M. Vidyasagar, âThe graph metric for unstable plants and robust- ness estimate for feedback ... M. Vidyasagar, âNormalized coprime factorizations for nonstrictly.
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G. Zames, “On the input-output stability of time-varying nonlinear feedback systems-part I: Conditions using concepts of loop gain, conicity and positivity,” ZEEE Trans. Automat. Confr., vol. AC-11, no. 2, pp. 228-238, Apr. 1966. M. G. Safonov and M. Athans, “On stability theory,” in Proceedings 1978 ZEEE Conf. Decision Confr.,San Diego, CA, Jan. 9-12, 1979. M. G. Safonov, Stability and Robustness of Multivariable Feedback Systems. Cambridge, MA: M.I.T. Press, 1980. M. Vidyasagar, “The graph metric for unstable plants and robustness estimate for feedback stability,” ZEEE Trans. Automat. Confr., vol. AC-29, no. 5, pp. 403-418, May 1984. A. K. El-Sakkary, “The gap metric: Robustness of stabilization of feedback systems,” ZEEE Trans. Automaf. Contr., vol. AC-30, no. 3, pp. 240-247, Mar. 1985. T. T. Georgiou, “On the computation of the gap metric,” Sysf. Confr. Lett., vol. 11, no. 4, pp. 253-257, Oct. 1988. T. T. Georgiou and M. C. Smith, “Optimal robustness in the gap metric,” ZEEE Trans. on Automat. Contr., vol. AC-35, no. 6, pp. 673-686, June 1990. K. Glover and D. McFarlane, “Robust stabilization of normalized coprime factor plant descriptions with H,-bounded uncertainty,” IEEE Trans. Automat. Contr., vol. 34, no. 8, Aug. 1989. M. Vidyasagar and H. Kimura, “Robust controllers for uncertain linear multivariable systems,” Automatica, vol. 22, no. 1, pp. 85-94, Jan. 1986. D. C. Youla, “On the factorization of rational transfer function matrices,” IRE Trans. Inf. Theory, IT-7, pp. 172-189, July 1961. D. Meyer and G. Franklin, “A connection between normalized coprime factorizations and linear quadratic regulator theory,” IEEE Trans. Automat. Confr.,vol. AC-32, no. 3, pp. 227-228, Mar. 1987. M. Vidyasagar, “Normalized coprime factorizations for nonstrictly proper systems,” IEEE Trans. Automat. Contr., vol. 33, no. 3, pp. 300-301, Mar. 1988. D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions. New York: Springer-Verlag, 1990. M. J. Englehart and M. C. Smith, “A four-block problem for H, design: properties and applications,”Aufomafica,vol. 27, no. 5, pp. 811-818, Sept. 1991.
Adaptive Hybrid Control Strategies for Constrained Robots Jong-Hann Jean and Li-Chen Fu Abstract-This note addresses the problem of adaptive hybrid controller design for constrained robots with the consideration of compntational efficiency. Two efficient control schemes respectively based on Lagrange and Newton-Euler dynamics formulation are presented. Detailed analyses on tracking properties of joint positions, velocities, and constrained forces are derived for both the Lagrange approach and the Newton-Euler approach. Although control laws in these two approaches
Manuscript received August 9, 1991; revised January 17, 1992. This work was supported by the National Science Council under Grant NSC 79-0404-E002-03. J.-H. Jean is with the Department of Electrical Engineering, National Taiwan University, Taiwan, Republic of China. L.-C. Fu is with the Department of Electrical Engineering and Computer Science and Information Engineering, National Taiwan University, Taiwan, Republic of China. IEEE Log Number 9204972.
are developed independently, a tight connection between them is signified, which highlights a possible bridge over different general adaptive approaches respectively based on the two dynamics formulations.
