# Stability Analysis of Trajectories on Manifolds with Applications to
Observer and Controller Design

Dongjun Wu1 Bowen Yi2 and Anders Rantzer1 *This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme under grant agreement No 834142
(ScalableControl).1 D. Wu and A. Rantzer are with the Department of Automatic
Control, Lund University, Box 118, SE-221 00 Lund, Sweden
<EMAIL_ADDRESS><EMAIL_ADDRESS>B. Yi is with
Robotics Institute, University of Technology Sydney, Sydney, NSW 2006,
Australia<EMAIL_ADDRESS>
###### Abstract

This paper examines the local exponential stability (LES) of trajectories for
nonlinear systems on Riemannian manifolds. We present necessary and sufficient
conditions for LES of a trajectory on a Riemannian manifold by analyzing the
complete lift of the system along the given trajectory. These conditions are
coordinate-free which reveal fundamental relationships between exponential
stability and incremental stability in a local sense. We then apply these
results to design tracking controllers and observers for Euler-Lagrangian
systems on manifolds; a notable advantage of our design is that it visibly
reveals the effect of curvature on system dynamics and hence suggests
compensation terms in the controller and observer. Additionally, we revisit
some well-known intrinsic observer problems using our proposed method, which
largely simplifies the analysis compared to existing results.

##  1 Introduction

Many physical systems are naturally modelled on Riemannian manifolds. The most
important example may refer to a class of mechanical systems with
configuration spaces being Riemannian manifolds, rather than Euclidean spaces
[1]. Another well known example appears in quantum systems [2], in which
systems state lives on Lie groups [3].

It is well known that local stability of equilibria for systems, whose state
space is on Riemannian manifolds, can be analyzed via linearization in local
coordinate—similar to the case in Euclidean space—known as the Lyapunov
indirect method. In many practically important control tasks, we are very
interested in the stability of _a particular solution_ $X(\cdot)$, the problem
of which widely arises in observer design, trajectory tracking [4], orbital
stabilization [5], and synchronization [6]. In Euclidean space, these tasks,
or equivalently the stability of a solution $X(\cdot)$, may be solved by
introducing an error variable and then studying the error dynamics, which is
usually a nonlinear time-varying system. In particular, the local exponential
stability (LES) of $X(\cdot)$ for a given nonlinear system can be
characterized by the linearization of its error dynamics near the trajectory.

A similar problem arises in contraction and incremental stability analysis [7,
8], in which we are interested in the attraction of any two trajectories to
each other, rather than a particular one $X(\cdot)$. The basic idea is to
explore the stability of a linearized dynamics, regarded as first-order
approximation, to obtain the incremental stability of the given system.
Indeed, studying the stability of a _particular_ solution via first-order
approximation has already been used, which, from the perspective of
incremental stability, is known as partial (or virtual) contraction [9, 8]. As
discussed above, some excitation conditions of the given trajectory may be
needed to continue stability analysis. A successful application may be found
in [10] for the stability of extended Kalman filter (EKF).

For the system evolving on Riemannian manifolds, however, the stability
analysis of a solution $X(\cdot)$ is much more challenging. The difficulty
arises from two aspects. On one hand, the “error dynamics” for such a case is
more involved—there are, indeed, no generally preferred definition of tracking
(or observation, synchronization) errors—the induced Riemannian distance on
manifolds can hardly be used to derive error dynamics directly. In practice,
one has to choose an error vector according to the structure of the manifold,
see [4, 11, 12] for examples. On the other hand, the alternative method, via
first-order approximation (or partial contraction), is non trivial to be
applied to Riemmanian manifolds, since it is usually a daunting task to
calculate the differential dynamics on Riemannian manifolds, and also some
complicated calculations of parallel transport are involved. Overcoming these
two major challenges is the main motivation of the paper.

To address this, we provide in this paper an alternative way to study LES of
trajectories on Riemannian manifolds, namely, LES will be characterized by the
stability of the _complete lift_ of the system along the trajectory, in this
way removing the need of obtaining error dynamics. Complete lift, or tangent
lift, has been used to study various control problems, see for example [13,
14, 1, 15, 16]. Among the listed references, the most relevant work to ours
are [16, 15]. In [16] the authors have remarked that the complete lift can be
seen as a linearization procedure. However, the verification of stability of
the complete lift system is challenging since it is a system living in the
tangent bundle and thus how to effectively use the aforementioned
characterization to guide controller and observer design is an open question.
We address this question in this paper.

The main contributions of the paper are three folds.

  * -

Establish the relationship between LES of a solution to the stability of the
complete lift along this solution on a Riemannian manifold, which can be seen
as the Lyapunov indirect method on manifolds. Then show that LES of a solution
is equivalent to local contraction near the solution $X(\cdot)$.

  * -

Propose an alternative approach for analysis of LES based on the
characterization of complete lift system. This novel approach obviates the
calculation of complete lift and hence facilitates the analysis of local
exponential stability and contraction. We demonstrate the efficiency of the
proposed methods by revisiting some well-known research problems.

  * -

Two main types of application problems are studied, namely, controller and
observer design, especially for mechanical systems on manifolds. These results
largely simplify the analysis in some existing works. In particular, the
proposed method is quite efficient for analyzing a class of systems called
Killing systems.

Notation. Throughout this paper we use rather standard notations from
Riemannian geometry [17, 18]. Denote $M$ the Riemannian manifold of dimension
$n$, $\langle\cdot,\cdot\rangle$ the metric, $\nabla$ the Levi-Civita
connection, $R(X,Y)Z$ the Riemannian curvature, $\pi:TM\rightarrow M$ the
natural projection of the tangent bundle. We use $\nabla$ and
$\operatorname*{grad}(\cdot)$ interchangeably to represent the gradient
operator. Let $\operatorname*{Hess}(\cdot)$ be the Hessian, $\exp(\cdot)$ the
exponential map, $P_{x}^{y}:T_{x}M\rightarrow T_{y}M$ the parallel transport
from $T_{x}M$ to $T_{y}M$, $d(x,y)$ the Riemannian distance between $x$ and
$y$, and $B_{c}(x)=\\{\xi\in M|d(\xi,x)\leq c\\}$ the Riemannian ball. Let
$\phi^{f}(t;t_{0},x_{0})$ be the flow of the equation $\dot{x}=f(x,t)$; and we
sometime write $\phi(\cdot)$ when clear from the context. The notation
$L_{f}Y$ stands for Lie derivative of $Y$ along $f$.

##  2 Local Exponential Stability on Riemannian Manifolds

###  2.1 Theory: LES and Complete Lift

Consider a system

$\dot{x}=f(x,t)$ (1)

with the system state $x$ on the Riemannian manifold $M$, and $X(\cdot)$ a
particular solution, i.e., $\dot{X}(t)=f(X(t),t)$ from the intial condition
$X(t_{0})=X_{0}\in M$. We study the local exponential stability of the
solution $X(t)$. Some definitions are recalled below.

###### Definition 1

The solution $X(\cdot)$ of the system (1) is _locally exponentially stable_
(LES) if there exist positive constants $c,K$ and $\lambda$, all independent
of $t_{0}$, such that

$d(x(t),X(t))\leq Kd(x(t_{0}),X(t_{0}))e^{-\lambda(t-t_{0})},\;$

for all $t\geq t_{0}\geq 0$ and $x(t_{0})$ satisfying
$d(x(t_{0}),X(t_{0}))<c$.

###### Remark 1

For the case that $X(t)$ is a trivial solution at an equilibrium, i.e.,
$X(t)\equiv X_{0},~{}\forall t\geq t_{0}$, Definition 2 coincides with the
standard definition of LES of an equilibrium. We should also notice the
peculiarity of this definition—it may happen that the union of LES solutions
forms into a dense set. For example, every solution of $\dot{x}=Ax$ is LES
when $A$ is Hurwitz.

We recall the definition of complete lift of a vector field, see [19, 20] for
more detailed discussions.

