A Globally Exponentially Convergent Immersion

and Invariance Speed Observer for n Degrees of

Freedom Mechanical Systems with constraints

Alessandro Astolﬁ

, Romeo Ortega

and Aneesh Venkatraman

Dept. of Electrical and Electronic Engineering, Imperial College London

London, SW7 2AZ, U.K.

and DISP, University of Roma ”Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy

Laboratoire des Signaux et Syst`emes, CNRS–SUPELEC, 91192, Gif–sur–Yvette, France

Institute of Mathematics and Computing Science, University of Groningen

PO Box 407, 9700 AK, Groningen, The Netherlands

Abstract. The problem of velocity estimation for mechanical systems is of great

practical interest. Although many partial solutions have been reported in the liter-

ature the basic question of whether it is possible to design a globally convergent

speed observer for general n degrees of freedom mechanical systems remains

open. In this paper an afﬁrmative answer to the question is given by proving the

existence of a 3n + 1–dimensional globally exponentially convergent speed ob-

server. Instrumental for the construction of the speed observer is the use of the

Immersion and Invariance technique, in which the observer design problem is re-

cast as a problem of rendering attractive and invariant a manifold deﬁned in the

extended state–space of the plant and the observer.

Notation. For general mappings S : R

× R

→ R

, (x, ζ) 7→ S we deﬁne

∇

S(x, ζ) :=

∂S(x,ζ)

∂x

and ∇

S(x, ζ) :=

∂S(x,ζ)

∂ζ

. For brevity, when clear from

the context, the subindex of ∇ and, in general, the arguments of all the functions

are omitted.

1 Problem Formulation

We consider general n degree of freedom mechanical systems with nonholonomic con-

straints described in Lagrangian form by [11], [13],

M(q)¨q + C(q, ˙q) ˙q + ∇U (q) = G(q)u + A(q)λ, (1)

⊤

(q) ˙q = 0, (2)

where q(t), ˙q(t) ∈ R

are the generalized positions and velocities, respectively, u(t) ∈

is the control input, A(q)λ are the constraint forces with A : R

→ R

n×k

, λ ∈ R

G : R

→ R

n×m

is the input matrix, M : R

→ R

n×n

is the mass matrix with

Astilﬁ A., Ortega R. and Venkatraman A. (2009).

A Globally Exponentially Convergent Immersion and Invariance Speed Observer for n–Degrees of Freedom Mechanical Systems.

In Proceedings of the International Workshop on Networked embedded and control system technologies: European and Russian R&D cooperation,

pages 134-145

 SciTePress

M = M

⊤

> 0 and U : R

→ R is the potential energy function. C(q, ˙q) ˙q is the

vector of Coriolis and centrifugal forces, with the (ik)–th element of the matrix C :

× R

→ R

n×n

deﬁned by

(q, ˙q) =

j=1

ijk

(q) ˙q

where C

ijk

: R

→ R are the Christoffel symbols of the ﬁrst kind deﬁned as

ijk

(q) :=



∂M

∂q

∂M

∂q

−

∂M

∂q



. (3)

We consider q(t) to be measurable and assume that the input u(t) is such that q(t), ˙q(t)

exist for all time, that is, the system is forward complete. Our objective is to design a

globally asymptotically convergent observer for ˙q(t).

Speed observation is a longstanding problem whose complete theoretical solution

has proven highly elusive. The ﬁrst results were reported in 1990 in the fundamental

paper [14], and many interesting partial solutions have been reported afterwards. Par-

ticular attention has been given to the case in which the system (1) can be rendered

linear in the unmeasurable velocities via partial changes of coordinates, see, e.g., [6,

16]. An intrinsic local observer, exploiting the Riemannian structure of the system, has

been recently proposed in [1] (see also [2] for a Lyapunov analysis and [7] for a gener-

alization). A solution for a class of two degrees of freedom systems has been recently

reported in [8]. The reader is referred to the recent books [5,?,?] for an exhaustive list

of references.

