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Elementwise stablewise finite reachability time natural tracking control of robots
14th World Congress oflFAC
ELEMENTWISE ST ABLEWISE FINITE REACHABILTTY TIME NA. ..
B-Id-OI-6
Copyright © 1999 IF AC
14th Triennial World Congress, Beijing, P.R. China
ELEMENTWISE STABLEWlSE FINITE REACHABILITY TIME NATURAL TRACKING CONTROL OF ROBOTS
William Pratt Mounfield, Jr.* and Lyubomir T. Gruyitch ....
*M & M Technologies, Inc.,
p.a. Box 399,
Lexington, se 29071-0399 USA [email protected] **Ecole Nationale d'Ingenieurs, Technopole Belfort; P.o. Box 525, 90016 Belfort Cedex, France. l;ubomir. [email protected]
Abstract: A high-perfonnance, robust control technology is proposed using a natural tracking control algorithms to force zero Cartesian tracking errors in robot systems in fmite time. Natural tracking control (Grujic and Mounfield, 1995), based upon the natural trackability condition, improves the performance of these robot systems by forcing the elementwise, stablewise fmite reachability time tracking of desired outputs without the knowledge of the internal dynamics of the plant or of the external disturbances. Natural tracking control compensates for friction, backlash, and other nonlinear effects. A simulation is shown for a class of non linear robots with internal friction and severe white noise torque disturbances. Copyright © 1999 IFAC Keywords: Nonlinear control system, Robust control, Robot Control, Tracking Systems
1. INTRODUCTION
Many of the advances in the basic control theory have a basis in the control of robotic systems. Two examples are: the technique in (Khorasani and Kokotovic, 1985), (Khorasani and Zheng, 1992) on feedback and control using perturbation methods that divide the dynamics of the deflection/gross motion into fast and slow dynamics in order to derive a deflection control algorithm; and adaptive control of poorly defined systems that must provide a desired performance in the face of changes in the environment (AstrOID and Whittenmark, 1972),
(Tsypkin, 1971). Yet these methods are extremely CPU intensive and have not been implemented in commodity robots. The literature is filled with other reports of the computation burden from the control of rigid body robots (Kuntz and Jacubasch, 1985) and flexible body robots (Asada and Tokumaru, 1990), (Gourdeau and Schwartz, 1993) (Khorasani, 1992). The cost of computers is prohibitive to solve problems in these ways unless the control task is optimized for a particular trajectory off-line and apriori; otherwise some other new control method must be found as herein.
517
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWISE FINITE REACHABILITY TIME NA. ..
14th World Congress oflFAC
h(q,q(l) )=
It is also recognized that robot system dynamics are often not precisely known. The tracking control
should be robust enough to cope effectively with such uncertainties. A useful tracking control would be one synthesized without any knowledge about a mathematical description of the plant internal dynamics and which operates in Cartesian space. Such a control is natural tracking control (Grujic and Mounfield, 1995). The basis for natural tracking control is the new qualitative system concept called natural trackability for linear and nonlinear MIMO systems. Natural trackability assures the existence of a control that can be synthesized without using any information about the plant internal dynamics. This control can force the output to reach and maintain the desired output in a finite time. Such a control guaranteeing either a prespecified exponential tracking quality or a prespecified elementwise stablewise fmite reachability time (ESFRT) tracking quality may be synthesized for naturally trackable plants. Elementwise exponential and ESFRT-tracking have been defmed for linear and nonlinearplants (Grujic and Mounfield, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993, 1994). Herein, it is defined for a class of nonlinear systems described by mixed, general differential equations. This class of systems includes robots described by second-order non linear vector differential equations in their joint space and with a nonlinear vector equations in Cartesian space. It is advantageous for a robot to follow (track) a desired trajectory in Cartesian space even though the initial condition of the robot injoint space prevents a zero initial tracking error. The ability to exponentially converge to the desired output for each individual component of the real output space of the robot denotes a "high-quality" tracking property; ESFRT-1racking is even higher tracking quality. Tn order to achieve this high-quality property without complete knowledge of the internal dynamics ofthe robot the system must exhibit a qualitative property called natural trackability. 2. SYSTEM DESCRIPTION A description of a robot and of its actuators (without electric circuits of their stators if the robots have electrical actuators) and sensors is given by (1) A(q)qW +h(q"fiJ) = Bb(u) +Fd (la) Y = g(q). (I b)
For example, a two-degree-of-freedom robot operating in a vertical plane, with normal gravitational effects, can be modeled as (1) with q(t)= (qJ,q~T Goint positions), q<"J(t)= (q/~,q/k»)~ k=l,2, Goint velocities and accelerations), and without loss of generality,
B(q)= B=
1.0 0.0] and b(u)= (u/,u) . [0.0 1.0 T
(2)
The internal robot dynamics including Coriolis and centripetal forces (3), may be described as where qP), i€(i,2) are the joint velocities. Note that the gravitational forces are modeled as a part of the unknown dynamics of the robot. The output of the robot in Cartesian space and an example Fare
~I] = [IICOS(ql)+12COS(ql +q2)], ~2 llsin(ql)+/2 sin(ql +q2)
[2.5 3.5]
F= 0.65 -.45'
(4)
The control algorithm uses the inertial matrix function A(q) in conjunction with the lacobian J(q) of the output functiong(q). The functions A(,), h(.), and gO in this robot example are independent of non gravitational disturbances. 3. FINITE-TIME NATURAL TRACKABILITY The trackability concept was introduced in (Grujic and Mounfield, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993, 1994) and is broadened by: Defmition 1. (a) The robot described by (1) is trackable over Sy if, and only if, there is wO, fFtr(S), and for every YdES" and every OEjO, +00[ there is a control u(.) such that Ileoll
(c) The preceding properties are global if, and only if, they hold fOT fF+ "". The trackability generalizes the concepts of output controllability Chen, 1993), output functional controllability (Bertram and Sarachik., 1960), (Rosenbrock, 1990), and output reproducibility
518
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWTSE FINITE REACHABILITY TIME NA. ..
