strict lyapunov functions for control of robot...
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Pergamon PIk !Moo5-lo!xq%)ool!M-x
Auromarica. Vol. 33, No. 4, pp. 675-682, 1997 0 1997 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
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Brief Paper
Strict Lyapunov Functions for Control of Robot Manipulators*
VICTOR SANT1BmZ-F and RAFAEL KELLYt
Key Words-Robot control; Lyapunov function; stability analysis; regulation; tracking; adaptive control; energy shaping.
AMract-We present a methodology based on the energy shaping framework to derive strict Lyapunov functions for a class of global regulators for robot manipulators. The class of controllers is described by control laws composed by the gradient of an artificial potential energy plus a linear velocity feedback. We provide explicit sufficient conditions on the artificial potential energy that allow to obtain in a straightforward manner strict Lyapunov functions ensuring directly global asymptotic stability of the closed-loop system. As an important consequence of this methodology, we also establish a framework for designing adaptive versions for this class of regulators. An explicit update law is proposed that guarantees closed-loop stability and global positioning. Finally, we characterize a class of tracking controllers for which global uniform asymptotic stability is ensured via strict Lyapunov functions. 0 1997 Elsevier Science Ltd.
1. Introduction One of the landmarks in robot control is the controller design methodology for robot manipulators introduced by Takegaki and Arimoto (1981). The main idea of this methodology is reshape the robot system’s natural energy via a suitable controller such that a regulation objective is reached. This approach has been developed by several researchers (Koditschek, 1984; Wen and Bayard, 1987a; Nijmeijer and van der Schaft, 1990; Wen, 1990; Wen et al., 1992, Berghuis and Nijmeijer, 1993; Arimoto, 1995a,b), who have offered extensions and improvements. For an interesting historical review of this idea, see Koditschek (1989).
The Takegaki and Arimoto methodology, also called the energy shaping plus damping injection technique, aims at modifying the potential energy of the closed-loop system, and provides the injection of the required damping. This is achieved by choosing a controller structure such that, first, the total potential energy of the closed-loop system due to gravity and the controller is a radially unbounded function with a unique minimum, which is global, at zero position error; and, second, it injects damping via velocity feedback. The resulting closed-loop system is an autonomous one, with the nice property that zero position error and zero velocity form the unique equilibrium point. By using the total energy, i.e. the kinetic plus total potential energies, as a Lyapunov function, it follows that this equilibrium is stable. In order to
*Received 25 January 1995; 8 March, 1996; received in final form 1 October 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Annaswamy under the direction of Editor C. C. Hang. Corresponding author Dr Rafael Kelly. Tel. +l 617 4 45 01, ext.2 53 25; Fax +l 617 4 51 55; E-mail [email protected].
t Divisidn de Ffsica Aplicada, CICESE, Apdo. Postal 2615, Adm. 1, Carretera Tijuana-Ensenada Km. 107, Ensenada, B.C., 22800, Mexico.
prove that the equilibrium is in fact globally asymptotically stable, the final key step is to exploit the autonomous nature of the closed loop system to invoke LaSalle’s invariance principle.
The above procedure has been the starting point to develop strict Lyapunov functions (Koditschek, 1988a; Wen and Bayard, 1987a; Wen, 1990, Tomei, 1991; Whitcomb et nl., 1991). By a strict Lyapunov function, we mean a radially unbounded and globally positive-definite function whose time derivative along the trajectories of the closed-loop system gives a globally negative-definite function. This allows one to conclude global asymptotic stability in agreement with Lyapunov’s direct method without invoking LaSalle’s invariance principle. In addition to their intrinsic interest, strict Lyapunov functions have three important applications, as pointed out by Wen (1990). First, since a strict Lyapunov function possesses a negative-definite time derivative, it is sometimes possible to conclude exponential stability instead of only asymptotic stability. This allows one to obtain exponential convergence rates. Second, following ideas reported by Koditschek (1984) it is easy, under certain conditions, to generalize the set-point controller to trajectory tracking. Third, strict Lyapunov functions play an initial role in the development of adaptive regulators.
Strict Lyapunov functions have usually been composed of the sum of the kinetic and total potential energies as in the energy shaping plus damping injection technique, but in addition there is a cross-term between position error and velocity. In turn, this cross-term can be normalized or not. Below, we summarize previous related work.
