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Adaptive high-speed resonant robot
Mechatronics Vol, 6, No. 8, pp. 897-913, 1996 Copyright © 1996 Elsevier Science Ltd. All rights reserved. Printed in Great Britain. 0957-4158196 $15.00+0.00
Pergamon
PII: S0957-4158 (96) 00030.X
ADAPTIVE HIGH-SPEED RESONANT ROBOT V. I. BABITSKY* and M. Y. CHITAYEV* *Department of Mechanical Engineering, Loughborough University, Loughborough, Leics LE11 3TU, U.K. and tMotorola Australia Software Centre, 2 Second Avenue, Technology Park, Adelaide, SA 5095, Australia
(Received 20 April 1996; accepted 25 June 1996) Abstract--The use of the natural modes of oscillation for efficient performing of complex space motions of a robot with changeable parameters and dynamically interacting degrees of freedom is discussed. For the realisation of high-speed motions in these conditions under the limitation of control torques, it is necessary to identify at first, through on-line analysis of motions in the process of manipulation, the real parameters of natural modes. Then the control torques should be brought into correlation with natural motions by the adaptation of control in accordance with the findings of identification. As a result, the processes of identification, adaptation and control have to be optimised in accordance with the dynamics of robot oscillation. The formulated concept is demonstrated by research and development of an adaptive high-speed manipulator MARS-3 realised by the above mentioned principles for operating in cylindrical co-ordinates. Copyright © 1996 Elsevier Science Ltd.
1. INTRODUCTION The creation of resonant robots [1-3] requires the development of motion control concepts for effective maintenance of autoresonant states [4] in spite of changes in parameters of the robot oscillating system. The changing of manipulated mass and interaction of dynamically coupled degrees of freedom have a major influence on natural modes of robot oscillation. The dynamical correction of control through the variation of programs or deviation of manipulator parameters during the robot's lifetime is also important. The above mentioned parameters and dynamic loads usually cannot be measured preliminarily or directly and have to be identified in an indirect way during the functioning of the system and evaluated in motion control. Therefore, the times of identification, adaptation and control have to be correlated with the real intervals of cyclic motions. This demands high quality and speed from the proper algorithmic and control stuctures. At the same time, the use of resonant control principles permits simplification of the control algorithms, to reduce greatly the energy consumption and to ensure, in this way, an additional resource for increasing control speed and quality. Figure 1 shows the design schematic of the resonant robot MARS [4]. This is a 897
898
V.I. BABITSKY and M. Y. CHITAYEV
3
,~
~
w
~ J i ''~ x
Fig. 1. Design schematic of resonant robot MARS. pick-and-place automatic manipulator intended for transportation of light payloads between fixed positions in cylindrical co-ordinates. The robot consists of the three modules assembled with the "outlet-base" principle in which the moving member of one module serves as attachment for the next one. There is a controlled gripper fixed on the end of the last module. Each module is an electromechanical auto-oscillating system. It consists of the carriage (1) oscillating along a base guide (2) and connected to the base by springs (3). The adjustable latches (4) mounted on the base can interrupt the motion at the reversing points. The carriage (1) is connected by gearing with a controllable electric motor (5). The module operates in the following way. The carriage is brought into an operating state by fixing one of the latches. To this end, the automatic self-exciting regime is selected by successive reversing of the electric motor according to the direction of movement. When a work stroke is to be made, the latch is automatically withdrawn and the carriage moves to the other extreme position under the force of the spring and motor torque, where it is again fixed by a latch. It reserves the energy accumulated by the spring. The next work stroke takes place after reversal of the electric motor and release of the latch. The programmed cycle of robot operation is realised by a controller which establishes the sequence of module moves. The redistribution of energy between spring and carriage takes place during the operation. The total mechanical energy of the module changes itself only through dissipation and realised work. Since the elastic characteristic of the spring is known, and since fulfilled motions are close to the natural modes of oscillation, it is possible to calculate preliminarily the mechanical energy of the module as a whole and to keep its level during the
Adaptive high-speed resonant robot
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motion by means of control in order to achieve the fixed position with a prescribed kinetic energy (which is necessary for bringing the latch into action). So, the problem of resonant module motion control can be transformed into the problem of stabilisation of the prescribed mechanical energy of the system at each point of the trajectory. Depending on the level of parameter uncertainty, and its influence on control quality, it is possible to develop both rough control, which is robust to parameter deviations, and adaptive control. In the latter case, the automatic tuning of the control system is fulfilled on-line, during the realisation of programmed motions. For the simultaneous operation of dynamically coupled modules, it is necessary to compensate additionally for the forces of interaction.
