# Safe Control of Euler-Lagrange Systems with Limited Model Information

Yujie Wang and Xiangru Xu Y. Wang and X. Xu are with the Department of
Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706,
USA. Email: {yujie.wang<EMAIL_ADDRESS>
###### Abstract

This paper presents a new safe control framework for Euler-Lagrange (EL)
systems with limited model information, external disturbances, and measurement
uncertainties. The EL system is decomposed into two subsystems called the
proxy subsystem and the virtual tracking subsystem. An adaptive safe
controller based on barrier Lyapunov functions is designed for the virtual
tracking subsystem to ensure the boundedness of the safe velocity tracking
error, and a safe controller based on control barrier functions is designed
for the proxy subsystem to ensure controlled invariance of the safe set
defined either in the joint space or task space. Theorems that guarantee the
safety of the proposed controllers are provided. In contrast to existing safe
control strategies for EL systems, the proposed method requires much less
model information and can ensure safety rather than input-to-state safety.
Simulation results are provided to illustrate the effectiveness of the
proposed method.

##  I Introduction

Safe-by-design control has received increasing interest because of its broad
applications. Control Barrier Functions (CBFs) and Barrier Lyapunov Function
(BLFs) are two widely investigated barrier type functions that can provably
ensure _safety_ expressed as the controlled invariance of a given set [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11]. By integrating the CBF constraint into a convex
quadratic program (QP), a CBF-QP-based controller is capable of serving as a
safety filter that minimally alters possibly unsafe control inputs. In
contrast, BLFs are Lyapunov-like functions defined in given open sets, such
that they can ensure safety and stability simultaneously.

Euler-Lagrange (EL) systems, which represent a large number of mechanical
systems including robot manipulators and vehicles, have been extensively
investigated in the literature [12, 13, 14, 15]. Recently, the safe control of
EL systems attracted significant attention because of the broad application of
robotic systems in safety-critical scenarios, such as human-robot interaction.
Many CBF-based control strategies have been developed for EL systems [16, 17,
18]. Although these methods are demonstrated by both theoretical analysis and
simulation/experimental results, they rely on model information of the EL
system (i.e., the exact forms of the inertia matrix, the Coriolis/centripetal
matrix, and the gravity term), which is hard to obtain precisely in practice.
Few research has been devoted to the safe control of EL systems with limited
model information [19, 20]. In [19], a novel CBF that integrates kinetic
energy with the classical form is proposed, resulting in reduced model
dependence and less conservatism; however, this method does not take account
of external disturbances, which are ubiquitous in practical applications. In
[20], a safe velocity is designed based on reduced-order kinematics and
tracked by a velocity tracking controller; nevertheless, only _input-to-state
safety_ [21, Definition 3] rather than safety is ensured when the model
information is unavailable, and the safe velocity is required to be
differentiable. On the other hand, various BLF-based controllers have been
developed for EL systems [8, 9], whereas these approaches require the desired
trajectory to stay inside the safe set and impose relatively strict structural
requirements on safety constraints.

Figure 1: Illustration of the proposed proxy-CBF-BLF control design scheme for
safe control design of EL systems in the joint space. The original EL system
is decomposed into the proxy subsystem and the virtual tracking subsystem. The
safe velocity for the virtual tracking subsystem is generated by the proxy
subsystem. A CBF-QP-based controller is designed for the proxy subsystem to
ensure safety, while an adaptive BLF-based control law is proposed for the
virtual tracking subsystem to constrain the safe velocity tracking error.

In this work, we propose a new control strategy for EL systems with limited
model information, external disturbances, and measurement uncertainties. The
original EL system is decomposed into two subsystems: the proxy subsystem,
which is a double integrator with a mismatched bounded disturbance, and the
virtual tracking subsystem, which corresponds to the dynamical model of the EL
system. A CBF-based controller is designed for the proxy subsystem to generate
the safe velocity, while an adaptive BLF-based controller is developed for the
virtual tracking subsystem to track the safe velocity and ensure the
boundedness of the tracking error. See Fig. 1 for illustration, where the
symbols will be introduced in Section III. Compared with existing results, the
proposed method has four main advantages as shown in the following:

  1. 1.

The proposed method does not rely on any model information except for the
upper bound of the inertia matrix’s norm, which implies that even the bounds
of the Coriolis-centrifugal and gravity matrices are not required in control
design because such bounds are estimated by adaptive laws online.

  2. 2.

The closed-loop system is guaranteed to be safe, instead of input-to-state
safe, in the presence of external disturbances and measurement uncertainties.

  3. 3.

The safe velocity’s differentiability, which is important for velocity
tracking control design, is guaranteed, and calculating its derivative is
straightforward.

  4. 4.

The proposed method takes measurement uncertainties into account, allowing its
use in robots where precise angular velocity measurements are not available.

The remainder of this paper is organized as follows. In Section II,
preliminaries and the problem statement are introduced; in Section III, the
joint space safe control strategy is presented; in Section IV, the task space
safe control scheme is shown; in Section V, numerical simulation results are
presented to validate the proposed method; and finally, the conclusion is
drawn in Section VI.

##  II Preliminaries and Problem Statement

Throughout the paper, we denote by $\mathbb{R}_{>0}$ and $\mathbb{R}_{\geq 0}$
the sets of positive real and nonnegative numbers, respectively. We denote
$\|\cdot\|$ the 2-norm for vectors and the induced 2-norm for matrices. We
denote by $\sigma_{\rm min}(A)$ the smallest eigenvalue of a square matrix
$A$. We consider the gradient $\frac{\partial h}{\partial
x}\in\mathbb{R}^{n\times 1}$ as a row vector, where $x\in\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\to\mathbb{R}$ is a function with respect to $x$.

###  II-A Control Barrier Functions & Barrier Lyapunov Functions

CBFs and BLFs are two types of barrier functions that are widely used to
ensure the controlled invariance of a given set [1, 5]. Our approach aims to
combine the advantages of both CBFs and BLFs, which are briefly reviewed
below.

####  II-A1 Control Barrier Functions

Consider a control-affine system given as $\dot{x}=f(x)+g(x)u$ where
$x\in\mathbb{R}^{n}$ is the state, $u\in U\subset\mathbb{R}^{m}$ is the
control input, and $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and
$g:\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$ are locally Lipchitz continuous
functions. Define a _safe set_ $\mathcal{C}=\\{x\in\mathbb{R}^{n}:h(x)\geq
0\\}$ where $h$ is a continuously differentiable function. The function $h$ is
called a _(zeroing) CBF_ of relative degree 1, if there exists a constant
$\gamma>0$ such that $\sup_{u\in U}\left[L_{f}h(x)+L_{g}h(x)u+\gamma
h(x)\right]\geq 0$ where $L_{f}h(x)=\frac{\partial h}{\partial x}f(x)$ and
$L_{g}h(x)=\frac{\partial h}{\partial x}g(x)$ are Lie derivatives [2]. In this
paper, we assume there is no constraint on the input $u$, i.e.,
$U=\mathbb{R}^{m}$. The following result that guarantees the forward
invariance of $\mathcal{C}$ is given in [2].

###### Lemma 1

[2, Corollary 7] If $h$ is a (zeroing) CBF on $\mathbb{R}^{n}$, then any
Lipschitz continuous controller $u:\mathbb{R}^{n}\to U$ such that $u(x)\in
K(x)\triangleq\\{u\in U\mid L_{f}h(x)+L_{g}h(x)u+\gamma h(x)\geq 0\\}$ will
guarantee the forward invariance of $\mathcal{C}$, i.e., the _safety_ of the
closed-loop system.

By including the CBF condition into a convex QP, the provably safe controller
is obtained by solving a CBF-QP online. The time-varying CBF and its safety
guarantee for a time-varying system are discussed in [22].

####  II-A2 Barrier Lyapunov Function

In contrast to CBFs, BLFs are positive definite functions that are more
tightly connected with Lyapunov functions.

###### Definition 1

[5, Definition 2] A barrier Lyapunov function is a scalar function $V(x)$,
defined with respect to the system $\dot{x}=f(x)$ on an open region
$\mathcal{D}$ containing the origin, that is continuous, positive definite,
has continuous first-order partial derivatives at every point of
$\mathcal{D}$, has the property $V(x)\to\infty$ as $x$ approaches the boundary
of $\mathcal{D}$, and satisfies $V(x(t))\leq b$ for any $t>0$ along the
solution of $\dot{x}=f(x)$ for $x(0)\in\mathcal{D}$ and some positive constant
$b$.