14th World Congress ofIFAC
over ShXSdXSy iff, it both exhibits elementwise stablewise tracking over SJrXSdXSy and there is a positive diagonal matrix Lt, .tl=J(d,U'Yd:h) for every (e",h,d,y)E9l.d, xS"xsdxSy so that the vector time 1"n is the fll'st vector instant such that Ig(t;e",'d,u,Ydr'hJ /< 1] for all I€(T,., +001), t=(t t
(Brockett and Mesarovic. 1965). TrackabiIity implies these, but versa does not hold. Theorem 1. In order for the robot described by (1) to be: a) natural1y trackable over ShXSdXSy it is necessary that for some {J>O, fl=jJ(S},.S"S), det[ J(q)A·1(q)B} "'0, V(t,q,h, d,y,/.)) E9l+X (jl(t;Yd.)) xSIr xSdxSy, i.e., (jl(t;y.f.)) =1Jr. (6)
globally naturally trackable over S"xSdxSy.ll is necessary and sufficient that a) holds for fl=+oo. Proof. Not shown due to page limitations. The proof for trackability of different systems is shown in (Grujic and Mounfield, ]990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993. 1994).
... IF,
d)
elementwise stablewise tracking with vector reachabilitv time 1"R over ShXSdXSy iff, it exhibits elementwise stablewise tracking over SlrxSdXSy and rR is the first vector instant such that leol
e)
These tracking properties are global iffthey are valid for N=diag{+ 00+ 00, .. +t%J}.
b)
4. ELEMENTWISE TRACKING QUALITY The classical notion of tracking (asymptotic tracking) has been used to describe the property of a system whose real output Y asymptotically approaches a time-varying desiredy./O as time t increases infinitely and without the influence of any disturbance. Controlled systems are most often under the influence of disturbances and are required to have their real outputs asymptotically follow their desired outputs. Such tracking should also have a prescribed quality. In the following we shall call 'RI 7:fIr=(rflrJo "!It], ... , "~ a vector instant or a vector (elementwise) reach ability time. Similarly, 1"'7=(1: 11)1 't2n2 ... 't", '1 )
T
is a vector (elementwise) settling time
with respect to a vector 1J=(TJt 1]2... 1]",Y, 1]€]O, +001[, 8f.:J={e:lel
elementwise stablewise tracking over ShXSdXSy iff, both there is a vector pO, 1/!=TjJ(d,YtP'h) such that leol
limllg.(t;e/J>·d,u,Yd;hJI: t~+f>O}=O and for every vector EE}O.+ocl[andevery (h,d,yd)E ShXSdXSy there is a vector oE(O, +"'). o=O(E,d.U,Yd;h), such that /e,,/< c5impJies Ig(t;eo;u,d,y.-fth)/
elementwise stablewise tracking with T] : vector settling time 1",., 1"'1=1",je",S,;SIv Syu).
By referring to this defmition and the well known properties oftime-invariant linear systems we can verifY the following statement for exponential tracking. Lemma 1. If the contrQllaw u(.) is such that the output error em obeys (7), T[e(O, efl)(oJ=K,iJ)(t;eu;u,d,Yd;h) (7a) +Ko e(t;e/p'u,d,Yrih),
T[e(t}, ef1)(O]=O. V(e",t,h,d.,y)E!!f"x31+xS"xSdx~.,
(7b)
where Ko and XI are such positive diagonal matrices that e = 0 is asymptotically stable state of (7) then the system (l) controlled by u(.) exhibits global elementwise exponential tracking over Sh XSdXSyThis lemma results from (7) and Def'mition 2. The (exponential) solution to (7) is given by (8), e(t;eo;u,d,YtP·A}={exp(-tK/KrJJeo.
(8)
Lemma 2. a) If the control algorithm is such that a control u() ensures existence of a positive diagonal matrix Lt guaranteeing that leol<.:j1 implies (9) for KE{21,3I, ...} and positive diagonal ZEIR+ mx""
i l ) (t; e,; d; U,Yd; h) + KZS'(e.) [S(e.)e(t;e,,;d;u,Yrih)] (/_K - J) = 0 for all (t,h,d,y)E3l+ XShXSdXSy,
(9)
then system (l) controlled by the control u(.) exhibits global elementwise stablewise tracking with both vector settling time T'7= (J12)ZI (/ +S(leol x- l -1J K -')}(]eo l K- I _ 1J K - I ) and with vectorreachability time .R(e.)=Z'lleol K- I QYg ShXSdXSy without oscillations. overshoot and undershoot. The equation (9) has the unique ESFRT-tracking solution defmed by g,(t;e,,;d, U'Ycbh) = (112)S(e.){J +S{\eo/ K- I -tZl}}{\eo l r l _t T1!', WE31+. (10)
519
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWISE FINITE REACHABILITY TIME NA. ..
Other tracking properties may be found for Lyapunov (Grujic, 1988a, 1988b, 1989), and naturally trackable systems in (Grujic and Mounfleld, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992,1993, 1994). The matrices Ko, K 1 , K, Z can be selected so that the solution ffJ)(.) (8) or (10) possesses a prespecified tracking property and quality (9) or (11) respectively. Their choice is independent of the system (1) and (h,d,yJESIt><'Sd><'Sy-
5. HIGH-GAIN NATURAL TRACKING CONTROL Definition 3. A control u(.) is a natural tracking control for the robot (1) if, and only if: a) it is continuous in time tE31+; b) it is synthesized without any information about hO, and d(.); and, c) it forces the robot to exhibit tracking. Theorem 2.
