object frame in each corresponding direction X, Y , and Z by rX = (rXx , rXy , rXz )T ,
r Y = (rY x , rY y , rY z )T , and r Z = (rZx , rZy , rZz )T as
shown in Fig.2. Next, let us denote the cartesian coordinates
of the object mass center Oc.m. by x = (x, y, z)T based
on the frame coordinates O-xyz and note that three mutually orthogonal unit vectors fixed at the object frame may
rotate dependently on the angular velocity vector ω of body
rotation. Then, it is well known that the 3 × 3 rotation matrix
R(t) = (r X , rY , rZ ) ∈ SO(3)
(7)
is subject to the first-order differential equation
d
R(t) = R(t)Ω(t)
dt
where
⎛
0
Ω(t) = ⎝ ωz
−ωy
−ωz
0
ωx
⎞
ωy
−ωx ⎠
0
(8)
(9)
Next, denote the position of the center of each hemispherical finger-end by x0i = (x0i , y0i , z0i )T . Then, it is possible
to notice that (see Fig.5)
xi = x0i −(−1)i (ri −Δxi )r X
(12)
(10)
x = x0i −(−1)i (ri −Δxi + li )r X −Yi rY −Zi rZ (11)
(13)
A rolling constraint between one finger-end and its contacted object surface can be expressed by the equality of
two contact point velocities expressed on either of fingerend spheres and its corresponding tangent plane (that is
coincident with one of object flat surfaces) as follows [10]:
(r1 − Δx1 ) {ωz − rZz (q̇11 + q̇12 + q̇13 )} = Ẏ1
(14)
(r1 − Δx1 ) {−ωy + rY z (q̇11 + q̇12 + q̇13 )} = Ż1
(r2 −Δx2
) −ωz+rZz q̇21z +rZx q̇21x +rT
Z21 r Zq̇22
=Ẏ2 (15)
(r2 −Δx2 ) ωy −rY z q̇21z −rY x q̇21x −rT
r
q̇
Z21 Y 22 =Ż2
The rolling constraint conditions expressed through Equations (14) and (15) are non-holonomic but linear and homogeneous with respect to velocity variables. Hence, Equations
(14) and (15) can be treated as Pfaffian constraints [2] that
can be expressed with accompaning Lagrange’s multipliers
{λY 1 , λZ1 } for Equation (14), and {λY 2 , λZ2 } for Equations
(15) in such forms as
⎧
T
T
⎪
ẋ+Y
Y
q
+Y
ϕ̇+Y
ψ̇+Y
θ̇
=0
λ
⎪
Y
i
ϕi
ψi
θi
q
i
i
x
i
⎨
T
T
(16)
λZi Z q i q i +Z xi ẋ+Zϕi ϕ̇+Zψi ψ̇+Zθi θ̇ = 0
⎪
⎪
⎩
i = 1, 2
where
⎧
i
i
⎪
Y q i = ∂∂Y
⎪
q i − (ri − Δxi ) (−1) rZz eiz + rZx eix
⎪
⎨ T
+rZ21 rZ eiZ21
∂Y
∂Y
∂Y
⎪
Y
xi = ∂ xi , Yϕi = ∂ϕi , Yψi = ∂ψi
⎪
⎪
⎩
i
i
Yθi = ∂Y
∂θ + (−1) (ri − Δxi ), i = 1, 2
and
⎧
i
Z q i = ∂∂Z
− (ri − Δxi ) (−1)i rY z eiz + rY x eix
⎪
⎪
q
i
⎪
⎨ T
+rZ21 r Y ei2
∂Zi
i
Z xi = ∂Z
⎪
⎪
∂ x , Zϕi = ∂ϕ ,
⎪
⎩Z = ∂Zi − (−1)i (r − Δx ), Z = ∂Zi , i = 1, 2
ψi
i
i
θi
∂ψ
∂θ
and q 1 = (q11 , q12 , q13 )T , q 2 = (q21x , q21z , q22 )T , e1z =
(1, 1, 1)T , e2z = (0, 1, 0)T , e1x = 03 , e2x = (1, 0, 0)T ,
e12 = 03 , and e22 = (0, 0, 1). To simplify notations, we
rewrite Equation (16) into
λY i Y T
i (dX/dt) = 0
, i = 1, 2
(17)
T
λZi Z i (dX/dt) = 0
T
T
T
where X = (q T
1 , q 2 , x , ϕ, φ, θ) ,
⎧
T
⎪
T
⎨ Y1= YT
q 1 , 02 , Y x1 , Yϕ1 , Yφ1 , Yθ1
