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Sensitivity Analysis of Robot Control Systems Based on Disturbance Observers
Copyright © IFAC Robot Control, Vienna, Austria, 1991
SENSITIVITY ANALYSIS OF ROBOT CONTROL SYSTEMS BASED ON DISTURBANCE OBSERVERS R. Gorez and D. Galardini Universite Catholique de Louvain. Laboracoire d·Aucomiltique. Dynamique et Analyse des Systemes. Bcilimenl MaxweiJ. Place du Levant 3. B-1348 Louvain-Ia-Neuve. Belgium
Abstract.. A decencralized concrol system consisting of discrete-time single input single output poD conrmllers and simple disturbance observers is proposed for the positional control of multi-degree of freedom manipulators. Stability of the conrml system is proved and its robustness is investigated through a theoretical analysis and simulation results.
Ke»',:ords. Concrol systems; decenrralized concrol; industrial robots; model reference concrol; position concrol; servomechanisms.
CONTROL WITH DISTURBANCE OBSERVER
INTRODUCTION
Figure I shows a block-diagram of a SISO control system consisting of a main feedback control loop and a disturbance observer. The latter is based on a model of the system to be controlled; the actual system and its model are driven by the same basic control variable calculated by the main controller. If there are no model mismatchings the difference between the output signals of the system and its model results from the effect of all the disturbances acting upon the actual system. After processing through a suitable "filter" or amplifier this "return difference" can be seen as an estimate of an equivalent global disturbance acting upon the system input. Then it can be superposed onto the basic conrml variable, at the actual system input, in order to counteract the effect of disturbances. This control scheme can be viewed as a secondary model reference control system imbedded inside the main feedback conrml loop. From Fig. 1 it appears clearly that this control scheme is the dual of a feedback control system based on a Luenberger observer (Luenberger, 1971; Gorez, Galardini and Zhu, 1991). In the latter the return difference is fedback to the model in order to force its state to track that of the actual system and to use the state of the model in the control system. Here the return difference is fed back to the actual system in order to force its state to track that of the model, thereby counteracting the effect of disturbances acting upon the system. As a consequence of this duality (Gorez, Galardini and Zhu. 1991) nice properties of state feedback control with Luenberger observer are retained, namely separation between the observer dynamics and the tracking dynamics; the latter is the same as if there was no observer.
In most industrial robots the motions of the various links are controlled by a bank of conventional single input-single output (SISO) proportional-derivative (P-D) or proportional-integralderivative (P-I-D) controllers, one of them being used for each axis. Other control algorithms for positional control of manipulators have been proposed in the literature. The best known of them is probably the "computed torque method" (Raibert and Horn, 1978); adaptive versions of the latter have been considered by various authors, see e.g. (Craig, 1988; Middleton, 1988; Ortega and Spong, 1989). Recently the use of disturbance observers providing an estimate of the disturbing torques has been suggested as a nice alternative for the "computed torque method" (Gorez and O'Shea, 1990). However, as control with disturbance observers is a kind of model-reference control scheme, its performances and robustness rely on the validity of the model which is used. It has been shown that a continuous-time control system consisting of SISO PoD controllers and simple disturbance observers is able to control a multi-degree of freedom (DOF) manipulator in spite of couplings and nonlinearities in the motions of the various arms (Gorez and Galardini, 1991). Similar results are obtained here in the case of a discrete-time implementation of the control system. The paper is organized as follows. Next section briefly resumes the theory of control with disturbance observers and its application to the positional control of a single DOF servomechanism, as presented in (Gorez and O'Shea, 1990; Gorez and Galardini, 1990). Sensitivity of this control system to deviations of the actual controlled system from its nominal model is investigated in the following section: thereafter robustness of the control scheme is considered from the viewpoint of stability and transient responses. Then the fourth section deals with the positional control of a multi-DOF robot by a bank of SISO PoD controllers and simple disturbance observers_ This implies two major differences with respect to the single DOF servomechanism considered in preceding sections: I) motions of different arms are coupled and dynamic equations are nonlinear and time-varying; 2) control algorithms are implemented in a digital computer so that sampling effects must be taken into account. Using an approach similar to that introduced by Middleton (Middleton, 1988), with a suitable Lyapunov-like function, it is shown that under natural assumptions (desired motions and deviations of parameters from their nominal values are bounded): I) the proposed control system is stable ; 2) steady-state position errors are null. Moreover relationships between parameter bounds, nominal closed-loop natural frequencies and observers bandwidths can be derived. Finally, in the last section, performances of the proposed control system are illuscrated by simulation results concerning a PUMA 562 manipulator the end effector of which has to follow a rectangular trajectory at fixed speeds.
Fig. 1. Block-diagram of a control system with disturbance observer.
123
Fig. 2. Control system with disturbance observer using an intermediate variable.
Moreover, as shown in Fig. 2, it is not necessary to set up a complete model of the controlled system if an intermediate state variable reflecting the effect of disturbances is available. Such a control scheme is appealing in positional control of single DOF mechanical systems where the manipulated variable is a driving effort. Then the loading effort is the external disturbance and velocity is a suitable intermediate state variable which can be measured easily. Assuming that friction is negligibly small (it could be considered as an additional disturbance as well. .. ) and giving the output device of the control system a gain equal to the system mass or moment of inertia lead to a model which is a plain integrator. Imbedding such a disturbance observer inside a main P-D feedback control loop results in the control system depicted in Fig. 3.
(b)
Fig. 4. Block-diagrams of a disturbance observer: a) Original diagram b) Equivalent diagram.
Transfer functions can be considered either in the Laplace domain for continuous-time systems or in the z-domain for discrete-time systems. In the nominal case where model matching is perfect, G s = G, so that (I) and (2) become respectively: (3)
1+ GGoC I _ GG o Fr - I + GGaC . 1 + GS .
Clearly, the tracking transfer function (3) is the same as if there was no disturbance observer. On the contrary, the regulation transfer function (4) is that without disturbance observer multiplied by (I - F) = (I + GS)- l, where F = GS . (1 + GS)-l is the transfer function relating the disturbance estimate t1 , namely the output of S, to the disturbance up' In other wordslf is the transfer function of the disturbance observer. Ideally F should be = I, but this is not realizable, except in steady-state if S or G includes an integrator. This is precisely the case for the mechanical system depicted in Fig. 3.
