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Development Rotating Workspace Observer based on Novel Kinematicsfor Two-link Manipulator with Biarticular Actuator Coordination and
its Application for Translational Motionpost AMC12 paper
Sehoon Oh, Yasuto Kimura, and Yoichi Hori
Abstract— This paper proposes totally novel kinematics,statics and dynamics of a two-link manipulator and developsa control algorithm based on them which enables two-linkmanipulators behave more like human or animals.
There are two main points: introduction of biarticularmuscle/actuator coordination and utilization of the rotatingreference frame instead of the fixed frame. These two points arealso related to the biomechanics and natural dynamics; humansand animals have the biarticular muscles playing importantroles in their motions, and the motion of humans’ extremitiesare usually modeled using the rotating reference frame. Thepaper reveals that these two points are also important inanalysis and control of a two link manipulator, particularlywhen the two link manipulator needs to behave like a human’sextremity.
The motions of a two-link manipulator are re-describedbased on these points, which provides simple and straightfor-ward understanding of the motions of an endeffector. Based onthis understanding, a rotating workspace observer is designedand found to be theoretically robust to the singular pointsince it does not have any function of the joint angles inthe denominators of the observer design. Using the rotatingworkspace observer, Two-degree-of-freedom control is proposedin the proposed reference frame.
Comparative simulations verifies the effectiveness of theproposed controller and the proposed reference frame.
I. I NTRODUCTION
Robots have the similar structure to the human/animals’structure such as two-link or multi-link manipulators. Eventhough there are a number of research to mimic the human’smotion using the structure, we still have difficulty controllingthe robots to conduct the same motion as humans.
This paper proposes that two points can address this diffi-culty: the introduction of the biarticular actuator coordinationand the utilization of the rotating reference frame.
In order to control a two-link manipulator to behave likehuman extremities, the motion of the endeffector needs tobe described in terms of the actuators that are implementedin the robots. The motion of the endeffector is usuallydescribed in the task-space or workspace, and the actuatorsinput and output are described in the joint space. Jacobianwhich relates two spaces is widely used and various research
Sehoon Oh is with Samsung Heavy Industries, MechatronicsCenter, Institute of Industrial Technology Tel: [email protected]
Y. Kimura and Yoichi Hori are with Department of Advanced Energy,School of Frontier Sciences, University of Tokyo, 5-1-5 KashiwanohaKashiwa-shi, Chiba, [email protected],[email protected]
have proposed the control algorithms using Jacobian basedon these two spaces [1], [2].
The proposal in this paper is to change these two spacesto better describe the endeffector motion and to designcontrollers in simpler ways.
Conventional joint space deals with the actuator coordinatethat is defined on each joint. However, this paper proposesto use the biarticular actuator coordinate which is derivedfrom the biarticular muscle concept that is well-known inthe human/animal musculoskeletal systems.
Due to the existence of the biarticular muscles, the mus-cle coordination problem has been an important issue inbiomechanics[3], [4] , while the robotics avoids the issueby ignoring the biarticular muscle/actuator. However, thebiarticular muscles have been said to play an important role[5], and there are still lots of research on this topic. .
Theory-wise, the redundancy caused by the biarticularmuscle has attracted researchers. Kumamoto proposed a 3-pair 6-muscle model as the most fundamental structure of atwo-link manipulator with the biarticular muscle, and showedthe way the muscle pairs co-work to generate forces at thehand using clinical experiments. The algorithm related thetorques in the biarticular coordination and the force at theendeffector, mostly focusing on the statics[6].
The algorithm is found to be the optimized solution for theforce generation at the endeffector by solving the∞−normof the muscle torques required to generate the forces [7], [8].
Practical-wise the biarticular muscle mechanism is widelyadopted in the robots for jumping and running [9], [10], [11].They adopt the biarticular muscle to improve the jumpingstability and performance, and there is a theoretical backupof this improvement [12], [13].