T
⎪
T
⎩ Y 2 = 0, Y T
q 2 , Y x2 , Yϕ2 , Yφ2 , Yθ2
2501
(18)
and
⎧
T
⎪
T
⎨ Z1 = ZT
q 1 , 02 , Z x1 , Zϕ1 , Zφ1 , Zθ1
T
⎪
T
⎩ Z 2 = 0, Z T
q 2 , Z x2 , Zϕ2 , Zφ2 , Zθ2
(19)
The reproducing force due to finger-tip deformation can
be described as (see [13])
fi (Δxi , Δẋi ) = f¯i (Δxi ) + ξi (Δxi )Δẋi
(20)
where H stands for the constant inertia matrix of the object
that must be evaluated on the basis of fixed body coordinates
Oc.m. -XY Z, that is,
⎛
⎞
IXX IXY IXZ
H = ⎝ IY X IY Y IY Z ⎠
(28)
IZX IZY IZZ
Similarly, H2i (i = 1, 2) is given in the following:
T
(t)
H2i = R2i (t)H̄2i R2i
where
f¯i (Δxi ) = ki Δx2i , i = 1, 2
(21)
with stiffness parameter ki > 0[N/m2 ] and ξi (Δxi ) is a
positive scalar function of Δxi .
The viscosities around rX at two contact points due to
spinning motion around the opposing axis is represented by
the following Rayleigh’s dissipation functions:
ct1
T
{ω − (q̇11 + q̇12 + q̇13 )ez } rX 2
(22)
Rt1 =
2
ct2
T
Rt2 =
(ω − q̇21z ez − q̇21x ex − q̇ 22 ) r X 2 (23)
2
Rt = Rt1 + Rt2
(24)
where Rt1 expresses the Rayleigh’s dissipation function at
the left contact point, Rt2 at the right contact point, both
ct1 and ct2 positive constant values, ex = (1, 0, 0)T , and
ez = (0, 0, 1)T . The function Rt is derived from taking
into considerations that the difference between the rotational
velocity of motion of the robot fingers and the rotational
velocity of motion of the object relative to the r X vector
generates viscosity, and no viscosity arises in the case of
equal velocities between the velocity of each robot finger
and the velocity of the object relative to the r X vector.
The Lagrangian for the overall fingers-object system can
be expressed by the scalar quantity L = K − P , where K
denotes the total kinetic energy expressed as
1
1 T
1
H1 q̇ 1 + m21 ẋT
K = q̇ T
m21 ẋm21 + q̇ 21 H̄21 q̇ 21
2 1
2
2
1
+ m22 ẋT
m22 ẋm22
2
1
T
+ (q̇ 21 + q̇ 22 ) H̄22 (q̇ 21 + q̇ 22 )
2
1
1 + M ẋ2 + ẏ 2 + ż 2 + ω T H0 ω
(25)
2
2
and P denotes the total potential energy expressed as
P = P1 + P2 − M gy +
PΔxi
(26)
where H̄21 similarly stands for the constant inertia matrix
of the first link of the robot finger 2 and H̄22 the constant
inertia matrix of its second link, that must be expressed on
the basis of fixed robot link coordinates Oc.m.2i -X2i Y2i Z2i
respectively, that is,
⎞
⎛
IXX2i IXY 2i IXZ2i
H̄2i = ⎝ IY X2i IY Y 2i IY Z2i ⎠ , i = 1, 2
(30)
IZX2i IZY 2i IZZ2i
where
R22 (t) = (rX22 , rY 22 , rZ22 ) ∈ SO(3)
H0 = R(t)HRT (t)
(27)
(31)
Thus, owing to the variational principle applied to the form
t1
t1 −
uT
δLdt =
i δq i
t0
t0
i=1,2
T
δX
dt
− λY i Y T
+
λ
Z
Zi
i
i
t1
∂Rt
∂Δẋi
−
δX
+
ξ
(Δx
)Δ
ẋ
δX
dt
i
i
i
T
T
t0 ∂ Ẋ
∂ Ẋ
i=1,2
(32)
