An Approach to Simultaneous Force/Position Control of Robot Manipulators

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

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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 filaret@pma.ru; 2zuev@iacp.dvo.ru

1

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 29

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Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

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 rigorously, 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

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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 ( i = 1, m , m is number of degrees of freedom of manipulator) should develop the torque which provides tracking of desired values of generalized coordinates qi , 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

from object to manipulator. This force will be acting in joints of the manipulator in the form of the external loading torque ni .

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X = AX + Bu + W , X (t 0 ) = X 0 ,

(1)

where X ∈ R k is a vector of phase coordinates, u ∈ R m is a vector of control signals, A ∈ R k ×k is a matrix of dynamic features of system, B ∈ R m×k is a matrix of amplification coefficients at the control signals, W ∈ R k is a vector of external influence on system.

System (1) can be rewritten with respect to errors vector ε = X − X d ( ε ∈ R k ) as

ε = Aε + Bu + Θ , ε (t 0 ) = X (t 0 ) − X d (t 0 ) , 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 Tm i can be divided into two components: motion J iω i

(2)

where Θ = AX d + W − X d , ( Θ ∈ R k ), X d ∈ R k is a vector of desirable values of phase coordinates.

The control law that minimizes the following quadratic functional ~ T

1 Y (u , ε ) = (ε Т Φε + u Т Ψu )dt , 2t

∫

(3)

0

(where J i is a moment of inertia of electric motor

on the given time interval, takes the form [Anderson, 2007; Merriam, 1964]:

shaft and rotating parts of reducers, ωi is a electric

u(t ) = −Ψ −1 B Т ( Kε + E ) ,

motor velocity) and force component ni . After that, we should simultaneously provide desirable values of these two components ( Jω d and nid 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.

(4)

where Φ ∈ R k×k is a positively semi-definite weighting matrix, Ψ ∈ R m×m is a positively-definite weighting ~ matrix, T is a transposing symbol, t 0 and T are the ~ initial and final time of integration ( T → ∞ ).

The matrix K ∈ R k ×k is a solution of Riccati differential equation ~ − K = KA + AТ K − KBΨ −1 B Т K + Φ , K (T ) = 0 .

(5)

Vector E ∈ R k is defined by solution of the following differential equation

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.

At this, the equations (5) and (6) should be solved in inverse time regime.

Method Description of Analytical Designing of Optimum Regulators

~ where E = −( A Т − KBΨ −1 B Т ) −1 .

Let the control object be described by system of the stationary linear differential equations

Taking into account equation (7) the control law (4) can be rewritten as

~  E = −( AТ − KBΨ −1 B Т ) E − KΘ , E (T ) = 0 .

(6)

If the components of vectors X d (t ) , W (t ) and Θ(t ) are changed sufficiently slowly, then vector E can be defined in the next form ~ E = EKΘ ,

(7)

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Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

~ u(t ) = −Ψ −1 B Т ( Kε + EKΘ) = − D λ ε + D en Θ = ~ = u λ + u en = −Ψ −1 B Т [ KX + ( EKA − K ) X d − ~ − EKX d ],

designations are used: k u is the power amplifier gain (8)

k m is the torque coefficient; k с is

coefficient;

the

~ where D λ = Ψ −1 B Т K and D en = −Ψ −1 B Т EK are matrixes of amplification coefficients ( D λ = R m×k , D en = R m×k ).

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; Tm is a total torque of motor; α , ω , ω

Note that the control law (8) is always stable [Merriam, 1964].

are the output rotation angle, velocity and acceleration at the reducer output shaft, respectively; ir is the

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] k

m

i =1

j =1

∑ϕii ε i2max = ∑ψ jj u 2j max ,( i = 1, k ; j = 1, m ), 2

(9)

reducer transmission ratio; n * is a external loading torque reduced to output shaft and determined be n * = n / i r . 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

2

 ε 1max  u   ϕ , ψ =  1max  ψ , ε where ϕ ii =  is jj 11  ε i max   u j max  11 i max     maximal admissible value of error of phase coordinate i, u j max is maximal admissible value of control

signal j .

