P. R. Ouyang
Associate Professor
Mem. ASME
e-mail: [email protected]
V. Pano
e-mail: [email protected]
Department of Aerospace Engineering,
Ryerson University,
350 Victoria Street,
Toronto, ON, M5B 2K3, Canada
Position Domain
Synchronization Control
of Multi-Degrees of Freedom
Robotic Manipulator
In this paper, a new position domain synchronization control (PDSC) law is proposed for
contour control of multi-DOF nonlinear robotic manipulators with the main goal of
improving contour tracking performance. The robotic manipulator is treated as a masterslave motion system, where the position of the master motion is used as an independent
reference via equidistant sampling, and the slave motions are described as functions of
the master motion. To build this relationship, the dynamics of the original system is transformed from time domain to position domain. The new control introduces synchronization
and coupled errors in the control law to further coordinate the master and slave motions.
Stability analysis is performed based on the Lyapunov method for the proposed PDSC,
and simulations are conducted to verify the effectiveness of the developed control system.
[DOI: 10.1115/1.4025755]
Keywords: position domain control, synchronization, contour tracking, robot
1
Introduction
Control of robotic manipulators has been an area of exhaustive
research, mainly due to its high complexity, nonlinearity, and wide
applications. Through the years, various control methods have been
developed with the purpose of achieving high tracking performances [1–5]. Inarguably, the most popular control scheme is proportional integral derivative (PID) control and its variants, such as PD
and PI control, whose innate simplicity, proven stability, and ease
of implementation have made quite appealing for industrial applications [6–10]. Nevertheless, this type of control is normally
employed in decoupled individual joints of a robotic manipulator,
and its feedback signal is based on the tracking errors of each joint.
The main motivation for this research is the development of a new
control system for high precision contour tracking of multi-DOF
robotic systems, considering the coupling feature of joints.
Contour tracking for a robotic manipulator can be defined as
the control of the motion of the end-effector following a predefined path in a precise and effective manner. Unfortunately, for a
serial robot manipulator with multiple joints, good contour tracking of the end-effector cannot be guaranteed by good trajectory
tracking performance of individual joints. Furthermore, independent joint motion is accompanied with asynchronous motion, which
results in deteriorated accuracy of contour tracking [1,2,10–12].
Thus, the synchronization of the motions of individual joints of a
robotic manipulator becomes an important factor for the improvement of contour tracking performance of the end-effector [10–14].
In recent decades, various control schemes of motion synchronization for contour tracking have been introduced. In 1980,
Koren [1] developed the concept of cross-coupling control (CCC).
This control scheme synchronizes two motion axes based on the
resulting contour tracking error and it is proven to efficiently
reduce tracking error for specific contours. Later, a model-free
variable gain CCC was proposed by Koren and Lo [2] for a general class of contours. Regardless of its good performance, CCC
features a number of disadvantages that limit its use in the control
of robotic manipulators. The complexity of the gain selection for
Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received
April 19, 2013; final manuscript received October 16, 2013; published online
December 16, 2013. Assoc. Editor: Alexander Leonessa.
the CCC loop can lead to a highly inaccurate control signal and
hence, result in diminished contour performance [3]. Also, most
of the concepts related to CCC are developed and applied to CNC
machining which are quite dissimilar to robotic manipulators in
term of complexities, operating conditions and requirements, and
objectives [4].
To avoid the problem of complicated contour estimation and
bring the problem back to the joint motion, a different control
approach was introduced by Sun in Ref. [10]. Position synchronization control (PSC) introduced the notion of synchronization
error and coupled error. The coupled error combines the synchronization errors and tracking errors, and the controller forces both
errors to converge to zero simultaneously. With the introduction
of the synchronization concept, classical contour errors can be
reinterpreted as position synchronization errors, significantly
improving the tracking performance of the robotic manipulator
[13,14]. More recently, a model-free version of the synchronization control was proposed in Ref. [10] that resembled a PD type
controller.
In our previous research [15], a position domain controller in
PD type (PDC-PD) was proposed for the control of a 2-DOF linear translational manipulator. This controller proposed the transformation of the system dynamics from time domain to position
domain. In that paper, the developed controller was based on the
master-slave [12] control scheme in order to achieve good contour
performance. Furthermore, in Refs. [16] and [17], position domain
PID control for CNC machining was proven to be better in contour tracking than CCC control.
The main purpose of this research is to combine the masterslave synchronization of the position domain controller with the
position synchronization controller to develop a new PD type synchronization control law and apply it for contour control of nonlinear multi-DOF nonlinear robotic manipulators. Therefore, this
research is an extension and further development of the previous
work for the purpose of further improving contour tracking
performance.
2
Relative Derivative and Dynamic Model
2.1 Relative Derivative. For a multi-DOF serial robotic manipulator, to facilitate the discussion in this paper, following the
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similar concept developed in Ref. [12], the motion of the first joint
is assumed to be the master motion, while the rest joint motions
are viewed as the slave motions. To develop the control law in the
position domain, the relationship between the master motion (qm )
and the ith slave motion (qsi ) has to be developed. For this reason,
a relative derivative concept was introduced in Refs. [15–17].