we obtain a set of Lagrange’s equations of motion of the
overall system:
1
∂Rt
T
Ḣi + Si q̇ i +
Hi q̈ i +
− (−1)i fi J0i
rX
2
∂ q̇ i
−λY i Y q i − λZi Z q i + g i = ui ,
i = 1, 2
(33)
M ẍ − (f1 − f2 )r X + (λY 1 + λY 2 )r Y
⎛ ⎞
0
+(λZ1 + λZ2 )r Z − M g ⎝ 1 ⎠ = 0
0
⎛
⎞
0
1
∂Rt
Ḣ0 + S ω+
+ f1 ⎝ −Z1 ⎠
H0 ω̇+
2
∂ω
Y1
⎛
⎞
⎛
⎞
⎛
⎞
0
Z1
Z2
+f2 ⎝ Z2 ⎠ − λY 1 ⎝ 0 ⎠ − λY 2 ⎝ 0 ⎠
−Y2
l1
−l2
⎞
⎞
⎛
⎛
−Y1
−Y2
−λZ1 ⎝ −l1 ⎠ − λZ2 ⎝ l2 ⎠ = 0
0
0
i=1,2
where H1 stands for the inertia matrix for finger 1, m21
the mass of the first link of the finger 2, m22 the mass
of the second link of it, M the mass of the object, Pi
the potential
Δx energy of finger i, g the gravity constant, and
PΔxi (= 0 i f¯i (ξ)dξ) the potential energy of reproducing
force for finger i, and H0 is given in the following:
(29)
(34)
(35)
where calculation of partial differentiations of Yi , Zi in ϕ,
ψ, θ is presented in Table I. Further, it is possible to see
2502
TABLE I
−1
N̂1z = γN
1z
PARTIAL DERIVATIVES OF CONSTRAINTS IN (ϕ, ψ, θ).
(37)
Finally, we interestingly notice six wrench vectors composed of accompaniments to f1 , f2 , λY 1 , λY 2 , λZ1 , and λZ2
in Equations (34) and (35). The six wrench vectors and the
external force composed of the last term in Equation (34)
work on the 3-dimensional object.
IV. C ONTROL SIGNALS
A control input realizing stable grasping under the gravity
effect without using object kinematics or external sensing,
which is called “blind grasping”, is proposed as follows:
fd
J T (x01 − x02 )
r1 + r2 0i
M̂ g ∂y0i
−ri N̂iz eiz
2 ∂q i
−ri N̂ix eix −ri N̂i2 ei2 , i = 1, 2
M̂ = M̂ (0) +
0
t
Differently from the case of rigid finger-ends [11], f0 is not
a constant but dependent on the magnitude of Δx1 + Δx2 .
Nevertheless, it is possible to find Δxdi (i = 1, 2) for a given
fd > 0 so that they satisfy
l1 +l2 −Δxd1 −Δxd2
¯
fd , i = 1,2 (45)
fi (Δxdi ) = 1 +
r1 + r2
because f¯i (Δx) is of the form of f¯i (Δx) = ki Δx2 (Equation
(21)). Substituting this control signals (Equation (39)) into
Equations (33), (34), and (35) yields
1
∂Rt
T
Ḣi + Si + Ci q̇ i +
− (−1)i J0i
Δfi r X
Hi q̈ i +
2
∂ q̇ i
ΔM g ∂y0i
−ΔλY iY q i −ΔλZi Z q i +
+ri ΔNizeiz
2
∂q i
+ri ΔNix eix +ri ΔNi2 ei2 +g = 0, i = 1, 2
(46)
M ẍ − (Δf1 − Δf2 )r X + (ΔλY 1 + ΔλY 2 )rY
+(ΔλZ1 + ΔλZ2 )r Z = 0
(39)
T
−1 gγM
∂y0i
q i dτ
2 i=1,2 ∂q i
−1
gγM
(y01 (t) + y02 (t)
2
−y01 (0) − y02 (0))
(43)
and γM , γN iz (i = 1, 2), γN 2x , and γN 22 are positive constants. The form is nothing but the control signal proposed
in the rigid contact case [11] [13]. The third term of the
right hand side of Equation (41) is a signal based upon the
opposable force between O01 and O02 (not between O1 and
O2 , because positions of O1 and O2 can not be measured).