(a)
Let a tracking algorithm
ill1 (11)
detennine le(t) I as elementwise non-increasing. Then, in order for robot (l) controlled by a high-gain natural tracking control u() to exhibit a sufficiently accurate requested tracking over Sh ><'SdXSy so that (8) or Cl 1) holds. it is sufficient that both (12) and (3) hold for some p>O for which a set (t(t;y'/')) ~ positive invariant and for Il sufficiently close to one, IlEJO,1]. det[ J[q(t)]A-I[q(t)]B] 04), V(t, q, d,y,f.)) E81+ X (j1(t;y,f.)) xSdxS)", (12)
b[u(t))= Ilb[u(r)] + (J[q(t)]A"l[q(t)]BrT[e(t)], V(t,q(J>h,d,y) ElH+ XQf1xShXSdXSy.
(13)
Natural (Grujic and Mounfield, 1990, 1992, 1993), and hight gain (Mounfield and Grujic, 1991, 1992, 1993, 1994) natural tracking control have been previously introduced. Comment 1. The tracking control algorithm (13) defines the configuration of the natural tracking controller rather than the closed fonn solution of the control. It is emphasized that there is no interest in the knowledge of a solution for the control, yet the controller defined by (13) will determine the control u(.) at every moment tE31+ Comment 2. The internal dynamics of the robot governed by h(.) and the external disturbance d(.) are compensated by the local positive feedback in u(.) without delay (e=O). This and global negative output feedback ensure the requested tracking quality for any h() ESI!, dO ESd •
14th World Congress oflFAC
6. ROBOT TRACKING CONTROL SYNTHESIS The tracking qualities compared by simulation are ESFRT-tracking and elementwise PD exponential tracking. Either may be represented by T[e(t),ffIYtJ]=O where T[e(t), efJ)(t)] describes the tracking quality irrespective of the plant model. The tracking property is multiplied by the time-varying, nonlinear gain A[q(t)]{(J[q(t)}/I, which is poorly known for this particular system. The controller output U, u(t)=pu(t-c} +A{q(t)]{J[q(t))fl T{e(t)), 1l=0.9999, is implemented using the (almost) unit positive feedback of u(t) through the time delay e, with e=0+ and high-gain pI, in combination with A{q(t)]{J[q(t)]rT(e). 7. ROBOT TRACKING CONTROL SIMULATION WITH IMPERFECT KNOWLEDGE OF A[q(t)] Simulations of the robot system with both a PD ESFRT-tracking and then a PD exponential natural tracking controller are shown in Figs. 1 through 6. The robot is simulated with link lengths I J =1. 75 rn, /2= 1.0 m, and link masses m J = 2.0 kg, m]= 1.0 kg. The initial position of the end of the robot was set at q(O)= (q}(O) q1(OyT= (0.6 rad, -0.5 rady The initial desired output was y,JO) = [(YdJ(O) Yt/2(OJ)T= (2.2 m, 0.70 mY. The desired path of the robot in Cartesian space is an arc of radius 0.2 meters. The controller had ±25% imperfect knowledge of A[q(t)] by modification of m} and m z. The disturbances in Fig. 1 for the robot were formed by the combination of two Omstein-Uhlenbeck band-limited white noise sources, d}(t) [Dl] with break: frequency ofO.25 Hz, and d2 (t) [D2] with break frequency of 0.35 Hz; these disturbances are very severe and the disturbances' magnitude is similar to the magnitude of h(.) filtered through the matrix F (4) above. In Fig. 2 the actual output and desired outputYdlt)= YdlO) +0.2sin(2t) [YIDES] andYd2(t) = Ym(O) +O.2cos(2t) [Y2DES] are plotted as Cartesian coordinates (the desired output defmes the true circular arc in the figure). The real tracking errors e }(t) [E 1] and e](O [E2] in Fig. 3 show the difference between ESFRT-tracking and exponential tracking. The (smooth, continuous) joint motions qJ(t) [QI] and q/t) [Q2] for both tracking qualities required to rrack the desired outputs are shown in Fig. 4. The natural tracking controls u,(t) [UI] and 1.1 2 (0 (U2] in Fig. 5 force the robot to track, even with substantial imperfections in its nonlinear model and extreme disturbances. The fmite reachability time control is no more dynamic than the exponential tracking control; both, however, are subjected to severe torque and friction disturbances. The friction disturbances for the fmite reachability time test are shown in Fig. 6; note the nonlinear and continuous but only piecewise differentiable nature of the friction disturbances.
520
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWTSE FINITE REACHABILITY TIME NA. ..
8. CONCLUSION Natural tracking control algorithms guarantee a prespecified tracking quality which may be elementwise exponential or stablewise tracking with finite reachability time. These have been successfully applied to a nonlinear robot system. The system natural trackability condition was used to form the controller. The natural trackability condition does not require the knowledge of the internal dynamics of the robot. The natural trackability condition sets the minimum requirements for knowledge of the robot to achieve tracking. An example simulation demonstrates natural tracking control as applied to a two-degree-of-freedom robot model. A tracking convergence rate was prespecified without the knowledge of the internal dynamics of the robot. It was shown that the natural trackability information required is the inverses of the product of the inertial matrix components, the output Jacobian, and the actuator connections. These components may be unknown individually and perhaps even poorly known collectively. Using the tracking quality, the internal (almost) unity feedback, and the track ability information, the tracking errors converged toward zero from an unknown initial joint configuration while following an a-priori unknown time varying trajectory. REFERENCES Asada, H., Ma, Z.D., and Tokumaru, H. (1990) Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajedory Cont. ASME JDSMC, 112, No. 2,177-185. Astrom, K.J., and Whttenmark, B. (1972) On Self Tuning Regulators, Automatica, 9, 185-199. Bertram, J.E., and Sarachik, P.E. (1960) On Optimal Computer Control. Proc. First Int. Congress ofIFAC, Moscow, 1,419-422. Brockett, RW., and Mesarovic, M.D. (1965) The Reproducibility of Multivariable Systems. J ofMath. Analysis and Apps., 1,548-563. Chen, C.T. (1984) Linear System Theory and Design, Holt, Reinehart, and Winston, NY. Gourdeau, R., and Schwartz, H.M. (1993) Adaptive Control of Robotic Manipulators Using an Extended Kalman Filter. ASME JDSMC, 115, No. 1,203-208. Grujic, L.T. (1988a) Tracking Versus Stability; Theory. Proc. IMACS 12th World Congress on Scientific Computation, 319-327. Grujic, L.T. (l988b) Tracking Control Obeying Prespecified Performance Index," ibid., 332-336. Grujic, L.T. (1989) On The Theory of Non linear Systems Tracking With Guaranteed Performance Index Bounds: Application To Robot Control. IEEE ICRA, 3, 1486-1490.