Similar to the partial derivative definition, we define the relative
derivative of slave joint s with respect to the master joint as
q0si ¼
q_ si
dqsi
¼
dqm
q_ m
for i ¼ 1; 2; 3; …
(1)
From Eq. (1), it can be easily understood that q0si is an angular
speed ratio between the slave and the master motions and it
describes a synchronized motional relationship between these two
motions. This relative derivative is called relative position velocity of slave joint si with respect to the master joint m.
In the same manner, we define the second relative derivative,
also called relative position acceleration [15–17]
q00si ¼
dq0si
dqm
for i ¼ 1; 2; 3; …
(2)
From Eq. (1), the velocity of joint s can be defined as
0
q_ si ¼ q_ m qsi
for i ¼ 1; 2; 3; …
(3)
If differentiating Eq. (3) again, we can express the acceleration of
joint si as
0
q€si ¼ q00si ðq_ m Þ2 þ q€m qsi
for i ¼ 1; 2; 3; …
q_ 2m Mss q00s ðqm Þ þ ðq€m Mss þ q_ m Css Þq0s ðqm Þ þ q€m Msm þ q_ m Csm þ Gs
þ Fs ¼ ss ðqm Þ
(7)
In Eq. (7), the dimensions for the position domain dynamic
model
fMsm ; Gs ; Fs ; ss g 2 Rn ,
fMss ; Css g 2 Rnn ,
are
qs ; q0s ; q00s 2 Rn .
Remark 1. Equation (7) represents the dynamic relationship
between the master motion, indicated by subscript m, and the
slave motions, indicated by subscript s, in the position domain via
the transformation described by Eqs. (3) and (4). Clearly, the nonlinearity of the time domain model in Eq. (6) is maintained in the
position domain.
Remark 2. From Eq. (7), it can be deduced that in order to
achieve accurate contour performance, high precision measurement of the master motion is required. Nonetheless, a high tracking precision of the master motion is not required for the position
domain control.
3
Position Domain Synchronization Control
3.1 Control Law. In order to take advantage of the synchronization control properties and the improved contour tracking performance of position domain control, a new control law is
introduced for the slave motions of a multi-DOF robotic manipulator described in Eq. (7). First, the new error concepts have to be
defined.
The tracking position error vector es ðqm Þ 2 Rn and the relative
derivative error vector e0s ðqm Þ 2 Rn in position domain are
defined as
2
(4)
If it is assumed that the robotic manipulator has n þ 1 DOF,
then the dynamic model can be rewritten in a submatrix formation
that indicates the master (1-DOF) and the slave motions (n-DOF)
as follows:
"
#" # "
#" # " # " # " #
cmm Cms q_ m
gm
fm
sm
mmm Mms q€m
þ
þ
þ
¼
Msm Mss
q€s
Csm Css
q_ s
Gs
Fs
ss
(6)
The dynamic model in Eq. (6) is a combination of the master
motion and slave motions. Subscript m refers to the master
motion, and subscript s refers to the rest of the links, or slave
motions. qm 2 R1 is the reference position and is used as the reference for tracking a defined contour, and qs 2 Rn is the joint
position vector of slave motions. Consequently, the dynamic
model for slave motions qs can be rewritten in position domain as
a function of the reference qm through a transformation from time
domain (t) to position domain (qm ).
Substituting Eqs. (3) and (4) to Eq. (6) for the slave motions, a
dynamic model for the slave motions in position domain is
derived in the following form:
2
qs1d ðqm Þ qs1 ðqm Þ
3
..
.
7
7
7
5
qsnd ðqm Þ qsn ðqm Þ
esn
2
e0s1
3
2
6 7 6
6 7 6
e0s ðqm Þ ¼ 6 ... 7 ¼ 6
4 5 6
4
2.2 Dynamic Model in Position Domain. A multi-DOF
robotic manipulator, consisted of revolute joints, has the following
dynamic model [7–9]
(5)
3
6 7 6
6 . 7 6
es ðqm Þ ¼ 6 .. 7 ¼ 6
4 5 4
Equations (3) and (4) show the relationships between absolute
and relative motions. Equation (3) relates the absolute velocity in
the time domain with the relative derivative in the position domain, whereas Eq. (4) relates the absolute acceleration and the relative acceleration. Both equations are used to transform the
dynamic model from time domain to position domain.
MðqÞ€
q þ Cðq; q_ Þq_ þ GðqÞ þ Fðt; q; q_ Þ ¼ sðtÞ
es1
e0sn
q0s1d ðqm Þ q0s1 ðqm Þ
..
.
q0snd ðqm Þ q0sn ðqm Þ
3
(8)
7
7
7
7
5
Synchronization is the coordination of events to operate a system in unison. The synchronization aims to enable each control
loop for each joint to receive feedback from itself as well as the
other joints for better coordination among all the joints to achieve
better contour performance. In this paper, the synchronization
error is the difference of the tracking errors between two neighbor
joints, and it can be easily obtained through a displacement sensor, such as encoder. The synchronization error vector
es ðqm Þ 2 Rn is expressed as
2
es1 es2
6
6 es2 es3
6
6
6
..
es ðqm Þ ¼ 6
.