2. SYNTHESIS OF REGULATOR The equation describing the dynamics of unidirectional movement of a resonant robot module, supplied with the spring accumulator of energy and a direct current motor with constant moment of inertia and invariable mass of transmission, has the following form:
m.~ + b.:c + F(x - Xo) = u + olmg
(1)
with the boundary conditions
x(O) = O,
:c(O) = O,
x ( T ) = xk,
:c(T) = .:ok,
where x is the co-ordinate of the carriage, Xo is the co-ordinate corresponding to the minimum of the spring accumulator energy, F(x) is the elastic characteristic of the spring, u = nuo is the driving force (Ud = control voltage, n = transmission ratio), m is the reduced mass, b is the damping coefficient, a: is the coefficient of the gravitational component and dry friction, g is the acceleration due to gravity and T is time of cycle. It is proposed that part of the elastic characteristic disposed symmetrically around the carriage equilibrium position x0 is realised only during the movement. For the calculation of control torque, the static characteristic of the electric motor can be used because its electric time delay is essentially less than T. The permissible velocity :~~ of approaching position Xk is defined as the velocity from an e-neighbourhood of nominal value :~k chosen from the conditions of reliable fixation. It is desired that the mechanism with one degree of freedom [Eqn (1)] realises a control which transfers the module carriage from the initial position to the target position with a prescribed velocity of final approach and which is invariant to changes in mechanism parameters. The total movement has to be fulfilled during the finite time interval defined by the natural period of carriage oscillation. The control torque is limited. Proposing that phase co-ordinates of the carriage x and :~ are measured permanently, it is possible to calculate preliminarily in the arbitrary position x the reference velocity ~r ensuring enough kinetic energy to reach the target position Xk with the prescribed final velocity ~k
:or= {22k + 2 [ p ( X k - -
Xo)-- P(x -- xo)]}
(2)
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V.I. BABITSKY and M. Y. CHITAYEV
where r~ is the estimation of moving mass and P ( x ) is the characteristic of spring energy. Thus, for the object [Eqn (1)] and model [Eqn (2)] it is desired to synthesise the control law ]u(x, Yc)l ~ Um,× so that, for prescribed tr and e > 0 , the conditions i k ( t ) - .fr(t)l < ~" are ensured for all t e (t,. T) and permissible variations of parameters. The control law for tracing the analytical model [Eqn (2)] was synthesised as a PI-regulator with an additional compensation for dissipation and outer excitation. This type of regulator was chosen due to its robustness. The equation of the regulator is the following:
u = i~2 + kt],(ar_
- - f ) d r + k:(.f, - 2) + r~g~,
(3)
where k I and k 2 are tuneable coefficients: r~, /~ and ~ are estimations of parameters received during the identification and tuning. A control schematic for the module is shown in Fig. 2. The prescribed velocity kk is introduced as a parameter of the tuneable analytical model [Eqn (2)]. The current co-ordinate x and velocity ~f are measured during the control time. The co-ordinate x is transmitted simultaneously to the input of the analytical model. On the basis of received information the model calculates reference velocity 2r corresponding to the current co-ordinate x. The error, 2d = 2 - - . f r , is transmitted to the input of the regulator that also receives the measured signal k, used for compensation of dissipation. The regulator then forms the control voltage Ud according to [Eqn (3)] for the electric motor. The coefficients kl and k 2 are tuned to achieve the condition ]2 - k~ I < e during t < T with an acceptable quality of transient motion and prescribed limitations of control. The results of the numerical simulation of horizontal translation module dynamics
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~k
Adaptive high-speed resonant robot
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are demonstrated below. The parameters of the module correspond to the real hardware: m = 0.4 kg, b = 0.1 kg/s, cr = 0.1. The elastic characteristic of the spring is linear with stiffness c = 57 N/m, final velocity Xk = 0 in the position Xk = 0.4 m. The control voltage is limited: Imax Udl <~ 40 V. Figure 3 shows the curves £d(t) and ud(t) corresponding to control [Eqn (3)] with the analytical model [Eqn (2)] for different values of kl and k2. If kl = 1000, k2 = 100 (curve 1), the process of regulation is very short, but the control exceeds the prescribed limitation (curve 1, Fig. 3b). With the decreasing of kl to the value kl = 300, the deflection of the process in the middle of a time interval occurs (curve 2, Fig. 3a). It is prompted by the high level of perturbation which the integral component has no time to suppress. Meanwhile, by approaching the final position, the velocity £d hits the e-neighbourhood. The control in this case does not exceed the prescribed limitation (curve 2, Fig. 3b). Further decrease in kl until kl = 20 leads to the loss of controllability (curve 3, Fig 3a).
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V. 1. BABITSKY and M. Y. CHITAYEV
The tuning of k2 is performed in a similar way. When k2 = 500 (kl = 300), the control process is an aperiodic one and the control voltage exceeds the prescribed limitation as in curve 1, Fig. 3. Very large decrease of k2 to k: = 20 (kl = 300) increases the overshoot (curve 4). As a result of tuning, values of parameters were chosen (kx = 300, k2 = 100) which permit realisation of a satisfactory control process in the framework of the prescribed limitations. The robustness of the chosen control strategy was investigated by analysis of the sensitivity of control to variation of model parameters. Figure 4 demonstrates the results of such analysis. The areas of feasible variations of parameters are limited by the permissible levels of control voltage. It is seen that the synthesised regulator is robust to deviation of mass, damping and tilting. Comparison of the results of simulation with the experimental data received by testing of the horizontal translational module of robot MARS is shown in Fig. 5. Figure 5a corresponds to comparison of phase trajectories and Fig. 5b to control voltages. It is seen that the difference between numerical simulation and results of measurement does not exceed 10%. The inaccuracies in the determination of parameter b (curves 2) and parameter o~ (curves 3) have little influence on the law of control and phase trajectories. At the same time, changing the mass by 10% (curve 4) deflects control and phase trajectories substantially.
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Adaptive high-speed resonant robot
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3. ADAPTIVE CONTROL The described regulator does not ensure that the module functions in the case of substantial deflection of parameters. During operation, the mass of the manipulated payload is usually unknown and can change significantly in every cycle of movement. It emphasises the importance of additional adaptive control that is capable of adjusting the control according to changes in mass. The adaptive control has to transfer the carriage from the initial to the final position with a prescribed final approach velocity in conditions of unknown moving mass. The total move has to be performed during the finite time defined by the natural period of carriage oscillation, and with limited control voltage. The general schematic of the development model reference adaptive control system is shown in Fig. 6. The process of adaptive control consists of two stages: identification of the unknown parameter ~ during initial uncontrollable motion and active control of the carriage in accordance with the results of the identification. The use of the separate stages of identification and control simplifies substantially the control process.