14th World Congress ofIFAC
Grujic, L.T., and Mounfield, W.P. Jr. (1990) Natural Tracking Control of Linear Systems. 13th IMACS World Congress on Computational Mathematics, Dublin, 3, 1269-1270. Grujic, L.T., and Mounfield, W.P. Jr. (1992) Natural Tracking PID Process Control for Exponential Tracking,. AIChE Journal, 38, No. 4, 555-562. Grujic, L.T., and Mounfield, W.P. Jr. {l993) PDControl for Stablewise Tracking with Finite Reachability Time: Linear Continuous Time MIMO Systems with State-Space Description. Int. J of Robust and Nonlinear Control, 3, No. 4, 341-360. Grujic, L.T., and Mounfield, W,P. Jr. (1995) Natural Tracking Controller. V.S. Patent No. 5,379,210, Qssued 113/95) Khorasani, K. (1992) Adaptive Control of FlexibleJoint Robots. IEEE TRA, 8, No. 2,250-267. Khorasani, K., and Kokotovic, P.V. (1985) Feedback Linearization of a Flexible Manipulator Near Its Rigid Body Manifold. System and Control Letters, 6, No. 3, 187-192. Khorasani, K., and Zheng, S. (1992) An Inner/Outer Loop ControJler for Rigid-Flexible Manipulators. ASME JDSMC, 114, No. 4, 580-587. Kuntz, H.B., and Jacubasch, A.K., (1985) Control Algorithms for Stiffening an Elastic Industrial Robot. IEEE Journal ojRobotics and Automation, RA-I, No. 2, 71-78. Mounfield, W.P. Jr. and Grujic, L.T. (1991) HighGain Natural Tracking Control of Linear Systems. 13th lMACS, Trinity College, Dublin, Ireland, 3, 1271-1272. Mounfield, W.P. Jr. and Grujic, L.T. (1992) High-Gain PI Natural Control for Exponential Tracking of Linear Single-Output Systems. Automatique, Productique biformatique Industrielle (AP/I), 26, No. 2, 125-146. Mounfield. W.P. Jr. and Grujic, L.T. (1993) PID Natural Tracking Control of a Robot: Application. IEEElSMC'93 Conj., Le Touquet, 4, 328-333. Mounfield, W.P. Jr. and Grujic, L.T. (1994) HighGain PI Natural Tracking Control for Exponential Tracking of Linear MIMO Systems with State-Space Description.. I. J. of Systems Science, 25, No. 11, 1793-1817. Rosenbrock, H.H. (1990) State-Space and Multivariable Theory, Nelson, London. Tsypkin, Y.Z., (l97J) Adaptation and Learning in Automatit; Systems, Academic Press, NY.
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Copyright 1999 IF AC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWISE FINITE REACHABTLTTY TIME NA. ..
14th World Congress ofTFAC
I(l
~~----~----r-----r----'
'~----r----'.----'-----'
~+-----+-----+-----+---t-4 le
No+-----+-----~----+_--~ >-
~ o;+-----+_----+-~~~--~
~,~----.-----.-----~----,
~ o+-----+-----+-----+---~
1.50
!oz.
~.so
T (SEC)
3.75
2.00
1.75
Z.2S
YI
2.50
Fig. 2 Desired circular y,ft) motion and actual motiony(t} in Cartesian space, ESFRT tracking (solid), exponential (dashed).
5.00
Fig. 1 Band limited external white noise disturbances d} and dJ •
~•.-----r-----~----r----'
8
~;~
____
~
____
~
_ _ _ _L -_ _
~
8.-----r-----r-----r----,
!i! ci~----r-----r-----r----'
51
o
., .........-
o+-~--+-----+-----~--~ 0.00 I. 2S T tSlt10 3.75 5.00
~
o~----~----~----~--~ le.DO 1.2'5 z.sn 3.15 5.00 T (SECl
Fig. 4 Joint angles q,[Ql] and qz [Q2], motion in Cartesian space, ESFRT· tracking (solid), exponential tracking (dashed).
Fig. 3 Cartesian tracking errors eJI)
[El] and e2 (t) [E2] ESFRT.tracking (solid), exponential tracking (dashed»
!i!
O·,-----,-----r-----r----,
~
9·~----~----~----~----J
c
gy-----,-----r-----r----.
., LL-
h
....0
o· 0-
'"g~----~----+-----~--~ 'O.OD
1.25
2.50 T (SEC)
3.75
I
!1!
5.00
C?'Q.oo
Fig. 5 Natural tracking control uJt} [Ui], u](t) [U2], ESFRT-tracking (solid), exponential tracking (dashed).