6
6
6
6 esðn1Þ esn
4
esn es1
3
2
7 6
7 6
7 6
7 6
7 6
7¼6
7 6
7 6
7 6
7 6
5 4
1
1
0
1
1
..
.
..
.
..
.
0
1
1
0
…
¼ Tes ðqm Þ
0
3
7
7
7
7
.. 7
. 7
7es ðqm Þ
7
7
1 7
5
1
(9)
where T 2 Rnn is called the synchronization matrix with special
constant elements.
Additionally, a coupled error vector 0 es ðqm Þ 2 Rn and the
coupled relative derivative error vector es ðqm Þ 2 Rn are defined
021017-2 / Vol. 136, MARCH 2014
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Transactions of the ASME
and used to combine the synchronization and tracking error
together
( es ðqm Þ ¼ es ðqm Þ þ Bes ðqm Þ
(10)
0
es ðqm Þ ¼ e0s ðqm Þ þ Be0s ðqm Þ
with Eqs. (5) and (7) are used in the following stability analysis.
These properties are described as follows:
P1: The inertia matrix MðqÞ 2 Rnþ1nþ1 is symmetric and positive definite. Mss 2 Rnn can easily be proven to be symmetric and positive definite.
_ is skew symmetric, and conseP2: The matrix M_ ðqÞ 2Cðq; qÞ
_ is skew symmetric as well.
quently M_ ss ðqÞ 2Css ðq; qÞ
P3: The inertia and centrifugal-coriolis matrices satisfy the following equations:
where B 2 Rnn is called the synchronization coupling gain matrix and it is a constant diagonal matrix, that is
B ¼ diagfb1 ; b2 ; …; bn g.
Matrix B is used to deal with the synchronization of different
joints to achieve better contour tracking performance as being
demonstrated in the following simulation study. If B is set to be a
zero matrix, then the developed PDSC becomes a position domain
control without the synchronization effect.
With the error definitions of Eqs. (8)–(10), the new PDSC law
is defined as
0
ss ðqm Þ ¼ Kps es ðqm Þ þ Kds es ðqm Þ
(12)
Remark 6. From Eq. (12), one can see that the proposed PDSC is
only related to the tracking errors and their derivative. Therefore,
it is a model-free feedback control and can be easily implemented.
Using Eqs (3) and (4), Eq. (12) can be transformed back to time
domain and expressed as
ss ðtÞ ¼ Kps es ðtÞ þ Kds q_ s ðtÞ=q_ m ðtÞ
(13)
Remark 7. Equation (13) shows that the derivative term of the
position domain synchronization controller is variable in nature
with its value depending on the value of the master motions speed.
If the master motion has a constant speed and is sampled equidistantly, the position domain controller can be viewed as a constant
gain controller similar to the position synchronization controller
with similar stability properties.
(14)
_ are all bounded. From P4
P4: MðqÞ; Cðq; q_ Þ; GðqÞ, and Fðt; q; qÞ
it can also be deduced that
(a)
(b)
(c)
(d)
Msm is bounded with kMsm k msm .
Csm is bounded with kCsm k csm .
Gs is bounded with kGs k gs .
Fs is bounded with kFs k fs .
In addition, the following notations are introduced. km ðMÞ and
kM ðMÞ represent the smallest and the largest eigenvalues of a matrix M. If a square matrix M is positive definite, then it is denoted
as M 0. If a square matrix M N is positive definite, then it is
denoted as M N 0.
For positive definite matrices, the following properties [18] will
be used in this paper
P5: If M 0; then M1 0.
P6: If M N 0, then M1 N 1 0.
P7: If M 0 and k > 0 is a real number, then kM 0.
P8: If M 0 and N 0, then M þ N 0, MNM 0, and
NMN 0.
Finally, the following reasonable assumptions are introduced in
this paper.
A1: The master motion qm is a monotonically increasing/
decreasing function with the continuous second order derivative for qm 2 ½qms ; qmn .
A2: The velocity q_ m and acceleration q€m of the master motion
are bounded in the desired trajectory region.
A3: The desired contour trajectory qsd ðqm Þ is second order continuous for qm .
4
0
M_ ðqÞ ¼ Cðq; q_ Þ þ CT ðq; q_ Þ
M_ ss ðqÞ ¼ Css ðq; q_ Þ þ CT ðq; q_ Þ
ss
(11)
where Kps 2 Rnn is the proportional control gain matrix and
Kds 2 Rnn is the derivative control gain matrix. Both are considered as constant diagonal gain matrices in this study. ss 2 Rn is
the control torque vector for the slave motions.
Remark 3. It is understood that the position domain control
structure requires the master motion control to operate in the time
domain. In that sense, PDSC is a hybrid system with two different
controllers in two different domains running in sequence.
Remark 4. For a master motion with complicated motion profiles, we divided the master motion into several segments to make
sure that each segment is monotonic increasing (positive speed) or
monotonic decreasing (negative speed), and the developed PDSC
can be used for the slave motion control in several contour segments where each segment has only one direction motion for the
master motion.