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Fig. 5. Comparisons of simulation with experiment. 1: nominal parameters; 2: increment of b; 3: increment of a; 4: increment of m. (a) Phase trajectories; (b) control voltage.
'm~,~ I' t
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,--
nalytical model
,
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Fig. 6. Adaptive controls with tuneable model.
Adaptive high-speed resonant robot
905
During the identification, the behaviour of the carriage with unknown mass is compared with the proper output from the model, and the model parameter (mass) is tuned according to comparison error, The value of the parameter which brings the model behaviour into conformity with the real movement of the carriage (with accepted accuracy) is identified as an unknown mass. Let us define the problem of identification. It is necessary to develop the algorithm of ~ = m ( x , Yc) estimation in such a manner that, for the prescribed small values A and tS, it would exist in such a moment t. when for all t > t* the conditions t
~(t) = f0[~ - ~s(~)]2d~ < A,
I~(t) - ml < &
t* < T
would be performed. Here xs and :is are parameters of the tuneable model described by the equation ...2
~ 2 + P ( x s - x0) + b ~ 2 d v + & g ~ s d z + P ( x , - x0), (4) 2 2 where xn and :t, are initial phase co-ordinates. The formulated problem was solved for the described object by the speed gradient method [5]. With this aim the functional g is assigned to the function d~ - ~(x~, ~ , ,~). (5) dt Then the gradient of this function along the tuneable parameters ~ is calculated and the algorithm of identification is composed according to the rule - --
12 = -yV~;t, where # = ~-~, ). = ~ and y is the gain coefficient ensuring the desirable speed of convergence. As a result, the algorithm of carriage mass identification for the resonant module will be the following: ^
t.2
f~ = -y(Yc - Y¢~) P(xs - x°) - P(xn - xo) - b f o x , d r , Yc~
(6)
where :t~ is the tuneable model variable given by the equation ~--
~+2.P(x~-x0)-P(xn-x0)-b
~d~ - S g ( x ~ - x 0 )
.
(7)
It is established that conditions of convergence and identification of this algorithm are satisfied, if the estimated parameter is constant, and characteristic P ( x ) is a piece-smooth monotonous function [5]. For the algorithm of Eqns (6) and (7) presented in the finite-difference form, the condition of the choice of coefficient g ensured that the desirable speed of convergence with prescribed accuracy of estimation was received. The accuracy is limited by the level of noises of measurement and quantisation. Figure 7 demonstrates the process of mass identification for the different gain coefficients 7 and set of payload masses ml < m2 < m3. For the low gain factor y
V. I. BABITSKY and M. Y. CHITAYEV
906
080 1 0"607 l'] 4 in3~..]14~,__~-~ _ ~
~.
, , ~r..---
1
040~" "II1 ,_t j_3 m
I
---
0.20
I
0.02
0.04
0.06
0.08
t(s)
Fig. 7. Identification of payload mass (influence of Y on convergence). (curve 1) the process of identification is slow and the sensibility to noise is weak. With the increasing gain factor (curves 2-4) the identification process is accelerated, though there is not total convergence due to the rise of the noise sensibility. It is seen that the duration of identification does not depend on the payload mass. The functioning of adaptive control was investigated for the parameters of the MARS horizontal translation module. The total algorithm of adaptive control includes initial identification for the tuning of module parameters and the following control with the tuned model. Figure 8 demonstrates the results of simulation of the phase trajectories (Fig. 8a) and control processes (Fig. 8b) for different durations of identification. It is seen that increasing the identification time in order to obtain a higher estimation accuracy (curve 3) leads to an increase in object deflection from the nominal trajectory (curve 1). It causes a later big jump of the control voltage achieving the limitation. The results of simulation (Fig. 8) were confirmed by experiments (Fig. 9).
4. SIMULTANEOUS CONTROL OF ROTATION AND TRANSLATION For the realisation of some manipulation functions, it is necessary to fulfil the simultaneous movements of modules. This leads to the dynamic interaction of rotation and horizontal translation and the control has to overcome this interaction. The equations in the case of the dynamically coupled motions will be the following [compare with Eqn (1)]: m £ + bx.ic + F x ( x - Xo) = m C v 2 x signet + Ux
(8)
(Jo + m x : ) ¢¢ + bw ¢v + F , , ( w - Wo) = -2mYcCvx + u,,
(9)
with the initial conditions x(0) = 0,
~(0) = 0,
w(0) = 0,
w(0) = 0.
Here x is the co-ordinate of the horizontal module carriage, measured from the axis of rotation, w is the co-ordinate of the turntable, m is the reduced mass of the
907
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0.05
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0.15
0.20
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0.25
0.30
t (s)
(b) Fig. 8. Simulation of adapted control. 1: Nominal motion; 2: time of identification 10%; 3: time of identification 20%. (a) Phase trajectories; (b) control voltage. carriage with the payload, J0 is the initial moment of inertia of the turntable, bx and bw are reduced coefficients of viscous friction, Fx(.) and Fw(') are force characteristics of the module spring elements. As in the case of independent module movements, let us solve the problem of positioning for every moving element of the module as a problem of tracing its program motion. Each regulator in this case has to bring into line the real and nominal trajectories during the finite time interval under limited control. The final velocity for each module is prescribed by proper values of kinetic energy for reliable response of latches. In accordance with the above described method of analytical model development, each of the modules has its own analytical model. For the horizontal module it will be ~, =
~
+ 2[edx~ m
and for the rotation module
- xo) - P x ( x - x0)l
,
(10)
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V.I. BABITSKY and M. Y. CHITAYEV 3.0.