A complete solution to the problem is given by the proposition below. As will be-

come clear in the proof, the construction of the observer relies on the use of the Im-

mersion and Invariance (I&I ) technique—ﬁrst reported in [4] and further developed in

[3,10]. In I&I the observer design is recast as a problem of rendering attractive a suit-

ably selected invariant manifold deﬁned in the extended state–space of the plant and

the observer. It should be mentioned that the observers in [8, 16] are also based on the

I&I approach.

2 Main Result

Proposition 1. Consider the system (1), and assume u is such that trajectories exist for

all t ≥ 0 . There exist smooth mappings A : R

3n−2k+1

× R

→ R

3n−2k+1

and

B : R

→ R

(n−k)×(3n−2k+1)

such that the dynamical system

˙χ = A(χ, q, u) (4)

with state χ(t) ∈ R

3n−2k+1

, inputs q(t) and u(t), and output

η = B(q)χ, (5)

has the following property.

All trajectories of the interconnected system (1), (2), (4) are such that

lim

t→∞

αt

[N (q) ˙q(t) − η(t)] = 0, (6)

for some α > 0 and for all initial conditions (q(0), ˙q(0), χ(0)) ∈ R

×R

3n−2k+1

where N (q) : R

→ R

n−k

× R

is a left invertible matrix. That is, (4), (5) is a globally

exponentially convergent speed observer for the mechanical system (1)-(2).

Remark 1. For the special case of a mechanical system with no nonholonomic con-

straints, it is clear that k = 0 and subsequently the matrix N (q) becomes an invertible

square matrix.

3 A Preliminary Lemma

Before giving the proof of the main result, we recall that the system (1)-(2) can be

written in the port-Hamiltonian form [13] as



˙q

˙p





0 I

−I 0



∇

H(q, p)

∇

H(q, p)





G(q)



u +



A(q)



λ, (7)

⊤

(q)λ = 0, (8)

where p = M (q) ˙q are the generalized momenta and

H(q, p) =

⊤

−1

(q)p + U (q)

represents the total energy stored in the system. Further, as per [13], the system (7)-(8)

when restricted to the constrained space

= {(q, ˙q)|A

⊤

(q) ˙q = 0},

takes the form



˙q

˜p





S(q)

−

⊤

(q) J(q, ˜p)



∇

H(q, ˜p)

∇

˜p

H(q, ˜p)





(q)



u +



A(q)



λ, (9)

H(q, ˜p) = V (q) +

˜p

⊤

−1

(˜q)˜p, (10)

with ˜p ∈ R

n−k

being given as ˜p =

⊤

(q)p where

S(q) ∈ R

n×n−k

is the full rank

annihilator of the matrix A(q) satisfying the condition A

⊤

(q)

S(q) = 0. The matrix

J(q, ˜p) is skew-symmetric and is given by

(q, ˜p) = −p

⊤

[

], (11)

where [

] denotes the standard Lie bracket of the column vectors S

, S

and the

matrix

M(q) ∈ R

n−k×n−k

is symmetric positive-deﬁnite.

In order to streamline the presentation in this section, we introduce a factorization

of the mass matrix

−1

(q) = T

⊤

(q)T (q), (12)

where T : R

→ R

n−k×n−k

is a full rank matrix

and deﬁne the mappings L : R

→

n×n−k

and F : R

× R

→ R

n−k

L(q) =

S(q)T

⊤

(q), (13)

F (q, u) = L

⊤

(q)[B

(q)u − ∇U(q)]. (14)

Notice that, since q and u are measurable, these mappings are known. We next state the

following proposition.