The following lemma is used for BLF control design to guarantee that
constraints on the output or state are satisfied.

###### Lemma 2

[5, Lemma 1] For any positive constants $k_{a_{1}}$, $k_{b_{i}}$, let
$\mathcal{Z}_{1}\triangleq\\{z_{1}\in\mathbb{R}:-k_{a_{1}}<z_{1}<k_{b_{1}}\\}\subset\mathbb{R}$
and
$\mathcal{N}\triangleq\mathbb{R}^{l}\times\mathcal{Z}_{1}\subset\mathbb{R}^{l+1}$
be open sets. Consider the system $\dot{\eta}=h(\eta,t)$ where
$\eta\triangleq[w,z_{1}^{\top}]\in\mathcal{N}$, and $h:\mathbb{R}_{\geq
0}\times\mathcal{N}\to\mathbb{R}^{l+1}$ is piecewise continuous in $t$ and
locally Lipschitz in $z$, uniformly in $t$, on $\mathbb{R}_{\geq
0}\times\mathcal{N}$. Suppose that there exist functions
$U:\mathbb{R}^{l}\to\mathbb{R}_{\geq 0}$ and
$V_{1}:\mathcal{Z}_{1}\to\mathbb{R}_{\geq 0}$, continuously differentiable and
positive definite in their respective domains, such that
$V_{1}(z_{1})\to\infty$ as $\ z_{1}\to-k_{a_{1}}$ or $z_{1}\to k_{b_{1}}$, and
$\gamma_{1}(\|w\|)\leq U(w)\leq\gamma_{2}(\|w\|)$, where $\gamma_{1}$ and
$\gamma_{2}$ are class $\mathcal{K}_{\infty}$ functions. Let
$V(\eta)\triangleq V_{1}(z_{1})+U(w)$, and $z_{1}(0)$ belong to the set
$z_{1}\in(-k_{a_{1}},k_{b_{1}})$. If the inequality $\dot{V}\leq\frac{\partial
V}{\partial\eta}h\leq 0$ holds, then $z_{1}(t)$ remains in the open set
$z_{1}\in(-k_{a_{1}},k_{b_{1}})$, $\forall t\in[0,\infty)$.

###  II-B Euler-Lagrange Systems

Consider an EL system given as follows [13, 23]:

$\displaystyle\dot{q}$ $\displaystyle=$ $\displaystyle\omega,$ (1a)
$\displaystyle\dot{\omega}$ $\displaystyle=$ $\displaystyle
M^{-1}(q)\left(\tau-C(q,\omega)\omega-G(q)+\tau_{d}\right),$ (2a)

where $q\in\mathbb{R}^{n}$ is the generalized coordinate,
$\omega\in\mathbb{R}^{n}$ is the generalized velocity, $\tau\in\mathbb{R}^{n}$
is the control input, $\tau_{d}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}$ is the
external disturbance, $M:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ is the
inertia matrix, $C:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n\times
n}$ is the Coriolis/centripetal matrix, and
$G:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the gravity term. We assume that the
exact knowledge of the velocity $\omega$ is not known, and denote the measured
generalized velocity as $\hat{\omega}$ (e.g., in some application scenarios,
$\omega$ is obtained by numerically differentiating $q$ such that it may be
contaminated by measurement noise); therefore the velocity measurement
uncertainty can be defined as

$\xi=\omega-\hat{\omega}.$

Furthermore, we assume that $\tau_{d}$, $\xi$, and $\dot{\xi}$ are all
bounded.

###### Assumption 1

The disturbance $\tau_{d}$ satisfies $\|\tau_{d}\|\leq D_{0}$ where $D_{0}>0$
is a positive constant.

###### Assumption 2

The measurement uncertainty $\xi$ and its derivative $\dot{\xi}$ are bounded
as $\|\xi\|\leq D_{1}$ and $\|\dot{\xi}\|\leq D_{2}$, where $D_{1}$ and
$D_{2}$ are positive constants.

Note that Assumption 1 is extensively used in the robust control literature,
and numerous state estimation techniques have been developed to ensure that
the state estimation error is bounded.

The system given in (1a) has the following properties that will be exploited
in the subsequent control design [24].

######  Property 1 (P1)

The matrix $M$ is positive definite, symmetric, and satisfies

$\displaystyle\lambda_{1}\|q\|^{2}\leq
q^{\top}M(q)q\leq\lambda_{2}\|q\|^{2},\quad\forall q\in\mathbb{R}^{n},$ (3)

where $\lambda_{1}$, $\lambda_{2}$ are positive constants.

######  Property 2 (P2)

The matrices $C(q,\omega)$ and $G(q)$ satisfy

$\displaystyle\|C(q,\omega)\|\leq\zeta_{c}\|\omega\|,\quad\|G(q)\|\leq\zeta_{g},\quad\forall
q,\omega\in\mathbb{R}^{n},$ (4)

where $\zeta_{c}$ and $\zeta_{g}$ are positive constants.

###  II-C Problem Statement

In this work, we consider provably safe control design for an EL system given
in (1a) with _limited information_. Specifically, we assume that the matrices
$M,C,G$ in (1a) are unknown and satisfy inequalities (3) and (4) but only
$\lambda_{2}$ is known. With such an EL system, the first problem we aim to
solve is to design a feedback controller based on the knowledge of $q$ and
$\hat{\omega}$ to ensure the safety of the system in the joint space.

###### Problem 1

Consider an EL system described by (1a) where the matrices $M,C,G$ are
unknown, and a joint space safe set $\mathcal{C}_{q}$ defined as

$\mathcal{C}_{q}=\\{q\in\mathbb{R}^{n}:h(q)\geq 0\\},$ (5)

where $h$ is a twice differentiable function. Suppose that Assumptions 1 and 2
hold with $D_{0}$, $D_{2}$ unknown, and $M,C,G$ satisfy inequalities (3) and
(4) with constant $\lambda_{2}$ known and constants
$\lambda_{1},\zeta_{c},\zeta_{g}$ unknown. Design a feedback control law
$\tau(q(t),\hat{\omega}(t),t)$ such that the closed-loop system is always safe
with respect to $\mathcal{C}_{q}$, i.e., $h(q(t))\geq 0,\forall t\geq 0$.

The second problem we aim to solve is about designing a safe controller in the
task space.

###### Problem 2

Consider an EL system described by (1a) where the matrices $M,C,G$ are
unknown, and the forward kinematics of the EL system:

$p=f(q),$ (6)

where $p\in\mathbb{R}^{k}$ denotes the variable of the task space and
$f:\mathbb{R}^{n}\to\mathbb{R}^{k}$ represents a continuously differentiable
function with $k\leq n$. Consider a task space safe set $\mathcal{C}_{p}$
defined as

$\mathcal{C}_{p}=\\{p\in\mathbb{R}^{p}:h(p)\geq 0\\},$ (7)

where $h$ is a twice differentiable function. Suppose that Assumptions 1 and 2
hold with $D_{0},D_{2}$ unknown, and $M,C,G$ satisfy inequalities (3) and (4)
with constant $\lambda_{2}$ known and constants
$\lambda_{1},\zeta_{c},\zeta_{g}$ unknown. Design a feedback control law
$\tau(p(t),q(t),\hat{\omega}(t),t)$ such that the closed-loop system is safe
with respect to $\mathcal{C}_{p}$, i.e., $h(p(t))\geq 0,\forall t\geq 0$.

The main difficulty of Problems 1 and 2 lies in the limited information of the
EL system: $\lambda_{1}$, $D_{0}$, $D_{2}$, $\zeta_{c}$, $\zeta_{g}$ are
assumed to be unknown in control design. The proposed controller in this work
is highly robust to model uncertainties and can be easily transferred between
different EL systems without re-designing the control laws. Existing safe
control design approaches for EL systems are not applicable to solve the
problems in this work because they rely on the exact forms of $M$, $C$, $G$ or
the values of $\lambda_{1}$, $D_{0}$, $D_{2}$, $\zeta_{c}$, $\zeta_{g}$; see
[14, 15, 16, 17, 18, 19, 20] for more details.