In an early work, Arimoto and Miyazaki (1984) performed a stability analysis of the PID control of manipulators; a Lyapunov function was proposed as the total energy function plus a cross-term bilinear in position and velocity, resulting in a local strict Lyapunov function. Later, Koditschek (1988a) presented in a formal framework the construction of a class of strict Lyapunov functions for dissipative mechanical systems; again the total energy plus a cross-term served as a strict Lyapunov function. In the important paper by Wen and Bayard (1987a), a local strict Lyapunov function for an extensive class of motion controllers was introduced. The extension to adaptive control of these motion controllers was presented by Wen and Bayard (1987b). Wen (1990) introduced a unified approach based on a strict Lyapunov function with cross-term between position and velocity errors to analyze a large class of PD-type controllers, including trajectory tracking controllers and their adaptive versions. Inspired by Koditschek (1988b), which dealt with the adaptive control of a rigid body for attitude tracking, Tomei (1991) proposed a strict Lyapunov function for the stability study of PD control with gravity compensation. This strict Lyapunov also has a cross-term, but now it is normalized by a function of the position error. Tomei (1991) also used this strict Lyapunov function to develop a PD controller with adaptive gravity compensation. Wen et nl. (1992) developed a new class of robot control algorithms using an energy-like Lyapunov function with a cross-term and error measures
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676 Brief Papers
reflecting the topology of the joint error space. Whitcomb et af. (1991) proposed a strict Lyapunov function with normalized bilinear cross-terms for the study of global asymptotic stability of the tracking controller PD+, whose nonadaptive control law was previously reported in Koditschek (1984) and Paden and Panja (1988). This strict Lyapunov function allowed the straightforward derivation of the first adaptive version of the PD+ controller. A strict Lyapunov function for the PD controller with desired gravity compensation, which is useful in obtaining its adaptive version, was proposed by Kelly (1993); again, a normalized cross-term was incorporated in the strict Lyapunov function. More recently, Arimoto et al. (1995) characterized a class of saturated-proportional plus derivative feedback for which strict Lyapunov functions were established; they used the latter functions to design adaptive and learning tracking controllers.
In this paper we broach the problem of finding strict global Lyapunov functions for a class of global regulators. The main objective of this paper is to characterize a class of regulators for which a strict Lyapunov function can be derived to show directly global asymptotic stability. The class of regulators is described by control laws composed of a gradient of an artificial potential energy plus a linear velocity feedback. We provide explicit sufficient conditions on the potential energy that allows us to obtain in a straightforward manner a strict Lyapunov function. Further, based on these strict Lyapunov functions, we also establish a framework to design adaptive versions of such a class of regulators; an explicit update law is proposed that guarantees closed-loop stability and global positioning. In addition, we characterize a class of tracking controllers for which global uniform asymptotic stability is ensured via strict Lyapunov functions.
Throughout this paper, we use the notation A,(A) and &{A} to indicate the smallest and largest eigenvalues respectively of a symmetric positive-definite bounded matrix A(x) for any x E R”. The norm of a vector x is defined as llxll = a and that of a matrix A as the corresponding induced norm IIA II= vAM{ATA}.
This paper is organized as follows. Section 2 describes the dynamics of rigid robots and its main properties. In Section 3 we present our main results on global regulation concerning the characterization of a class of regulators for which a strict Lyapunov function can be derived. The adaptive version of these regulators is addressed in Section 4, and trajectory tracking control via strict Lyapunov function is presented in Section 5. The stability analysis of the well-known PD control with gravity compensation and PD control with desired gravity compensation, as well as two new controllers, are given in Section 6. Finally, we offer some concluding remarks in Section 7.
2. Dynamics of rigid robots In the absence of friction and other disturbances, the
dynamics of a serial n-link robot with revolute joints can be written as (Spong and Vidyasagar, 1989):
d q dtcj= [I [ M-w’ - CL? ihi - g(q)1 1 ’ (1)
where q is the n X 1 vector of joint displacements, q is the n X 1 vector of joint velocities, r is the n X 1 vector of applied torques, M(q) is the II x n symmetric positive- definite manipulator inertia matrix, C(q, $4 is the n X 1 vector of centripetal and Coriolis torques, and g(q) is the n X 1 vector of gravitational torques obtained as the gradient of the robot potential energy Q(q) due to gravity, i.e.
g(q)=y.
The following are two important properties of the robot dynamics.
Property 1. (See e.g. Koditschek (1984)). The matrix C(q, tj) and the time derivative M(q) of the inertia matrix satisfy
qy:ni(q) - C(q, q)]q = 0 vq, lj E R”.
Property 2. There exist positive constants kc and k, such that for all x, y, z l W” we have
l IIW,Y)~I Sk llrll 1141.
l k&W - &Y)II 5k.q Ilx- yll.
3. Global regulation The goal in global regulation is to ensure that the robot
joint position q(t) tends asymptotically to a desired constant position qd regardless of the initial condition, i.e.
!iiI q(t) = q.j Vq(0) E w.
In this paper we consider energy shaping-based controllers whose control laws can be written as (Takegaki and Arimoto, 1981)
where K, is an n x n symmetric positive definite matrix, ij = qd - q E R” is the joint position error, and %!&(%, 8) is the so-called artificial potential energy provided by the controller, whose properties will be established later.
