Robust Global Exponential Stabilization on the $n$-Dimensional Sphere with Applications to Trajectory Tracking for Quadrotors

In this paper, we design a hybrid controller that globally exponentially stabilizes a system evolving on the $n$-dimensional sphere, denoted by $\sphere n$. This hybrid controller is induced by a <span>"synergistic"</span> collection of potential functions on $\sphere n$. We propose a particular construction of this class of functions that generates flows along geodesics of the sphere, providing convergence to the desired reference with minimal path length. We show that the proposed strategy is suitable to the exponential stabilization of a quadrotor vehicle.

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Hybrid control systems, Synergistic potential functions, Lyapunov-based control.

Introduction
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Motivation and Problem Statement
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In this paper, we design a hybrid controller for global exponential stabilization of a setpoint for a system evolving on the $n$-dimensional sphere, given by $\sphere n\ceq\{x\in\R{n+1}:x\tp x=1\}$. The dynamics of this system can be described by $$\label{eq:kinematics}
\begin{aligned}
\dot{x}&=\PTSn(x) \omega & x&\in\sphere n,
\end{aligned}$$ where $\omega\in\R{n+1}$ is the input and $\PTSn(x)=\eye{n}-x x\tp$ projects $\omega$ onto the tangent space to $\sphere n$ at $x$, given by $\TS[x]{n}\ceq\{z\in\R{n+1}:z\tp x=0\}.$ Even though there exist controllers that globally asymptotically stabilize a setpoint on the $n$-dimensional sphere (c.f. @mayhew_global_2013) and others that globally exponentially stabilize a setpoint on the special orthogonal group of order 3 (c.f. @lee_global_2015 and @berkane_hybrid_2017), to the best of our knowledge, the problem of global exponential stabilization of a setpoint in $\sphere n$ has not been addressed before, despite the fact that it has many meaningful applications, such as visual servoing (c.f. @triantafyllou_constrained_2018); control of robotic manipulators (c.f. @chaturvedi_asymptotic_2009); exoskeleton tracking (c.f. @brahmi_cartesian_2017); multi-agent synchronization (c.f. @markdahl_almost_2018); formation control (c.f. @zhao_bearing_2016); rigid-body stabilization (c.f. @chaturvedi_rigid-body_2011) and trajectory tracking for multi-rotor aerial vehicles (c.f. @mahony_multirotor_2012), which we also explore in this paper. To see how the proposed controller for global exponential stabilization on the $n$-dimensional sphere applies to <span>trajectory tracking for a multi-rotor aerial vehicle</span>, consider the following: the position dynamics of a multi-rotor vehicle can be described by $$\label{eq:quadpv}
\begin{aligned}
\dot{p}&=v&
\dot{v}&=x u+\grv
\end{aligned}$$ where $\grv$ is the acceleration of gravity, $p\in\R 3$ and $v\in\R 3$ denote the position and the velocity of the vehicle with respect to the inertial reference frame, <span>$u$ denotes the magnitude of the thrust</span> and $x\in\sphere 2$ denotes the direction of the thrust (c.f. @hamel_dynamic_2002). Given a reference trajectory with acceleration $\ddot p_d$ and a control law $w:\R 3\to\R 3$ that exponentially stabilizes the double integrator, if the controller for $x$ exponentially stabilizes the commanded thrust direction, given by $\Td(\pe,\ve,\ddot p_d)\ceq\frac{w(\pe,\ve)-\grv+\ddot p_d}{\norm{w(\pe,\ve)-\grv+\ddot p_d}}$ for each $(\pe,\ve,\ddot p_d)\in\{(\pe,\ve,\ddot p_d)\in\R 9: w(\pe,\ve)-\grv+\ddot p_d\neq 0\}$ where $\pe$ and $\ve$ denote the position and velocity tracking errors, respectively, then exponential tracking is attained. This, however, cannot be achieved through continuous feedback because it is not possible to globally asymptotically stabilize a given setpoint on a compact manifold by means of continuous feedback (c.f. @bhat_topological_2000). Moreover, if a dynamical system cannot be globally asymptotically stabilized through continuous feedback, it cannot be robustly stabilized by discontinuous feedback either, as shown in @mayhew_topological_2011. Overcoming these limitations is particularly important for multi-rotor aerial vehicles due to their popularity and the wide range of applications in which they are used, such as surveillance, tracking, search and rescue, infrastructure inspection, agriculture <span>and</span> disaster mitigation (see e.g @fink_planning_2011, @mellinger_trajectory_2012, @jiang_inverse_2013, @augugliaro_dance_2013, @lupashin_platform_2014, @floreano_science_2015 and @liang_dynamics_2018). In this paper, we present a hybrid controller that not only provides global exponential stability, but also confers a quantifiable robustness margin to perturbations.

<span>Recent</span> developments on hybrid control theory overcame the topological obstructions to global stabilization on compact manifolds with the introduction of synergistic potential functions in @mayhew_global_2010. A potential function on a compact manifold is a continuously differentiable function that is positive definite relative to a given point, thus it induces a gradient vector field which asymptotically stabilizes the given reference from every initial condition except a set of measure zero. To see this, consider the closed-loop system resulting from the interconnection of the gradient-based feedback law of a potential function $\height{r}$ that is positive definite relative to $r\in \sphere n$ and , given by: $$\label{eq:cfeed}
\begin{aligned}
\dot x=-\PTSn(x)\grad \height{r}(x).
\end{aligned}$$ <span>The equilibrium points of </span> are the critical points of $\height{r}(x)$, denoted by $$\label{eq:critV}
\begin{aligned}
\crit\height{r}\ceq\{x\in\sphere n: \PTSn(x)\grad\height{r}(x)=0\},
\end{aligned}$$ <span>which correspond to the set of points $x$ where $\grad \height{r}(x)$ is orthogonal to $\TS[x]{n}$. Since the set  includes</span> the maximum and the minimum of $\height{r}$ on $\sphere n$, it follows that $r\in\sphere n$ is not globally asymptotically stable for . On the other hand, synergistic potential functions are collections of potential functions that <span>enable a controller to achieve robust global asymptotic stabilization of</span> the given setpoint, because, at the undesired equilibria, there exists another function in the collection with a lower value that we can switch to.

<span>The concept of synergistic potential functions has been introduced to address the problem of stabilizing a three-dimensional pendulum in @mayhew_global_2010 and later used in full attitude stabilization in @mayhew_synergistic_2013, @lee_global_2015; attitude synchronization @mayhew_quaternion-based_2012; partial attitude stabilization (c.f. @mayhew_global_2013), and stabilization by hybrid backstepping (c.f. @Mayhew2011). More recently, the interest in this control technique has spawned the design of new synergistic potential functions on $\SO(3)$, such as the ones by @berkane_design_2015, @berkane_construction_tac2016 and @berkane_hybrid_2017.</span> It was shown by the authors in @casau_robust_2015 that synergistic potential functions can be used for global asymptotic stabilization of a reference trajectory for a multi-rotor aerial vehicle, but the extent to which exponential stabilization is possible was not addressed.

Contributions
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The contributions in this paper are as follows. *Extension to the concept of synergistic potential functions:* in Section \[sec:synergistic\], we show that the existence of a centrally synergistic potential function $V:\sphere n\x\yset\to\Rnneg$<span>, where $\yset$ is a compact set,</span> induces a gradient-based control law that renders $$\label{eq:Amc}
\begin{aligned}
\Amc\ceq\{(x,y)\in\sphere n\x\yset:x=r\}
\end{aligned}$$ globally asymptotically stable for the closed-loop system. This nomenclature is inherited from @mayhew_global_2013, where synergistic potential functions satisfying $V(r,y)=0$ for all $y\in\yset$ are said to be central because they share are a common minimum at $r$. In this paper, we extend the previous notion of centrally synergistic potential functions, because we consider that $\yset$ is compact rather than finite, which adds flexibility to the design of synergistic potential functions. The proposed controller is significantly different from the one in @mayhew_synergistic_2013 because the focus is on global exponential stabilization of $\sphere n$ rather than $\SO(3)$ and it further expands the work in @mayhew_global_2010 from global asymptotic stabilization on $\sphere 2$ to global exponential stabilization on $\sphere n$. Interestingly, the controller that we propose may be used for global stabilization on $\sphere 3$ which is the universal cover of $\SO(3)$. However, the resulting controller would be more complex than that of @mayhew_quaternion-based_2011 if used for rigid-body stabilization, because it would not take advantage of the fact that $\sphere 3$ is a double cover of $\SO(3)$. *Global exponential stability of a setpoint on $\sphere n$:* in Theorem \[theorem:expStability\], we show that, if $V$ is bounded from above and below by a polynomial function of the distance to $\Amc$ and if $V$ converges exponentially fast to $0$, then $\Amc$ is globally exponentially stable for the closed-loop system, in the sense that, for all initial conditions, the state of the system converges exponentially to $\Amc$. This is different from the controllers proposed in @lee_global_2015 and @berkane_hybrid_2017, because these address the problem of global exponential stabilization on $\SO(3)$ rather than $\sphere n$. *Optimal switching:* in Section \[sec:construct\], we construct a synergistic potential function on the $n$-dimensional sphere that meets the requirements for exponential stability and has an optimal switching law, in the sense that it guarantees that solutions to the closed-loop system follow geodesics whenever a jump of the hybrid controller is triggered. *Robustness to perturbations:* the proposed hybrid controller satisfies the so-called hybrid basic conditions, therefore it is endowed with nominal robustness properties that are outlined in @goebel_hybrid_2012. In addition, the switching rule introduces a hysteresis gap that prevents chattering. *Saturated thrust feedback for exponential tracking of a reference trajectory for a multi-rotor aerial vehicle:* in Section \[sec:quad\], we employ the hybrid controller for global exponential stabilization on $\sphere n$ in trajectory tracking for a multi-rotor aerial vehicle. We show that, for each compact set of initial position and velocity tracking errors and for all initial orientations, the reference trajectory is exponentially stable. This is particularly difficult, because in addition to the topological constraints to global attitude stablilization, multirotor aerial vehicles are subject to underactuation constraints that prevent stabilization of the vehicle when the commanded thrust is zero (c.f. @lizarraga_osbtructions_2004). This issue has been widely acknowledged but mostly overlooked due to its singular nature. For example, in @lee_nonlinear_2013 this flight condition is assumed to not occur and in @hua_control_2009 the controller is turned off when the commanded thrust approaches zero. Similarly to the work of @hua_control_2015, we assume that the reference trajectory does not lead to a situation where the commanded thrust is zero, but, unlike the aforementioned approach, we explicitly build this limitation into the control design procedure so to achieve semi-global exponential stability with respect to the position and velocity errors. <span>A video of experimental runs using this controller can be found in @Casau2017.</span>