I. INTRODUCTION Many industrial applications of robotic manipulator systems may involve tasks in which the manipulator end-effectors are required to make contact with the environment. Typical examples of these tasks are such as contour following, grinding, scribing, writing, deburring, as well as all assembly related tasks. By a proper design for such tasks, robot end-effectors can be viewed as being constrained by the environment [1]-[9] and contact forces will build up to maintain satisfaction of the constraints. As a result, not only robot positions but also contact forces should be taken into account for the success of task execution. From a practical point of view, the inertial properties and gravitational loads of a robot system may vary from one task to another and, hence, may not be precisely known in advance. Several adaptive control schemes [ 101-[13] for robot motion control have been developed to assure the stability of the overall system in spite of the imprecise knowledge of the system dynamics. Although these schemes can be shown to achieve trajectory tracking, the computation complexity for their implementation is considerably high. Therefore, more recently, several researchers addressed the problem of computational efficiency for realizing the above-mentioned adaptive control schemes. Walker [14] utilized the method proposed in [ l l ] to develop an efficient recursive control algorithm for manipulators containing closed kinematics loops. Sadegh and Horowitz [15] presented a modified version of their previous work [12] to allow off-line computation of a major part of the control signals using the desired joint positions and velocities instead of the actual measurements. Both of these works lead the adaptive controller design toward a more practical direction. This note addresses the problem of adaptive hybrid controller design for constrained robots with the consideration of computational efficiency. The contributions of this note can be listed as follows: By incorporating similar conceptual development of the schemes proposed in [14] and [15], two efficient control schemes respectively based on Lagrange and Newton-Euler dynamics formulation are derived. Analyses on asymptotic tracking properties of joint positions, velocities, and constrained forces are given for both the Lagrange approach and the Newton-Euler approach. Although control laws in these two approaches are developed independently, a tight connection between them is signified, which highlights a possible bridge over different general adaptive approaches respectively based on the two dynamics formulations. This facilitates robot control theorists to realize the spine of any adaptive control methodology regardless of dynamics formulation of the target system. The note is organized as follows: In Section 11, the dynamics formulation of a constrained robot is introduced. In Section 111, the adaptive hybrid control schemes using Lagrange approach and the Newton-Euler approach are developed and the asymptotic tracking properties of the overall system are presented. The connection between the two approaches is also described in
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section 111. Finally, some discussions are stated and conclusions are provided in Section IV. 11. DYNAMIC MODELOF A CONSTRAINED ROBOT
Consider a rigid robot whose end-effector is in contact with the environment modeled as a rigid frictionless surface. To restrict the robot end-effector to keep contact with the surface amounts to imposing kinematic constraints on the robot. It can be shown [l], [16] that if the constraints are nonredundant, then the robot joint coordinate, q E R " , can be partitioned into q = [q: ,q;JT such that the kinematic constraints can be represented as q1 = R(q2) for all q2 E 0
(2.1)
where R is a nonlinear map from R"' to R"-" and 0 is an open set in Rm. The constraints imposed on joint velocities and accelerations of the robot can be obtained by differentiating (2.1) into the form 41 =
E(q2)42,
q1
'(q2)qZ + F ( q 2 7
(2.2)
q2)42?
where E ( q 2 ) = dRn(q,)/dq, and F ( q , , q,) = d E ( q , ) / d t . In the following, the dynamics of a constrained robot are described by using two different formulations, namely, Lagrange formulation and Newton-Euler formulation. In the Lagrange formulation, the equations of motion are derived by employing the Lagrange equation and are represented in a closed form. On the other hand, in the Newton-Euler formulation the equations of motion are obtained by using the Newton-Euler equation and are represented in a recursive form.
where
M
=
ETMllE
+ E T M 1 2+ M Z 1 E+ MZ2
c=ETMllF+ETC1X+MzlF+C2X
G = E T G , + G,, X
=
[
Later in the text, force control loop and motion control loop are designed respectively based on (2.5) and (2.6), which after following the generalization done by McClamroch and Wang [l] gives the hybrid position/force controller [MI.
B. Newton-Euler Formulation The presentation to be given here will be based on the spatial notation [19]. Let the ith joint variable and its associated actuator input be denoted as 8, and p, respectively, where the lower order joint is closer to the base and the higher order joint is closer to the end-effector. Let I, denote the spatial inertia matrix of link i with respect to the coordinate associated with link i. It can be shown [14] that I, can be represented as a linear combination of ten parameters m r k ,k = l;.., 10, 10 Rkmrk
k= 1
where R , is a 6 x 6 matrix containing only zeros and ones. Let the spatial transformation matrix from link i to the base then the spatial inertia matrix of link i with respect to be the base can be computed through the relation
4 'OX,
I,,XO.