Fig. 3. Positional control with P-D controller and disturbance observer.
It must be noted that this control scheme can be extended to servomechanisms controlled by d-c electrical motors driven through their armature voltage, especially if the armature inductance can be neglected (Gorez and Galardini, 1990). Indeed, replacing the observer block-diagram of Fig. 2 by equivalent block-diagrams such as in Fig. 4 shows that the use of a disturbance observer is equivalent to the combination of a secondary feedback loop with a parallel compensator connected to the main controller. This feedback loop can be realized inside the control system or it may be provided by an internal feedback inside the actual controlled system itself, for example the bem! in a d-c motor controlled through its armature voltage. Obviously, in such cases, the return transfer function is imposed by the controlled system and cannot be selected freely by the control system designer.
For the servomechanism of Fig. 3 assuming no friction, no restoring effort and compensation of the mass or moment of inertia by the gain of the final control element yields: G(s) = Gis) =
+ GS) . GsGaC Ft - I + Gs(S + GaC) + GGsGaCS '
sI '
(5)
where s is the Laplace variable (continuous-time system). Taking a pure amplifier for the secondary feedback loop the transfer functions of the basic P-D controller and the secondary feedback loop are respectively:
Referring to Fig. 2, straightforward calculations give the following expressions for the tracking and regulation transfer functions, respectively Ft and F f' relating the controlled variable y to the desired signal Yd and to the equivalent external disturbance up: _
(4)
where wr is the bandwidth of the disturbance observer, wn and ~ being respectively the closed-loop natural frequency and damping factor of the servomechanism. Then the nominal tracking and regulation transfer functions are respectively:
(I
(I) (2)
124
~t ~r
r =as'
(7)
I
I
~. W~ + 21;wn s + S2 . S
s + W '
(8)
11
where denotes nominal values or expressions. The last factor in (8) shows the low frequency disturbance rejection capacity of this control system. Note that it is also possible to get ~ t = I thanks to an additional feed forward of the desired acceleration. SENSITIVITY ANALYSIS In case of model mismatchings: deviation of the mass or moment of inertia M from its nominal value ~, viscous friction andlor restoring efforts, respectively Bq and/or Gq, the two transfer functions (7) and (8) have to be multiplied by a common factor: F
Pc(s)
c
(S + w r) . (s2 + 21;wn s + Pc(s)
w;)
=----~----~~~----~
= (s + wr)·(s2+ 21;w ns + W~) + s·(8s 2+ ~wns + yw~)
Fig. 5. poD position control with disturbance observer. Rootloci with respect to 8
(9)
= M~ ~
(full lines) and
W
= w/wn (dotted lines).
(10)
being the characteristic polynomial of the complete system, with adimensional coefficients:
8
M-~
=-~- '
(11)
Then it is obvious that the disturbance rejection capacity of the control system is retained in spite of parameter deviations. As for the robustness of the control system , in other words its sensitivity to parameter deviations, it can be easily checked through an analysis of the roots of (10). First of all, it can be seen that, with a high gain disturbance observer (wr-t 00), the dominant characteristic roots of the closed-loop system will be close to those of the nominal polynomial (s2 + 21;wn s + W~) . Besides, using Routh-Hurwitz criterion, it can be easily verified that stability is guaranteed if y> -1 - 21;v (y = -I would mean a
Fig. 6. PoD position control with disturbance observer. Envelopes ofroot-Ioci with respect to 8
negative internal stiffness cancelling the servo stiffness!), 8> - 1 (8 = -I would mean M = 0, in other words no inertia in the mechanical system!), and : 8 < 21;v + 21;(21; +~) + y+ (I + y)(21; + ~)Iv.
M-~
= -~-
and ~
=~
B
wn .
global external disturbance all the gravitational, frictional, centrifugal and Coriolis efforts due to the motions of the manipulator as well as some variations of the inertia coefficients resulting from the changes of the manipulator configuration (Gorez and O'Shea, 1990). In the previous section it has been shown that such a control system is robust when applied to a SISO continuous-time time-invariant system. Here a bank of SISO discrete-time controllers is used for decentralized control of a nonlinear time-varying multi variable system. Then it is necessary to check at least the stability of the complete system, in other words, to check if, under reasonable assumptions, displacements and velocities of the manipulator are bounded. This involves a suitable mathematical description of the different pans of the system and the use of a Ljapunov-like technique for stability analysis.
(12)
Figure 5 shows root-loci with respect to 0 in the range (-D.5, I) corresponding to ~/2 :s; M :s; 2~, and W = W!W n in the range (0.5,4). Figure 6 gives the envelopes of root-loci with respect to 8 in the same range and ~ in the range (0, I). Both Fig. have been obtained for y = 0 and a nominal damping factor 1; = I. They show a deterioration of the dynamics due to model mismatchings. However the actual damping factor is always better than 0.5 and the response time better than twice its nominal value. Besides it appears that sensitivity to parameter deviations is much lower for 8 < 0, which means M < ~ . Therefore, if the mass of the mechanical system is not well known, robustness is better if the mass or inertia of the mechanical system is overestimated rather than underestimated in the design of the control system.
'mt----------------------------,
ROBOT POSmONAL CONTROL wrrn DISTURBANCE OBSERVERS Industrial robots andlor manipulators are composed of several links connected by rotary or prismatic joints, each of them being controlled by a SISO control system. The control system proposed here consists of a poD main controller combined with a simple individual disturbance observer, both in discrete-time, the driving effort on the joint being the control variable manipulated by the control system (Fig. 7). The latter treats as a
Fig. 7. Control of the i th joint of a multi-DOF manipulator by a poD controller with disturbance observer. N.!. = Numerical Integrator.