The fourth term stands for compensation for the object mass
based upon its estimator. The fifth term is introduced for
saving excess movements of finger joints from the initial
pose. Next define
l1 + l2 − Δx1 − Δx2
(44)
f0 = fd 1 +
r1 + r2
−
where
(42)
0
From taking inner products between q̇ i and Equation (33),
ẋ and Equation (34), and ω and Equation (35), we obtain
the relation
d
q̇ T
(K + P ) + 2Rt +
ξ(Δxi )Δẋ2i (38)
i ui =
dt
i=1,2
i=1,2
ui = −Ci q̇ i + (−1)i
T r1 e1z q̇ 1 dτ
−1
= γN
22 r2 (q22 (t) − q22 (0))
from PΔxi (i = 1, 2) and Equations (12) and (13) that
⎧ ∂(PΔx1 +PΔx2 )
= −(−1)i fi JiT rX , i = 1, 2
⎪
∂qi
⎪
⎪
⎨ ∂(PΔx1 +PΔx2 )
= − (f1 − f2 ) r X
∂x
(36)
∂(PΔx1 +PΔx2 )
+PΔx2 )
⎪
= 0, ∂(PΔx1∂ψ
=−f1 Z1 +f2 Z2
⎪
⎪
⎩ ∂(PΔx1∂ϕ
+PΔx2 )
= f1 Y1 − f2 Y2
∂θ
Y xi = ∂Yi /∂x = −rY
Z xi = ∂Zi /∂x = −rZ
0
t
−1
T
= rN
1z ri e1z (q 1 (t) − q 1 (0))
t
T −1
r2 e2z q̇ 2 dτ
N̂2z = γN
2z
0
t
−1
N̂2x = γN
(ri q̇21x ) dτ
02
0
t
−1
N̂22 = γN
(r2 q̇22 ) dτ
22
∂PΔx1
Δx2
= ∂P∂ϕ
=0
∂ϕ
∂Yi
T ∂rY = (x − x)T r = Z
=
(x
−
x)
0i
0i
i
Z
∂ϕ
∂ϕ
∂rY
∂Yi
T
= (x0i − x) ∂ψ = 0
∂ψ
∂Yi
Y
= (x0i − x)T ∂r
= −(x0i −x)TxX = −(−1)i (ri −Δxi +li )
∂θ
∂θ
∂rZ
∂Zi
T
= (x0i − x) ∂ϕ = (x0i − x)T rY = −Yi
∂ϕ
∂Zi
Z
= (x0i − x)T ∂r
= −(x0i −x)TrX = (−1)i(ri −Δxi +li )
∂ψ
∂ψ
∂Zi
Z
= (x0i − x)T ∂r
=0
∂θ
∂θ
∂Yi
i
Yϕi = ∂ϕ = Zi , Zϕi = ∂Z
= −Yi
∂ϕ
∂Yi
∂Zi
Yψi = ∂ψ = 0, Zψi = ∂ψ − (−1)i (ri − Δxi ) = (−1)i li
i
Yθi = ∂Y
+ (−1)i (ri − Δxi ) = −(−1)i li , Zθi = 0
∂θ
= M̂ (0) +
(40)
(41)
2503
(47)
⎛
⎞
0
1
∂Rt
+ Δf1 ⎝ −Z1 ⎠
H0 ω̇ + Ḣ0 + S ω +
2
∂ω
Y1
⎛
⎞
⎛
⎞
⎛
⎞
0
Z1
Z2
+Δf2 ⎝ Z2 ⎠ − ΔλY 1 ⎝ 0 ⎠ − ΔλY 2 ⎝ 0 ⎠
−Y2
l1
−l2
⎞
⎞ ⎛
⎛
⎛
⎞
Sx
−Y1
−Y2
−ΔλZ1 ⎝ −l1 ⎠ −ΔλZ2 ⎝ l2 ⎠ − ⎝ Sy ⎠ = 0 (48)
0
0
Sz
TABLE II
where
P HYSICAL PARAMETERS OF THE FINGERS AND OBJECT.
Mg
rXy
(49)
Δfi = f¯i − f0 − (−1)i
2
fd
Mg
rY y (50)
ΔλY i = λY i + (−1)i
(Y1 − Y2 ) −
r1 + r2
2
fd
Mg
rZy (51)
(Z1 − Z2 ) −
ΔλZi = λZi + (−1)i
r1 + r2
2
ΔM = M̂ − M, ΔNiz = N̂iz − Niz , i = 1, 2
ΔN2x = N̂2x − Nx2 , ΔN22 = N̂22 − N22
fd (ri − Δxi )
(Y1 − Y2 )rZz
r1 + r2
ri
−(Z1 − Z2 )rY z
i (ri − Δxi ) M g
−(−1)
rY y rZz
r
2
i
−rZy rY z , i = 1, 2
fd (r2 − Δx2 )
N2x =
(Y1 − Y2 )rZx
r1 + r2
r2
−(Z1 − Z2 )rY x
(r2 − Δxi ) M g
−
rY y rZx
r2
2
−rZy rY x
fd (r2 − Δx2 )
(Y1 − Y2 )r T
N22 =
Z21 r Z
r1 + r2
r2
−(Z1 − Z2 )rT
Z21 r Y
(r2 − Δxi ) M g T
−
r Z21 r Z rY y
r2
2
−r T
Z21 r Y rZy
l11 = l12 = l22
l13
l21
m11
m12
m13
m21
m22
r0
ri (i = 1, 2)
L
M
li (i = 1, 2)
h
ki (i = 1, 2)
cΔi (i = 1, 2)
ct1 = ct2
(52)
Niz =
length
length
length
weight
weight
weight
weight
weight
link radius
radius
base length
object weight
object width
object height
stiffness
viscosity
viscosity
0.040 [m]
0.030 [m]
0.060 [m]
0.043 [kg]
0.031 [kg]
0.020 [kg]
0.060 [kg]
0.040 [kg]
0.005 [m]
0.010 [m]
0.063 [m]
0.040 [kg]
0.015 [m]
0.050 [m]
3.000 × 105 [N/m2 ]
1000.0[Ns/m2 ]
5.0 × 10−4 [Nms]
TABLE III
PARAMETERS OF CONTROL SIGNALS .
fd
c
γM
γNiz (i = 1, 2)
γN2x
γN22
M̂ (0)
N̂iz (0)(i = 1, 2)
N̂2x (0)
N̂22 (0)
q21x (0) = q21z (0)
(53)
internal force
damping coefficient
regressor gain
regressor gain
regressor gain
regressor gain
initial estimate value
initial estimate value
initial estimate value
initial estimate value
initial virtual angle
1.000 [N]
0.001 [Nms]
0.050 [m2 /kgs2 ]
5.000 × 10−4 [s2 /kg]
5.000 × 10−4 [s2 /kg]
5.000 × 10−4 [s2 /kg]
0.020 [kg]
0.000 [N]
0.000 [N]
0.000 [N]
0.000[deg]
(54)
(55)
Mg
{(Z1 + Z2 )rY y − (Y1 + Y2 )rZy }
(56)
2
fd
Sy =
(r1 − Δx1 + r2 − Δx2 )(Z1 − Z2 )
r1 + r2
Mg
{rXy (Z1 + Z2 ) + rZy (l1 − l2 )}
−
(57)
2
fd
Sz = −
(r1 − Δx1 + r2 − Δx2 )(Y1 − Y2 )
r1 + r2
Mg
{rXy (Y1 + Y2 ) + rY y (l1 − l2 )}
+
(58)
2
V. N UMERICAL SIMULATION RESULTS AND I NITIAL
Sx =
and δq21z have a meaning as a virtual displacement but
quantity of δq21x and δq21z do not have physical meanings
of rotational angels. Hence we set the virtual angles q21x (0)
and q21z (0) to zero respectively as shown Table III and the
virtual angles do not directly affect numerical simulations
at all. The results of numerical simulations, representing key
variables of fingers-object system, converge to some constant
values as shown in Fig.7. It is confirmed that spinning motion
arises but eventually it ceases after 10 seconds, that is,
ω → 0 as t → ∞ seen through Figs.6 and 7. On the
other hand, if we set the damping parameters ct1 and ct2 to
zero respectively, we can see that spinning motion continues
eternally. Therefore it is confirmed that motion of the overall
fingers-object system is stabilized from the viewpoint of
simulations.