Then the equations of motion for a constrained robot can be described by a recursive inverse dynamics algorithm 1141. Sl) Given the desired positions, velocities, and accelerations of the independent joint variables, q 2 , calculate the corresponding ones of the dependent joint variables, ql, through equations M ( q ) q + C ( q , 4 ) q + G ( q ) = T + AT A (2.3) (2.1) and (2.2). S2) For each link, determine p,, the ith actuator input needed where M ( q ) denotes the generalized inertia matrix, C(q, q ) q to achieve the desired motion, from two recursions as follows. denotes the vector of centrifugal and Coriolis forces, G ( q ) Outward loop: denotes the vector of gravitational forces, T is the vector of i = l;.. , n nonconservative generalized forces, A E Rn-" is the vector of uO= 0, Lagrange's multipliers associated with the constraints, A = [ I , U, = y - 1 + s L q - E(q,)], and AT A is the constraint force vector. Conceivably, (2.3) would not be adequate for one to analyze U, = + + v, x s , q the dynamic behavior of the system since components in the
A. Lagrange Formulation
According to D'Alembert's principle and Lagrange's multiplier rule 1171, the equations of motion for the constrained robot system appear with the following form
A
joint coordinate q are dependent and the Lagrange's multipliers are not determined yet. Hence, we intend to find an alternative more amiable to controller design and stability analysis. First, decompose the equations of motion (2.3) into the following form:
.
where s, denotes the spatial vector representation of the ith joint axis. Inward loop:
f,,,
=
j
0,
=
n;.., 1
5 = I.V. + vi x I j U j + zjg I 1 fj=< It can be shown [16] that, after suitably transforming the coordinate, the equation of motion (2.3) can be decomposed into two parts, namely, force part and motion part, i.e.,
M(qz)qz
+ c(q2,q2)q2 + G(q,) = X T T
T~
(2.6)
+fj+1
p. = S*f. I
1
1
where g = [0 0 9.8 0 0 OIT and the symbols * and 2 represent the spatial transpose and the spatial cross operator 1193, respectively. S3) Determine the vectors T~ and 7 2 from the computation results of pi, i = 1;..,n. Then the vector T~ can be obtained through the transformation defined in (2.6).
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111. EFFICIENT ADAPTIVE HYBRID CONTROL STRATEGIES
A complete task description for a constrained robot includes not only a trajectory that the robot end-effector should follow but also the evolution of constrained forces along that trajectory. Suppose the desired motion of a robot can be described in joint coordinate by a vector function qd(t) = [ q l d ( t I T&, & I T I T , where q l d ( t )= Cl(q2Jt))for consistency with the imposed constraints. Furthermore, the desired constrained forces fd(t) can be characterized by some multiplier function A&) as fd(t) = A T (qd(t))Ad(t), where the multiplier function can be physically interpreted as the desired contact forces in the constraint coordinate [l].Our objective is then to design a hybrid controller to achieve asymptotic tracking of both joint positions and constrained forces, that is, to yield q ( [ )j qd(t) and f(l> -)fd(t) as
"'.
---*
where K = K&A - A d ) then designed as
-
A. The parameter adaptation law is
e^ = - r - l 4 q 2 , 4 2 ,
> 0.
42r, q2r)$2r,
(3.7)
Theorem 3.2: Consider the constrained robot system whose
dynamics are governed by (2.3). Given the bounded desired trajectory qZd such that its first and second derivatives are also bounded almost everywhere. Then, the following will be true provided the adaptive control laws (3.5)-(3.7) are used.
1) a) Signals q2, q2 and
e^ are bounded and lim ( q - qd)
=
0
t--r cc
b) Force errors are bounded by the norm of the inverse of the gain matrix K f .
IlA - hdll I CXIIK;~IIfor some a > 0.
