Hybrid Feedback for Global Tracking on Matrix Lie Groups $SO(3)$ and $SE(3)$

The attitude tracking control problem of rigid body systems has been widely investigated in the literature with many applications related to robotics, aerospace and marine engineering, for instance [2, 3, 4, 5, 6, 7]. In particular, geometric control design on Lie groups $SO(3)$ and $SE(3)$, has generated a great deal of research work [8, 9, 10, 11, 6, 12, 13]. It is well known that achieving global stability results with feedback control schemes designed on Lie groups such as $SO(3)$ and $SE(3)$, is a fundamentally difficult task due to the topological obstruction of the motion space induced by the fact that these manifolds are not homeomorphic to $\mathbb{R}^{n}$ and that there is no smooth vector field that can have a global attractor [8]. To achieve almost global asymptotic stability (AGAS), a class of suitable “navigation functions” has been introduced in [8]. In [9], the Riemannian structure of the configuration manifold for a class of mechanical systems, is used to derive a local exponential control law, while in [11], an almost global tracking controller has been proposed for a general class of Lie groups via intrinsic globally-defined error dynamics. It is well known that for any smooth potential function on $SO(3)$, there exist at least four critical points where its gradient vanishes [14], and as such, almost global stability is the strongest result that one can achieve in this case.

Using the hybrid dynamical systems framework of [15, 16], a unit-quaternion based hybrid control scheme for global attitude tracking was first proposed in [17], which led thereafter to other variants such as [18, 19, 20]. The use of a “synergistic” family of potential functions to overcome the topological obstruction to global asymptotic stability (GAS) on $SO(3)$ has been introduced in [21]. A family of potential functions is “centrally” synergistic, if the identity is the common critical point of all the potential functions in the family. This synergistic hybrid approach was successfully applied to the rigid body attitude control problem in [22], where a hysteresis-based switching mechanism was introduced to avoid all the undesired critical points and ensure some robustness to measurement noise. However, only numerical examples were provided to construct such a synergistic family of potential functions via angular warping on $SO(3)$. Inspired by this, a number of hybrid controllers and observers on Lie groups $SO(3)$, $SE(3)$ and $SE_{2}(3)$ have been proposed in the literature [23, 24, 25, 26, 27, 28, 29, 30, 31]. The work in [26], provides a systematic and comprehensive procedure for the construction of synergistic potential functions on $SO(3)$, which is then applied to design velocity-free hybrid attitude stabilization and tracking control schemes [26, 27]. Moreover, a hybrid attitude control scheme on $SO(3)$ using an “exp-synergistic” family of potential functions, leading to global exponential stability, has been proposed in [28]. Alternatively, a “non-central” synergistic family of potential functions has been considered in [23] to relax the centrality condition. A similar idea can be found in [25]. Recently, a hybrid control approach on smooth manifolds, relying on a switching logic between local coordinates, has been proposed in [32].

Contributions: In this paper, we propose new hybrid feedback control strategies for the tracking problems on matrix Lie groups $SO(3)$ and $SE(3)$, leading to GAS guarantees. Some extensions are also proposed to smooth out the discontinuities induced by the control input switching, and to handle the lack of angular velocity measurements. The main novelty and strength of our approach is the fact that it can overcome the compactness condition required in the synergistic hybrid approaches. Therefore, it can be applied to a general class of non-compact matrix Lie groups to generate globally asymptotically stabilizing hybrid feedback laws as demonstrated through the design of a geometric hybrid tracking control scheme on the non-compact manifold $SE(3)$. The key idea of our hybrid attitude control strategy consists in using a suitable potential function on $SO(3)\times\mathbb{R}$ involving an auxiliary scalar variable with hybrid dynamics. This scalar variable is governed by some appropriately designed flow dynamics when the state in $SO(3)\times\mathbb{R}$ is away from the undesired critical points, and is governed by an appropriately designed jump strategy when the state is in the neighborhood of the undesired critical points. The flow and jump strategies are designed to ensure a decrease of the potential function over the flows and jumps. In contrast with the synergistic hybrid approaches where a synergistic family of potential functions on $SO(3)$ is used [22, 21, 23, 26, 27], our hybrid approach relies on a single potential function on $SO(3)\times\mathbb{R}$ parameterized by the hybrid auxiliary scalar variable. As it is going to be shown later, our approach on top of the hybrid control design simplification, allows to overcome the set compactness assumption (stemming from the diffeomorphism condition of the transformation map) required in the synergistic hybrid approaches. This important advantage, makes our approach a good candidate for the design of hybrid controllers on non-compact manifold such as $SE(3)$ where the existing hybrid approaches (for instance, [22, 21, 25, 26, 28, 27, 32]) are not applicable in a straightforward manner. A preliminary version of this work has been presented in [1] without the semi-global exponential stability proof, the control smoothing mechanism and the velocity-free hybrid tracking scheme presented in the present paper.

