Two Globally Convergent Adaptive Speed Observers for Mechanical Systems
aa r X i v : . [ m a t h . D S ] J a n Two Globally Convergent Adaptive Speed Observers forMechanical Systems
Jose Guadalupe Romero* a , Romeo Ortega b , a Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier, 161 Rue Ada, 34090 Montpellier, France b Laboratoire des Signaux et Systémes, Supélec, Plateau du Moulon, 91192 Gif-sur-Yvette, France
Abstract
A globally exponentially stable speed observer for mechanical systems was recently reported in the literature, under theassumptions of known (or no) Coulomb friction and no disturbances. In this note we propose and adaptive version of thisobserver, which is robust vis–à–vis constant disturbances. Moreover, we propose a new globally convergent speed observerthat, besides rejecting the disturbances, estimates some unknown friction coefficients for a class of mechanical systems thatcontains several practical examples.
Key words:
Mechanical systems, adaptive observers, robustness.
The design of speed observers for mechanical systems is aproblem of great practical importance that has attractedthe attention of researchers for over years—the readeris referred to the recent books [1,3] for an exhaustive listof references. The first globally exponentially convergentspeed observer for general, simple mechanical systemswas recently reported in [2], where the Immersion andInvariance (I&I) techniques developed in [1] were used.Although the observer of [2] considers the case of sys-tems with non–holonomic constraints, it relies on the as-sumptions of no friction and the absence of disturbances.In [13] this speed observer was redesigned to accommo-date the presence of known Coulomb friction and it wasused to design a uniformly globally exponentially stabletracking controller using only position feedback for me-chanical systems (without non–holonomic constraints).In this paper we propose two new robust velocity ob-servers for mechanical systems (without non–holonomicconstraints). First, we add an adaptation stage to theobserver of [2] to reject constant input disturbances. Sec-ond, for mechanical systems with zero Riemann symbols ⋆ Corresponding author J.G. Romero Tel. +33 4 67 14 9568 .
Email addresses:
[email protected] (Jose Guadalupe Romero* ), [email protected] (Romeo Ortega). (ZRS) [11,14,16], we propose a new adaptive speed ob-server that, besides rejecting the disturbances, estimatessome of the unknown friction coefficients . It should benoted that in this case there are products of unmea-surable states and unknown parameters, a situation forwhich very few results are available in the observer de-sign literature—even for the case of linear systems. Asimilar transformation has been presented in [4,5], wherea coordinate transformation that removes the quadraticterms in velocity is found to solve challenging position–feedback tracking problems for surface ships and mobilerobots.The paper is organized as follows. The two adaptive ob-servation problems addressed in the paper are presentedin Section 2. The standing assumptions and a prelim-inary lemma are given in Section 3. In Section 4 wepresent an adaptive observer for systems with unknownfriction and disturbances. In Section 5 the I&I observerof [13] is redesigned to accommodate the possible pres-ence of disturbances and known friction. Some physicalexamples are given in Section 6. The paper is wrapped–up with some future research in Section 7.
Notation.
To avoid cluttering the notation, throughoutthe paper κ and α are generic positive constants. I n is the n × n identity matrix and n × s is an n × s matrix of zeros, n is an n –dimensional column vector of zeros. Given a i ∈ R , i ∈ ¯ n := { , . . . , n } , we denote with col ( a i ) the n –dimensional column vector with elements a i . For any Preprint submitted to Automatica 22 August 2018 atrix A ∈ R n × n , ( A ) i ∈ R n denotes the i –th column, ( A ) i the i –th row and ( A ) ij the ij –th element. That is,with e i ∈ R n , i ∈ ¯ n , the Euclidean basis vectors, ( A ) i := Ae i , ( A ) i := e ⊤ i A and ( A ) ij := e ⊤ i Ae j . For x ∈ R n , S ∈ R n × n , S = S ⊤ > , we denote the Euclidean norm | x | := x ⊤ x , and the weighted–norm k x k S := x ⊤ Sx .Given a function f : R n → R we define the differentialoperators ∇ f := (cid:18) ∂f∂x (cid:19) ⊤ , ∇ x i f := (cid:18) ∂f∂x i (cid:19) ⊤ , where x i ∈ R p is an element of the vector x . For a map-ping g : R n → R m , its Jacobian matrix is defined as ∇ g := ( ∇ g ) ⊤ ... ( ∇ g m ) ⊤ , where g i : R n → R is the i -th element of g . In the paper we consider n –degrees of freedom, per-turbed , simple, mechanical systems described in port–Hamiltonian (pH) form [15] by " ˙ q ˙ p = " I n − I n − R ∇ H ( q, p )+ " G ( q ) u + " d (1)with total energy function H : R n × R n → R H ( q, p ) = 12 p ⊤ M − ( q ) p + V ( q ) , (2)where q, p ∈ R n are the generalized positions and mo-menta, respectively, u ∈ R m is the control input, G : R n → R n × m is the input matrix, the inertia matrix M : R n → R n × n verifies M ( q ) = M ⊤ ( q ) > and V : R n → R is the potential energy function. As cus-tomary in the observer literature, it is assumed that thecontrol signal u ( t ) is such that trajectories exist for all t ≥ .The system is subject to two different perturbations.- Unknown constant disturbances d = col ( d i ) ∈ R n .- Coulomb friction captured by R = diag { r , r , ., r n } ∈ R n × n , (3)with unknown r i ≥ , i ∈ ¯ n . The problem is to design a globally convergent robustadaptive observer for the momenta p . The main contri-butions of the paper are the following.(i) For systems with ZRS design a new observer that isglobally convergent in spite of the presence of the dis-turbance d and some unknown friction coefficients r i .(ii) If the friction is known robustify the observer of [2]to reject the disturbance d . Remark 2.1
The qualifier “some" in item (i) is essen-tial because, as will become clear later, except for thecase of constant inertia matrix, we will not be able toconsider the presence of unknown friction in all general-ized coordinates.