Remark 5. It should be noted that the PDSC controller in Eq.
(11) is used for the synchronization of the slave motions and the
master motion does not participate in the process. This is because
the master to slave synchronization is already achieved by the
position domain control structure itself. Consequently, the synchronization error for the master motion is eliminated and the
master motion controller is reduced to a simple PD controller.
Taking account of the error definitions in Eq. (10), the proposed
PDSC law in Eq. (11) can also be expressed in the following
form:
ss ðqm Þ ¼ Kps es ðqm Þ þ Kds es ðqm Þ
Kps ¼ Kps ðI þ BTÞ
Kds ¼ Kds ðI þ BTÞ
(
Stability Analysis
4.1 Theorem. For a rigid robotic manipulator described in
the position domain of Eq. (7), if the PDSC law in Eq. (12) is
applied to control contour tracking of the robotic manipulator, and
the following conditions in Eq. (15) are satisfied, then the controlled robotic manipulator is globally asymptotical stable for contour tracking with bounded tracking errors
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
Kps 0
Kds 0
K þ q_ m CTss q€m Mss > 0
(15)
km Kps > q_ 2m kM ðMss Þ > 0
km Kps > 12 kM K þ q_ m CTss q€m Mss
km Kds þ q€m q_ 2m Mss > 12 kM K þ q_ m CTss q€m Mss
where K 2 Rnn is a user defined constant positive definite
matrix.
3.2 Properties and Assumptions of the Dynamic System. A
list of properties of a rigid robotic manipulator [7–9] associated
4.2 Property of Synchronization Matrix. The synchronization matrix T can be proven to be positive semidefinite as follows:
Journal of Dynamic Systems, Measurement, and Control
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2
zT Tz ¼ ½ a1 a2 an1
6
6
6
6
6
an 6
6
6
6
6
4
1
1
0
1
..
.
..
.
0
..
.
1
0
0
0
32
a1
3
7
76
7
6
1 0 7
76 a2 7
7
76
..
.. 76 . 7
6 . 7
.
. 7
76 . 7
7
76
7
76
6 an1 7
0 1 1 7
5
54
0
1
an
¼ a1 ða1 a2 Þ þ a2 ða2 a3 Þ þ þ an1 ðan1 an Þ
þ an ðan a1 Þ
i
1h
¼ ða1 a2 Þ2 þ þ ðan1 an Þ2 0
2
(16)
where a1 ; a2 ; …; an are any real values.
By choosing Kps , Kds , and B to be positive definite, we can
prove that Kps and Kds are also positive definite.
Then, according to P1 and reorganizing Eq. (24)
T 1 2
S ¼ q_ 2m Mss ðq_ 2m Mss
ÞKps ðq_ m Mss Þ 0
Hence, according to the Proposition and Eq. (15), it is proved
that matrix L is symmetric and positive definite.
4.4 Stability Analysis. Using Eq. (8), the dynamic model in
Eq. (7) can be rewritten in an error function format as follows:
8
00
0
>
q ¼ q_ 2m Mss qsd ðqm Þ þ q€m Mss qsd ðqm Þ
>
>
>
>
<
þ Msm q€m q_ m þ Csm q_ m þ Gs þ Fs
(26)
00
0
2
>
q_ m Mss es ðqm Þ þ ðKds þ Mss q€m þ Css q_ m Þes ðqm Þ
>
>
>
>
:
þKps es ðqm Þ ¼ q
where the defined vector q 2 Rn . According to the properties and
the assumptions, we have
00
4.3 Proposition. Assume a matrix Q is a symmetric matrix
expressed as
"
#
A B
(17)
Q¼
BT C
Let S be the Schur complement [18] of a matrix A in Eq. (17), that
is
S ¼ C BT A1 B
(18)
Then the matrix Q is positive definite if and only if A and S are
both positive definite, i.e., if A 0 and S 0, then Q 0. The
proof of this proposition can be found in Ref. [18].
To prove the stability of the new PDSC law, we first
prove the following matrix L 2 R2n2n is symmetric positive
definite:
"
#
q_ 2m Mss
Kps
(19)
L¼
q_ 2m Mss q_ 2m Mss
Proof. The synchronization control gains are symmetric diagonal
matrices with positive constant elements. From Eq. (15), we have
Kps 0. From P1, we know that Mss is a symmetric positive defiT
and Mss 0. Therefore, L is a symmetric
nite, i.e., Mss ¼ Mss
matrix
From Eq. (15), we have
(20)
km Kps > q_ 2m kM ðMss Þ > 0
From Eq. (20), it can be concluded
Kps q_ 2m Mss 0
(21)
(22)
According to the definition of a positive definite matrix,
Eq. (22) can be rewritten as
1
1
q_ 2m Kps
0
Mss
(23)
Furthermore, based on Eq. (23) and Mss 0, according to P8,
we have
1
1
q_ 2m Kps
ÞMss Þ 0
q_ 2m ðMss ðMss
0
q kq_ 2m Mss qsd ðqm Þ þ q€m Mss qsd ðqm Þ þ Msm q€m þ Csm q_ m þ Gs þ Fs k
00
0
qm qsd k þ msm k€
qm k þ csm kq_ m þ kgs k þ kfs k
mss kq_ 2m qsd k þ mss k€
¼ kqk
(27)
Equation (27) indicated that the parameter q is bounded with q.
