Hybrid path-lifting algorithm and Equivalence of Stability results for MRP-based control strategies

Abstract

The modified Rodrigues parameters (MRP) consist of two numerically different triplets that, by switching between them, yield a minimal globally non-singular attitude description with advantageous properties. The MRP space results from the Alexandroff compactification of the three-dimensional Euclidean space and is a double cover of $\mathrm{SO(3)}$. By capitalizing on instrumental properties of the covering map, this paper proposes a novel hybrid dynamic path-lifting mechanism to unambiguously and robustly extract the MRP from the attitude space. This hybrid solution allows applying an MRP-based feedback controller to the attitude dynamics in the base space while preserving its asymptotic and exponential stability properties. Furthermore, by profiting from the distinct characteristics of the MRP, the resulting interconnection is impervious to the unwinding phenomenon. The design and validation of an MRP-based controller exemplify the application of the proposed algorithm alongside the novel results for equivalence of stability between spaces. The solution renders the attitude space tracking dynamics robustly globally exponentially stable, demonstrating the potential of this novel methodology.

Figures (5)

• Figure 1: Scheme of the maps between the topological spaces $\mathbb{\bar{R}}^3$, $\mathbb{S}^3$ and $\mathrm{SO}(3)$ and of the respective elements describing the associated attitude representation for a given time instant.
• Figure 2: Depiction of the MRP switching logic: when $\|\boldsymbol{\vartheta}_{\mathbf{2}}\|$ is equal or greater than $1 + \delta$, a), the state jumps to the shadow MRP representation through the map $\boldsymbol{\Upsilon}(\boldsymbol{\vartheta}_{\mathbf{2}})$. The resulting MRP description, b), has a norm equal or lower than $(1 + \delta)^{-1}$, and corresponds to the shortest principal rotation available junkins_2009. The switch is hysteretic due to the inclusion of the parameter $\delta$.
• Figure 3: Attitude responses obtained in simulation. From top to bottom, left to right: \ref{['fig:SimPosAtt_a']} roll, \ref{['fig:SimPosAtt_b']} pitch, \ref{['fig:SimPosAtt_c']} yaw, and \ref{['fig:SimPosAtt_d']} attitude error in MRP in conjunction with the discrete state $m$.
• Figure 4: MRP error norm and actuation obtained during the attitude tracking test in simulation. From left to right: \ref{['fig:SimErrorNormInput_a']}$\|\boldsymbol{\Tilde{\vartheta}}\|$, \ref{['fig:SimErrorNormInput_b']}$\boldsymbol{\tau}$.
• Figure 5: Scheme of the approach used to determine $T'$. For ease of comprehension, this visual representation was obtained considering $\mathbf{e_2^{\!\top}}\mathbf{q_1} = \mathbf{e_3^{\!\top}}\mathbf{q_1} = 0$. The variable $\alpha$ and the memory state $\mathbf{\hat{q}_1}$ were set with the values $1 - \cos(75^\circ)$ and $(\cos(30^\circ), \sin(30^\circ), 0, 0)$, respectively.

Theorems & Definitions (13)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3