######  Definition 2 (Complete Lift)

Consider the time-varying vector field $f(x,t)$. Given a point $v\in TM$, let
$\sigma(t,s)$ be the integral curve of $f$ with $\sigma(s,s)=\pi(v)$. Let
$V(t)$ be the vector field along $\sigma$ obtained by Lie transport of $v$ by
$f$. Then $(\sigma,V)$ defines a curve in $TM$ through $v$. For every $t\geq
s$, the _complete lift_ of $f$ into $TTM$ is defined at $v$ as the tangent
vector to the curve $(\sigma,V)$ at $t=s$. We denote this vector field by
$\tilde{f}(v,t)$ , for $v\in TM$.

###### Definition 3

Given the system (1), and a solution $X(t)$. Define the complete lift of the
system (1) along $X(t)$ as

$\dot{v}=\tilde{f}(v,t),\;v(t)\in T_{X(t)}M$ (2)

where $\tilde{f}$ is the complete lift of $f$ as in Definition 2.

The most important property of the complete lift system is linearity at a
fixed fibre. We refer the reader to [15] for coordinate expression of (2).
From this definition, one can easily verify that the solution to (2), i.e.,
$v(t)$ has the property that $\pi v(t)=X(t)$. Hence we say that (2) defines a
dynamical system along the particular solution $X(t)$.

The following simple characterization is the theoretical basis of this paper.
It can be viewed as an analogue of the Lyapunov indirect method on Riemannian
manifolds.

###### Theorem 1

Assume the system (1) is forward complete for $t\geq 0$. If the solution
$X(t)$ is LES, then the complete lift of the system (1) along $X(\cdot)$ is
exponentially stable. If the solution $X(\cdot)$ is bounded, the converse is
also true.

######  Proof 2.2.

($\Longrightarrow$) Assume that the solution $X(t)$ is LES. Denote the
minimizing normalized (_i.e._ with unit speed) geodesic joining $X(t_{0})$ to
$x(t_{0})$ as $\gamma:[0,\hat{s}]\rightarrow M$, with
$\gamma(0)=X(t_{0}),\quad\gamma(\hat{s})=x(t_{0})$ and
$0\leq\hat{s}=d(X(t_{0}),x(t_{0}))$. Let $v_{0}\in TM$ with
$\pi(v_{0})=X(t_{0})$ and $v_{0}=\gamma^{\prime}(0)$, and $v(t)$ be the
solution to the complete lift system (2). Then

$\hat{s}\left|v(t)\right|=d\left(\exp_{X(t)}\left(\hat{s}v(t)\right),X(t)\right),$
(3)

where $\exp_{x}:TM\rightarrow M$ is the exponential map, by choosing $\hat{s}$
sufficiently small such that $\exp$ is defined. Using the metric property of
$d$, we have

$\displaystyle d\left(\exp_{X(t)}\left(\hat{s}v(t)\right),X(t)\right)$
$\displaystyle\leq$ $\displaystyle
d\left(\exp_{X(t)}\left(\hat{s}v(t)\right),x(t)\right)+d(x(t),X(t))$ (4)
$\displaystyle\leq$ $\displaystyle
d\left(\exp_{X(t)}\left(\hat{s}v(t)\right),x(t)\right)+K\hat{s}e^{-\lambda(t-t_{0})},$
(5)

where the second inequality follows from Definition 1. Fixing $t$ at the
moment and invoking (3) and (5) we get

$\displaystyle\left|v(t)\right|$
$\displaystyle\leq\kappa(\hat{s})+Ke^{-\lambda(t-t_{0})}$ (6)
$\displaystyle\kappa(\hat{s})$
$\displaystyle:=\frac{d\left(\exp_{X(t)}\left(\hat{s}v(t)\right),x(t)\right)}{\hat{s}}.$

Note that more precisely $\kappa$ is a function of both $t$ and $\hat{s}$. But
omitting the $t$ argument does not affect the following analysis.

Now we need to show the term $\kappa(\hat{s})$ is of order $O(\hat{s})$. Since
$x(t_{0})=\gamma(\hat{s})$, this term can be rewritten as

$\kappa(s)=\frac{d\left(\exp_{X(t)}\left(sv(t)\right),\phi(t;t_{0},\gamma(s))\right)}{s}$

where we have replaced $\hat{s}$ by $s$. To this end, we consider two
functions $\alpha_{1}(s)=\exp_{X(t)}\left(sv(t)\right)$,
$\alpha_{2}(s)=\phi(t;t_{0},\gamma(s))$. Similarly, we have omitted the $t$
argument which does not affect the proof. We have
$\alpha_{1}(0)=\alpha_{2}(0)=X(t)$ and
$\alpha_{1}^{\prime}(0)=\alpha_{2}^{\prime}(0)=v(t).$ Thus

$\kappa(s)=\frac{1}{s}d(\alpha_{1}(s),\alpha_{2}(s))=O(s),$

where we have used Lemma 6.26 given in Appendix. Now letting
$\hat{s}\rightarrow 0$ in ( 6) and noticing that the geodesic is unit speed,
we have

$|v(t)|\leq K|v(t_{0})|e^{-\lambda(t-t_{0})},$

for any $v(t_{0})\in T_{X(t_{0})}M$.

($\Longleftarrow$) A consequence of Proposition 2.6 (see Section 2.2): If the
complete lift along $X(\cdot)$ is ES, then the proof of Proposition 2.6 shows
that the system is contractive on a bounded set $B_{c}$ and thus the LES of
$X(\cdot)$.

######  Remark 2.3.

Theorem 1 provides a characterization for LES of trajectories on manifolds via
complete lift. Unfortunately, the original form of this theoretical result
lacks practical utility for applications. The main reason is that the complete
lift on manifolds is difficult to obtain, and quite often, its calculation of
it relies on local coordinates, which is in conflict with the purpose
(coordinate-free design) of this paper. To circumvent this issue, we propose
an alternative approach in Section 2.3 based on Theorem 1, which will be much
more efficient to use. But we must emphasize that Theorem 1 plays the
fundamental role for the rest of the paper.

From Theorem 1, we can derive the following interesting corollary which says
that there no unbounded LES solution exists for _autonomous_ systems.

######  Corollary 2.4.

For a time invariant system $\dot{x}=f(x)$, a LES solution $X(t)$ should
always be bounded and non-periodic.

######  Proof 2.5.

The complete lift of $\dot{x}=f(x)$ is $\dot{v}=\frac{\partial f}{\partial
x}v,\;v\in T_{x}\mathbb{R}^{n}.$ Clearly, $v=\dot{x}$ is a solution to the
complete lift system. Then by Theorem 1, $|\dot{X}(t)|\leq
k|\dot{X}(0)|e^{-\lambda t},$ hence $X(t)$ cannot be periodic. Further more,
$|X(t)|\leq|X(0)|+\int_{0}^{t}k|\dot{X}(0)|e^{-\lambda
s}ds=|X(0)|+\frac{k|\dot{X}(0)|}{\lambda}(1-e^{-\lambda
t})<|X(0)|+\frac{k|\dot{X}(0)|}{\lambda}.$

In [21, Lemma 1], the authors obtain a similar result for autonomous systems,
i.e., there is a unique attractive equilibrium in an invariant set, in which
the system is incrementally exponentially stable.

###  2.2 Contraction and LES

Contraction theory has become a powerful tool for analysis and design of
control systems, see [7, 8, 6, 22, 23, 24] and the references therein. In
Section 2.1, we have studied LES of solutions to the system (1). In this
subsection, we will show the close connection between the proposed result and
contraction analysis on manifolds [25, 15]. The reader may refer to [22, 26]
for the case on Euclidean space.

We say that the system (1) is contractive on a set $C$ if there exist positive
constants $K,\lambda$, independent of $t_{0}$ such that

$d(\phi(t;t_{0},x_{1}),\phi(t;t_{0},x_{2}))\leq
Kd(x_{1},x_{2})e^{-\lambda(t-t_{0})},$ (7)

for all $x_{1},x_{2}\in C,t\geq t_{0}\geq 0$. For technical ease, we have
slightly modified the definition of contraction by allowing the set $C$ to be
not forward invariant. Based on Theorem 1, we have the following proposition,
which can be viewed as a bridge from LES to local contraction.

######  Proposition 2.6.

A bounded solution $X(t)$ to the system (1) is LES if and only if there exists
a constant $c$ such that the system (1) is contractive on a bounded set
$B_{c}$ whose interior contains $X(\cdot)$.