For the dynamic model defined in position domain in Eq. (7),
we define the following Lyapunov function:
1 T 0 es
1
0
es e Ts L 0 þ eTs ðK þ Kds Þes
Vðes ðqm Þ; es ðqm ÞÞ ¼
2
2
es
"
# es
Mss q_ 2m
Kps
1 T 0
e e Ts
¼
0
2
2
2 s
es
Mss q_ m Mss q_ m
1
þ eTs ðK þ Kds Þes
2
(28)
From the above discussion, we know that matrix L is symmetric
positive definite, and K þ Kds is also positive definite from Eq.
(15). Therefore, the Lyapunov function in Eq. (28) is a positive
definite function
Vðes ðqm Þ;
0
es ðqm ÞÞ > 0
(29)
In position domain control, the reference angular position qm is
an independent variable that has the similar meaning of t in time
domain. es and e0s are functions of the independent variable qm .
Therefore, the derivative of V is related to variable qm in this stability analysis.
From Eq. (28), the derivative of V along the master motion of
the system is given by
0
dV
q_ m _
0
2
_
es
¼ eTs Kps þ Kds þ K þ q_ m M_ ss es þ e0T
q
M
þ
M
ss
ss
s
m
dqm
2
2
00
_
þ eTs þ e0T
q
M
e
(30)
ss
s
s
m
Considering P2 and P3, and applying Eqs. (15), (25), (26) to
Eq. (30), we obtain
As Kps 0 and Mss 0, according to P5–P7, we have
1
1
q_ 2m Kps
0
Mss
(25)
(24)
dV
€m q_ 2m Mss e0s
¼ eTs Kps es e0T
s Kds þ q
dqm
þ eTs K þ q_ m CTss q€m Mss e0s þ eTs þ e0T
s q
(31)
Since K þ q_ m CTss q€m Mss is positive definite from Eq. (15), we
have
1 eTs K þ q_ m CTss q€m Mss e0s kM K þ q_ m CTss q€m Mss
2
0
ðeTs es þ e0T
s es Þ
021017-4 / Vol. 136, MARCH 2014
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(32)
Transactions of the ASME
Applying Eq. (32) to Eq. (31), we get
dV 1 kes kþ e0s kqkeTs Kps km K þ q_ m CTss q€m Mss I e0s
dqm
2
1
T 2
T
€
_
_
€
es Kds þ qm qm Mss þ km K þ qm Css qm Mss I es
2
(33)
Now, we define the following two parameters:
8
1 >
T
>
_
€
k
¼
k
K
þ
q
C
q
M
q
I
K
>
m
ps
M
m ss
m ss
e
>
>
2
>
>
>
<
q0e ¼ km Kds þ q€m q_ 2m Mss :
>
>
>
>
>
>
1 >
>
:
þ kM K þ q_ m CTss q€m Mss I
2
(34)
Fig. 1
Scheme of 3-DOF serial robotic manipulator
According to Eq. (15), we have qe > 0 and qe0 > 0. Applying
Eq. (34), Eq. (33) can be rewritten as
dV
qe kes k2 þ kqkkes k qe0 ke0s k2 þ kqkke0s k
dqm
(35)
strate the effectiveness of the PDSC for various contour
trackings.
5.1 Simulation Setup
Applying another inequality
a2 1
az bz bz2
b 4
2
for a > 0;
b>0
(36)
8
qe
kqk2
>
2
2
>
>
< kqkkes k qe kes k 4 kes k þ q
e
2
>
>
q
kqk
0
>
: kqkke0s k qe0 ke0s k2 e ke0s k2 þ
4
qe0
(37)
We have
Applying Eq. (37) to Eq. (35), we finally get
dV
q
q 0 2
1
1
e kes k2 e e0s þ
þ 0 kqk2
dqm
qe qe
4
4
(38)
Based on the Lyapunov theorem, we conclude that the robotic
manipulator controlled by the position domain synchronization
control law is globally ultimately bounded. The bounded errors
for the tracking error and the relative derivative of the tracking
error can be obtained as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
>
>
> kes k 2 1 þ 1 kqk
>
>
<
q2e qe qe0
(39)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
1
1
>
0
>
>
: kes k 2 q2 þ q q 0 kqk
e e
e0
From Eq. (39), it can be seen that the tracking errors can be
controlled in a very small boundary layer through proper selection
of the proportional and derivative gains. It is shown that the two
eigenvalues in Eq. (34), associated with the two control gains,
have significant contributions in the control of tracking errors.
From Eq. (34), one can see that a large gain Kps will increase the
value of qe , while a large control gain Kds will increase the value
of qe0 . From Eqs. (34) and (39), we conclude that the increase for
both control gains will reduce the tracking errors. Such a conclusion is very similar with the synchronization control in the time
domain [14].