There have been several approaches to utilize the theoryrelated to the biarticular muscle coordination [14] with thepractical robot motions such as jumping and swimming [12],[15]. Those approaches, however, were limited to specificcases and were not general enough. This paper extendsand generalizes those approaches by introducing the rotatingworkspace.
The motions of human’s extremity are often depictedas the translational motion. In order to describe human’swalking and running, the biomechanics utilizes the SpringLoaded Inverted Pendulum (SLIP) model [16], [17], [18],[19], [20], [21].
In this model, the motion of endeffector is comprised of
the translational motion from the base and the rotation aroundthe base [16], [17], which is different from the workspacedefinition that is used for the industrial robot [1], [2]. As areference frame that is appropriate to deal with this motion,this paper proposes a rotating reference frame and defines themotion of the endeffector in the proposed reference frame.This reference frame can describe the translational motionand the rotational motion of the inverted pendulum as twoindependent coordinates.
This paper shows the rotating reference frame along withthe biarticular actuator coordination can simplify the kine-matics, statics, and the dynamics of a two-link manipulator.The controller for the two-link manipulator can be designedbased on the proposed simplified kinematics, statics and thedynamics in a more straightforward way, and the rotatingworkspace observer and Two-degree-of-freedom control[22]for the manipulator are designed in this way.
[23] can be similar approach to the proposed observer de-signing the disturbance observer in the workspace. Howeverthe proposed observer uses the rotating workspace ratherthan the conventional Cartesian workspace. The proposedobserver can decouple the inertia coupling caused by theasymmetric mass distribution of the two link manipulator.This decoupling is significant for a two-link manipulator tobehave like a SLIP model [19], [20], [21].
The paper is organized as follows; Section II explainshow the biarticular actuator coordination and the rotatingworkspace definition can simplify the kinematics and thestatics of the two-link manipulator comparing it with theconventional ones. In Section III, the dynamics of the two-link manipulator is derived in the rotating workspace, andRotating Workspace Observer is designed. In Section IV,several simulations are conducted to verify the effectivenessof the proposed observer.
The basic idea of this paper was presented at the 12th In-ternational Workshop on Advanced Motion Control [24]. Thederivation of the kinematics, the statics, and the dynamics aredescribed in totally different ways, and simpler expressionsare presented in this paper. The novel workspace which wasnamed Leg Space is represented as Rotating Workspace togeneralize the concept. Adding to the disturbance observerdesign, Two-degree-of-freedom control in the proposed ref-erence frame is designed and simulated in this paper.
II. K INEMATICS AND STATICS SIMPLIFICATION IN THE
ROTATING REFERENCE FRAME BY BIARTICULAR
ACTUATOR COORDINATION
A. Introduction of the rotating reference frame
Figure 1 explains the rotating reference frame which isdepicted as the dashed line, comparing with the conventionalCartesian reference frame depicted as the solid line. In therotating reference frame, thexR axis is set along the lineconnecting the first joint and the end effector, and theyR axisis set orthogonal to thexR. Since the endeffector is not fixed,the reference frame also is not fixed. The translational motionof the endeffector does not change the reference frame, butthe rotation of the endeffector affects the reference frame,
which is the reason why the proposed reference frame is therotating reference frame.
θ2
θR
x
y
xR
yR
l1
l2
lm
θ1
Fig. 1. Two Reference FramesΣ andΣR
l1, l2 are the length of the first link and the secondlink, θ1 is the angle of the first link from the Cartesianreference frame, andθ2 is the angle of the second linkfrom the first link.θR is the angle of thexR axis from theCartesian reference frame. In the remainder of this paper,the kinematics, dynamics analysis and the control design aredone in this rotating reference frame.
B. Kinematics in the rotating reference frame
Jacobian used for the kinematics and inverse kinematicsin the convetional reference frame is defined as follows.
J =
(−l1 sin θ1 − l2 sin(θ1 + θ2) −l2 sin(θ1 + θ2)l1 cos θ1 + l2 cos(θ1 + θ2) l2 cos(θ1 + θ2)
)(1)
This Jacobian can be decomposed as follows.