The closed-loop system equation obtained by substituting the control law (3) into the robot dynamics (1) leads to
dii= dtq [I r -4 1
$ [%(qd, @ + Q(qd - @I- Kv4 - C(q, il)il (4)
where (2) has been used. Let us define the total potential energy function %,(Q, i@
of the closed-loop system as the sum of the potential energy Q(q) due to gravity plus the artificial potential energy %.Jq,,, ij) induced by the controller, i.e.
%T(qd, 4) = %(qd? 9) + %(qd - 4). (5)
Conditions for global asymptotic stability of this system were established by Takegaki and Arimoto (1981), and are summarized in the following proposition.
Proposifion 1. Consider the closed-loop system equation (4). Assume that the closed-loop total potential energy is a %’ function satisfying
Al. w(qd,q) is radially unbounded with respect to q, and q = 0 E R” is its unique and global minimum for all pd.
Then the equilibrium [q’ qTIT = 0 E W2” of the closed-loop system (4) is globally asymptotically stable.
Proofi For the sake of completeness, we provide an outline of the proof. Assumption Al implies the existence and uniqueness of the equilibrium at the origin of the state space. Also, because of Al and the positive-definiteness of M(q), the following Lyapunov function (total energy) is globally positive-definite and radially unbounded:
v(q, 4) = :iiTM(q)q + %LT(qd, 4) - %“hdv Oh (6)
By virtue of Property 1, the time derivative along the trajectories of the closed-loop system yields
v(fj, Q) = -qTK,4, (7)
which is a globally negative-semidefinite function. Therefore, from Lyapunov’s direct method, stability is established. Finally, the Krasovskii-LaSalle theorem (see e.g. Vid- yasagar, 1993) allows the conclusion of global asymptotic stability. cl
In this paper, by a strict Lyapunov function we mean a globally positive-definite function whose time derivative along the trajectories of the closed-loop system yields a globally negative-definite function. Strict Lyapunov functions
Brief Papers 677
allow the conclusion of global asymptotic stability in Proposition 2. Consider the class of controllers (3) that agreement with Lyapunov’s direct method, and therefore produce a closed-loop total potential energy satisfying (8) avoid the need to use the Krasovskii-LaSalle theorem. Such and (9) for some strict positive constants /?, p’, E, and E’. functions also have important applications pointed out by Then the equilibrium [qT qqT = 0 E R*” of the closed-loop Wen (1990) in the easy derivation of adaptive version of system (4) is globally asymptotically stable, and a strict these controllers. Lyapunov function to prove this is given by
Remark 1. It is worth noting that the Lyapunov function (6) is not a strict Lyapunov function, because its time derivative (7) is only globally negative-semidefinite instead of globally negative-definite.
where
v(% il) = +ilTM(&i + %tqd, ii)
- %(qd, 0) - uf(~)TM(+i? (10)
The main objective of this paper is to characterize a class of controllers (3) for which a strict Lyapunov function can be derived. This characterization is given in terms of conditions on the closed-loop total potential energy ‘?&(qd,Q), which is related via (5) to the artificial potential energy ‘?&(q.,,ij) induced by the controller.
qq,=L- 1 + II411 q
(11)
and (Y is a constant such that
min ~,VLl
A,IM)‘m2[k, + 2A,{M} >a>O. (12)
In order to cover a broad spectrum of total potential energies, we conisder not only those lower-bounded by global quadratic functions but also those that grow lower-bounded by a local quadratic function and a linear function outside a ball centered at the origin. This latter class covers total potential energies having bounded gradients that induce saturated position error feedback. The usefulness of a subclass of such a functions (%? total potential functions) arises in the design of PID-type global regulators without compensation of the gravity forces (Arimoto er al., 1995). Further, we also include ‘%‘I total potential energies leading to hard-saturated position error feedback, and VZ2 total potential energies that have a unique and global minimum but that are nonconvex functions.
Proof The Lyapunov-function candidate (10) corresponds to the total energy of the closed-loop system by taking the sum of the kinetic energy &=M(q)ip with the potential function introduced by the controller ?La(qd,ij), the potential energy of the robot %(qd - q) due to gravity, a cross-term bilinear in position error and velocity, and an appropriate additive constant. The normalized function f(q) was first suggested by Whitcomb et al. (1991) in the context of globally stable adaptive tracking.
The Lyapunov-function candidate (10) can be rewritten as
v(ij, q) = :[i - af($]=M(q)bi - ar(q)l + %TtT(qd, 4)
- QT(qd, 0) - :JfWM(qMl)~ (13) In this section we show that strict Lyapunov functions can
be constructed for the dass of controllers whose associated potential energy leads to a %’ total potential energy satisfying the following two conditions:
thus it will be a radially unbounded positive-definite function provided that
%TT(qdr @ - %T(qd, 0) - :~~WWMTI) (14)
is also a radially unbounded positive-definite function in 4.
where p, p’, E and E’ are strictly positive ConStantS.
