An Exponentially Convergent Adaptive Sliding Mode Control of Robot Manipulators
An Approach to Simultaneous Force/Position Control of Robot Manipulators
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Transcript of An Approach to Simultaneous Force/Position Control of Robot Manipulators
Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013 www.as-se.org/ccse
29
An Approach to Simultaneous Force/Position Control of Robot Manipulators Vladimir Filaretov1, Alexander Zuev2
Institute of Automation and Control Processes, Far Eastern Federal University 5, Radio str., 690041, Vladivostok, Russia [email protected]; [email protected]
Abstract
In this article, the new approach to high-quality simultaneous force/position control for robot manipulators is proposed based on the division of actuator's total torque of manipulator for each degree of freedom onto two components: position (which provides the position and orientation of the end-effector) and force (which provides the force and moment from end-effector on environment), and on the further simultaneous minimization of the errors of this two component with the help of the quadratic functional. Therefore, it is possible to synthesize the force/position control systems for manipulators which allow controlling precisely both the position of end-effector of manipulator and the force exerted by its end-effector on some objects.
Keywords
Force/Position Control; Manipulator
Introduction
There are many types of manufacturing operations performed by the manipulators, which require that robot interacts with some objects or its environment. For example: deburring, scraping, grinding, polishing, twisting, cutting, excavating, etc. Fulfillment of these operations requires that a manipulator provides the desired motion of its end-effector with desired orientation and simultaneously creates the necessary force on work objects.
To date, there are several approaches for robot force/position control [Yoshikawa, 2000; Zeng, 1997] which can be divided into two groups: fundamental force control and advanced force control, and algorithms of the first based on application of the relationship between position and applied force or between velocity and applied force or the application of direct force feedback, or their combinations involve following methods: stiffness control, impedance control, hybrid position/force control, hybrid impedance control, inner-outer position/force control
and parallel control; while algorithms of the second group based on adaptive or robust control methods combined with the fundamental methods, include: adaptive compliant motion control, adaptive impedance (or admittance) control, adaptive force/position control. In this section, we will briefly consider the given methods and their features.
Stiffness control can be passive or active. In passive stiffness control, the end-effector of manipulator is equipped with a mechanical device composed of springs (or springs and dampers). Application of this method is successful only in assembling. By contrast, active stiffness control [Salisbury, 1980] can be regarded as a programmable spring, since through a force feedback, the stiffness of the closed-loop system is altered. However, the changing stiffness of manipulator mechanism can lead to low accuracy and uncontrolled vibrations.
The basis of impedance control [Hogan, 1985; Anderson, 1988] is that the manipulator control system should be designed not to track a motion trajectory alone, but rather to regulate the mechanical impedance (relationship between the velocity and the applied force) of the manipulator. Task of impedance control is to provide the behavior of the controlled system by an equivalent mass-spring-damping system. The disadvantages of this approach are the higher flexibility of manipulator and low accuracy, necessity to use contact model but it is not always available and practical. Also, the field of using impedance control is bounded.
Hybrid control approach [Raibert, 1981; Kwan, 1995] selects joints in which position of end-effector of manipulator and directions in which the force exerted by the end-effector on some object should be controlled for a given task, and performs some control to make the position and force follow the given desired trajectories simultaneously. Usually, a control system based on this approach has two feedback loops
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for separated control of position and force. Current position in position-control directions and current force in force-controlled directions are measured by complex force/moment sensors. Corrections based on these measurements are applied by joint actuators to make the manipulator trace the desired position and force trajectories. Normally, the position control law consists of a PD action, and the force control law consists of a PI action. The disadvantages of hybrid control method are poor accuracy of motion with high speed in view of great errors in force-controlled directions and some difficulty with matrix of switching for complex manipulators. In addition, if the dynamics of the manipulator is not considered rig-orously, the hybrid control may cause some unstable response.
An inner-outer position/force control [DeSantis, 1996] is strategy for interaction with objects of unknown model where an outer force control loop is closed around the inner motion control loop which is typically available in a robot manipulator. In order to embed the possibility of controlling motion along the unconstrained task directions, the desired end-effector motion can be input to the inner loop of an inner/outer motion/force control scheme. The resulting parallel control is composed of a force control action and a motion control action, where the former is designed so as to dominate the latter in order to ensure force control along the constrained task directions. Low speed and accuracy are disadvantage of the given approach.