##  III Joint Space Safe Control

In this section, a novel proxy-CBF-BLF-based method will be presented to solve
Problem 1 for the EL system with limited information, external disturbances,
and measurement uncertainties. We will show the main idea of the method in
Subsection III-A, propose an adaptive BLF-based control design approach for
the virtual tracking subsystem in Subsection III-B, and a CBF-based control
design strategy for the proxy subsystem in Subsection III-C.

###  III-A Method Overview

The main idea of our method is to decompose an EL system into two subsystems,
called the proxy111The term “proxy” is inspired by proxy-based sliding mode
control [25] and haptic rendering [26]. subsystem and the virtual tracking
subsystem, and use the CBF and BLF to design safe controllers for the two
subsystems, respectively, such that the overall controller will ensure the
safety of the EL system (see Fig. 1 for illustration).

The proxy subsystem is given as:

$\displaystyle\dot{q}$ $\displaystyle=$ $\displaystyle\mu+e_{q}+\xi,$ (8a)
$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle\nu,$ (9a)

where $\mu$ is the virtual safe velocity with $\mu(0)=\hat{\omega}(0)$,
$e_{q}$ is the virtual velocity tracking error defined as

$\displaystyle e_{q}=\hat{\omega}-\mu,$ (10)

and $\nu$ is the virtual control input to be designed. Note that (8a) is
equivalent to (1a) augmented with an integrator.

The virtual tracking subsystem is given as:

$\dot{e}_{q}=M(q)^{-1}(\tau-C(q,\omega)\omega-G(q)+\tau_{d})-\dot{\xi}-\nu$
(11)

where $\tau$ is the control input to be designed and $\nu$ is from the proxy
subsystem (8a).

With this decomposition, Problem 1 can be solved by accomplishing two tasks
shown as follows.

###### Task 1

For the virtual tracking subsystem (11), design a controller $\tau$ to
guarantee

$\displaystyle\|e_{q}(t)\|<L,\forall t\geq 0,$ (12)

where $L>0$ is an arbitrary positive constant.

###### Task 2

For the proxy subsystem (8a), design a control law $\nu$ to ensure
$h(q(t))\geq 0,\forall t\geq 0$, under the assumption that
$\|e_{q}(t)\|<L,\forall t\geq 0$.

###### Remark 1

In [20], a safe velocity is designed based on reduced-order kinematics, which
is similar to (8a) in our proxy subsystem. However, including an additional
integrator as shown in (9a) is important because $\dot{\mu}$, which is equal
to $\nu$, is required in the virtual tracking subsystem (11) and $L$ can be
selected to be arbitrarily small, thereby reducing the potential conservatism
of the safe controller (see Remark 4). Nevertheless, the added integrator will
result in a system with a mismatched virtual disturbance, $e_{q}+\xi$; a new
CBF-based safe control scheme will be proposed for such a system in Section
III-C.

###  III-B BLF-based Control For the Virtual Tracking Subsystem

In this subsection, an adaptive BLF-based controller will be presented to
accomplish Task 1. The BLF-based method is suitable for this task because it
does not rely on the bounds of the unknown parameters and the external
disturbances.

Inspired by our previous work [27], the following theorem presents a
controller $\tau$ for the virtual tracking subsystem to ensure
$\|e_{q}(t)\|<L,\forall t\geq 0$.

###### Theorem 1

Consider the virtual tracking subsystem (11) where the matrices $M,C,G$ are
unknown. Suppose that Assumptions 1 and 2 hold with $D_{0}$, $D_{2}$ unknown,
and $M,C,G$ satisfy inequalities (3) and (4) with constant $\lambda_{2}$ known
and constants $\lambda_{1},\zeta_{c},\zeta_{g}$ unknown. Suppose that the
controller $\tau$ is designed as

$\displaystyle\tau$ $\displaystyle=-\lambda_{2}e_{q}\mathcal{N}$ (13)

where

$\displaystyle\mathcal{N}\\!$ $\displaystyle=$
$\displaystyle\\!k_{1}\\!+\\!\frac{(\hat{\theta}_{1}\varphi)^{2}}{\hat{\theta}_{1}\varphi\|e_{q}\|\\!+\\!\epsilon_{1}}\\!+\\!\frac{\hat{\theta}_{2}^{2}}{\hat{\theta}_{2}\|e_{q}\|\\!+\\!\epsilon_{2}}\\!+\\!\frac{\|\nu\|^{2}}{\|e_{q}\|\|\nu\|\\!+\\!\epsilon},$
(14a) $\displaystyle\dot{\hat{\theta}}_{1}$ $\displaystyle=$
$\displaystyle-\gamma_{\theta}\hat{\theta}_{1}+\frac{\|e_{q}\|\varphi}{L^{2}-\|e_{q}\|^{2}},$
(15a) $\displaystyle\dot{\hat{\theta}}_{2}$ $\displaystyle=$
$\displaystyle-\gamma_{\theta}\hat{\theta}_{2}+\frac{\|e_{q}\|}{L^{2}-\|e_{q}\|^{2}},$
(16a)

and $\varphi=(\|\hat{\omega}\|+D_{1})^{2}$, with positive constants
$\epsilon,\epsilon_{1},\epsilon_{2},\gamma_{\theta}>0$,
$k_{1}>\frac{\Lambda}{L^{2}}$, and
$\Lambda=\epsilon+\epsilon_{1}+\epsilon_{2}$. If
$\hat{\theta}_{1}(0),\hat{\theta}_{2}(0)>0$, then $\|e_{q}(t)\|<L$ for any
$t\geq 0$.

###### Proof:

From (15a)-(16a),
$\dot{\hat{\theta}}_{1}\geq-\gamma_{\theta}\hat{\theta}_{1},\dot{\hat{\theta}}_{1}\geq-\gamma_{\theta}\hat{\theta}_{2}$
hold in the open set
$\mathcal{Z}_{L}\triangleq\\{e_{q}\in\mathbb{R}^{n}\mid\|e_{q}\|<L\\}$. Since
$\hat{\theta}_{1}(0)>0,\hat{\theta}_{2}(0)>0$, it is easy to see that
$\hat{\theta}_{1}(t)\geq 0$ and $\hat{\theta}_{2}(t)\geq 0$ for any $t\geq 0$
by the Comparison Lemma [28, Lemma 2.5].

Define $\theta_{1}=\zeta_{c}\lambda_{1}^{-1}$ and
$\theta_{2}=\lambda_{1}^{-1}(\zeta_{g}+D_{0})+D_{2}$, which are unknown
parameters because $\lambda_{1},\zeta_{c},\zeta_{g},D_{2}$ are unknown. Define
a candidate BLF as

$V=\frac{1}{2}\log\left(\frac{L^{2}}{L^{2}-\|e_{q}\|^{2}}\right)+\frac{1}{2}\tilde{\theta}_{1}^{2}+\frac{1}{2}\tilde{\theta}_{2}^{2},$
(17)

where $\tilde{\theta}_{1}=\theta_{1}-\hat{\theta}_{1}$,
$\tilde{\theta}_{2}=\theta_{2}-\hat{\theta}_{2}$. The derivative of $V$ in the
open set $\mathcal{Z}_{L}$ can be expressed as

$\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\frac{e_{q}^{\top}}{L^{2}-\|e_{q}\|^{2}}(M^{-1}(\tau-C(q,\omega)\omega-G(q)+\tau_{d})$
(18)
$\displaystyle-\dot{\xi}-\nu)-\tilde{\theta}_{1}\dot{\hat{\theta}}_{1}-\tilde{\theta}_{2}\dot{\hat{\theta}}_{2}$
$\displaystyle\leq$
$\displaystyle\frac{e_{q}^{\top}M^{-1}\tau}{L^{2}-\|e_{q}\|^{2}}+\frac{\|e_{q}\|}{L^{2}-\|e_{q}\|^{2}}(\|M^{-1}\|(\|C(q,\omega)\omega\|$
$\displaystyle+\|G\|+\|\tau_{d}\|)+\|\dot{\xi}\|+\|\nu\|)-\tilde{\theta}_{1}\dot{\hat{\theta}}_{1}-\tilde{\theta}_{2}\dot{\hat{\theta}}_{2}$
$\displaystyle\leq$
$\displaystyle\frac{e_{q}^{\top}M^{-1}\tau}{L^{2}-\|e_{q}\|^{2}}+\frac{\|e_{q}\|}{L^{2}-\|e_{q}\|^{2}}(\lambda_{1}^{-1}(\zeta_{c}(\|\hat{\omega}\|+D_{1})^{2}$
$\displaystyle+\zeta_{g}+D_{0})+D_{2}+\|\nu\|)-\tilde{\theta}_{1}\dot{\hat{\theta}}_{1}-\tilde{\theta}_{2}\dot{\hat{\theta}}_{2}$
$\displaystyle=$
$\displaystyle\frac{e_{q}^{\top}M^{-1}\tau}{L^{2}-\|e_{q}\|^{2}}+\frac{\|e_{q}\|}{L^{2}-\|e_{q}\|^{2}}(\hat{\theta}_{1}\varphi+\hat{\theta}_{2}+\|\nu\|)$
$\displaystyle-\tilde{\theta}_{1}\\!\left(\\!\dot{\hat{\theta}}_{1}\\!-\\!\frac{\|e_{q}\|\varphi}{L^{2}\\!-\\!\|e_{q}\|^{2}}\\!\right)\\!-\\!\tilde{\theta}_{2}\\!\left(\\!\dot{\hat{\theta}}_{2}\\!-\\!\frac{\|e_{q}\|}{L^{2}\\!-\\!\|e_{q}\|^{2}}\\!\right),$