125
Basic EQuations
torque method". In the latter a complete expression of 't d ", should be calculated along the desired or the actual trajectory,
Manipulator dynamics Assuming rigid links, the manipulator dynamics can be described by the following equation (Middleton, 1988; Onega and Spong, 1989): M(q)·
q + C(q, q). q
+ g(q) = 't m
-
't L
,
using data on qd' qd and Qd and/or available measurements on q, q and q in the assumed expressions of M(q). C(q. q). g(q) and 't L' Here only a panial estimate 't dk needs to be possibly calculated, for example an estimate of the gravitational effons, but it could be omitted as well , as it is completed by an additional estimate ~'" = ~ "'. K r . r", provided by the disturbance observers. Control law (17) indeed can be rewritten as follows:
(13)
where: · q =(ql' q2"'" qn)T and 't =('t\, 't 2, ... , 'tn)T are respectively sets of joint displacements (generalized coordinates) and (generalized) effons acting upon the manipulator, n being the number of links of the latter and T denoting the transpose of a vector or a matrix;
(18)
· . and - denote time-derivatives, q and i:j being respectively joint velocities and accelerations;
In the right-hand side of (18) the first term is that provided by the basic P-D controller, including a feedforward term related to the desired acceleration, the second term is an a-priori estimate of all disturbing torques, which could be omitted, and the last term is a fuller estimate provided by disturbance observers.
· M(q), C(q, q) and g(q) are respectively the inenia matrix of the manipulator, the matrix expressing centrifugal and Coriolis effons as functions of generalized velocities. and the vector of gravitional effons;
Stability Analysis · subscripts m and L refer respectively to the driving effons applied by the motors controlling the different joints and to the loading effons (including all the effons related to possible unmodelled dynamics, for instance dry and/or viscous friction effons, effons due a possible external load .... ).
Stability will be analyzed in discrete-time. which requires the substitution of a discrete-time model to the continuous-time equation of the manipulator. As shown in (Middleton, 1988; Ortega and Spong, 1989). the term representing Coriolis and centrifugal effons in equation (13) comes from:
Control system. The discrete-time control system consists of three parts: a P-D controller including a feedforward acceleration term, the disturbance observer and the final control element. They are described by the following recursive algorithm:
3",
e vk
qd'" - q""
ep'"
= Kp' epk
=
qdk
qk'
.
. . = M(q). q-
I 'TdM .
'2 q
act
q ,
(19)
which allows the rewritting of (13) as follows: d
(14)
•
I 'TdM .
dt [M(q).q] - '2 q dq q + g(q)
(15)
+ Kv' evk + qdl<'
.
C(q, q) . q
where M(q).q is a generalized momentum. Then integrating (20) on the sampling interval (t""tk+l) yields:
vk+1
v k + Ts'
't m",
~'" • (3", + K,. . r",)
3""
rk + 't dk '
vk -
qk'
(16) (17)
1 'TdM • where tk' gk' c k are mean values of t L • g(q). - '2 q dq q over the sampling interval. Combining (20) and (14-17) results in an equation involving the observers residuals, more convenient for the purpose of stability analysis:
where: · e p' e v' 3. v, r are n-vectors representing the sets of position errors, velocity errors, driving accelerations calculated by the main controllers, velocities calculated by the models in the disturbance observers. observers residuals (differences between model velocities and actual velocities);
(21)
· Kt>' Kv and Kr are n x n diagonal matrices containing the (poSitive) gains of the different P-D controllers and observers; · subscript d refer to the desired trajectory (with corresponding data stored in the control system memory). subscript", denotes a value sampled or calculated at the sampling time tk = to + kT s' Ts being the sampling interval.
Expression (22) can be seen as an estimation error on the effons. Following assumptions are quite natural: I) desired displacements. velocities and accelerations are bounded; 2) loading effons are bounded; 3) estimates ~k and tdk are bounded wathever the way they are obtained; 4) permissible position and velocity errors as well as observers residuals are restricted to a convex set ne' permissi ble displacements and velocities of the manipulator and its model are restricted to a convex set n. n and ne being parallelipipedons defined by the bounds on the corresponding variables.
Symbol" denotes an estimate of the corresponding function or parameter. There are some uncenainties on the geometrical and kinematical parameters of the manipulator as well as on its mass distribution; then M(q), C(q, q), g(q) are not exactly known, moreover they are time-varying due to changes in the manipulator configuration, so that only estimates of them can be used in the control system. These estimates may be predetermined constant values or they may be calculated along the desired trajectory and updated from time to time or they may be estimated via a discrete time parameter estimator and a hybrid adaptive control scheme (Middleton, 1988). Here a diagonal matrix Mic is used for the estimate of M(q), 't d", is an estimate of the gravitational, Coriolis and centrifugal effons which may include some known loading effons.
"
Using the above assumptions it can be assumed that M(q), C(q. q) and g(q) are bounded and bounds on the residual effons and on the actual acceleration of the manipulator can be calculated:
The above remarks and the previous relationships exhibit the difference between this approach and the usual "computed
126
IIq 11 ~ a.
(23) (31)
Then defining a Ljapunov-Iike function: where coefficients!3i' 'Ili' 'Ili, i and rM .
where Plo can be seen as a residual momentum, and using standard inequalities of linear algebra (Strang, 1988) yield two upper bounds on the difference ~Vk = Vk+l - Vk:
= 1,2, depend on
~, ~, T s'
a
From (27) and (31) it may be concluded that, for small lIep,oll,
ne
lIev,oll and IIroll, {e p' e v' r) and Iq, q, v) will remain in and n respectively, in other words that the complete system is stable in the previously defined sense. Constraints: (32) and expressions of !3 i , 'Ili' 'Ill' for each axis, allow the derivation of relationships between parameter bounds, observers bandwidths and nominal natural frequencies of the manipulator control loops.