The key concept of parallel force/position control strategy [Chiaverini, 1998; Siciliano, 1999] is to combine the simplicity and robustness of the impedance and inner-outer position/force control schemes with the ability of controlling both position and force. In order to embed the possibility of controlling motion along the unconstrained task space directions, a desired position can be input to the inner loop of an inner-outer position/force control scheme. The result is two control actions working in parallel, namely, a force control action and a position control action. In order to ensure force control along the constrained task space directions, the force action is designed so as to dominate the position action. Drawbacks of this method include low accuracy and bounded application.
The basic objective of adaptive and robust force control [Fanaei, 2005; Roy, 2002; Mut, 2000; Chien Ming-Chih, 2004; Saadia, 2001] is to maintain consistent performance of control system at the
presence of variable parameters of robot and unknown environment. Based on the existing definitions of the fundamental force control methods, robot adaptive force controller incorporates certain adaptive strategy into controller in order to maintain the proper desired stiffness, impedance, admittance and so forth when unknown parameters of robot and contact environment exist. This method has a similar drawbacks as well as that and fundamental force control methods which underlies. Further, adaptive and robust force control methods have difficulties in realization.
In almost all these methods, the complex force/moment sensors are used to determine a magnitude and direction of current force. And then the force control law is formed on the basis of this information. These force/moment sensors are mounted near end-effector and have a heavy load [Gorinevsky, 1997]. For this reason, a strong interaction between all degrees of freedom of manipulator are arising and it is necessary to apply high-power actuators. Also force/moment sensors have the following disadvantages: the limited effective range, impossibility to define some components of force vector, essential sensitivity to changing of environment's parameters, high cost. Therefore it will be topical to develop approach which not using force/moment sensors in control.
For qualitative force/position control, it is necessary to provide the constant stiffness of manipulator and the simultaneous control, when each degree of freedom of manipulator is controlled by position and force. As well, it is important to avoid the use of force/moment sensors. In this article, the problem is solved by means of creation of the synthesis method of simultaneous force/position control systems which meet the given requirements.
The Description of the Proposed Method
At the manipulator position control, the actuator of its i-th joint ( mi ,1= , m is number of degrees of freedom of manipulator) should develop the torque which provides tracking of desired values of generalized coordinates iq , which in turn will provide motion of end-effector on the program trajectory with the given orientation.
Let the end-effector of manipulator start to exert some generalized force F on object (see Fig. 1). At this, the force equal on module but return in direction will act
Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013 www.as-se.org/ccse
31
from object to manipulator. This force will be acting in joints of the manipulator in the form of the external loading torque in .
FIG. 1. MANIPULATOR EXERTED FORCE ON SURFACE
Thereby, to provide the motion of end-effector on program's trajectory and simultaneously to create the force on surface, it is necessary to ensure simultaneously the desirable torque for movement and the desirable torque for force in each actuator. This is possible when the total torque of each actuator
imT can be divided into two components: motion iiJ ω
(where iJ is a moment of inertia of electric motor shaft and rotating parts of reducers, iω is a electric motor velocity) and force component in . After that, we should simultaneously provide desirable values of these two components ( dJω and d
in respectively) for each joint for qualitative simultaneous force/position control.
This division of the total torque allows simultaneously controlling motion of the end-effector and exerting force on a surface without force/moment sensors. At this for formation of feedback of force, the current sensors can be applied in each joint of manipulator.
We propose to solve the problem of division of the total torque of actuator into two components by means of methods of the optimum control theory namely using analytical designing of optimum regulators by quadratic criterion [Athans, 2006; Naidu, 2002]. At the expense of choice of phase coordinates of actuators and their desirable values, it is possible to divide the total torque into two components and then by using the quadratic criterion of quality, it is possible to minimize errors on these coordinates.
Method Description of Analytical Designing of Optimum Regulators
Let the control object be described by system of the stationary linear differential equations
WBuAXX ++= , 00 XtX =)( , (1)
where kRX ∈ is a vector of phase coordinates, mRu∈ is a vector of control signals, kkRA ×∈ is a matrix of dynamic features of system, kmRB ×∈ is a matrix of amplification coefficients at the control signals,
kRW ∈ is a vector of external influence on system.