1.25
1
2.50
5.00
(SEC)
Fig. 6 Internal friction disturbances tqlt) [TQIF] and lq,(t) [TQ2Fl, for circular Cartesian motion.
522
Copyright 1999 IF AC
ISBN: 008 0432484
ELEMENTWISE ST ABLEWISE FINITE REACHABILTTY TIME NA. ..
B-Id-OI-6
Copyright © 1999 IF AC
14th Triennial World Congress, Beijing, P.R. China
ELEMENTWISE STABLEWlSE FINITE REACHABILITY TIME NATURAL TRACKING CONTROL OF ROBOTS
William Pratt Mounfield, Jr.* and Lyubomir T. Gruyitch ....
*M & M Technologies, Inc.,
p.a. Box 399,
Lexington, se 29071-0399 USA [email protected] **Ecole Nationale d'Ingenieurs, Technopole Belfort; P.o. Box 525, 90016 Belfort Cedex, France. l;ubomir. [email protected]
Abstract: A high-perfonnance, robust control technology is proposed using a natural tracking control algorithms to force zero Cartesian tracking errors in robot systems in fmite time. Natural tracking control (Grujic and Mounfield, 1995), based upon the natural trackability condition, improves the performance of these robot systems by forcing the elementwise, stablewise fmite reachability time tracking of desired outputs without the knowledge of the internal dynamics of the plant or of the external disturbances. Natural tracking control compensates for friction, backlash, and other nonlinear effects. A simulation is shown for a class of non linear robots with internal friction and severe white noise torque disturbances. Copyright © 1999 IFAC Keywords: Nonlinear control system, Robust control, Robot Control, Tracking Systems
1. INTRODUCTION
Many of the advances in the basic control theory have a basis in the control of robotic systems. Two examples are: the technique in (Khorasani and Kokotovic, 1985), (Khorasani and Zheng, 1992) on feedback and control using perturbation methods that divide the dynamics of the deflection/gross motion into fast and slow dynamics in order to derive a deflection control algorithm; and adaptive control of poorly defined systems that must provide a desired performance in the face of changes in the environment (AstrOID and Whittenmark, 1972),
(Tsypkin, 1971). Yet these methods are extremely CPU intensive and have not been implemented in commodity robots. The literature is filled with other reports of the computation burden from the control of rigid body robots (Kuntz and Jacubasch, 1985) and flexible body robots (Asada and Tokumaru, 1990), (Gourdeau and Schwartz, 1993) (Khorasani, 1992). The cost of computers is prohibitive to solve problems in these ways unless the control task is optimized for a particular trajectory off-line and apriori; otherwise some other new control method must be found as herein.
517
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ELEMENTWISE STABLEWISE FINITE REACHABILITY TIME NA. ..
14th World Congress oflFAC
h(q,q(l) )=
It is also recognized that robot system dynamics are often not precisely known. The tracking control
should be robust enough to cope effectively with such uncertainties. A useful tracking control would be one synthesized without any knowledge about a mathematical description of the plant internal dynamics and which operates in Cartesian space. Such a control is natural tracking control (Grujic and Mounfield, 1995). The basis for natural tracking control is the new qualitative system concept called natural trackability for linear and nonlinear MIMO systems. Natural trackability assures the existence of a control that can be synthesized without using any information about the plant internal dynamics. This control can force the output to reach and maintain the desired output in a finite time. Such a control guaranteeing either a prespecified exponential tracking quality or a prespecified elementwise stablewise fmite reachability time (ESFRT) tracking quality may be synthesized for naturally trackable plants. Elementwise exponential and ESFRT-tracking have been defmed for linear and nonlinearplants (Grujic and Mounfield, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993, 1994). Herein, it is defined for a class of nonlinear systems described by mixed, general differential equations. This class of systems includes robots described by second-order non linear vector differential equations in their joint space and with a nonlinear vector equations in Cartesian space. It is advantageous for a robot to follow (track) a desired trajectory in Cartesian space even though the initial condition of the robot injoint space prevents a zero initial tracking error. The ability to exponentially converge to the desired output for each individual component of the real output space of the robot denotes a "high-quality" tracking property; ESFRT-1racking is even higher tracking quality. Tn order to achieve this high-quality property without complete knowledge of the internal dynamics ofthe robot the system must exhibit a qualitative property called natural trackability. 2. SYSTEM DESCRIPTION A description of a robot and of its actuators (without electric circuits of their stators if the robots have electrical actuators) and sensors is given by (1) A(q)qW +h(q"fiJ) = Bb(u) +Fd (la) Y = g(q). (I b)
For example, a two-degree-of-freedom robot operating in a vertical plane, with normal gravitational effects, can be modeled as (1) with q(t)= (qJ,q~T Goint positions), q<"J(t)= (q/~,q/k»)~ k=l,2, Goint velocities and accelerations), and without loss of generality,
B(q)= B=
1.0 0.0] and b(u)= (u/,u) . [0.0 1.0 T
(2)
The internal robot dynamics including Coriolis and centripetal forces (3), may be described as where qP), i€(i,2) are the joint velocities. Note that the gravitational forces are modeled as a part of the unknown dynamics of the robot. The output of the robot in Cartesian space and an example Fare
~I] = [IICOS(ql)+12COS(ql +q2)], ~2 llsin(ql)+/2 sin(ql +q2)
[2.5 3.5]
F= 0.65 -.45'
(4)
The control algorithm uses the inertial matrix function A(q) in conjunction with the lacobian J(q) of the output functiong(q). The functions A(,), h(.), and gO in this robot example are independent of non gravitational disturbances. 3. FINITE-TIME NATURAL TRACKABILITY The trackability concept was introduced in (Grujic and Mounfield, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993, 1994) and is broadened by: Defmition 1. (a) The robot described by (1) is trackable over Sy if, and only if, there is wO, fFtr(S), and for every YdES" and every OEjO, +00[ there is a control u(.) such that Ileoll
(c) The preceding properties are global if, and only if, they hold fOT fF+ "". The trackability generalizes the concepts of output controllability Chen, 1993), output functional controllability (Bertram and Sarachik., 1960), (Rosenbrock, 1990), and output reproducibility
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over ShXSdXSy iff, it both exhibits elementwise stablewise tracking over SJrXSdXSy and there is a positive diagonal matrix Lt, .tl=J(d,U'Yd:h) for every (e",h,d,y)E9l.d, xS"xsdxSy so that the vector time 1"n is the fll'st vector instant such that Ig(t;e",'d,u,Ydr'hJ /< 1] for all I€(T,., +001), t=(t t
(Brockett and Mesarovic. 1965). TrackabiIity implies these, but versa does not hold. Theorem 1. In order for the robot described by (1) to be: a) natural1y trackable over ShXSdXSy it is necessary that for some {J>O, fl=jJ(S},.S"S), det[ J(q)A·1(q)B} "'0, V(t,q,h, d,y,/.)) E9l+X (jl(t;Yd.)) xSIr xSdxSy, i.e., (jl(t;y.f.)) =1Jr. (6)
globally naturally trackable over S"xSdxSy.ll is necessary and sufficient that a) holds for fl=+oo. Proof. Not shown due to page limitations. The proof for trackability of different systems is shown in (Grujic and Mounfield, ]990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992, 1993. 1994).