######  Proof 2.7.

Assume that $X(t)$ is LES. Then the complete lift system along $X(t)$ is
exponential stable (ES) by Theorem 1. By converse Lyapunov theorem, there
exists a $\mathcal{C}^{1}$ function $V(t,v)$, quadratic in $v$ satisfying

$c_{1}|v|^{2}\leq V(t,v)\leq c_{2}|v|^{2},\ \forall v\in T_{X(t)}M$ (8)

and

$\dot{V}(t,v)=\frac{\partial V}{\partial t}(t,v)+L_{\tilde{f}}V(t,v)\leq-
c_{3}|v|^{2},\ \forall v\in T_{X(t)}M,$ (9)

for all $t\geq t_{0}\geq 0$ and three positive constants $c_{1},c_{2},c_{3}$.

Due to the smoothness of $V$, we have

$|\dot{V}(t,P_{X(t)}^{x(t)}v)-\dot{V}(t,v)|\leq
c_{4}d_{TM}(P_{X(t)}^{x(t)}v,v)=c_{4}d_{M}(x(t),X(t)).$

Thus

$\displaystyle\sup_{\tiny{\begin{subarray}{c}|w|=1,\\\ w\in
T_{x(t)}M\end{subarray}}}\dot{V}(t,w)$
$\displaystyle=\sup_{\tiny{\begin{subarray}{c}|v|=1,\\\ v\in
T_{X(t)}M\end{subarray}}}\dot{V}(t,P_{X(t)}^{x(t)}v)$
$\displaystyle=\sup_{\tiny{\begin{subarray}{c}|v|=1,\\\ v\in
T_{X(t)}M\end{subarray}}}\dot{V}(t,v)+\dot{V}(t,P_{X(t)}^{x(t)}v)-\dot{V}(t,v)$
$\displaystyle\leq-c_{3}+c_{4}d(x(t),X(t))<-c_{5}<0,$

for $c$ small enough such that $d(x(t),X(t))$ will be small enough for all
$t\geq t_{0}$ when $x(t_{0})\in B_{c}({X(t_{0})})$. Since $\dot{V}$ is
quadratic in $v$ (due to the linearity of the complete lift system and that
$V(t,v)$ is quadratic in $v$), this implies

$\dot{V}(t,v)\leq-c_{5}|v|^{2},\ \forall v\in T_{x(t)}M,\ t\geq t_{0}$

for all $x(t_{0})\in B_{c}({X(t_{0})})$. Then the system (1) is contractive on
$B_{c}:=\bigcup_{t_{0}\geq 0}B_{c}(X(t_{0}))$ which is bounded as is
$X(\cdot)$ (use Theorem 2 [15]). The converse is obvious, and hence the proof
is completed.

The following corollary is a straightforward consequence.

######  Corollary 2.8.

Assume that the system (1) has an equilibrium point $x_{\star}\in M$. Then
$x_{\star}$ is LES if and only if there exists an open neighborhood of
$x_{\star}$ on which the system is contractive.

In [27], the authors proved similar result to this corollary for _autonomous_
systems in Euclidean space. The paper [22] focuses on asymptotic stability and
asymptotic contraction, also in Euclidean space.

###  2.3 A More Usable Form

As remarked earlier, Theorem 1 is not suitable for practical applications due
to the difficulty of calculating the complete lift system. In this subsection,
we propose a more usable version of Theorem 1 (still intrinsic) which will
make the analysis of LES a routine task.

For reasons that will be clear later, we rename the state $x$ in the system
(1) as $q$. Fig. 1 is drawn to illustrate our idea. In Fig. 1, the solid curve
represents a trajectory of the system system (1), say $q:\mathbb{R}_{\geq
0}\to M$, whose velocity vectors are drawn as black arrows, denoted $\dot{q}$.
The dashed curves are flows of the initial curve $\gamma:s\mapsto\gamma(s)\in
M$. The blue arrows emanating from the curve $q$ are the (transversal)
velocities of the dashed curve, denoted as $q^{\prime}$, or in precise
language, $q^{\prime}=\frac{\partial q(s,t)}{\partial s}$ for the
parameterized curve $(s,t)\mapsto q(s,t)$. We call $q^{\prime}$ a variation
along $q(\cdot)$.

Figure 1: Illustration of $\dot{q}$ and $q^{\prime}$.

Two important observations can be made from the figure:

  * -

By construction, $q^{\prime}$ is the solution to the complete lift of the
system (1) along the trajectory $q(\cdot)$. Thanks to this, the Lie bracket
$[\dot{q},q^{\prime}]$ vanishes for all $t\geq t_{0}$ along $q(\cdot)$. 111
Recall that
$[X,Y]=\left.\frac{d}{dt}\right|_{t_{0}}(\phi^{X}_{t})^{*}Y(t_{0})$, thus
$[\dot{q},q^{\prime}]=\left.\frac{d}{dt}\right|_{t_{0}}(\phi^{f}_{t})^{*}(\phi_{t}^{f})_{*}q^{\prime}(t_{0})=\left.\frac{d}{dt}\right|_{t_{0}}q^{\prime}(t_{0})=0$.

  * -

The map $(s,t)\mapsto q(s,t)$ forms a parameterized surface in $M$. Then due
to the torsion-free property of Levi-Civita connection, there holds
$\frac{D}{dt}\frac{\partial q}{\partial s}=\frac{D}{ds}\frac{\partial
q}{\partial t}$ (see (35) and [17, Lemma 3.4]), which implies that
$\nabla_{\dot{q}}q^{\prime}=\nabla_{q^{\prime}}{\dot{q}}=\nabla_{q^{\prime}}{f}$.

Now that $q^{\prime}$ is the solution to the complete lift system, it is
sufficient to analyze the dynamics of $q^{\prime}$. This may seem naïve at the
first thought and that the novelty seems to be only at a notational level. The
fact is, however, due to the above two observations, we now have access to
rich results in Riemannian geometry. In particular, we will see how LES on
Riemannian manifold is affected by curvature – the most important ingredient
of a Riemannian manifold.

###  2.4 Revisit of some existing results

####  2.4.1 Contraction on Riemannian manifolds [25]

The following result is obtained in [25] (the contraction version):

######  Theorem 2.9 ([25]).

Let $q(\cdot)$ be a solution to the system (1), if

$\langle\nabla_{v}f,v\rangle\leq-k\langle v,v\rangle,\;\forall v\in
T_{q(t)}M,\ t\geq 0,$

for some positive constant $k$, then the solution $q(t)$ is LES.

The proof of this theorem will now simplify to a few lines:

######  Proof 2.10.

It suffices to show the exponential convergence of the metric $\langle
q^{\prime},q^{\prime}\rangle$. Indeed,

$\frac{1}{2}\frac{d}{dt}\langle
q^{\prime},q^{\prime}\rangle=\left<\nabla_{\dot{q}}q^{\prime},q^{\prime}\right>=\left<\nabla_{q^{\prime}}f,q^{\prime}\right>\leq-k\left<q^{\prime},q^{\prime}\right>.$

Thus $\left<q^{\prime},q^{\prime}\right>$ converges exponentially.

Notice that we have used the fact that $[q^{\prime},\dot{q}]=0$.

####  2.4.2 Intrinsic reduced observer [28]

The following lemma was among the key results in [28]:

######  Lemma 2.11 ([28]).

Let $M$ be a smooth Riemannian manifold. Let $P\in M$ be fixed. On the
subspace of $M$ defined by the injectivity radius at $P$, we consider

$\dot{q}=-\frac{1}{2\lambda}\operatorname*{grad}d(q,P)^{2},\quad\lambda>0.$
(10)

If the sectional curvature is non-positive, the dynamics is a contraction in
the sense of [7], i.e., if $\delta x$ is a virtual displacement at fixed $t$,
we have

$\frac{d}{dt}\left<\delta q,\delta q\right>\leq-\frac{2}{\lambda}\left<\delta
q,\delta q\right>.$ (11)

If the sectional curvature is upper bounded by $A>0$, then (12) holds for
$d(q,P)<\pi/(4\sqrt{A})$.