5
Simulation Study
In this section, the performance of the proposed control
is studied and compared with existed control to demonJournal of Dynamic Systems, Measurement, and Control
5.1.1 Robotic Manipulator and Controller. For the purposes
of simulations, a 3-DOF planar robotic manipulator is used that is
set on a vertical plane. The robotic manipulator consists of three
revolute joints [19,20] as shown in Fig. 1. The structural parameters of the robot are listed in Table 1.
For the given robotic manipulator, the control scheme in position domain is defined as follows:
(
sm ¼ Kpm em ðtÞ þ Kdm e_ m ðtÞ
(40)
0
ss ¼ Kps es ðqm Þ þ Kds es ðqm Þ
Clearly, the first joint motor is used to produce the master
motion, which is controlled with a conventional PD controller.
The rest of the motors are used to generate slave motions of the
robotic system where the proposed PDSC is used as described in
Eq. (40).
Furthermore, the PDSC controller’s performance will be compared with that of a standard PD controller, the model-free PSC as
developed in Ref. [14], and the PDC-PD controller introduced in
Ref. [15].
5.1.2 Trajectory Planning. For the trajectory planning of the
end-effector in time domain, a fifth-order polynomial is used as
described in Eq. (41)
t 3
t 4
t 5
15
þ6
r ðtÞ ¼ 10
T
T
T
(41)
where t is time and T is the total time duration required for each
segment in the contour of the end-effector. The linear and nonlinear trajectories are then defined parametrically as functions of rðtÞ
as described in Ref. [21], and detailed trajectory planning method
will be discussed in Secs. 5.1.3 and 5.1.4. Then, the inverse kinematics analysis is used to plan the motion of all joints.
Table 1 Structural parameters of a serial robotic manipulator
Link
1
2
3
Mass mi (kg)
Length li (m)
Center ri (m)
Inertia Ii (kg m2)
1.00
1.00
0.50
0.50
0.50
0.30
0.25
0.25
0.15
0.10
0.10
0.05
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Table 2 Friction parameters
Parameter
Table 3 Initial and final end-effector positions for linear
contours
Value
Segment 1
Coulomb friction ðf c Þ
Stribeck friction ðf t Þ
Static friction ðf s Þ
Viscous friction ðbÞ
d
3
100
5
1:5
2
5.1.3 Friction Model. In an effort to make a realistic simulation and demonstrate the robustness of the new controller, the friction of the robotic manipulator joints is added to the simulated
model. The friction was modeled based on the following equation
[22]:
8
d 9
q_ =
<
f i (42)
sgnðq_ i Þ þ bq_ i
Fðq_ i Þ ¼ fc þ ðfs fc Þe t
:
;
where fc is the Coulomb friction force, ft is the magnitude of the
Stribeck friction, b is the viscous friction parameter, fs is the static
friction coefficient, and d is an empirically determined parameter
[22].
The friction model in Eq. (42) efficiently predicts the static, viscous, and break-away friction values for the joint actuators of the
robotic manipulators [22]. Table 2 shows the friction parameters
used for the simulations. Since the joint actuators are assumed to
be of the same type, the same parameters are used for each joint.
5.1.4 Gain Selection. The following control gains were chosen for the simulations for both time domain and position domain
controllers:
Linear contour
KP ¼ diagf½9090; 8300; 6720g
Kd ¼ diagf½9870; 7900; 7475g
Ks ¼ diagf½9:6; 4:9; 1:2g
B ¼ diagf½6:5; 6:7; 8:5g
Nonlinear contour
Kp ¼ diagf½6895; 7500; 7530g
Kd ¼ diagf½4850; 7350; 8580g
Ks ¼ diagf½580; 765; 690g
B ¼ diagf½13:1; 13:4; 12:9g
It should be noted that Ks is only used in PSC as it can be seen in
Ref. [14]. Additionally, due to the nature of PDSC, position domain controller gains, Kps ; Kds , and B, utilized only the second
and third entries of the corresponding gains listed above.
A number of different contours will be simulated, both linear
and nonlinear. For the linear contour types, zigzag and diamond
contours motions will be simulated, and for the nonlinear type, the
simulation will consist of a circular contour and an epitrochoidal
contour.
For the purposes of trajectory planning, the lookup table
method is used for the transformation of the desired joint trajectories. More specifically, the desired trajectory of each joint is initially defined as a function of time. Afterwards, the resulting
trajectories of the slave motions are interpolated with respect to
the real trajectory of the master motion. In this way, the desired
slave motions are found with the equivalent master motion real
trajectory via equidistant sampling.
For the time domain controllers, a sampling frequency of 1000
Hz was used for all simulations. Due to the complicated
Segment 2
Segment 3
Segment 4
pf
pi
pf
pi
pf
pi
pf
Zigzag contour
xðmÞ
0.50
yðmÞ
0.10
qðradÞ
1.047
0.60
0.40
1.047
0.60
0.40
1.047
0.70
0.10
1.047
0.70
0.10
1.047
0.80
0.40
1.047
0.80
0.40
1.047
0.90
0.10
1.047
Diamond contour
xðmÞ
0.70
yðmÞ
0.70
qðradÞ
1.047
0.90
0.50
1.047
0.90
0.50
1.047
0.70
0.30
1.047
0.70
0.30
1.047
0.50
0.50
1.047
0.50
0.50
1.047
0.70
0.70
1.047
pi
kinematics of the robotic manipulator, the above described trajectory planning method is used to obtain the desired and real values
of the trajectories and to make sure that the total data are the same
for both time domain control and the position domain control.