... IF,
d)
elementwise stablewise tracking with vector reachabilitv time 1"R over ShXSdXSy iff, it exhibits elementwise stablewise tracking over SlrxSdXSy and rR is the first vector instant such that leol
e)
These tracking properties are global iffthey are valid for N=diag{+ 00+ 00, .. +t%J}.
b)
4. ELEMENTWISE TRACKING QUALITY The classical notion of tracking (asymptotic tracking) has been used to describe the property of a system whose real output Y asymptotically approaches a time-varying desiredy./O as time t increases infinitely and without the influence of any disturbance. Controlled systems are most often under the influence of disturbances and are required to have their real outputs asymptotically follow their desired outputs. Such tracking should also have a prescribed quality. In the following we shall call 'RI 7:fIr=(rflrJo "!It], ... , "~ a vector instant or a vector (elementwise) reach ability time. Similarly, 1"'7=(1: 11)1 't2n2 ... 't", '1 )
T
is a vector (elementwise) settling time
with respect to a vector 1J=(TJt 1]2... 1]",Y, 1]€]O, +001[, 8f.:J={e:lel
elementwise stablewise tracking over ShXSdXSy iff, both there is a vector pO, 1/!=TjJ(d,YtP'h) such that leol
limllg.(t;e/J>·d,u,Yd;hJI: t~+f>O}=O and for every vector EE}O.+ocl[andevery (h,d,yd)E ShXSdXSy there is a vector oE(O, +"'). o=O(E,d.U,Yd;h), such that /e,,/< c5impJies Ig(t;eo;u,d,y.-fth)/
elementwise stablewise tracking with T] : vector settling time 1",., 1"'1=1",je",S,;SIv Syu).
By referring to this defmition and the well known properties oftime-invariant linear systems we can verifY the following statement for exponential tracking. Lemma 1. If the contrQllaw u(.) is such that the output error em obeys (7), T[e(O, efl)(oJ=K,iJ)(t;eu;u,d,Yd;h) (7a) +Ko e(t;e/p'u,d,Yrih),
T[e(t}, ef1)(O]=O. V(e",t,h,d.,y)E!!f"x31+xS"xSdx~.,
(7b)
where Ko and XI are such positive diagonal matrices that e = 0 is asymptotically stable state of (7) then the system (l) controlled by u(.) exhibits global elementwise exponential tracking over Sh XSdXSyThis lemma results from (7) and Def'mition 2. The (exponential) solution to (7) is given by (8), e(t;eo;u,d,YtP·A}={exp(-tK/KrJJeo.
(8)
Lemma 2. a) If the control algorithm is such that a control u() ensures existence of a positive diagonal matrix Lt guaranteeing that leol<.:j1 implies (9) for KE{21,3I, ...} and positive diagonal ZEIR+ mx""
i l ) (t; e,; d; U,Yd; h) + KZS'(e.) [S(e.)e(t;e,,;d;u,Yrih)] (/_K - J) = 0 for all (t,h,d,y)E3l+ XShXSdXSy,
(9)
then system (l) controlled by the control u(.) exhibits global elementwise stablewise tracking with both vector settling time T'7= (J12)ZI (/ +S(leol x- l -1J K -')}(]eo l K- I _ 1J K - I ) and with vectorreachability time .R(e.)=Z'lleol K- I QYg ShXSdXSy without oscillations. overshoot and undershoot. The equation (9) has the unique ESFRT-tracking solution defmed by g,(t;e,,;d, U'Ycbh) = (112)S(e.){J +S{\eo/ K- I -tZl}}{\eo l r l _t T1!', WE31+. (10)
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Other tracking properties may be found for Lyapunov (Grujic, 1988a, 1988b, 1989), and naturally trackable systems in (Grujic and Mounfleld, 1990, 1992, 1993), and (Mounfield and Grujic, 1991, 1992,1993, 1994). The matrices Ko, K 1 , K, Z can be selected so that the solution ffJ)(.) (8) or (10) possesses a prespecified tracking property and quality (9) or (11) respectively. Their choice is independent of the system (1) and (h,d,yJESIt><'Sd><'Sy-
5. HIGH-GAIN NATURAL TRACKING CONTROL Definition 3. A control u(.) is a natural tracking control for the robot (1) if, and only if: a) it is continuous in time tE31+; b) it is synthesized without any information about hO, and d(.); and, c) it forces the robot to exhibit tracking. Theorem 2.