The proof provided in [28] is a bit technical. We now give a simplified proof
using the methods developed in this paper and provide a new estimation of the
convergence rate.

######  Lemma 2.12.

Let $M$ be a smooth Riemannian manifold whose curvature is upper bounded by
$A\geq 0$. Let $P\in M$ be fixed. Then the dynamics (10) is globally
contractive if $A=0$, and locally contractive otherwise, with contraction rate
$\gamma(q)=\frac{2\sqrt{A}d(q,P)}{\lambda\tan(\sqrt{A}d(q,P))}$,222$\gamma(P)$
is understood as $\lim_{d(q,P)\to 0}\gamma(q)=\frac{2}{\lambda}$. Notice that
$\gamma$ is monotone decreasing and strictly positive on the interval
$[0,\frac{\pi}{2})$ i.e.,

$\frac{d}{dt}\left<\delta q,\delta q\right>\leq-\gamma\left<\delta q,\delta
q\right>.$ (12)

######  Proof 2.13.

Let $F(q)=\frac{1}{2}d(q,P)^{2}$ and we estimate

$\displaystyle\frac{d}{dt}\left<q^{\prime},q^{\prime}\right>$
$\displaystyle=2\left<-\frac{1}{\lambda}\nabla_{q^{\prime}}\nabla
F,q^{\prime}\right>$
$\displaystyle=-\frac{2}{\lambda}\operatorname*{Hess}F(q^{\prime},q^{\prime})$

where the last equality follows from the definition of the Hessian operator,
see (36). The conclusion follows invoking comparison for the Hessian of square
distance (e.g., [29, Theorem 6.6.1]):

$\operatorname*{Hess}F\geq\sqrt{A}d(q,P)\cot(\sqrt{A}d(q,P))\text{Id}$

for all $q\in\text{inj}(P)$ if $A>0$ and for all $q\in M$ if $A=0$.

######  Remark 2.14.

The second part of the Lemma 2.11 [28] seems incorrect: by Rauch comparison
(see [18, Theorem 6.4.3]), for manifold with sectional curvature lower bounded
by $k>0$, there holds $\operatorname*{Hess}F\leq(1-kF)g$, where $g$ is the
Riemannian metric. Therefore, the contraction rate is strictly less than
$\frac{2}{\lambda}$ in any neighborhood of $P$.

######  Remark 2.15.

Since $\operatorname*{Hess}F|_{P}=g$, if the Hessian is continuous at $P$,
then the dynamics (10) is always locally contractive without assumptions on
curvature, which also implies that $P$ is an LES equilibrium.

The above method is not limited to study contraction of distance, in fact, it
can be easily adapted to study $k$-contraction [30] (Hausdorff measure such as
area and volume) on Riemannian manifolds. As an example, let us consider the
contraction of volume. Suppose that
$\\{q^{\prime}_{1},\cdots,q^{\prime}_{n}\\}$ forms a frame at $q$ and denote
$\operatorname*{vol}(q_{1}^{\prime},\cdots,q^{\prime}_{n})$ the signed volume
of the parallelepiped spanned by this frame and we study the change of the
volume under the dynamics (12):

$\displaystyle\frac{d}{dt}\operatorname*{vol}(q^{\prime}_{1},q^{\prime}_{2},\cdots,q^{\prime}_{n})$
(13) $\displaystyle=$ $\displaystyle-(\operatorname*{div}\nabla
F/\lambda)\operatorname*{vol}(q^{\prime}_{1},q^{\prime}_{2},\cdots,q^{\prime}_{n})$
$\displaystyle=$ $\displaystyle-\frac{\Delta
F}{\lambda}\operatorname*{vol}(q^{\prime}_{1},q^{\prime}_{2},\cdots,q^{\prime}_{n})$

where $\Delta$ is the Laplace-Beltrami operator [29]. Now $\Delta
F=\operatorname*{tr}(G^{-1}\operatorname*{Hess}F)$, with $G$ the Riemannian
metric, we can conclude that the condition in Lemma 2.12 implies exponential
contraction of volume (on non-positive curvature manifold). Since $\Delta F$
is controlled by $\operatorname*{tr}(G^{-1}\operatorname*{Hess}F)$, non-
positive curvature assumption is too restrictive. In fact, $\Delta
F=1+H(q,P)d(q,P)$, where $H(q,P)$ is the mean curvature, thus the same
conclusion can be drawn for manifold with non-positive mean curvature.

######  Remark 2.16.

From the proof of Lemma 2.12 we see that the function $F$ need not be the
square distance. It can be replaced by any function whose Hessian has the
required property, as the next example shows.

####  2.4.3 Filtering on ${\it SO}(3)$

Consider first the attitude control problem

$\dot{R}=Ru$ (14)

where $R\in SO(3)$ and the control input $u\in\mathfrak{so}(3)$. The control
objective is to exponentially stabilize a solution $R_{*}(t)\in SO(3)$, which
verifies $\dot{R_{*}}(t)=R_{*}(t)\Omega(t)$, where $\Omega(t)$ is some known
signal. The Lie group $SO(3)$ is a Riemannian manifold with the bi-invariant
metric $\langle X,Y\rangle=\operatorname*{tr}(X^{\top}Y)$. Due to the bi-
invariance of the metric, the Levi-Civita connection is simply
$\nabla_{X}Y=\frac{1}{2}[X,Y]$, see (38). Consider the function

$F(R,R_{*})=\frac{1}{2}||R-R_{*}||^{2},$

where $||\cdot||$ is the Frobenius norm ($F$ is not the square distance). The
gradient and Hessian of $F$ can be calculated as $\nabla
F=\frac{1}{2}R(R_{*}^{\top}R-R^{\top}R_{*})$,
$\operatorname*{Hess}F(RY,RZ)=\frac{1}{4}\operatorname*{tr}(Z^{\top}YR^{\top}_{*}R)$,
with $X,Y\in\mathfrak{so}(3)$ respectively. Clearly, $R_{*}(\cdot)$ is the
solution to

$\displaystyle\dot{R}$ $\displaystyle=-k\nabla F(R,R_{*})+R\Omega(t)$
$\displaystyle=-\frac{k}{2}R(R_{*}^{\top}(t)R-R^{\top}R_{*}(t))+R\Omega(t).$

Let us check the LES of the $R_{*}(\cdot)$. For $T_{R}SO(3)\ni R^{\prime}=RX$
for some $X\in\mathfrak{so}(3)$, we calculate

$\displaystyle\frac{1}{2}\frac{d}{dt}\left<R^{\prime},R^{\prime}\right>=$
$\displaystyle-k\operatorname*{Hess}F(R^{\prime},R^{\prime})+\left<\nabla_{R^{\prime}}(R\Omega(t)),R^{\prime}\right>$
$\displaystyle=$
$\displaystyle-k\operatorname*{Hess}F(R^{\prime},R^{\prime})+\frac{1}{2}\left<[R^{\prime},R\Omega(t)],R^{\prime}\right>$
$\displaystyle=$
$\displaystyle-k\operatorname*{Hess}F(R^{\prime},R^{\prime})+\frac{1}{2}\operatorname*{tr}\\{(X^{\top}X-XX^{\top})\Omega\\}$
$\displaystyle=$ $\displaystyle-k\operatorname*{Hess}F(R^{\prime},R^{\prime})$

since $X^{\top}X-XX^{\top}$ is symmetric. Note that the Hessian of $F$ is
positive definite at $R=R_{*}$. Hence the controller

$u=-\frac{k}{2}(R_{*}^{\top}(t)R-R^{\top}R_{*}(t))+\Omega(t).$

renders the trajectory $R_{*}(\cdot)$ LES as expected.

The extension to the design of a low pass filter becomes straightforward: the
following dynamics

$\dot{\hat{R}}=-\frac{k}{2}\hat{R}(R^{\top}\hat{R}-R^{\top}\hat{R})+\hat{R}\Omega$
(15)

is a locally exponential observer (filter) for $\dot{R}=R\Omega$. This result
has been obtained in [11], see also [12].