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Fig. 9. Experimental verification of adaptive control. 1: nominal motion; 2: time of identification 10% ; 3: time of identification 20%. (a) Phase trajectories; (b) control voltage.
=
2
+ Pw(wk- Wo)- Pw(w-
w0)|}
,
(11)
where J(x) = Jo + rnx2, J~ = J(x). It is seen from Eqn (11) that, during the calculation of the rotation module velocities, the changing of moment of inertia is taken into consideration. For the synthesis of the control laws in this case, the methods of decomposition, linearisation and compensation of the nonlinearities were used [6]. The equation of the horizontal movement module is l
ux = k~fo($q- . f ) d r + k ~ ( 2 r - .f) + bx2 - ~ i , 2 x sign~,
(12)
and of the rotational movement module it is w f "t .
w
.
uw = k t Jo(wr - w ) d r + k2(wr - ¢v) + f)wCV + ~YcxCv,.
(13)
Adaptive high-speed resonant robot
909
Here k~, k~, k~' and k~' are regulator coefficients for horizontal and rotation modules; ~ , bx and bw are estimations of module mechanism parameters. It is seen from Eqns (12) and (13) that in addition to the terms which ensure the conformity between model and object output, there are the addends for compensation of interaction forces: centrifugal and Coriolis ones. These forces depend on the motion velocities and substantially exceed the dissipation forces. Figure 10 demonstrates the schematic of the developed control system. Each regulator includes additionally the unit for compensation of interaction forces. The analytical model of rotation now uses the signal of horizontal displacement. The areas of permissible parameter variations were investigated by direct numerical integration, as in the one-dimensional case. The robot parameters were chosen according to the real ones of the robot modification MARS-3 with the full payload: m = 1.9 kg, bx = 0.1 kg/s, Cx = 57 N/m, J0 = 2.4 kgm 2, bw = 0.1 kgm2/s, cw = 48 Nm. Control voltages were limited in this case as follows: Imax u~l <~ 100 V and Imax u~l <~ 100 V.
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V.I. BABITSKY and M. Y. CHITAYEV
The regulator coefficients were chosen to ensure control quality: to diminish the transient time and control voltage. As a result, they are as follows: k~ = 350, k+ = 150, k~' = 280 and k~' = 130. In Fig. 11 the simulation of simultaneous control is shown. Curve 1 corresponds to translation co-ordinate x and curve 2 to rotation co-ordinate w. After the choice of the regulator coefficients the sensitivity of control to variation of model parameters was investigated. As in the case of one-dimensional control, an increased sensitivity to variation of payload mass was observed. In Fig. 12 are shown the variations of controls u~(t) and u~(t) depending on the parameter deviation from the nominal values. The deflections of dissipation coefficients bx and bw (Fig. 12a) lead to the change of control voltage (curves 2, 2'). The deviation of mass ~ (Fig. 12b) transforms the control and substantially raises its level of voltage. For the compensation of this sensitivity, the adaptation of analytical model parameters to the variation of mass r~ was performed, as in the one-dimensional control. Figure 13 demonstrates the phase trajectories and control voltages in the case of 10% (curves 1) and 15% (curves 2) identification duration. During the initial
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Adaptive high-speed resonant robot
911
(a) 20.
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uncontrolled motions deflection from the nominal partial motions is observed. Switching on the active control gives jumps of control voltages. Increasing the identification time leads to increase of these jumps and additional consumption of control energy.
5. CONCLUSION The use of adaptive control principles in the design of resonant robots permits the expansion of their abilities in speed, accuracy and efficiency. The simultaneous motion of dynamically coupled degrees of freedom increases speed of manipulation. The permanent adaptation of control to variations of natural dynamic characteristics
912
V.I. BABITSKY and M. Y. CHITAYEV .
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Fig. 13. Adaptive simultaneous control of rotation and translation. 1: time of identification 10%; 2: time of identification 15%. (a) Phase trajectory of translation; (b) phase trajectory of rotation; (c) control voltage. of the object leads to the reduction of drive energy consumption. The maintenance of the strict fixation conditions raises the accuracy of positioning and reliability of the robot. There is a possibility to recognise, through the identification, the type of payload and to change properly the program of its manipulation.
Adaptive high-speed resonant robot
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REFERENCES 1. Akinfiev T., manipulator. 2. Babitsky V., consumption 3. Babitsky V.,
Babitsky V., Kondratiev V. and Yurchenkov., High-speed resonant Sov. Eng. Res. 2, 9-16 (1986). Kovaleva A. and Shipilov A., Resonant drive control with minimum of energy. Tech. Cybernetics 4,210-213 (1989). Kovaleva A. and Shipilov A., Relieved control in cyclic systems.
Arch. Appl. Mech. 65, 488-495 (1995). 4. Babitsky V., Autoresonant mechatronic 5. Fomin V., Fradkov A. and Jakubovich
systems. Mechatronics 5, 483-495 (1995). V., Adaptive Control of Dynamic Objects.
