Adaptive Passive Velocity Field Control - Enet

Proceedings of the AmericanControl Conference San Diego, California June 1999 ●

Adaptive Passive Velocity Field Control 1 Perry Y. Li Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis MN 55455. E-mail: pli@me. umn. edu

Velocity field to draw’a circie

Abstract Passive velocity field control (PVFC) was previously developed for mechanical systems which have strong coordination and must interact with the physical Applications include teleoperated environment. manipulators, contouring in machining and smart The methodology encodes tasks exercise machines. using time invariant desired velocity fields instead of the more traditional method of timed trajectories and guarantees that the closed loop system behave passively with environment power as the supply rate. By maintaining the passivity property of the closed loop system, stability and robustness will be enhanced, especially when interacting with uncertain environments. The present paper extends PVFC to situations where the inertia parameters of the mechanical system are unknown. A direct adaptive control scheme is proposed which preserves the passivity of the closed loop system and ensures that the asymptotic convergence of the velocity to the direction of desired velocity field. 1 Introduction Passive velocity field control (PVFC) was developed in [1, 2, 3] with two objectives in mind: 1) to ensure that mechanical systems with strong coordination requirements are well coordinated; 2) to enhance the safety, robustness and stability of such systems when they interact mechanically with humans and other physical environments. To these ends, the formulation of PVFC has two distinct features: 1) the task is encoded using a velocity jield on the configuration space of the sys-

tem; 2) controllers are constrained so that the closed loop system appears to the physical environment to be a passive system. In PVFC, instead of requiring the motion of the system to track a desired timed trajectory, as is traditionally the case, the task is encoded using a velocity jield on the configuration space of the system, i.e. specifying a desired velocity V(q) at each configuration q. By 1Research supported by National Science Foundation under grant CMS-9870013

0-7803-4990-6/99

$10.0001999

AACC

774

:2

Figure

.1.5

-1

-0.5 0 0.5 X - ccerdlnate

1

1.5

2

1: A velocity field that encodes a circular contour following task

requiring that the velocity of the system converges to a scalar multiple of the encoding field, the system will be guided to satisfy the task in an expedient and coordinated manner. For example, in the important application of contour following, a desired velocity field can be defined such as in Figure 1 for the task of tracing a circular contour so that each arrow represents the desired velocity V(q) at configuration q. If the actual velocity q(t) is controlled to satisfy q = ctV(q), Q >0, then the mechanical system will be guided to follow the contour. The speed at which the manipulator follows the contour is proportional to the scalar ct. On the other hand, if the the task is formulated as a timed trajectory, a situation can arise that the actual position and the desired location is just step by a time interval, At. Since timing is often not critical compared to coordination, it may not be necessary to keep up with the desired timed trajectory. However, if a timed trajectory based controller tries to keep up with the desired trajectory, it can risk leaving the desired contour. This causes a phenomenon known as radial reduction in which the actual path traced out has a smaller radius

.. .. .... .. .... .. .... ... ... .... ... ... ...... T

F: %!

linear parameterization. An adaptation scheme for the unknown inertia parameters is then derived based on Lyapunov / Barbalat’s technique to ensure that the velocity converges asymptotically to the scaled version of the desired velocity field. In addition, the structure of the controller will also maintain the passivity of the closed loop system.

n

?9 (--’J Control

Mechmicd System

law

(q,4)

(q,q)

[. . .Fedbacksystem . . . . . . . . . ------

Figure

------

------

2: Passivity

-------

Environment

The rest of the paper is organized as follows. Section 2 presents the modified PVFC algorithm that is amenable to linear paramet erizat ion. Section 3 presents the adaptive algorithm and its properties. Simulation and experimental results are presented in section 4. Section 5 contains concluding remarks.

-----’

of theclosed

loop system

than the one specified by the desired trajectory.

2 Modified

The second feature of Passive Velocity Field Control is that the closed loop system must maintain a passive relationship with its physical environment. A dynamic system with input u = U and output v E ~ is passive with respect to the supply rate .s : U x ~ -+ $? if, for any u : $2+ ~ U and any t ~ O, ~~ s(u(~), y(~))d~ ~ –cz, for some c 6 32 which may depend on the system’s initial condition [4].