Under the conditions (8) and (9), it can be shown that w(q.,,$ satisfies Assumption Al of Proposition 1. To see this, first note from (8) that QT(q,, $) - 41,(%, 0) is a radially unbounded and globally positive-definite function, and therefore has a global minimum at q = 0. Radial unbounded- ness and the existence of a global minimum at Q = 0 are also features of &(qd, q). It only remains to prove that there are no other minima besides the origin. This is ensured by (9), which states that the gradient of 91,(q,,ij) does not vanish, and therefore there are no critical points for any 4 # 0.
Remark 2. Wen and Bayard (1987a) have considered the class of controller described by (3) for which the total potential energy QT(qdrij) is assumed to have a global minimum at ij = 0 and
for some v > 0. The latter condition on &(q.,, ij) is indeed stronger than (9); thus there is a family of controllers that can be analyzed via the approach presented in this paper but that arise from Wen and Bayard’s formulation.
Remark 3. Recently, Arimoto er nl. (1995) have introduced a class of artificial potential energies that induce total potential energies in agreement with (8) and (9). These total potential energies are Z2 functions quadratic in the vicinity of 4 = 0 and linear outside a ball centered around the origin. For this class of functions, Arimoto et al. (1995) proposed strict Lyapunov functions, making it easy to derive learning and adaptive tracking controllers.
Our main result is presented in the following proposition.
Since (14) vanishes at Q = 4, only remains to show that it is radially unbounded and positive for all ij # 0 E W”. To this end, we use the following important expression:
1 b*hAW lVl12 if II411 <E, t15j 5 b2AMIMI INill if llijll 2 E,
which holds for all E > 0. Next, by taking the same E in (15) as in (8), we obtain
%TT((ld, $ - %T(qd* O) - :cr2p (@“6-,)f(@
> (P- t
b2MW) 14112 if II411 <E, t16j (P - b2MW) lliill if II911 2 E.
Obviously, since a satisfies (12), we have ensured that (14) is a positive-definite function in ij. In summary, the Lyapunov-function candidate (10) is a radially unbounded and globally positive-definite function. It is important to note that the constant (2 is only required for purposes of analysis, and we do not need to know its numerical value.
The time derivative of the Lyapunov-function candidate (10) along the trajectories of the closed-loop equation (4) can be written as
%, il) = q= 1
$ [“u(qd - @ + %,(qd, q)]
- Kvil - C(q, Vi)
- ni(Cj)=M(q)tj - af(Q)Tni(q)a
- @i)= l
$ [O”(qd - 4) + %(qd, @I
- Kvil - C(q, 414 I
. (17)
678 Brief Papers
By virtue of Property 1, the time derivative of the Lyapunov function becomes
fi(Q, 4) = -qTKvQ - cd(ij)TM(q)tj
- af(@T a%(qdy q) 4
+ ar(QTKv4
- afwC(q, 41=il. (18) We now provide upper bounds on the following terms:
-ar(B)‘C(q, U’ci 5 ok, llil12, (19)
-(rf(d)FM(q)q 5 2ahMiM) ~~~~~*~ (20)
-4TfLil 5 - :il=Kvil - t&nIKl Ilil12, (21)
where we have used Property 2 and the following two properties of f@j):
Ilr(ii)ll 5 1,
lleI,ll 5 I&il + I/ - (* + ,G2 llqll iill52 11611. (22)
It now follows from the inequalities (19)-(21) that the time derivative V(& 4) in (18) satisfies
%, il) 5 - :rjT&q - tA,IKJ Ml2 + ~h,bW lli1112
+ a& lItill - 4NT a%(qdv 4)
aQ
+ af(4)TKVb
which can be rewritten as
I%, 4) 5 - $I - 4WGh - afW1 - &L#Gl- ak - 2@A~@f]) 111112
aqTK4 -~ 1 20 + liill) .
Since a satisfies (12), to prove that v(fj,@ negative-definite function, it only remains to show that
qr a%(qdv 4) _ QijTK”ij
4 w + 11411)
is positive-definite in 4. To do this, first note that
aijTKvq
-20 + lliillf - &MKI 11~112 if llqll <E’, - bh,.&~ II411 if II411 = E’,
(23)
is a
(24)
for all E’ > 0. From this, and taking E’ as in (9) and using the assumption (12) on a, we obtain
qT a%+h a ZP 114112 > hdL1 lMl12 2
alTKdfj
4 20 + 11~11)
for all E’ > I]@/ >O, and we have
for all llfj/l 2~‘. This allows us to conclude that (24) is a positive-definite function; thus we can guarantee that V(q, 4) satisfies
2 - W&l - 6 - 2ahIMl) IIQII ,
which is a negative-definite function. Finally, by invoking Lyapunov’s direct method (see e.g.
Vidyasagar, 1993), we conclude global asymptotic stability of the equilibrium point [ijT q7’ = 0 E R2” for the closed-loop equation (4).