Nauka, Moscow (1981) [in Russian]. 6. Vukobratovic M., Stokic D. and Kirchancky N., Nonadaptive Control of Robotic ~unipu~ators. Mir, Moscow (1989) [in Russian].
and Adaptive
Pergamon
PII: S0957-4158 (96) 00030.X
ADAPTIVE HIGH-SPEED RESONANT ROBOT V. I. BABITSKY* and M. Y. CHITAYEV* *Department of Mechanical Engineering, Loughborough University, Loughborough, Leics LE11 3TU, U.K. and tMotorola Australia Software Centre, 2 Second Avenue, Technology Park, Adelaide, SA 5095, Australia
(Received 20 April 1996; accepted 25 June 1996) Abstract--The use of the natural modes of oscillation for efficient performing of complex space motions of a robot with changeable parameters and dynamically interacting degrees of freedom is discussed. For the realisation of high-speed motions in these conditions under the limitation of control torques, it is necessary to identify at first, through on-line analysis of motions in the process of manipulation, the real parameters of natural modes. Then the control torques should be brought into correlation with natural motions by the adaptation of control in accordance with the findings of identification. As a result, the processes of identification, adaptation and control have to be optimised in accordance with the dynamics of robot oscillation. The formulated concept is demonstrated by research and development of an adaptive high-speed manipulator MARS-3 realised by the above mentioned principles for operating in cylindrical co-ordinates. Copyright © 1996 Elsevier Science Ltd.
1. INTRODUCTION The creation of resonant robots [1-3] requires the development of motion control concepts for effective maintenance of autoresonant states [4] in spite of changes in parameters of the robot oscillating system. The changing of manipulated mass and interaction of dynamically coupled degrees of freedom have a major influence on natural modes of robot oscillation. The dynamical correction of control through the variation of programs or deviation of manipulator parameters during the robot's lifetime is also important. The above mentioned parameters and dynamic loads usually cannot be measured preliminarily or directly and have to be identified in an indirect way during the functioning of the system and evaluated in motion control. Therefore, the times of identification, adaptation and control have to be correlated with the real intervals of cyclic motions. This demands high quality and speed from the proper algorithmic and control stuctures. At the same time, the use of resonant control principles permits simplification of the control algorithms, to reduce greatly the energy consumption and to ensure, in this way, an additional resource for increasing control speed and quality. Figure 1 shows the design schematic of the resonant robot MARS [4]. This is a 897
898
V.I. BABITSKY and M. Y. CHITAYEV
3
,~
~
w
~ J i ''~ x
Fig. 1. Design schematic of resonant robot MARS. pick-and-place automatic manipulator intended for transportation of light payloads between fixed positions in cylindrical co-ordinates. The robot consists of the three modules assembled with the "outlet-base" principle in which the moving member of one module serves as attachment for the next one. There is a controlled gripper fixed on the end of the last module. Each module is an electromechanical auto-oscillating system. It consists of the carriage (1) oscillating along a base guide (2) and connected to the base by springs (3). The adjustable latches (4) mounted on the base can interrupt the motion at the reversing points. The carriage (1) is connected by gearing with a controllable electric motor (5). The module operates in the following way. The carriage is brought into an operating state by fixing one of the latches. To this end, the automatic self-exciting regime is selected by successive reversing of the electric motor according to the direction of movement. When a work stroke is to be made, the latch is automatically withdrawn and the carriage moves to the other extreme position under the force of the spring and motor torque, where it is again fixed by a latch. It reserves the energy accumulated by the spring. The next work stroke takes place after reversal of the electric motor and release of the latch. The programmed cycle of robot operation is realised by a controller which establishes the sequence of module moves. The redistribution of energy between spring and carriage takes place during the operation. The total mechanical energy of the module changes itself only through dissipation and realised work. Since the elastic characteristic of the spring is known, and since fulfilled motions are close to the natural modes of oscillation, it is possible to calculate preliminarily the mechanical energy of the module as a whole and to keep its level during the
Adaptive high-speed resonant robot
899
motion by means of control in order to achieve the fixed position with a prescribed kinetic energy (which is necessary for bringing the latch into action). So, the problem of resonant module motion control can be transformed into the problem of stabilisation of the prescribed mechanical energy of the system at each point of the trajectory. Depending on the level of parameter uncertainty, and its influence on control quality, it is possible to develop both rough control, which is robust to parameter deviations, and adaptive control. In the latter case, the automatic tuning of the control system is fulfilled on-line, during the realisation of programmed motions. For the simultaneous operation of dynamically coupled modules, it is necessary to compensate additionally for the forces of interaction.