J =
(− sin θ1 − sin(θ1 + θ2)cos θ1 cos(θ1 + θ2)
)(l1 00 l2
)(1 01 1
)(2)
In order to make the whole development simple, two lengthsl1 and l2 are assumed to be identical, which is still practicalsince most of robot manipulators have this configuration(l1 = l2 = l).
The joint angular velocity (θ1, θ2) can be replaced the(θ1, θ12) as Equation (3), which is derived from the mul-tiplication with the last matrix in Equation (2)(
1 01 1
)(θ1θ2
)=
(θ1
θ1 + θ2
)=
(θ1θ12
), (3)
whereθ12 representsθ1 + θ2. θ12 is considered the velocityoutput that corresponds to the biarticular muscle torque input[14].
The velocity of the endeffector is transformed to thevelocity in the rotating reference frame as follows.(
xR
yR
)=
(cos θR sin θR− sin θR cos θR
)(xy
)(4)
Based on the geometry in Figure 1,θR becomesθ1+θ22 with
the assumptionl1 = l2.
Equation (2), (3) and (4) transform the relation between(x, y)T and(θ1, θ2)T to the relation between(xR, yR)T and(θ1, ˙θ12)
T as follows.(xR
yR
)= l
(cos θR sin θR− sin θR cos θR
)(− sin θ1 −sin θ12cos θ1 cos θ12
)(θ1θ12
)= l
(sin θ2
2 −sin θ22
cos θ22 cos θ2
2
)(θ1θ12
)= JR
(θ1θ12
)
=
l sin θ22
(θ1 − θ12
)l cos θ2
2
(θ1 + θ12
) , (5)
JR is the new Jacobian that relates the endeffector velocityin the rotating reference frame and the angular velocity ofthe biarticular actuator coordination.
Notice that the differential mode and common mode ofθ1 and θ12 correspond toxR and yR respectively in thiskinematics.
Based on this kinematics in Equation (5), the inversekinematics is derived as follows.(
θ1θ12
)= J−1
R
(xR
yR
)=
1
l sin θ2
(cos θ2
2 sin θ22
−cos θ22 sin θ2
2
)(xR
yR
)
=1
2
(1 1−1 1
) xR
l sinθ22
yR
l cosθ22
(6)
Equation (5) and (6) are the kinematics and the inversekinematics of the two-link manipulator with the biarticu-lar actuator coordination and the rotating reference frame.Compared to the conventional kinematics using Jacobian inEquation (1), the relationship becomes much simpler andmakes a respective connection betweenθ1− θ12 andxR andalso betweenθ1+ θ12 andyR. There is a scale factorl sin θ2
2
betweenθ1− θ12 and xR, andl cos θ22 betweenθ1+ θ12 and
yR.
C. Statics based on the proposed kinematics
Statics between two joint torquesτ1, τ2 that acts on twojoints respectively and the force at the endeffectorfx, fy thatis defined in the conventional Cartesian reference frame isdescribed by the transpose of JacobianJT . Using the similarprocess used in the kinematics, the statics can be simplified.The transpose of Jacobian is decomposed as Equation (7).