Consider a fully actuated n DOF mechanical system given by:

and linear parameterization

M(q)q+

C(q, q) = T+

(1)

F

where M(q) ~ R“xn E.the inertia matrix, C(q, q) contains the Coriolis and centripetal terms satisfying M(q(t))

Mechanical systems are subject to two types of inputs: control inputs T (torques or forces generated by actuators) and environment forces F (disturbances, contact forces). While it is well known that mechanical systems are passive with respect to the supply rate defined by TtOt~ where TtOL= T + F, the closed loop system with

PVFC

- 2c(q,

q) = - [M(q(t))- 2c(q,

q)]T ,

T E $2” is the control force, and F ● R“ is the environment force. We make the following assumptions: Assumption

the environment force F as the input and the velocity q being the output (figure 2) is generally not passive with the environment power input F~q as the supply rate. If closed loop passivity can be maintained, by virtue of the passivity theorem, the interconnection of a passive system with a strictly passive system is necessary stable. Thus contact stability and safety can be enhanced when such systems interact mechanically with their physical environment. This can be useful in applications like machining (cutting and deburring), and machines that cooperate and interact closely with humans such as teleoperated surgical robots, and smart exercise machines. Passive Velocity Field Control laws that have these features developed in [1, 2, 3], assuming that the model of the mechanical system is known. It has been successfully applied to various applications including the control of smart exercise machines [5, 6], contour following [7, 8], and bilateral teleoperated manipulators [9]. The application to robotic deburring is currently underway. The present paper deals with the situation where the inertia parameters of the plant are unknown. The main difficulty in making the control algorithm in [1, 2, 3] adaptive is that it does not lend itself directly to linear parameterization with respect to the unknown inertia parameters. To resolve this difficulty, the control law in [1, 3] is first generalized so that it is amenable to

775

1

There

exist

&’(q) ~ @xk, i,~ = 1, ~~,n, vez subset 63 c !Rk so that A’fij (q) = $fJf(q)o,

for some O c E). Thus, M(q)

regressor

functions

and a CO~PaCt, COn-

Vi,

j=l,

. . ..n

can be afinely parame-

trized.

■

that since C(q, q) depends linearly on the spatial derivatives of M(q), the elements of C(q, q) can similarly be linearly parameterized: Cij (q, q) =

Notice

4:(%

ti)e

Assumption 2 A constant ~ G %+ can be chosen so that for each q ~ g and ~ E e, ; vi (q)v~ (CI)4: (q)@ < ~, where the LHS is the kinetic energy of the system if the parameter vector is 6 and the velocity is given by the desired velocity field V(q). 8 Assumption 1 is a standard linear parameterization assumption. It is reasonable for El to be compact and convex since it is often possible to know the approximate magnitude of the components in M“ (q). Assumption 2 states that for every O E @, the kinetic energy associat ed with the desired velocity field is finite. This can always be satisfied if the configuration space is compact, e.g. revolute jointed robot.

to be the desired velocity field for the augmented chanical system (7).

We now describe the modified PVFC algorithm. The key ideas (as originally proposed in [1, 3, 2] and in the present paper) are that the controller dynamics mimic a fictitious energy storage such as a flywheel, and that energy can be exchanged conservatively between the flywheel and the plant via a coupling torque. Let us consider a “fictitious flywheel” with configuration qf G Y?,

and inertia

To design the flywheel inertia &ff (q), we ensure that,

for all qa = [qT, gf ], the following energy conservation condition is satisfied: ~ = ;V~(qJM”(q(JV.

Define qa = [q; q~] ~ R“x 1

f14f(q).

to be the configuration of the augmented system - i.e. the product system consisting of the plant (1) and the “fictitious flywheel”.

=; V~(q)M(q)V(q)

–D(qa,

(2)

qa)qa + T“

=

where T“ ~ R“+l is the coupling torque to be defined later, and D(q~, q. ) is a matrix defined so that

+M’(q.)

+ ;Mf(q)V;

(q),

(8)

(9)

= a~(d + @f(q)@

Mf(q)

(3)

– 2C”(%, 6.)

(qJ

where the constant ~ is the same as in Assumption 2. (8) implies that the augmented system (7) is energetically balanced if qa = aVa (qa ) for any scalar constant cr. Noticed that V.(. ) is a completely known quantity even if the inertia parameter O is unknown. Also, Assumption 1 implies that Mf (q) depends afinely on O. Specifically,

Dynamic augmentation The control law that mimics the flywheel is of the form:

(G)9f)

me-

where

is skew symmetric; where

Ca(qa, qa) :=

(lo)

C(q, q)

Onxl

OIXTZ

o )

(

+ D(q., qa).

(4)

n

df(cl) = *

The flywheel inertia fvf~(q) which, in the original PVFC design can be chosen arbitrarily, needs to be defined carefully for the control law to be amenable to linear parameterization. How Mf (q) should be defined will be described later.

M“(q.)

