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Adaptive robust approximate constraint-following control for mechanical systems
ARTICLE IN PRESS
Journal of the Franklin Institute 347 (2010) 69–86 www.elsevier.com/locate/jfranklin
Adaptive robust approximate constraint-following control for mechanical systems Ye-Hwa Chena,, Xinrong Zhangb a
The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA b Department of Mechanical and Electronic Engineering, Chang’an University, Xian, PR China
Received 24 August 2009; received in revised form 6 October 2009; accepted 12 October 2009
Abstract We formulate control problems of mechanical systems as constraint following. The system contains uncertainty which is (possibly fast) time-varying. It is unknown but bounded. The bound is, however, unknown. The objective is to design control which renders approximate constraint following. Adaptive laws are constructed. The controls are then based on the adaptive parameters. We are able to demonstrate guaranteed system performance, regardless of the uncertainty. & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Mechanical system; Constraint; Motion control; Robust control; Adaptive control
1. Introduction The most important feature of mechanical system motion, in connecting to real world practices, according to Joseph-Louis Lagrange, is constraint following. It is because of this, the Lagrange’s form of d’Alembert’s principle was postulated. This is what Lagrange asserted what the Nature would do [1]. In the past, the majority of the efforts in this branch of mechanics can be divided into two categories: the passive constraint problem and the servo constraint problem. In the passive constraint problem, which is along with the original Lagrange aspect, the main focus is to investigate what the Nature will do in order to assure that the constraints are Corresponding author. Tel.: +1 404 894 3210; fax: +1 404 894 9342.
E-mail address: [email protected] (Y.-H. Chen). 0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.10.012
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(strictly) obeyed. Many important contributions have been made (for a survey, see, for example, [1]). In the servo constraint problem, which constructively utilizes the Lagrange aspect, the main focus is to find what the engineer should do, so that the constraints are followed. Most of the emphasis is on precise model-based control design. A survey on the use of geometric and algebraic approaches can be found in Bloch [2] and Bullo and Lewis [3]. There are also efforts which deal with uncertainty (e.g., [4–7] and their bibliographies). The scope of this paper falls into the servo approach. Several classes of control problems are formulated as constraint following. As a result, the constraint following task is to be pursued for solving the control problems. Since the engineer’s knowledge of the system is sometimes limited, we consider the presence of modelling uncertainty. The uncertainty is (possibly fast) time varying. No further information except it is bounded is assumed. Furthermore, the bound is unknown. We propose to design control which renders the system to follow a class of pre-specified constraints approximately. For the control design, we follow the framework established in Udwadia and Kalaba [8–10]. First, a nominal control is proposed to address the system dynamics while no uncertainty is involved. Second, adaptive laws, which emulate a constant parameter vector, are constructed. This parameter vector may be relevant (but not necessarily identical) to the uncertainty bound, which is unknown. Third, two classes of adaptive robust controls are designed. The controls are able to guarantee either convergence to zero (hence strictly constraint following) or uniform ultimate boundedness, regardless of the uncertainty. 2. Mechanical system subject to constraints Consider the following mechanical system [11,12]: € þ CðqðtÞ; qðtÞ; _ _ þ gðqðtÞ; sðtÞ; tÞ ¼ tðtÞ: MðqðtÞ; sðtÞ; tÞqðtÞ sðtÞ; tÞqðtÞ n
ð2:1Þ n
Here t 2 R is the independent variable, q 2 R is the coordinate, q_ 2 R is the velocity, q€ 2 Rn is the acceleration, s 2 S Rp is the uncertain parameter, and t 2 Rn is the control input. Here S Rp is compact but unknown, which stands for the possible bounding of s. _ s; tÞq_ is the Coriolis/centrifugal force, Furthermore, Mðq; s; tÞ is the inertia matrix, Cðq; q; _ s; tÞ, and and gðq; sðtÞ; tÞ is the gravitational force. The matrices/vector Mðq; s; tÞ, Cðq; q; gðq; s; tÞ are of appropriate dimensions. We assume that the functions MðÞ, CðÞ, and gðÞ are continuous (this can be generalized to be Lebesgue measurable in t). Remark. The coordinate q can be selected based on the specifics of the problem and does not need to be the generalized coordinate. The following constraints are proposed: n X Ali ðq; tÞq_ i ¼ cl ðq; tÞ; l ¼ 1; . . . ; m;
ð2:2Þ
i¼1
_ Ali ðÞ and cl ðÞ are both C 1 , mrn. They are the first where q_ i is the i th component of q, order form of the constraints. The constraints may not be integrable; and may be nonholonomic in general. The constraints can be put in matrix form Aðq; tÞq_ ¼ cðq; tÞ; where A ¼ ½Ali mn , c ¼ ½c1 c2 cm T .
ð2:3Þ
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There are two ways of interpreting the constraints. First, they may be passive. That is, the environment (or the structure) is to supply the constraint force in order for the system to comply with the constraint. Second, they may be active. That is, the system’s control input supplies the required force so that the constraints are met. In this study, we shall adopt the second view. We now convert the first order form into second order form [13]. Differentiating the constraint equations (2.2) with respect to t yields n X d i¼1
n X d Ali ðq; tÞ q_ i þ Ali ðq; tÞq€ i ¼ cl ðq; tÞ; dt dt i¼1
ð2:4Þ
where n X d @Ali ðq; tÞ @Ali ðq; tÞ q_ k þ Ali ðq; tÞ ¼ ; dt @qk @t k¼1
ð2:5Þ
n X d @cl ðq; tÞ @cl ðq; tÞ cl ðq; tÞ ¼ : q_ k þ dt @q @t k k¼1
ð2:6Þ
Eq. (2.4), the second order form of the constraints, can be rewritten as n X i¼1
Ali ðq; tÞq€ i ¼
n X d i¼1
d Ali ðq; tÞ q_ i þ cl ðq; tÞ dt dt
_ tÞ; ¼: bl ðq; q;
ð2:7Þ
l ¼ 1; . . . ; m, or in matrix form _ tÞ; Aðq; tÞq€ ¼ bðq; q;
ð2:8Þ
where b ¼ ½b1 b2 bm T . Remark. In Chen [14], it has been demonstrated that various control problems, including stabilization, trajectory following, and optimality, can be cast into the form (2.8).
3. Constraint force when uncertainty is known We show the constraint force when the uncertainty is known. Assumption 1. For each ðq; tÞ 2 Rn R, s 2 S, Mðq; s; tÞ40. Remark. The assumption on the positive definiteness of the inertia matrix will be vital in later development. In the past, it was often believed that this was always true, and therefore a fact rather than an assumption. However, there are counter examples, as listed in [15], when q is not selected to be the generalized coordinate. Definition 1. For given A and b, the constraint (2.8) is called consistent if there exists at € least one solution q. Assumption 2. The constraint (2.8) is consistent.
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Theorem 1 (Udwadia and Kalaba [8, p. 233]). Consider the system (2.1) and the constraint (2.8). Subject to Assumptions 1 and 2, the constraint force Qc ¼ M 1=2 ðq; s; tÞðAðq; tÞM 1=2 ðq; s; tÞÞþ _ tÞ þ Aðq; tÞM 1 ðq; s; tÞðCðq; q; _ s; tÞq_ þ gðq; s; tÞÞ ½bðq; q;
ð3:1Þ
obeys the Lagrange’s form of d’Alembert’s principle [1] and renders the system to meet the constraint. Here ‘‘ þ ’’ stands for the Moore–Penrose generalized inverse [16, p. 337]. Remark. The Lagrange’s form of d’Alembert’s principle renders the constraint force (3.1) to be the one with minimum norm, out of all possible alternative forces which can also meet Eq. (2.8) [8]. Note that Qc 2 RðAT Þ (with the understanding that in this d’Alembert’s principle, the virtual displacement dq 2 N ðAÞ; in addition RðAT Þ ? N ðAÞ). Remark. Theorem 1 shows the strategy the Nature will undertake to meet the constraint. The constraint force is model-based. That is, it is based on the exact model information. Based on the theorem, one could apply the control input t ¼ Qc to drive the system to meet Eq. (2.8), if the uncertainty were known. A more realistic design, from the engineer’s point of view, when the uncertainty is unknown, is investigated in the next section.
4. Adaptive robust servo control design I We now take the uncertainty into account while designing the control t. Decompose the M, C, and g as follows: Mðq; s; tÞ ¼ M ðq; tÞ þ DMðq; s; tÞ; _ s; tÞ ¼ C ðq; q; _ tÞ þ DCðq; q; _ s; tÞ; Cðq; q; gðq; s; tÞ ¼ gðq; tÞ þ Dgðq; s; tÞ:
ð4:1Þ
Here M , C , and g denote the ‘‘nominal’’ portions with M 40 (this is always feasible since it is the designer’s discretion), while DM, DC and Dg are the uncertain portions. The functions M ðÞ, DMðÞ, C ðÞ, DCðÞ, gðÞ, and DgðÞ are all continuous. Let Dðq; tÞ :¼ 1
M ðq; tÞ, DDðq; s; tÞ :¼ M 1 ðq; s; tÞ M (hence DDðq; s; tÞ ¼ Dðq; tÞEðq; s; tÞÞ.