JT =
(1 10 1
)(l1 00 l2
)(− sin θ1 cos θ1− sin θ12 cos θ12
)(7)
Biarticular muscle/actuator torque inputτb works as aninput that acts on the two joints at the same time. The firstmatrix in Equation (7) relatesτ1, τ2 to a different torquepair τm, τb, whereτm is the monoarticular torque on the firstjoint, andτb is the biarticular torque. The following equationdescribes this characteristic.(
τ1τ2
)=
(1 10 1
)(τmτb
)=
(τm + τb
τb
)(8)
The force at the endeffector(fx, fy) can be transformedto fR
x , fRy in the rotating reference frame as Equation (9).(
fxfy
)=
(cos θR − sin θRsin θR cos θR
)(fRx
fRy
)(9)
Equation (7), (8) and (9) with the assumptionl1 = l2 = lprovide the simplified statics as follows.(τmτb
)= l
(− sin θ1 cos θ1− sin θ12 cos θ12
)(cos θR − sin θRsin θR cos θR
)(fRx
fRy
)= l
(sin θ2
2 cos θ22
− sin θ22 cos θ2
2
)(fRx
fRy
)=JT
R
(fRx
fRy
)=
(1 1−1 1
)(l sin θ2
2 fRx
l cos θ22 f
Ry
)(10)
The inverse statics also can be rederived in the rotatingreference frame as follows.(
fRx
fRy
)=(JTR
)−1(τmτb
)=
1
l sin θ2
(cos θ2
2 − cos θ22
sin θ22 sin θ2
2
)(τmτb
)
=1
2
(τm−τb)
l sinθ22
(τm+τb)
l cosθ22
(11)
Note that the force at the end effector along thexR andyR
axes also are described by the common mode and differentialmode independently;τm − τb determinesfR
x with the scalefactor of 1
2l sin θ2/2, andτm+τb determinesfR
y with the scalefactor of 1
2l cos θ2/2.
III. D ESIGN OFROTATING WORKSPACEOBSERVER
Even though the torques can generate force at the endeffector in thexR, yR direction independently, the two-linkmanipulator cannot work like a SLIP model, since it hasa different inertia structure. The inertia matrix is different;there is a coupling between the inertias in thexR directionand theyR direction.
A disturbance observer that can decouple the inertiastructure to make a two-link manipulator work like a SLIPmodel is proposed in this section. The proposed observer candecouple the inertia not only for the actuator torque but alsofor the external force.
In order to design the observer, the dynamics is re-described along in rotating reference frame also using thebiarticular actuator coordination. Based on the derived dy-namics, the disturbance observer is designed to decouple theinertia matrix.
A. Inertia coupling problem in two-link manipulators
Equation (12) is the conventional description of the dy-namics of a two-link manipulator, whereτ1, τ2 are the twotorque inputs, andθ1, θ2 are the acceleration outputs.g isthe acceleration of gravity,di is the distance from the centerof a joint i to the center of the gravity point of the linki,
mi is the weight of the linki, Ji is the moment of inertiaabout an axis through the center of mass of linki.(
J1+J2+m2l21+2m2l1d2 cos θ2 J2+m2l1d2 cos θ2
J2 +m2l1d2 cos θ2 J2
)(θ1θ2
)+
(−m2l1d2 sin θ2(θ
22 + 2θ1θ2)
m2l1d2 sin θ2θ21
)+
(g(m1d1+m2l1) cos θ1+gm2d2 cos(θ1+θ2)
gm2d2 cos(θ1 + θ2)
)=
(τ1τ2
)(12)
The dynamics can be simplified using the biarticularactuator. Elements of the inertia matrix can be grouped into
A1 = J1 +m2l21, A2 = J2, A12 = m2l1d2 cos θ2. (13)
Using this description, the inertia matrix in Equation (12)can then be written as(
J1+J2+m2l21+2m2l1d2 cos θ2 J2+m2l1d2 cos θ2
J2 +m2l1d2 cos θ2 J2
)=
(A1 +A2 + 2A12 A12 +A2
A12 +A2 A2
)=
(1 10 1
)(A1 A12
A12 A2
)(1 01 1
)(14)
Note that the first and the last matrix are the conver-sion matrices that project the conventional joint coordinates(τ1, τ2) and (θ1, θ2) to the biarticular actuator coordinates(τm, τb) and(θ1, θ12). The following equations illustrate theconversion.(
θ1θ12
)=
(1 01 1
)(θ1θ2
),
(θ1θ2
)=
(1 0−1 1
)(θ1θ12
)(15)(
τ1τ2
)=
(1 10 1
)(τmτb
),
(τmτb
)=
(1 −10 1
)(τ1τ2
)(16)
Using the above conversions, (12) can be expressed asfollows.(τmτb
)=
(1 −10 1
)(A1+A2+2A12 A2+A12
A2+A12 A2
)(1 0−1 1
)(θ1θ12
)+
(1 −10 1
)(−m2l1d2 sin θ2(θ
22 + 2θ1θ2)
m2l1d2 sin θ2θ21
)+
(1 −10 1
)(g(m1d1+m2l1) cos θ1+gm2d2 cos(θ1+θ2)
gm2d2 cos θ12
)=
(J1 +m2l
21 m2l1d2 cos θ2
m2l1d2 cos θ2 J2
)(θ1θ12
)+
(−m2l1d2 sin θ2θ
212
m2l1d2 sin θ2θ21
)+
(g(m1d1+m2l1) cos θ1
gm2d2 cos θ12
)(17)
With this inertia matrix and the kinematics derived in theprevious session, the operational inertia matrix in the rotating
reference frame is derived as follows.