where AM and Am are respectively upper and lower bounds of the largest and the smallest eigenvalues of the inertia matrix and those of its estimate, and ~ and (Or are upper and lower bounds of the disturbance observers bandwidths. Each of the above expressions is a second degree polynomial in IIPkll, having two real roots with opposite signs if the leading coefficient of the polynomial is negative. The latter condition is achieved for both polynomials if:
Moreover, from (29), it is clear that ep,k is necessarily equal to zero in steady-state (rk+ t = rio' ea.k = 0). Therafter, the proposed control system guarantees stability of the manipulator closed-loop dynamics and cancelling of position errors in steady-state, which implies small position errors for relatively slow displacements. SIMULATION RESULTS
(26)
The expected performances of the proposed control system have been confirmed by a simulation analysis of the motions of a PUMA 562 manipulator the end effector of which has to follow a rectangular path. The displacement along any side of this path was perfonned at constant velocity except for short acceleration and deceleration periods after and before the end-corners. Some simulation results are shown in Fig. 8 and 9, which give the position errors for the x-y-z coordinates in the work space of the manipulator. Figure 8 shows that the use of high-gain disturbance observers (~ = 1000, Ts = I ms) results in errors slightly smaller than those resulting from exact feed forward compensation of the gravity effects; in both cases the accuracy is lOO times better than with P-D control alone. On Fig. 9 it can be seen that reducing the observer gains by a factor of 7 (timeconstant of the disturbance observers approximately equal to the mechanical time-constant of motors driven through their armature voltage) does not decrease significantly the performances, except for short larger "pulses" at the very beginning of the transients; those "pulses" are due to the initial adaptation of the model outputs. On the same Fig. the effect of an increase of the sampling interval (for the disturbance observers only) is also examined; setting Ts = 2 ms (stability limit of the secondary velocity feedback loops) leads to short relatively large transients at each corner of the trajectory, but does not decrease the tracking accuracy. Simulation results have also shown that the control system is unsensitive to overestimation (+ 20%) or underestimation (- 20%) of the inertia matrix, and fairly robust versus the load (up to 10 kgs) carried by the robot end-effector.
those inequalities being nothing else than stability conditions for the sampled-data secondary velocity feedback loops introduced by the disturbance observers. Then any polynomial (25) will be negative if IIPkll is above its positive root, so that Vk or IIPkll is a decreasing function of k (of time) if IIPkll is above the smallest of the positive roots of (25a) and (25b). Then it can be said that 'V k ~ 0:
and again through standard inequalities of linear algebra (Strang, 1988), that IIrkll is bounded by:
Afterwards, combining (IS) and (16) yields for any manipulator axis a recursive relationship such as follows:
+ ev.k+l-evk = rk+ 1 - rk. K e +K e v v,le Ts Ts p p,le
(28)
Using the standard Taylor's formula (Kitchen, 1968) allows the transformation of (28) in a second-order difference equation:
CONCLUSIONS The above simulation results show that introducing high gain disturbance observers in control systems of robots results in a position accuracy which is of the same order of magnitude as that which is provided by exact compensation of the disturbing torques. Moreover, from simulation results and from a theoretical analysis, both on a linear SISO servomechanism and on a nonlinear multi-DOF manipulator, it can be said that such an augmented control system is stable and robust versus errors in the estimation of the manipulator parameters. Besides its performances are only slightly influenced by a decrease of the observer gains. As the additional computational load is relatively low disturbance observers can be introduced in fast sampling digital controllers. Then they can be considered as a nice alternative to the conventional computer torque method for the position control of fast manipulators. However, for such applications, their robustness versus un modelled dynamics, vibrational motions, ... , must still be investigated.
where ea,k is a linear combination of the differences between the desired and actual accelerations at several unknown times within the sampling intervals before and after tie:
The right-hand side of (29) is bounded as a consequence of (23) and (27) . Then, through the transient response of (29), it is possible to obtain bounds on ep,lc and ev,lc:
127
0.05
r---------- --- - - - -----------------,
4 xl(}3
0.04 0.03
.,.
0 .02
'
. . '. J4.
"
0 .01
f,. ----------i~ 1>------\
-0.01
3 1
:iI\.--------Lf""- --lL_L~' ____ " OH'
-----1
er--
.1 o~==;;~~~--~~~~~_=~-__~------.J 1000 2000 3000 4000 5000 6000 7000 8000
4 xl(}3
0 .05 0.04 0.03 0.02 0.01
\
0
- - - - - --
.' '
o
-0.01 -0.02
0
1000
2000
3000
4000
5000
6000
7000
.1 0~-----;-; 1000 =--~2000 =---:;3~000 ;:;:-~4c::000 ;:;:----:5:-:::000 ::::---6:-:0=---::: 00 7000 c=--~ 8000
8000
Fig. 8. Position errors in the x-y-z frame of the robot work space. poD control with disturbance observer (lOO x error). Full lines: Broken lines: P-D control with compensation of gravitational efforts (100 x error). Dotted lines: P-D control alone. ACKNOWLEDGMENTS
Fig. 9. Position errors in the x-y-z frame of the robot work space. P-D control with disturbance observer. Full lines: observer bandwidth = 1000 s-l. sampling interval = I ms. Broken lines: observer bandwidth = 143 s-l. sampling interval = 1 ms. Dotted lines: observer bandwidth = 1000 s-l. sampling interval = 2 ms.
This research is supported by the Belgian National IncentiveProgram for Fundamental Research in Artificial Intelligence. Prime Minister's Office - Science Policy Programming. The scientific responsibility is assumed by its authors.
Gorez. R.• D. Galardini and K. Y. Zhu (1991). Model based control systems. To be presented at 13 th I.M.A.C.S. World Congress on Computation and Applied Mathematics. Dublin. Kitchen. 1. W. (1968). Calculus of one variable. AddisonWesley Pub. Cy. Reading. Ma. Luenberger. D. G. (1971). An inttoduction to observers. IEE.E. Transactions on Automatic Control. AC-16 596-602. • Middleton. R. H. (1988). Hybrid adaptive control for robot manipulators. Proc. 27 th IEEE Conference on Decision and Control. pp. 1592-1597. Ortega. R. and M. W. Spong (1989). Adaptive motion control of rigid robots: a tutorial. Automatica. 25 . 877-888. Raibert. M. H. and B. K. P. Horn (1978). Manipulator control using the configuration space method. The industrial robot. 5 (2). 69-73. Strang, G. (1988). Linear Algebra and its Applications. 3rd ed. Harcourt Brace Jovanovitch. San Diego. Ca.