1
ðq; tÞ,
Eðq; s; tÞ :¼ M ðq; tÞM 1 ðq; s; tÞ I
Assumption 3. For each ðq; tÞ 2 Rn R, Aðq; tÞ is of full rank. This means Aðq; tÞAT ðq; tÞ is invertible. Assumption 4. Under the provision of Assumption 3, for given P 2 Rmm , P40, let W ðq; s; tÞ :¼ PAðq; tÞDðq; tÞEðq; s; tÞM ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 :
ð4:2Þ
There exists a (possibly unknown) constant rE 4 1 such that for all ðq; tÞ 2 Rn R, 1 min lm ðW ðq; s; tÞ þ W T ðq; s; tÞÞZrE : 2 s2S
ð4:3Þ
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Remark. The constant rE is in general unknown since the uncertainty bound S is unknown. In the special case that M ¼ M (i.e., no uncertainty), E ¼ 0, W ¼ 0 and hence one can choose rE ¼ 0. Thus by continuity this assumption imposes the effect of uncertainty on the possible deviation of M from M to be within a certain threshold. We also stress that this threshold is unidirectional (that is, it is not bounded in one direction). Let _ tÞ :¼ M p1 ðq; q;
1=2
ðq; tÞðAðq; tÞM
1=2
_ tÞ þ Aðq; tÞM ½bðq; q;
ðq; tÞÞþ
1
_ tÞq_ þ gðq; tÞÞ; ðq; tÞðC ðq; q;
_ tÞ :¼ kM ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 ðAðq; tÞq_ cðq; tÞÞ: p2 ðq; q;
ð4:4Þ ð4:5Þ
Assumption 5. (1) There exists an unknown constant vector a 2 ð0; 1Þk and a known _ tÞ 2 Rn Rn R, s 2 S, function PðÞ : ð0; 1Þk Rn Rn R-Rþ such that for all ðq; q; _ s; tÞq_ gðq; s; tÞ þ p1 ðq; q; _ tÞ ð1 þ rE Þ1 max½JPAðq; tÞDDðq; s; tÞðCðq; q; s2S
_ tÞÞ PAðq; tÞDðq; tÞðDCðq; q; _ s; tÞq_ þ Dgðq; s; tÞÞJrPða; q; q; _ tÞ: þp2 ðq; q;
ð4:6Þ
_ tÞ 2 Rn Rn R, the function Pð; q; q; _ tÞ : ð0; 1Þk -Rþ is (i) C 1 , (2) For each ðq; q; k (ii) concave; that is, for any a1;2 2 ð0; 1Þ , _ tÞ Pða2 ; q; q; _ tÞr Pða1 ; q; q;
@P _ tÞða1 a2 Þ; ða2 ; q; q; @a
ð4:7Þ
and (iii) non-decreasing with respect to each component of its argument a. Remark. Note that 1 þ rE 40 if rE 4 1. The function PðÞ may be interpreted the uncertainty bound. The constant vector a may be relevant to the bounding set S. It is _ tÞ is linear in a, then unknown since S is unknown. In the special case that Pð; q; q; Assumption 5(2) is met. In a sense, what Assumption 5 does is the parameterization of the worst case effect of the uncertainty, which will be further elaborated in the proof of Theorem 2. We now propose the control ^ _ _ _ qðtÞ; qðtÞ; tÞ; tÞ þ p2 ðqðtÞ; qðtÞ; tÞ þ p3 ðaðtÞ; tðtÞ ¼ p1 ðqðtÞ; qðtÞ;
ð4:8Þ
with ^ q; q; ^ q; q; ^ q; q; ^ q; q; _ tÞ ¼ ½M ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 gða; _ tÞmða; _ tÞPða; _ tÞ; p3 ða;
ð4:9Þ where
8 > > > <
1 ^ q; q; _ tÞJ4eðtÞ; if Jmða; ^ q; q; _ tÞJ Jmða; ^ q; q; _ tÞ ¼ gða; > > 1 ^ q; q; _ tÞJreðtÞ; if Jmða; > : eðtÞ
ð4:10Þ
^ q; q; ^ q; q; _ tÞ; _ tÞ ¼ ðAðq; tÞq_ cðq; tÞÞPða; mða;
ð4:11Þ
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k 2 R, k40, e_ ðtÞ ¼ leðtÞ;
eðt0 Þ40; l40:
ð4:12Þ
_ tÞ :¼ Aðq; tÞq_ cðq; tÞ. The parameter a^ is governed by the following adaptive law: Let bðq; q; T
@P ^ q; q; _ tÞJbðq; q; _ tÞJ; a^_ ¼ L ða; @a
ð4:13Þ
^ i ¼ 1; . . . ; k), L 2 Rkk 40, each a^ i ðt0 Þ40 (where a^ i is the i-th component of the vector a, entry of L is non-negative. Remark. According to the design, eðtÞ40, a^ i ðtÞ40 for all i and tZt0 . Remark. We consider the approximate constraint following problem. That is, it is _ possible that ba0 (hence Aqac). This may be due to modelling uncertainty (and hence Eq. (3.1) cannot be implemented by the designer; while it can be by the Nature). In addition, the system may not start with the constraint manifold in the beginning (i.e., ba0 as t ¼ t0 ). ^ q; q; _ tÞ is the adaptive robust action which is based on the Remark. The control p3 ða; ^ In a sense, the parameter a^ intends to emulate the unknown adaptive parameter a. parameter a. ^ aÞT 0 2 Rmþkþ1 . Subject to Assumptions 1–5, _ Theorem 2. Let dðtÞ :¼ ½bT ðqðtÞ; qðtÞ; tÞðaðtÞ consider the system (2.1). The control (4.8) renders the following performance: (i) Uniform stability: For each z40, there exists x40 such that if dðÞ is any solution with Jdðt0 ÞJox, then JdðtÞJoz for all tZt0 . (ii) Convergence to zero: For any given trajectory dðÞ, lim b ¼ 0:
ð4:14Þ
t-1
Proof. Let V ðb; a^ a; eÞ ¼ bT Pb þ ð1 þ rE Þða^ aÞT L1 ða^ aÞ þ
1 þ rE e: 2l
ð4:15Þ
This is a legitimate Lyapunov function candidate [17]. For a given uncertainty sðÞ and the ^ of the controlled system, the derivative of V is _ corresponding trajectory qðÞ, qðÞ, and aðÞ evaluated as (in the proof, for simplicity, arguments of functions are omitted when no confusions are likely to arise, except for a few critical ones): 1 þ rE V_ ¼ 2bT Pb_ þ 2ð1 þ rE Þða^ aÞT L1 a^_ þ e_ : ð4:16Þ 2l We shall analyze each term separately. First, 2bT Pb_ ¼ 2bT PðAq€ bÞ ¼ 2bT PfA½M 1 ðC q_ gÞ þ M 1 ðp1 þ p2 þ p3 Þ bg:
ð4:17Þ
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Decompose M 1 ¼ D þ DD, C q_ g ¼ ðC q_ gÞ þ ðDC q_ DgÞ. Then we have A½M 1 ðC q_ gÞ þ M 1 ðp1 þ p2 þ p3 Þ b ¼ A½ðD þ DDÞðC q_ g DC q_ DgÞ þ ðD þ DDÞðp1 þ p2 þ p3 Þ b ¼ A½DðC q_ gÞ þ Dðp1 þ p2 Þ þ DðDC q_ DgÞ þ DDðC q_ g þ p1 þ p2 Þ þ ðD þ DDÞp3 b: By Eq. (4.4) with the special case that s 0 (hence M 1 ¼ M Qc ¼ p1 Þ,
ð4:18Þ 1
¼ D, C ¼ C , g ¼ g,
A½DðC q_ gÞ þ Dp1 b ¼ 0:
ð4:19Þ
Next, by Eq. (4.6), 2bT PA½DDðC q_ g þ p1 þ p2 Þ þ DðDC q_ DgÞ r2JbJJPA½DDðC q_ g þ p1 þ p2 Þ þ DðDC q_ DgÞJ _ tÞ: r2JbJð1 þ rE ÞPða; q; q;
ð4:20Þ
Based on Eq. (4.5) and performing matrix cancellation yield 2bT PADp2 ¼ 2bT PAD½kM AT ðAAT Þ1 P1 ðAq_ cÞ ¼ 2bT ðkÞðAq_ cÞ ¼ 2kJbJ2 :
ð4:21Þ
By Eq. (4.9) and with DD ¼ DE, ^ q; q; _ tÞg 2bT PAðD þ DDÞp3 ¼ 2bT PADf½M AT ðAAT Þ1 P1 gmPða; ^ q; q; _ tÞg: þ 2bT PADEf½M AT ðAAT Þ1 P1 gmPða;
ð4:22Þ
^ q; q; _ tÞ ¼ m) By a direct algebra, we can show that (by using bPða; ^ q; q; _ tÞg 2bT PADf½M AT ðAAT Þ1 P1 gmPða; T ^ q; q; _ tÞÞ gm ¼ 2ðbPða; ¼ 2mT gm ¼ 2gJmJ2 :
ð4:23Þ
Adopting the Rayleigh’s principle [16, p. 431], we have ^ q; q; _ tÞg 2bT PADEf½M AT ðAAT Þ1 P1 gmPða; T 1 1 T T ¼ 2m ½PADEM A ðAA Þ P gm ¼ 2g12m½PADEM AT ðAAT Þ1 P1 þ P1 ðAAT ÞT AM E T DAT Pm r 2g12lm ðW þ W T ÞJmJ2 r 2grE JmJ2 :
ð4:24Þ
Combining Eqs. (4.23) and (4.24), 2bT PAðD þ DDÞp3 r 2gð1 þ rE ÞJmJ2 :
ð4:25Þ
If JmJ4e, by Eq. (4.10), 2gð1 þ rE ÞJmJ2 ¼ 2ð1 þ rE Þ
1 JmJ2 ¼ 2ð1 þ rE ÞJmJ: JmJ
ð4:26Þ
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If JmJre, then by Eq. (4.10), 1 JmJ2 : 2gð1 þ rE ÞJmJ2 ¼ 2ð1 þ rE Þ JmJ2 ¼ 2ð1 þ rE Þ e e With Eqs. (4.19)–(4.27), we have, for JmJ4e, _ 2kJbJ2 2ð1 þ r ÞJmJ þ 2ð1 þ r ÞJbJPða; q; q; _ tÞ 2bT Pbr E
ð4:27Þ
E
^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: ¼ 2kJbJ2 þ ð1 þ rE Þf2JbJPða;
ð4:28Þ
As JmJre, 2
_ 2kJbJ2 2ð1 þ r Þ JmJ þ 2JbJð1 þ r ÞPða; q; q; _ tÞ 2bT Pbr E E e JmJ2 zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ ^ q; q; _ tÞ þ 2JbJPða; ¼ 2kJbJ2 þ ð1 þ rE Þ 2 e zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞ þð1 þ rE Þ 2JbJPða; r 2kJbJ2
1 ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞ þ ð1 þ rE Þ 2JbJPða; 4=e e 2 ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: ¼ 2kJbJ þ ð1 þ rE Þ þ ð1 þ rE Þf2JbJPða; 2 ð4:29Þ þ2ð1 þ rE Þ
Here the first equality is simply due to adding and subtracting the same term. Next, by Assumption 5(2), ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞr2JbJ 2JbJPða;
@P ^ q; q; ^ _ tÞða aÞ: ða; @a
ð4:30Þ