2. SYNTHESIS OF REGULATOR The equation describing the dynamics of unidirectional movement of a resonant robot module, supplied with the spring accumulator of energy and a direct current motor with constant moment of inertia and invariable mass of transmission, has the following form:
m.~ + b.:c + F(x - Xo) = u + olmg
(1)
with the boundary conditions
x(O) = O,
:c(O) = O,
x ( T ) = xk,
:c(T) = .:ok,
where x is the co-ordinate of the carriage, Xo is the co-ordinate corresponding to the minimum of the spring accumulator energy, F(x) is the elastic characteristic of the spring, u = nuo is the driving force (Ud = control voltage, n = transmission ratio), m is the reduced mass, b is the damping coefficient, a: is the coefficient of the gravitational component and dry friction, g is the acceleration due to gravity and T is time of cycle. It is proposed that part of the elastic characteristic disposed symmetrically around the carriage equilibrium position x0 is realised only during the movement. For the calculation of control torque, the static characteristic of the electric motor can be used because its electric time delay is essentially less than T. The permissible velocity :~~ of approaching position Xk is defined as the velocity from an e-neighbourhood of nominal value :~k chosen from the conditions of reliable fixation. It is desired that the mechanism with one degree of freedom [Eqn (1)] realises a control which transfers the module carriage from the initial position to the target position with a prescribed velocity of final approach and which is invariant to changes in mechanism parameters. The total movement has to be fulfilled during the finite time interval defined by the natural period of carriage oscillation. The control torque is limited. Proposing that phase co-ordinates of the carriage x and :~ are measured permanently, it is possible to calculate preliminarily in the arbitrary position x the reference velocity ~r ensuring enough kinetic energy to reach the target position Xk with the prescribed final velocity ~k
:or= {22k + 2 [ p ( X k - -
Xo)-- P(x -- xo)]}
(2)
900
V.I. BABITSKY and M. Y. CHITAYEV
where r~ is the estimation of moving mass and P ( x ) is the characteristic of spring energy. Thus, for the object [Eqn (1)] and model [Eqn (2)] it is desired to synthesise the control law ]u(x, Yc)l ~ Um,× so that, for prescribed tr and e > 0 , the conditions i k ( t ) - .fr(t)l < ~" are ensured for all t e (t,. T) and permissible variations of parameters. The control law for tracing the analytical model [Eqn (2)] was synthesised as a PI-regulator with an additional compensation for dissipation and outer excitation. This type of regulator was chosen due to its robustness. The equation of the regulator is the following:
u = i~2 + kt],(ar_
- - f ) d r + k:(.f, - 2) + r~g~,
(3)
where k I and k 2 are tuneable coefficients: r~, /~ and ~ are estimations of parameters received during the identification and tuning. A control schematic for the module is shown in Fig. 2. The prescribed velocity kk is introduced as a parameter of the tuneable analytical model [Eqn (2)]. The current co-ordinate x and velocity ~f are measured during the control time. The co-ordinate x is transmitted simultaneously to the input of the analytical model. On the basis of received information the model calculates reference velocity 2r corresponding to the current co-ordinate x. The error, 2d = 2 - - . f r , is transmitted to the input of the regulator that also receives the measured signal k, used for compensation of dissipation. The regulator then forms the control voltage Ud according to [Eqn (3)] for the electric motor. The coefficients kl and k 2 are tuned to achieve the condition ]2 - k~ I < e during t < T with an acceptable quality of transient motion and prescribed limitations of control. The results of the numerical simulation of horizontal translation module dynamics
Analyticalmodel x
r
[
~ __~,r(~k,X )
i t
I
I [ I I I I I I
L-
Regulator Fig. 2. Model reference control system.
~k
Adaptive high-speed resonant robot
901
are demonstrated below. The parameters of the module correspond to the real hardware: m = 0.4 kg, b = 0.1 kg/s, cr = 0.1. The elastic characteristic of the spring is linear with stiffness c = 57 N/m, final velocity Xk = 0 in the position Xk = 0.4 m. The control voltage is limited: Imax Udl <~ 40 V. Figure 3 shows the curves £d(t) and ud(t) corresponding to control [Eqn (3)] with the analytical model [Eqn (2)] for different values of kl and k2. If kl = 1000, k2 = 100 (curve 1), the process of regulation is very short, but the control exceeds the prescribed limitation (curve 1, Fig. 3b). With the decreasing of kl to the value kl = 300, the deflection of the process in the middle of a time interval occurs (curve 2, Fig. 3a). It is prompted by the high level of perturbation which the integral component has no time to suppress. Meanwhile, by approaching the final position, the velocity £d hits the e-neighbourhood. The control in this case does not exceed the prescribed limitation (curve 2, Fig. 3b). Further decrease in kl until kl = 20 leads to the loss of controllability (curve 3, Fig 3a).
(a)
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/1
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Fig. 3. Tuning of regulator. 1: kl -- 1000, k2 = 100; 2: kl = 300, k2 = 100; 3: kl = 20, k2 = 100; 4: kl = 300, k2 --- 20. (a) Error of velocity control; (b)control voltage.
902
V. 1. BABITSKY and M. Y. CHITAYEV
The tuning of k2 is performed in a similar way. When k2 = 500 (kl = 300), the control process is an aperiodic one and the control voltage exceeds the prescribed limitation as in curve 1, Fig. 3. Very large decrease of k2 to k: = 20 (kl = 300) increases the overshoot (curve 4). As a result of tuning, values of parameters were chosen (kx = 300, k2 = 100) which permit realisation of a satisfactory control process in the framework of the prescribed limitations. The robustness of the chosen control strategy was investigated by analysis of the sensitivity of control to variation of model parameters. Figure 4 demonstrates the results of such analysis. The areas of feasible variations of parameters are limited by the permissible levels of control voltage. It is seen that the synthesised regulator is robust to deviation of mass, damping and tilting. Comparison of the results of simulation with the experimental data received by testing of the horizontal translational module of robot MARS is shown in Fig. 5. Figure 5a corresponds to comparison of phase trajectories and Fig. 5b to control voltages. It is seen that the difference between numerical simulation and results of measurement does not exceed 10%. The inaccuracies in the determination of parameter b (curves 2) and parameter o~ (curves 3) have little influence on the law of control and phase trajectories. At the same time, changing the mass by 10% (curve 4) deflects control and phase trajectories substantially.
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Fig. 4. Robustness of control (area of feasible parameter variations and control limitations).