Lemma 1. The system dynamics (9)-(10) when expressed in the newcoordinates(y, x) =

(q, T (q)˜p), admits a state space representation of the form

˙y = L(y)x, (15)

˙x = S(y, x)x + F (y, u), (16)

where

S = T JT

⊤

i=1

({

∂T

∂y

−1

x}{L

⊤

}

⊤

− {L

⊤

}{

∂T

∂y

−1

⊤

), (17)

and e

is the i

basis vector of R

n−k

Proof. We directly obtain (15) by differentiating y and by using (9), (10), (13). We next

compute the following,

˙x =

T ˜p + T

˜p, (18)

T ˜p − T

⊤

∂

∂y

(

˜p

⊤

−1

(˜q)˜p) − T

⊤

∇U + T

⊤

u + T JT

⊤

x, (19)

T ˜p − L

⊤

∂

∂y

(

˜p

⊤

−1

(˜q)˜p) + F + T JT

⊤

x, (20)

where we have made use of (10), (13) and (14). We now compute that,

T ˜p =

i=1

(

∂T

∂y

˜p)(e

⊤

−1

˜p),

i=1

(

∂T

∂y

˜p)(e

⊤

L)x, (21)

and further obtain

∂

∂y

{

˜p

⊤

−1

˜p} =

∂

∂y

{

˜p

⊤

T p}

i=1

{

∂T

∂y

˜p}

⊤

x. (22)

Since M is positive deﬁnite this factorization always exists. It may be taken to be the (univo-

cally deﬁned) Cholesky factorization, as proposed in [9].

Substituting (21) and (22) in (20) we obtain the dynamics of x as,

˙x =

i=1

({

∂T

∂y

−1

x}{L

⊤

}

⊤

− {L

⊤

}{

∂T

∂y

−1

⊤

)x + T JT

⊤

x + F,

= Sx + F, (23)

where we have used (17) to obtain the equation (23). This concludes the proof.

Remark 2. It can be veriﬁed easily that matrix S(y, x) deﬁned in (17) satisﬁes the fol-

lowing properties:

(i) S is skew–symmetric, that is,

S + S

⊤

= 0.

(ii) S is linear in the second argument, that is,

S(y, a

x + a

¯x) = a

S(y, x) + a

S(y, ¯x),

for all y, x, ¯x ∈ R

, and a

, a

∈ R.

(iii) There exists a mapping

S : R

× R

→ R

n×n

such that

S(y, x)¯x =

S(y, ¯x)x,

for all y, x, ¯x ∈ R

Remark 3. Lemma 1 implies that the speed observer problem for system (1)-(2) can be

recast as an observer problem for system (15)-(16) with output y.

Remark 4. For the special case of no nonholonomic constraints, we have k = 0 and

J(q, ˜p) = 0,

S(q) = I,

M(q) = M(q), L(q) = T

⊤

(q).

This subsequently simpliﬁes the expression for S(y, x) as

S(y, x) =

i=1

({

∂T

∂y

−1

x}{T e

}

⊤

− {T e

}{

∂T

∂y

−1

⊤

). (24)

It can be shown (refer to [16]) that the (jk)–th element of S is S

= −{T

−1

⊤

, T

4 Proof of the Main Result

The observer is constructed in four steps.

(S1) Following the I&I procedure [3], we deﬁne a manifold (in the extended state-space

of the plant and the observer) that should be rendered attractive and invariant

. As

is well–known, to achieve the latter objective a partial differential equation (PDE)

should, in principle, be solved.

The manifold should be such that the unmeasurable part of the state can be reconstructed from

the function that deﬁnes the manifold.

(S2) To avoid the need to solve the PDE the “approximation” technique proposed in

[10] is adopted. Using this approximation induces some errors in the observer error

dynamics.

(S3) Always borrowing from [10], we introduce a dynamic scaling that dominates—

in a Lyapunov–like analysis—the effect of the aforementioned disturbance terms,

proving that the scaled observer error converges to zero.

(S4) To prove that the dynamic scaling factor is bounded and, consequently, that the ac-

tual observer error converges,exponentially, to zero, high gain terms are introduced

in the observer dynamics to, again, dominate sign–indeﬁnite terms in a Lyapunov–

like analysis.

Step 1. (Deﬁnition of the manifold) For the system (15)-(16), we propose the manifold

M := {(y, x, ξ, ˆy, ˆx) : ξ − x + β(y, ˆy, ˆx) = 0} ⊂ R

5n−3k

, (25)

where ξ ∈ R

n−k

, ˆy ∈ R

n−k

, ˆx ∈ R

are (part of) the observer state, the dynamics of

which are deﬁned below, and the mapping β : R

3n−2k

→ R

n−k

is also to be deﬁned.

To prove that the manifold M is attractive and invariant it is shown that the off–

the–manifold coordinate

z = ξ − x + β(y, ˆy, ˆx), (26)

the norm of which determines the distance of the state to the manifold M, is such that:

(C1) z(0) = 0 ⇒ z(t) = 0, for all t ≥ 0 (invariance);

(C2) z(t) asymptotically (exponentially) converges to zero (attractivity).

Notice that, if z(t) → 0, an asymptotic estimate of x is given by ξ + β.

To obtain the dynamics of z differentiate (26), yielding

˙z =

ξ − ˙x +

ξ − S(y, x)x − F + ∇

β ˙y + ∇

ˆy

ˆy + ∇

ˆx

ˆx.

Let

ξ = F − ∇

ˆy

ˆy − ∇

ˆx

ˆx + S(y, ξ + β)(ξ + β) − ∇

βL(y)(ξ + β), (27)

where

ˆy and

ˆx are to be deﬁned. Replacing (27) in the equation of ˙z above, and invoking

properties (ii) and (iii) of Lemma 1, yields

˙z = −S(y, ξ + β − z)(ξ + β − z) +

+S(y, ξ + β)(ξ + β) − ∇

βL(y)z

= S(y, x)z + S(y, z)(ξ + β) − ∇

βL(y)z

= S(y, x)z +

S(y, ξ + β)z − ∇

βL(y)z. (28)

From (28) it is clear that condition (C1) above is satisﬁed. On the other hand, condition

(C2) would be satisﬁed if we could ﬁnd a function β that solves the PDE

∇

β = [k

I +

S(y, ξ + β)]L

−1

(y), (29)

with k

> 0, where L

−1

(y) : R

→ R

n−k×n

is the full rank left inverse of the

matrix L(y). Indeed, in this case, the z–dynamics reduce to ˙z = (S − k

)z, achieving

the desired exponential convergence property. Unfortunately, solving the PDE (29) is a

daunting task, and we don’t even know if a solution exists. Therefore, in the next step

of the design we proceed to “approximate” its solution.

Step 2. (“Approximate solution” of the PDE) Deﬁne the “ideal ∇

β” as

H(y, ξ + β) := [k

I +

S(y, ξ + β)]L

−1

(y), (30)

and denote the columns of this n − k × n matrix by H

: R

× R

n−k

→ R

n−k

for

i = 1, . . . , n, that is,

H(y, ξ + β) =



(y, ξ + β) | · · · | H

(y, ξ + β)



Now, mimicking [10], deﬁne

β(y, ˆy, ˆx) :=

([s, ˆy

, . . . , ˆy

], ˆx)ds + · · · +

([ˆy

, . . . , ˆy

n−1

, s], ˆx)ds. (31)

From the deﬁnition of the mapping β, and adding and substracting H(y, ξ + β), we

have that ∇

β can be written as

∇

β(y, ˆy, ˆx) = H(y, ξ + β) −

H(y, ξ + β)−



, ˆy

, . . . , ˆy

, ˆx) . . . H

(ˆy

, . . . , ˆy

n−1

, y

, ˆx)



Since the term in brackets is equal to zero if ˆy = y and ˆx = ξ + β, and all functions are

smooth, there exist mappings ∆

: R

× R

n−k

× R

→ R

n−k×n

, ∆

: R

× R

n−k

→ R

n−k×n

such that

∇

β(y, ˆy, ˆx) = H(y, ξ + β) − ∆

(y, ˆx, e

) − ∆

(y, ˆx, e

), (32)

with

:= ˆy − y, e

:= ˆx − (ξ + β), (33)

and such that

∆

(y, ˆx, 0) = 0, ∆

(y, ˆx, 0) = 0, (34)

for all y, ˆy, ∈ R

and x, ˆx ∈ R

n−k

Replacing (30) and (32) in (28) yields

˙z = (S − k

)z + (∆

+ ∆

)L(y)z. (35)

We attract the readers attention to the particular selection of the arguments used in the inte-

grands. Namely that, with some abuse of notation, the vector ˆy has been spelled out into its

components.

Recalling that S is skew–symmetric and k

> 0, it is clear that the mappings ∆

and

∆

play the role of disturbances that we will try to dominate with a dynamic scaling in

the next step of the design.

Step 3. (Dynamic scaling) Deﬁne the scaled off–the–manifold coordinate

η =

z, (36)

with r a scaling dynamic factor to be deﬁned below. Differentiating (36), and using

(35), yields

˙η =

˙z −

˙r

= (S − k

)η + (∆

+ ∆

)L(y)η −

˙r

η.

Consider the function

(η) =

|η|

and note that its time derivative is such that

= −(k

˙r

)|η|

− η

⊤

(∆

+ ∆

)L(y)η

≤ −



˙r

−

k[∆

+ ∆

]Lk



|η|

≤ −



˙r

−



k∆

+ k∆





|η|

(37)

where k·k is the matrix induced 2—norm and we have applied Young’s inequality (with

the factor k

) to get the second bound. Let

˙r = −

(r − 1 ) +



k∆

+ k∆



, r(0) ≥ 1. (38)

Notice that the set {r ∈ R | r ≥ 1} is invariant for the dynamics (38). Replacing (38)

in (37) yields the bounds

≤ −



−

r − 1



|η|

≤ −

|η|

, (39)

where the property

r−1

≤ 1 has been used to get the second bound. From (39) we

conclude that η(t) converges to zero exponentially.

Step 4. (High–gain injection) From (36) and the previous analysis it is clear that z(t) →

0 if we can provethat r ∈ L

∞

, which is the property established in this step. To enhance

readability the procedure is divided into two parts. First, we make the function

(η, e

, e

) = V

(η) +

(|e

+ |e

a strict Lyapunov function. Then, the derivative of the function

(η, e

, e

, r) = V

(η, e

, e

) +

, (40)

is shown to be non–positive—establishing the desired boundedness of r. In both steps

the objectives are achieved adding, via a suitable selection of the observer dynamics,

negative quadratic terms in η, e

, e

in the Lyapunov function derivative. We recall that

and e

are measurable quantities, deﬁned in (33).

Towards this end, deﬁne

ˆy = L(y)(ξ + β) − ψ

(y, r)e

, (41)

with ψ

: R

× R

→ R

a gain function to be deﬁned. The error dynamics, obtained

combining (15) and (41), are

˙e

= Lz − ψ

. (42)

Now, select

ˆx = F + S(y, ξ + β)(ξ + β) − ψ

(y, r)e

, (43)

with ψ

: R

× R

→ R

a gain function to be deﬁned. Recalling (27) the error

dynamics for e

become

˙e

= ∇

βLz − ψ

. (44)

Using (39), (42) and (44) and doing some basic bounding, yields

≤ −

|η|

+ re

⊤

Lη − ψ

+re

⊤

∇

βLη − ψ

(45)

≤ −(

− 1)|η|

−



−

kLk



−



−

k∇

βk

kLk



. (46)

Selecting

= k

+ ψ

kLk

= k

+ ψ

k∇

βk

kLk

, (47)

with k

, k

> 0 and ψ

, ψ

: R

× R

→ R

to be deﬁned, we conclude that

≤ −

− 2)|η|

− k

which, selecting k

> 4, establishes that η, e

, e

∈ L

∩ L

∞

and the origin of the

(non-autonomous) subsystem with state η, e

, e

is uniformly globally exponentially

stable.