Using Eq. (4.30) in Eqs. (4.28) and (4.29), we have that for all JmJ, _ 2kJbJ2 þ ð1 þ r Þ e þ 2JbJð1 þ r Þ @P ða; ^ ^ q; q; _ tÞða aÞ: ð4:31Þ 2bT Pbr E E 2 @a Regarding the second term on the RHS of Eq. (4.16), by using the adaptive law (4.13), we have 2ð1 þ r Þða^ aÞT L1 a^_ E
¼ 2ð1 þ rE Þða^ aÞT L1 L ¼ 2ð1 þ rE Þða^ aÞT
@PT ^ q; q; _ tÞJbJ ða; @a
@PT ^ q; q; _ tÞJbJ ða; @a
@P ^ q; q; _ tÞða^ aÞJbJ; ð4:32Þ ða; @a where the last equality is due to that the RHS is a scalar, hence is equal to its transpose. Regarding the third term on the RHS of Eq. (4.16), by using Eq. (4.12), ¼ 2ð1 þ rE Þ
ð1 þ rE Þ ð1 þ rE Þ ð1 þ rE Þ e_ ¼ ðleÞ ¼ e: 2l 2l 2
ð4:33Þ
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Using Eqs. (4.31)–(4.33) in Eq. (4.16), we conclude that V_ r 2kJbJ2 :
77
ð4:34Þ
Since the Lyapunov derivative is non-positive, we have the uniform stability. In addition, by Barbalat’s lemma [17], we conclude the convergence of b to 0 as t-1. & Remark. With the uncertainty in presence and no restrictions on the initial condition, it is only reasonable to expect approximate constraint following, which is shown in the (worst case) sense that ba0 at any finite time. In the special case when there is no uncertainty, i.e., _ tÞ ¼ 0 and hence p3 ¼ 0. This DD 0, DC 0, and Dg 0, one may choose Pða; q; q; means t ¼ p1 þ p2 . If, in addition, we choose p2 ¼ 0, then V_ ¼ 0. This means if b ¼ 0 initially (i.e., the constraint is met initially), then b ¼ 0 for all tZt0 . This special case falls into Theorem 1, which implies the perfect constraint following. 5. Adaptive robust servo control design II We next introduce an alternative control design. The following assumption is proposed. _ tÞ, Pða; q; q; _ tÞ Assumption 5. ð2Þ0 Under the provision of Assumption 5(1), for each ða; q; q; ~ : Rn Rn can be linearly factorized with respect to a: There exists a function PðÞ R-Rkþ such that ~ _ tÞ: _ tÞ ¼ aT Pðq; q; Pða; q; q; Remark. Assumption 5ð2Þ0 is just a special case of Assumption 5(2). Let 8 1 > > ^ q; q; _ tÞJ4^e ; if Jmða; < Jmða; ^ q; q; _ tÞJ ^ q; q; _ tÞ ¼ g^ ða; 1 > > : ^ q; q; _ tÞJr^e ; if Jmða; e^
ð5:1Þ
ð5:2Þ
where e^ 40 is a scalar constant. We now propose the control ~ _ _ _ tðtÞ ¼ p1 ðqðtÞ; qðtÞ; qðtÞ; qðtÞ; tÞ; tÞ þ p2 ðqðtÞ; qðtÞ; tÞ þ p4 ðaðtÞ;
ð5:3Þ
where ~ q; q; ~ q; q; ~ q; q; ~ q; q; _ tÞ ¼ ½M ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 ^g ða; _ tÞmða; _ tÞPða; _ tÞ: p4 ða;
ð5:4Þ Here the adaptive parameter vector a~ 2 Rk is governed by the following adaptive law: ~ ~ _ tÞJbðq; q; _ tÞJ k2 a; q; a~_ ¼ k1 Pðq; ð5:5Þ ~ i ¼ 1; . . . ; k, k1;2 2 R, k1;2 40. a~ i ðt0 Þ40 (where a~ i is the i-th component of the vector a), Remark. The adaptive law (5.5) is of leakage type, while the second term on the RHS is the leak. Note that if the initial condition a~ i ðt0 Þ is selected to be strictly positive, then a~ i ðtÞ40 for all tZt0 . This is since the first term on the RHS is always non-negative and the second term alone will render an exponentially decaying (to zero) solution from above.
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Remark. The major difference between (4.10) and (5.2) is that in the former, it is possible that control chattering may occur when eðtÞ becomes very small (note that it converges to 0 exponentially, according to (4.12)); while in the latter this will not happen since e^ is a finite constant. Theorem 3. Let d~ :¼ ½bT ða~ aÞT T 2 Rmþk . Subject to Assumptions 1–4, 5(1), and 5ð2Þ0 , consider the system (2.1). The control (5.3) renders the following performance: ~ 0 ÞJrr, (i) Uniform boundedness: For any r40, there is a dðrÞo1 such that if Jdðt ~ then JdðtÞJrdðrÞ for all tZt0 ; ~ 0 ÞJrr, there exists a d 40 such that (ii) Uniform ultimate bounded: For any r40 with Jdðt ~ JdðtÞJrd for any d 4d as tZt0 þ Tðd ; rÞ, where Tðd ; rÞo1. Proof. Let ~ ¼ bT Pb þ k11 ð1 þ rE Þða~ aÞT ða~ aÞ V ðb; aÞ
ð5:6Þ
be a legitimate Lyapunov function candidate. Again, in the proof, arguments of functions are largely omitted except for a few critical ones. The derivative of V is given by ~_ V_ ¼ 2bT Pb_ þ 2k1 ð1 þ r Þða~ aÞT a: ð5:7Þ 1
E
First, similar to Eq. (4.28), for JmJ4^e , _ 2kJbJ2 þ ð1 þ r Þf2JbJPða; ~ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: 2bT Pbr E
ð5:8Þ
Also, similar to Eq. (4.29), for JmJr^e , _ 2kJbJ2 þ ð1 þ r Þ e^ 2bT Pbr E 2 ~ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: þ ð1 þ rE Þf2JbJPða;
ð5:9Þ
By Assumption 5ð2Þ0 , ~ ~ ~ q; q; _ tÞ þ 2JbJaT Pðq; _ tÞ _ tÞ þ 2JbJPða; q; q; _ tÞ ¼ 2JbJa~ T Pðq; q; q; 2JbJPða; ~ ~ T Pðq; _ tÞ: ¼ 2JbJða aÞ q;
ð5:10Þ
Using Eq. (5.10) in Eqs. (5.8) and (5.9), we have that for all JmJ, _ 2kJbJ2 þ ð1 þ r Þ e^ þ 2JbJð1 þ r Þða aÞ ~ ~ T Pðq; _ tÞ: 2bT Pbr q; E E 2
ð5:11Þ
By using the adaptive law (5.5), we have ~ ~ _ tÞJbJ k2 aÞ q; 2k11 ða~ aÞT a~_ ¼ 2k11 ða~ aÞT ðk1 Pðq; T ~ 1 ~ ~ _ tÞJbJ 2k1 ða aÞT k2 a~ ¼ 2ða aÞ Pðq; q; ¼0
zfflfflfflffl}|fflfflfflffl{ ~ ~ _ tÞJbJ 2k11 k2 ða~ aÞT ðaa ¼ 2ða~ aÞ Pðq; þ aÞ q; T ~ T 1 ~ ~ ~ _ ¼ 2ða aÞ Pðq; q; tÞJbJ 2k1 k2 ða aÞ ða aÞ 2k11 k2 ða~ aÞT a T
~ _ tÞJbJ 2k11 k2 Ja~ aJ2 þ 2k11 k2 Ja~ aJJaJ: r2ða~ aÞT Pðq; q; ð5:12Þ
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~ 2¼ Multiplying Eq. (5.12) by 1 þ rE and with Eq. (5.11), Eq. (5.7) becomes (note that JdJ ~ JbJ2 þ Ja~ aJ2 , Ja~ aJrJdJ) e^ V_ r 2kJbJ2 þ ð1 þ rE Þ 2k11 k2 ð1 þ rE ÞJa~ aJ2 þ 2k11 k2 ð1 þ rE ÞJa~ aJJaJ 2 ~ 2 þ k JdJ ~ þk ; r k 1 JdJ ð5:13Þ 2 3 where k 1 ¼ minf2k; 2k11 k2 ð1 þ rE Þg; k 2 ¼ 2k11 k2 ð1 þ rE ÞJaJ; k 3 ¼ ð1 þ rE Þ^e =2: Upon invoking the standard arguments as in Chen [18] and Khalil [17], we conclude the uniform boundedness with 8 > rffiffiffiffiffi > > > g2 > > < g R if rrR; 1 ð5:14Þ dðrÞ ¼ > > rffiffiffiffiffi > > g2 > > : r if r4R; g1 R¼
1 ðk þ 2k 1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 22 þ 4k 1 k 3 Þ;
ð5:15Þ
where g1 ¼ minflmin ðPÞ; k11 ð1 þ rE Þg, g2 ¼ maxflmax ðPÞ; k11 ð1 þ rE Þg. Furthermore, uniform ultimate boundedness also follows with rffiffiffiffiffi g2 R; d¼ g1
Tðd ; rÞ ¼
8 > > >0 > < > > > > :
if rrd
ð5:16Þ
rffiffiffiffiffi g1 ; g2
g2 r2 ðg21 =g2 Þd
ð5:17Þ
2
2
k 1 d ðg1 =g2 Þ k 2 d ðg1 =g2 Þ1=2 k 3
otherwise:
&
6. Illustrative example Consider a two-link R–R mechanical manipulator, whose equation of motion is in the form of Eq. (2.1) with [19, p. 177]
2 l2 m2 þ 2l1 l2 m2 c2 þ l12 ðm1 þ m2 Þ l22 m2 þ l1 l2 m2 c2 M¼ ; ð6:1Þ l22 m2 l22 m2 þ l1 l2 m2 c2 " # 2 m2 l1 l2 s2 y_ 2 2m2 l1 l2 s2 y_ 1 y_ 2 _ tÞq_ ¼ Cðq; q; ; ð6:2Þ 2 m2 l1 l2 s2 y_ 1
m2 l2 Gc12 þ ðm1 þ m2 Þl1 Gc1 ; gðqÞ ¼ m2 l2 Gc12
ð6:3Þ
where q ¼ ½y1 y2 T , t ¼ ½t1 t2 T , l1 (l2 ) is the length of link 1 (2), m1 (m2 ) is the mass of link 1 (2), ci ¼ cosðyi Þ, i ¼ 1; 2; s2 ¼ sinðy2 Þ, c12 ¼ cosðy1 þ y2 Þ, G is the gravitational
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constant. As a simple illustration, it is desired to have the links’ angular velocities to be constrained by y_ 1 þ y_ 2 ¼ 0:
ð6:4Þ
The constraint can be cast into the form of Eqs. (2.3) and (2.8) with A ¼ ½1 1;
c ¼ 0;
b ¼ 0:
ð6:5Þ T
We consider the masses are uncertain parameters (hence s ¼ ½m1 m2 ): m1 ¼ m 1 þ Dm1 ðtÞ, m2 ¼ m 2 þ Dm2 ðtÞ: Here Dm1;2 ðtÞ are unknown, possibly time-varying variations of
Fig. 1. System performance comparison (red: adaptive robust control (4.8); dashed: nominal control). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Adaptive parameter history in (4.13).