where the second inequality comes from

$\|C(q,\omega)\omega\|\stackrel{{\scriptstyle{\rm(P1)}}}{{\leq}}\zeta_{c}\|\omega\|^{2}=\zeta_{c}\|\hat{\omega}+\xi\|^{2}\leq\zeta_{c}(\|\hat{\omega}\|+D_{1})^{2},$

and the third inequality arises from the fact
$\lambda_{2}^{-1}\leq\|M(q)^{-1}\|\leq\lambda_{1}^{-1}$ for any
$q\in\mathbb{R}^{n}$, according to Property 1. Substituting (14a) into (18)
yields

$\displaystyle\dot{V}$ $\displaystyle\leq$
$\displaystyle\frac{1}{L^{2}\\!-\\!\|e_{q}\|^{2}}\bigg{(}\\!-\lambda_{2}\underbrace{(e_{q}^{\top}\\!M^{-1}e_{q})}_{\geq\lambda_{2}^{-1}\|e_{q}\|^{2}}\bigg{(}\\!k_{1}\\!+\\!\underbrace{\frac{(\hat{\theta}_{1}\varphi)^{2}}{\hat{\theta}_{1}\varphi\|e_{q}\|\\!+\\!\epsilon_{1}}}_{\geq
0}$ (19)
$\displaystyle\\!+\\!\underbrace{\frac{(\hat{\theta}_{2})^{2}}{\hat{\theta}_{2}\|e_{q}\|\\!+\\!\epsilon_{2}}}_{\geq
0}\\!+\\!\underbrace{\frac{\|\nu\|^{2}}{\|e_{q}\|\|\nu\|\\!+\\!\epsilon}}_{\geq
0}\bigg{)}\\!+\\!\|e_{q}\|(\hat{\theta}_{1}\varphi\\!+\\!\hat{\theta}_{2}\\!+\\!\|\nu\|)\bigg{)}$
$\displaystyle+\gamma_{\theta}(\tilde{\theta}_{1}\hat{\theta}_{1}+\tilde{\theta}_{2}\hat{\theta}_{2})$
$\displaystyle\leq$
$\displaystyle\frac{1}{L^{2}-\|e_{q}\|^{2}}\bigg{(}-k_{1}\|e_{q}\|^{2}+\left(\hat{\theta}_{1}\varphi\|e_{q}\|-\frac{(\hat{\theta}_{1}\varphi\|e_{q}\|)^{2}}{\hat{\theta}_{1}\varphi\|e_{q}\|+\epsilon_{1}}\right)$
$\displaystyle+\left(\\!\hat{\theta}_{2}\|e_{q}\|\\!-\\!\frac{(\hat{\theta}_{2}\|e_{q}\|)^{2}}{\hat{\theta}_{2}\|e_{q}\|\\!+\\!\epsilon_{2}}\right)\\!+\\!\left(\\!\|e_{q}\|\|\nu\|\\!-\\!\frac{(\|e_{q}\|\|\nu\|)^{2}}{\|e_{q}\|\|\nu\|\\!+\\!\epsilon}\\!\right)\bigg{)}$
$\displaystyle+\gamma_{\theta}(\tilde{\theta}_{1}\hat{\theta}_{1}+\tilde{\theta}_{2}\hat{\theta}_{2})$
$\displaystyle\leq$
$\displaystyle\frac{1}{L^{2}-\|e_{q}\|^{2}}\left(-k_{1}\|e_{q}\|^{2}+\Lambda\right)+\gamma_{\theta}(\tilde{\theta}_{1}\hat{\theta}_{1}+\tilde{\theta}_{2}\hat{\theta}_{2}),$

where the last inequality comes from the fact that for any $A\geq
0,\epsilon>0$,
$A-\frac{A^{2}}{A+\epsilon}=\frac{A\epsilon}{A+\epsilon}\leq\epsilon$ holds
true. Noting that
$\frac{\Lambda}{L^{2}-\|e_{q}\|^{2}}=\frac{\Lambda}{L^{2}}+\frac{\Lambda}{L^{2}}\frac{\|e_{q}\|^{2}}{L^{2}-\|e_{q}\|^{2}}$
and
$\tilde{\theta}_{i}\hat{\theta}_{i}=\tilde{\theta}_{i}(\theta_{i}-\tilde{\theta}_{i})\leq\frac{\theta_{i}^{2}-\tilde{\theta}_{i}^{2}}{2},\
i=1,2$, we have

$\displaystyle\dot{V}$ $\displaystyle\leq$
$\displaystyle-\frac{\chi\|e_{q}\|^{2}}{L^{2}-\|e_{q}\|^{2}}-\frac{\gamma_{\theta}}{2}\sum_{i=1}^{2}\tilde{\theta}_{i}^{2}+\frac{\Lambda}{L^{2}}+\frac{\gamma_{\theta}}{2}\sum_{i=1}^{2}\theta_{i}^{2}$
$\displaystyle\leq$
$\displaystyle-\chi\log\left(\frac{L^{2}}{L^{2}-\|e_{q}\|^{2}}\right)-\frac{\gamma_{\theta}}{2}\sum_{i=1}^{2}\tilde{\theta}_{i}^{2}+\frac{\Lambda}{L^{2}}+\frac{\gamma_{\theta}}{2}\sum_{i=1}^{2}\theta_{i}^{2}$
$\displaystyle\leq$ $\displaystyle-\kappa V+K,$

where $\chi=k_{1}-\frac{\Lambda}{L^{2}}$,
$K=\frac{\Lambda}{L^{2}}+\frac{\gamma_{\theta}}{2}\sum_{i=1}^{2}\theta_{i}^{2}$,
$\kappa=\min\\{2\chi,\gamma_{\theta}\\}$, and the second inequality comes from
the fact that
$\log\frac{L^{2}}{L^{2}-\|e_{q}\|^{2}}\leq\frac{\|e_{q}\|^{2}}{L^{2}-\|e_{q}\|^{2}}$
holds in the open set $\mathcal{Z}_{L}$ [6, Lemma 2]. Thus, $V(t)$ is bounded,
which implies $\|e_{q}(t)\|<L$ for any $t\geq 0$, according to Lemma 2. ∎

###  III-C CBF-based Control For the Proxy Subsystem

In this subsection, a CBF-based control law is presented to solve Task 2. Note
that designing a CBF-based controller for accomplishing Task 2 is challenging
because the term $e_{q}+\xi$ is considered as a bounded mismatched disturbance
to the proxy subsystem and its derivative, $\dot{e}_{q}+\dot{\xi}$, is not
necessarily bounded.

Since in Task 2 we assume $\|e_{q}\|<L$ holds, which is ensured by Theorem 1,
the term $e_{q}+\xi$ is bounded as

$\|e_{q}+\xi\|<D_{1}+L.$ (20)

The following theorem provides a CBF-based controller $\nu$ that ensure
$h(q(t))\geq 0,\forall t\geq 0$.