Having chosen one value ψ jj any (for example ψ 11 =1) we can calculate coefficients ψ jj ( j ≠ 1 ) and then from equation (9) coefficients ϕ ii can be discovered. Thereby

L

dI ~ + RI + k cω = k uU , Tm = Jω + n * = k m I . dt

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 X = [α , ω , Tm ]Т .

Concerning this vector, the differential equation of actuator in Cauchy normal form will be

we will define values of weight matrixes Φ and Ψ .

x1 = a12 x 2 ,

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.

x 2 = a 23 x3 − a 23 n * ,

In the next section, we apply this approach to actuator with DC motor and synthesize force/position regulator for it.

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(11)

~ x 3 = a32 x 2 + a33 x3 + b3U ,

where a12 = b3 =

−k k 1 −R 1 , a 23 = , a32 = m c , a33 = , ir J L L

−k u k m .

L

Relatively the errors ε = x − x d , system (11) can be rewritten in the form

ε1 = a12ε 2 + a12 x2d − x1d ,

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

(10)

ε2 = a 23ε 3 + a 23 x3d − a 23 n * − x 2d ,

~ ε3 = a32ε 2 + a32 x2d + a33ε 3 + a33 x3d − x3d + b3U .

Taking x 2d

=

a 23 x3d

becomes

into − a 23 n

account *d

,

x3d

=n

that *d

+

Jx 2d

x1d = a12 x2d

,

system

(12)

, (12)

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ε1 = a12ε 2 , ε2 = a 23ε 3 + a 23 ( n * d − n * ), ~

ε3 = a32ε 2 + a33ε 3 + b3U + a32 x2d + a33 Jx 2d −

(13)

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which general view is shown on Fig. 3. In this section, the results of the spent experimental researches will be presented.

− Jx2d + a33 n * d − n * d .

Matrix notation of system (13) has a form (3), where 0 a12  A = 0 0 0 a32

0  0    a 23  , B =  0  , b3  a33 

0     *d * a 23 ( n − n ) Θ= . a32 x2d + a33 Jx 2d − Jx2d + a33 n * d − n * d 

(14)

Taking into account the equations (8) and (14), the control law minimizing functional (3) will be ~ U (t ) = − D1λ ε 1 − D2λ ε 2 − D3λ ε 3 + D2en a 23 ( n * d − − n * ) + D3en ( a32 x2d + a33 Jx 2d − Jx2d + a33 n * d − n * d ),

where

(15)

D1λ , D2λ , D3λ , D2en , D3en are corresponding

elements of vectors D λ and D en . They are calculated by (8).

As far as, n d = Ik m − Jω and also taking into account that x1d = α d , x 2d = ω d , x 2d = ω d we can rewrite (15) as

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 inputoutput 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.

~ U (t ) = − D1λ (α − α d ) − D2λ ω + D2en a 23 Jω − − ( D3λ + D2en a 23 ) Ik m + ( D2λ + D3en a32 )ω d + + ( D3λ + D3en a33 ) Jω d + ( D3λ + D2en a 23 +

(16)

+ D3en a33 )n * d − D3en n * d .

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.

FIG. 4. AN DC MOTOR ERROR BY POSITION AT SIMULTANEOUS FORCE/POSITION CONTROL

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

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 n * d (t ) = 2.6 Nm (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 ε 1 of an output shaft of the DC motor does not exceed 0.014 radian at simultaneous force/position control, and maximum value of error ε 3 on the created external 2.8 ⋅ 10 −4 Nm .

loading

torque

is

less

than

Maximum value of position error at tracing of only desirable angle of rotation without creation of the external moment no exceed 6.7 ⋅ 10 −4 radian .