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m1;2 , with the possible bounds of the variations unknown. Assumptions 1–4 can be verified. _ and g are either constant, trigonometric in positions, or Note that the terms in M, C q, quadratic in velocities, Assumption 5(1), (2), and ð2Þ0 are met by choosing (note also that the constraint (6.4) is linear in velocities) _ 2 þ a2 JqJ _ þ a3 _ tÞ ¼ a1 JqJ Pða; q; q; 2 3 _ 2 JqJ 6 7 ¼ ½a1 a2 a3 4 JqJ _ 5 1 ~ _ tÞ; q; ¼: a Pðq; T
Fig. 3. Adaptive robust control history in (4.8).
ð6:6Þ
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where a1;2;3 40 are unknown constant parameters. It is interesting to note that we can also meet Assumption 5(1), (2), and ð2Þ0 by an alternative choice of P as follows: _ 2 þ a2 JqJ _ þ a3 raðJqJ _ 2 þ 2JqJ _ þ 1Þ a1 JqJ _ þ 1Þ2 ¼ aðJqJ ~ _ tÞ; ¼: aPðq; q;
ð6:7Þ
where a ¼ maxfa1 ; a2 =2; a3 g. This shows the flexibility of the control design and adaptive laws. The control t can be selected according to Eqs. (4.8) and (5.3). The adaptive laws
Fig. 4. Nominal control history.
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_ ¼ (4.13) and (5.5) are given by, according to Eq. (6.7), respectively (note that JqJ 2 2 1=2 ðy_ þ y_ Þ , b ¼ y_ 1 þ y_ 2 Þ 1
2
2 2 a^_ ¼ Lððy_ 1 þ y_ 2 Þ1=2 þ 1Þ2 Jy_ 1 þ y_ 2 J;
^ 0 Þ40; aðt
2 2 ~ a~_ ¼ k1 ððy_ 1 þ y_ 2 Þ1=2 þ 1Þ2 Jy_ 1 þ y_ 2 J k2 a;
~ 0 Þ40; aðt
ð6:8Þ ð6:9Þ
where L and k1;2 are all scalars. It is interesting to note that the adaptive laws are only velocity (hence not position) dependent.
Fig. 5. System performance comparison (red: adaptive robust control (5.3); dashed: nominal control). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Adaptive parameter history in (5.5).
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Figs. 1–7 show the simulations by using m1 ¼ 1, m2 ¼ 0:5, l1 ¼ 1, l2 ¼ 2, G ¼ 9:8, ^ ¼ 0:2, y1 ð0Þ ¼ 0, y2 ð0Þ ¼ 0, y_ 1 ð0Þ ¼ 0:4, y_ 2 ð0Þ ¼ 0:1, k ¼ L ¼ 1, eð0Þ ¼ 0:1, l ¼ 0:01, að0Þ ~ e^ ¼ 0:1, k1 ¼ 1, k2 ¼ 0:1, að0Þ ¼ 0:2. Fig. 1 shows the constraint time history by using control (4.8). For comparison, the use of nominal control (i.e., (4.8) but with p3 0) is shown. This corresponds to Udwadia–Kalaba-like control for asymptotic convergence. By using (4.8), the constraint approaches to a desirable neighborhood before t ¼ 0:2 (while the ^ nominal control does not result in any finite time settling). The adaptive parameter aðtÞ history is shown in Fig. 2. It increases to a neighborhood close to 1. The adaptive robust control history is shown in Fig. 3. For comparison, we also show the nominal control history in Fig. 4. It is interesting to note that, despite the significant difference in the resulting system performance, the maximum magnitude of the nominal control is almost
Fig. 7. Adaptive robust control history in (5.3).
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twice as much as that of the adaptive robust control. Figs. 5–7 show the corresponding comparison by using (5.3). Fig. 5 shows the system performance with or without the adaptive robust control. Fig. 6 shows the adaptive parameter a~ history. Since there is a leakage, the parameter decreases its value after some time; while in Fig. 2 a^ does not. Fig. 7 shows the control (5.3). 7. Conclusions We consider a mechanical system subject to a class of (possibly nonholonomic) constraints. The system contains uncertainty. The bound of uncertainty is unknown. The objective is to design a control which renders the system to follow the constraint sufficiently close, even in the presence of uncertainty. Two adaptive laws are proposed. The first one is able to render the convergence of b to zero at the expense of possibly incurring chattering. The second one renders the combined state d~ uniform bounded and uniform ultimate bounded, while avoid the chattering phenomena. Acknowledgment The research of Xinrong Zhang is sponsored by the Chinese Scholarship Council during a sabbatical leave at Georgia Tech. References [1] J.G. Papastavridis, Analytical Mechanics, Oxford University Press, New York, NY, 2002. [2] A.M. Bloch, Nonholonomic Mechanics and Control, Springer, New York, NY, 2003. [3] F. Bullo, A.D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer, New York, NY, 2005. [4] Z. Song, D. Zhao, J. Yi, X. Li, Robust motion control for nonholonomic constrained mechanical systems: sliding mode approach, in: American Control Conference, 2005, pp. 2883–2888. [5] C.S. Tseng, B.S. Chen, A mixed H2 =H1 adaptive tracking control for constrained non-holonomic systems, Automatica 39 (2003) 1011–1018. [6] Z.P. Wang, S.S. Ge, T.H. Lee, Robust motion/force control of uncertain holonomic/nonholonomic mechanical systems, IEEEA/SME Transactions on Mechatronics 9 (2004) 118–123. [7] J. Wang, X. Zhu, M. Oya, C.Y. Su, Robust motion tracking control of partially nonholonomic mechanical systems, Automatica 54 (2006) 332–341. [8] F.E. Udwadia, R.E. Kalaba, Analytical Approach: A New Approach, Cambridge University Press, Cambridge, UK, 1996. [9] F.E. Udwadia, R.E. Kalaba, Equations of motion for constrained mechanical systems and the extended D’Alembert’s principle, Quarterly of Applied Mathematics (1997) 321–331. [10] F.E. Udwadia, R.E. Kalaba, What is the general form of the explicit equations of motion for constrained mechanical systems?, ASME Transactions on Journal of Applied Mechanics (2002) 335–339. [11] L.A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965 (reprinted 1979 by Ox Bow Press, Woodbridge, Connecticut). [12] R.M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum, New York, NY, 1977. [13] Y.H. Chen, Second order constraints for equations of motion of constrained systems, IEEE/ASME Transactions on Mechatronics 3 (1998) 240–248. [14] Y.H. Chen, Constraint-following servo control design for mechanical systems, Journal of Vibration and Control (2009) 369–389. [15] Y.H. Chen, G. Leitmann, J.S. Chen, Robust control for rigid serial manipulators: a general setting, in: Proceedings of the 1998 American Control Conference, 1998, pp. 912–916. [16] B. Noble, J.W. Daniel, Applied Linear Algebra, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1977.
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[17] H.K. Khalil, Nonlinear Systems, third ed., Prentice-Hall, Upper Saddle River, NJ, 2002. [18] Y.H. Chen, On the deterministic performance of uncertain dynamical systems, International Journal of Control (1986) 1557–1579. [19] J.J. Craig, Introduction to Robotics: Mechanics and Control, third ed., Pearson Prentice-Hall, Upper Saddle River, NJ, 2005.
Journal of the Franklin Institute 347 (2010) 69–86 www.elsevier.com/locate/jfranklin
Adaptive robust approximate constraint-following control for mechanical systems Ye-Hwa Chena,, Xinrong Zhangb a
The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA b Department of Mechanical and Electronic Engineering, Chang’an University, Xian, PR China
Received 24 August 2009; received in revised form 6 October 2009; accepted 12 October 2009
Abstract We formulate control problems of mechanical systems as constraint following. The system contains uncertainty which is (possibly fast) time-varying. It is unknown but bounded. The bound is, however, unknown. The objective is to design control which renders approximate constraint following. Adaptive laws are constructed. The controls are then based on the adaptive parameters. We are able to demonstrate guaranteed system performance, regardless of the uncertainty. & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Mechanical system; Constraint; Motion control; Robust control; Adaptive control
1. Introduction The most important feature of mechanical system motion, in connecting to real world practices, according to Joseph-Louis Lagrange, is constraint following. It is because of this, the Lagrange’s form of d’Alembert’s principle was postulated. This is what Lagrange asserted what the Nature would do [1]. In the past, the majority of the efforts in this branch of mechanics can be divided into two categories: the passive constraint problem and the servo constraint problem. In the passive constraint problem, which is along with the original Lagrange aspect, the main focus is to investigate what the Nature will do in order to assure that the constraints are Corresponding author. Tel.: +1 404 894 3210; fax: +1 404 894 9342.
E-mail address: [email protected] (Y.-H. Chen). 0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.10.012
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(strictly) obeyed. Many important contributions have been made (for a survey, see, for example, [1]). In the servo constraint problem, which constructively utilizes the Lagrange aspect, the main focus is to find what the engineer should do, so that the constraints are followed. Most of the emphasis is on precise model-based control design. A survey on the use of geometric and algebraic approaches can be found in Bloch [2] and Bullo and Lewis [3]. There are also efforts which deal with uncertainty (e.g., [4–7] and their bibliographies). The scope of this paper falls into the servo approach. Several classes of control problems are formulated as constraint following. As a result, the constraint following task is to be pursued for solving the control problems. Since the engineer’s knowledge of the system is sometimes limited, we consider the presence of modelling uncertainty. The uncertainty is (possibly fast) time varying. No further information except it is bounded is assumed. Furthermore, the bound is unknown. We propose to design control which renders the system to follow a class of pre-specified constraints approximately. For the control design, we follow the framework established in Udwadia and Kalaba [8–10]. First, a nominal control is proposed to address the system dynamics while no uncertainty is involved. Second, adaptive laws, which emulate a constant parameter vector, are constructed. This parameter vector may be relevant (but not necessarily identical) to the uncertainty bound, which is unknown. Third, two classes of adaptive robust controls are designed. The controls are able to guarantee either convergence to zero (hence strictly constraint following) or uniform ultimate boundedness, regardless of the uncertainty. 2. Mechanical system subject to constraints Consider the following mechanical system [11,12]: € þ CðqðtÞ; qðtÞ; _ _ þ gðqðtÞ; sðtÞ; tÞ ¼ tðtÞ: MðqðtÞ; sðtÞ; tÞqðtÞ sðtÞ; tÞqðtÞ n
ð2:1Þ n
Here t 2 R is the independent variable, q 2 R is the coordinate, q_ 2 R is the velocity, q€ 2 Rn is the acceleration, s 2 S Rp is the uncertain parameter, and t 2 Rn is the control input. Here S Rp is compact but unknown, which stands for the possible bounding of s. _ s; tÞq_ is the Coriolis/centrifugal force, Furthermore, Mðq; s; tÞ is the inertia matrix, Cðq; q; _ s; tÞ, and and gðq; sðtÞ; tÞ is the gravitational force. The matrices/vector Mðq; s; tÞ, Cðq; q; gðq; s; tÞ are of appropriate dimensions. We assume that the functions MðÞ, CðÞ, and gðÞ are continuous (this can be generalized to be Lebesgue measurable in t). Remark. The coordinate q can be selected based on the specifics of the problem and does not need to be the generalized coordinate. The following constraints are proposed: n X Ali ðq; tÞq_ i ¼ cl ðq; tÞ; l ¼ 1; . . . ; m;
ð2:2Þ
i¼1
_ Ali ðÞ and cl ðÞ are both C 1 , mrn. They are the first where q_ i is the i th component of q, order form of the constraints. The constraints may not be integrable; and may be nonholonomic in general. The constraints can be put in matrix form Aðq; tÞq_ ¼ cðq; tÞ; where A ¼ ½Ali mn , c ¼ ½c1 c2 cm T .
ð2:3Þ
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There are two ways of interpreting the constraints. First, they may be passive. That is, the environment (or the structure) is to supply the constraint force in order for the system to comply with the constraint. Second, they may be active. That is, the system’s control input supplies the required force so that the constraints are met. In this study, we shall adopt the second view. We now convert the first order form into second order form [13]. Differentiating the constraint equations (2.2) with respect to t yields n X d i¼1
n X d Ali ðq; tÞ q_ i þ Ali ðq; tÞq€ i ¼ cl ðq; tÞ; dt dt i¼1
ð2:4Þ
where n X d @Ali ðq; tÞ @Ali ðq; tÞ q_ k þ Ali ðq; tÞ ¼ ; dt @qk @t k¼1
ð2:5Þ
n X d @cl ðq; tÞ @cl ðq; tÞ cl ðq; tÞ ¼ : q_ k þ dt @q @t k k¼1
ð2:6Þ
Eq. (2.4), the second order form of the constraints, can be rewritten as n X i¼1
Ali ðq; tÞq€ i ¼
n X d i¼1
d Ali ðq; tÞ q_ i þ cl ðq; tÞ dt dt
_ tÞ; ¼: bl ðq; q;
ð2:7Þ
l ¼ 1; . . . ; m, or in matrix form _ tÞ; Aðq; tÞq€ ¼ bðq; q;
ð2:8Þ
where b ¼ ½b1 b2 bm T . Remark. In Chen [14], it has been demonstrated that various control problems, including stabilization, trajectory following, and optimality, can be cast into the form (2.8).
3. Constraint force when uncertainty is known We show the constraint force when the uncertainty is known. Assumption 1. For each ðq; tÞ 2 Rn R, s 2 S, Mðq; s; tÞ40. Remark. The assumption on the positive definiteness of the inertia matrix will be vital in later development. In the past, it was often believed that this was always true, and therefore a fact rather than an assumption. However, there are counter examples, as listed in [15], when q is not selected to be the generalized coordinate. Definition 1. For given A and b, the constraint (2.8) is called consistent if there exists at € least one solution q. Assumption 2. The constraint (2.8) is consistent.
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Theorem 1 (Udwadia and Kalaba [8, p. 233]). Consider the system (2.1) and the constraint (2.8). Subject to Assumptions 1 and 2, the constraint force Qc ¼ M 1=2 ðq; s; tÞðAðq; tÞM 1=2 ðq; s; tÞÞþ _ tÞ þ Aðq; tÞM 1 ðq; s; tÞðCðq; q; _ s; tÞq_ þ gðq; s; tÞÞ ½bðq; q;
ð3:1Þ
obeys the Lagrange’s form of d’Alembert’s principle [1] and renders the system to meet the constraint. Here ‘‘ þ ’’ stands for the Moore–Penrose generalized inverse [16, p. 337]. Remark. The Lagrange’s form of d’Alembert’s principle renders the constraint force (3.1) to be the one with minimum norm, out of all possible alternative forces which can also meet Eq. (2.8) [8]. Note that Qc 2 RðAT Þ (with the understanding that in this d’Alembert’s principle, the virtual displacement dq 2 N ðAÞ; in addition RðAT Þ ? N ðAÞ). Remark. Theorem 1 shows the strategy the Nature will undertake to meet the constraint. The constraint force is model-based. That is, it is based on the exact model information. Based on the theorem, one could apply the control input t ¼ Qc to drive the system to meet Eq. (2.8), if the uncertainty were known. A more realistic design, from the engineer’s point of view, when the uncertainty is unknown, is investigated in the next section.
4. Adaptive robust servo control design I We now take the uncertainty into account while designing the control t. Decompose the M, C, and g as follows: Mðq; s; tÞ ¼ M ðq; tÞ þ DMðq; s; tÞ; _ s; tÞ ¼ C ðq; q; _ tÞ þ DCðq; q; _ s; tÞ; Cðq; q; gðq; s; tÞ ¼ gðq; tÞ þ Dgðq; s; tÞ:
ð4:1Þ
Here M , C , and g denote the ‘‘nominal’’ portions with M 40 (this is always feasible since it is the designer’s discretion), while DM, DC and Dg are the uncertain portions. The functions M ðÞ, DMðÞ, C ðÞ, DCðÞ, gðÞ, and DgðÞ are all continuous. Let Dðq; tÞ :¼ 1
M ðq; tÞ, DDðq; s; tÞ :¼ M 1 ðq; s; tÞ M (hence DDðq; s; tÞ ¼ Dðq; tÞEðq; s; tÞÞ.
1
ðq; tÞ,
Eðq; s; tÞ :¼ M ðq; tÞM 1 ðq; s; tÞ I
Assumption 3. For each ðq; tÞ 2 Rn R, Aðq; tÞ is of full rank. This means Aðq; tÞAT ðq; tÞ is invertible. Assumption 4. Under the provision of Assumption 3, for given P 2 Rmm , P40, let W ðq; s; tÞ :¼ PAðq; tÞDðq; tÞEðq; s; tÞM ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 :
ð4:2Þ
There exists a (possibly unknown) constant rE 4 1 such that for all ðq; tÞ 2 Rn R, 1 min lm ðW ðq; s; tÞ þ W T ðq; s; tÞÞZrE : 2 s2S
ð4:3Þ
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Remark. The constant rE is in general unknown since the uncertainty bound S is unknown. In the special case that M ¼ M (i.e., no uncertainty), E ¼ 0, W ¼ 0 and hence one can choose rE ¼ 0. Thus by continuity this assumption imposes the effect of uncertainty on the possible deviation of M from M to be within a certain threshold. We also stress that this threshold is unidirectional (that is, it is not bounded in one direction). Let _ tÞ :¼ M p1 ðq; q;
1=2
ðq; tÞðAðq; tÞM
1=2
_ tÞ þ Aðq; tÞM ½bðq; q;
ðq; tÞÞþ
1
_ tÞq_ þ gðq; tÞÞ; ðq; tÞðC ðq; q;
_ tÞ :¼ kM ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 ðAðq; tÞq_ cðq; tÞÞ: p2 ðq; q;
ð4:4Þ ð4:5Þ
Assumption 5. (1) There exists an unknown constant vector a 2 ð0; 1Þk and a known _ tÞ 2 Rn Rn R, s 2 S, function PðÞ : ð0; 1Þk Rn Rn R-Rþ such that for all ðq; q; _ s; tÞq_ gðq; s; tÞ þ p1 ðq; q; _ tÞ ð1 þ rE Þ1 max½JPAðq; tÞDDðq; s; tÞðCðq; q; s2S
_ tÞÞ PAðq; tÞDðq; tÞðDCðq; q; _ s; tÞq_ þ Dgðq; s; tÞÞJrPða; q; q; _ tÞ: þp2 ðq; q;
ð4:6Þ
_ tÞ 2 Rn Rn R, the function Pð; q; q; _ tÞ : ð0; 1Þk -Rþ is (i) C 1 , (2) For each ðq; q; k (ii) concave; that is, for any a1;2 2 ð0; 1Þ , _ tÞ Pða2 ; q; q; _ tÞr Pða1 ; q; q;
@P _ tÞða1 a2 Þ; ða2 ; q; q; @a
ð4:7Þ
and (iii) non-decreasing with respect to each component of its argument a. Remark. Note that 1 þ rE 40 if rE 4 1. The function PðÞ may be interpreted the uncertainty bound. The constant vector a may be relevant to the bounding set S. It is _ tÞ is linear in a, then unknown since S is unknown. In the special case that Pð; q; q; Assumption 5(2) is met. In a sense, what Assumption 5 does is the parameterization of the worst case effect of the uncertainty, which will be further elaborated in the proof of Theorem 2. We now propose the control ^ _ _ _ qðtÞ; qðtÞ; tÞ; tÞ þ p2 ðqðtÞ; qðtÞ; tÞ þ p3 ðaðtÞ; tðtÞ ¼ p1 ðqðtÞ; qðtÞ;
ð4:8Þ
with ^ q; q; ^ q; q; ^ q; q; ^ q; q; _ tÞ ¼ ½M ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 gða; _ tÞmða; _ tÞPða; _ tÞ; p3 ða;
ð4:9Þ where
8 > > > <
1 ^ q; q; _ tÞJ4eðtÞ; if Jmða; ^ q; q; _ tÞJ Jmða; ^ q; q; _ tÞ ¼ gða; > > 1 ^ q; q; _ tÞJreðtÞ; if Jmða; > : eðtÞ
ð4:10Þ
^ q; q; ^ q; q; _ tÞ; _ tÞ ¼ ðAðq; tÞq_ cðq; tÞÞPða; mða;
ð4:11Þ
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k 2 R, k40, e_ ðtÞ ¼ leðtÞ;
eðt0 Þ40; l40:
ð4:12Þ
_ tÞ :¼ Aðq; tÞq_ cðq; tÞ. The parameter a^ is governed by the following adaptive law: Let bðq; q; T
@P ^ q; q; _ tÞJbðq; q; _ tÞJ; a^_ ¼ L ða; @a
ð4:13Þ
^ i ¼ 1; . . . ; k), L 2 Rkk 40, each a^ i ðt0 Þ40 (where a^ i is the i-th component of the vector a, entry of L is non-negative. Remark. According to the design, eðtÞ40, a^ i ðtÞ40 for all i and tZt0 . Remark. We consider the approximate constraint following problem. That is, it is _ possible that ba0 (hence Aqac). This may be due to modelling uncertainty (and hence Eq. (3.1) cannot be implemented by the designer; while it can be by the Nature). In addition, the system may not start with the constraint manifold in the beginning (i.e., ba0 as t ¼ t0 ). ^ q; q; _ tÞ is the adaptive robust action which is based on the Remark. The control p3 ða; ^ In a sense, the parameter a^ intends to emulate the unknown adaptive parameter a. parameter a. ^ aÞT 0 2 Rmþkþ1 . Subject to Assumptions 1–5, _ Theorem 2. Let dðtÞ :¼ ½bT ðqðtÞ; qðtÞ; tÞðaðtÞ consider the system (2.1). The control (4.8) renders the following performance: (i) Uniform stability: For each z40, there exists x40 such that if dðÞ is any solution with Jdðt0 ÞJox, then JdðtÞJoz for all tZt0 . (ii) Convergence to zero: For any given trajectory dðÞ, lim b ¼ 0:
ð4:14Þ
t-1
Proof. Let V ðb; a^ a; eÞ ¼ bT Pb þ ð1 þ rE Þða^ aÞT L1 ða^ aÞ þ
1 þ rE e: 2l
ð4:15Þ
This is a legitimate Lyapunov function candidate [17]. For a given uncertainty sðÞ and the ^ of the controlled system, the derivative of V is _ corresponding trajectory qðÞ, qðÞ, and aðÞ evaluated as (in the proof, for simplicity, arguments of functions are omitted when no confusions are likely to arise, except for a few critical ones): 1 þ rE V_ ¼ 2bT Pb_ þ 2ð1 þ rE Þða^ aÞT L1 a^_ þ e_ : ð4:16Þ 2l We shall analyze each term separately. First, 2bT Pb_ ¼ 2bT PðAq€ bÞ ¼ 2bT PfA½M 1 ðC q_ gÞ þ M 1 ðp1 þ p2 þ p3 Þ bg:
ð4:17Þ
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Decompose M 1 ¼ D þ DD, C q_ g ¼ ðC q_ gÞ þ ðDC q_ DgÞ. Then we have A½M 1 ðC q_ gÞ þ M 1 ðp1 þ p2 þ p3 Þ b ¼ A½ðD þ DDÞðC q_ g DC q_ DgÞ þ ðD þ DDÞðp1 þ p2 þ p3 Þ b ¼ A½DðC q_ gÞ þ Dðp1 þ p2 Þ þ DðDC q_ DgÞ þ DDðC q_ g þ p1 þ p2 Þ þ ðD þ DDÞp3 b: By Eq. (4.4) with the special case that s 0 (hence M 1 ¼ M Qc ¼ p1 Þ,
ð4:18Þ 1
¼ D, C ¼ C , g ¼ g,
A½DðC q_ gÞ þ Dp1 b ¼ 0:
ð4:19Þ
Next, by Eq. (4.6), 2bT PA½DDðC q_ g þ p1 þ p2 Þ þ DðDC q_ DgÞ r2JbJJPA½DDðC q_ g þ p1 þ p2 Þ þ DðDC q_ DgÞJ _ tÞ: r2JbJð1 þ rE ÞPða; q; q;
ð4:20Þ
Based on Eq. (4.5) and performing matrix cancellation yield 2bT PADp2 ¼ 2bT PAD½kM AT ðAAT Þ1 P1 ðAq_ cÞ ¼ 2bT ðkÞðAq_ cÞ ¼ 2kJbJ2 :
ð4:21Þ
By Eq. (4.9) and with DD ¼ DE, ^ q; q; _ tÞg 2bT PAðD þ DDÞp3 ¼ 2bT PADf½M AT ðAAT Þ1 P1 gmPða; ^ q; q; _ tÞg: þ 2bT PADEf½M AT ðAAT Þ1 P1 gmPða;
ð4:22Þ
^ q; q; _ tÞ ¼ m) By a direct algebra, we can show that (by using bPða; ^ q; q; _ tÞg 2bT PADf½M AT ðAAT Þ1 P1 gmPða; T ^ q; q; _ tÞÞ gm ¼ 2ðbPða; ¼ 2mT gm ¼ 2gJmJ2 :
ð4:23Þ
Adopting the Rayleigh’s principle [16, p. 431], we have ^ q; q; _ tÞg 2bT PADEf½M AT ðAAT Þ1 P1 gmPða; T 1 1 T T ¼ 2m ½PADEM A ðAA Þ P gm ¼ 2g12m½PADEM AT ðAAT Þ1 P1 þ P1 ðAAT ÞT AM E T DAT Pm r 2g12lm ðW þ W T ÞJmJ2 r 2grE JmJ2 :
ð4:24Þ
Combining Eqs. (4.23) and (4.24), 2bT PAðD þ DDÞp3 r 2gð1 þ rE ÞJmJ2 :
ð4:25Þ
If JmJ4e, by Eq. (4.10), 2gð1 þ rE ÞJmJ2 ¼ 2ð1 þ rE Þ
1 JmJ2 ¼ 2ð1 þ rE ÞJmJ: JmJ
ð4:26Þ
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If JmJre, then by Eq. (4.10), 1 JmJ2 : 2gð1 þ rE ÞJmJ2 ¼ 2ð1 þ rE Þ JmJ2 ¼ 2ð1 þ rE Þ e e With Eqs. (4.19)–(4.27), we have, for JmJ4e, _ 2kJbJ2 2ð1 þ r ÞJmJ þ 2ð1 þ r ÞJbJPða; q; q; _ tÞ 2bT Pbr E
ð4:27Þ
E
^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: ¼ 2kJbJ2 þ ð1 þ rE Þf2JbJPða;
ð4:28Þ
As JmJre, 2
_ 2kJbJ2 2ð1 þ r Þ JmJ þ 2JbJð1 þ r ÞPða; q; q; _ tÞ 2bT Pbr E E e JmJ2 zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ ^ q; q; _ tÞ þ 2JbJPða; ¼ 2kJbJ2 þ ð1 þ rE Þ 2 e zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞ þð1 þ rE Þ 2JbJPða; r 2kJbJ2
1 ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞ þ ð1 þ rE Þ 2JbJPða; 4=e e 2 ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: ¼ 2kJbJ þ ð1 þ rE Þ þ ð1 þ rE Þf2JbJPða; 2 ð4:29Þ þ2ð1 þ rE Þ
Here the first equality is simply due to adding and subtracting the same term. Next, by Assumption 5(2), ^ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞr2JbJ 2JbJPða;
@P ^ q; q; ^ _ tÞða aÞ: ða; @a
ð4:30Þ
Using Eq. (4.30) in Eqs. (4.28) and (4.29), we have that for all JmJ, _ 2kJbJ2 þ ð1 þ r Þ e þ 2JbJð1 þ r Þ @P ða; ^ ^ q; q; _ tÞða aÞ: ð4:31Þ 2bT Pbr E E 2 @a Regarding the second term on the RHS of Eq. (4.16), by using the adaptive law (4.13), we have 2ð1 þ r Þða^ aÞT L1 a^_ E
¼ 2ð1 þ rE Þða^ aÞT L1 L ¼ 2ð1 þ rE Þða^ aÞT
@PT ^ q; q; _ tÞJbJ ða; @a
@PT ^ q; q; _ tÞJbJ ða; @a
@P ^ q; q; _ tÞða^ aÞJbJ; ð4:32Þ ða; @a where the last equality is due to that the RHS is a scalar, hence is equal to its transpose. Regarding the third term on the RHS of Eq. (4.16), by using Eq. (4.12), ¼ 2ð1 þ rE Þ
ð1 þ rE Þ ð1 þ rE Þ ð1 þ rE Þ e_ ¼ ðleÞ ¼ e: 2l 2l 2
ð4:33Þ
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Using Eqs. (4.31)–(4.33) in Eq. (4.16), we conclude that V_ r 2kJbJ2 :
77
ð4:34Þ
Since the Lyapunov derivative is non-positive, we have the uniform stability. In addition, by Barbalat’s lemma [17], we conclude the convergence of b to 0 as t-1. & Remark. With the uncertainty in presence and no restrictions on the initial condition, it is only reasonable to expect approximate constraint following, which is shown in the (worst case) sense that ba0 at any finite time. In the special case when there is no uncertainty, i.e., _ tÞ ¼ 0 and hence p3 ¼ 0. This DD 0, DC 0, and Dg 0, one may choose Pða; q; q; means t ¼ p1 þ p2 . If, in addition, we choose p2 ¼ 0, then V_ ¼ 0. This means if b ¼ 0 initially (i.e., the constraint is met initially), then b ¼ 0 for all tZt0 . This special case falls into Theorem 1, which implies the perfect constraint following. 5. Adaptive robust servo control design II We next introduce an alternative control design. The following assumption is proposed. _ tÞ, Pða; q; q; _ tÞ Assumption 5. ð2Þ0 Under the provision of Assumption 5(1), for each ða; q; q; ~ : Rn Rn can be linearly factorized with respect to a: There exists a function PðÞ R-Rkþ such that ~ _ tÞ: _ tÞ ¼ aT Pðq; q; Pða; q; q; Remark. Assumption 5ð2Þ0 is just a special case of Assumption 5(2). Let 8 1 > > ^ q; q; _ tÞJ4^e ; if Jmða; < Jmða; ^ q; q; _ tÞJ ^ q; q; _ tÞ ¼ g^ ða; 1 > > : ^ q; q; _ tÞJr^e ; if Jmða; e^
ð5:1Þ
ð5:2Þ
where e^ 40 is a scalar constant. We now propose the control ~ _ _ _ tðtÞ ¼ p1 ðqðtÞ; qðtÞ; qðtÞ; qðtÞ; tÞ; tÞ þ p2 ðqðtÞ; qðtÞ; tÞ þ p4 ðaðtÞ;
ð5:3Þ
where ~ q; q; ~ q; q; ~ q; q; ~ q; q; _ tÞ ¼ ½M ðq; tÞAT ðq; tÞðAðq; tÞAT ðq; tÞÞ1 P1 ^g ða; _ tÞmða; _ tÞPða; _ tÞ: p4 ða;
ð5:4Þ Here the adaptive parameter vector a~ 2 Rk is governed by the following adaptive law: ~ ~ _ tÞJbðq; q; _ tÞJ k2 a; q; a~_ ¼ k1 Pðq; ð5:5Þ ~ i ¼ 1; . . . ; k, k1;2 2 R, k1;2 40. a~ i ðt0 Þ40 (where a~ i is the i-th component of the vector a), Remark. The adaptive law (5.5) is of leakage type, while the second term on the RHS is the leak. Note that if the initial condition a~ i ðt0 Þ is selected to be strictly positive, then a~ i ðtÞ40 for all tZt0 . This is since the first term on the RHS is always non-negative and the second term alone will render an exponentially decaying (to zero) solution from above.
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Remark. The major difference between (4.10) and (5.2) is that in the former, it is possible that control chattering may occur when eðtÞ becomes very small (note that it converges to 0 exponentially, according to (4.12)); while in the latter this will not happen since e^ is a finite constant. Theorem 3. Let d~ :¼ ½bT ða~ aÞT T 2 Rmþk . Subject to Assumptions 1–4, 5(1), and 5ð2Þ0 , consider the system (2.1). The control (5.3) renders the following performance: ~ 0 ÞJrr, (i) Uniform boundedness: For any r40, there is a dðrÞo1 such that if Jdðt ~ then JdðtÞJrdðrÞ for all tZt0 ; ~ 0 ÞJrr, there exists a d 40 such that (ii) Uniform ultimate bounded: For any r40 with Jdðt ~ JdðtÞJrd for any d 4d as tZt0 þ Tðd ; rÞ, where Tðd ; rÞo1. Proof. Let ~ ¼ bT Pb þ k11 ð1 þ rE Þða~ aÞT ða~ aÞ V ðb; aÞ
ð5:6Þ
be a legitimate Lyapunov function candidate. Again, in the proof, arguments of functions are largely omitted except for a few critical ones. The derivative of V is given by ~_ V_ ¼ 2bT Pb_ þ 2k1 ð1 þ r Þða~ aÞT a: ð5:7Þ 1
E
First, similar to Eq. (4.28), for JmJ4^e , _ 2kJbJ2 þ ð1 þ r Þf2JbJPða; ~ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: 2bT Pbr E
ð5:8Þ
Also, similar to Eq. (4.29), for JmJr^e , _ 2kJbJ2 þ ð1 þ r Þ e^ 2bT Pbr E 2 ~ q; q; _ tÞ þ 2JbJPða; q; q; _ tÞg: þ ð1 þ rE Þf2JbJPða;
ð5:9Þ
By Assumption 5ð2Þ0 , ~ ~ ~ q; q; _ tÞ þ 2JbJaT Pðq; _ tÞ _ tÞ þ 2JbJPða; q; q; _ tÞ ¼ 2JbJa~ T Pðq; q; q; 2JbJPða; ~ ~ T Pðq; _ tÞ: ¼ 2JbJða aÞ q;
ð5:10Þ
Using Eq. (5.10) in Eqs. (5.8) and (5.9), we have that for all JmJ, _ 2kJbJ2 þ ð1 þ r Þ e^ þ 2JbJð1 þ r Þða aÞ ~ ~ T Pðq; _ tÞ: 2bT Pbr q; E E 2
ð5:11Þ
By using the adaptive law (5.5), we have ~ ~ _ tÞJbJ k2 aÞ q; 2k11 ða~ aÞT a~_ ¼ 2k11 ða~ aÞT ðk1 Pðq; T ~ 1 ~ ~ _ tÞJbJ 2k1 ða aÞT k2 a~ ¼ 2ða aÞ Pðq; q; ¼0
zfflfflfflffl}|fflfflfflffl{ ~ ~ _ tÞJbJ 2k11 k2 ða~ aÞT ðaa ¼ 2ða~ aÞ Pðq; þ aÞ q; T ~ T 1 ~ ~ ~ _ ¼ 2ða aÞ Pðq; q; tÞJbJ 2k1 k2 ða aÞ ða aÞ 2k11 k2 ða~ aÞT a T
~ _ tÞJbJ 2k11 k2 Ja~ aJ2 þ 2k11 k2 Ja~ aJJaJ: r2ða~ aÞT Pðq; q; ð5:12Þ
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~ 2¼ Multiplying Eq. (5.12) by 1 þ rE and with Eq. (5.11), Eq. (5.7) becomes (note that JdJ ~ JbJ2 þ Ja~ aJ2 , Ja~ aJrJdJ) e^ V_ r 2kJbJ2 þ ð1 þ rE Þ 2k11 k2 ð1 þ rE ÞJa~ aJ2 þ 2k11 k2 ð1 þ rE ÞJa~ aJJaJ 2 ~ 2 þ k JdJ ~ þk ; r k 1 JdJ ð5:13Þ 2 3 where k 1 ¼ minf2k; 2k11 k2 ð1 þ rE Þg; k 2 ¼ 2k11 k2 ð1 þ rE ÞJaJ; k 3 ¼ ð1 þ rE Þ^e =2: Upon invoking the standard arguments as in Chen [18] and Khalil [17], we conclude the uniform boundedness with 8 > rffiffiffiffiffi > > > g2 > > < g R if rrR; 1 ð5:14Þ dðrÞ ¼ > > rffiffiffiffiffi > > g2 > > : r if r4R; g1 R¼
1 ðk þ 2k 1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 22 þ 4k 1 k 3 Þ;
ð5:15Þ
where g1 ¼ minflmin ðPÞ; k11 ð1 þ rE Þg, g2 ¼ maxflmax ðPÞ; k11 ð1 þ rE Þg. Furthermore, uniform ultimate boundedness also follows with rffiffiffiffiffi g2 R; d¼ g1
Tðd ; rÞ ¼
8 > > >0 > < > > > > :
if rrd
ð5:16Þ
rffiffiffiffiffi g1 ; g2
g2 r2 ðg21 =g2 Þd
ð5:17Þ
2
2
k 1 d ðg1 =g2 Þ k 2 d ðg1 =g2 Þ1=2 k 3
otherwise:
&
6. Illustrative example Consider a two-link R–R mechanical manipulator, whose equation of motion is in the form of Eq. (2.1) with [19, p. 177]
2 l2 m2 þ 2l1 l2 m2 c2 þ l12 ðm1 þ m2 Þ l22 m2 þ l1 l2 m2 c2 M¼ ; ð6:1Þ l22 m2 l22 m2 þ l1 l2 m2 c2 " # 2 m2 l1 l2 s2 y_ 2 2m2 l1 l2 s2 y_ 1 y_ 2 _ tÞq_ ¼ Cðq; q; ; ð6:2Þ 2 m2 l1 l2 s2 y_ 1
m2 l2 Gc12 þ ðm1 þ m2 Þl1 Gc1 ; gðqÞ ¼ m2 l2 Gc12
ð6:3Þ
where q ¼ ½y1 y2 T , t ¼ ½t1 t2 T , l1 (l2 ) is the length of link 1 (2), m1 (m2 ) is the mass of link 1 (2), ci ¼ cosðyi Þ, i ¼ 1; 2; s2 ¼ sinðy2 Þ, c12 ¼ cosðy1 þ y2 Þ, G is the gravitational
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constant. As a simple illustration, it is desired to have the links’ angular velocities to be constrained by y_ 1 þ y_ 2 ¼ 0:
ð6:4Þ
The constraint can be cast into the form of Eqs. (2.3) and (2.8) with A ¼ ½1 1;
c ¼ 0;
b ¼ 0:
ð6:5Þ T
We consider the masses are uncertain parameters (hence s ¼ ½m1 m2 ): m1 ¼ m 1 þ Dm1 ðtÞ, m2 ¼ m 2 þ Dm2 ðtÞ: Here Dm1;2 ðtÞ are unknown, possibly time-varying variations of
Fig. 1. System performance comparison (red: adaptive robust control (4.8); dashed: nominal control). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Adaptive parameter history in (4.13).
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m1;2 , with the possible bounds of the variations unknown. Assumptions 1–4 can be verified. _ and g are either constant, trigonometric in positions, or Note that the terms in M, C q, quadratic in velocities, Assumption 5(1), (2), and ð2Þ0 are met by choosing (note also that the constraint (6.4) is linear in velocities) _ 2 þ a2 JqJ _ þ a3 _ tÞ ¼ a1 JqJ Pða; q; q; 2 3 _ 2 JqJ 6 7 ¼ ½a1 a2 a3 4 JqJ _ 5 1 ~ _ tÞ; q; ¼: a Pðq; T
Fig. 3. Adaptive robust control history in (4.8).
ð6:6Þ
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where a1;2;3 40 are unknown constant parameters. It is interesting to note that we can also meet Assumption 5(1), (2), and ð2Þ0 by an alternative choice of P as follows: _ 2 þ a2 JqJ _ þ a3 raðJqJ _ 2 þ 2JqJ _ þ 1Þ a1 JqJ _ þ 1Þ2 ¼ aðJqJ ~ _ tÞ; ¼: aPðq; q;
ð6:7Þ
where a ¼ maxfa1 ; a2 =2; a3 g. This shows the flexibility of the control design and adaptive laws. The control t can be selected according to Eqs. (4.8) and (5.3). The adaptive laws
Fig. 4. Nominal control history.
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_ ¼ (4.13) and (5.5) are given by, according to Eq. (6.7), respectively (note that JqJ 2 2 1=2 ðy_ þ y_ Þ , b ¼ y_ 1 þ y_ 2 Þ 1
2
2 2 a^_ ¼ Lððy_ 1 þ y_ 2 Þ1=2 þ 1Þ2 Jy_ 1 þ y_ 2 J;
^ 0 Þ40; aðt
2 2 ~ a~_ ¼ k1 ððy_ 1 þ y_ 2 Þ1=2 þ 1Þ2 Jy_ 1 þ y_ 2 J k2 a;
~ 0 Þ40; aðt
ð6:8Þ ð6:9Þ
where L and k1;2 are all scalars. It is interesting to note that the adaptive laws are only velocity (hence not position) dependent.
Fig. 5. System performance comparison (red: adaptive robust control (5.3); dashed: nominal control). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Adaptive parameter history in (5.5).
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Figs. 1–7 show the simulations by using m1 ¼ 1, m2 ¼ 0:5, l1 ¼ 1, l2 ¼ 2, G ¼ 9:8, ^ ¼ 0:2, y1 ð0Þ ¼ 0, y2 ð0Þ ¼ 0, y_ 1 ð0Þ ¼ 0:4, y_ 2 ð0Þ ¼ 0:1, k ¼ L ¼ 1, eð0Þ ¼ 0:1, l ¼ 0:01, að0Þ ~ e^ ¼ 0:1, k1 ¼ 1, k2 ¼ 0:1, að0Þ ¼ 0:2. Fig. 1 shows the constraint time history by using control (4.8). For comparison, the use of nominal control (i.e., (4.8) but with p3 0) is shown. This corresponds to Udwadia–Kalaba-like control for asymptotic convergence. By using (4.8), the constraint approaches to a desirable neighborhood before t ¼ 0:2 (while the ^ nominal control does not result in any finite time settling). The adaptive parameter aðtÞ history is shown in Fig. 2. It increases to a neighborhood close to 1. The adaptive robust control history is shown in Fig. 3. For comparison, we also show the nominal control history in Fig. 4. It is interesting to note that, despite the significant difference in the resulting system performance, the maximum magnitude of the nominal control is almost
Fig. 7. Adaptive robust control history in (5.3).
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twice as much as that of the adaptive robust control. Figs. 5–7 show the corresponding comparison by using (5.3). Fig. 5 shows the system performance with or without the adaptive robust control. Fig. 6 shows the adaptive parameter a~ history. Since there is a leakage, the parameter decreases its value after some time; while in Fig. 2 a^ does not. Fig. 7 shows the control (5.3). 7. Conclusions We consider a mechanical system subject to a class of (possibly nonholonomic) constraints. The system contains uncertainty. The bound of uncertainty is unknown. The objective is to design a control which renders the system to follow the constraint sufficiently close, even in the presence of uncertainty. Two adaptive laws are proposed. The first one is able to render the convergence of b to zero at the expense of possibly incurring chattering. The second one renders the combined state d~ uniform bounded and uniform ultimate bounded, while avoid the chattering phenomena. Acknowledgment The research of Xinrong Zhang is sponsored by the Chinese Scholarship Council during a sabbatical leave at Georgia Tech. References [1] J.G. Papastavridis, Analytical Mechanics, Oxford University Press, New York, NY, 2002. [2] A.M. Bloch, Nonholonomic Mechanics and Control, Springer, New York, NY, 2003. [3] F. Bullo, A.D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer, New York, NY, 2005. [4] Z. Song, D. Zhao, J. Yi, X. Li, Robust motion control for nonholonomic constrained mechanical systems: sliding mode approach, in: American Control Conference, 2005, pp. 2883–2888. [5] C.S. Tseng, B.S. Chen, A mixed H2 =H1 adaptive tracking control for constrained non-holonomic systems, Automatica 39 (2003) 1011–1018. [6] Z.P. Wang, S.S. Ge, T.H. Lee, Robust motion/force control of uncertain holonomic/nonholonomic mechanical systems, IEEEA/SME Transactions on Mechatronics 9 (2004) 118–123. [7] J. Wang, X. Zhu, M. Oya, C.Y. Su, Robust motion tracking control of partially nonholonomic mechanical systems, Automatica 54 (2006) 332–341. [8] F.E. Udwadia, R.E. Kalaba, Analytical Approach: A New Approach, Cambridge University Press, Cambridge, UK, 1996. [9] F.E. Udwadia, R.E. Kalaba, Equations of motion for constrained mechanical systems and the extended D’Alembert’s principle, Quarterly of Applied Mathematics (1997) 321–331. [10] F.E. Udwadia, R.E. Kalaba, What is the general form of the explicit equations of motion for constrained mechanical systems?, ASME Transactions on Journal of Applied Mechanics (2002) 335–339. [11] L.A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965 (reprinted 1979 by Ox Bow Press, Woodbridge, Connecticut). [12] R.M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum, New York, NY, 1977. [13] Y.H. Chen, Second order constraints for equations of motion of constrained systems, IEEE/ASME Transactions on Mechatronics 3 (1998) 240–248. [14] Y.H. Chen, Constraint-following servo control design for mechanical systems, Journal of Vibration and Control (2009) 369–389. [15] Y.H. Chen, G. Leitmann, J.S. Chen, Robust control for rigid serial manipulators: a general setting, in: Proceedings of the 1998 American Control Conference, 1998, pp. 912–916. [16] B. Noble, J.W. Daniel, Applied Linear Algebra, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1977.
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