4. Adaptive regulation The control law (3) is a function of the artificial potential
energy Qa(qd,Q), which in turn can depend on some robot and payload parameters such as masses and centers of mass. In this section we assume that these parameters are constant but some of them are uncertain. By this, we mean that their exact values are unknown but upper bounds on them are
available. With this information, it is possible to obtain upper bounds on A,,,{M}, kc and k, required in the design of the adaptive regulators introduced below.
It is convenient to remember that the robot dynamics can be linearly reparametrized into a parameter vector 0* E RP containing only the unknown robot and payload dynamic parameters (Khosla and Kanade, 1985; Koditschek, 1985). In particular, the gravitational torque vector g(qd - Q) = g(Q - 4; e*) can be expressed as
fi(qd - @ e*) = &(qd - ii) + @(qd, @e*t (25)
where &(qd - ij) represents the known part of g(q,, - ij; O*), and @(Q, fj) E Rnxr is a regressor matrix that contains nonlinear but known functions of ij. Since we have assumed that upper bounds on the entries of fi* are known, say f3:>0, we can define the set ~:={OEWP:O=~$I&+, with i=l , . . . , p}. Obviously the true (unknown) parameter vector 0* belongs to the set 0.
To denote the deoendence of the artificial potential energy on some robot and payload parameters rewrite the control law (3) as
explicitly, let us
7= a%(qd? 4; O*) _
4
K q Y .
In the remainder of this section we dependence of the artificial potential
assume that the energy on the
parameter vector 0* is such that its gradient with respect to the joint positions satisfies
(26)
a%(qd? iii e*)
4 = h(‘ld, a) + @(qd, $@*T (27)
where h(qd, ij) E R” is a continuous function of qd and q. We propose the following adaptive version of the control
law (26):
(28)
where 4 E f@ is the adaptive parameter vector given by the following update law:
O(t) = r [*(qd? g(o))T[af(q(n)) 0
-g(a)] do + h(O), (29)
where I E RfxP is the symmetric positive-definite adaptation gain matrix, O(0) E I@ is any vector (but is usually selected in practice as the ‘best’ a priori approximation available on the unknown parameter vector @*) and a is a constant such that (12) is satisfied. Notice that, in contrast with the nonadaptive regulation case, we now require a for update-law implementation.
Before we obtain the closed-loop equation, let us define the parameter error vector as 6 = h- O*. Thus, using (27), the control law (28) can be written as
7= a%(qd, 4: e*)
4 - hj + @(qd, 4% (30)
On the other hand, from the definition of 6 and since the parameter vector 8* is assumed to be constant, we have s=i% Using this together with the update law (29) and substituting the control action (30) into the robot dynamics, we obtain the following closed-loop equation:
_ _
I -4
zz M(q)-I[ $ ‘t&(qd, q; @*) - Kvq - c(q, $14 + *(qdv q)e t
r*(qd? $‘Laf($) - I]
(31)
where qlT(qd, ij; @*) = “u,(q& $; e*) + Q(qd - 8; @*). The a-
bility result is established in the following proposition.
Brief Papers 679
Proposirion 3. Consider the class of adaptive regulators (28), (29) with an artificial potential energy 9&(qd, ij; @) satisfying (8) and (9) with some strictly positive constants /3, P’,_E and E’, for all 8 E 6. Then the equilibrium [4’ qT t37’= 0 E R”‘+p of the closed-loop system (31) is stable and lim,,, a(t) = 0. A Lyapunov function to prove this is given by
v(ii, q, 6) = :qWq)i + Qr(qd, 4; 0*) - %(qd, 0; e*) - af(q)=M(q)4 + :e’r-‘e. (32)
Proof. This follows closely that of Proposition 2. To carry out the stability analysis, we propose the Lyapunov-function candidate (32). Note that this is just the strict Lyapunov function (10) used to show asymptotic stability in the nonadaptive case plus the nonnegative quadratic term $OTT-‘ft. Since it has previously been proved that the Lyapunov function (10) is a positive-definite function, we have (32) is in turn a positive-definite function in ]qr Q eqT.
Following the steps in the proof of Proposition 2, we obtain the following expression for the time derivative of the Lyapunov function (32) along the trajectories of the closed-loop equation (31):
V(ij, q, 6) c - :[q - af(ij)]TK”[lj - af(cj)]
- (:4nwJ - & - 244~~1) llill12
a
[
iiT a%(qd, 4) _ CYQK$j
1 + 11q11 871 2u + 11w I . (33)
The right-hand side of this equation is identical to (23) which has been shown to be a negative-definite function in fj and q. Thus we have (33) in turn a negative-semidefinite function in the full state. Therefore, using Lyapunov’s direct method, we immediately conclude Lyapunov stability of the origin of the closed-loop equation.