P = H ( q)q + h( q , q )q + Mˆ ( q , q , q) ,

(17)

where H (q ) is a diagonal element of matrix of inertia of manipulator; h( q , q ) is a element of the vector of Coriolis and centrifugal forces; Mˆ ( q , q , q) = M ( q , q , q) + n( q ) , M ( q , q , q) 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] L( H * + J )ω + [ L( 2h * + k v ) + R( H * + J )]ω + + [ L h * + R( h * + k ) + k k ]ω + R( n * + M * + v

c m

~ + M df ) + L( n + M ) = k u k mU , *

(18)

*

where k v is a viscous friction coefficient; M df is a

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.

Coulomb friction moment; H * , h * , M * , reduced to motor shaft value of H , h , M determined by equations H * = H / ir2 , M * = M / i , h * = h / i 2 , M * = M / i .

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.

It should be noted that the derivative M df is zero at

Below we will consider the synthesis of such selfadjusted 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 qi 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]

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r

r

h * , M * are , h , M and h * = h / i r2 ,

r

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] L( J n + n * )ω + R( J n + n * )ω + kc k mω = ku k mU ,

(19)

where J n is a nominal value of moment of inertia

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

(usually J n choose equal J ); U is a control signal coming to input of self-adjusted correcting unit.

Thereby, the self-adjusted control law will be [Zuev, 2009] L( 2h * + k v ) 1 ~ H* + J U = U+ [ Lh * + ω + Jn ku k m ku k m

× [n * (1 −

L H* + J × ) + M * + M dfi ] + Jn ku k m

× [n * (1 −

H* + J ) + M * ] .

Jn

system (13) can be rewritten in the form

ε1 = a12ε 2 , ε2 = a 23ε 3 + a 23 ( n * d − n * ), ~

ε3 = a32ε 2 + a33ε 3 + b3U + a32 x 2d + a33 n * d + + a33 ( H * + J ) x 2d + a33 ( h * + k v ) x 2d + a33 ( M * +

(21)

+ M df ) − n * d − ( H * + J ) x2d − ( 2h * + k v ) x 2d − − h * x 2d − M * .

On this basis of the simultaneous force/position control law (16) for manipulator’s actuator will have the next form U (t ) = − D1λ α − [ D2λ − D2en a 23 ( h * + k v )]ω +

(20)

The signal n * is obtained from equation of torques of actuator

n * = k m I − ( H * + J )ω − ( h * + k v )ω − M *− M df .

Taking into account, that component with ω can be disregarded [Filaretov, 2006], equation for n * will be n * =

x3d = n * d + ( H * + J ) x 2d + ( h * + k v ) x2d + M * + M df , x 3d = n * d + ( H * + J ) x2d + ( 2h * + k v ) x 2d + h * x2d + M * ,

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.

H* + J R + R( h * + k v ) + k c k m (1 − × )]ω + Jn ku k m

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km ~ ( k uU − RI + k cω ) − ( 2h * + k v )ω − h * ω − M * .

L

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.

+ D2en a 23 ( H * + J )ω − ( D3λ + D2en a 23 )k m I + + D λ α d + [ D λ + D en ( a − h * ) + ( D λ +

1 2 3 23 3 λ en * d en + D3 a33 )( h + k v )]ω + [( D3 + D3 a33 ) × × ( H * + J ) − D3en ( 2h * + k v )]ω d + [ D3λ + + D2en a 23 + D3en a33 ]n * d − D3en n * d − D3en M * + [ D3λ + D2en a 23 + D3en a33 ]( M * + M df ).

(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: α d , ω d , ω d , n * d , n * d . Laws of change α d = q d are defined from the decision of a return problem of kinematics for concrete manipulators. If laws of change α d and n * d are previously known then the acquisition of the laws of change of coordinates ω d , ω d , n * d also does not represent difficulties. All these signals are formed before the start of manipulator work.

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 ε = x − x d and taking into account that

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 n * d 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 θ i , α i , ai , and  d i , as shown in Fig. 7.