###  2.5 Killing system

####  2.5.1 Low-pass filter for Killing system

Consider a system defined by a time-varying Killing field [18, Chapter 8] on a
Riemannian manifold $(M,g)$:

$\dot{q}=f(t,q)$ (16)

i.e., $L_{f}g=0$, see also (39) in Appendix. We call such system a Killing
system. When the system (16) is perturbed by some noise, it is tempting to
design a low pass filter to reconstruct the system state from the corrupted
data $q$. For that, we propose the simple filter

$\dot{\hat{q}}=f(t,\hat{q})-k\nabla F(\hat{q},q),$ (17)

where $F(q,p)=\frac{1}{2}d(q,p)^{2}$ and $k$ is a positive constant. To verify
the convergence of this filter, we calculate as before

$\displaystyle\frac{1}{2}\frac{d}{dt}\left<\hat{q}^{\prime},\hat{q}^{\prime}\right>$
$\displaystyle=\left<\nabla_{\hat{q}^{\prime}}(f-k\nabla
F),\hat{q}^{\prime}\right>$
$\displaystyle=\left<\nabla_{\hat{q}^{\prime}}f,\hat{q}^{\prime}\right>-k\left<\nabla_{\hat{q}^{\prime}}\nabla
F,\hat{q}^{\prime}\right>$
$\displaystyle=-k\left<\nabla_{\hat{q}^{\prime}}\nabla
F,\hat{q}^{\prime}\right>\text{ (by \eqref{eq:Killing}) }$
$\displaystyle=-k\operatorname*{Hess}F(\hat{q}^{\prime},\hat{q}^{\prime}).$

Since $\operatorname*{Hess}$ is locally positive definite, the filter
converges at least locally. If in addition, the manifold has non-positive
curvature, then the convergence is global recalling that
$\operatorname*{Hess}F\geq g$ for manifold with non-positive curvature. This
is the case for the manifold of symmetric positive definite matrices: ${\rm
SPD}:=\\{P\in\mathbb{R}^{n\times n}:P=P^{\top}>0\\}$ equipped with the metric
$\left<X,Y\right>:=\operatorname*{tr}(XP^{-1}YP^{-1})$ for $X,Y\in T_{P}{\rm
SPD}$, see e.g., [31].

####  2.5.2 Discretization

Let us now consider the discretization of this filter. First, notice that $f$
is Killing, thus the discretization of (16) may be written as
$x_{k+1}=\tau\cdot x_{k}$ for some isometric action $\tau\in\text{Iso}(M)$
($\tau$ is time-invariant since (16) is autonomous). This model has been
considered in for example [32], where the manifold is the set of symmetric
positive definite matrices and it is called a “linear system” on Riemannian
manifolds. Next, viewing $-k\nabla F(\hat{q},q)$ as disturbance, we may
discretize (17) as

$\hat{q}_{k+1}=\exp_{\tau\cdot\hat{q}_{k}}(k\Delta t\log_{\hat{q}_{k}}q_{k})$

where $\Delta t$ is the sampling time and we use the standard notation
$\log_{x}y=\exp_{x}^{-1}y$, see e.g. [33]. Indeed, fix $q$ and let
$r(x)=d(x,q)$, then $\nabla r(x)=-\frac{\exp_{x}^{-1}q}{||\exp_{x}^{-1}q||}$,
see [29, Lemma 5.1.3]. Thus $\nabla F=-\log_{\hat{q}}q$. For example, in
Euclidean space $-\log_{\hat{q}}q=\hat{q}-q$, which is in accordance with
$\nabla_{\hat{q}}\frac{1}{2}||\hat{q}-q||^{2}$. Let
$e_{k}=\log_{\hat{q}_{k}}q_{k}$ be the error, then

$\displaystyle e_{k+1}$ $\displaystyle=\log_{\hat{q}_{k+1}}q_{k+1}$
$\displaystyle=\log_{\exp_{\tau\cdot\hat{q}_{k}}(k\Delta
t\log_{\hat{q}_{k}}q_{k})}\tau\cdot q_{k}$
$\displaystyle\approx\log_{\tau\cdot\hat{q}_{k}}\tau\cdot q_{k}-k\Delta
t\log_{\hat{q}_{k}}q_{k}$
$\displaystyle=D\tau\cdot\log_{\hat{q}_{k}}q_{k}-k\Delta
t\log_{\hat{q}_{k}}q_{k}$ $\displaystyle=(D\tau-k\Delta t\,{\rm Id})e_{k}$

By assumption, $D\tau$ is a linear isometric mapping (unitary), therefore,
$e_{k}$ tends to zero exponentially for all sufficiently small sampling time.

####  2.5.3 Revisit of filter on ${\it SO}(3)$

For a Lie group $G$ equipped with a left- (resp. right-) invariant metric $g$,
it is known that any right- (resp. left-) invariant vector fields are Killing
fields, see for example [18, Chapter 8]. Indeed, equip $G$ with a right-
invariant metric and consider a left-invariant vector field $V(x):=dL_{x}(v)$
for $v\in\mathfrak{g}$ and $x\in G$, whose flow reads
$F^{t}(x)=R_{\exp(tv)}x$; here $L$ and $R$ represent left and right action
respectively. Thus $DF^{t}=dR_{\exp(tv)}$. Hence for any right-invariant
vector fields $W_{1}(x)=dR_{x}(w_{1}),W_{2}(x)=dR_{x}(w_{2})$, for
$w_{1},w_{2}\in\mathfrak{g}$, we have

$\displaystyle\left<DF^{t}(W_{1}),DF^{t}(W_{2})\right>$
$\displaystyle=\left<dR_{\exp(tv)}(W_{1}),dR_{\exp(tv)}(W_{2})\right>$
$\displaystyle=\left<W_{1},W_{2}\right>_{x}\text{ (right-invariant metric)}$
$\displaystyle=\left<w_{1},w_{2}\right>,$

which is constant for all $t\geq 0$. Summarizing, a system defined by left-
invariant vector field (right-invariant is similar) e.g.,

$\dot{x}=dL_{x}(v(t)),\quad x\in G,\;v(t)\in\mathfrak{g},\,\forall t\geq 0$
(18)

is a Killing system when the underling metric of $G$ is right-invariant.
Therefore, a low-pass filter can be designed using formula (17). In
particular, for the system (14) on $SO(3)$, when $u$ is known, it reads
$\dot{\hat{R}}=\hat{R}u+k\log_{\hat{R}}R=\hat{R}u+k\log{\hat{R}^{\top}R}$ when
$SO(3)$ is equipped with the standard bi-invariant metric. Thus we obtain
another low-pass filter for left-invariant dynamics on $SO(3)$.

##  3 Application to Euler-Lagrangian Systems

In this section, we utilize the proposed methods to study LES of trajectories
of Euler-Lagrangian (EL) systems. As pointed out in Section 1, trajectory
tracking and observer design are two typical important applications which
involve the analysis of LES of trajectories. Compared to the analysis of
stability of equilibrium, these tasks are generally much harder on manifolds.
Most existing results rely on calculations in local coordinates, which is
usually a daunting task. We demonstrate in this section that, the proposed
approach can be efficiently used to design and analyze controllers and
observers for mechanical systems, while obviating the calculation in local
coordinates.

There are two pervasive approaches – Lagrangian and Hamiltonian – for
modelling of mechanical systems [34]. These two different approaches have led
to different design paradigms. Amongst the vast literature, we mention two
books that include some of the most important results in the two fields: the
book of R. Ortega et. al. [35] (Lagrangian approach) and the book of van der
Schaft [36] (Hamiltonian approach).

In this paper, we focus on the Lagrangian approach. Since we will work on
manifolds (the configuration space is a manifold rather than Euclidean), we
adopt the geometric modelling which is well documented in [1]. Briefly
speaking, one starts with a configuration space $Q$ and then calculates the
kinetic energy $\left<v_{q},v_{q}\right>$ and potential energy $V(q)$. The
kinetic energy thus defines a Riemannian metric on the configuration space,
which depends only on the inertial of system. Using principles of classical
mechanics (e.g., d’Alembert principle), one can derive the following so called
Euler-Lagrangian (EL) equation:

$\nabla_{\dot{q}}\dot{q}=-\operatorname*{grad}V(q)+\sum_{i=1}^{m}u_{i}B_{i}(q)$
(19)

where $\dot{q}$ is the velocity, $\nabla$ is the Levi-Civita connection
associated with the metric, $B_{i}$ are some tangent vectors, and $u_{i}$ are
external forces (viewed as input in our setting) taking values in
$\mathbb{R}$.