Remark 2.2
Notice that the objective is to observeonly the momenta (equivalently, the velocity) not toensure consistent estimation of the parameters d and r := col ( r i ) . As is well–known in the identification liter-ature, a necessary condition for parameter convergenceis that the signals satisfy a persistency of excitation con-dition [10]. In this respect, we notice that the system (1),(2), which can be represented in the state space form ˙ q = v ˙ v = U ( q, v ) − R v + G ( q ) u + d ˙ r = 0˙ d = 0 , for some U : R n × R n → R n , with measurement q doesnot satisfy the observability rank condition [7] at zerovelocity, hampering the observation of the parameters r . Remark 2.3
See Remark 1 in [12] for a physical inter-pretation of the disturbances d that, we underscore, en-ter at the level of the momenta. As shown in [16], the change of coordinates ( q, p ) ( q, T ⊤ ( q ) p ) , where T : R n → R n × n is a full rank factorization of theinverse inertia matrix, that is, M − ( q ) = T ( q ) T ⊤ ( q ) , (4) The authors thank Prof. Witold Respondek for this in-sightful remark. " ˙ q ˙ p = " T ( q ) − T ⊤ ( q ) J ( q, p ) − R ( q ) ∇ W ( q, p )+ " T ⊤ ( q ) G ( q ) u + " T ⊤ ( q ) d , (5)with new Hamiltonian W : R n × R n → R W ( q, p ) = 12 | p | + V ( q ) , the jk element of the skew–symmetric matrix J : R n × R n → R n × n given by J jk ( q, p ) = − p ⊤ [( T ) j , ( T ) k ] , (6)with [ · , · ] the standard Lie bracket [14] and the trans-formed friction matrix R ( q ) := T ⊤ ( q ) R T ( q ) ≥ . (7) Remark 3.1
One possible choice of the factorization(4) is the Cholesky factorization [6,8]. But, as will be-come clear below, other choices may prove more suitablefor the solution of the problem.
In this section we solve the problem of robust observationof momenta in the presence of disturbances and someunknown friction coefficients for a class of mechanicalsystems. M ( q ) and a preliminary lemma Assumption 4.1 M − ( q ) admits a factorization (4)with a factor T ( q ) verifying h(cid:16) T ( q ) (cid:17) i , (cid:16) T ( q ) (cid:17) j i = 0 , i, j ∈ ¯ n. (8)Instrumental for the developments of this paper is thefollowing result, whose proof may be found in [16]. Lemma 4.1
The following statements are equivalent:(i) M ( q ) satisfies Assumption 4.1.(ii) The Riemann symbols of M ( q ) are all zero. See equations (6) and (7) of [16] for the definition of thesesymbols. (iii) There exists a mapping Q : R n → R n such that ∇ Q ( q ) = T − ( q ) . (9) Remark 4.1
Mechanical systems verifying Assump-tion 4.1 have been extensively studied in analyticalmechanics and have a deep geometric significance—stemming from Theorem 2.36 in [11]. They belong tothe class of systems that are partially linearizing viachange of coordinates studied in [16]—see that paperfor some additional references.