(a)
Let a tracking algorithm
ill1 (11)
detennine le(t) I as elementwise non-increasing. Then, in order for robot (l) controlled by a high-gain natural tracking control u() to exhibit a sufficiently accurate requested tracking over Sh ><'SdXSy so that (8) or Cl 1) holds. it is sufficient that both (12) and (3) hold for some p>O for which a set (t(t;y'/')) ~ positive invariant and for Il sufficiently close to one, IlEJO,1]. det[ J[q(t)]A-I[q(t)]B] 04), V(t, q, d,y,f.)) E81+ X (j1(t;y,f.)) xSdxS)", (12)
b[u(t))= Ilb[u(r)] + (J[q(t)]A"l[q(t)]BrT[e(t)], V(t,q(J>h,d,y) ElH+ XQf1xShXSdXSy.
(13)
Natural (Grujic and Mounfield, 1990, 1992, 1993), and hight gain (Mounfield and Grujic, 1991, 1992, 1993, 1994) natural tracking control have been previously introduced. Comment 1. The tracking control algorithm (13) defines the configuration of the natural tracking controller rather than the closed fonn solution of the control. It is emphasized that there is no interest in the knowledge of a solution for the control, yet the controller defined by (13) will determine the control u(.) at every moment tE31+ Comment 2. The internal dynamics of the robot governed by h(.) and the external disturbance d(.) are compensated by the local positive feedback in u(.) without delay (e=O). This and global negative output feedback ensure the requested tracking quality for any h() ESI!, dO ESd •
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6. ROBOT TRACKING CONTROL SYNTHESIS The tracking qualities compared by simulation are ESFRT-tracking and elementwise PD exponential tracking. Either may be represented by T[e(t),ffIYtJ]=O where T[e(t), efJ)(t)] describes the tracking quality irrespective of the plant model. The tracking property is multiplied by the time-varying, nonlinear gain A[q(t)]{(J[q(t)}/I, which is poorly known for this particular system. The controller output U, u(t)=pu(t-c} +A{q(t)]{J[q(t))fl T{e(t)), 1l=0.9999, is implemented using the (almost) unit positive feedback of u(t) through the time delay e, with e=0+ and high-gain pI, in combination with A{q(t)]{J[q(t)]rT(e). 7. ROBOT TRACKING CONTROL SIMULATION WITH IMPERFECT KNOWLEDGE OF A[q(t)] Simulations of the robot system with both a PD ESFRT-tracking and then a PD exponential natural tracking controller are shown in Figs. 1 through 6. The robot is simulated with link lengths I J =1. 75 rn, /2= 1.0 m, and link masses m J = 2.0 kg, m]= 1.0 kg. The initial position of the end of the robot was set at q(O)= (q}(O) q1(OyT= (0.6 rad, -0.5 rady The initial desired output was y,JO) = [(YdJ(O) Yt/2(OJ)T= (2.2 m, 0.70 mY. The desired path of the robot in Cartesian space is an arc of radius 0.2 meters. The controller had ±25% imperfect knowledge of A[q(t)] by modification of m} and m z. The disturbances in Fig. 1 for the robot were formed by the combination of two Omstein-Uhlenbeck band-limited white noise sources, d}(t) [Dl] with break: frequency ofO.25 Hz, and d2 (t) [D2] with break frequency of 0.35 Hz; these disturbances are very severe and the disturbances' magnitude is similar to the magnitude of h(.) filtered through the matrix F (4) above. In Fig. 2 the actual output and desired outputYdlt)= YdlO) +0.2sin(2t) [YIDES] andYd2(t) = Ym(O) +O.2cos(2t) [Y2DES] are plotted as Cartesian coordinates (the desired output defmes the true circular arc in the figure). The real tracking errors e }(t) [E 1] and e](O [E2] in Fig. 3 show the difference between ESFRT-tracking and exponential tracking. The (smooth, continuous) joint motions qJ(t) [QI] and q/t) [Q2] for both tracking qualities required to rrack the desired outputs are shown in Fig. 4. The natural tracking controls u,(t) [UI] and 1.1 2 (0 (U2] in Fig. 5 force the robot to track, even with substantial imperfections in its nonlinear model and extreme disturbances. The fmite reachability time control is no more dynamic than the exponential tracking control; both, however, are subjected to severe torque and friction disturbances. The friction disturbances for the fmite reachability time test are shown in Fig. 6; note the nonlinear and continuous but only piecewise differentiable nature of the friction disturbances.
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8. CONCLUSION Natural tracking control algorithms guarantee a prespecified tracking quality which may be elementwise exponential or stablewise tracking with finite reachability time. These have been successfully applied to a nonlinear robot system. The system natural trackability condition was used to form the controller. The natural trackability condition does not require the knowledge of the internal dynamics of the robot. The natural trackability condition sets the minimum requirements for knowledge of the robot to achieve tracking. An example simulation demonstrates natural tracking control as applied to a two-degree-of-freedom robot model. A tracking convergence rate was prespecified without the knowledge of the internal dynamics of the robot. It was shown that the natural trackability information required is the inverses of the product of the inertial matrix components, the output Jacobian, and the actuator connections. These components may be unknown individually and perhaps even poorly known collectively. Using the tracking quality, the internal (almost) unity feedback, and the track ability information, the tracking errors converged toward zero from an unknown initial joint configuration while following an a-priori unknown time varying trajectory. REFERENCES Asada, H., Ma, Z.D., and Tokumaru, H. (1990) Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajedory Cont. ASME JDSMC, 112, No. 2,177-185. Astrom, K.J., and Whttenmark, B. (1972) On Self Tuning Regulators, Automatica, 9, 185-199. Bertram, J.E., and Sarachik, P.E. (1960) On Optimal Computer Control. Proc. First Int. Congress ofIFAC, Moscow, 1,419-422. Brockett, RW., and Mesarovic, M.D. (1965) The Reproducibility of Multivariable Systems. J ofMath. Analysis and Apps., 1,548-563. Chen, C.T. (1984) Linear System Theory and Design, Holt, Reinehart, and Winston, NY. Gourdeau, R., and Schwartz, H.M. (1993) Adaptive Control of Robotic Manipulators Using an Extended Kalman Filter. ASME JDSMC, 115, No. 1,203-208. Grujic, L.T. (1988a) Tracking Versus Stability; Theory. Proc. IMACS 12th World Congress on Scientific Computation, 319-327. Grujic, L.T. (l988b) Tracking Control Obeying Prespecified Performance Index," ibid., 332-336. Grujic, L.T. (1989) On The Theory of Non linear Systems Tracking With Guaranteed Performance Index Bounds: Application To Robot Control. IEEE ICRA, 3, 1486-1490.