Organization: The remainder of this paper is organized as follows: Section II provides the preliminaries that will be used throughout this paper. In Section III, we formulate our attitude tracking control problem. In Section IV, a new hybrid mechanism using a potential function on $SO(3)\times\mathbb{R}$ is presented. In Sections V-VII, we present our three hybrid attitude tracking controllers. In Section VIII, we provide a systematic procedure for the construction of the potential function on $SO(3)\times\mathbb{R}$ satisfying all the requirements for the design of our hybrid attitude tracking controllers. In Section IX, our hybrid control strategy is extended to the pose tracking problem on the non-compact Lie group $SE(3)$. Simulation results are presented in Section X.

The dynamical equations of motion of a rigid body on $SO(3)$ are given by

where the rotation matrix $R$ denotes the attitude of the rigid body, $\omega\in\mathbb{R}^{3}$ is the body-frame angular velocity, $J=J^{\top}\in\mathbb{R}^{3\times 3}$ is the inertia matrix (constant and known), and $\tau\in\mathbb{R}^{3}$ is the control torque to be designed.

Let $m>0$ and let $\mathcal{W}_{d}\subset SO(3)\times\mathbb{R}^{3}$ be a compact (closed and bounded) subset. Let the desired reference trajectory be generated by the following dynamical system [23]:

where $R_{r}$ and $\omega_{r}$ are the desired rotation and angular velocity, and $m\mathbb{B}:=\{x\in\mathbb{R}^{3}:\|x\|\leq m\}$ is the closed ball in $\mathbb{R}^{3}$. As shown in [23], every maximal solution to (6) is complete, and any possible solution component $\omega_{r}$ of (6) is Lipschitz continuous with Lipschitz constant $m$, but not necessarily differentiable.

Let us consider the left-invariant attitude tracking error $R_{e}=R_{r}^{\top}R$ and the angular velocity tracking error $\omega_{e}=\omega-R_{e}^{\top}\omega_{r}$. From (2)-(6), one obtains the following error dynamics [23]:

where the functions $\Upsilon:SO(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ and $\Sigma:SO(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathfrak{so}(3)$ are given by

Note that $\Sigma$ is skew-symmetric, and as such, for each $u\in\mathbb{R}^{3}$ one has $u^{\top}\Sigma(R_{e},\omega_{e},\omega_{r})u=0$. It is clear that $\Upsilon(R_{e},\omega_{r},z)$ is known, and $\Upsilon(R_{e},\omega_{r},z)=0$ if $R_{r}$ is constant.

The control objective consists in designing hybrid feedback laws such that the desired equilibrium of the error dynamics (7a)-(8b), i.e., $(R_{e}=I_{3},\omega_{e}=0)$, is globally asymptotically stable.