5.2 Linear Motion of End-Effector. The initial and final
positions of the end-effector for linear contours are listed in
Table 3. The time duration T is 1 s for each segment of the zigzag
contour motion and 1.5 s for each segment of the diamond contour
motion, respectively.
Using Eq. (41), for the linear motion of the end-effector, the trajectory at the end-effector level for each segment in time domain
is defined as
(
xðtÞ ¼ ðpfx pix ÞrðtÞ
yðtÞ ¼ pfy piy rðtÞ
5.2.1 Zigzag Contour. All four control schemes, time domain
controller in PD type (TDC-PD), PSC, PDC-PD, and PDSC
achieved good axial tracking performance for a zigzag motion.
The position domain controllers were able to produce less tracking
errors than the time domain controllers with the PDSC controller
being approximately 30% more accurate than the PDC-PD controller. All the tracking errors are also displayed in Fig. 2. Concerning synchronization in Fig. 3, PDSC outperformed the PSC
controller by achieving 12.5% (joint 2) and 50% (joint 3) lower
synchronization error with equally lower standard deviations.
Regarding the contour tracking accuracy, the position domain
controllers were able to perform much better than the time domain
controllers, as previous research had also demonstrated [15,16].
The PDC-PD controller had 27% and 52% lower contour errors
than the TDC-PD and PSC controllers, respectively, but the
PDSC controller outperformed the TDC-PD by 88.4% as seen in
Table 4. Hence, it is clear that the PDSC controller achieved the
best contour performance with the least deviation from the desired
contour, demonstrated in Fig. 4.
5.2.2 Diamond Contour. Similar to the zigzag motion tracking, the diamond contour simulations produced good trajectory
tracking results for all four controllers. Still, the PDSC controller
yielded better results than the rest of the controllers as it can be
seen in Fig. 5. More specifically, the PDSC controller produced
approximately 70% less error than the PSC controller and approximately 32% less error than the PDC-PD controller.
As for the synchronization error, the PDSC controller outperformed the PSC controller on the reduction of synchronization
error of the second joint by 27:6% and 18% on the third joint with
the standard deviation following the same trend, seen in Fig. 6.
Once again, the position domain controllers performed better
than their time domain counterparts, featuring mean errors and
standard deviations an order of magnitude lower than the time domain as shown in Fig. 7. Furthermore, the PDSC controller produced 25.6% lower mean error than the PDC-PD controller with
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Fig. 2
Tracking error for zigzag contour
Fig. 3 Synchronization error for zigzag motion
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Table 4 Mean and standard deviation of contour tracking for
zigzag motion
TDC-PD
PSC
PDC-PD
PDSC
Mean (m)
SD (m)
0:00022
0:00015
0:00010
1:285 105
0:00022
0:00014
0:00011
1:199 105
the standard deviation being also lower by 33.4%, as shown in
Table 5.
5.3 Nonlinear Motion of End-Effector. For the nonlinear
contour motion, Table 6 summarizes the initial and final endeffector positions and the maximum velocities for the two simulated nonlinear contours. The time duration T is 8 s for the circular
contour motion and 4 s for the epitrochoidal contour motion,
respectively.
The trajectory at the end-effector level for the nonlinear motion
in time domain is defined as
8
R1 þ R2
>
>
rðtÞ þ a
> xðtÞ ¼ ðR1 þ R2 Þ cosð2prðtÞÞ d cos 2p
>
<
R2
>
>
R1 þ R2
>
>
rðtÞ þ b
: yðtÞ ¼ ðR1 þ R2 Þ sinð2prðtÞÞ d sin 2p
R2
where the parameters are selected as follows:
For the circular contour
a ¼ b ¼ d ¼ 0;
R1 ¼ 0:5;
R2 ¼ 0:1
For the epitrochoidal contour
a ¼ b ¼ 0:3;
Fig. 4 Zigzag motion contour tracking performance
d ¼ 0:08;
R1 ¼ 0:3;
R2 ¼ 0:1
5.3.1 Circular Contour. The circular contour simulations also
yielded good tracking error results, as shown in Fig. 8. On the second joint, TDC-PD produced 49% lower mean error than PDSC.
Fig. 5 Tracking error for diamond contour
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Fig. 6
Synchronization error for diamond contour
Table 6 Initial and final end-effector positions for nonlinear
contours
Initial Position
Fig. 7
Contour tracking error for diamond contour
Table 5 Mean and standard deviation for diamond contour
TDC-PD
PSC
PDC-PD
PDSC
Mean (m)
SD (m)
0:00028
0:00015
2:947 105
2:181 105
0:00027
0:00019
3:973 105
2:652 105
On the third joint, PSC had the lowest tracking error with PDSC
coming second. Similar comments can be made for the standard
deviation of the tracking errors of these four controllers. The PSC
controller was proven to produce more stable tracking error
Journal of Dynamic Systems, Measurement, and Control
Final Position
Max velocity
Circular contour
xðmÞ
0:60 (m)
yðmÞ
0:0 ðmÞ
qðradÞ
1:047 ðradÞ
0:60 ðmÞ
0:0 ðmÞ
1:047 ðradÞ
0:768 ðm=sÞ
0:884 ðm=sÞ
0 ðrad=sÞ
Epitrochoidal contour
xðmÞ
0:62 ðmÞ
yðmÞ
0:30 ðmÞ
qðradÞ
0:524 ðradÞ
0:62 ðmÞ
0:30 ðmÞ
0:524 ðradÞ
1:561 ðm=sÞ
2:120 ðm=sÞ
0:0 ðrad=sÞ
having 10.8% and 32% lower standard deviation for the second
and third joints when compared with the TDC-PD controller
which was second best.
On the contrary, the PDSC controller proves more efficient in
the synchronization errors since its mean error and their standard
deviations are lower than the PSC’s by 4% and 47% for the second and third joint, respectively, as indicated in Fig. 9.
As presented in Fig. 10, the PSC and PDSC controllers generated lower mean contour error than the rest of the controllers.
Nonetheless, PDSC yielded 32% lower mean contour error than
PSC while having a 68.5% difference from the mean error of
PDC-PD which was the lowest performer. Additionally, the standard deviation of the error was equally small for all the controllers
as shown in Table 7, but the PDSC controller yielded 52.9% lower
deviation than the TDC-TD controller which was the second best
in that category.
5.3.2 Epitrochoidal Contour. Interestingly, the time domain
controllers, TDC-PD and PSC, showed better tracking
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Fig. 8 Tracking error for circular motion
Fig. 9
Synchronization error for circular contour
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Fig. 10 Circular motion contour tracking performance
Table 7 Mean and standard deviation of contour tracking error
or circular contour
TDC-PD
PSC
PDC-PD
PDSC
Mean (m)
SD (m)
0:00012
7:98118 105
0:00016
5:34616 105
4:80021 105
5:04345 105
7:06575 105
2:27127 105
performances than the position domain controllers, as shown
in Fig. 11. Particularly, the TDC-PD controller featured 18%
lower tracking error than the PDSC for the second joint, and the
PSC controller achieved 14% lower error than the PDSC for the
third joint. Similarly, the standard deviation of TDC-PD for the
second joint was 53% lower than the standard deviation of PDSC
with the equivalent standard deviation for the PSC being 63%
lower.
In terms of the synchronization shown in Fig. 12, the
PDSC controller outperformed the PSC controller. From Fig. 12,
one can see that, for the second joint, the mean synchronization
error was approximately 27% lower for the PDSC than the
PSC controllers. Similarly, for the third joint, synchronization
error for PDSC was approximately 56% lower than the error of
the PSC.
Despite the PDSC controller being overtaken by the other
controllers on the tracking error in the joint level, Fig. 13
shows that it still outperformed the other controllers in terms
of the reduction of contour tracking error. Table 8 lists the mean
and standard deviation of contour errors controlled by four controllers. In particular, the contour error of PDSC was 52.4%
lower than the error of the second best controller, PSC.
Comparing it with the lower contour error performer, the PDSC
produced 70.6% lower mean error than the PD controller. The
standard deviation follows identical trends, as it can be seen in
Table 8.
Furthermore, the contouring performance of each controller can
be seen in Fig. 14, where the amplified contour errors are shown
and it clearly demonstrated the super performance of the PDSC
compared with other controllers.
Fig. 11 Tracking error for epitrochoidal contour
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Fig. 12 Epitrochoidal contour synchronization error
Fig. 13 Epitrochoidal motion contour tracking performance
6
Fig. 14 Contour performance for epitrochoidal contour (error
amplified by a factor of 60)
Conclusion
In this paper, position domain synchronization control is proposed as an alternative to time domain synchronization control for
performance improvement of contour tracking. A position domain
dynamic model for a multi-DOF serial robotic manipulator is
developed via transformation of the original dynamic equations
from time domain to position domain. Stability analysis is performed and the system is proven to be globally stable based on the
Lyapunov theorem. Lastly, different types of motions in the endeffector lever were simulated to test the effectiveness of the proposed controller.
It is proven that the proposed position domain synchronization
control law can perform better in contour tracking when compared
Table 8 Mean and standard deviation of contour tracking error
of epitrochoidal contour
TDC-PD
PSC
PDC-PD
PDSC
Mean (m)
SD (m)
0:00024
0:00021
0:00034
9:984 105
0:00018
0:00017
0:00034
6:243 105
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Journal of Dynamic Systems, Measurement, and Control
MARCH 2014, Vol. 136 / 021017-13
with its time domain counterpart PSC as well as classic TDC-PD
and PDC-PD. The performance of the new control law should be
further studied and external disturbance or noise should be considered. As a future work, experimental tests must be performed for
the proposed position domain synchronization control.
Acknowledgment
This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant.
References
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