=

(

Onxl

Coupling

T“(qa, q.) =

D(q.,

&) :=

(

G(%,qa)

+ Ca(qa, qa)qa

= T“ + F“

P“(qa) (6)

[ P“p”~

– p’P

= M+(q.)V.

pa(q. ) 6.) = M+q. )&

)

w(qa> &) = Ma(q~)V~(qa(t))

(7)

where Ta E $?”+l is the coupling control to be defined, F“ = [F~, 0]~ is the environment force acting on the augmented system, C“ (qa, q. ) is given by (4) with an appropriate choice of D (q,,, &), and Ml” (q. ) is given by (5). Augmented desired velocity field and Flywheel inertia Let Vf (q) be an arbitrarily defined desired velocity field for the flywheel. Define Vm(qa) := [v(q)~,

G(qa, &)&

:= [w’(q”)6”)P”~ pat.

Wq., tia) :=

The plant (1) together with the dynamic augmentation (2) forms an augmented system with dynamics: M“(qa)qa

(q.,

q.

) in

(12)

+ yR(q., &)& W+lX

“+1

are

w(q. )4. )T]

skew-

~13)

-pa aT

1 ,

(14)

where

0 i E:=l %$%’

The coupling control T“

q.) E where G(q., q.), R(q~, symmetric and are defined by:

Alternatively, D(q~, &) can be defined simply by: 0.’. ~

control

(2),(7) is given by:

(5)

Mf(q) ) “

(11)

i,j=l

Supposed that f14f(q) has been designed, one can define D(qa, qa) E !W+lx”+l so that CR(q”, q“) corresponds to the Christoffel symbols of the first kind. This has the aesthetic appeal that the augmented system corresponds to a simple n + 1 dimensional Lagrangian mechanical system with inertia given by M(q) ~lXn

&j(o’;(q)v~(dG iR1xk

~

Vf (q)]T e W+l

776

(15) (16) + C“ (q. ) ti.)v~(q~) (17)

and M+ (qa ) is any family of positive definite matrices that one may choose. Notice that M“ (qa ) depends affinely on O and so is the q.) is coupling matrix G(q~, q.). The matrix R(q~, not a function of 0. The design procedure above differs from that in [1, 3, 2] in several ways: 1) In the original PVFC, the flywheel inertia &ff (q) is defined arbitrarily (normally a constant) and then the desired augmented velocity field v. (q.) is defined according to (8). This make v. (q.) a highly nonlinear function of the inertia parameters

@. In contrast, in the present paper, V. (q.) is defined first, and Mf (qa) is then obtained by solving (8); 2) In the original PVFC, the matrix M+ (qa) used to define G(, ),R(., ) in (13)-(14) is the same as the augmented inertia matrix M“ (qa ). This makes the coupling control quadratic in the inertial parameter vector O. In the modified PVFC, M+(qa) needs not be the same as M’ (qa ). These modifications make the coupling control T“ amenable to affine parameterization w.r.t. .9.

where e.(t) = q.(t)–aV~ (q. (t)). Utilizing Lemma 2.2 and the skew symmetry of R(qa, qa ), it can be shown that

= –a~

-

The passivity and convergence properties of the modified PVFC algorithm given in the following theorem are exactly the same as the original algorithm given in [1, 2], Theorem

2.1

q; M+(q.

)q.

V.(q.

)M+(q.

)V. (q.)

(i!’M+(dV&J)2]

(18)

The quantity inside the bracket on the RHS is positive by the Schwartz inequality. This shows that the trajectories (q. (t), q.(t)) that make e~ (t)vanish, are stable when aT >0.

Under the modified PVFC described in

this section, 1. The closed loop system is passive with respect to the supply rate FTq. 2. For each CI so that a ~ > 0, the solution & = aV. (qa) is a stable solution when F = O. 5’. suppose that F = O. Then, from almost eve~ initial condition (q. (0), &(0)), foT every a E 3? such that a 7 > 0, q.(t) converges to /Wa (qa (t))exponentially where ~ = s’gn(~)~ The proof Lemma.

[

of this theorem

relies on the following

dynamic” function “inverse Lemma 2.2 The w(qo, q.) defined in (17) has the property that for each qa E 3?’+1, WT (qo, qa)” v~(q~) = O. This implies that G(qa,qa) V~(q~) = w(q~,ti~). where G(q., qa) is given by (13).

This result is obtained by differentiating the Proof: energy conservation condition (8) w.r.t. time. ■

In the case when F(t) 0, the RHS of (18) is uniformly continuous. Thus, by applying Barbalat’s lemma, we have lim~+~ {qa (t) – ~(t)v~(q~(t))} = O for some function ~(t).However, isconserved, since fia (qa, qa) = ~q~ (t)M” (q. (t))q.(t)

and ~Va(t)M”(qa(t)) Va(t) = ~ is a constant. must also converge to a constant given by:

/3(t)

The sign of @ is determined based on stability consideration according to conclusion 2. By a following a proof similar to that in [1, 3], it is also possible to -+ 9V~ (q.(t)) exponentially from show that &(t) almost every initial conditions. The excluded initial conditions are those such that q. (t) = –pv~ (qa (t)) which is a set of measure O. ■ 3 Adaptive

velocity

field control

algorithm

The modified control law in section 2 admits tine parameterization of the control law. This in turn enables us to develop stable adaptation algorithm using certainty equivalence and direct Lyapunov technique. Suppose that d(t)is the estimate of the unknown parameter O in Assumption 1.