Let $V(R)$ be a potential function on $SO(3)$ with respect to $I_{3}$. Let $\nabla_{R}V:R\to T_{R}SO(3)$ denote the gradient of $V$ at point $R$. According to Lusternik-Schnirelmann theorem [34] and Morse theory [14], a smooth vector field on $SO(3)$ can not have a global attractor, and any smooth potential function on $SO(3)$ must have at least four critical points. Let the set of all critical points of $V(R)$ be denoted by $\Psi_{V}=\{R\in SO(3)|\nabla_{R}V(R)=0\}$, and the set of all undesired critical points be denoted by $\Psi_{V}\setminus\{I_{3}\}$.

One possible way to make the desired critical point $I_{3}$ a global attractor, consists in generating a non-smooth gradient-based vector field on $SO(3)$ through a switching mechanism between a family of potential functions as done in [22, 21, 26]. The potential functions are constructed using a modified trace function and a transformation map $\mathcal{T}:SO(3)\to SO(3)$ such that all the potential functions share only the desired critical point $I_{3}$. For instance, a transformation map $\mathcal{T}(R)=\exp(\vartheta(R)u^{\times})R$ with $\vartheta(R)=kV(R),k\in\mathbb{R},u\in\mathbb{S}^{2}$, known as the “angular warping”, is considered in [21]. As shown in [21, Theorem 8], $\mathcal{T}$ is required to be a diffeomorphism, and to be as such, $k$ has to be chosen sufficiently small in magnitude (i.e., $\sqrt{2}|k|\max\|\nabla_{R}V(R)\|_{F}<1$ for all $R\in SO(3)$). A similar design of the central synergistic family of potential functions can be found in [26], where a different transformation map $\mathcal{T}(R)=R\exp(\vartheta(R)u^{\times})$ with $\vartheta(R)=2\arcsin(kV(R))$ and $k$ sufficiently small, has been considered. Note that the existence of the parameter $k$ in [22, 21, 26] is guaranteed mainly due to the fact that $SO(3)$ is compact. Alternatively, in [23], a non-central synergistic family of potential functions has been designed based on a modified trace function through constant translation, scaling and biasing. Unfortunately, it is not straightforward to explicitly construct such a family of potential functions, especially when dealing with systems evolving on non-compact manifolds.

To overcome the above mentioned problems, we propose a new approach that does not require the generation of a family of potential functions via a diffeomorphism map, leading to a much simpler design of hybrid control systems on $SO(3)$ or other non-compact manifolds. The key idea consists in using a single potential function $U:SO(3)\times\mathbb{R}\to\mathbb{R}_{\geq 0}$, with respect to the $\mathcal{A}_{o}:=(I_{3},0)$, parameterized by a scalar variable $\theta\in\mathbb{R}$ that has flow and jump dynamics. In contrast with the previously mentioned synergistic potential functions approaches, where the potential functions are parameterized by a discrete variable, our single potential function $U$ is adjusted through the continuous flows and the discrete jumps of the auxiliary variable $\theta$ such that the resulting non-smooth gradient-based vector field yields a single attractor $\mathcal{A}_{o}$. The details of the construction of the potential function $U$ and its properties will be presented later. Let us define the set of all the critical points of $U$ as

with $\nabla_{R}U(R,\theta)$ and $\nabla_{\theta}U(R,\theta)$ denoting the gradients of $U(R,\theta)$ with respect to $R$ and $\theta$, respectively. Let $\Theta\subset\mathbb{R}$ be a nonempty and finite real set, and consider the following basic assumption for our potential function $U(R,\theta)$:

Once a nonempty finite set $\Theta\subset\mathbb{R}$ and a potential function $U$ satisfying the basic Assumption 1 are constructed, as it will be shown later, the flow and jump dynamics governing the evolution of $\theta$ and in turn of $U$ will be designed to avoid the undesired critical points, leaving $\mathcal{A}_{o}$ as the unique attractor. In fact, $\theta$ flows when the state $(R,\theta)$ is away from the set $\Psi_{U}\setminus\{\mathcal{A}_{o}\}$, and jumps to some ${\theta}^{\prime}\in\Theta$ leading to minimum value of $U(R,\theta^{\prime})$) when the state $(R,\theta)$ is in the neighborhood of the set $\Psi_{U}\setminus\{\mathcal{A}_{o}\}$. We propose the following hybrid dynamics $\mathcal{H}_{\theta}$ for $\theta$:

with $\theta(0,0)\in\mathbb{R}$. The flow and jump sets are defined as

with $\mu_{U}$ defined in (11), and the flow map $f:SO(3)\times\mathbb{R}\to\mathbb{R}$ and jump map $g:SO(3)\times\mathbb{R}\rightrightarrows\Theta$ are defined as

with constant scalar $k_{\theta}>0$. By Assumption 1, the design of the jump set $\mathcal{J}$ implies that all the undesired critical points are located in the jump set, i.e., $\Psi_{U}\setminus\{\mathcal{A}_{o}\}\subset\mathcal{J}$. Note that the flow map $f$ in (14) is nothing else but the negative gradient of $U$ with respect to $\theta$ contributing to driving the state $(R,\theta)$ towards the critical points of $U$. The jump map $g$ in (15) is designed to drive (through jumps) the state $(R,\theta)$ away from the undesired critical points. From the definitions of the jump set $\mathcal{J}$ and jump map $g$, it is clear that every $(R,\theta)\in\mathcal{J}$ , one has $U(R,\theta)-U(R,g(R,\theta))=U(R,\theta)-\min\nolimits_{{\theta}^{\prime}\in\Theta}U(R,{\theta}^{\prime})=\mu_{U}(R,\theta)\geq\delta$, which guarantees a minimum decrease of the potential function $U$ by $\delta$, after each jump.

We propose the following hybrid feedback tracking control scheme:

where function $\Upsilon$ is defined in (8a), the flow and jump sets $\mathcal{F}$ and $\mathcal{J}$ are defined in (13a) and (13b), respectively, the function $\kappa$ is given by

with constant scalars $k_{R},k_{\omega}>0$, and the maps $f$ and $g$ (in terms of $R_{e}$ and $\theta$) are defined in (14) and (15), respectively. The main difference between the proposed hybrid feedback (21)-(22) with respect to the ones proposed in [23, 27] is the extended hybrid dynamics of the auxiliary variable $\theta$ which modifies (continuously) the potential function in the flow set $\mathcal{F}$ and modifies (through jumps) the potential function in the jump set $\mathcal{J}$ (i.e., in the neighborhood of the undesired critical points of $U$). The gradient of the potential function $U$, with the extended hybrid dynamics of $\theta$, is used in the control to force $\mathcal{A}_{o}$ to be a global attractor. Fig. 2 illustrates the proposed hybrid feedback strategy.

Define the new state space $\mathcal{S}:=SO(3)\times\mathbb{R}\times\mathbb{R}^{3}\times\mathcal{W}_{d}$ and the new state $x:=(R_{e},\theta,\omega_{e},R_{r},\omega_{r})\in\mathcal{S}$. In view of (7), (12)-(13) and (21), one has the following hybrid closed-loop system:

where the flow and jump maps are given by

with $\Sigma,f,g$ and $\kappa$ defined in (8b), (14), (15) and (22), respectively. One can verify that $\mathcal{F}_{c}\cup\mathcal{J}_{c}=\mathcal{S}$, $\mathcal{F}_{c}$ and $\mathcal{J}_{c}$ are closed, and the hybrid closed-loop system (23) is autonomous and satisfies the hybrid basic conditions [16, Assumption 6.5]. Now, one can state one of our main results:

Now, under the following additional assumptions on the potential function $U$, we will show that the proposed hybrid controller achieves exponential stability.