To design our robust adaptive observer, besides Assump-tion 4.1, a restriction on the friction coefficients is im-posed. Namely, we assume that there are s , with s ≤ n ,unknown coefficient and decompose the friction matrix R (3) as R = R k + R u where R k , R u are n × n diagonal matrices containing the known and the unknown friction coefficients respectively.As a working example consider the case n = 3 and s = 2 with R k = diag { , r , } , R u = diag { r , , r } Similarly, with an obvious definition, we decompose thetransformed friction matrix (7) into R ( q ) = R k ( q ) + R u ( q ) . To streamline the presentation all friction coefficients aregrouped in a vector r = col ( r i ) ∈ R n with the unknownand known coefficients in vectors r u ∈ R s and r k ∈ R n − s ,respectively. Thus, for our working example we have r = col ( r , r , r ) , r k = r , r u = col ( r , r ) . We also define a set of integers κ ⊂ ¯ n that contains theindices of the unknown coefficients of r , which in theexample is κ = { , } .Finally, we define a matrix C ∈ R n × s such that C ⊤ r = r u . (10)Clearly, the matrix C verifies: • rank { C } = s . • For j ∈ κ , ( C ) j = e κ j .In our example C = . Assumption 4.2
The i –th row of factor T ( q ) is inde-pendent of q for i ∈ κ . Lemma 4.2
Under Assumption 4.2, there exists con-stant matrices Y j ∈ R n × s , j ∈ ¯ n, such that, for all vec-tors z = col ( z i ) ∈ R n we have R u ( q ) z = ( n X j =1 Y j z j ) r u . (11) Proof 4.1
From (10) it follows that R u = n X i =1 e i e ⊤ i ( e ⊤ i Cr u ) . (12)Using the definition of R u ( q ) we get R u ( q ) z = T ⊤ ( q ) " n X i =1 e i e ⊤ i ( e ⊤ i Cr u ) T ( q ) z, = n X i =1 T ⊤ ( q ) e i e ⊤ i T ( q ) ze ⊤ i Cr u = n X i =1 n X j =1 T ⊤ ( q ) e i e ⊤ i T ( q ) e j z j e ⊤ i Cr u . (13)Hence, (11) follows swapping the sums and defining Y j := n X i =1 T ⊤ ( q ) e i e ⊤ i T ( q ) e j e ⊤ i C = X L i e j e ⊤ i C, j ∈ ¯ n (14)with matrices L i defined as L i := T ⊤ ( q ) e i e ⊤ i T ( q ) , i ∈ ¯ n (15)It only remains to prove that the matrices Y j and L i areconstant. Towards this end we refer to (14) and noticethat, in view of Assumption 4.2, the term e ⊤ i T ( q ) is con-stant for i ∈ κ while the term e ⊤ i C is an × s zero vectorfor i / ∈ κ . Completing the proof. Remark 4.2
As indicated in Remark 2.1, except forthe case when M (and, consequently, the factor T ) areconstant, to satisfy Assumption 4.2 we have to assumethat some of the elements of r are known. See Section 6for some physical examples. Consequently, the unknown friction coefficients are locatedin these rows.
Proposition 1
Consider the system (1), (2) where theinertia matrix M ( q ) and the friction matrix R verifyAssumptions 4.1 and 4.2. The n + s dimensional I & Iadaptive momenta observer ˙ p I = − T ⊤ ( q )[ ∇ V − G ( q ) u − ˆ d ] − ( n X i =1 Y i ˆ p i )ˆ r u − [ λQ ( q ) + R k ( q )]ˆ p ˙ r u I = ( n X i =1 Y ⊤ i ˆ p i )( ˙ p I + λ ˆ p )˙ d I = T ( q )ˆ p ˆ p = p I + λQ ( q )ˆ r u = r u I + 12 λ ( s X i =1 ˆ p ⊤ L i ˆ p ) e i ˆ d = d I + q ˆ p = T −⊤ ( q )ˆ p with the constant n × n matrices L i given by (15), Q ( q ) given in (9), Y i ∈ R n × s , i ∈ ¯ n given in (14) and λ > afree parameter, ensures boundedness of all signals and lim t →∞ [ˆ p ( t ) − p ( t )] = 0 . (16)for all initial conditions ( q (0) , p (0)) ∈ R n × R n . Proof 4.2
Let the observation and parameter estima-tion errors be defined as ˜ p = ˆ p − p ˜ r u = ˆ r u − r u (17) ˜ d = ˆ d − d. (18)Following the I&I adaptive observer procedure [1] wepropose to generate the estimates as the sum of a pro-portional and an integral term, that is, ˆ p = p I + p P ( q )ˆ r u = r u I + r u P (ˆ p ) (19) ˆ d = d I + d P ( q ) , (20)where the mappings p P : R n → R n r u P : R n → R s d P : R n → R n , and the observer states p I , d I ∈ R n and r u I ∈ R s will bedefined below. The reason for the particular selection of the argumentsof the proportional terms will become clear below. ˜ p and compute ˙˜ p = ˙ p I + ∇ p P T ( q ) p + T ⊤ ( q )[ ∇ V − G ( q ) u ]+ [ R k ( q ) + R u ( q )](ˆ p − ˜ p ) − T ⊤ ( q )( ˆ d − ˜ d ) , where we have invoked Assumption 4.1 that ensures—via (6) and (8)—that J ( q, p ) = 0 , and used (17) to obtainthe terms in the second row. Invoking (11) we can write R u ( q )ˆ p = ( n X i =1 Y i ˆ p i ) r u . Hence, proposing ˙ p I = −∇ p P T ( q )ˆ p − T ⊤ ( q )[ ∇ V − G ( q ) u ] − (cid:16) n X i =1 Y i ˆ p i (cid:17) ˆ r u − R k ( q )ˆ p + T ⊤ ( q ) ˆ d, (21)yields ˙˜ p = − [ R ( q ) + ∇ q p P T ( q )]˜ p − ( n X i =1 Y i ˆ p i )˜ r u + T ⊤ ( q ) ˜ d = − [ R ( q ) + λI n ]˜ p − ( n X i =1 Y i ˆ p i )˜ r u + T ⊤ ( q ) ˜ d, (22)where to obtain the second equations we have selected p P ( q ) = λQ ( q ) , with Q ( q ) given in (9).Now, the time derivative of ˜ r u is given as ˙˜ r u = ˙ r u I + ∇ r u P ˙ˆ p = ˙ r u I + ∇ r u P [ ˙ p I + ∇ p P T ( q ) p ]= ˙ r u I + ∇ r u P [ ˙ p I + λ (ˆ p − ˜ p )] . Hence, choosing ˙ r u I = −∇ r u P ( ˙ p I + λ ˆ p ) , yields ˙˜ r u = − λ ∇ r u P ˜ p. (23)Finally, the time derivative of ˜ d is given as ˙˜ d = ˙ d I + ∇ d P T ( q ) p = ˙ d I + ∇ d P T ( q )(ˆ p − ˜ p ) . Hence, choosing ˙ d I = −∇ d P T ( q )ˆ p, yields ˙˜ d = −∇ d P T ( q )˜ p. (24)We will now analyze the stability of the error model (22), (23) and (24) with the aid of the proper Lyapunovfunction candidate V (˜ p, ˜ d, ˜ r u ) = 12 ( | ˜ p | + | ˜ d | + | ˜ r u | ) . (25)Taking its time-derivative we obtain ˙ V = − ˜ p ⊤ h R ( q ) + λI n i ˜ p − ˜ p ⊤ (cid:2) ( n X i =1 Y i ˆ p i )˜ r u − T ⊤ ( q ) ˜ d (cid:3) − (cid:2) λ ˜ r ⊤ u ∇ r u P + ˜ d ⊤ ∇ d P T ( q ) (cid:3) ˜ p. (26)Clearly, if the mappings r u P (ˆ p ) and d P ( q ) solve the par-tial differential equations (PDEs) ∇ r u P = − λ ( n X i =1 Y ⊤ i ˆ p i ) ∇ d P = I n , (27)one gets ˙ V = − ˜ p ⊤ [ R ( q ) + λI n ]˜ p ≤ − λ | ˜ p | . (28)From (25), (28) we conclude that ˜ p ∈ L ∩ L ∞ and ˜ d, ˜ r u ∈ L ∞ . Doing some standard signal chasing it isstraightforward to prove from here that (16) holds.Motivated by the conclusion above let us now study thePDEs (27). The second one has the trivial solution d P = q . Regarding the first one, it is clear that the elementsof the mapping r u P (ˆ p ) must be of the quadratic form ( r u P (ˆ p )) i = 12 λ ˆ p ⊤ L i ˆ p, i ∈ ¯ s, with constant, symmetric matrices L i ∈ R n × n . Replac-ing the expression above in the PDE (27) yields ˆ p ⊤ L ... ˆ p ⊤ L s = − n X i =1 Y ⊤ i ˆ p i . That lead us to the solution h L ⊤ e j . . . L ⊤ s e j i = − Y j , j ∈ ¯ n. (29)It only remains to show that the resulting matrices L i − L ⊤ j = − h L ⊤ j e L ⊤ j e . . . L ⊤ j e n i = h Y e j Y e j . . . Y n e j i Clearly, the matrix L j is symmetric if and only if e ⊤ i Y k = e ⊤ k Y i , ∀ i, k ∈ ¯ n, i = k. This fact can be easily verified using (14) e ⊤ k Y i = e ⊤ k n X j =1 T ⊤ ( q ) e j e ⊤ j T ( q ) e i e ⊤ j C = e ⊤ i n X j =1 T ⊤ ( q ) e j e ⊤ j T ( q ) e k e ⊤ j C = e ⊤ i Y k . Replacing all the derivations above in ˙ p I , ˙ d I and ˙ r u I gives the equations given in the proposition completingthe proof. Remark 4.3
If Assumption 4.1 is not imposed a term J ( q, p ) p appears in the error equation (22) and (23).Even though this term is quadratic in the unknown state p , the properties of J ( q, p ) can be used to handle thisterm in the first error equations—this is done in thesecond observer in the next section. However, there is noobvious way to create a suitable error term for the seconderror equation styming the relaxation of Assumption 4.1. Remark 4.4
If the matrices Y i are not constant thefirst PDE in (27) does not admit a solution, hence As-sumption 4.2 is required. It can be shown that making r u P function of q does not solve the problem, because aquadratic function of ˆ p will appear in ˙ V . Remark 4.5
As shown in Proposition 6 of [16] the dy-namics of mechanical systems satisfying Assumption 4.1expressed in the coordinates ( Q, p ) take the form ˙ Q = p ˙ p = − ˜ R ( Q ) p − ˜ T ⊤ ( Q )[ ∇ ˜ V ( Q ) − ˜ G ( Q ) u + d ] , (30)where ˜( · )( Q ) := ( · )( Q I ( Q )) , with Q I : R n → R n a leftinverse of Q ( q ) , that is, Q ( Q I ( z )) = z for all z ∈ R n .Although the construction of an observer for (30) whenthe friction is known is straightforward, the case of un-known friction is far from trivial. Applying the I & I pro-cedure used in Proposition 1 leads, for the definition of d P ( q ) , to a PDE of the form ∇ S ( Q ) = ˜ T ( Q ) , whose solution is not obvious. In this section we redesign the I&I speed observer of [13],see also [2], to ensure its global convergence in spite ofthe presence of the unknown disturbances d and known friction forces in all coordinates. Proposition 2
Consider the system (1), (2) with known friction matrix R . There exist smooth mappings A : R n × R ≥ × R n × R n → R n +1 B : R n × R ≥ × R n → R n such that the interconnection of (1), (2) with ˙X = A (X , q, u )ˆ p = B (X , q ) , (31)where X ∈ R n × R ≥ , ˆ p ∈ R n , ensures (16) holds forall initial conditions ( q (0) , p (0) , X(0)) ∈ R n × R n × R n × R ≥ . This implies that, in spite of the presence of the unknowndisturbances d , (31) is a globally convergent momentaobserver for the mechanical system with friction (1), (2). Proof 5.1
The construction of the observer follows veryclosely the one reported in [13] with the only differenceof the inclusion of an adaptation law for the unknowndisturbance parameters d . However, for the sake of com-pleteness, a detailed derivation of all the steps is given.Define the estimation errors ˜ p = ˆ p − p ˜ d = ˆ d − d. (32)Following the I&I adaptive observer procedure [1] wepropose to generate the estimates as ˆ p := p I + p P ( q, q| , |p )ˆ d := d I + d P ( q, r ) (33)where the mappings p P : R n × R n × R n → R n and d P ∈ R n , and the observer states d I ∈ R n , p I ∈ R n and r ∈ R are defined such that (16) holds.6e, therefore, study the dynamic behavior of ˜ p and com-pute ˙˜ p = ˙ p I + ∇ q p P ˙ q + ∇ q| p P ˙ q| + ∇ |p p P ˙ |p −− J ( q, p ) p + T ⊤ ( q )[ ∇ V − G ( q ) u ] + R ( q ) p − T ⊤ ( q ) d. In [2] it has been shown that the mapping J ( q, p ) definedin (6) verifies the following properties:(P.i) J ( q, p ) is linear in the second argument, that is J ( q, α p + α ¯ p ) = α J ( q, p ) + α J ( q, ¯ p ) for all q , p , ¯ p ∈ R n , and α , α ∈ R .(P.ii) There exists a mapping ¯ J : R n × R n → R n × n satisfy-ing J ( q, p )¯ p = ¯ J ( q, ¯ p ) p. Hence, proposing ˙ p I := −∇ q| p P ˙ q| − ∇ |p p P ˙ |p + J ( q, ˆ p )ˆ p − R ( q )ˆ p −− T ( q ) ⊤ ( q ) ∇ V + v − ∇ q p P T ( q )ˆ p + T ⊤ ( q ) ˆ d, (34)together with Properties (P.i) and (P.ii) yields ˙˜ p = [ J ( q, p ) + ¯ J ( q, ˆ p ) − R ( q ) − ∇ q p P T ( q )]˜ p + T ⊤ ( q ) ˜ d. (35)It is clear that if the mapping p P solves the PDE ∇ q p P = [ ψI n + ¯ J ( q, ˆ p )] T − ( q ) , with ψ > a design constant, the ˜ p –dynamics reduces to ˙˜ p = [ J ( q, p ) − ψI n − R ( q )]˜ p + T ⊤ ( q ) ˜ d. Recalling that J ( q, p ) is skew–symmetric and R ( q ) ≥ the unperturbed part of the error dynamics above, i.e. when ˜ d = 0 , is exponentially stable.Similarly to [13], to avoid the solution of the PDE, thedynamic scaling technique is used. Towards this end,define the mapping H ( q, ˆ p ) := [ ψI n + ¯ J ( q, ˆ p )] T − ( q ) . (36)and define p P as p P ( q, q| , |p ) := H ( q| , |p ) q. (37)The choice above yields ∇ q p P = H ( q| , |p ) , which may bewritten as ∇ q p P = H ( q, ˆ p ) − [ H ( q, ˆ p ) − H ( q| , |p )] . (38) Now, since the term in brackets in (38) is equal to zeroif |p = ˆ p and q| = q , there exist mappings ∆ q , ∆ p : R n × R n × R n → R n × n verifying ∆ q ( q, |p ,
0) = 0 , ∆ p ( q, |p ,
0) = 0 , (39)and such that H ( q, ˆ p ) − H ( q| , |p ) = ∆ q ( q, q| , e q ) + ∆ p ( q, |p , e p ) , (40)where e q := q| − q, e p := |p − ˆ p. (41)Substituting (36), (38) and (40) in (35), yields ˙˜ p = [ J ( q, p ) − ψI n − R ]˜ p + (cid:16) ∆ q ( q, q| , e q ) + ∆ p ( q, |p , e p ) (cid:17) T ( q )˜ p + T ⊤ ( q ) ˜ d. The mappings ∆ q , ∆ p play the role of disturbances thatare dominated with a dynamic scaling and a properchoice of the observer dynamics. For, define the dynam-ically scaled off–the–manifold coordinate η = 1 r ˜ p, (42)where r is a scaling factor to be defined. The dynamicbehavior of η is given by ˙ η = ( J − R − ψI ) η + (∆ q + ∆ p ) T η + 1 r T ⊤ ˜ d − ˙ rr η, (43)where, for brevity, the arguments of the mappings areomitted.Mimicking [2] select the dynamics of q| , |p as ˙ q| = T ( q )ˆ p − ψ e q ˙ |p = − T ⊤ ( q ) ∇ V + v + J ( q, ˆ p )ˆ p − R ˆ p − ψ e p + T ⊤ ( q ) ˆ d (44)where ψ , ψ are some positive functions of the statedefined later. Using (44), together with (41), we get ˙ e q = T ( q ) ηr − ψ e q ˙ e p = ∇ q p P T ( q ) ηr − ψ e p . (45)Moreover, select the dynamics of r as ˙ r = − ψ r − rψ ( k ∆ p T k + k ∆ q T k ) , r (0) ≥ , (46)7ith k · k the matrix induced –norm. At this point wemake the important observation that the set { r ∈ R : r ≥ } is invariant for the dynamics (46). Hence, r ( t ) ≥ , ∀ t ≥ .On other hand, taking the time-derivative of ˜ d , we get ˙˜ d = ˙ d I + ∇ q d P T ( q ) p + ∇ r d P ˙ r = ˙ d I + ∇ q d P ( q, r ) T ( q )(ˆ p − rη ) + ∇ r d P ˙ r and choosing ˙ d I = −∇ q d P T ( q )ˆ p − ∇ r d P ˙ r, (47)the ˜ d –dynamics take the form ˙˜ d = − r ∇ q d P T ( q ) η (48)We now analyze the error system (42), (45), (46), (48)—with the coordinate ˜ r = ( r − . For, define the properLyapunov function candidate. V ( η, e q , e p , ˜ r, z a ) := 12 (cid:16) | η | + | e q | + | e p | + ˜ r + | ˜ d | (cid:17) . (49)Taking its time-derivative we obtain ˙ V ≤ − (cid:18) ψ − (cid:19) | η | − (cid:18) ψ − r k T ( q ) k (cid:19) | e q | − (cid:18) ψ − r k∇ q p P k k T ( q ) k (cid:19) | e p | + ˜ r ˙ r ++ ˜ d ⊤ (cid:18) r − r ∇ q d P (cid:19) T ( q ) η Clearly, if we set ψ = 4(1 + ψ ) , ψ = 12 r k T ( q ) k + ψ (50)and ψ = 12 r k∇ q p P k k T ( q ) k + ψ , where ψ , ψ , ψ are positive functions of the state de-fined below, one gets ˙ V ≤ − ψ | η | − ψ | e q | − ψ | e p | + ˜ r ˙ r + ˜ d ⊤ (cid:18) r − r ∇ q d P ( q, r ) (cid:19) T ( q ) η. To eliminate the cross term appearing in the last termof the right hand side above we select d P ( q, r ) = 1 r q, (51) which clearly solves the PDE r − r ∇ q d P ( q, r ) = 0 . It only remains to study the term ˜ r ˙ r , which is given by ˜ r ˙ r = − ψ r + ˜ r rψ ( k ∆ p T k + k ∆ q T k ) . Now, (39) ensures the existence of mappings ¯∆ q , ¯∆ p : R n × R n × R n → R n × n such that k ∆ q ( q, |p , e q ) k ≤ k ¯∆ q ( q, |p , e q ) k | e q |k ∆ p ( q, |p , e p ) k ≤ k ¯∆ p ( q, |p , e p ) k | e p | . Hence k ∆ p T k + k ∆ q T k ≤ k T k | ( k ¯∆ p k | e p | + | ¯∆ q k | e q | ) . Setting ψ = κψ = r ˜ r ψ ) k T k k ¯∆ q k + κψ = r ˜ r ψ ) k T k k ¯∆ p k + κ, one gets ˙ V ≤ − κ ( | η | + | e q | + | e p | + ˜ r ] (52)for some positive constant κ .The proof is completed invoking the arguments of [13],selecting the observer state as X := ( q| , |p , p I , d I , r − , and defining A (X , q, u ) from (34), (44), (46) and (47)and setting B (X , q ) via (33). Remark 5.1
It is clear from the proof that the key stepto reject the disturbances is to make the proportionalterm of the parameter estimator, d P a function of thedynamic scaling factor r , see (51). In this section we present three physical mechanical sys-tems that satisfy the conditions of Proposition 1. Con-sequently, robust adaptive speed observation is possiblefor them.8 .1 Constant inertia matrix
In the case of constant inertia matrix Assumption 4.1is trivially satisfied, because the factor T can be takento be constant. Assumption 4.2 is also satisfied with dim( q ) = n and C = I n , hence all friction coefficientscan be identified.Given any constant factor T , the vector field Q ( q ) thatsolves (9) is given by Q ( q ) = T − q. Finally, from (14) and (15) we get Y j = n X i =1 T ⊤ ( q ) e i e ⊤ i T ( q ) e j e ⊤ i j ∈ ¯ nL i = [( T ) i ] ⊤ ( T ) i , i ∈ ¯ n. This is a 4-dof underactuated mechanical system de-picted in 1. The inverse inertia matrix is given by mM q1q2 q3q4 L
Fig. 1. Planar redundant manipulator with one elastic degreeof freedom. M − ( q ) = I − I ∗ a + I − mℓ S
12 1 mℓ C ∗ ∗ a ∗ ∗ ∗ a , where we defined S := sin( q + q ) , C := cos( q + q ) a := √ M mℓ √ m + M , a := √ M + m, and the definition of all constants may be found in [16].The Cholesky factorization is given as T ( q ) = √ I − √ I a − q M m a S
12 1 a q M m a C a , and the vector field Q ( q ) that solves (9) is Q ( q ) = √ Iq a ( q + q ) − a q M m C + a q − a q M m S + a q . A matrix C that satisfies Assumption 4.2 is C = " I × . Hence, we can consider as unknown the frictions in theelastic coordinate r and the revolute joint r .Finally, Y ⊤ = I I − a √ I Y ⊤ = − a √ I a L = " I
00 0 × × × L = I − a √ I − a √ I a × × × This is a 3-dof underactuated mechanical system de-picted in Fig. 2. The inertia matrix is M ( q ) = m r + m mL C ∗ m r + m mL S ∗ ∗ mL , y F x q L m r m ( x r , x y ) XY Fig. 2. 2D-Spider crane gantry cart with inverse M − ( q ) = m r + mC ( m r + m ) m r mC S ( m r + m ) m r − C L m r ∗ m r + m − mC ( m r + m ) m r − S m r L ∗ ∗ m r + mm r L m where, to simplify the notation, we have defined S := sin( q ) , C := cos( q ) , and the definition of all constants may be found in [9].An upper triangular factorization of M − ( q ) is given as T ( q ) = a − bC a − bS c , (53)where we defined the constants a := 1 √ m r + m , b := 1 cL m r , c := s m r + mmL m r . We can check that the columns of T ( q ) satisfy (8) andthus the system verifies Assumption 1. Taking the in-verse of T ( q ) we get T − ( q ) = a aL mC a aL mS c , From the equation above it is clear that a mapping Q ( q ) that solves (9) is Q ( q ) = a q + aL mS a q − aL mC c q . From the definition of T ( q ) in (53) it is clear that C ⊤ = h i satisfies Assumption 4.2. Hence, we can consider as un-known the friction parameter r .Finally, from (14) and (15) we get Y ⊤ = h c i ,L = c . We show several simulations of the proposed observer.The system has the two forces shown in Fig. 2 as con-trol inputs, that is, u = col ( F x , F y ) and constant inputmatrix G of the form G = . The parameters are taken as m r = 0 . kg for ring mass, m = 1 kg for payload mass and L = 0 . m for the cablelength. We fix the control inputs as F x = 1 .