The paper is organized as follows: in Section \[sec:not\], we present the notation and the framework of hybrid dynamical systems that is used in this paper. In Section \[sec:synergistic\], we develop the notions of synergistic potential potential functions on the $n$-dimensional sphere. In Section \[sec:quad\], we apply the given controllers to the tracking of a reference trajectory for a vectored-thrust vehicle, and in Section \[sec:conclusions\], we present the conclusions of this work. <span>In @Casau2015 we reported the results that are presented in Section \[sec:synergistic\] without the proofs. In @casau_exponential_2016, we presented some preliminary results on work reported in Section \[sec:quad\] without the constructive controller synthesis that is presented in this paper for the quadrotor application.</span>

Preliminaries {#sec:not}
=============

A hybrid system $\Hmc$ defined on $\R n$ can be represented by <span>$$\label{eq:hybrid}
\Hmc:\left\{
\begin{aligned}
\dot x &\in F(x) & x&\in C \\
x\pl &\in G(x) & x&\in D 
\end{aligned}
\right .$$</span> where $C\subset\R n$ is the flow set, $F:\R n\tto \R n$ is the flow map, $D\subset\R n$ and $G:\R n\tto \R n$ with $D\subset \dom G$ is the jump map, as defined in @goebel_hybrid_2012. The maps $F$ and $G$ are set-valued maps satisfying $C\subset \dom F$ and $D\subset\dom G$, <span>respectively.</span> Loosely speaking, solutions to hybrid systems are hybrid arcs that are compatible with the data of the hybrid system, i.e., functions $(t,j)\mapsto\sol(t,j)$ defined on a hybrid time domain $\htd\subset\Rnneg\x\N_0$ that satisfy $\dot\sol(t,j)\in F(\sol(t,j))$ for almost all $t\geq 0$ and $\sol(t,j)\in D$, $\sol(t,j+1)\in G(\sol(t,j))$ for every $\sol(t,j)$ and $\sol(t,j+1)$ belonging to $\dom\sol$. We say that a solution $\sol$ to  is maximal if it cannot be extended, it is complete if its domain is unbounded, it is discrete if $\dom\phi\subset\{0\}\x\N$ and it is Zeno if it is complete and $\sup_t\dom\phi<+\infty$ (for a more rigorous description of solutions to hybrid systems see @goebel_hybrid_2012). Under the assumption of completeness of maximal solution to $\Hmc$, a compact set $\Amc$ is said to be: stable for $\Hmc$, if for each $\epsilon>0$ there exists $\delta>0$ such that for each solution $\sol$ to $\Hmc$ with $\norm[\Amc]{\sol(0,0)}\leq\delta$ satisfies $\norm[\Amc]{\sol(t,j)}\leq\epsilon$ for each $(t,j)\in\dom\sol$; attractive for $\Hmc$ if $\lim_{t+j\rightarrow\infty}\norm[\Amc]{\sol(t,j)}=0$, where $\norm[\Amc]{x}:=\min_{y\in\Amc}\norm{x-y}$ and $\norm{x}\ceq \sqrt{\inner{x}{x}}$ for each $x\in\R{n}$. A set $\Amc$ is globally asymptotically stable for $\Hmc$ if it is both globally attractive and globally stable for $\Hmc$. We say that a compact set $\Amc\subset\R n$ is exponentially stable <span>in the $t$-direction</span> from $U$ if there exists $k,\lambda>0$ such that $\sup_t\dom\phi = \infty$ and $\norm[\Amc]{\phi(t,j)}\leq k\exp(-\lambda t)\norm[\Amc]{\phi(0,0)}$ for each maximal solution $\sol$ to the hybrid system with $\sol(0,0)\in U$ and $(t,j)\in\dom\phi$.