It remains to prove that the position error fj vanishes asymptotically. Toward this end, we invoke standard adaptive control arguments. First, since (33) is a globally negative-semidefinite function in the full state and the Lyapunov function (32) is a globally positive-definite function, we can guarantee that q, q and 8 are bounded vector functions. Next, by integrating both sides of the inequality (33) we conclude that q and q - af(fj) are square-integrable functions; hence f(q) is a square-integrable vector function as well. From this and using the previous conclusion on the boundedness i& we ensure that fj is in turn a square-integrable function. But a bounded and square- integrable function whose derivative ($ = -4) is bounded must tend to zero (Desoer and Vidyasagar, 1975); hence we finally have lim,,, q(r) = 0, as desired. 0
5. Trajectory tracking control The purpose of this section is to exploit the methodology
above described to derive strict Lyapunov functions for a class of trajectory-tracking controllers. Now the goal is to find $1) such that
lim q(r) = 0, ,-=
where $ = q,,(t) - q(t) E R” again denotes the position error, but now the desired joint position qd(t) is a V* vector function of time. Thus the vector 4 = r&(t) - q(r) E R” stands for the velocity error. We assume that the desired velocity norm has a known upper bound IlqdllMax.
Motivated by Koditschek (1984) let us consider the following extension of the control law (3):
+) = a%(qd? a ~ + Kv$ + M(q)iid + c(% @iPd, aq (34)
where the terms M(q)& and C(q,q)k have been added. Note that when qd is a constant vector, we have the regulation setting, and the control law (34) becomes the regulator (3).
In the trajectory-tracking control problem we must assume that the artificial potential energy %a(qd, 4) can be written as
“U,(Q, Q) = h(q) - Q(q, - Q); hence the total potential energy will be given by
“llT($ = %tqdv %) + %d - 4) = h(%), (35)
which is assumed to satisfy (8) and (9). Note that Q&j) depends explicitly only on q but not on
q+ This restriction does not allow us to generalize to tracking controllers those global regulators having a desired gravity compensation g(qd) term.
Since
the control law (34) can be written as
$1) = af(q) - + K,3 + h’f(q)ijd + c(q, i)& + g(q). da
(36)
Remark 4. The PD+ controller, introduced by Koditschek (1984) and subsequently analyzed by Paden and Panja (1988) using Matrosov’s theorem, is a particular case of the control law (36) with h(g) = $jTK,@
The closed-loop system equation is obtained by substitut- ing the control law (36) into the robot dynamics (1). By using (35), this can be written in terms of the state vector [$ 4’1’ as
4
- $ [“llT(@] - &$ - c(q, @b (37)
The main result of this section is established in the following proposition.
Pruposition 4. Consider the class of tracking controllers (36) with a total potential energy QT(qd, 4) = h(q) satisfying (8) and (9) for some strictly positive constants p, p’, E and E’. Then the equilibrium [ij’ 4TT= 0 E R*” of the closed-loop system (37) is globally uniformly asymptotically stable, and a strict Lyapunov function to prove this is given by
v(@ $) = :4TM(q)4 + qT(q) - %(O) + fl(q)Twh (38)
where f(ij) was defined in (11) and y is a constant such that
Proof Following the same steps as in the proof of Proposition 2, it is easy to show that the Lyapunov-function candidate (38) is a radially unbounded and positive-definite function.
The time derivative of the Lyapunov-function candidate (38) along the trajectories of the closed-loop equation (37) can be written as
Kv4 - C(q> 414 1
a%(q) Tz + ;ij?%qq)i$ + ~ [ I aq q
+ YmTwl)B + rfWTMq)4
- K,q - C(q, $41. (40)
680 Brief Papers
Using Property 1, the time derivative of the Lyapunov function becomes
I+j, ij) = -i.j’K,lj + y+(i(a)%f(q)ij
_ yfiq)Ty _ yf(fi)TK4
+ Yf(il)TC(q, VT4 (41)
We now provide upper bounds on the following terms;
Yf(il)TC(q, i)TG 5 Ykc lisllZ
+ Ykc IIMM,, IlfWIl ll4ll~ (42)
ri(q)TM(q)a 5 2YAMIM) 111112, (43)
-ij’&lj 5 - $ijTK”ij - $A,{&} ~~lj~~2. (44)
where we have used Property 2 and (22). It now fo!lows from the inequalities (42)-(44) that the time
derivative V($ 4) in (41) gives
Q, $) 5 - (:A,K] - Yk, - 2YhdMH 114112 - t$TK,Q
+ Ykc IMMax IIWII II111
- yf(ij)T gp - yf(ij)TK,ij.