VALUES
We carry out numerical simulations based on the physical
parameters of the fingers-object system given in Table II and
the parameters of control gains given in Table III in order
to confirm the stability of motion of the overall dynamics
applied by the proposed control input. It is noted that δq21x
2504
(a) Initial pose
Fig. 6.
(b) After 10 seconds
Motions of pinching a 3-D object under the gravity effect
2
2
2
f1[N]
f2[N]
λ Y 1[N]
1
1
f1
0
0
5
1
f2
0
10
time[s]
0
5
0
10
time[s]
(a) f1
0
(b) f2
λY 2
λZ1
1
0
5
0
10
time[s]
0
5
1
q12[rad/s]
1
0.5
.
0.03
<
5
.
q11
The value of zero
10
time[s]
0
5
(g) M̂
0
1
q21z[rad/s]
.
.
q13
The value of zero
0
5
.
.
q21x
The value of zero
-0.5
10
time[s]
0.5
0
0
(j) q̇13
5
0
(k) q̇21x
ω y[rad/s]
ω x[rad/s]
1
0.5
0
.
q22
The value of zero
5
time[s]
(m) q̇22
0
ωx
T he value of zero
-0.5
0
10
10
(l) q̇21z
0.5
0
5
time[s]
1.5
1
1
0.5
0
.
q21z
The value of zero
10
time[s]
1.5
1.5
-0.5
0
-0.5
5
10
time[s]
(n) ωx
0.003
ωy
T he value of zero
-0.5
0
5
10
time[s]
(o) ωy
0.003
1.5
1
ωz
The value of zero
-0.5
0
5
time[s]
(p) ωz
Fig. 7.
10
Δ x2[m]
0
0.002
Δ x1[m]
ω z[rad/s]
0.002
0.5
0.001
0.001
Δx1
0
0
5
time[s]
Δx2
0
10
(q) Δx1
0
5
time[s]
R EFERENCES
10
(i) q̇12
0.5
0
5
time[s]
1.5
1
q21x[rad/s]
1
q13[rad/s]
-0.5
1.5
0.5
q22[rad/s]
0
(h) q̇11
1.5
VII. ACKNOWLEDGMENTS
This work was partially supported by Japan Society for
the Promotion of Science (JSPS), Grant-in-Aid for Scientific
Research (B) (20360117).
10
time[s]
ferently from the previous paper [10], the overall fingersobject dynamics are derived as a full-variables model. It is
shown through computer simulations that spinning motion
between finger tips and the object arises but eventually stops
by virtue of the incorporated viscosity model exerted on
rotational motion of the object around rX vector at two
contact points between a left or right finger tip and the object
surface. Numerical simulation results showed the stability
of motion of the overall fingers-object dynamics to which
the proposed control input is applied. However, to confirm
the effectiveness of our proposed control input in a sound
meaning, the analysis of its stability is indispensable and
should be carried out from the mathematically reasonable
viewpoint. Finally it is shown that the proposed viscosity
model plays a crucial role in pinching a 3-D object.
.
q12
The value of zero
0.5
.
0
-0.5
M
0.02
10
(f) λZ2
1.5
q11[rad/s]
M[kg]
0.04
-0.5
5
time[s]
(e) λZ1
<
.
0
1.5
0.05
.
10
time[s]
(d) λY 2
0
λZ2
λ Z2[N]
1
0
10
2
λ Z1[N]
1
5
time[s]
(c) λY 1
2
λY 2[N]
2
0
λY 1
10
(r) Δx2
Transient responses of physical variables
VI. C ONCLUSION
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A saddle joint model with orderless actuations and rolling
constraints have been treated in a faithful manner and incorporated into the overall fingers-object system. It is shown
that 3-D pinching by two robot fingers, one is a 3-DOFs
planar finger (resembles the index finger) and the other is a
3-DOFs finger with a saddle joint (the thumb), is performed
numerically by using the derived computational model. Dif-
2505