Adaptive high-speed resonant robot
903
3. ADAPTIVE CONTROL The described regulator does not ensure that the module functions in the case of substantial deflection of parameters. During operation, the mass of the manipulated payload is usually unknown and can change significantly in every cycle of movement. It emphasises the importance of additional adaptive control that is capable of adjusting the control according to changes in mass. The adaptive control has to transfer the carriage from the initial to the final position with a prescribed final approach velocity in conditions of unknown moving mass. The total move has to be performed during the finite time defined by the natural period of carriage oscillation, and with limited control voltage. The general schematic of the development model reference adaptive control system is shown in Fig. 6. The process of adaptive control consists of two stages: identification of the unknown parameter ~ during initial uncontrollable motion and active control of the carriage in accordance with the results of the identification. The use of the separate stages of identification and control simplifies substantially the control process.
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904
V . I . BABITSKY and M. Y. C H I T A Y E V (b) 5 t4
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Fig. 5. Comparisons of simulation with experiment. 1: nominal parameters; 2: increment of b; 3: increment of a; 4: increment of m. (a) Phase trajectories; (b) control voltage.
'm~,~ I' t
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,--
nalytical model
,
11
Fig. 6. Adaptive controls with tuneable model.
Adaptive high-speed resonant robot
905
During the identification, the behaviour of the carriage with unknown mass is compared with the proper output from the model, and the model parameter (mass) is tuned according to comparison error, The value of the parameter which brings the model behaviour into conformity with the real movement of the carriage (with accepted accuracy) is identified as an unknown mass. Let us define the problem of identification. It is necessary to develop the algorithm of ~ = m ( x , Yc) estimation in such a manner that, for the prescribed small values A and tS, it would exist in such a moment t. when for all t > t* the conditions t
~(t) = f0[~ - ~s(~)]2d~ < A,
I~(t) - ml < &
t* < T
would be performed. Here xs and :is are parameters of the tuneable model described by the equation ...2
~ 2 + P ( x s - x0) + b ~ 2 d v + & g ~ s d z + P ( x , - x0), (4) 2 2 where xn and :t, are initial phase co-ordinates. The formulated problem was solved for the described object by the speed gradient method [5]. With this aim the functional g is assigned to the function d~ - ~(x~, ~ , ,~). (5) dt Then the gradient of this function along the tuneable parameters ~ is calculated and the algorithm of identification is composed according to the rule - --
12 = -yV~;t, where # = ~-~, ). = ~ and y is the gain coefficient ensuring the desirable speed of convergence. As a result, the algorithm of carriage mass identification for the resonant module will be the following: ^
t.2
f~ = -y(Yc - Y¢~) P(xs - x°) - P(xn - xo) - b f o x , d r , Yc~
(6)
where :t~ is the tuneable model variable given by the equation ~--
~+2.P(x~-x0)-P(xn-x0)-b
~d~ - S g ( x ~ - x 0 )
.
(7)
It is established that conditions of convergence and identification of this algorithm are satisfied, if the estimated parameter is constant, and characteristic P ( x ) is a piece-smooth monotonous function [5]. For the algorithm of Eqns (6) and (7) presented in the finite-difference form, the condition of the choice of coefficient g ensured that the desirable speed of convergence with prescribed accuracy of estimation was received. The accuracy is limited by the level of noises of measurement and quantisation. Figure 7 demonstrates the process of mass identification for the different gain coefficients 7 and set of payload masses ml < m2 < m3. For the low gain factor y
V. I. BABITSKY and M. Y. CHITAYEV
906
080 1 0"607 l'] 4 in3~..]14~,__~-~ _ ~
~.
, , ~r..---
1
040~" "II1 ,_t j_3 m
I
---
0.20
I
0.02
0.04
0.06
0.08
t(s)
Fig. 7. Identification of payload mass (influence of Y on convergence). (curve 1) the process of identification is slow and the sensibility to noise is weak. With the increasing gain factor (curves 2-4) the identification process is accelerated, though there is not total convergence due to the rise of the noise sensibility. It is seen that the duration of identification does not depend on the payload mass. The functioning of adaptive control was investigated for the parameters of the MARS horizontal translation module. The total algorithm of adaptive control includes initial identification for the tuning of module parameters and the following control with the tuned model. Figure 8 demonstrates the results of simulation of the phase trajectories (Fig. 8a) and control processes (Fig. 8b) for different durations of identification. It is seen that increasing the identification time in order to obtain a higher estimation accuracy (curve 3) leads to an increase in object deflection from the nominal trajectory (curve 1). It causes a later big jump of the control voltage achieving the limitation. The results of simulation (Fig. 8) were confirmed by experiments (Fig. 9).
4. SIMULTANEOUS CONTROL OF ROTATION AND TRANSLATION For the realisation of some manipulation functions, it is necessary to fulfil the simultaneous movements of modules. This leads to the dynamic interaction of rotation and horizontal translation and the control has to overcome this interaction. The equations in the case of the dynamically coupled motions will be the following [compare with Eqn (1)]: m £ + bx.ic + F x ( x - Xo) = m C v 2 x signet + Ux
(8)
(Jo + m x : ) ¢¢ + bw ¢v + F , , ( w - Wo) = -2mYcCvx + u,,
(9)
with the initial conditions x(0) = 0,
~(0) = 0,
w(0) = 0,
w(0) = 0.
Here x is the co-ordinate of the horizontal module carriage, measured from the axis of rotation, w is the co-ordinate of the turntable, m is the reduced mass of the
907
Adaptive high-speed resonant robot 3.0
2.5
/!
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0.1
0
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0.3
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| . . . . . . . . .
0
i ,,,
0.05
i .....