System (1) can be rewritten with respect to errors vector dXX −=ε ( kR∈ε ) as
Θ++= BuAεε , )()()( 000 tXtXt d−=ε , (2)
where dd XWAX −+=Θ , ( kR∈Θ ), kd RX ∈ is a vector of desirable values of phase coordinates.
The control law that minimizes the following quadratic functional
dtuuuYT
t∫ Ψ+Φ=
~
ТТ )(),(0
21 εεε , (3)
on the given time interval, takes the form [Anderson, 2007; Merriam, 1964]:
)()( Т EKBtu +Ψ−= − ε1 , (4)
where kkR ×∈Φ is a positively semi-definite weighting matrix, mmR ×∈Ψ is a positively-definite weighting matrix, T is a transposing symbol, 0t and T~ are the
initial and final time of integration ( ∞→T~ ).
The matrix kkRK ×∈ is a solution of Riccati differential equation
Φ+Ψ−+=− − KBKBKAKAK ТТ 1 , 0=)~(TK . (5)
Vector kRE ∈ is defined by solution of the following differential equation
Θ−Ψ−−= − KEBKBAE )( ТТ 1 , 0=)~(TE . (6)
At this, the equations (5) and (6) should be solved in inverse time regime.
If the components of vectors )(tX d , )(tW and )(tΘ are changed sufficiently slowly, then vector E can be defined in the next form
Θ= KEE ~ , (7)
where 11 −−Ψ−−= )(~ ТТ BKBAE .
Taking into account equation (7) the control law (4) can be rewritten as
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],~)~([
)~()(Т
Т
d
den
en
XKE
XKKAEKXBuu
DDKEKBtu
−
−−+Ψ−=+=
=Θ+−=Θ+Ψ−=−
−
1
1
λ
λεε
(8)
where KBD Т1−Ψ=λ and KEBDen ~Т1−Ψ−= are matrixes of amplification coefficients ( kmRD ×=λ ,
kmen RD ×= ).
Note that the control law (8) is always stable [Merriam, 1964].
If it is assumed that maximum deviations of all phase coordinates during each moment of time bring into quadratic criterion of quality, the identical contribution, and their full contribution is equal to the total contribution of control signals, each of which also brings the identical contribution in specified quadratic criterion of quality then we can write down [Merriam, 1964]
∑∑==
=m
jjjj
k
iiii u
1
2
1
2maxmax ψεϕ ,( ki ,1= ; mj ,1= ), (9)
where 11
21 ϕε
εϕ
=
max
max
iii , 11
21 ψψ
=
max
max
jjj u
u, maxiε is
maximal admissible value of error of phase coordinate i, maxju is maximal admissible value of control
signal j .
Having chosen one value jjψ any (for example 11ψ =1)
we can calculate coefficients jjψ ( 1≠j ) and then from
equation (9) coefficients iiϕ can be discovered. Thereby we will define values of weight matrixes Φ and Ψ .
It's possible to see that using the method of analytical designing of optimum regulators, we can divide control into several component by choosing phase coordinates and its desirable values. And after this, minimizing quadratic criterion of quality, we can simultaneously eliminate errors on this components.
In the next section, we apply this approach to actuator with DC motor and synthesize force/position regulator for it.
Force-Position Controller for Actuator with DC Motor
Let's consider the actuator with DC motor and solve a task of simultaneously control by angle of rotation of output shaft and the external loading torque developed by it. A block diagram of of such actuator is presented on Fig. 2in which, the following
designations are used: uk is the power amplifier gain coefficient; mk is the torque coefficient; сk is the counter-electromotor force coefficient; J is the total moment of the inertia reduced to the motor shaft; R , L , I are resistance, inductance and current of electric motor rotor circuits, accordingly; U~ is the rotor voltage; mT is a total torque of motor; α , ω , ω
are the output rotation angle, velocity and acceleration at the reducer output shaft, respectively; ri is the
reducer transmission ratio; *n is a external loading torque reduced to output shaft and determined be
rinn /* = . Here and below we won't write index i for simplicity.