Proof: (Theorem 2.1) To show the passivity result, consider the total kinetic energy:

1.

Estimate

~he flywheel inertia to be $If (q, t) =

a~(d + @f(q)@(t) wherea~(.)and #f() ~~e@ve~ in (10)-(11).

Then recognizing the matrix in (3) together with G(qa, q.) and R(qa, qa) defined in(13) and (14) me skew symmetric, we obtain ~~~ (q.(t), q. (t)) = @’T6. The passivity property of the closed loop system results after integration. To show the convergence and stability result when F(t) = O for all t ~ O, for each a E J?, define the Lyapunov function W~ (q.(t),

q.(t))

= ~e$(t)M”(q.

(t))e~ (t)

777

2. Let the dynamic augmentation to be of the form

(19) where the D(, ., t) is defined as given in section 2 (e.g. using (6)) using the estimate fif (q, t) based on b(t) The additional term is needed in instead of Mf (q). (19) toAmaintain the passivity of the augmented system when Mf (q, t)is time varying.

3. Choose a positive dejinite metric M+(qa) the coupling control T“ in (19) to be

and define

Choose a positive scalar d and define the d- velocity field tracking error ee := q= – @Va(q. ),

G(qa,qa,t)

:=

and consider the Lyapunov function,

[w(q”’q”’’)paT - Pawqa,qa,tr] W’(qa)M+(dva(q.) (21)

R(q.,

q.) :=

[

p“p”~

– p“P”T]

Suppose that the environment force F = O. Then, following the proof of Theorem 2.1, we obtain:

(22)

,

where

Wfi = –~e~

P“(q. ) = M+(q.)V.;

pa(qa, q.) = M+(q. )%

w%, Ga,t)= Ma(%, wa(%) + c“(%,

[

WV.

+ CaV~ – GV~

1

+ ti-yVZR.q~.

(23) Let us define

%,t)va(%), (24)

fi(%,q~~t)

‘= ~“(%>

‘)v~ +~(%]

d~]t)va(%). (27)

t)and C“(qa, qa, t) are the estimate of M”(q~) and C“(q~, q.) based on the estimate ~(t) Ofo.

M“(qa,

Then, utilizing Lemma 3.2 and by expanding R(q~, q.) in (22),

4. The adaptation algorithm for e(t) will be presented

$Wa=@e~(w–w)

after some preliminary Lyapunov analysis. Let us define ~a(q~,t) C“(q.,qa,t)

:=

:=

(c(:’q) :)

+D(qa, qa, t)+

Onxl

M(q)

(Olxn

iff(q, t))

(

O?txn

Olxn

– 67 and

Va(qa)M+(q.

)va(qa) (28)

If 67>0, then, by the Schwartz inequality, the second term on the RHS is negative.

Onxl ++(q., t) )

We are now ready to present the adaptation algorithm. Let ;(t) = e(t)– 8

‘)4.

=G(qa, qa, t)qa + TR(q., da)tia + J?a(~).

be the parameter

estimation error. Notice that the difference between w(q~, q., t)and w(q~, q., t) can be ex-

(25)

pressed as

The associated kinetic energy function for the system (25) is

zc(qa,

q~M+(qa)&

- (q;M+(qcJVNkJ)2]

Then the closed loop system is given by:

Tr(qa , ‘kia + =(%,4,

[

1 .1’-. G., ‘) = p M (qa,’)ti.

W%!ti.

)t) – W%>tiQlt)

= @w(qa) 6.) ~(’)

for some known function @W(qa,q=)

(26)

(29)

: Y?k+ !R”+l.

Consider a new Lyapunov function Proposition

3.1 Let e(t) E Y?k be a trajectory of the estimate of the unknown parameter 0. The mechanical system (1) under the control of the adaptive algorithm given by (19) -(24) is passive with respect to the supply rate FTq.

where r is a positive definite gain matrix, and @ is the scalar chosen earlier. Let the adaptation algorithm be:

Proof: This is obtained by differentiating (26) with respect to time and the fact that G (.,., t)and R(,. ) are skew symmetric. ■

~e(t)

where ProJi is the usual projection

$W~

&I

‘)VQ(%)

{ -r@wT(qa,q.)e@}

(30)

algorithm which

ensures that ~(t) lies wit hin the compact, convex set El C Rk. Using the fact that @ is a convex set, we obtain

Lemma 3.2 The estimated “inverse dynamic” function %T(qu, qa) in (24) satisfy

‘T(%)

= Proje