The reference system i+1 is related to the i reference system by means of the rotation matrix i Ri +1 and the position vector i pOi ,Oi +1 :

i

Ri +1

  cos θ i +1    = cos α i +1 sin θ i +1   sin α sin θ i +1 i +1 

 − sin θ i +1   cos α i +1 cos θ i +1   sin α i +1 cos θ i +1

  , − sin α i +1   cos α i +1  0

 ai +1       i pOi ,Oi +1 =  d i +1 sin α i +1  .  − d i +1 cos α i +1    The force of reaction f i ( f i ∈ R 3 ) or the moment of   force ni ( ni ∈ R 3 ) (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 nid in each joint of any manipulator which provides force and moment by end-effector on object will be    f i = i Ri +1 f i +1 , f i +1 = F , (i = n ,1),     ni = i Ri +1ni +1 + i pOi ,Oi +1 ×( i Ri +1 f i +1 ), ni +1 = N , (23)   nid = eiТ ( f iδ i + niδ i ), where δ i = 1 if joint i is progressive and δ i = 0 if it is rotative; δ i = 1 − δ i ; eiТ = (0 , 0 ,1) is a unit vector connected and directed along the Z i 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 nid 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 F = ζf z the resulting vector of power acting F on df

object, it is necessary to deflect from a normal to a surface on some corner of a friction ϕ Fig. 8 see). Thus vector of components F should be formed in the form: f x = ζF sin(γ ), f y = ζF cos(γ ),

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 Zaxis is located on the axis in the same node, which connects links i-1 and i.

36

(24)

f z= F ,

where γ = artg (

dxtr ) ; xtr ytr is coordinates of a dytr

trajectory of movement of the tool on a processed

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

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of end-effector on trajectory (see Fig. 10) in forward and reverse direction was 3 sec.

surface; ζ - dynamic factor of a sliding friction.

2) in simultaneous force/position control mode at motion on the same trajectory with constant force | F |= 100 N 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. 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. 10. MOTION TRAJECTORY OF END-EFFECTOR

Typical parameters of for all manipulator’s actuators were taken: J = 10 −4 kg ⋅ m ⋅ s 2 , ir = 100 , k u = 100 , L = 0.004 H

,

,

R = 0.4 Ω

k m = k c = 0.02 V ⋅ s

,

M df = 0.06 sign(ω ) , k v = 0.005 N ⋅ m ⋅ s rad .

The following amplifier coefficients of corresponding connections of synthesized control law (22) were λ

λ

calculated: D 1 = 1.224 × 10 , D 2 = 0.1 , D3λ = 1.8 , 3

D2en = 4 × 10 −4 , D3en = 2 ⋅10 −3 .

In Fig. 11, the desired values of generalized coordinates of manipulator at the motion of its endeffector 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.9. SIX - DEGREES OF FREEDOM PUMA MANIPULATOR

Thus, the following parameters of manipulator links were taken, link’s weight: m1 = 35.8 kg , m2 = 3.5 kg , m3 = 3 kg ; l3 = 0.4 m ,

link’s

length l1 = 0.2 m , l 2 = 0.5 m ,

l t = 0.08 m

;

end-effector’s

weight:

m t = 0.5 kg .

A simulation was produced in two modes of the manipulator work: 1) in position tracking mode only when time of motion

FIG. 11. DESIRED VALUES OF THE GENERALIZED COORDINATES OF MANIPULATOR

37

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Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

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. 15. DYNAMIC ERRORS OF MOTION OF END-EFFECTOR ALONG TRAJECTORY (FIG.13) AND ERROR OF EXERTED FORCE

The maximum values of position error ( e1 ) and error of exerted force to surface ( e3 ) 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 ( e2 ) 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

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 e1 and e3 accordingly), and as well as the dynamic error (see curve e2 ) of spatial movement of this endeffector at the mode of position tracking are presented.

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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

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 3, July 2013

and Govenmant of Russia (Grant № 02.G25.31.0025).

Hogan,

N.

www.as-se.org/ccse

“Impedance

Control:

An

Approach

to

Ma¬nipulation: Part I-Theory; Part II -Implementation;

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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

40

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