Next, since the total potential energy Ur(fj) is assumed to satisfy (9) and y satisfies (39), we can rewrite the time derivative of the Lyapunov-function candidate as
I%, 4) 5 - :A,%] 11~112 + Y(~#LI
+ kc Ilbll~ax)~ ll4ll -yLq$ (45)
for /lfjll C E’ and
k(Q, 8) 5 - :&,,{&) 11$112 + Y~UL~
+ kc IliidllMax) 1 + llgll mlI~l,_y~y_l& (46)
for ~~ij~~ 2 E’, where we have used
Note that the only difference between (45) and (46) is in the final right-hand term. It only remains to show that V&t) in (45) and (46) are negative-definite functions. Toward this end, first note that for all x, y E R and strictly positive constants a, b, c and Y, the functions
IYI a lxj2 - yb 1x1 - Iy12
1+ IYI +Ycl+lyl (47)
and
IYI 4.42-y~kl-
lyl 1+ IYI +Ycl+lyl (48)
are both positive-definite provied that 4uc/b2 > y. Therefore, by taking a = $A,,,{K,}, b = AM(KV} + kc ll&llMax and c = p’, the above condition becomes
2P’&K] (AM{%) + kc Ilbllmd2 ’ ”
which is satisfied from (39). Thus we conclude that p’(& $) is a negative-definite function. At this point we conclude global asymptotic stability of the origin of the state space for the closed-loop system.
To prove uniform asymptotic stability, first note that we have considered a class of controllers whose associated artificial potential energy satisfies the conditions (8) and (9). For this class of controllers there always exists a function & of class X such that
Note also that
Y@i)TM(q)a = Y~{M) llr(l)II ll$ll
5 ~b.&f} II411 ll~ii
5 yb.,{M) (11%112 + ll$l12).
Thus we can find an upper bound for V(@ $) = $aj’M(q)4 +
@h(q) - %TT(O) + Yf(ij)TM(q)6:
W, #I :&&I 11$112 + ~~~~~~(11411~ + 11~112) + PdllQll)
= ~h&fI lbjl12 + 32~ + lh.dW ll~ll’ + P3(114ll)
where we have used iI_ II II 1 - llijll’+ llij112. Thus the Lyapunov
function V(ij, $) is upI?er-bounded by a class-X function of the norm of the state vector. Therefore, by invoking Lyapunov’s direct method (see e.g. Vidyasagar, 1993) we conclude global uniform asymptotic stability of the equilibrium point [ijT fiTIT= E R*” for the closed-loop equation (37). 0
Finally, it is worth noting that the Lyapunov-function candidate (38) can be considered as an extension of that proposed by Whitcomb et al. (1991) in the sense that it qualifies as a Lyapunov function for the PD+ controller as well as for the class of tracking controllers satisfying the assumptions of the proposition.
6. Examples of application In this section we illustrate the application of Proposition 2
to derive strict Lyapunov functions for global regulators. 6.1. PD control with gravity compensation. The PD control
with gravity compensation (Takegaki and Arimoto, 1981) is given by
T = K& - Kyq + g(q). (49)
It is well known that global asymptotic stability of the closed-loop system is ensured provided that K,,, and K, are symmetric positive-definite matrices. This claim is usually proved by using LaSalle’s invariance principle together with the following (nonstrict) Lyapunov function: V(q, 4) = 1QTWcl)4 + hiTKp4.
From the control law (3) we can see that the gradient of the artificial potential energy provided by the controller (49) is given by
a%d%, 4) = K
aq P $ + g(qd _ a)
Therefore we can find the following expressions for the potential-energy functions:
_ - - %&,d, ‘4) = - %d - ‘4) + :qTK,s
%@,d. 4) = %(qd, # + %tqd - ii) = :qTfQ.
It is easy to show that the total potential energy satisfies
%tqd, $ - %(qd, 0) = t$K,q z P 114112 if II411 <G
P II411 if II411 2 6
_Ta%qd9ti)=qTK q1 4
P’ ~~~112 if iiqil <E’,
aq P
P’ 1141 if IlQll 2 E’s
where ,9 = iA,( p’ = A,{K,,} and E = E’ = 1. Therefore, from Proposition 2, we can conclude that a strict Lyapunov function is given by
V(fj, q) = ;gTM(q)4 + $iTKpij - 44)TWq)il,
where a must satisfy (12), i.e.
min hn{Kvl
AM{M} ’ AM{Kv} ’ 2(kc + 2A,{M}) >a “’
Remark 5. It should be pointed out that a strict Lyapunov
Brief Papers 681
function for PD control with gravity compensation has been previously proposed by Tomei (1991).
6.2. PD control with desired gravity compensation. Among the set-point controllers for robot manipulators, perhaps the simplest is the so-called PD control with desired gravity compensation proposed by Takegaki and Arimoto (1981). The control law is given by
(50)
It has been reported by Takegaki and Arimoto (1981) that global asymptotic stability of the closed-loop system is achieved provided that K, + dg(q)/dq and K, are symmetric positive-definite matrices. A further step was given by Tomei (1991), who derived the following sufficient condition for global asymptotic stability: h,,,{K,} > k, and A,{K.,) > 0. Takegaki and Arimoto (1981) have invoked LaSalle’s invariance principle and the following (nonstrict) Lyapunov function:
V(Q, q) = &i”M(q)q + :fjTQj + Q(q)
- a(%) + g(GT4
With regard to the control law (50), we can straightfor- wardly derive the expressions
%(q,, q) = gT(s&j + tQKd>
%(qdr ?i) = Q(q, - ii) + g-Yq&I + :IiTK,4.