###### Theorem 2

Consider the proxy subsystem given in (8a) and a joint space safe set
$\mathcal{C}_{q}$ defined in (5). Suppose that $h(q(0))>0$, $\|e_{q}(t)\|<L$
for any $t\geq 0$, and there exist positive constants $\lambda,\gamma,\beta>0$
such that
(i) $\frac{\partial h}{\partial
q}(q(0))\mu(0)-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
q}(q(0))\right\|^{2}-\frac{\beta(D_{1}+L)^{2}}{2}+\lambda h(q(0))\geq 0$;
(ii) the set
$K_{BF}^{q}(q,\mu)=\\{{\mathfrak{u}}\in\mathbb{R}^{n}:\Psi_{0}+\Psi_{1}{\mathfrak{u}}\geq
0\\}$ is not empty for any $q\in\mathcal{C}_{q}$ and $\mu\in\mathbb{R}^{n}$,
where

$\displaystyle\Psi_{0}$ $\displaystyle=$
$\displaystyle\mathcal{M}\mu-\left\|\mathcal{M}\right\|(D_{1}+L)+\gamma\bar{h},$
(21a) $\displaystyle\Psi_{1}$ $\displaystyle=$ $\displaystyle\frac{\partial
h}{\partial q},$ (22a)

with $\mathcal{M}=\mu^{\top}{\rm H}_{h}-\frac{1}{\beta}\frac{\partial
h}{\partial q}{\rm H}_{h}+\lambda\frac{\partial h}{\partial q}$, ${\rm
H}_{h}=\frac{\partial^{2}h}{\partial q^{2}}$ denotes the Hessian, and
$\bar{h}=\frac{\partial h}{\partial
q}\mu-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
q}\right\|^{2}-\frac{\beta(D_{1}+L)^{2}}{2}+\lambda h$.
Then, any Lipschitz continuous control input $\nu\in K_{BF}^{q}(q,\mu)$ will
make $h(q(t))\geq 0$ for any $t\geq 0$.

###### Proof:

First, we show that $\nu\in K_{BF}^{q}(q,\mu)\implies\bar{h}(t)\geq 0$ for any
$t\geq 0$. Note that Condition (i) indicates that $\bar{h}(q(0),\mu(0))\geq
0$. Meanwhile, one can observe that $\dot{\bar{h}}$ can be expressed as

$\displaystyle\dot{\bar{h}}$ $\displaystyle=$ $\displaystyle\frac{\partial
h}{\partial q}\nu+\left(\mu^{\top}{\rm H}_{h}-\frac{1}{\beta}\frac{\partial
h}{\partial q}{\rm H}_{h}+\lambda\frac{\partial h}{\partial
q}\right)(\mu+e_{q}+\xi)$ $\displaystyle=$ $\displaystyle\frac{\partial
h}{\partial q}\nu+\mathcal{M}\mu+\mathcal{M}(e+\xi)$
$\displaystyle\stackrel{{\scriptstyle\eqref{exi}}}{{\geq}}$
$\displaystyle\Psi_{1}\nu+\mathcal{M}\mu-\left\|\mathcal{M}\right\|(D_{1}+L).$

Selecting $\nu\in K_{BF}^{q}(q,\mu)$ yields
$\dot{\bar{h}}\geq-\Psi_{0}+\mathcal{M}\mu-\left\|\mathcal{M}\right\|(D_{1}+L)=-\gamma\bar{h},$
which indicates $\bar{h}(q(t),\mu(t))\geq 0,\forall t\geq 0$ because
$\bar{h}(q(0),\mu(0))\geq 0$. Since

$\displaystyle\dot{h}+\lambda h$ $\displaystyle=$ $\displaystyle\frac{\partial
h}{\partial q}(\mu+e_{q}+\xi)+\lambda h$ $\displaystyle\geq$
$\displaystyle\frac{\partial h}{\partial
q}\mu-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
q}\right\|^{2}-\frac{\beta}{2}\|e_{q}+\xi\|^{2}+\lambda h$
$\displaystyle\stackrel{{\scriptstyle\eqref{exi}}}{{\geq}}$
$\displaystyle\frac{\partial h}{\partial
q}\mu-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
q}\right\|^{2}-\frac{\beta(D_{1}+L)^{2}}{2}+\lambda h$ $\displaystyle=$
$\displaystyle\bar{h}(q,\mu)\geq 0,$

one can conclude that $h(q(t))\geq 0,\forall t$ since $h(q(0))\geq 0$. ∎

The safe virtual controller proposed in Theorem 2 is obtained by solving the
following CBF-QP:

$\displaystyle\min_{\nu}\quad$ $\displaystyle\|\nu-\nu_{d}\|^{2}$ (23) s.t.
$\displaystyle\Psi_{0}+\Psi_{1}\nu\geq 0,$

where $\Psi_{0},\Psi_{1}$ are given in (21a) and $\nu_{d}$ is any given
nominal control law.

The safe feedback control law $\tau(q(t),\hat{\omega}(t),t)$ to the EL system
(1a) consists of the control law $\tau$ given in (13) and the control law
$\nu$ given in (23). By Theorems 1 and 2, the safe controller will ensure that
the closed-loop system is always safe with respect to $\mathcal{C}_{q}$, i.e.,
$h(q(t))\geq 0$ for all $t\geq 0$.

###### Remark 2

The nominal control law $\nu_{d}$ can be designed as
$\nu_{d}=-\alpha_{1}E_{q}-\alpha_{2}E_{\mu}+\ddot{q}_{d}$, where
$E_{q}\triangleq q-q_{d}$, $E_{\mu}\triangleq\mu-\dot{q}_{d}$, $q_{d}$ denotes
the reference trajectory, and $\alpha_{1},\alpha_{2}\in\mathbb{R}$ are
selected such that

$\sigma_{\rm min}\left(\begin{bmatrix}0_{n\times n}&I_{n\times n}\\\
-\alpha_{1}I_{n\times n}&-\alpha_{2}I_{n\times
n}\end{bmatrix}\right)\triangleq-\alpha<-\frac{1}{2}.$

Define a Lyapunov candidate function as
$V=\frac{1}{2}\varepsilon^{\top}\varepsilon$, where
$\varepsilon=[E_{q}^{\top}\ E_{\mu}^{\top}]^{\top}$. Since $\dot{V}$ satisfies

$\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\varepsilon^{\top}\begin{bmatrix}0_{n\times n}&I_{n\times n}\\\
-\alpha_{1}I_{n\times n}&-\alpha_{2}I_{n\times
n}\end{bmatrix}\varepsilon+E_{q}^{\top}(e_{q}+\xi)$ $\displaystyle\leq$
$\displaystyle-\left(2\alpha-1\right)V+\frac{(D_{1}+L)^{2}}{2},$

the tracking error is uniformly ultimately bounded [28].

###### Remark 3

Suppose that $q^{*}$ is the unique zero of $\Psi_{1}$ in $\mathcal{C}_{q}$. We
claim that if

${\rm H}_{h}^{*}\triangleq{\rm H}_{h}(q^{*})\succ aI_{n\times n},\
h^{*}\triangleq h(q^{*})>0,$ (24)

where $a$ is an arbitrary positive constant, then one can always find
$\gamma,\beta,\lambda>0$ such that Condition (i) and (ii) in Theorem 2 hold
true. Indeed, one can easily select $\beta$ and $\lambda$ such that Condition
(i) is fulfilled. Meanwhile, from (21a) one can observe that
$\Psi_{0}^{*}\triangleq\Psi_{0}(q^{*},\mu)=\mu^{\top}{\rm
H}_{h}^{*}\mu-\|\mu^{\top}{\rm H}_{h}^{*}\|(D_{1}+L)+\gamma\lambda
h-\frac{\beta(D_{1}+L)^{2}}{2}$ satisfies

$\displaystyle\Psi_{0}^{*}$ $\displaystyle\geq$ $\displaystyle
a\|\mu\|^{2}\\!-\\!\|{\rm
H}_{h}^{*}\|(D_{1}\\!+\\!L)\|\mu\|\\!+\\!\gamma\lambda
h^{*}\\!-\\!\frac{\beta(D_{1}\\!+\\!L)^{2}}{2}$ $\displaystyle=$
$\displaystyle a\left(\|\mu\|-\frac{\|{\rm
H}_{h}^{*}\|(D_{1}+L)}{2a}\right)^{2}+\gamma\lambda h^{*}-\Xi$
$\displaystyle\geq$ $\displaystyle\gamma\lambda h^{*}-\Xi,$

where $\Xi=\frac{\|{\rm
H}_{h}^{*}\|^{2}(D_{1}+L)^{2}}{4a}+\frac{\beta(D_{1}+L)^{2}}{2}$. It is
obvious that selecting $\gamma\geq\frac{\Xi}{\lambda h^{*}}$ will yield
$\Psi_{0}\geq 0$, such that $K_{BF}^{q}$ is not empty when $q=q^{*}$, which
shows the correctness of the claim. Furthermore, it is obvious that if
$\Psi_{1}$ has finite zeros in $\mathcal{C}_{q}$ and each zero satisfies (24),
then one can always select appropriate $\gamma,\lambda,\beta$ such that
Conditions (i) and (ii) of Theorem 2 are satisfied. Nevertheless, it should be
noticed that (24) is not the unique criterion for verifying the conditions in
Theorem 2. Developing systematic methods to design $h$ satisfying these
conditions will be our future work.