In the following, we will propose efficient adaptive control strategies to achieve the control objective. Controller designs based on both Lagrange formulation and Newton-Euler formulation are separately discussed. A . Lagrange Approach
First, we will summarize some useful dynamical properties of the constrained robot system in the following proposition. The proof of the proposition is straightforward and, hence, is omitted here. Proposition 3.1: For the constrained robot system described in the previous section, the following properties hold: i) a(q2) is symmetric and positive definite; ii) by a p r o p c choice of C ( q , 4 ) to define &?(q,) - 2C(q2,q 2 )is skew symmetric; and iii) the following decompositions will hold:
2) If the thira-order derivatives of qZdis also bounded almost everywhere, then a) lim,,Jq - q d ) = 0 and lim,+-(q - q d ) = 0, b) lim,+,wT ( q 2 ,q 2 ,42r,q z r ) 6= 0. 3) If w ( q 2 d ,cjZd, q 2 d ) , which is obtained by replacing ( q 2 ,q,, L& ii) in (3.3) with ( q Z dq, 2 d r42d,q Z d )respectively, is persistently exciting (PE) [15], [20], [21], then lim e'(t) t-
=
0,
m
lim h ( t )
=
0.
t+m
Proof of Theorem 3.2: From (3.3), (3.51, and (3.61, we can obtain the position error dynamics:
c, the matrix
(M,,E
Define the Lyapunov function V ( t ) as
+ M12)ii + (M,,F + C,X)U
+ Gl
=
w: ( q 2 , 4 2 , U , Q P l
(3.1)
and then take its time derivative along trajectories of (3.7) and (3.8) to yield the following with the help of Proposition 3.1: d -V(t) dt
and, hence,
=
w T ( q 2 ,q 2 ,U , i i p , (3.3)
where U , , w 2 , and w are known functions of their arguments and el, e,, and, hence, 0 consist of dynamic parameters of the constrained robot. Now, we are ready to introduce an adaptive hybrid controller for position/force tracking of a constrained robot. Define the auxiliary signals q2r and as
G2r
42r = 4 2
42, = 4 2 d - krG2,
- 42,
(3.4)
where q2 q2 - q2d is the positional tracking error and k , is a positive constant. Design the control signals as 71
~:(q2t42,42r,q~r)O+ 1 K
7 2 = WZ' ( q 2 , 4 2 , 4 2 r , 9 2 r ) e ^ 2
-
K
-~
6
(3.5)
2 -r ~ p 4 , (3.6)
=
-$ZrKu$2r- k,@: K,@, I 0.
(3.9)
Therefore, all signals in V ( t ) are bounded and $ z r is square integrable. Consequently, the results in 1)-a) hold by the definitions of (3.4) in [ll]. This, in addition, implies the boundedness of qt,?, i = 1, 2, and, hence, @$): as a result from (3.8) and Proposition 3.1. Now, the boundedness of the force errors can be seen from the following equation derived from (2.51, (3.11, and (3.5)
(M,,E
+ M12)& + (M,,F + C,X)$,,
-
w:
e',
= Kfh.
(3.10) Since all terms on the left-hand side of (3.10) are bounded, the result in 1)-b) can thus be concluded. is bounded leads to the boundedNote that the fact that ness of cji2) again through definitions (3.41, which along with the hypothesis that q$y is also bounded almost everywhere further implies almost everywhere boundedness of 4 4 : ) . Now, by differentiating both sides of (3.81, we can easily see that @$): and, are almost everywhere bounded. Then, it readily hence, implies that @ t ) ( t )and, hence, &)(t), i = 1, 2, tend to zero as t goes to infinity. This therefore concludes the result 2)-a) and the result 2)-b) from (3.8).
@i3)
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Finally, if d q 2 d 9 4 2 d , 42d)is PE, then d q 2 ,q 2 ,iZr, i Z rcan ) also be shown to be PE [15]. Consequently, from the standard arguments of adaptive control [15], [20], and [21] the parameter error will converge to zero which, in turn, implies the convergence of the force errors. For the consideration of computational efficiency, we modify the control law as =
T2 =
w: ( q 2 d 3 4 2 d , 4 2 d ) e ^ l
( q 2 d ? q Z d ,42d)i2
-
ET
+
(3.11)
- Ku42r - Kp@2
- a1@212d2r
(3.12)
601
Parameter adaptation law:
Aj =
?=
(3.13)
where the function matrix w ( q 2 d ,q Z dq, Z d )is off-line computed prior to the control, then similar results can be obtained. The following corollary will summarize these results. The proof can be found in [22]. Corollary 3.3: Consider the constrained robot system whose dynamics are governed by (2.3). If the adaptive control laws (3.111, (3.121, and (3.13) are used, and K,, K p , and CT are sufficiently positive, then all the results in Theorem 3.2 hold.