We are now ready for the coup de gr

ace, namely the selection of ψ

and ψ

guarantee that r ∈ L

∞

. For, recall (34), which ensures the existence of mappings

∆

× R

n−k

× R

→ R

n−k×n

∆

: R

× R

n−k

× R

n−k

→ R

n−k×n

such that

k∆

(y, ˆx, e

)k ≤ k

∆

(y, ˆx, e

)k |e

k∆

(y, ˆx, e

)k ≤ k

∆

(y, ˆx, e

)k |e

|. (48)

Now, evaluate the time derivative of V

, deﬁned in (40), replace (47) in (46), and use

the bounds (48) to get

≤ −(

− 1)|η|

−



−

∆

kLk



−



−

∆

kLk



Fixing

∆

kLk

∆

kLk

ensures

≤ 0, which ensures r ∈ L

∞

To prove condition (6) note that equation (39) implies

|η(t)| ≤ |η(0)|e

−

hence

|z(t)| ≤

r(t)

r(0)

|z(0)|e

−

≤ sup

t≥0

{r(t)}|z(0)|e

−

which yields the claim, by boundedness of r(t).

The proof is completed deﬁning the state vector of the observer as χ = (ˆx, ˆy, ξ, r),

obtaining A(χ, q, u) from (43), (41), (27), and (38), and deﬁning

B(y) :=



−1

(y) 0 0 0



Remark 5. The four components ˆx, ˆy, ξ and r of the state vector of the observer can be

given the following interpretation. The component ˆx is the estimate of x and a ﬁltered

version of ξ + β. The component ˆy is a ﬁltered version of the measured variable y.

The ξ-dynamics render the set z = 0 invariant, regardeless of the selection of the other

dynamics, and ξ can be regarded as the state of a reduced order observer

. Finally, the

r-dynamics are used to trade stability of the nominal design for robustness against the

disturbances ∆

and ∆

To clarify this point note that, ideally, the PDE (29)should have a solution β which is a function

of y alone. In this case the variable ξ would play the role of the state of the (reduced) order

observer(see the examples in [8]).

Remark 6. Although the analysis of the performance of the proposed observer in the

presence of noise is not within the scope of the paper, it is worth noting the following.

The Lyapunov argument establishing uniform asymptotic stability of the zero equilib-

rium of the (η, e

, e

)-subsystem yields robustness againsts small additiveperturbations

on the measured variables u and y. In the presence of such perturbations the variables

and e

do not converge to zero. Nevertheless, as long as they are sufﬁciently small,

equation (38) can be regarded as describing a linear (non-autonomous) scalar differen-

tial equation in which, by equations (34), the coefﬁcient of the linear term is uniformly

negative. This ensures boundedness of r(t) for all t.

5 Conclusions

A deﬁnite afﬁrmative answer has been given to the question of existence of a globally

convergent speed observer for general mechanical systems of the form (1). No assump-

tion is made on the existence of an upperbound for the inertia matrix, hence the result

is applicable for robots with prismatic joints. Also, no conditions are imposed on the

potential energy function. The only requirement is that the system is forward complete,

i.e., that trajectories of the system exist for all times t ≥ 0—which is a rather weak

condition.

In some sense, our contribution should be interpreted more as an existence result

than an actual, practically implementable, algorithm. Leaving aside the high complex-

ity of the observer dynamics, that can be easily retraced from the proof of Section 4, the

difﬁculty stems from the fact that the key function β is deﬁned via the integrals (31),

whose explicit analytic solution cannot be guaranteed a priori. Of course, the (scalar)

integrations can always be numerically performed leading to a numerical implementa-

tion of the observer. Given the recent spectacular advances in computational technology

this does not seem to constitute an unsurmountable difﬁculty.

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