REFERENCES Craig. 1. 1. (1988). Adaptive control of mechanical manipulators. Addison-Wesley Pub. Cy. Reading. Ma. Gorez, R. and J. O'Shea (1990). Robots positioning control revisited. Journal of intelligent and robotic systems. 3. 213-231. Gorez. R. and D. Galardini (1990). Robust positional control for robotic applications. Proc. 2e Congres National de Mecanique Tht?orique et Appliquee, Brussels. pp. 189192. Gorez. R. and D. Galardini (1991). Robot control with disturbance observers. To be presented at IEEE 91-ICAR (5 th International Conference on Advanced Robotics). Plsa. 128
SENSITIVITY ANALYSIS OF ROBOT CONTROL SYSTEMS BASED ON DISTURBANCE OBSERVERS R. Gorez and D. Galardini Universite Catholique de Louvain. Laboracoire d·Aucomiltique. Dynamique et Analyse des Systemes. Bcilimenl MaxweiJ. Place du Levant 3. B-1348 Louvain-Ia-Neuve. Belgium
Abstract.. A decencralized concrol system consisting of discrete-time single input single output poD conrmllers and simple disturbance observers is proposed for the positional control of multi-degree of freedom manipulators. Stability of the conrml system is proved and its robustness is investigated through a theoretical analysis and simulation results.
Ke»',:ords. Concrol systems; decenrralized concrol; industrial robots; model reference concrol; position concrol; servomechanisms.
CONTROL WITH DISTURBANCE OBSERVER
INTRODUCTION
Figure I shows a block-diagram of a SISO control system consisting of a main feedback control loop and a disturbance observer. The latter is based on a model of the system to be controlled; the actual system and its model are driven by the same basic control variable calculated by the main controller. If there are no model mismatchings the difference between the output signals of the system and its model results from the effect of all the disturbances acting upon the actual system. After processing through a suitable "filter" or amplifier this "return difference" can be seen as an estimate of an equivalent global disturbance acting upon the system input. Then it can be superposed onto the basic conrml variable, at the actual system input, in order to counteract the effect of disturbances. This control scheme can be viewed as a secondary model reference control system imbedded inside the main feedback conrml loop. From Fig. 1 it appears clearly that this control scheme is the dual of a feedback control system based on a Luenberger observer (Luenberger, 1971; Gorez, Galardini and Zhu, 1991). In the latter the return difference is fedback to the model in order to force its state to track that of the actual system and to use the state of the model in the control system. Here the return difference is fed back to the actual system in order to force its state to track that of the model, thereby counteracting the effect of disturbances acting upon the system. As a consequence of this duality (Gorez, Galardini and Zhu. 1991) nice properties of state feedback control with Luenberger observer are retained, namely separation between the observer dynamics and the tracking dynamics; the latter is the same as if there was no observer.
In most industrial robots the motions of the various links are controlled by a bank of conventional single input-single output (SISO) proportional-derivative (P-D) or proportional-integralderivative (P-I-D) controllers, one of them being used for each axis. Other control algorithms for positional control of manipulators have been proposed in the literature. The best known of them is probably the "computed torque method" (Raibert and Horn, 1978); adaptive versions of the latter have been considered by various authors, see e.g. (Craig, 1988; Middleton, 1988; Ortega and Spong, 1989). Recently the use of disturbance observers providing an estimate of the disturbing torques has been suggested as a nice alternative for the "computed torque method" (Gorez and O'Shea, 1990). However, as control with disturbance observers is a kind of model-reference control scheme, its performances and robustness rely on the validity of the model which is used. It has been shown that a continuous-time control system consisting of SISO PoD controllers and simple disturbance observers is able to control a multi-degree of freedom (DOF) manipulator in spite of couplings and nonlinearities in the motions of the various arms (Gorez and Galardini, 1991). Similar results are obtained here in the case of a discrete-time implementation of the control system. The paper is organized as follows. Next section briefly resumes the theory of control with disturbance observers and its application to the positional control of a single DOF servomechanism, as presented in (Gorez and O'Shea, 1990; Gorez and Galardini, 1990). Sensitivity of this control system to deviations of the actual controlled system from its nominal model is investigated in the following section: thereafter robustness of the control scheme is considered from the viewpoint of stability and transient responses. Then the fourth section deals with the positional control of a multi-DOF robot by a bank of SISO PoD controllers and simple disturbance observers_ This implies two major differences with respect to the single DOF servomechanism considered in preceding sections: I) motions of different arms are coupled and dynamic equations are nonlinear and time-varying; 2) control algorithms are implemented in a digital computer so that sampling effects must be taken into account. Using an approach similar to that introduced by Middleton (Middleton, 1988), with a suitable Lyapunov-like function, it is shown that under natural assumptions (desired motions and deviations of parameters from their nominal values are bounded): I) the proposed control system is stable ; 2) steady-state position errors are null. Moreover relationships between parameter bounds, nominal closed-loop natural frequencies and observers bandwidths can be derived. Finally, in the last section, performances of the proposed control system are illuscrated by simulation results concerning a PUMA 562 manipulator the end effector of which has to follow a rectangular trajectory at fixed speeds.
Fig. 1. Block-diagram of a control system with disturbance observer.
123
Fig. 2. Control system with disturbance observer using an intermediate variable.
Moreover, as shown in Fig. 2, it is not necessary to set up a complete model of the controlled system if an intermediate state variable reflecting the effect of disturbances is available. Such a control scheme is appealing in positional control of single DOF mechanical systems where the manipulated variable is a driving effort. Then the loading effort is the external disturbance and velocity is a suitable intermediate state variable which can be measured easily. Assuming that friction is negligibly small (it could be considered as an additional disturbance as well. .. ) and giving the output device of the control system a gain equal to the system mass or moment of inertia lead to a model which is a plain integrator. Imbedding such a disturbance observer inside a main P-D feedback control loop results in the control system depicted in Fig. 3.
(b)
Fig. 4. Block-diagrams of a disturbance observer: a) Original diagram b) Equivalent diagram.
Transfer functions can be considered either in the Laplace domain for continuous-time systems or in the z-domain for discrete-time systems. In the nominal case where model matching is perfect, G s = G, so that (I) and (2) become respectively: (3)
1+ GGoC I _ GG o Fr - I + GGaC . 1 + GS .
Clearly, the tracking transfer function (3) is the same as if there was no disturbance observer. On the contrary, the regulation transfer function (4) is that without disturbance observer multiplied by (I - F) = (I + GS)- l, where F = GS . (1 + GS)-l is the transfer function relating the disturbance estimate t1 , namely the output of S, to the disturbance up' In other wordslf is the transfer function of the disturbance observer. Ideally F should be = I, but this is not realizable, except in steady-state if S or G includes an integrator. This is precisely the case for the mechanical system depicted in Fig. 3.