0.10
0.15
0.20
,,,,I
0.25
0.30
t (s)
(b) Fig. 8. Simulation of adapted control. 1: Nominal motion; 2: time of identification 10%; 3: time of identification 20%. (a) Phase trajectories; (b) control voltage. carriage with the payload, J0 is the initial moment of inertia of the turntable, bx and bw are reduced coefficients of viscous friction, Fx(.) and Fw(') are force characteristics of the module spring elements. As in the case of independent module movements, let us solve the problem of positioning for every moving element of the module as a problem of tracing its program motion. Each regulator in this case has to bring into line the real and nominal trajectories during the finite time interval under limited control. The final velocity for each module is prescribed by proper values of kinetic energy for reliable response of latches. In accordance with the above described method of analytical model development, each of the modules has its own analytical model. For the horizontal module it will be ~, =
~
+ 2[edx~ m
and for the rotation module
- xo) - P x ( x - x0)l
,
(10)
908
V.I. BABITSKY and M. Y. CHITAYEV 3.0.
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~ 2"0 i ~ °~ 1.0 0.5
-0.10
0
0.10
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0.30
0.40
0.50
0.15
0.20
0.25
0.30
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8. >
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2i 0!
t (s) (b)
Fig. 9. Experimental verification of adaptive control. 1: nominal motion; 2: time of identification 10% ; 3: time of identification 20%. (a) Phase trajectories; (b) control voltage.
=
2
+ Pw(wk- Wo)- Pw(w-
w0)|}
,
(11)
where J(x) = Jo + rnx2, J~ = J(x). It is seen from Eqn (11) that, during the calculation of the rotation module velocities, the changing of moment of inertia is taken into consideration. For the synthesis of the control laws in this case, the methods of decomposition, linearisation and compensation of the nonlinearities were used [6]. The equation of the horizontal movement module is l
ux = k~fo($q- . f ) d r + k ~ ( 2 r - .f) + bx2 - ~ i , 2 x sign~,
(12)
and of the rotational movement module it is w f "t .
w
.
uw = k t Jo(wr - w ) d r + k2(wr - ¢v) + f)wCV + ~YcxCv,.
(13)
Adaptive high-speed resonant robot
909
Here k~, k~, k~' and k~' are regulator coefficients for horizontal and rotation modules; ~ , bx and bw are estimations of module mechanism parameters. It is seen from Eqns (12) and (13) that in addition to the terms which ensure the conformity between model and object output, there are the addends for compensation of interaction forces: centrifugal and Coriolis ones. These forces depend on the motion velocities and substantially exceed the dissipation forces. Figure 10 demonstrates the schematic of the developed control system. Each regulator includes additionally the unit for compensation of interaction forces. The analytical model of rotation now uses the signal of horizontal displacement. The areas of permissible parameter variations were investigated by direct numerical integration, as in the one-dimensional case. The robot parameters were chosen according to the real ones of the robot modification MARS-3 with the full payload: m = 1.9 kg, bx = 0.1 kg/s, Cx = 57 N/m, J0 = 2.4 kgm 2, bw = 0.1 kgm2/s, cw = 48 Nm. Control voltages were limited in this case as follows: Imax u~l <~ 100 V and Imax u~l <~ 100 V.
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910
V.I. BABITSKY and M. Y. CHITAYEV
The regulator coefficients were chosen to ensure control quality: to diminish the transient time and control voltage. As a result, they are as follows: k~ = 350, k+ = 150, k~' = 280 and k~' = 130. In Fig. 11 the simulation of simultaneous control is shown. Curve 1 corresponds to translation co-ordinate x and curve 2 to rotation co-ordinate w. After the choice of the regulator coefficients the sensitivity of control to variation of model parameters was investigated. As in the case of one-dimensional control, an increased sensitivity to variation of payload mass was observed. In Fig. 12 are shown the variations of controls u~(t) and u~(t) depending on the parameter deviation from the nominal values. The deflections of dissipation coefficients bx and bw (Fig. 12a) lead to the change of control voltage (curves 2, 2'). The deviation of mass ~ (Fig. 12b) transforms the control and substantially raises its level of voltage. For the compensation of this sensitivity, the adaptation of analytical model parameters to the variation of mass r~ was performed, as in the one-dimensional control. Figure 13 demonstrates the phase trajectories and control voltages in the case of 10% (curves 1) and 15% (curves 2) identification duration. During the initial
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Fig. 11. Simultaneous movement of modules. 1: rotation module; 2: translation module.
Adaptive high-speed resonant robot
911
(a) 20.
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15 10 5
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Ud
-15 -20 0
0.2
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uncontrolled motions deflection from the nominal partial motions is observed. Switching on the active control gives jumps of control voltages. Increasing the identification time leads to increase of these jumps and additional consumption of control energy.
5. CONCLUSION The use of adaptive control principles in the design of resonant robots permits the expansion of their abilities in speed, accuracy and efficiency. The simultaneous motion of dynamically coupled degrees of freedom increases speed of manipulation. The permanent adaptation of control to variations of natural dynamic characteristics
912
V.I. BABITSKY and M. Y. CHITAYEV .
44
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.... '.'n 1.20
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IC)
Fig. 13. Adaptive simultaneous control of rotation and translation. 1: time of identification 10%; 2: time of identification 15%. (a) Phase trajectory of translation; (b) phase trajectory of rotation; (c) control voltage. of the object leads to the reduction of drive energy consumption. The maintenance of the strict fixation conditions raises the accuracy of positioning and reliability of the robot. There is a possibility to recognise, through the identification, the type of payload and to change properly the program of its manipulation.
Adaptive high-speed resonant robot
913
REFERENCES 1. Akinfiev T., manipulator. 2. Babitsky V., consumption 3. Babitsky V.,
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