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Grujic, L.T., and Mounfield, W.P. Jr. (1990) Natural Tracking Control of Linear Systems. 13th IMACS World Congress on Computational Mathematics, Dublin, 3, 1269-1270. Grujic, L.T., and Mounfield, W.P. Jr. (1992) Natural Tracking PID Process Control for Exponential Tracking,. AIChE Journal, 38, No. 4, 555-562. Grujic, L.T., and Mounfield, W.P. Jr. {l993) PDControl for Stablewise Tracking with Finite Reachability Time: Linear Continuous Time MIMO Systems with State-Space Description. Int. J of Robust and Nonlinear Control, 3, No. 4, 341-360. Grujic, L.T., and Mounfield, W,P. Jr. (1995) Natural Tracking Controller. V.S. Patent No. 5,379,210, Qssued 113/95) Khorasani, K. (1992) Adaptive Control of FlexibleJoint Robots. IEEE TRA, 8, No. 2,250-267. Khorasani, K., and Kokotovic, P.V. (1985) Feedback Linearization of a Flexible Manipulator Near Its Rigid Body Manifold. System and Control Letters, 6, No. 3, 187-192. Khorasani, K., and Zheng, S. (1992) An Inner/Outer Loop ControJler for Rigid-Flexible Manipulators. ASME JDSMC, 114, No. 4, 580-587. Kuntz, H.B., and Jacubasch, A.K., (1985) Control Algorithms for Stiffening an Elastic Industrial Robot. IEEE Journal ojRobotics and Automation, RA-I, No. 2, 71-78. Mounfield, W.P. Jr. and Grujic, L.T. (1991) HighGain Natural Tracking Control of Linear Systems. 13th lMACS, Trinity College, Dublin, Ireland, 3, 1271-1272. Mounfield, W.P. Jr. and Grujic, L.T. (1992) High-Gain PI Natural Control for Exponential Tracking of Linear Single-Output Systems. Automatique, Productique biformatique Industrielle (AP/I), 26, No. 2, 125-146. Mounfield. W.P. Jr. and Grujic, L.T. (1993) PID Natural Tracking Control of a Robot: Application. IEEElSMC'93 Conj., Le Touquet, 4, 328-333. Mounfield, W.P. Jr. and Grujic, L.T. (1994) HighGain PI Natural Tracking Control for Exponential Tracking of Linear MIMO Systems with State-Space Description.. I. J. of Systems Science, 25, No. 11, 1793-1817. Rosenbrock, H.H. (1990) State-Space and Multivariable Theory, Nelson, London. Tsypkin, Y.Z., (l97J) Adaptation and Learning in Automatit; Systems, Academic Press, NY.
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I(l
~~----~----r-----r----'
'~----r----'.----'-----'
~+-----+-----+-----+---t-4 le
No+-----+-----~----+_--~ >-
~ o;+-----+_----+-~~~--~
~,~----.-----.-----~----,
~ o+-----+-----+-----+---~
1.50
!oz.
~.so
T (SEC)
3.75
2.00
1.75
Z.2S
YI
2.50
Fig. 2 Desired circular y,ft) motion and actual motiony(t} in Cartesian space, ESFRT tracking (solid), exponential (dashed).
5.00
Fig. 1 Band limited external white noise disturbances d} and dJ •
~•.-----r-----~----r----'
8
~;~
____
~
____
~
_ _ _ _L -_ _
~
8.-----r-----r-----r----,
!i! ci~----r-----r-----r----'
51
o
., .........-
o+-~--+-----+-----~--~ 0.00 I. 2S T tSlt10 3.75 5.00
~
o~----~----~----~--~ le.DO 1.2'5 z.sn 3.15 5.00 T (SECl
Fig. 4 Joint angles q,[Ql] and qz [Q2], motion in Cartesian space, ESFRT· tracking (solid), exponential tracking (dashed).
Fig. 3 Cartesian tracking errors eJI)
[El] and e2 (t) [E2] ESFRT.tracking (solid), exponential tracking (dashed»
!i!
O·,-----,-----r-----r----,
~
9·~----~----~----~----J
c
gy-----,-----r-----r----.
., LL-
h
....0
o· 0-
'"g~----~----+-----~--~ 'O.OD
1.25
2.50 T (SEC)
3.75
I
!1!
5.00
C?'Q.oo
Fig. 5 Natural tracking control uJt} [Ui], u](t) [U2], ESFRT-tracking (solid), exponential tracking (dashed).
1.25
1
2.50
5.00
(SEC)
Fig. 6 Internal friction disturbances tqlt) [TQIF] and lq,(t) [TQ2Fl, for circular Cartesian motion.
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