Fig. 3. Positional control with P-D controller and disturbance observer.
It must be noted that this control scheme can be extended to servomechanisms controlled by d-c electrical motors driven through their armature voltage, especially if the armature inductance can be neglected (Gorez and Galardini, 1990). Indeed, replacing the observer block-diagram of Fig. 2 by equivalent block-diagrams such as in Fig. 4 shows that the use of a disturbance observer is equivalent to the combination of a secondary feedback loop with a parallel compensator connected to the main controller. This feedback loop can be realized inside the control system or it may be provided by an internal feedback inside the actual controlled system itself, for example the bem! in a d-c motor controlled through its armature voltage. Obviously, in such cases, the return transfer function is imposed by the controlled system and cannot be selected freely by the control system designer.
For the servomechanism of Fig. 3 assuming no friction, no restoring effort and compensation of the mass or moment of inertia by the gain of the final control element yields: G(s) = Gis) =
+ GS) . GsGaC Ft - I + Gs(S + GaC) + GGsGaCS '
sI '
(5)
where s is the Laplace variable (continuous-time system). Taking a pure amplifier for the secondary feedback loop the transfer functions of the basic P-D controller and the secondary feedback loop are respectively:
Referring to Fig. 2, straightforward calculations give the following expressions for the tracking and regulation transfer functions, respectively Ft and F f' relating the controlled variable y to the desired signal Yd and to the equivalent external disturbance up: _
(4)
where wr is the bandwidth of the disturbance observer, wn and ~ being respectively the closed-loop natural frequency and damping factor of the servomechanism. Then the nominal tracking and regulation transfer functions are respectively:
(I
(I) (2)
124
~t ~r
r =as'
(7)
I
I
~. W~ + 21;wn s + S2 . S
s + W '
(8)
11
where denotes nominal values or expressions. The last factor in (8) shows the low frequency disturbance rejection capacity of this control system. Note that it is also possible to get ~ t = I thanks to an additional feed forward of the desired acceleration. SENSITIVITY ANALYSIS In case of model mismatchings: deviation of the mass or moment of inertia M from its nominal value ~, viscous friction andlor restoring efforts, respectively Bq and/or Gq, the two transfer functions (7) and (8) have to be multiplied by a common factor: F
Pc(s)
c
(S + w r) . (s2 + 21;wn s + Pc(s)
w;)
=----~----~~~----~
= (s + wr)·(s2+ 21;w ns + W~) + s·(8s 2+ ~wns + yw~)
Fig. 5. poD position control with disturbance observer. Rootloci with respect to 8
(9)
= M~ ~
(full lines) and
W
= w/wn (dotted lines).
(10)
being the characteristic polynomial of the complete system, with adimensional coefficients:
8
M-~
=-~- '
(11)
Then it is obvious that the disturbance rejection capacity of the control system is retained in spite of parameter deviations. As for the robustness of the control system , in other words its sensitivity to parameter deviations, it can be easily checked through an analysis of the roots of (10). First of all, it can be seen that, with a high gain disturbance observer (wr-t 00), the dominant characteristic roots of the closed-loop system will be close to those of the nominal polynomial (s2 + 21;wn s + W~) . Besides, using Routh-Hurwitz criterion, it can be easily verified that stability is guaranteed if y> -1 - 21;v (y = -I would mean a
Fig. 6. PoD position control with disturbance observer. Envelopes ofroot-Ioci with respect to 8
negative internal stiffness cancelling the servo stiffness!), 8> - 1 (8 = -I would mean M = 0, in other words no inertia in the mechanical system!), and : 8 < 21;v + 21;(21; +~) + y+ (I + y)(21; + ~)Iv.
M-~
= -~-
and ~
=~
B
wn .
global external disturbance all the gravitational, frictional, centrifugal and Coriolis efforts due to the motions of the manipulator as well as some variations of the inertia coefficients resulting from the changes of the manipulator configuration (Gorez and O'Shea, 1990). In the previous section it has been shown that such a control system is robust when applied to a SISO continuous-time time-invariant system. Here a bank of SISO discrete-time controllers is used for decentralized control of a nonlinear time-varying multi variable system. Then it is necessary to check at least the stability of the complete system, in other words, to check if, under reasonable assumptions, displacements and velocities of the manipulator are bounded. This involves a suitable mathematical description of the different pans of the system and the use of a Ljapunov-like technique for stability analysis.
(12)
Figure 5 shows root-loci with respect to 0 in the range (-D.5, I) corresponding to ~/2 :s; M :s; 2~, and W = W!W n in the range (0.5,4). Figure 6 gives the envelopes of root-loci with respect to 8 in the same range and ~ in the range (0, I). Both Fig. have been obtained for y = 0 and a nominal damping factor 1; = I. They show a deterioration of the dynamics due to model mismatchings. However the actual damping factor is always better than 0.5 and the response time better than twice its nominal value. Besides it appears that sensitivity to parameter deviations is much lower for 8 < 0, which means M < ~ . Therefore, if the mass of the mechanical system is not well known, robustness is better if the mass or inertia of the mechanical system is overestimated rather than underestimated in the design of the control system.
'mt----------------------------,
ROBOT POSmONAL CONTROL wrrn DISTURBANCE OBSERVERS Industrial robots andlor manipulators are composed of several links connected by rotary or prismatic joints, each of them being controlled by a SISO control system. The control system proposed here consists of a poD main controller combined with a simple individual disturbance observer, both in discrete-time, the driving effort on the joint being the control variable manipulated by the control system (Fig. 7). The latter treats as a
Fig. 7. Control of the i th joint of a multi-DOF manipulator by a poD controller with disturbance observer. N.!. = Numerical Integrator.