ΛR = (JTR )−1AJ−1
R (18)
=1
l sin θ2
(cos θ2
2 − cos θ22
sin θ22 sin θ2
2
)(A1 A12
A12 A2
)·
1
l sin θ2
(cos θ2
2 sin θ22
−cos θ22 sin θ2
2
)
=1
2l2
JR1
sin2 θ22
JRm
sin θ2
JRm
sin θ2
JR2
cos2θ22
, (19)
where the inertia elements are defined as follows.
JR1 = A1−2A12+A2=J1+m2l
2+J2−2m2d2l cos θ2
(20)
JR2 = A1+2A12+A2=J1+m2l
2+J2+2m2d2l cos θ2
(21)
JRm = A1−A2=J1+m2l
2−J2 (22)
JRm is the coupling inertia term making the motions the
xR and yR directions mutually dependent and thus makesthe motion of a two-link manipulator different from the SLIPdynamics.
B. Design of rotational workspace disturbance observer
Decoupling of the inertia is not significant when thetracking performance is the only concern, when two actuatorscontrol the 2 dimensional position of endeffector suppressingall disturbances including external forces.
However, when it comes to the motion or dynamics bythe external forces, the intrinsic inertia coupling can bea significant problem. Humans and animals achieve theirmotion such as walking and running utilizing the externalforce such as the contact with the ground. The SLIP modelwhich describes human’s locomotion is also the case whenthe reaction against the gravity plays an important role. Fora two-link manipulator to work like this SLIP model, theinertia should be decoupled so that the external force appliedin thexR does not affect the motion inyR. The disturbanceobserver which is designed in the rotating reference framecan remove this inertia coupling as disturbance.
Equation (23) is the design of the rotating workspacedisturbance observer, where the disturbances, the controlinput and the system output are defined in the rotatingreference frame.(
dRxdRy
)=
1
τs+1
((fRx
fRy
)− ΛR
n
(xR
yR
)),(23)
In order to implement this observer, the control inputfRx , fR
y that are provided by the actuators to the endeffectorand the system outputxR, yR of the endeffector need tobe replaced by the actuator torquesτm, τb and the outputsθ1, θ12 that can be directly measured or obtained.
Equation (24) is the disturbance observer design re-described using these values.Q(s) is a low pass filter with
a time constantτ (Q(s) = 1τs+1 ).(
dRxdRy
)= Q(s)(JT
R )−1
(τmτb
)−Q(s)ΛR
nJR
(θ1θ12
)(24)
To remove the inertia coupling,ΛRn in this design is set as
follows so that the coupling in Equation (19) is dealt withas the lumped disturbance and removed by the feedback ofthe lumped disturbance.
ΛRn =
(ΛR11n 00 ΛR
22n
)(25)
ΛR11n =
JR1n
2l2 sin2 θ22
,ΛR22n =
JR2n
2l2 cos2 θ22
(26)
With this decoupled nominal matrix in addition toJTR and
JR in Equation (5) and (10), the disturbance observer inEquation (24) is simplified as follows.(
dRxdRy
)=
Q(s)
2l
( 1
sinθ22
(τm−τb)1
cosθ22
(τm+τb)
)−
JR1
sinθ22
(θ1−θ12
)JR2
cosθ22
(θ1+θ12
)(27)
Lastly, the estimated disturbancesdRx,y also should betransformation to the actuator space so that it can be fedback to the torque inputsτm, τb. This can be done usingJT
R
as Equation (28).(dRxm
dRy
b
)= JT
R
(dRxdRy
)(28)
This transformation cancels outsin θ22 , cos
θ22 in the de-
nominators in Equation (27). Equation (29) and (30) are thefinal design of the rotating workspace observer. Notice thatthere is no division bysin θ2 or cos θ2 that causes the singularposition. This shows that the proposed rotating workspaceobserver is robust to the singular points.