###  3.1 Tracking controllers for EL systems

Suppose that $(q_{*}(\cdot),\dot{q}_{*}(\cdot),u_{*}(\cdot))$ is a feasible
pair of the system (19), i.e., $\nabla_{\dot{q}_{*}}\dot{q}_{*}=-\nabla
V(q_{*})+\sum_{i=1}^{m}u_{i*}(t)B_{i}(q_{*})$. The objective is to design a
controller $u(\cdot)$ such that $(q_{*}(\cdot),\dot{q}_{*}(\cdot))$ is LES.
Before moving on however, we must stop for a moment to clarify the statement
that “$(q_{*}(\cdot),\dot{q}_{*}(\cdot))$ is LES”. Unlike $q(\cdot)$, the
curve $(q_{*}(\cdot),\dot{q}_{*}(\cdot))$ lives in the tangent bundle $TM$,
which is not equipped with a distance function a priori. Thus in order to talk
about convergence, a topology should be defined on $TM$. This is achieved via
the so-called Sasaki metric [19]. Due to the importance of this metric, we
briefly recall its construction in the following.

Let $V,W\in TTM$ be two tangent vectors at $(p,v)\in TM$ and

$\alpha:t\mapsto(p(t),v(t)),\,\beta:t\mapsto(q(t),w(t)),$

are two curves in $TM$ with $p(0)=q(0)=p$, $v(0)=w(0)=v$,
$\alpha^{\prime}(0)=V$, $\beta^{\prime}(0)=W$. Define the inner product on
$TM$ by

$\left<V,W\right>_{\rm
s}:=\left<p^{\prime}(0),q^{\prime}(0)\right>+\left<v^{\prime}(0),w^{\prime}(0)\right>$
(20)

in which we write $v^{\prime}(0)=\left.\frac{Dv(t)}{dt}\right|_{0}$. The
Sasaki metric is well-known to be a bona fide Riemannian metric on $TM$. For
details, see [19].

For a curve $w(s)=(c(s),v(s))$ lying in $TM$, we can calculate its length
under the Sasaki metric as:

$\displaystyle\ell(w)=$ $\displaystyle\int\sqrt{\langle
w^{\prime}(s),w^{\prime}(s)\rangle_{\rm s}}ds$ $\displaystyle=$
$\displaystyle\int\sqrt{\langle c^{\prime}(s),c^{\prime}(s)\rangle+\langle
v^{\prime}(s),v^{\prime}(s)\rangle}ds$

in which $v^{\prime}(s)$ is understood as the covariant derivative of
$v(\cdot)$ along $c(\cdot)$.

###### Assumption 1

In the sequel we assume that for each pair of points $(q,v)$ and $(p,w)$ in
$TM$, the minimizing geodesic that joins $(q,v)$ to $(p,w)$ always exists.

Now that the EL equation (19) defines a system on $TM$, it seems that to
analyze LES of solutions of the EL equation, one has to consider variation
(see Section 2.3) of the form $(q^{\prime},v^{\prime})$, with $v^{\prime}\in
TTM$. The next theorem shows that this is not needed.

######  Theorem 3.17.

Consider a dynamical system on a Riemannian manifold $(M,g)$:

$\nabla_{\dot{q}}\dot{q}=f(q,\dot{q})$ (21)

where $f$ is smooth. Let $(q(\cdot),\dot{q}(\cdot))$ be a trajectory of the
system and $q^{\prime}$ any variation along $q(\cdot)$. Then the system (21)
is contractive if the following system

$\frac{D}{dt}\begin{bmatrix}q^{\prime}\\\
\frac{Dq^{\prime}}{dt}\end{bmatrix}=F\left(\begin{bmatrix}q^{\prime}\\\
\frac{Dq^{\prime}}{dt}\end{bmatrix}\right)$ (22)

is exponentially stable along any $q(\cdot)$.

######  Remark 3.18.

Notice that $(q^{\prime},Dq^{\prime}/dt)\in T_{q}M\times T_{q}M$, thus
exponential stability can be defined in the obvious way for the system (22)
using Sasaki metric.

######  Proof 3.19.

Given a point $(q_{1},v_{1})\in TM$, and the integral curve
$\eta_{1}(t)=(q_{1}(t),\dot{q}_{1}(t))$ of the system (21) passing through it
at time $t=0$. Let $\eta_{0}(t)=(q(t),\dot{q}(t))$ be another integral curve
with initial condition $(q_{0},v_{0})$. By assumption, there exists a
minimizing geodesic $\gamma(s)=(q(s),v(s)),\ s\in[0,1]$ joining
$(q_{0},v_{0})$ to $(q_{1},v_{1})$, that is, $\gamma(0)=(q_{0},v_{0}),\
\gamma(1)=(q_{1},v_{1})$. Let $q(s,t)$ be the solution to the system (21) with
initial condition $\gamma(s)$, then the parameterized curve
$s\mapsto(q(s,t),\frac{\partial q(s,t)}{\partial t})$ forms a variation
between the curves $\eta_{0}(\cdot)$ and $\eta_{1}(\cdot)$. Therefore, the
following estimation of the distance between the two points $\eta_{0}(t)$ and
$\eta_{1}(t)$ is obvious:

$\displaystyle d_{TM}(\eta_{0}(t),\eta_{1}(t))$
$\displaystyle\leq\int_{0}^{1}\sqrt{\left|\frac{\partial q}{\partial
s}(s,t)\right|^{2}+\left|\frac{D}{ds}\frac{\partial q}{\partial
t}\right|^{2}}ds$ (23) $\displaystyle=\int_{0}^{1}\sqrt{\left|\frac{\partial
q}{\partial s}(s,t)\right|^{2}+\left|\frac{D}{dt}\frac{\partial q}{\partial
s}\right|^{2}}ds$

The conclusion follows immediately after replacing $\frac{\partial q}{\partial
s}$ by $q^{\prime}$.

As we have remarked earlier, due to Theorem 3.17, the analysis of LES and
contraction does not require variation of the form $(q^{\prime},v^{\prime})$
and that $q^{\prime}$ alone is sufficient. This observation is crucial for the
rest of this section.

With the preceding preparations, we are now in a position to study tracking
controller for the EL system. We focus on fully-actuated system:

$\nabla_{\dot{q}}\dot{q}=-\operatorname*{grad}V(q)+u$ (24)

and assume $(q_{*}(\cdot),\dot{q}_{*}(\cdot),u_{*}(\cdot)\equiv 0)$ is a
bounded feasible solution to the EL equation, i.e.,
$\nabla_{\dot{q}_{*}}\dot{q}_{*}=-\nabla V(q_{*})$ (solution with non-zero
$u_{*}$ is similar). We propose a controller with structure
$u=u_{P}+u_{V}+u_{R}$ to locally exponentially stabilize
$(q_{*}(\cdot),\dot{q}_{*}(\cdot)$, where

$\displaystyle u_{P}(q)=-k_{2}\nabla F(q,q_{*}),$ (25) $\displaystyle
u_{D}(q,\dot{q})=-k_{1}(\dot{q}-P_{q_{*}}^{q}\dot{q}_{*}),$ $\displaystyle
u_{R}(q,\dot{q}_{*})=R(\dot{q},\nabla F(q,q_{*}))\dot{q}$

As before, $F$ is half of the square distance function. $k_{1}$ and $k_{2}$
are constants to be determined and $P_{q}^{p}$ is the parallel transport from
$q$ to $p$, $R(\cdot,\cdot)\cdot$ is the curvature tensor. Heuristically, this
can be seen as a PD-controller [35], with a curvature compensation term. By
construction, $(q_{*}(\cdot),\dot{q}_{*}(\cdot))$ is a solution to the closed
loop system since $u(q_{*},\dot{q}_{*})\equiv 0$. Hence it remains to show the
LES of this solution.