125
Basic EQuations
torque method". In the latter a complete expression of 't d ", should be calculated along the desired or the actual trajectory,
Manipulator dynamics Assuming rigid links, the manipulator dynamics can be described by the following equation (Middleton, 1988; Onega and Spong, 1989): M(q)·
q + C(q, q). q
+ g(q) = 't m
-
't L
,
using data on qd' qd and Qd and/or available measurements on q, q and q in the assumed expressions of M(q). C(q. q). g(q) and 't L' Here only a panial estimate 't dk needs to be possibly calculated, for example an estimate of the gravitational effons, but it could be omitted as well , as it is completed by an additional estimate ~'" = ~ "'. K r . r", provided by the disturbance observers. Control law (17) indeed can be rewritten as follows:
(13)
where: · q =(ql' q2"'" qn)T and 't =('t\, 't 2, ... , 'tn)T are respectively sets of joint displacements (generalized coordinates) and (generalized) effons acting upon the manipulator, n being the number of links of the latter and T denoting the transpose of a vector or a matrix;
(18)
· . and - denote time-derivatives, q and i:j being respectively joint velocities and accelerations;
In the right-hand side of (18) the first term is that provided by the basic P-D controller, including a feedforward term related to the desired acceleration, the second term is an a-priori estimate of all disturbing torques, which could be omitted, and the last term is a fuller estimate provided by disturbance observers.
· M(q), C(q, q) and g(q) are respectively the inenia matrix of the manipulator, the matrix expressing centrifugal and Coriolis effons as functions of generalized velocities. and the vector of gravitional effons;
Stability Analysis · subscripts m and L refer respectively to the driving effons applied by the motors controlling the different joints and to the loading effons (including all the effons related to possible unmodelled dynamics, for instance dry and/or viscous friction effons, effons due a possible external load .... ).
Stability will be analyzed in discrete-time. which requires the substitution of a discrete-time model to the continuous-time equation of the manipulator. As shown in (Middleton, 1988; Ortega and Spong, 1989). the term representing Coriolis and centrifugal effons in equation (13) comes from:
Control system. The discrete-time control system consists of three parts: a P-D controller including a feedforward acceleration term, the disturbance observer and the final control element. They are described by the following recursive algorithm:
3",
e vk
qd'" - q""
ep'"
= Kp' epk
=
qdk
qk'
.
. . = M(q). q-
I 'TdM .
'2 q
act
q ,
(19)
which allows the rewritting of (13) as follows: d
(14)
•
I 'TdM .
dt [M(q).q] - '2 q dq q + g(q)
(15)
+ Kv' evk + qdl<'
.
C(q, q) . q
where M(q).q is a generalized momentum. Then integrating (20) on the sampling interval (t""tk+l) yields:
vk+1
v k + Ts'
't m",
~'" • (3", + K,. . r",)
3""
rk + 't dk '
vk -
qk'
(16) (17)
1 'TdM • where tk' gk' c k are mean values of t L • g(q). - '2 q dq q over the sampling interval. Combining (20) and (14-17) results in an equation involving the observers residuals, more convenient for the purpose of stability analysis:
where: · e p' e v' 3. v, r are n-vectors representing the sets of position errors, velocity errors, driving accelerations calculated by the main controllers, velocities calculated by the models in the disturbance observers. observers residuals (differences between model velocities and actual velocities);
(21)
· Kt>' Kv and Kr are n x n diagonal matrices containing the (poSitive) gains of the different P-D controllers and observers; · subscript d refer to the desired trajectory (with corresponding data stored in the control system memory). subscript", denotes a value sampled or calculated at the sampling time tk = to + kT s' Ts being the sampling interval.
Expression (22) can be seen as an estimation error on the effons. Following assumptions are quite natural: I) desired displacements. velocities and accelerations are bounded; 2) loading effons are bounded; 3) estimates ~k and tdk are bounded wathever the way they are obtained; 4) permissible position and velocity errors as well as observers residuals are restricted to a convex set ne' permissi ble displacements and velocities of the manipulator and its model are restricted to a convex set n. n and ne being parallelipipedons defined by the bounds on the corresponding variables.
Symbol" denotes an estimate of the corresponding function or parameter. There are some uncenainties on the geometrical and kinematical parameters of the manipulator as well as on its mass distribution; then M(q), C(q, q), g(q) are not exactly known, moreover they are time-varying due to changes in the manipulator configuration, so that only estimates of them can be used in the control system. These estimates may be predetermined constant values or they may be calculated along the desired trajectory and updated from time to time or they may be estimated via a discrete time parameter estimator and a hybrid adaptive control scheme (Middleton, 1988). Here a diagonal matrix Mic is used for the estimate of M(q), 't d", is an estimate of the gravitational, Coriolis and centrifugal effons which may include some known loading effons.
"
Using the above assumptions it can be assumed that M(q), C(q. q) and g(q) are bounded and bounds on the residual effons and on the actual acceleration of the manipulator can be calculated:
The above remarks and the previous relationships exhibit the difference between this approach and the usual "computed
126
IIq 11 ~ a.
(23) (31)
Then defining a Ljapunov-Iike function: where coefficients!3i' 'Ili' 'Ili, i and rM .
where Plo can be seen as a residual momentum, and using standard inequalities of linear algebra (Strang, 1988) yield two upper bounds on the difference ~Vk = Vk+l - Vk:
= 1,2, depend on
~, ~, T s'
a
From (27) and (31) it may be concluded that, for small lIep,oll,
ne
lIev,oll and IIroll, {e p' e v' r) and Iq, q, v) will remain in and n respectively, in other words that the complete system is stable in the previously defined sense. Constraints: (32) and expressions of !3 i , 'Ili' 'Ill' for each axis, allow the derivation of relationships between parameter bounds, observers bandwidths and nominal natural frequencies of the manipulator control loops.