dRxm =
Q(s)
2((τm − τb) + (τm + τb))
−Q(s)
2
(JR1
(θ1−θ12
)+JR
2
(θ1+θ12
))(29)
dRym =
Q(s)
2(− (τm − τb) + (τm + τb))
−Q(s)
2
(−JR
1
(θ1−θ12
)+JR
2
(θ1+θ12
))(30)
The design of this rotating workspace disturbance observeris illustrated as the block diagram in Figure 2. The only timevarying component is the nominal inertias which need theangleθ2 in the numerators.
C. Two-degree-of-freedom control design in the rotating ref-erence frame
Two-degree-of-freedom control [22] can be designed inthe rotating reference frame. Figure 3 is the block diagramof Two-degree-of-freedom control in the rotating referenceframe.
The controller focuses on the tracking ofθ1 − θ12 andθ1 + θ12, and the references are given as(θ1 − θ12)
∗ and
τs+1
τs+1
1
1
τs+1
J1 s
τs+1
J2 s
R
+
+ + +
+ +
+
+
+ +
+
+
+
+
-
-
-
-
--
Manipulator
Dynamics
dτm
dτb
τm
τb*
*ω1
ω12
+
+ +
-
θ2
21
21
R
Fig. 2. Disturbance Observer in the Rotating Reference Frame
τs+1
τs+1
1
1
τs+1
J1 s
τs+1
J2 sR
R
+
+
+
+
+
+
+
+
+
+
-
-
-
-
--
Manipulator
Dynamics
(θ1-θ12) +
+
+-
θ2
21
21
. .
-
θ1
θ12
.
.
(θ1+θ12). .
C_diff
C_sum
*
*
C_diff
C_sumff
ff
fb
fb
+
+
Fig. 3. Two-degree-of-freedom Control Designed in the Rotating ReferenceFrame
(θ1 + θ12)∗. The velocity of the endeffectorxR∗
, yR∗
in therotating reference frame are converted to these referencesusing the inverse kinematics of Equation (6).
(θ1 − θ12)∗ =
xR∗
l sin θ22
(31)
(θ1 + θ12)∗ =
yR∗
l cos θ22
(32)
Inverse dynamics derived using the nominal inertia matrixin Equation (25) can be used for the feedforward controllers.Equation (33) shows the design ofCff
sum, Cffdiff .
Cffsum =
ΛR11ns
τffs+ 1, Cff
diff =ΛR22ns
τffs+ 1(33)
The tracking performance by this Two-degree-of-freedomcontrol is verified in the following section.
IV. V ERIFICATION OF THE ROTATING WORKSPACE
OBSERVER BY SIMULATIONS
The effectiveness of the proposed rotating workspaceobserver is verified by simulations. Two kinds of simula-tions are performed: tracking performance verification anddecoupling betweenxr and yr directional motions againstthe external force. The parameters used in the simulation areshown in Table I. Two simulation tools are used; MatlabR⃝
is used in Section IV-A, and Adams is used Section IV-B.
TABLE I
PARAMETER OF SIMULATION
m1 2.64 kg m2 2.64 kgl1 0.3 m l2 0.3 md1 0.15 m d2 0.15mJ1 0.0853 kg·m2 J2 0.0853 kg·m2
θ1 π/4 θ2 −π/2
A. Tracking control of the endeffector
The performance of Two-degree-of-freedom control inFigure 3 is verified by simulations. Two points are evaluated:tracking performance by Two-degree-of-freedom control anddecoupling by the rotating workspace observer.