Thanks to Theorem 3.17 and Proposition 2.6, we need only check the exponential
stability of the system (22) along $q_{*}(\cdot)$. For this we calculate

$\displaystyle\nabla_{q^{\prime}}\nabla_{\dot{q}}\dot{q}$
$\displaystyle=\nabla_{\dot{q}}\nabla_{q^{\prime}}\dot{q}+R(\dot{q},q^{\prime})\dot{q}$
(26)
$\displaystyle=\nabla_{\dot{q}}\nabla_{\dot{q}}q^{\prime}+R(\dot{q},q^{\prime})\dot{q}$
$\displaystyle=\frac{D^{2}q^{\prime}}{dt^{2}}+R(\dot{q},q^{\prime})\dot{q}$

where we used the basic fact about the curvature tensor:
$\frac{D}{dt}\frac{D}{ds}X-\frac{D}{ds}\frac{D}{dt}X=R(\dot{q},q^{\prime})X$,
see e.g., [17, Lemma 4.1]. The following calculations are in order (notice
that we calculate along $q_{*}(\cdot)$, otherwise these are invalid):

$\displaystyle\nabla_{q^{\prime}}u_{P}$
$\displaystyle=-k_{2}\nabla_{q^{\prime}}\nabla F=k_{2}q^{\prime}$ (27)
$\displaystyle\nabla_{q^{\prime}}u_{D}$
$\displaystyle=-k_{1}\nabla_{q^{\prime}}(\dot{q}-P_{q_{*}}^{q}\dot{q}_{*})=-k_{1}\nabla_{\dot{q}}q^{\prime}$
$\displaystyle\nabla_{q^{\prime}}u_{R}$
$\displaystyle=\nabla_{q^{\prime}}R(\dot{q},\nabla F)\dot{q}$
$\displaystyle=(\nabla_{q^{\prime}}R)(\dot{q},\nabla
F)\dot{q}+R(\nabla_{q^{\prime}}\dot{q},\nabla F)\dot{q}$
$\displaystyle\quad+R(\dot{q},\nabla_{q^{\prime}}\nabla
F)\dot{q}+R(\dot{q},\nabla F)\nabla_{q^{\prime}}\dot{q}$
$\displaystyle=R(\dot{q},\nabla_{q^{\prime}}\nabla F)\dot{q}$
$\displaystyle=R(\dot{q},q^{\prime})\dot{q}$

where we have used the fact that $\nabla_{q^{\prime}}\nabla
F(q,\dot{q}_{*})|_{q=q_{*}(t)}=q^{\prime}$, $\nabla F(q_{*},q_{*})=0$. The
second line of (27) holds because one can take $s\mapsto q(s,t)$ as a
geodesic. Substituting (26) and (27) into the EL equation we immediately get

$\frac{D^{2}q^{\prime}}{dt}=-k_{1}\frac{Dq^{\prime}}{dt}-k_{2}q^{\prime}-\nabla_{q^{\prime}}\nabla
V.$ (28)

######  Theorem 3.20.

Let $(q_{*}(\cdot),\dot{q}_{*},u_{*}\equiv 0)$ be a bounded feasible solution
to the fully-actuated Euler-Lagrangian system (24). If the Hessian of the
potential function $V$ is bounded along $q_{*}(\cdot)$, then the controller
(25) renders $(q_{*}(\cdot),\dot{q}_{*}(\cdot)$ LES for $k_{1}>0$ and
$k_{2}>0$ large enough.

######  Proof 3.21.

If the Hessian of $V$ is bounded along $q_{*}(\cdot)$, then it is obvious that
the “linear system” (28) is exponentially stable setting $k_{1}>0$ and
choosing $k_{2}>0$ large enough. The theorem follows invoking Theorem 3.17.

######  Remark 3.22.

Note that the assumption of Theorem 3.20 holds if $V\in\mathcal{C}^{2}$ as
$q_{*}(\cdot)$ is bounded. If $V$ is (weakly) convex, then the Hessian of $V$
is positive semi-definite, hence the same holds true for arbitrary positive
constants $k_{1},k_{2}$.

######  Remark 3.23.

In equation (28), we have in fact obtained the celebrated Jacobi equation by
setting $u=0$ and $V=0$:

$\frac{D^{2}q^{\prime}}{dt^{2}}=-R(\dot{q},q^{\prime})\dot{q}.$ (29)

Since in this case the EL equation reads $\nabla_{\dot{q}}\dot{q}=0$ (geodesic
equation), equation (29) characterizes the effect of curvature to the geodesic
flow. The Jacobi equation plays a significant role in Riemannian geometry and
has many important implications. In order to help readers from the control
community appreciate this equation, we now provide a control flavour to it.

For (29), choose a “Lyapunov function”

$V(\dot{q},q^{\prime})=\langle\frac{Dq^{\prime}}{dt},\frac{Dq^{\prime}}{dt}\rangle+\langle
R(\dot{q},q^{\prime})\dot{q},q^{\prime}\rangle.$

Since we work only locally, let us consider a constant curvature manifold,
that is

$\langle
R(\dot{q},q^{\prime})\dot{q},q^{\prime}\rangle=K\langle\dot{q},\dot{q}\rangle\langle
q^{\prime},q^{\prime}\rangle,\quad\forall\dot{q},q^{\prime}$

for some constant $K$. The time derivative of $V$ reads

$\displaystyle\dot{V}$
$\displaystyle=2\langle\frac{D^{2}q^{\prime}}{dt^{2}},\frac{Dq^{\prime}}{dt}\rangle+\langle
R(\dot{q},\frac{Dq^{\prime}}{dt})\dot{q},q^{\prime}\rangle+\langle
R(\dot{q},q^{\prime})\dot{q},\frac{Dq^{\prime}}{dt}\rangle$
$\displaystyle=2\langle-R(\dot{q},q^{\prime})\dot{q},\frac{Dq^{\prime}}{dt}\rangle+2\langle
R(\dot{q},q^{\prime})\dot{q},\frac{Dq^{\prime}}{dt}\rangle$ $\displaystyle=0,$

where we have used the fact that $\frac{D\dot{q}}{dt}=0$. Remember that
$q(\cdot)$ is a geodesic, we may assume $|\dot{q}|=1$, then it follows that

$V(\dot{q},q^{\prime})=|Dq^{\prime}/dt|^{2}+K|q^{\prime}|^{2}=\text{
constant}.$

Therefore, we can draw the following non-rigorous conclusions:

  * •

$K>0$: along a given geodesic, nearby geodesics oscillate around it (see Fig.
2).

  * •

$K<0$: along a given geodesic, nearby geodesics have a trend to diverge.

  * •

$K=0$: the geodesics neither converge nor diverge.

Figure 2: For $K>0$, the geodesics oscillate near a given geodesic.

In the above we have studied tracking controller design for fully-actuated EL
systems. This problem becomes more involved for under-actuated systems. In
that case, we may apply energy shaping method to obtain some matching
conditions and then try to solve some PDEs on the manifolds [37], see also
[38] and the references therein.

###  3.2 Speed Observer for EL Systems

Consider the EL system without input

$\nabla_{\dot{q}}\dot{q}=-\nabla V(q)$ (30)

where $V(q)$ is the potential energy. The objective is to design a speed
observer for $\dot{q}(\cdot)$ knowing $q(\cdot)$. In [39], Aghannan and
Rouchon proposed the following intrinsic speed observer for the system (30)
when there is no potential energy in the EL equation:

$\left\\{\begin{aligned} \dot{\hat{q}}&=\hat{v}-\alpha\nabla F(\hat{q},q)\\\
\nabla_{\dot{\hat{q}}}\hat{v}&=-\beta\nabla F(\hat{q},q)+R(\hat{v},\nabla
F)\hat{v}.\end{aligned}\right.$ (31)

where $F$ is half of the square distance as before. The convergence of this
observer was analyzed in local coordinates via contraction analysis [39],
which was, in our opinion, quite tedious.