where AM and Am are respectively upper and lower bounds of the largest and the smallest eigenvalues of the inertia matrix and those of its estimate, and ~ and (Or are upper and lower bounds of the disturbance observers bandwidths. Each of the above expressions is a second degree polynomial in IIPkll, having two real roots with opposite signs if the leading coefficient of the polynomial is negative. The latter condition is achieved for both polynomials if:
Moreover, from (29), it is clear that ep,k is necessarily equal to zero in steady-state (rk+ t = rio' ea.k = 0). Therafter, the proposed control system guarantees stability of the manipulator closed-loop dynamics and cancelling of position errors in steady-state, which implies small position errors for relatively slow displacements. SIMULATION RESULTS
(26)
The expected performances of the proposed control system have been confirmed by a simulation analysis of the motions of a PUMA 562 manipulator the end effector of which has to follow a rectangular path. The displacement along any side of this path was perfonned at constant velocity except for short acceleration and deceleration periods after and before the end-corners. Some simulation results are shown in Fig. 8 and 9, which give the position errors for the x-y-z coordinates in the work space of the manipulator. Figure 8 shows that the use of high-gain disturbance observers (~ = 1000, Ts = I ms) results in errors slightly smaller than those resulting from exact feed forward compensation of the gravity effects; in both cases the accuracy is lOO times better than with P-D control alone. On Fig. 9 it can be seen that reducing the observer gains by a factor of 7 (timeconstant of the disturbance observers approximately equal to the mechanical time-constant of motors driven through their armature voltage) does not decrease significantly the performances, except for short larger "pulses" at the very beginning of the transients; those "pulses" are due to the initial adaptation of the model outputs. On the same Fig. the effect of an increase of the sampling interval (for the disturbance observers only) is also examined; setting Ts = 2 ms (stability limit of the secondary velocity feedback loops) leads to short relatively large transients at each corner of the trajectory, but does not decrease the tracking accuracy. Simulation results have also shown that the control system is unsensitive to overestimation (+ 20%) or underestimation (- 20%) of the inertia matrix, and fairly robust versus the load (up to 10 kgs) carried by the robot end-effector.
those inequalities being nothing else than stability conditions for the sampled-data secondary velocity feedback loops introduced by the disturbance observers. Then any polynomial (25) will be negative if IIPkll is above its positive root, so that Vk or IIPkll is a decreasing function of k (of time) if IIPkll is above the smallest of the positive roots of (25a) and (25b). Then it can be said that 'V k ~ 0:
and again through standard inequalities of linear algebra (Strang, 1988), that IIrkll is bounded by:
Afterwards, combining (IS) and (16) yields for any manipulator axis a recursive relationship such as follows:
+ ev.k+l-evk = rk+ 1 - rk. K e +K e v v,le Ts Ts p p,le
(28)
Using the standard Taylor's formula (Kitchen, 1968) allows the transformation of (28) in a second-order difference equation:
CONCLUSIONS The above simulation results show that introducing high gain disturbance observers in control systems of robots results in a position accuracy which is of the same order of magnitude as that which is provided by exact compensation of the disturbing torques. Moreover, from simulation results and from a theoretical analysis, both on a linear SISO servomechanism and on a nonlinear multi-DOF manipulator, it can be said that such an augmented control system is stable and robust versus errors in the estimation of the manipulator parameters. Besides its performances are only slightly influenced by a decrease of the observer gains. As the additional computational load is relatively low disturbance observers can be introduced in fast sampling digital controllers. Then they can be considered as a nice alternative to the conventional computer torque method for the position control of fast manipulators. However, for such applications, their robustness versus un modelled dynamics, vibrational motions, ... , must still be investigated.
where ea,k is a linear combination of the differences between the desired and actual accelerations at several unknown times within the sampling intervals before and after tie:
The right-hand side of (29) is bounded as a consequence of (23) and (27) . Then, through the transient response of (29), it is possible to obtain bounds on ep,lc and ev,lc:
127
0.05
r---------- --- - - - -----------------,
4 xl(}3
0.04 0.03
.,.
0 .02
'
. . '. J4.
"
0 .01
f,. ----------i~ 1>------\
-0.01
3 1
:iI\.--------Lf""- --lL_L~' ____ " OH'
-----1
er--
.1 o~==;;~~~--~~~~~_=~-__~------.J 1000 2000 3000 4000 5000 6000 7000 8000
4 xl(}3
0 .05 0.04 0.03 0.02 0.01
\
0
- - - - - --
.' '
o
-0.01 -0.02
0
1000
2000
3000
4000
5000
6000
7000
.1 0~-----;-; 1000 =--~2000 =---:;3~000 ;:;:-~4c::000 ;:;:----:5:-:::000 ::::---6:-:0=---::: 00 7000 c=--~ 8000
8000
Fig. 8. Position errors in the x-y-z frame of the robot work space. poD control with disturbance observer (lOO x error). Full lines: Broken lines: P-D control with compensation of gravitational efforts (100 x error). Dotted lines: P-D control alone. ACKNOWLEDGMENTS
Fig. 9. Position errors in the x-y-z frame of the robot work space. P-D control with disturbance observer. Full lines: observer bandwidth = 1000 s-l. sampling interval = I ms. Broken lines: observer bandwidth = 143 s-l. sampling interval = 1 ms. Dotted lines: observer bandwidth = 1000 s-l. sampling interval = 2 ms.
This research is supported by the Belgian National IncentiveProgram for Fundamental Research in Artificial Intelligence. Prime Minister's Office - Science Policy Programming. The scientific responsibility is assumed by its authors.
Gorez. R.• D. Galardini and K. Y. Zhu (1991). Model based control systems. To be presented at 13 th I.M.A.C.S. World Congress on Computation and Applied Mathematics. Dublin. Kitchen. 1. W. (1968). Calculus of one variable. AddisonWesley Pub. Cy. Reading. Ma. Luenberger. D. G. (1971). An inttoduction to observers. IEE.E. Transactions on Automatic Control. AC-16 596-602. • Middleton. R. H. (1988). Hybrid adaptive control for robot manipulators. Proc. 27 th IEEE Conference on Decision and Control. pp. 1592-1597. Ortega. R. and M. W. Spong (1989). Adaptive motion control of rigid robots: a tutorial. Automatica. 25 . 877-888. Raibert. M. H. and B. K. P. Horn (1978). Manipulator control using the configuration space method. The industrial robot. 5 (2). 69-73. Strang, G. (1988). Linear Algebra and its Applications. 3rd ed. Harcourt Brace Jovanovitch. San Diego. Ca.
REFERENCES Craig. 1. 1. (1988). Adaptive control of mechanical manipulators. Addison-Wesley Pub. Cy. Reading. Ma. Gorez, R. and J. O'Shea (1990). Robots positioning control revisited. Journal of intelligent and robotic systems. 3. 213-231. Gorez. R. and D. Galardini (1990). Robust positional control for robotic applications. Proc. 2e Congres National de Mecanique Tht?orique et Appliquee, Brussels. pp. 189192. Gorez. R. and D. Galardini (1991). Robot control with disturbance observers. To be presented at IEEE 91-ICAR (5 th International Conference on Advanced Robotics). Plsa. 128