###### Remark 4

The bound $L$ for $\|e_{q}\|$ as given in (12) should be carefully selected to
achieve a trade-off between the control performance and the maximum magnitude
of the control input. If $L$ is selected to be very small, the control input
tends to be significant because the state is more likely to approach the
boundary of the output constraint; if $L$ is chosen to be large, unnecessary
conservatism (i.e., the system only operates in a subset of the original
safety set) may be introduced because in Theorem 2 the worst-case of
$e_{q}+\xi$ is considered.

If the proxy subsystem is not augmented with an additional integrator, the
requirement of $L$ would be more restrictive, i.e.,
$L\geq|\hat{\omega}(0)-\mu(q(0))|$ is required. In practice, this may
necessitate the selection of a larger $L$, which could result in unnecessarily
conservatism. Meanwhile, $L$ is used in the design of $\mu$ to guarantee
safety, which implies that $\mu(q(0))$ implicitly relies on $L$. Thus, in some
cases, it may be difficult to find an appropriate $L$ that satisfies
$L\geq|\hat{\omega}(0)-\mu(q(0))|$.

##  IV task space Safe Control

In this section, we will utilize the idea presented in the preceding section
to solve the task space safe control problem for the EL system with limited
information, external disturbances, and measurement uncertainties. The proxy
subsystem in task space is more complicated to control than that in joint
space; therefore, a different CBF-based control scheme is proposed.

Invoking (6), one can see

$\dot{p}=J(q)\omega=J(q)(\hat{\omega}+\xi),$ (25)

where $J=\frac{\partial f}{\partial q}$ denotes the Jacobian [23].
Substituting (25) into (1a) yields

$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
J(q)(\hat{\omega}+\xi),$ (26a) $\displaystyle\dot{\hat{\omega}}$
$\displaystyle=$ $\displaystyle
M^{-1}(q)(\tau-C(q,\omega)\omega-G(q)+\tau_{d})-\dot{\xi}.$ (27a)

System (26a) can be decomposed into the proxy subsystem and the virtual
tracking subsystem similar to Section III. The proxy subsystem is given as:

$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
J(q)\eta+J(q)(e_{p}+\xi),$ (28a) $\displaystyle\dot{\eta}$ $\displaystyle=$
$\displaystyle\upsilon,$ (29a)

where $\eta$ is the virtual state with $\eta(0)=\hat{\omega}(0)$,
$e_{p}\triangleq\hat{\omega}-\eta$, and $\upsilon$ denotes the virtual control
input to be designed. The virtual tracking subsystem is given as:

$\dot{e}_{p}=M^{-1}(\tau-C(q,\omega)\omega-G(q)+\tau_{d})-\dot{\xi}-\upsilon.$
(30)

Note that system (30) corresponds to system (11), for which the adaptive BLF-
based controller developed in Theorem 1 is still applicable. On the other
hand, the CBF-based controller presented in Theorem 2 is inapplicable to the
proxy subsystem given in (28a) because (28a) is different from (8a). We will
design a new CBF-based safe control law for (28a) to ensure the forward
invariance of $\mathcal{C}_{p}$. To that end, we first design a nominal
tracking controller for the proxy subsystem (28a) based on backstepping [29]
as shown in the following proposition.

###### Proposition 1

Consider the proxy subsystem (28a) and a desired trajectory $p_{d}$. Suppose
that $\|e_{p}\|<L$ and the Jacobian $J$ has full row rank, i.e., there exists
$J^{\dagger}$ such that $JJ^{\dagger}=I_{k\times k}$. If the desired control
input $\upsilon_{d}$ is designed as

$\displaystyle\delta$ $\displaystyle=$ $\displaystyle
J^{\dagger}\left(-l_{1}\epsilon_{d}+\dot{p}_{d}-\frac{\|J\|^{2}}{2}\epsilon_{d}\right),$
(31a) $\displaystyle\upsilon_{d}$ $\displaystyle=$ $\displaystyle-
l_{2}\epsilon_{\eta}\\!+\\!\frac{\partial\delta}{\partial
p}J\eta\\!+\\!\frac{\partial\delta}{\partial
t}\\!-\\!\frac{1}{2}\left\|\frac{\partial\delta}{\partial
p}J\right\|^{2}\epsilon_{\eta}\\!-\\!J^{\top}\epsilon_{d},$ (32a)

where $\epsilon_{d}=p-p_{d}$, $\epsilon_{\eta}=\eta-\delta$, and
$l_{1},l_{2}>0$ are arbitrary positive constants, then the tracking error
$\epsilon_{d}$ is uniformly ultimately bounded.

###### Proof:

Define a Lyapunov candidate function as
$V_{1}=\frac{1}{2}\epsilon_{d}^{\top}\epsilon_{d}$. The derivative of $V_{1}$
satisfies

$\displaystyle\dot{V}_{1}$ $\displaystyle=$
$\displaystyle\epsilon_{d}^{\top}(J\delta+J\epsilon_{\eta}+J(e_{p}+\xi)-\dot{p}_{d})$
$\displaystyle\leq$
$\displaystyle\epsilon_{d}^{\top}(J\delta+J\epsilon_{\eta}-\dot{p}_{d})+\frac{\|\epsilon_{d}\|^{2}\|J\|^{2}}{2}+\frac{(D_{1}+L)^{2}}{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eta}}}{{\leq}}$ $\displaystyle-
l_{1}\|\epsilon_{d}\|^{2}+\epsilon_{d}^{\top}J\epsilon_{\eta}+\frac{(D_{1}+L)^{2}}{2}.$

Then, an augmented Lyapunov candidate function is designed as
$V_{2}=V_{1}+\frac{1}{2}\epsilon_{\eta}^{\top}\epsilon_{\eta}$, whose
derivative can be expressed as

$\displaystyle\dot{V}_{2}$ $\displaystyle\leq$ $\displaystyle-
l_{1}\|\epsilon_{d}\|^{2}+\epsilon_{\eta}^{\top}J^{\top}\epsilon_{d}+\frac{(D_{1}+L)^{2}}{2}$
$\displaystyle+\epsilon_{\eta}^{\top}\left(\upsilon_{d}-\frac{\partial\delta}{\partial
p}J(\eta+e_{p}+\xi)-\frac{\partial\delta}{\partial t}\right)$
$\displaystyle\leq$ $\displaystyle-
l_{1}\|\epsilon_{d}\|^{2}+\epsilon_{\eta}^{\top}J^{\top}\epsilon_{d}+(D_{1}+L)^{2}$
$\displaystyle+\epsilon_{\eta}^{\top}\left(\upsilon_{d}-\frac{\partial\delta}{\partial
p}J\eta-\frac{\partial\delta}{\partial
t}\right)+\frac{1}{2}\|\epsilon_{\eta}\|^{2}\left\|\frac{\partial\delta}{\partial
p}J\right\|^{2}$ $\displaystyle\stackrel{{\scriptstyle\eqref{nud}}}{{\leq}}$
$\displaystyle-
l_{1}\|\epsilon_{d}\|^{2}-l_{2}\|\epsilon_{\eta}\|^{2}+(D_{1}+L)^{2}.$

Therefore, the tracking error $\epsilon_{d}$ is uniformly ultimately bounded
[28]. ∎

A CBF-based safe control law is proposed for the proxy subsystem (28a) in the
following theorem.

###### Theorem 3

Consider the proxy subsystem (28a) and the set $\mathcal{C}_{p}$ defined in
(7). Suppose that $h(p(0))\geq 0$, $\|e_{p}(t)\|<L,\forall t\geq 0$, and there
exist constants $\lambda,\gamma,\beta>0$ such that
(i) $\frac{\partial h}{\partial
p}(p(0))J(q(0))\eta(0)-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
p}(p(0))J(q(0))\right\|^{2}-\frac{\beta(D_{1}+L)^{2}}{2}+\lambda h(p(0))\geq
0$;
(ii) the set
$K_{BF}^{p}(p,q,\mu)=\\{{\mathfrak{u}}\in\mathbb{R}^{n}:\Psi_{0}+\Psi_{1}{\mathfrak{u}}\geq
0\\}$ is not empty for any $p\in\mathcal{C}_{p}$ and $\eta\in\mathbb{R}^{n}$,
where

$\displaystyle\Phi_{0}$ $\displaystyle=$
$\displaystyle\frac{\partial\bar{h}}{\partial
q}(J\eta+\hat{\omega})-\left\|\frac{\partial\bar{h}}{\partial
q}J\right\|(D_{1}+L)$ (34a)
$\displaystyle-\left\|\frac{\partial\bar{h}}{\partial
q}\right\|D_{1}+\gamma\bar{h},$ $\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial h}{\partial p}J,$ (35a)

with $\bar{h}=\frac{\partial h}{\partial
p}J\eta-\frac{1}{2\beta}\left\|\frac{\partial h}{\partial
p}J\right\|^{2}-\frac{\beta(D_{1}+L)^{2}}{2}+\lambda h$.
Then, any Lipschitz continuous control input $\upsilon\in K_{BF}^{p}$ will
make $h(p(t))\geq 0$ for any $t\geq 0$.

###### Proof:

We only show the sketch of the proof due to space limitation and the
similarity of the proof to that of Theorem 2. One can see that selecting
$\upsilon\in K_{BF}^{p}$ ensures $\dot{\bar{h}}\geq-\gamma\bar{h}$; therefore,
$\bar{h}(t)\geq 0$ for any $t\geq 0$ since Condition (i) implies
$\bar{h}(p(0),q(0),\eta(0))\geq 0$. Then, it can be proved that
$\bar{h}(t)\geq 0\implies h(t)\geq 0$ for any $t\geq 0$. ∎

Based on Proposition 1 and Theorem 3, the safe virtual controller $\nu$ can be
obtained by solving a CBF-QP:

$\displaystyle\min_{\nu}\quad$ $\displaystyle\|\upsilon-\upsilon_{d}\|^{2}$
(36) s.t. $\displaystyle\Phi_{0}+\Phi_{1}\upsilon\geq 0,$

where $\Phi_{0},\Phi_{1}$ are given in (34a) and $\upsilon_{d}$ is presented
in (31a).

The safe feedback control law $\tau(p(t),q(t),\hat{\omega}(t),t)$ to the EL
system (1a) consists of the control law $\tau$ given in (13) and the control
law $\nu$ given in (36). By Theorems 1 and 3, the control law
$\tau(p(t),q(t),\hat{\omega}(t),t)$ will ensure the safety of the closed-loop
system with respect to $\mathcal{C}_{p}$, i.e., $h(p(t))\geq 0$ for all $t\geq
0$.

##  V Simulation

In this section, numerical simulation results are presented to demonstrate the
effectiveness of the proposed method. Consider a two-linked robot manipulator,
whose dynamics can be described by (1a) with

$M(q)\\!=\\!\begin{bmatrix}\frac{m_{1}l^{2}}{3}\\!+\\!\frac{4m_{2}l^{2}}{3}\\!+\\!m_{2}l^{2}\cos
q_{2}&\frac{m_{2}l^{2}}{3}\\!+\\!\frac{m_{2}l^{2}}{2}\cos q_{2}\\\
\frac{m_{2}l^{2}}{3}\\!+\\!\frac{m_{2}l^{2}}{2}\cos
q_{2}&\frac{m_{2}l^{2}}{3}\end{bmatrix},$
$C(q,\omega)\\!=\\!\begin{bmatrix}-\frac{m_{2}l^{2}}{2}\dot{q}_{2}\sin
q_{2}&-\frac{m_{2}l^{2}}{2}(\dot{q}_{1}+\dot{q}_{2})\sin q_{2}\\\
\frac{m_{2}l^{2}}{2}\dot{q}_{1}\sin q_{2}&0\end{bmatrix},$
$G(q)=\begin{bmatrix}\frac{m_{1}gl}{2}\cos
q_{1}\\!+\\!\frac{m_{2}gl}{2}\cos(q_{1}\\!+\\!q_{2})\\!+\\!m_{2}gl\cos
q_{1}\\\ \frac{m_{2}gl}{2}\cos(q_{1}\\!+\\!q_{2})\end{bmatrix},$

where $m_{1}=m_{2}=1\ kg$, $l=1\ m$, $q=[q_{1}\ q_{2}]\in\mathbb{R}^{2}$
denotes the joint angles, and $\omega=[\dot{q}_{1}\
\dot{q}_{2}]\in\mathbb{R}^{2}$ are joint angular velocities [30]. We emphasize
that $M$, $C$, and $G$ are assumed to be unknown, and only $\lambda_{2}=5$ in
Property 1 and $D_{1}=0.2$ in Assumption 2 are available in our control
design.

###  V-A Joint Space Safe Control

In this subsection, simulation results of the joint space safe control are
presented. The reference trajectories are $q_{1d}=q_{2d}=3\sin(t)$; four CBFs
are selected as $h_{1}=2.5-q_{1}$, $h_{2}=q_{1}+2.5$, $h_{3}=2-q_{2}$, and
$h_{4}=q_{2}+1$, which aim to ensure $-2.5\leq q_{1}\leq 2.5$ and $-1\leq
q_{2}\leq 2$; the control parameters are selected as $\beta=2$, $\gamma=10$,
$\lambda=16$, $\epsilon=\epsilon_{1}=\epsilon_{2}=0.01$, $L=0.3$,
$\gamma_{\theta}=1$, and $k_{1}=0.1$; the initial conditions are
$q_{1}(0)=q_{2}(0)=1$ and $\dot{q}_{1}(0)=\dot{q}_{2}(0)=0$; the measurement
uncertainty and disturbance are selected as $\xi=[0.2\sin(2t)\
0.2\sin(2t)]^{\top}$ and $\tau_{d}=10\sin(t)$, from which one can see that
Assumption 1 and 2 are satisfied. It is easy to check that Conditions (i) and
(ii) of Theorem 2 are fulfilled with the given parameters and CBFs. The
simulation results are presented in Fig. 2.

From the simulation results one can see that the safety of $\mathcal{C}_{q}$
is guaranteed as the trajectories of $q_{1}$ and $q_{2}$ always stay inside
the safe region whose boundaries are represented by the dashed red line, and
the reference trajectory is well-tracked within the safe set. Moreover, from
Fig. 2(c) one can observe that $\|e_{q}(t)\|<L$ is satisfied for any $t\geq
0$, which indicates that the adaptive BLF-based controller proposed in Theorem
1 is effective.

Figure 2: Simulation results of the joint space safe control. From (a) and (b)
it can be seen that the proposed controller can ensure safety of
$\mathcal{C}_{q}$ as the trajectories of $q_{1}$ and $q_{2}$ never cross the
boundary of the safe region represented by the dash red lines, with good
tracking performance inside the safe region. Moreover, from (c) one can
conclude that the adaptive BLF-based controller developed in Theorem 1 is
effective since the constraint on $e_{q}$ is not violated.

###  V-B Task Space Safe Control

In this subsection, simulation results for task space safe control are
presented. The forward kinematics can be expressed as

$\begin{bmatrix}x\\\
y\end{bmatrix}=\begin{bmatrix}l_{1}\cos(q_{1})+l_{2}\cos(q_{1}+q_{2})\\\
l_{1}\sin(q_{1})+l_{2}\sin(q_{1}+q_{2})\end{bmatrix},$

and the Jacobian is

$J(q)=\begin{bmatrix}-l_{1}\sin(q_{1})-l_{2}\sin(q_{1}+q_{2})&-l_{2}\sin(q_{1}+q_{2})\\\
l_{1}\cos(q_{1})+l_{2}\cos(q_{1}+q_{2})&l_{2}\cos(q_{1}+q_{2})\end{bmatrix}.$

Note that the measurement uncertainties and disturbance are the same as those
in Section V-A such that Assumption 1 and 2 are satisfied. To demonstrate the
effectiveness of the proposed method, three cases are considered.

  * •

Case 1: The CBF is $h=x^{2}+y^{2}-0.25$; the initial conditions are
$x(0)=1.59$, $y(0)=0.11$; the reference trajectories are $x_{d}(t)=1.5-0.3t$,
$y(t)=0$; and the control parameters are chosen as
$\epsilon=\epsilon_{1}=\epsilon_{2}=0.01$, $L=0.05$, $\gamma_{\theta}=1$,
$\beta=2$, $\lambda=100$, $l_{1}=l_{2}=20$, $\gamma=1000$, and $k_{1}=3$.

  * •

Case 2: The CBF is $h=1+x-y^{2}$; the initial conditions are $x(0)=1.8$,
$y(0)=0$; the reference trajectories are $x_{d}(t)=1.5\cos(t)$,
$y(t)=1.5\sin(t)$; and the control parameters are the same as those in Case 1
except for $\gamma=300$.

  * •

Case 3: The CBF is $h=1+x+y$; the initial conditions are $x(0)=1.8$, $y(0)=0$;
the reference trajectories are $x_{d}(t)=1.5\cos(t)$, $y(t)=1.5\sin(t)$; and
the control parameters are the same as those in Case 1 except for
$l_{1}=l_{2}=40$, $\lambda=100$, and $\gamma=500$.

The simulation results are presented in Fig. 3, from which one can see that in
all three cases the safety of $\mathcal{C}_{p}$ is ensured by the proposed
controller as the trajectories of $x$ and $y$ always stay inside the safe
region whose boundary is represented by the dash red lines, and the tracking
performance inside the safe region is satisfactory.

Figure 3: Simulation results of the task space safe control. One can observe
that proposed controller can ensure safety of $\mathcal{C}_{p}$ as the
trajectories of $x$ and $y$ always stay inside the safe region whose boundary
is represented by the dash red lines, and the tracking performance inside the
safe region is satisfactory.

##  VI Conclusion

In this paper, a novel proxy CBF-BLF-based control design approach is proposed
for EL systems with limited information by decomposing an EL system into the
proxy subsystem and the virtual tracking subsystem. A BLF-based controller is
designed for the virtual tracking subsystem to ensure the boundedness of the
safe velocity tracking error. Based on that, a CBF-based controller is
designed for the proxy subsystem to ensure safety in the joint space or task
space. Simulation results are given to verify the effectiveness of the
proposed method. Future work includes conducting experimental studies and
generalizing the results to ensure safety and stability simultaneously for EL
systems.

## References

  * [1] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 8, pp. 3861–3876, 2017.
  * [2] X. Xu, P. Tabuada, A. Ames, and J. Grizzle, “Robustness of control barrier functions for safety critical control,” in _IFAC Conference on Analysis and Design of Hybrid Systems_ , vol. 48, no. 27, 2015, pp. 54–61.
  * [3] M. Jankovic, “Robust control barrier functions for constrained stabilization of nonlinear systems,” _Automatica_ , vol. 96, pp. 359–367, 2018.
  * [4] Q. Nguyen and K. Sreenath, “Robust safety-critical control for dynamic robotics,” _IEEE Transactions on Automatic Control_ , vol. 67, no. 3, pp. 1073–1088, 2021.
  * [5] K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov functions for the control of output-constrained nonlinear systems,” _Automatica_ , vol. 45, no. 4, pp. 918–927, 2009.
  * [6] B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function,” _IEEE Transactions on Neural Networks_ , vol. 21, no. 8, pp. 1339–1345, 2010.
  * [7] D. Panagou, D. M. Stipanović, and P. G. Voulgaris, “Distributed coordination control for multi-robot networks using Lyapunov-like barrier functions,” _IEEE Transactions on Automatic Control_ , vol. 61, no. 3, pp. 617–632, 2015.
  * [8] X. Jin and J.-X. Xu, “A barrier composite energy function approach for robot manipulators under alignment condition with position constraints,” _International Journal of Robust and Nonlinear Control_ , vol. 24, no. 17, pp. 2840–2851, 2014.
  * [9] I. Salehi, G. Rotithor, D. Trombetta, and A. P. Dani, “Safe tracking control of an uncertain Euler-Lagrange system with full-state constraints using barrier functions,” in _59th Conference on Decision and Control_. IEEE, 2020, pp. 3310–3315.
  * [10] Y. Wang and X. Xu, “Observer-based control barrier functions for safety critical systems,” in _American Control Conference_. IEEE, 2022, pp. 709–714.
  * [11] ——, “Disturbance observer-based robust control barrier functions,” in _American Control Conference_. IEEE, 2023, pp. 2681–3687.
  * [12] J.-J. E. Slotine and W. Li, “On the adaptive control of robot manipulators,” _The International Journal of Robotics Research_ , vol. 6, no. 3, pp. 49–59, 1987.
  * [13] R. Ortega, J. A. L. Perez, P. J. Nicklasson, and H. J. Sira-Ramirez, _Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications_. Springer Science & Business Media, 2013.
  * [14] W. S. Cortez and D. V. Dimarogonas, “Safe-by-design control for Euler–Lagrange systems,” _Automatica_ , vol. 146, p. 110620, 2022.
  * [15] B. Capelli, C. Secchi, and L. Sabattini, “Passivity and control barrier functions: Optimizing the use of energy,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 1356–1363, 2022.
  * [16] F. S. Barbosa, L. Lindemann, D. V. Dimarogonas, and J. Tumova, “Provably safe control of Lagrangian systems in obstacle-scattered environments,” in _59th Conference on Decision and Control_. IEEE, 2020, pp. 2056–2061.
  * [17] F. Ferraguti, C. T. Landi, A. Singletary, H.-C. Lin, A. Ames, C. Secchi, and M. Bonfè, “Safety and efficiency in robotics: The control barrier functions approach,” _IEEE Robotics & Automation Magazine_, vol. 29, no. 3, pp. 139–151, 2022.
  * [18] A. W. Farras and T. Hatanaka, “Safe control with control barrier function for Euler-Lagrange systems facing position constraint,” in _SICE International Symposium on Control Systems_. IEEE, 2021, pp. 28–32.
  * [19] A. Singletary, S. Kolathaya, and A. D. Ames, “Safety-critical kinematic control of robotic systems,” _IEEE Control Systems Letters_ , vol. 6, pp. 139–144, 2021.
  * [20] T. G. Molnar, R. K. Cosner, A. W. Singletary, W. Ubellacker, and A. D. Ames, “Model-free safety-critical control for robotic systems,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 944–951, 2021.
  * [21] S. Kolathaya and A. D. Ames, “Input-to-state safety with control barrier functions,” _IEEE Control Systems Letters_ , vol. 3, no. 1, pp. 108–113, 2018.
  * [22] X. Xu, “Constrained control of input–output linearizable systems using control sharing barrier functions,” _Automatica_ , vol. 87, pp. 195–201, 2018.
  * [23] M. W. Spong, S. Hutchinson, and M. Vidyasagar, _Robot Modeling and Control_. Wiley: New York, 2006.
  * [24] W. E. Dixon, “Adaptive regulation of amplitude limited robot manipulators with uncertain kinematics and dynamics,” _IEEE Transactions on Automatic Control_ , vol. 52, no. 3, pp. 488–493, 2007.
  * [25] R. Kikuuwe, S. Yasukouchi, H. Fujimoto, and M. Yamamoto, “Proxy-based sliding mode control: A safer extension of PID position control,” _IEEE Transactions on Robotics_ , vol. 26, no. 4, pp. 670–683, 2010.
  * [26] D. C. Ruspini, K. Kolarov, and O. Khatib, “The haptic display of complex graphical environments,” in _Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques_ , 1997, pp. 345–352.
  * [27] Y. Wang and X. Xu, “Adaptive safety-critical control for a class of nonlinear systems with parametric uncertainties: A control barrier function approach,” _arXiv preprint arXiv:2302.08601_ , 2023.
  * [28] H. K. Khalil, _Nonlinear Systems_. Prentice Hall Upper Saddle River, NJ, 2002.
  * [29] P. V. Kokotovic, “The joy of feedback: Nonlinear and adaptive,” _IEEE Control Systems Magazine_ , vol. 12, no. 3, pp. 7–17, 1992.
  * [30] T. Sun, H. Pei, Y. Pan, H. Zhou, and C. Zhang, “Neural network-based sliding mode adaptive control for robot manipulators,” _Neurocomputing_ , vol. 74, no. 14-15, pp. 2377–2384, 2011.