######  Remark 3.24.

Using the notation introduced in Section 2.5, we may rewrite (31) as

$\left\\{\begin{aligned} \dot{\hat{q}}&=\hat{v}+\alpha\log_{\hat{q}}q\\\
\nabla_{\dot{\hat{q}}}\hat{v}&=\beta\nabla\log_{\hat{q}}q-R(\hat{v},\log_{\hat{q}}q)\hat{v}.\end{aligned}\right.$

obviating the use of the square distance function.

In this subsection, we provide a much simpler proof using the methods
developed in this paper. Note that our model contains non-vanishing potential
energy function, thus it is an extension to the free Lagrangian case in [39].

To cope with the potential energy, we consider a slightly modified version of
(31):

$\left\\{\begin{aligned} \dot{\hat{q}}&=\hat{v}-\alpha\nabla F(\hat{q},q)\\\
\nabla_{\dot{\hat{q}}}\hat{v}&=-\beta\nabla F(\hat{q},q)+R(\hat{v},\nabla
F)\hat{v}-P_{q}^{\hat{q}}\nabla V(q).\end{aligned}\right.$ (32)

Note that by construction, $(q(\cdot),\dot{q}(\cdot))$ is a solution to the
observer. Hence it suffices to study LES of $(q(\cdot),\dot{q}(\cdot))$.

Substituting $\hat{v}=\dot{\hat{q}}+\alpha\nabla F(\hat{q},q)$ into the second
line of (32), we get

$\displaystyle\nabla_{\dot{\hat{q}}}(\dot{\hat{q}}+\alpha\nabla F)=$
$\displaystyle-\beta\nabla F+R(\dot{\hat{q}}+\alpha\nabla F,\nabla
F)(\dot{\hat{q}}+\alpha\nabla F)$ $\displaystyle\;-P_{q}^{\hat{q}}\nabla V(q)$

or

$\displaystyle\nabla_{\dot{\hat{q}}}\dot{\hat{q}}=-\alpha\nabla_{\dot{\hat{q}}}\nabla
F-$ $\displaystyle\beta\nabla F+R(\dot{\hat{q}},\nabla
F)(\dot{\hat{q}}+\alpha\nabla F)$ $\displaystyle-P_{q}^{\hat{q}}\nabla V(q)$

Taking covariant derivative along $q(\cdot)$ on both sides yields

$\nabla_{{q}^{\prime}}\nabla_{\dot{\hat{q}}}\dot{\hat{q}}=\frac{D^{2}{q}^{\prime}}{dt^{2}}+R(\dot{\hat{q}},{q}^{\prime})\dot{\hat{q}},$
(33)

on the left, and

$\displaystyle-\alpha\nabla_{{q}^{\prime}}\nabla_{\dot{\hat{q}}}\nabla
F-\beta\nabla_{{q}^{\prime}}\nabla
F+\nabla_{{q}^{\prime}}[R(\dot{\hat{q}},\nabla F)(\dot{\hat{q}}+\alpha\nabla
F)]$ $\displaystyle=$
$\displaystyle-\alpha\nabla_{\dot{\hat{q}}}\nabla_{{q}^{\prime}}\nabla
F-\alpha R(\dot{\hat{q}},{q}^{\prime})\nabla
F-\beta\nabla_{{q}^{\prime}}\nabla F$
$\displaystyle\;+\nabla_{{q}^{\prime}}[R(\dot{\hat{q}},\nabla
F)(\dot{\hat{q}}+\alpha\nabla F)]$ $\displaystyle=$
$\displaystyle-\alpha\nabla_{\dot{\hat{q}}}{q}^{\prime}-\beta{q}^{\prime}+R(\dot{\hat{q}},\nabla_{{q}^{\prime}}\nabla
F)\dot{\hat{q}}$ $\displaystyle=$
$\displaystyle-\alpha\nabla_{\dot{\hat{q}}}{q}^{\prime}-\beta{q}^{\prime}+R(\dot{\hat{q}},{q}^{\prime})\dot{\hat{q}},$

on the right, where we have used the relations $\nabla F|_{\hat{q}=q}=0$,
$\nabla_{q^{\prime}}\nabla F|_{\hat{q}=q}=q^{\prime}$ and
$\nabla_{q^{\prime}}P_{q}^{\hat{q}}\nabla V(q)=0$ (be $q^{\prime}$ tangent to
a geodesic). Combining this with (33) yields

$\frac{D^{2}q^{\prime}}{dt^{2}}+\alpha\frac{Dq^{\prime}}{dt}+\beta
q^{\prime}=0.$ (34)

This, together with Theorem 3.17 shows the local exponential convergence of
the observer.

######  Remark 3.25.

Notice that in both the tracking controller and observer design, we have to
calculate the geodesic distance. Although there are efficient computation
schemes, it is still tempting to avoid computing geodesics. This may be solved
by embedding the system into Euclidean space and use equivalent distance
functions in Euclidean spaces. The example of observer design on $SO(3)$ in
Section 2.4 has used this method.

##  4 Conclusion

In this paper, we have proposed a novel intrinsic approach for analyzing local
exponential stability of trajectories and contraction. The advantages of our
approach have been justified by applications and improved analysis of some
existing works in the literature. We leave studies of concrete examples
including under-actuated mechanical systems for future research.

##  5 Acknowledgement

We thank Prof. Antoine Chaillet, who gave important comments and suggestions
through the writing of the paper.

##  6 Appendix

We collect some elementary formulas in Riemannian geometry as a reference for
the reader. They can be found in standard texts such as [17, 18]. Let $(M,g)$
be a smooth Riemannian manifold. The Levi-Civita connection on $M$ is
compatible with the metric $g$: for any three vector fields
$X,Y,Z\in\Gamma(M)$,
$X\left<Y,Z\right>=\left<\nabla_{X}Y,Z\right>+\left<Y,\nabla_{X}Z\right>$. The
Levi-Civita connection is torsion-free in the sense that
$\nabla_{X}Y-\nabla_{Y}X=[X,Y]$, where $[X,Y]$ is the Lie bracket. Given a
curve $q:t\mapsto q(t)$ in $M$ and a vector field $v(t)$ along $q(\cdot)$, the
covariant derivative of $v(\cdot)$ along $q(\cdot)$ is defined as
$\frac{Dv(t)}{dt}:=\nabla_{\dot{q}(t)}v(t)$. Given a $2$-surface parameterized
by $(s,t)\mapsto q(s,t)$, then there holds

$\frac{D}{ds}\frac{\partial q}{\partial t}=\frac{D}{dt}\frac{\partial
q}{\partial s}.$ (35)

The gradient of a scalar function $f$ on $M$ is defined as the unique vector
$\nabla f$ satisfying $\left<\nabla f,X\right>=df(X)$. The Hessian of a scalar
function is a symmetric bilinear form on $TM$ defined as

$\operatorname*{Hess}f(X,Y):=\left<\nabla_{X}\nabla f,Y\right>,\;\forall
X,Y\in\Gamma(M).$ (36)

For a parameterized surface $(s,t)\to q(s,t)$ and a vector field along the
surface, there holds

$\frac{D}{ds}\frac{DX}{dt}-\frac{D}{dt}\frac{DX}{ds}=R\left(\frac{\partial
q}{\partial t},\frac{\partial q}{\partial s}\right)X.$ (37)

A metric on a Lie group $G$ is bi-invariant if it is both left-invariant,
i.e., $dL_{x}\left<v,w\right>=\left<v,w\right>$ and right-invariant. For a bi-
invariant metric, the Levi-Civita connection admits a simple formula

$\nabla_{X}Y=\frac{1}{2}[X,Y].$ (38)

A vector field $X$ on is called a Killing field (w.r.t. $g$) if $L_{X}g=0$.
Consequently, if $X$ is Killing, $Y$ an arbitrary vector field, there holds

$g(\nabla_{Y}X,Y)=0.$ (39)

######  Lemma 6.26.

Given $\gamma_{1},\gamma_{2}\in\mathcal{C}^{1}(\mathbb{R}_{+};M)$, where $M$
is a Riemannian manifold. If $\gamma_{1}(0)=\gamma_{2}(0)=x$ and
$\gamma_{1}^{\prime}(0)=\gamma_{2}^{\prime}(0)=v$, then
$d(\gamma_{1}(s),\gamma_{2}(s))=O(s^{2})$ when $s>0$ is sufficiently small.

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