535 cos( t ) and F y = 7 .
67 sin( t ) . The disturbances are taken as d = col (0 . , . , . and the friction coefficients r = col (0 , , . , with r being unknown.The transient behavior of the error signals ˜ p, ˜ r and ˜ d with the tuning parameter λ = 0 . and different initialconditions of d I , r I and P I are shown in Fig.3 and Fig.4.As seen from the figures, besides the convergence to zeroof the momenta estimate predicted by the theory, we alsoobserve that the estimated parameters converge to theirtrue value—assessing the fact that the signals chosen forthe simulation are persistently exciting.To evaluate the effect of the tuning parameter λ on thetransient behavior we also show in Fig.5 and Fig.6 thetransient behavior of the error signals for different valuesof λ .Finally to illustrate the robustness of the adaptive ob-server, we carried out a simulation considering that theinput disturbance d is subject to step changes. The tra-jectories of d ( t ) and its estimation, depicted in Fig. 7,clearly illustrate the tracking capability of the proposedobserver. Remark 6.1
It is interesting to note that the standard(lower triangular) Cholesky factorization of M − ( q ) does not satisfy (8).10 ˜ p ˜ p time (sec) ˜ p Fig. 3. Transient behavior of ˜ p for λ = 0 . and differentinitial conditions of d I , r I and P I ˜ r ˜ d ˜ d time (sec) ˜ d Fig. 4. Transient behavior of ˜ d and ˜ r for λ = 0 . and differ-ent initial conditions of d I , r I and P I ˜ p ˜ p time (sec) ˜ p l =0.3 l =0.9 l =2 Fig. 5. Transient behavior of ˜ p for different values of λ The design of the observer in Proposition 1 requires the explicit solution of the PDE (9). This requirement re- ˜ r ˜ d ˜ d time (sec) ˜ d l =0.3 l =0.9 l =2 Fig. 6. Transient behavior of ˜ d and ˜ r for different values of λ time (sec) d v s ˆ d Fig. 7. Transient behavior of d and ˆ d with λ = 2 stricts the practical applicability of the approach. In-deed, this PDE has no free parameters and its explicitsolution may be even impossible. As indicated in [16]this is the case of the classical cart–pendulum example.Current research is under way to extend the realm of ap-plication of the observer in Proposition 1. In particular,it is possible to consider the following generalization ofthe Lyapunov function candidate (25) | ˜ p | + ˜ r ⊤ u P − ˜ r u + | ˜ d | , with P > a constant matrix. It is straightforward toshow that the second PDE in (27)—that cancels thecross term appearing in the derivative of the new Lya-punov function—becomes ∇ r u P = − λ ( n X i =1 P Y ⊤ i ˆ p i ) , where we underscore the presence of the matrix P infront of Y i . There are inertia matrices where the corre-sponding Y i are not constant but there exists positivedefinite P that will make P Y ⊤ i constant—hence relaxingAssumption 4.2. We are currently investigating whether11here exist physical systems for which such propertyholds.Another, quite challenging, task is the extension ofProposition 1 to systems that do not have ZRS. Onepossibility is to look into the next class of systems par-tially linearizable via coordinate changes characterizedin [16]. Acknowledgements
This work was supported by the Ministry of Edu-cation and Science of Russian Federation (Project14.Z50.31.0031).
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