Synergistic Potential Functions on $\sphere n$ {#sec:synergistic}
==============================================

In this section, we design a hybrid controller that *globally* exponentially stabilizes a given reference <span>$r\in\sphere n\ceq\{x\in\R{n+1}:x\tp x=1\}$</span> for the system  using the notion of centrally synergistic potential functions given next.

A function $h_r:\sphere n\to \R{}$ is said to be a potential function on $\sphere n$ relative to $r$ if it is continuously differentiable and positive definite relative to $r\in\sphere n$, i.e., $h_r(x)\geq 0$ for each $x\in\sphere n$ and $h_r(x)=0$ if and only if $x=r$. We denote the collection of potential functions on $\sphere n$ relative to $r$ by $\pfc[r]$.$\square$

\[dfn:synergisticFunc\] Given a compact set $\yset$, a function $V:\sphere n\x\yset\to\R{}$ is said to be a centrally synergistic potential function on $\sphere n$ relative to $r$ if the following hold:

For each $y\in\yset$, $V^y$ is a potential function on $\sphere n$ relative to $r$, where $V^y(x)\ceq V(x,y)$ for each $x\in\sphere n$;

There exists $\delta>0$ such that $$\label{eq:mu}
 \begin{aligned}
 \gap(x,y)\ceq V(x,y)-\minVy(x) >\delta
 \end{aligned}$$ for each

We denote the collection of centrally synergistic potential functions on $\sphere n$ relative to $r$ by $\spf[r]{\yset}$.$\square$

Given a centrally synergistic potential function $V$, the function $(x,y)\to\gap(x,y)$ is referred to as the *synergy gap* of $V$ at $(x,y)$. The synergy gap measures the difference between the current value of the function and all other potential functions in the collection. If  holds, we say that $V$ has *synergy gap exceeding $\delta$*. Given $V\in\spf[r]{\yset}$, we define the hybrid controller with output $\omega\in\R{n+1}$, state $y\in\yset$ and input $x\in\sphere n$ as follows:

\[eq:controller\] $$\begin{aligned}
\begin{rcases}
\omega = -\grad{V^y(x)}\\ 
\dot{y} = 0
\end{rcases}\  &(x,y)\in C\ceq\{(x,y)\in\Xmc:\gap(x,y)\leq\delta\}\\ 
y^+ \in\yminV(x)\phantom{-\grad{V^y}}
 &(x,y) \in D\ceq\{(x,y)\in\Xmc:\gap(x,y)\geq\delta\},\end{aligned}$$

where $\Xmc:=\sphere n\x\yset$ and $$\begin{aligned}
\label{eq:yminV}\yminV(x)&:=\argmin_{\bar y\in \yset}V(x,\bar y).
\end{aligned}$$ The interconnection between  and  is represented by the closed-loop hybrid system $\Hmc:=(C,F,D,G)$, given by $$\label{eq:closed_loop}
\begin{aligned}
\pmtx{\dot{x}\\ \dot{y}}&= F(x,y) = 
\pmtx{-\PTSn(x)\grad{V^y(x)}\\ 
 0} & (x,y)&\in C \\
\pmtx{x^+\\ y^+}&\in G(x,y)=\pmtx{x\\ \yminV(x)} & (x,y)&\in D.
\end{aligned}$$ The parameter $\delta$ that is used in the hybrid controller  is central to the construction of synergistic potential functions and it defines a hysteresis gap which prevents chattering.

<span>This construction is similar across many earlier works on synergistic hybrid feedback (see e.g., @mayhew_global_2013, @mayhew_synergistic_2013) and it guarantees global asymptotic stability of $x=r$ for the hybrid system , as shown next.</span>

\[theorem:gas\] Given $r\in\sphere n$ and a compact set $\yset$, if there exists $\delta>0$ such that $V\in\spf[r]{\yset}$ has synergy gap exceeding $\delta$, then the set $\Amc$ in  is globally asymptotically stable for the hybrid system $\Hmc$ in .

<span>It follows from Proposition \[prop:muIsNice\] that@goebel_hybrid_2012 [Assumption 6.5] is satisfied.</span>

<span>From computations similar to @mayhew_global_2013,</span> we conclude that the growth of $V$ along solutions to  is bounded by $u_C,u_D$, where

\[eq:uCuD\] $$\begin{aligned}
u_C(x,y)&\ceq\begin{cases}
-\norm{\PTSn(x)\grad{V^y(x)}}^2 & \text{ if } (x,y)\in C\\
-\infty & \text{ otherwise} 
\end{cases},\\
u_D(x,y)&\ceq\begin{cases}
-\delta & \text{ if } (x,y)\in D\\
-\infty & \text{ otherwise} 
\end{cases}\end{aligned}$$

