Geometric sliding mode control of mechanical systems on Lie groups

The geometric approach to control designs has achieved significant advances for mechanical systems on nonlinear manifolds, for recent developments in this topic, see, for instance, Bullo & Lewis (2005) and the references therein. As recognized in Koditschek (1989); Bullo et al. (1995); Maithripala et al. (2006), a key point in control design is how to define the tracking error. The tracking error defined on a Riemannian manifold relying on an error function and a transport map in Bullo & Murray (1999) may be simplified if the manifold is endowed with a Lie group structure (Maithripala et al., 2006), where the error notion can be globally defined explicitly and is easier to be manipulated for stability analysis of the closed-loop system (Maithripala & Berg, 2015; Saccon et al., 2013; De Marco et al., 2018; Lee, 2012; Sarlette et al., 2010). A similar situation is encountered in observer designs using an estimation error defined on a Riemannian manifold (Aghannan & Rouchon, 2003) versus an estimation error defined by the group operation on Lie groups bonnabel2009non. The ability to define a global error on Lie groups provides a powerful tool for treating the error as an object in the state space globally and controlling it as a physical system so that the tracking problem can be reduced to stabilizing the error dynamics to the group identity (Bullo et al., 1995; Maithripala et al., 2006; Spong & Bullo, 2005). Moreover, a separation principle can be proved in the geometric approach to control designs (Maithripala et al., 2006; Maithripala & Berg, 2015) when part of the state in the control law is estimated by an exponentially convergent observer designed on the Lie group (Bonnabel et al., 2009), similar to an LTI system. This opens a wide field of applications for systems on Lie groups, such as rigid body motion control and trajectory tracking in $2$D and $3$D spaces, given the significant advances in both geometric control designs (Bullo & Lewis, 2005; Spong & Bullo, 2005; Lee, 2011; Akhtar & Waslander, 2020; Rodríguez-Cortés & Velasco-Villa, 2022) and observer designs (Aghannan & Rouchon, 2003; Bonnabel et al., 2009; Mahony et al., 2008; Lageman et al., 2009; Zlotnik & Forbes, 2018).

GSMC on Lie groups has been considered using two main approaches: developing the SMC in the underlying Lie algebra or developing it on the Lie group itself. The main idea in the former approach is first expressing the tracking error defined on the Lie group in its Lie algebra through the locally diffeomorphic logarithmic map (Bullo et al., 1995). Since the Lie algebra is a vector space, a sliding surface can be designed as in the Euclidean case (Culbertson et al., 2021; Liang et al., 2021; Espíndola & Tang, 2022). In the latter approach, the sliding subgroup is designed directly on the Lie group. Since the topological structures of the configuration space (a Lie group) and the tangent space (a vector space) are very different when the underlying Lie group is not diffeomorphic to an Euclidean space, an important question arises as to how to ensure the sliding surface to be indeed a subgroup of the state space formed by the tangent bundle to guarantee the existence of the sliding mode and thus to inherit the salient features of SMC in the Euclidean space.

SMC designs using the second approach have been reported for Lie groups such as $SO(3)$, $\mathcal{S}^{3}$ for attitude control, and $SE(3)$ for motion controls (Ghasemi et al., 2020; Lopez & Slotine, 2021). However, the issue of whether the tangent bundle is a Lie group and whether the sliding subgroup is properly immersed on the tangent bundle was not addressed in these works. Therefore, the potential problem of lack of robustness due to the nonexistence of the sliding mode might appear. Recently, Gómez et al. (2019) brought this issue to the attention of the control community, and proposed an SMC on the rotation group $SO(3)$ with a sliding surface which was ensured to be a Lie subgroup immersed in the tangent bundle $SO(3)\times\mathbb{R}^{3}$, and a finite-time convergent controller was devised for attitude control. This design method was applied in Meng et al. (2023) to design a second-order SMC for fault-tolerant control designs.

We generalize the conventional sliding mode control designs for systems in Euclidean spaces to fully-actuated simple mechanical systems whose configuration space is a Lie group for the trajectory-tracking problem. The main contributions can be summarized as follows: (1) we endow the state space formed by the tangent bundle of the error dynamics with a Lie group structure by defining a group operation that is based on a generic kinematic control designed in the Lie algebra of the configuration Lie group; (2) we design a smooth sliding subgroup and show it to be a Lie subgroup of the tangent bundle, therefore, inheriting the Lie group structure of the state space; and (3) we design a coordinate-free geometric sliding mode controller for a fully-actuated mechanical system on a Lie group which drives the error dynamics to the sliding subgroup globally exponential at the reaching stage, the error dynamics then converges to the identity of the tangent bundle almost globally asymptotically at the sliding stage. In addition, rigid body tracking in $3$D space is addressed on the special orthogonal groups $SO(3)$ and on the unit sphere $\mathcal{S}^{3}$, respectively, by applying the proposed geometric sliding mode control.

The rest of the paper is organized as follows. Section 2 presents the notation and background materials for simple mechanical systems with Lie groups as the configuration space. Section 3 first endows the state space formed by the tangent bundle with a Lie group structure under a group operation, which is defined based on a generic kinematic control law in the Lie algebra of the configuration space. Then, a smooth sliding subgroup is defined, which is a Lie subgroup immersed in the tangent bundle. The convergence to the identity of the tangent bundle of the reduced-order dynamics constrained on the sliding surface is analyzed based on Lyapunov stability. Section 4 gives the design of the GSMC, composed of a reaching law to the sliding subgroup and the convergence property of the sliding subgroup. Attitude tracking of a rigid body in $3$D space is addressed in Section 5 respectively on the rotational group $SO(3)$ and the unit sphere $\mathcal{S}^{3}$, and simulation results under the GSMC developed on $SO(3)$ are presented in Section 6 for illustration and comparison. Conclusions are drawn in Section 7.

The purpose of this subsection is to design a control law $\nu(t)\in\mathfrak{g}$ for the kinematics (3) to render $g(t)\to e$, the group identity. Let $V:G\to\mathbb{R}_{\geq 0}$ be an infinitely differentiable proper Morse function, which satisfies $V(g)>0$, $\forall g\in G\backslash\{e\}$, $\mathrm{d}V(g)=0$ and $V(g)=0\iff g=e$. Morse functions, a class of error functions (Koditschek, 1989; Bullo et al., 1995), are guaranteed to exist on many Lie groups of practical interest considered in this paper (Maithripala & Berg, 2015; Bullo & Lewis, 2005). They represent potential energy that can be used to measure the distance between the configuration $g$ and the identity $e$ on $G$. The following definition specifies the class of kinematic controls considered in the paper.

Some comments on the class of kinematic controls are in order. Properties (i)-(ii) are instrumental to building a particular Lie subgroup on the tangent bundle. Properties (iii)-(iv) represent the sliding (convergence) property of the reduced-order dynamics on the sliding subgroup (Lemma 5 below). In particular, Property (iii) states the almost-global asymptotic stability for system (3) in closed loop with the kinematic control law $\nu(t)=-\nu_{u}(g(t))$. Note that since $\mathcal{O}_{u}$ is the set of closed-loop equilibria other than $g(t)=e$, they are critical points of $V(g)$. Since $V(g)$ is a Morse function, the set $\mathcal{O}_{u}$ consists of a finite number of isolated points. In addition, this set is nowhere dense, which means that it cannot separate the configuration space. Therefore, the complement $G\backslash\mathcal{O}_{u}$ is open and dense, i.e., $G\backslash\mathcal{O}_{u}$ is a submanifold of $G$ (Maithripala et al., 2006). Finally, Property (iv) establishes the local exponential stability of the closed-loop system, where the existence of the neighborhood $\mathcal{U}$ is immediate because $V(g)$ is a Morse function, which has a unique minimum at $e\in G$ by definition.

Note that both $V(g)$ and $\nu_{u}(g)$ are of free design, provided that the properties in Definition 3.1 hold. However, it is worth considering the kinematic control law in the logarithmic coordinate, that is, $\nu_{u}(g)=\mathrm{log}(g)$, or some parallel vectors to $\mathrm{log}(g)$ (Akhtar & Waslander, 2020), as this map has been found to provide the strongest stability results, for example, almost global and local exponential convergence to the identity through a geodesic path (Bullo et al., 1995).

For systems described in (3) and (5), the state space is the tangent bundle $TG\simeq G\times\mathfrak{g}$. To endow it with a Lie group structure, we consider the binary operation $\star:\ TG\times TG\mapsto TG$ defined in the following

$\forall h_{1}=(g_{1},\nu_{1}),\;h_{2}=(g_{2},\nu_{2})\in TG$, and $\lambda\in\mathbb{R}_{>0}$.

Being $TG$ a smooth manifold with (3.2) a smooth operation, it only remains to verify the group axioms as follows.

1. 1.

   $\forall h=(g,\nu)\in TG$, it satisfies

   	$\displaystyle h\star f$	$\displaystyle=\left(ge,\;\nu+0+\lambda\nu_{u}(g)+\lambda\nu_{u}(e)-\lambda\nu_{u}(ge)\right)$
   		$\displaystyle=f\star h=h,$

   where Property of Definition 3.1(i) is used.

2. 2.

   The group operation between $h=(g,\nu)\in TG$ and its inverse $h^{-1}=\left(g^{-1},-\nu\right)\in TG$ verifies

   	$\displaystyle h^{-1}\star h$
   	$\displaystyle=\big{(}g^{-1}g,-\nu+\nu+\lambda\nu_{u}(g^{-1})+\lambda\nu_{u}(g)-\lambda\nu_{u}(g^{-1}g)\big{)}$
   	$\displaystyle=\big{(}gg^{-1},\nu-\nu+\lambda\nu_{u}(g)+\lambda\nu_{u}(g^{-1})-\lambda\nu_{u}(gg^{-1})\big{)}$
   	$\displaystyle=h\star h^{-1}=f,$

   where Properties (i)-(ii) of Definition 3.1 are used.

3. 3.

   The associativity $h_{1}\star\left(h_{2}\star h_{3}\right)=\left(h_{1}\star h_{2}\right)\star h_{3}$ is proved straightforwardly by substitution, using the properties of Definition 3.1.

$\blacksquare$

In this subsection, we define a smooth sliding subgroup on the tangent bundle. The following lemma shows that $H\subset TG$ is an immersed submanifold of $TG$ that inherits the topology and smooth structure of the tangent bundle $TG$ (Lee, 2013).

The smoothness of $H$ is immediate, because the map defined in (8) is smooth. The proof consists thus in showing that subset $H$ inherits the group structure of the Lie group $TG$, by verifying the following:

1. i.

   Identity: The identity of the tangent bundle $f=(e,0)\in H$. This is immediate by Definition 3.1(i) since $s(f)=0+\lambda\nu_{u}(e)=0$.

2. ii.

   Inverse. $\forall h=(g,\nu)\in H$, $s(h)=0\implies\nu=-\lambda\nu_{u}(g)$. By Definition 3.1(ii) it follows that

   	$\displaystyle s\left(h^{-1}\right)$	$\displaystyle=-\nu+\lambda\nu_{u}\left(g^{-1}\right)$
   		$\displaystyle=-\left(-\lambda\nu_{u}(g)\right)+\lambda\nu_{u}\left(g^{-1}\right)=0.$

   This proves that $h^{-1}\in H$ for all $h\in H$.

3. iii.

   Closure. Given $h_{1}=(g_{1},\nu_{1}),\ h_{2}=(g_{2},\nu_{2})\in H$, then $s(h_{1})=0\implies\nu_{1}=-\lambda\nu_{u}(g_{1})$ and $s(h_{2})=0\implies\nu_{2}=-\lambda\nu_{u}(g_{2})$. By (3.2) $h_{1}\star h_{2}=\big{(}g_{1}g_{2},-\lambda\nu_{u}(g_{1}g_{2})\big{)}$. Thus, $s\left(h_{1}\star h_{2}\right)=-\lambda\nu_{u}(g_{1}g_{2})+\lambda\nu_{u}(g_{1}g_{2})=0$. That is, $H$ is closed under the group operation.

$\blacksquare$

The following lemma shows that once a trajectory reaches the sliding subgroup it will stay on it and converges to the group identity.

Consider a differentiable curve $g:I\to G$ of the dynamics (3). Let $V:G\to\mathbb{R}$ be a proper Morse function with the unique minimum at $e\in G$. Then, along the trajectory $g(t)$ and $\forall t\in I$, it yields

Assume that $h(t_{r})\in H$, for some $t_{r}\in I$. Then $s(h)=0$ gives $\nu(t)=-\lambda\nu_{u}(g(t))$. Therefore,

In light of Definition 3.1(iii), it follows that $\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)<0$, for all $g(t_{r})\in G\backslash\mathcal{O}_{u}$, and $\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)=0\iff g=e$, where $\mathcal{O}_{u}$ is a nowhere-dense set with a finite number of points given in Definition 3.1(iii). Therefore, $h(t)$ will remain on $H$ for all $t\geq t_{r}$, and the equilibrium $g(t)=e$ of (3) is almost globally asymptotically stable for all $g(t_{r})\in G\backslash\mathcal{O}_{u}$ and locally exponentially stable $\forall g(0)\in\mathcal{U}$, according to Definition 3.1(iv).

$\blacksquare$

The Euler-Lagrange dynamics (4), ignoring disturbance forces $\Delta_{d}$, is expressed as

which is defined on $TG$, being $h(t)=\left(g(t),\nu(t)\right)$ the state variable.

The intrinsic acceleration for the sliding variable (8) is calculated, by using (9), through the covariant derivative of $s(h)\in\mathfrak{g}$ with respect to itself as

Substituting the Euler-Poincaré equation (5) yields

where the skew-symmetry of the Lie bracket $[\cdot,\cdot]\in\mathfrak{g}$ in (1) is used. The reaching law is then proposed as follows

with $k_{s}>0$ a design parameter.

Consider the function $W:TG\to\mathbb{R}$ defined below

Its time evolution along trajectories of (4.1) is given by

which in closed loop with the controller (11) yields

By Lemma 12 in Appendix A the term $\mathbb{I}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\eta)\right),\;\zeta\right)=0$, for any $\zeta,\eta\in\mathfrak{g}$. Therefore,

It follows from Proposition 6.26 of Bullo & Lewis (2005) that $W(h(t))\to 0$ exponentially. $\blacksquare$

Let $g_{r}:I\to G$ be a twice differentiable configuration reference, with the corresponding reference body velocity $\nu_{r}:I\to\mathfrak{g}$ given by $\nu_{r}(t)\triangleq g^{-1}_{r}(t)\cdot\dot{g}_{r}(t)$. The problem is to design a control law $f_{u}$ to track the reference. The Lie group structure of the configuration space $G$ enables to define the following intrinsic configuration error

By left invariance the body velocity error is defined as

with $\eta_{r}(t)=\mathrm{Ad}_{g^{-1}_{e}}\nu_{r}(t)$. Then, the error dynamics evolving on $TG$ is described by

being the state variable $h_{e}(t)=\left(g_{e}(t),\nu_{e}(t)\right)\in TG$.

The tracking problem, therefore, boils down to stabilizing the identity $f=(e,0)$ on $TG$. By using the sliding-model control strategy, the error state is first driven to the sliding subgroup in the reaching stage, and then on the sliding subgroup, the reduced-order dynamics converges to the identity $f$ ensured by Lemma 5.

In terms of the error state $h_{e}$ the sliding variable (8) is given by

and, its covariant derivative, by using (1)-(2), is

Ignoring disturbance $\delta_{d}$ it follows from the Euler-Poincaré equation (5) that

We proposed the following tracking controller

where $k_{s}>0$ is a design parameter. The following theorem establishes the stability of the equilibrium $h_{e}=f$ in the closed-loop system (16)-(4.2).

Substituting the controller (4.2) in the error dynamics (16) yields the closed-loop dynamics

which has an equilibrium point at $s(h_{e})=0$. The results follow as a consequence of Theorem 6 (the reaching stage) and Lemma 5 (the sliding mode). $\blacksquare$

The group of rotations on $\mathbb{R}^{3}$ is the Lie group $SO(3)=\left\{R\in\mathbb{R}^{3\times 3}\;|\;RR^{T}=R^{T}R=I_{3},\;\mathrm{det}(R)=+1\right\}$, with the usual multiplication of matrices as the group operation. The identity of the group is the identity matrix $I_{3}$ of $3\times 3$, and the inverse is the transpose $R^{T}\in SO(3)$ for any $R\in SO(3)$. The Lie algebra is given by the set of skew-symmetric matrices $\mathfrak{so}(3)=\left\{S\in\mathbb{R}^{3\times 3}\;|\;S^{T}=-S\right\}$, which is isomorphic to $\mathbb{R}^{3}$, i.e., $\mathfrak{so}(3)\simeq\mathbb{R}^{3}$. The Lie bracket in $\mathbb{R}^{3}$ is defined by the cross product $[\zeta,\eta]=\mathrm{ad}_{\zeta}\eta\triangleq\zeta\times\eta$, $\forall\zeta,\eta\in\mathbb{R}^{3}$. Denote the isomorphism $\cdot^{\wedge}\vcentcolon\mathbb{R}^{3}\to\mathfrak{so}(3)$, and respectively the inverse map $\cdot^{\vee}\vcentcolon\mathfrak{so}(3)\to\mathbb{R}^{3}$. Then for a differentiable curve $R\vcentcolon I\to SO(3)$ with left-invariant dynamics $\dot{R}(t)\in T_{R}SO(3)$, the body angular velocity is given by