The advantage of the Newton-Euler formulation over the Lagrange formulation is that if the inertial parameters are known and the desired motion is given, the computation to obtain the necessary torque input is done only within the two recursions. Therefore, there is no need to generate the whole dynamic model explicitly and, hence, a lot of computation time will be saved especially when the number of joints of the robot is large. In this section, we will start with the Newton-Euler formulation to speed up the calculation of the robot dynamics. Subsequently, we will show that the former Lagrange approach can be modified into a Newton-Euler approach but with a similar concept in designing the controller. The adaptive hybrid controller to be proposed here is given in terms of an algorithmic procedure involving three steps. C1) Given the desired position trajectory q 2 d , the desired force trajectory A d , and the auxiliary signals q2, and GZr which are defined in (3.4) respectively, we compute the following signals 41r = E ( q 2 ) 4 2 r (3.14) 41r ='(q2)42r
+'(q2,
(3.15)
42)42r.
i = l;.., n , as follows:
C2) Generate the control signals Outward loop: uo,r =
i
0,
=
l;.. , n
v i . r = u L - ~ , r+ S i + , r =
Gi&r,r + S i q , r
+ ui x S&. A
.
Inward loop: fn+l,r =
F.
1.r
i = a,-.., 1
0,
= Z.(V. j . r - p'j,r j
r1 =
5,r+h+l,r syh, estimate of 4, p
vi X
n
'j9,r
where
q,
=
4. is an vi
-
9,r ,
r
(3.16)
el+
K
(3.17)
+ ( g Z l F+ e 2 X ) 4 , , + 6, - E T
K
(3.18) where v
= q2r
-
P42r
~
Remark: Clearly, Lemma 3.4 points out the connection between the Lagrange approach and the Newton-Euler approach, which is useful in the subsequent analysis as well as for a better understanding of the physical meaning of the underlying approach. As a matter of fact, several adaptive control schemes based on Lagrange dynamics have their counterparts in the domain of approaches based on Newton-Euler dynamics. The basic difference between them lies in the way to update the estimated parameters. Using the symbols defined in (3.1) and (3.2), the positional error dynamics can be fomulated from (3.17), (3.18), and (2.51, (2.6) as the following: W(q2)(92r
+ p d 2 r ) + '(q2742Id2r
=
wT
(92,42,42r9~)6.
(3.19) Note that the vector of parameter errors 6 is derived in a way that is different from the one in Subsection 111-A. The following theorem will summarize the properties of the constrained robot system after this Newtpn-Euler approach is applied. Theorem 3.5: Consider the same constrained robot system as the one in Theorem 3.2. Then all the results of Theorem 3.2 will still hold provided the adaptive hybrid control law Cl)-C3) is applied. Proof of Theorem 3.5: Through some modifications of the arguments in [141, the following properties can be shown:
C
v;Tr(F,
f=1
n
is a positive constant, and
[
( g l 1 E + k 1 2 ) u+ ( k l l F + e l X ) q 2 r +
n A
+ g) +
+
=
Although the form of this Newton-Euler approach is quite different from that of Lagrange approach given in Section 111-A, the concepts in designing respective controllers are, in fact, quite similar and the following lemma will signify their relationship. The proof of the lemma is provided in Appendix. Lemma 3.4: The control law (3.16) can be rewritten as
fi,r =
Pj,r =
[ ::]
r 2 = (kz1E-I-h?22)u
B. N e w t o n - E u l e r Approach
5>0
+)wj,
where %I and wl are 10 x 1 vectors with components hIkand w l k and U,, 6J,r, U,, r , and gJ are the vectors 9, GI,,, V I , . , and g with respect E lifik coordinate, respectively. C3) Determine the vectors r l , r and r 2 , r according to the computation results of A, i = l;.., n . Then the actuator inputs T are implemented as follows:
and the adaptation law as
e = -r-l W ( q 2 d , 4 2 d , 4 2 d ) @ 2 r
(
-F,,r) = 0
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Define the function V(t) as 1 . . 1 “ V ( t )= ~ @ ~ ~ M + ( q2 ~ )&;hl, 4 ~ ~
,=
h1= m, - A,.
1
From S2) and C2), it is easy to verify that
5
+ U, 2 z ] q , r )
- q , r = (z,i,,r
+ ({q,r+ I;g + U, x