The above total potential energy satisfies the following inequalities
%(qd, 4) - %(qd, O) = O”(qd - ii) - q(qd)
+ gT(qd)% + :?I%4
~ Plliill
1
* if IIQII <E, P IliTll if 11411 2 6
‘T ‘%T(qd, il) = $K q Jq P
q + gT[g(qd)
t-2
-g(qd-@lr[;, i/;;; if llijll <E’, if Ilqll 2 E’,
where p = $(h,,,{K,} - k,), /.i’ = A,{Kd - k,, E = 1 and E’ = 1, and Property 2 has been used.
Thus we conclude from Proposition 2 that if h,{K,) > k, then the following is a strict Lyapunov function for the closed-loop system:
v(& il) = :qTM(q)i + 4fITK& + Q(qd - @
- %U(qd) + ,$qdjTq - af(q)TM(@it
where (Y must satisfy (12) i.e.
h,WJ 2(kc + W~I,(M))
>a>o.
Remark 6. This strict Lyapunov function was previously suggested by Kelly (1993) to study the controller (50). and was used to derive a PD controller with adaptive desired gravity compensation.
6.3 Tanh D control with desired gravity compensation. In this subsection we introduce a modification to the regulator suggested by Cai and Song (1993) where gravity compensa- tion is now replaced by desired gravity compensation. We propose the following control law:
tanh 4,
r=K, tanh Qz
[!1
- K4 + g(qdh (51)
tanh 4,
where K, and K, are diagonal positive-definite matrices.
We can derive the following expressions for the artificial and total potential energies:
%(qd, 4) = O”(qd - 4) + gT(qdh
It is interesting to note that above total potential energy cannot be proved to be a strict convex function through analysis of its Hessian matrix; however, as will be shown below, conditions are given so that it satisfies (8) and (9), meaning that it has a unique and global minimum at 9 = 0.
After some manipulations and using Property 2, it can be shown that
-T J%(‘Id, fj) = qrK 4
tanh 42
Jii H P :
tanh gn
+ qTk(qd) - k!(qd - 111 z { ;: ;/i/i2 ;; ;;;// = ;:;
where p = $(A,{K,) tanh 1 -k,), p’ = A,{K,) tanh 1 -k,, E = 1 and E’ = 1. The above conditions were obtained by using the inequalities
tanh E if 1x1 < 8,
In cash x 2
1
7 IbY
!+ if 1x1 2 E,
xtanhxz
1
!+ lx\2 if (x(<E,
tanh E 1x1 if IxI~E.
Therefore if A,{K,) > k,/tanh 1, then, from Proposition 2, we can obtain the following strict Lyapunov function:
w i) = biTWq)i
+
+ Qu(qd - 4) - %U(qd) + gT(qd)q - af(@TM(q)il?
where * must satisfy (IZ), i.e.
min{~~,2(id~~~~flI-ks,
A,,%) 2(k, + 2AM{M)) I ’ a “.
6.4. A saturated-proportional derivative controller with gravity compensation. In this subsection we propose the following controller induced by a ‘%‘I artificial potential energy that has hard saturation in position error feedback
sat (4,) ‘T = K sat (42)
[.I
p : - Kil + g(q), (52)
sat Gd
where K, and K, are diagonal positive-definite matrices: and
Brief Papers
sat(x) denotes the well known saturation function:
1
1 if ~21, sat(x)= x if Ixl<l,
-1 ifxs-1.
By comparing the controller (52) with the control law (3) we can identify the gradient of the artificial potential energy induced by the controller as
rsat(qJl + g(qd - ii).
From this, we obtain the artificial and total potential energies
%(qdr 4) = %(qd, ii) + %L(qd - 4) = 2 h(41)? ,=I
where
The above total potential energy satisfies (8) and (9) with P = :A,{&), P’ = &nIK,), ~=l and E’ =l. Therefore we derive from Proposition 2 the following strict Lyapunov function:
with (Y such that
min hn{Kv} &,{M}’ AM{KV} ’ 2(ke + 2A,{M}) ‘o ” 1
7. Concluding remarks We have shown in a unified framework a systematic
methodology for deriving strict Lyapunov functions in terms of the total energy of the closed-loop system for a large class of global regulators for robot manipulators. The class of regulators are energy-shaping-based, and are described by control laws composed of the gradient of an artificial potential energy plus a linear velocity feedback. We have provided explicit sufficient conditions on the total potential energy allowing the derivation in a straightforward manner of a strict Lyapunov function. Based on this methodology, we have also addressed the design of adaptive regulators, taking as a starting point the corresponding nonadaptive strict Lyapunov function. Finally, we have generalized to the tracking-control case our results of regulation; a charac- terization for a family of trajectory tracking controllers together with their strict Lyapunov function have been established.
Acknowledgemenr-This work was partially supported by CONACYT, Mexico.
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