Several sinusoidal references are given toxR∗while the
reference toyR∗
is set to 0. These references are convertedusing Equation (31) and (32). PI controller is used as thefeedback controllersCfb
sum, Cfbdiff , and the gains are set as
KP = 6.8,Ki = 40 in both controllers. The time constantτ of Q filters of the disturbance observers are set to12π·10 .The time constantτff in the feedforward control of Equation(33) is set to 1
2π·30 .First, the tracking performance between the results with
and without feedforward control is compared. Figure 4 (a)shows the result with the reference signal of 2Hz and 5Hz.The result shows that the feedforward control using theproposed dynamics can improve the tracking performanceat high frequency.
Figure 4 (b) shows the coupling result with two referencesignals of 0.5Hz and 2Hz. Even though 0 is set as thereference for theyR, there are fluctuations inyR output.This is due to the coupling byxR motion that tracks thereference signalsxR∗
. However, the proposed observer canremove the coupling at low frequency as shown in Figure 4.
B. Reaction against contact force
The proposed rotating workspace observer can design thereaction against the external force independently in thexR
andyR directions. For example, the proposed algorithm canabsorb the impact force at the endeffector by folding thejoint in thexR direction as humans do by kneeling when wejump. However, humans keep the posture or orientation ofthe trunk while folding their knees, which can be interpretedas the position control of the endeffector in theyR direction.This motion can be achieved by designing a strong feedbackcontroller in theyR direction while setting a weak feedbackcontroller for thexR direction.
In order to verify this independent reaction force, theimpact force is given in thexR direction, and the reaction ofthe endeffector is examined. The disturbance observer inxR
direction is removed and the gains of the feedback controllerCfb
diff are set low, while the disturbance observer and thefeedback gains are set same as the previous session in theyR direction.
In order to emphasize the effectiveness of the proposedalgorithm, another type of the disturbance observer to the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
time(sec)
velo
city
(m/s
)
Referencewith FB onlywith FF and FF
0 0.1 0.2 0.3 0.4 0.5 0.6−1
−0.5
0
0.5
1
time(sec)
velo
city
(m/s
)
Referencewith only FBwith FB and FF
(a) Endeffector Velocity in thexR Direction (upper: 2Hz, lower: 5Hz)
0 0.5 1 1.5 2 2.5 3−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
time(sec)
velo
city
(m/s
)
w/o DOBwith DOB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1
−0.05
0
0.05
0.1
0.15
time(sec)
velo
city
(m/s
)
w/o DOBwith DOB
(b) Endeffector Velocity in theyR Direction (upper: 0.5Hz, lower: 2Hz)
Fig. 4. Simulation Results. (a) Tracking performance with two differentfrequency sinusoidal references given in thexR direction - 2Hz and 5Hz(b) Coupling in theyR axis velocity whenxR velocity is controlled withtwo sinusoidal references - 0.5Hz and 2Hz
manipulator [25] is adopted and compared with the proposedalgorithm. In [25], two disturbance observers are set on eachjoint suppressing the external forces in the joint space. Thisconventional disturbance observer is only good for suppres-sion in all directions, but it cannot design the direction ofthe stiffness at the endeffector independently.
In the following simulations, three cases are compared:1) the rotating workspace observer and a high gain velocitycontrol are set onθ1 + θ12, which corresponds to theyR
direction, while only a low gain velocity control set onθ1 − θ12 which corresponds to thexR direction (DOB inyR). 2) the joint space disturbance observer and a high gain
velocity control are set on theθ1 joint while only a lowgain velocity control are set on theθ2 (DOB on θ1). 3) thejoint space disturbance observer and a high gain velocitycontrol are implemented on theθ2 joint while only a lowgain velocity control are set on theθ1 joint (DOB on θ2).All the simulations are conducted using Adams.