FIG. 2. ACTUATOR WITH DC MOTOR
The equations of electric and mechanical chains of such actuator have the following form
.,~ * IknJTUkkRIdtdIL mmuc =+==++ ωω (10)
To simultaneously control both the position and the torque, it is necessary to choose phase coordinates containing output rotation angle and developed torque [Filaretov, 2006; Zuev, 2006]. Therefore, let's the vector of phase coordinates be Т],,[ mTX ωα= . Concerning this vector, the differential equation of actuator in Cauchy normal form will be
,~,
,*
Ubxaxax
naxax
xax
33332323
233232
2121
++=
−=
=
(11)
where ri
a 112 = ,
Ja 1
23 = , L
kka cm−=32 ,
LRa −
=33 ,
.Lkkb mu−
=3
Relatively the errors dxx −=ε , system (11) can be rewritten in the form
.~,
,*
Ubxxaaxaa
xnaxaa
xxaa
ddd
dd
dd
333333332322323
2233233232
12122121
+−+++=
−−+=
−+=
εεε
εε
εε
(12)
Taking into account that dd xax 2121 = , ddd naxax *
233232 −= , ddd xJnx 23 += * , system (12) becomes
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.
~),(
,
**
**
ddd
dd
d
nnaxJ
xJaxaUbaa
nnaa
a
−+−
−++++=
−+=
=
332
23323233332323
233232
2121
εεε
εε
εε
(13)
Matrix notation of system (13) has a form (3), where
.)(
,,
**
**
−+−+−=Θ
=
=
ddddd
d
nnaxJxJaxanna
bB
aaa
aA
332233232
23
33332
23
12
0
00
000
00
(14)
Taking into account the equations (8) and (14), the control law minimizing functional (3) will be
),()
()(~
***
*
ddddden
den
nnaxJxJaxaDn
naDDDDtU
−+−++−
−+−−−=
3322332323
232332211 εεε λλλ (15)
where λ1D , λ
2D , λ3D , enD2 , enD3 are corresponding
elements of vectors λD and enD . They are calculated
by (8).
As far as, ωJIkn md −= and also taking into account
that ddx α=1 , ddx ω=2 , ddx ω =2 we can rewrite (15) as
.)
()(
)()(
)()(~
** denden
enden
denm
en
end
nDnaD
aDDJaDD
aDDIkaDD
JaDDDtU
3333
23233333
32322323
23221
−+
+++++
++++−
−+−−−=
λλ
λλ
λλ
ω
ω
ωωαα
(16)
It should be noted that simultaneous force/position control law (16) do not contain signal with (its have neglected at the derivation of control law) because the researches [Zuev, 2006] have shown that all errors of actuators increase only a maximum of 0.8%.
Thus, the generated control law (16) allows thesimultaneous operation of both the position of output shaft of the actuator and external loading torque created by actuator because simultaneous minimization of errors is provided by position and external torque.
Experiments Researches of Force/Position Controller for Actuator with DC Motor
For the purpose of check of working capacity of the simultaneous force/position control systems (16) synthesised on the basis of a developed method for separate actuators of manipulators with constant parametres, experimental modelling was madeand conducted with use of the electromechanical stand in
which general view is shown on Fig. 3. In this section, the results of the spent experimental researches will be presented.
FIG. 3. THE GENERAL VIEW OF THE ELECTROMECHANICAL
STAND
The given stand contains two servodrives AeroTech BA30 of comparable power connected by the belting. The digital control is carried out by the stand through the PC by means of a multipurpose board of input-output Sensoray Model 626. The structure of a sensory subsystem of the stand includes following sensors: encoder for definition of an angle of rotation of an output shaft of the DC motor, tachogenerator for definition of a velocity of rotation of this shaft and the current sensor. The Matlab is used for control of the stand.
FIG. 4. AN DC MOTOR ERROR BY POSITION AT SIMULTANEOUS FORCE/POSITION CONTROL
FIG. 5. AN DC MOTOR ERROR ON THE DEVELOPED EXTERNAL
LOADING TORQUE
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The results of research of the synthesised force/position control system (16) of the DC motor at tracing of desirable angle of rotation (the desirable angle was set linearly by an increasing signal) with simultaneous creation on it, desired external torques
Nmtn d 62.)(* = (thus external loading was modelled
by special system of subscales) are shown in Fig. 4 and Fig. 5 .
These figures show that maximum value of positional error 1ε of an output shaft of the DC motor does not exceed 0.014 radian at simultaneous force/position control, and maximum value of error 3ε on the created external loading torque is less than
Nm41082 −⋅. .
Maximum value of position error at tracing of only desirable angle of rotation without creation of the external moment no exceed radian41076 −⋅. .
Thus, results of experimental modelling of the synthesised force/position control system of DC motor with constant parametres completely have confirmed high efficiency in various operating modes of this motor.
Therefore, we successfully have applied approach of division of actuator's total torque to two component for only actuator and have received clear result in the form of force/position control law. But if we have dealt with actuators mounted in manipulator, we can not apply this approach in the obvious form because each actuator at the manipulator motion will have considerable variable parameters due to interaction. To eliminate interactions, we will apply self-adjusted correcting units stabilizing variable parameters of the manipulator's actuators at nominal level and therefore eliminating acting of interaction and friction.
Below we will consider the synthesis of such self-adjusted correcting units.
Synthesis of the Self-Adjusted Correcting Units for Manipulator’s Actuators
It is expediently to use the second form Lagrange equation or other methods to obtain the torques P in each degree of freedom of the manipulator while moving its end-effector along the complex spatial trajectory with force acting on surface if laws of change manipulator’s generalized coordinates iq are known. The expressions for generalized torques acting in each degree of freedom of manipulator can be represented in the form [Filaretov, 2008; Zuev, 2009]
),,(ˆ),()( qqqMqqqhqqHP ++= , (17)
where )(qH is a diagonal element of matrix of inertia of manipulator; ),( qqh is a element of the vector of Coriolis and centrifugal forces;
)(),,(),,(ˆ qnqqqMqqqM += , ),,( qqqM is a torque which represents interactions acting to i-th degree of freedom from other manipulator links and also action of the gravitational forces.
Tacking into account equations (10), (17) and also Coulomb and viscous frictions, the differential equation of loaded manipulator’s actuator can be written as [Zuev, 2009]
,~)()
(])([
)]()([)(
**
****
***
UkkMnLM
MnRkkkhRhL
JHRkhLJHL
mudf
mcv
v
=+++
+++++++
++++++
ω
ωω 2
(18)
where vk is a viscous friction coefficient; dfM is a
Coulomb friction moment; *H , *h , *M , *h , *M are reduced to motor shaft value of H , h , M , h , M and determined by equations 2
riHH /* = , 2rihh /* = ,
riMM /* = , 2rihh /*
= , riMM /* = .
It should be noted that the derivative dfM is zero at
the motion of actuators because the Coulomb friction at motion is constant. Therefore, this derivative is absent in equation (18).
However, it can be seenthat the equation (18) has significantly changing coefficients because of sufficient changing in parameters *H , *h , *M , *h and *M . Hence, electric servo actuators described by equation (18) have changing dynamic properties and changing quality parameters of their work. Therefore, it is not possible to use the proposed force/position method.
To solve this problem, it is proposed to apply synthesis method of self-adjusted correcting units [Filaretov, 1993; Filaretov, 1995]. However, this method should be modified for force/position control namely, and it is necessary to stabilize variable parameters but to pass without change external loading torque signal i.e. no compensation for it.
Therefore, let the differential equation describe dynamics of servo actuator with constant nominal parameters and stable dynamic properties at force/position, and control will be [Zuev, 2009]
UkkkknJRnJL mumcnn =++++ ωωω )()( ** , (19)
where nJ is a nominal value of moment of inertia
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(usually nJ choose equal J ); U is a control signal coming to input of self-adjusted correcting unit.
By equation (19), we specify the desired dynamic properties of actuator at force/position control. Equations (19) and (18) will be equal if the senior derivatives of these equations will be equal. To provide it, it's necessary to express the highest order derivative from equation (19) and be substituted to (18). After the substitution, we can obtain the signal U~ providing the transformation of equation (18) with sufficiently changing parameters to equation (19) with constant nominal parameters.
Thereby, the self-adjusted control law will be [Zuev, 2009]
.])([
])([
)]()(
[)(~
**
*
**
*
**
***
MJ
JHn
kkL
MMJ
JHn
kkR
JJHkkkhR
hLkkkk
khLU
JJHU
n
mudfi
n
munmcv
mumu
v
n
++
−×
×++++
−×
×++
−+++
+++
++
=
1
1
1
12
ω
ω
(20)
The signal *n is obtained from equation of torques of actuator
dfvm MMkhJHIkn −−+−+−= **** )()( ωω .
Taking into account, that component with ω can be disregarded [Filaretov, 2006], equation for *n will be
**** )()~( MhkhkRIUkL
kn vcum
−−+−+−= ωωω 2 .
Components analysis of equation (20) shows that realization of self-adjusted correction units for each actuator does not present principal difficulties.
After stabilization of variable parameters of actuators on some nominal level, we can use method of division of total torque for synthesis force/position regulator of manipulator's actuators. The features of this synthesis will be considered in the next section.
Synthesis Features of Force-Position Controller for Actuators of Manipulator
Manipulator actuators after stabilization of their variable parameters will be described in Cauchy normal form by equation (11).
Relatively, the errors dxx −=ε and taking into account that
,)()(
,)()(*****
****
MxhxkhxJHnx
MMxkhxJHnxdd
vddd
dfd
vddd
++++++=
++++++=
2223
223
2
system (13) can be rewritten in the form
.
)()()
()()(
~),(
,
**
***
***
*
**
Mxh
xkhxJHnM
MaxkhaxJHa
naxaUbaa
nnaa
a
d
dv
dddf
dv
d
dd
d
−−
−+−+−−+
++++++
+++++=
−+=
=
2
22
33233233
3323233332323
233232
2121
2
εεε
εε
εε
(21)
On this basis of the simultaneous force/position control law (16) for manipulator’s actuator will have the next form
).]([
]
[)]()(
)[()])(
()([
)()(
)]([)(
*
***
**
*
*
*
*
dfenen
endendenen
dv
en
endv
en
endm
enenv
en
MMaDaDD
MDnDnaDaD
DkhDJH
aDDkhaD
DhaDDD
IkaDDJHaD
khaDDDtU
++++
+−−++
+++−+×
×++++
++−+++
++−++
++−−−=
3332323
33333232
33
3333333
323321
2323232
23221
2
λ
λ
λ
λλλ
λ
λλ
ω
ω
α
ω
ωα
(22)
The block diagram of the force/position regulator (22) is shown on Fig. 6.
It can be observed that for realization of this force/position regulator, it is necessary to enter feedback on phase coordinates: α , ω , ω , I and also connections on the desired values: dα , dω , dω , dn* ,
dn* . Laws of change dd q=α are defined from the
decision of a return problem of kinematics for concrete manipulators. If laws of change dα and dn* are previously known then the acquisition of the laws of change of coordinates dω , dω , dn*
also does not represent difficulties. All these signals are formed before the start of manipulator work.
FIG. 6. FORCE/POSITION REGULATOR
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It is necessary to note that control laws (20) and (22) contain the components proportional to angular accelerations of motor’s rotors of the manipulator. These components are present always when sizes of inductances of motor rotor circuits L of used DC motors are considerable. However, these inductances are very insignificant in many modern motors and it is possible simply to disregard them. In this case, the specified components in equations (20) and (22) vanish.
Thus the obtained control law (22) allows simultaneously accurately control over both the position of output shaft of actuators and external load torques created by this actuators since the minimization of functional (3) containing errors of two this component is guaranteed.
At last, it is necessary to calculate the desired value of external loading torque dn* in each degrees of freedom of manipulator which provide given generalized force. Generally for this purpose relation of the external torque vector with generalized force through the Jacobian matrix of manipulator is used. But in this case, there are some difficulties related with compound computing for complex manipulators. Therefore, we propose the recurrence equations for calculation of the desired external torque for manipulators with any kinematics in real time.
Recurrence Equations for Calculation of Desirable External Torque
To describe manipulator’s mechanism, it is possible to use the Denavit-Hartenberg modified notation [Poul, 1981], which considers four parameters iθ
, iα , ia , and
id
, as shown in Fig. 7.
FIG. 7. SYSTEMS OF COORDINATES AND THE MAIN VECTORS
OF THE MANIPULATOR’S LINK
In the mentioned notation, the reference system corresponding to link i is located on joint i, and the Z-axis is located on the axis in the same node, which connects links i-1 and i.
The reference system i+1 is related to the i reference system by means of the rotation matrix 1+i
i R and the position vector
1+ii OOi p , :
−−
=
+++++
+++++
++
+
11111
11111
11
1
0
iiiii
iiiii
ii
ii R
αθαθααθαθα
θθ
coscossinsinsinsincoscossincos
sincos,
−=
++
++
+
+
11
11
1
1
ii
ii
i
OOi
dd
ap
ii
αα
cossin, .
The force of reaction if
( 3Rfi ∈
) or the moment of force in ( 3Rni ∈
) (according to type of joint) will be acting in joint i of manipulator when its end-effector will exerts some generalized force on objects.
Therefore, the recurrence allowing calculation of desirable (program) values of external loading torques
din in each joint of any manipulator which provides
force and moment by end-effector on object will be
),(
,),(
),,(,,
Т
,
iiiiidi
iiii
OOi
iii
i
iiii
i
nfen
NnfRpnRn
niFffRf
ii
δδ
+=
=×+=
===
+++++
+++
+ 11111
111
1
1
(23)
where 1=iδ if joint i is progressive and 0=iδ if it is rotative; ii δδ −=1 ; ),,(Т 100=ie is a unit vector connected and directed along the iZ axis; F , N are the force and moment (parts of generalized force) exerted be end-effector on surface.
Thereby, the recurrences (23) allow calculation on the values d
in in real time for any multilink manipulators.
For qualitative performance of all forced operations with objects of works, it is necessary to consider force of a dry friction between this tool and a processed surface. At contact motion of the working tool on this surface taking into account force of a dry friction the zdf fF ζ= the resulting vector of power acting F on
object, it is necessary to deflect from a normal to a surface on some corner of a frictionϕ Fig. 8 see). Thus vector of components F should be formed in the form:
,
),cos(),sin(
z Ff
FfFf
y
x
=
==
γζγζ
(24)
where )(tr
tr
dydxartg=γ ; trx try is coordinates of a
trajectory of movement of the tool on a processed
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surface; ζ - dynamic factor of a sliding friction.
FIG. 8. FORMATION OF A VECTOR OF DESIRABLE FORCE
ACTING ON OBJECT OF WORKS TAKING INTO ACCOUNT THE DRY FRICTION FORCE
Thus, in the given section for any multilink manipulators, the problem of formation of the desirable moments at which realisation provides creation of desired force acting of the tool on a processed surface taking into account that real-life force of a friction is solved.
Work Simulation of PUMA Manipulator with Synthesized Force/Position Control System
To estimate the efficiency of proposed method, the simulation of force/position control system (see (20), (22), (23) and (24)) for six - degrees of freedom PUMA manipulator (see Fig. 9) has been performed [Zuev, 2008].
FIG.9. SIX - DEGREES OF FREEDOM PUMA MANIPULATOR
Thus, the following parameters of manipulator links were taken, link’s weight: kgm 8351 .= , kgm .532 = ,
kgm 33 = ; link’s length ml 201 .= , ml 502 .= , ml 403 .= , mlt 080.= ; end-effector’s weight: kgm t .50= .
A simulation was produced in two modes of the manipulator work:
1) in position tracking mode only when time of motion
of end-effector on trajectory (see Fig. 10) in forward and reverse direction was 3 sec.
2) in simultaneous force/position control mode at motion on the same trajectory with constant force
NF 100=|| on environment (perpendicularly to this surface). Thus frictional force between working tool and the surface was taken into account as well. Dynamic factor of a sliding friction is ζ = 0.12.
FIG. 10. MOTION TRAJECTORY OF END-EFFECTOR
Typical parameters of for all manipulator’s actuators were taken: 2410 smkgJ ⋅⋅= − , 100=ri , 100=uk ,
H0.004=L , Ω0.4=R , sVkk cm ⋅== 0.02 ,
)sign( 0.06 ω=dfM , radsmNkv ⋅⋅= 0.005 .
The following amplifier coefficients of corresponding connections of synthesized control law (22) were calculated: 3λ
1 10224.1 ×=D , 1.0λ2=D , 813 .λ =D ,
410−×= 42enD , 3
3 102 −⋅=enD .
In Fig. 11, the desired values of generalized coordinates of manipulator at the motion of its end-effector along trajectory (Fig. 10) are shown. The desired values of the external torques for each joint actuator calculated with help of expressions (23) are shown on Fig. 12.
FIG. 11. DESIRED VALUES OF THE GENERALIZED
COORDINATES OF MANIPULATOR
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FIG. 12. DESIRED VALUES OF EXTERNAL TORQUES
Position errors of target shaft of each actuator and errors of the external torques at the simultaneous force/position control mode taking into account interaction between all degrees of freedom are presented in Fig. 13 and Fig. 14, respectively.
FIG. 13. POSITION ERRORS OF THE MANIPULATOR'S
ACTUATORS
FIG. 14. ERRORS OF THE EXTERNAL TORQUES, DEVELOPED BY
THE MANIPULATOR'S ACTUATORS
The maximum values of the position errors of target shaft for all actuators the maximum values of errors of external torques do not exceed 0.018 degree and 0.0135 Nm, repectively. (see Fig. 14).
In Fig. 15, the dynamic errors of the spatial motion of end-effector and the force exerted on surface at simultaneous force/position control mode (see curves
1e and 3e accordingly), and as well as the dynamic error (see curve 2e ) of spatial movement of this end-effector at the mode of position tracking are presented.
FIG. 15. DYNAMIC ERRORS OF MOTION OF END-EFFECTOR
ALONG TRAJECTORY (FIG.13) AND ERROR OF EXERTED FORCE
The maximum values of position error ( 1e ) and error of exerted force to surface ( 3e ) at simultaneous force/position control mode do not exceed 3.2×10-4 meters and 7×10-2 N accordingly (see Fig. 15). The maximum value of dynamical error of spatial movement of the end-effector ( 2e ) only at the mode of position tracking does not exceed 2.1×10-4 meters.
Thus, the results of simulation have completely confirmed high efficiency (dynamic accuracy) of the force/position control systems synthesized on the basis of the proposed method.
Conclusions
In article, the new synthesis method of high-quality force/position control systems of robot manipulators is proposed. These control systems without using force/moment sensors and keeping the stiffness of mechanism allows controlling precisely both the position of end-effector of manipulator and the force (its may be variable) exerted by its end-effector on some object (or environment). The given method is based on a approach allowing dividing the torque developed by each actuator of manipulator onto two components: a component of motion and force component. Having minimized errors of these two components by means of quadratic functiona,l it is possible to synthesize high-quality simultaneous force/position control of manipulator.
Results of mathematical simulation have shown high efficiency of the synthesized control systems at the various operating modes of the manipulator. This control systems can be realized by serial microprocessors.
ACKNOWLEDGMENT
This work is supported by RFBR (Grants № 12-08-31026 and № 11-07-98505), Grant of President of Russia № 1523.2012.5
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and Govenmant of Russia (Grant № 02.G25.31.0025).
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Vladimir Filaretov. In 1976 he was awarded the degree of candidate of sciences (engineering) and in 1990 the degree of doctor of sciences in the field of automatic control. In 1992 Mr. Filaretov was confirmed in Professor’s degree. In 1995 he was elected the member of an Russian and in 1996 the member of an International Engineering Academy. At
present, he is professor of the Far Eastern State Technical University, Head of Robotic Laboratory of the Institute of Automatics and Control Process of Russian Academy of Sciences, President of Far Eastern Branch Russian Engineering Academy and Vice-president of Russian Engineering Academy; and he received the Professor’s degree in Electrical Engineering from Best University in 2003. His research interests include nonlinear control, adaptive control, robotics, underwater vehicles.
Alexander Zuev received the PhD degree in Institute of Automation and Control Processes in 2010. His research interests include nonlinear control, force/position control and system identification