<span>for each $(x,y)\in\sphere n\x\yset$. It follows from the assumption that $V$ is a synergistic potential function relative to $r$ that $\gap(x,y)>\delta$ for each $(x,y)\in\sticky(V)$, hence $u_C(x,y)<0$ for each $(x,y)\in\sphere n\x\yset\minus\Amc$. Since $u_D(x,y)<0$ for all $(x,y)\in\sphere n\x\yset$, it follows from @goebel_hybrid_2012 [Corollary 8.9] that $\Amc$ is globally pre-asymptotically stable for . Since $G(D)\subset C\cup D=\sphere n\x\yset$, $\sphere n\x\yset$ is compact and, for each $(x,y)\in C\minus D$, $F(x,y)$ belongs to the tangent cone to $C$ at $(x,y)$, it follows from @goebel_hybrid_2012 [Proposition 6.10] that each maximal solution to  is complete.</span>

We are also able to show that, under some additional conditions, the set  is globally exponentially stable in the $t$-direction for the hybrid system .

\[theorem:expStability\] Given $r\in\sphere n$ and a compact set $\yset$, if there exists $V\in\spf[r]{\yset}$ with synergy gap exceeding $\delta>0$ satisfying the following conditions

\[eq:expConditions\] $$\begin{aligned}
\label{eq:Vbounds}&\lbar{\alpha}\norm{x-r}^p\leq V(x,y)\leq \ubar{\alpha}\norm{x-r}^p & &\forall (x,y)\in C\cup D,\\
\label{eq:Vgamma}&\inner{\grad{V}(x,y)}{F(x,y)}\leq-\lambda V(x,y) & &\forall (x,y)\in C,\end{aligned}$$

for some $p,\lbar{\alpha},\ubar{\alpha},\lambda>0$, then the set  is globally exponentially stable in the $t$-direction for the hybrid system .

<span>It follows from the proof of Theorem \[theorem:gas\] that  satisfies @goebel_hybrid_2012 [Assumption 6.5], each of its maximal solutions is complete and $V$ is nonincreasing during jumps. Since $\gap(g)=0$ for each $g\in G(D)$, we have that $G(D)\cap D=\emptyset$. It follows from  that the conditions of @casau_hybrid_2017 [Theorem 1] are satisfied, thus $\Amc$ is globally exponentially stable in the $t$-direction for the hybrid system .</span>

Construction of centrally synergistic potential function on $\sphere n$ {#sec:construct}
-----------------------------------------------------------------------

Given $r\in\sphere n$, we construct a centrally synergistic potential function on $\sphere n$ using the height function: $\height{r}(x)\ceq 1-r\tp x$ for all $x\in\sphere n.$ Let $k > 0$, $\Vdom \ceq \Snsqr \minus \{(r,r)\}$, and define $V: \Vdom \to \R{}$ for all $(x,y) \in \Vdom$ as $$\label{eqn:V}
\begin{split}
V(x,y) &= \frac{\height{r}(x)}{h_{r}(x) + k\height{y}(x)} = \frac{1-r\tp  x}{1-r\tp  x + k\left(1-y\tp  x\right)}.
\end{split}$$

We now provide some differential properties of $V$, which follow from elementary calculation and some tedious manipulation, so they are presented without proof.

\[lem:gradV\] The function $V: \Vdom \to [0,1]$ defined in satisfies

$$\begin{aligned}
\label{eqn:Vgradx}
\grad V^y(x) &= \frac{kV(x,y)y - (1-V(x,y))r}{1-r\tp  x + k\left(1-y\tp  x\right)}\\
\label{eqn:normVgradx}
\left| \PTSn(x)\grad V^y(x) \right|^{2} &= \frac{2kV(x,y)\left(1-V(x,y)\right)\left(1-r\tp  y\right)}
{\left(1-r\tp  x + k\left(1-y\tp  x\right)\right)^{2}}\end{aligned}$$

Given $\yrbnd \in \R{}$ satisfying $-1 <\yrbnd < 1,$ we define the set $\yset \subset \sphere n$ as $$\label{eqn:yset}
\yset = \left\{ y \in \sphere n: r\tp  y \leq \yrbnd\right\}.$$ The boundary of $\yset$ is $\partial\yset = \left\{y \in \sphere n: r\tp  y = \yrbnd\right\}.$

\[lem:yminV\] Given $r\in\sphere n$ and $\yrbnd\in[-1,1)$, let $\yset$ be given by . The following holds for the function $V$ given in 

$$\begin{aligned}
\label{eqn:yminV}
\yminV(x) &= \begin{cases}
\yset & \text{if }x=r\\
-x & \text{if } r\tp  x \geq -\yrbnd \\
\frac{\sigma\left(r\tp  x\right)\PTSn(x)r}{\left|\PTSn(x)r\right|} + \alpha\left(r\tp  x\right)x & \text{if } -1 < r\tp  x < -\yrbnd \\
\partial\yset & \text{if } r\tp  x = -1.
\end{cases}\\
\label{eqn:cyminV}
\minVy(x) &= \begin{cases}
\displaystyle\frac{1-r\tp  x}{1-r\tp  x+2k} & \text{if } r\tp  x \geq -\yrbnd \\
\displaystyle\frac{1-r\tp  x}{1-r\tp  x +k\left(1-\alpha\left(r\tp  x\right)\right)} & \text{if } r\tp  x < -\yrbnd 
\end{cases}
 \end{aligned}$$

for each $x \in \sphere n$, where for each $v\in[-1,1]$, and $\minVy$, $\yminV$ are defined in  and , respectively.$\square$

Suppose that $x = r$. Then, according to of Lemma \[lem:V\], $V(x,y) = 0$ for all $y \in \yset$. Thus, any $y \in \yset$ attains the minimum of $y \mapsto V(r,y) = 0$. If $r\tp  x \geq -\yrbnd$, it follows that $-x \in \yset$. In this case, of Lemma \[lem:V\] yields that the unconstrained minimizer of $y \mapsto V(x,y)$, which is $-x$, is also the minimizer of $y \mapsto V(x,y)$ when $y$ is constrained to $y \in \yset$. When $x = -r$, we have that $V(-r,y) = 2/(2 + k(1+r\tp  y)).$ Clearly, $y \mapsto V(-r,y)$ is minimized by maximizing $r\tp  y$. When $y$ is constrained to $\yset$, the maximum value of $r\tp  y$ is $\yrbnd$, which is attained by any $y$ satisfying $r\tp  y = \yrbnd$, or equivalently, $y \in \partial \yset$. We now examine the case when $-1 < r\tp  x \leq -\yrbnd$. Since $-x \notin \yset$, it suffices to study the solutions to the constrained minimization problem by means of Lagrange multipliers.

Using , we may compute , from which the next result follows.

For any given $r\in\sphere n$ and <span>$\yrbnd\in (-1,1)$,</span> the function  is a centrally synergistic potential function relative to $r$ with synergy gap exceeding $\delta$, for any $\delta\in \left(0,(1+\yrbnd)/(2/k+1+\yrbnd)\right).$

This result follows from the fact that

In addition to global asymptotic stability of $\Amc$ for , we also show below that the function  satisfies , thus it follows from Theorem \[theorem:expStability\] that global $\Amc$ is also globally exponentially stable in the $t$-direction for . It follows from the fact that $\sticky(V)=V\inv (1)$ (c.f. Lemma \[lem:V\]) and from $\sticky(V)\subset D$, with $D$ given in , that $\maxVflow\in\R{}$, given by $\maxVflow = \max_{(x,y) \in C} V(x,y),$ with $C$ given in  satisfies $0 < \maxVflow < 1$. We note that $\maxVflow$ exists since $V$ is continuous on the compact set $C$. The next theorem follows naturally from these considerations

\[thm:VexpConditions\] The function $V\in\C^1(\sphere n\x\yset)$ given in  satisfies  with

\[eq:aagg\] $$\begin{aligned}
\label{eq:laa}\lbar{\alpha}&=(2(1+k+\sqrt{1+2k\yrbnd+k^2}))^{-1},\\
\label{eq:uaa}\ubar{\alpha}&=(2(1+k-\sqrt{1+2k\yrbnd+k^2}))^{-1}\\
\label{eq:gg}\lambda&=\frac{2k(1-\maxVflow)(1-\yrbnd)}{\left(1+k+\sqrt{1+2k\yrbnd+k^2}\right)^{2}}.\end{aligned}$$

Therefore, the set  is globally exponentially stable in the $t$-direction for .

Since $h_r(x)=1-r\tp   x=\norm{x-r}^2/2$ for every $x\in\sphere n$, it follows from  that  satisfies  with $\lbar{\alpha}$ and $\ubar{\alpha}$ given by  and , respectively. Since $r\tp   y\leq\yrbnd$ for each $(x,y)\in\sphere n\x\yset$, then $1-r\tp   y\geq 1-\yrbnd$. It follows from the fact that $V(x,y)\leq \maxVflow$ for each $(x,y)\in C$ and from Lemma \[lem:globaldenombnd\] that $$\label{eqn:gradVboundflow}
\left| \PTSn(x)\grad V^y (x) \right|^{2} \geq \frac{2k(1-\maxVflow)(1-\yrbnd)}{\left(1+k+\sqrt{1+2k\yrbnd+k^{2}}\right)^{2}}V(x,y),$$ which proves that  is satisfied. Global exponential stability of  for  follows from Theorem \[theorem:expStability\].

Minimal Geodesics
-----------------

We have shown that the proposed synergistic potential function  ensures not only global asymptotic stability of $\Amc$ for , but also global exponential stability. Moreover, there is a third property of this function that is worth mentioning: for each $x_0\in D$, the flows generated by the gradient vector field $\PTSn(x)\grad V^{\yminV(x_0)}(x)$ converge to $r$ through the minimal geodesic, i.e., the path of minimum distance between $x_0$ and $r$, as proved next. For every pair of points $p,q\in M$, where $M$ is a compact manifold, there is $\geo :\R{}\rightarrow M$ with $\geo(a)=p$ and $\geo(b)=q$ for some $a,b\in R{}$ such that the length of $\geo$, given by $\length{\geo}=\int_a^b\norm{\frac{dc}{dt}(\tau)}d\tau,$ is the lowest among all other paths with endpoints $p$ and $q$ (c.f. @burns_differential_2005). If a path $p:[0,\infty)\rightarrow M$ is Lebesgue integrable, then its length is given by $\length[\infty]{p}=\int_0^\infty\norm{\frac{dp}{dt}(\tau)}d\tau.$ Considering the Riemannian manifold $\sphere n$ with the standard Euclidean metric, its (unit-velocity) geodesics are of the form $x(t)=a\cos t+b\sin t,$ for each $t\in\R{}$, with $a,b\in\sphere n$ satisfying $\inner{a}{b}=0$, as shown in . Given $r\in\sphere n$, one verifies that the minimal geodesic between $r$ and any given point $x_0\in\sphere n\backslash\{r,-r\}$ is given by $\geo (t)=x_0\cos t+\PTSn(x_0)r\sin t/\norm{\PTSn(x_0)r},$ for each . If $x_0=-r$, then each minimal geodesic from $x_0$ to $r$ is given by $\geo(t)=x_0\cos t+x^\bot\sin t,$ for each $t\in[0,\pi]$, where $x^\bot\in\{y\in\sphere n: \inner{y}{x_0}=0\}$. <span>In particular, if $x_0$ is antipodal to $r$ and $n>1$ then there are uncountably many minimal geodesics from $x_0$ to $r$. </span>

It is possible to verify that, for every solution $(x,y)$ of  and each $(t,j),(t,j+1)\in\dom{(x,y)}$, the following holds: $x(t,j)=x(t,j+1).$ Therefore, $x{\downarrow_t} (t):=x(t,J(t)),$ is defined for each $t\in [0,\sup_t\dom{(x,y)})$, for each solution $(x,y)$ to , where with $J(t):=\max\{j:(t,j)\in\dom (x,y)\}.$ Moreover, it is absolutely continuous, hence $\length[\infty]{x{\downarrow_t}}$ is well-defined. Choosing $V$ in , we show next that the solutions to the hybrid system  that start in the jump set have minimal length.

\[lem:geo\] Consider the function $V\in\C^1(\sphere n\x\yset)$ given in . For each solution $(x,y)$ to  with initial condition $(x_0,y_0)\in D$, we have that $\length[\infty]{x_t}=\length{\geo_{x_0,r}}$, where $\geo_{x_0,r}$ denotes the minimal geodesic from $x_0$ to $r$.

This result follows from the fact that the vector field $\PTSn(x)\grad V^{\yminV(x_0)}(x)$ is tangent to the geodesic connecting $r$ and $x_0$ and from Theorem \[thm:VexpConditions\] which shows that solutions converge to $r$.

Note that function $\height{r}$ induces a gradient-vector field that generates flows along geodesics for almost all initial conditions. However, unlike the aforementioned continuous feedback law, the controller presented in this section exponentially renders a reference point globally asymptotically stable, which is not possible with continuous feedback. Moreover, if we resolve the ambiguity at $x=-r$ by means of some discontinuity, we still are left without any guarantees of robustness to small measurement noise. Since the hybrid controller presented in this section satisfies the hybrid basic conditions @goebel_hybrid_2012 [Assumption 6.5], the property of global asymptotic stability of $\Amc$ for  is endowed with robustness to small measurement noise, as discussed in @goebel_hybrid_2012.

Application to Trajectory Tracking for a Quadrotor {#sec:quad}
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Simulation Results {#sec:sim}
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![image](flowfields.eps){width="80.00000%"}

Conclusions {#sec:conclusions}
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In this paper, we have demonstrated that the existence of synergistic potential functions on $\sphere n$ is a sufficient condition for the asymptotic stabilization of systems evolving on the $n$-dimensional sphere. Moreover, if these functions and their derivatives satisfy some additional bounds, it is possible to achieve global exponential stability. We provided a construction of synergistic potential functions which generates flows along geodesics upon switching. The proposed controller was then applied to trajectory tracking for a vectored thrust vehicle.

Auxiliary results
=================

\[prop:muIsNice\] Let $r\in\sphere n$ and let $\yset$ be a compact set. Given $V\in\spf[r]{\yset}$, the following holds:

1.  \[item:Vpd\] $r\in \crit(V^y)$ for each $y\in\yset$;

2.  \[item:muC\]The function  is continuous and the map  is outer semicontinuous.

\[lem:V\] The function $V : \Vdom \to \R{}$ satisfies $$\begin{aligned}
\label{eqn:argminV}
\begin{split}
\argmin_{(x,y) \in \Vdom} V(x,y) &= V^{-1}(0) = \{(r,y) \in \Vdom\}
\end{split} \\
\label{eqn:argmaxV}
\begin{split}
\argmax\limits_{(x,y) \in \Vdom} V(x,y) &= V^{-1}(1) = \{(x,x) \in \Vdom\}
\end{split}\end{aligned}$$ Moreover, $V(x,y)$ is positive definite on $\Vdom$ relative to $\{(r,y) \in \Vdom\}$ and for each $x \in \sphere n$, $$\begin{aligned}
\label{eqn:argminyV}
\argmin_{y \in \sphere n} V(x,y) &= \begin{cases} -x & \text{if } x \neq r \\
\sphere n\minus\{r\} & \text{if } x = r
\end{cases} \\
\label{eqn:minyV}
\min_{y \in \sphere n} V(x,y) &= \frac{1-r\tp  x}{2k+1 - r\tp  x}.\end{aligned}$$

Since $\height{r}(x)\geq 0$ for all $x\in\sphere n$ and $\height{r}(x)=0$ if and only if $x=r$, it follows that $h_{r}(x) + kh_{y}(x) > 0$ on $\Vdom$. Setting $V(x,y) \leq 0$, we find that $h_{r}(x) \leq 0$, which can only be satisfied (with equality) when $x = r$. That is, $V$ attains its minimum value of zero on the set $\{(r,y) \in \Vdom\}$. Similarly, $V(x,y) \geq 1$ if and only if $h_{y}(x) \leq 0$, which is again satisfied (with equality) only when $x = y$ and thus, $V$ attains its maximum value of one on the set $\{(x,x) \in \Vdom\}$. To prove and , we note—from our previous observations—that $$\label{eqn:minyV2}
\begin{split}
\min_{\bar y \in \sphere n} V(x,\bar y) &= \frac{1-r\tp  x}{1-r\tp  x + k\max_{\bar y \in \sphere n} (1-\bar y\tp  x)}.
\end{split}$$ Since, for each $x \in \sphere n$, $1-y\tp  x$ attains its maximal value of two at $y = -x$, this choice minimizes $y \mapsto V(x,y)$ for each $x \in \sphere n\setminus\{r\}$; however, when $x = r$, $V(x,y) = 0$ for any $y \in \sphere n\setminus\{r\}$, thus proving . Eq.  follows from evaluating $V(x,y)$ with $y = -x$.

\[cor:critV\] Given $y \in \sphere n$, define $V^y: \sphere n \to \R{}$ for each $x \in \sphere n$ as $V^y(x) = V(x,y)$, with $V$ given by . Then, $$\label{eqn:critV}
\crit V^y= \begin{cases}
\{r,y\} & \text{if } r \neq y \\
\sphere n & \text{otherwise}.
\end{cases}$$

By definition, $x \in \crit V^y$ if and only if $\PTSn(x)\grad V^y(x) = 0$, or equivalently, $\left|\PTSn(x)\grad V^y(x)\right| = 0$. Noting that $V^y(x) = V(x,y)$ for each $x \in \sphere n$, it follows that $\grad V^y(x) = \gradx V(x,y)$ for each $x \in \sphere n$. By of Lemma \[lem:gradV\], it follows that $x \in \crit V^y$ if and only if $$\label{eqn:critVpoints}
V^y(x)(1-V^y(x))(1-r\tp  y) = 0.$$ Clearly, is satisfied in three cases: $V^y(x) = 0$, $1-V^y(x) = 0$, or $1-r\tp  y = 0$. When $1-r\tp  y = 0$, or equivalently, when $r = y$, it follows that every point in $\sphere n$ is a critical point, since is satisfied for every $x \in \sphere n$. In fact, when $r = y$, $V^y(x) = 1/(1+k)$, so that for all $x \in \sphere n$, $\grad V^y(x) = 0$ and obviously $\PTSn(x)\grad V^y(x) = 0$. We now examine the remaining cases. When $V^y(x) = 0$, it follows from the definition of $V^y(x) = V(x,y)$ that $1-r\tp  x = 0$, or equivalently, $x = r$. If $V^y(x) = 1$, a short calculation yields $1-y\tp  x = 0$, or equivalently, $x = y$. Thus, when $r \neq y$, $\crit V^y= \{r,y\}$.

\[lem:globaldenombnd\] The following holds $$\label{eqn:globaldenombnd}
\begin{multlined}
0< 1+k-\sqrt{1+2k\yrbnd+k^2} \leq 1-r\tp   x + k(1-y\tp   x) \\ \leq 1+k+\sqrt{1+2k\yrbnd+k^2}.
\end{multlined}$$ for all $(x,y) \in \sphere n\x\yset$.

The upper and lower bounds on  follow from the solution to the optimization problem $\min/\max\{J(x,y):1-y\tp   y = 0,1-x\tp   x = 0,r\tp   y - \yrbnd = 0\}$ with $J(x,y) = 1-r\tp   x + k(1-y\tp   x)$ for each $(x,y) \in \sphere n\x\yset$, by means of Lagrange multipliers.

{'source': 'arXiv_out/1910'}