In the following simulations, the time constant of thedisturbance observer is set to12π·10 , and the feedback gainsare set to form the characteristic polynomial of the closedloop system likes2 + 2ξωs + ω2. In the high gain controlcase,ω is set to2π · 5, andξ is set to 10, while in the lowgain control case,ω is set to2π · 3, andξ to 4. Two typesof the initial posture configurations are used:θ02 = π
100 andθ02 = π
6 .
As shown in Figure 5, a 30kg ball contacts the tip of theendeffector (the gravity is to the right of the figure), and thereaction against this contact force is examined in the abovethree cases. An additional link is added at the endeffector toprovide a surface enough for the contact, and the orientationof the link is set fixed since its motion is irrelevant here.Other parameters of the manipulator are set same as Table I.
Figure 5 shows the difference in reaction between the con-ventional observer and the proposed observer. Figure 5 (b)shows that the proposed observer can keep the position of theendeffector in theyR direction while it moves compliantly inthe xR direction. However the conventional observer eitherreacts rigidly in all directions (DOB onθ2, the right figure ofFigure 5 (a) ) or cannot keep the position of the endeffector(DOB on θ1, the left figure of Figure 5 (a) )
The detailed results are shown in Figure 6 and Figure 7.The position and the velocity of the endeffector in theyR
direction and the contact force are shown. Figure 6 is theresult with the initialθ2 angle set toπ6 , and in Figure 7, itis set to π
100 .
In both simulations, DOB inyR keeps the position of theendeffector suppressing the velocity in theyR direction. Eventhough DOB onθ2 can keep the position, it suppresses themotion in thexR direction too, showing high contact forcesat the endeffector. DOB onθ1 cannot keep the position ofthe endeffector.
The proposed observer can reduce the contact force; DOBon yR shows the smallest contact force in Figure 6 (c), andin Figure 7, even though the first contact forces are same, thesecond contact force is reduced while others are not. This isdue to the compliance designed by the low gain feedback inthe xR direction.
These results verify that the proposed observer and thefeedback controller designed based on the rotating referenceframe and the biarticular actuator coordination can designthe reaction force in thexR andyR directions independently.It is true that this contact force reduction by the proposedalgorithm assumes no friction and high back-drivability onthe joints, this feature can play an important role in thehumanoid and other legged robot application.
(a) Reactions by the Conventional DOB
(Left: DOB on θ1, Right: DOB onθ2)
(B) Reactions by the Proposed DOB (DOB onxR)
(Left: θ02 = π6
, Right: θ02 = π100
)
Fig. 5. Contact Reaction Simulation Results
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
time(sec)
End
effe
ctor
Vel
ocity
in th
e Y
dire
ctio
n (m
/s)
DOB in yR
DOB on θ1
DOB on θ2
(a) Endeffector Velocity in theyR Direction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
time(sec)
End
effe
ctor
Pos
ition
in th
e Y
dire
ctio
n (m
)
DOB in yR
DOB on θ1
DOB on θ2
(b) Endeffector Position in theyR Direction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1000
2000
3000
4000
5000
6000
time(sec)
Con
tact
For
ce (
N)
DOB in yR
DOB on θ1
DOB on θ2
(c) Contact Force in thexR Direction
Fig. 6. Simulation Results (θ02 = π6
)
V. CONCLUSION
The rotating reference frame and the biarticular actuatorcoordination simplified kinematics, statics and dynamics ofa two-link manipulator, and a control algorithm based onthem was designed in this paper. Theoretic development andsimulation results verified the effectiveness of the proposedkinematics, dynamics and control.
Even though the SLIP model was mentioned as the mainapplication of the proposed algorithm, it can be applied toany robot whose motions consist of the translational motionand the rotational motion.
The proposed analysis and control design can providesignificant and theoretic insight to the biomechanics, sincethe biarticular actuator/muscle coordination is common inthe animals. And it also can help biomechanics research tobe applied to the robotics.
Design of more elaborate impedance control in the refer-ence frame is the future work.
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
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End
effe
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Vel
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DOB on θ1
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(a) Endeffector Velocity in theyR Direction
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−20
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(b) Endeffector Position in theyR Direction
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