Assumptions 2 and 3 impose some bounds on the gradients of the potential function and the time derivative of $\psi(R_{e}^{\top}\nabla_{R_{e}}U(R_{e},\theta))$. This assumptions are not very restrictive as it is going to be shown later once we present the construction of the potential function in Section VIII.
Let $|x|_{\mathcal{A}}^{2}:=U(R_{e},\theta)+\|\omega_{e}\|^{2}$. Since $U$ is a potential function on $SO(3)\times\mathbb{R}$ with respect to $\mathcal{A}_{o}$, it follows from the definitions of $\mathcal{S}$ and $\mathcal{A}$ that $|x|_{\mathcal{A}}=0$ for all $x\in\mathcal{A}$, and $|x|_{\mathcal{A}}>0$ for all $x\in\mathcal{S}\setminus\mathcal{A}$. Now, one can state the following result:

In order to remove the discontinuities in the control input $\tau$ (caused by the discrete jumps of $\theta$), we propose the following modified hybrid feedback tracking scheme:

where $\theta(0,0)\in\mathbb{R},\zeta(0)\in\mathbb{R}^{3}$, constants $k_{R},k_{\omega}>0$, the maps $f$ and $g$ are defined in (14) and (15), the function $\Upsilon$ is defined in (8a), and the function $h$ is given by

with constant $k_{\zeta}>0$. In this case, the flow and jump sets are given by

where $0<\delta^{\prime}<\delta$, $\mho:=SO(3)\times\mathbb{R}\times\mathbb{R}^{3}$ and

with some $\varrho>0$ to be designed later. The main difference between this hybrid control scheme and the previous one in (21), is the use of an dynamical variable $\zeta$ that bears the hybrid jumps of $\theta$ resulting in a jump-free control signal. As shown in (37)-(38), the dynamics of $\zeta$ allow to relocate the jumps one integrator away from the control torque. This mechanism leads to a continuous torque input since $\zeta$ is continuous (not necessary differentiable due to the discrete jumps of $\theta$ in the gradient-based term).

Since $U$ is a potential function on $SO(3)\times\mathbb{R}$ with respect to $\mathcal{A}_{o}$ and $\psi(R_{e}^{\top}\nabla_{R_{e}}U(R_{e},\theta))=0$ as $(R_{e},\theta)=\mathcal{A}_{o}$, one can show that $W$ defined in (42) is a potential function on $\mho$ with respect to $\widehat{\mathcal{A}}_{o}:=(R_{e}=I_{3},\theta=0,\zeta=0)$, and the set of its critical points is given by $\Psi_{W}:=\{(R_{e},\theta,\zeta)\in\mho:(R_{e},\theta)\in\Psi_{U},\zeta=0\}$. To ensure that all the undesired critical points of $W$ are located in the jump set $\widehat{\mathcal{J}}$ in (40), we consider the following assumption:

The proof of Lemma 1 is given in Appendix -C. From the definition of $\widehat{\mathcal{J}}$ in (40), Lemma 1 implies that all the undesired critical points of $W$ in (42) are located in the jump set $\widehat{\mathcal{J}}$ (i.e., $\Psi_{W}\setminus\{\widehat{\mathcal{A}}_{o}\}\subset\widehat{\mathcal{J}}$) under Assumption 1, 4 and some small enough positive constant $\varrho$.

Define the new state space $\widehat{\mathcal{S}}:=\mathcal{S}\times\mathbb{R}^{3}$ and the new state $\hat{x}:=(x,\zeta)\in\widehat{\mathcal{S}}$. In view of (7), (12)-(13) and (37)-(38), one has the following hybrid closed-loop system:

where the flow and jump maps are given by

with $\Sigma,f$ and $g$ defined in (8b), (14) and (15), respectively. One can verify that $\widehat{\mathcal{F}}_{c}\cup\widehat{\mathcal{J}_{c}}=\widehat{\mathcal{S}}$, $\widehat{\mathcal{F}}_{c}$ and $\widehat{\mathcal{J}}_{c}$ are closed, and the hybrid closed-loop system (44) satisfies the hybrid basic conditions [16, Assumption 6.5]. The properties of the set $\widehat{\mathcal{A}}:=\{\hat{x}\in\widehat{\mathcal{S}}:(R_{e},\theta,\zeta)=\widehat{\mathcal{A}}_{o},\omega_{e}=0\}$ for the closed-loop system (44) are stated in the following theorem: