Soulaimane Berkane

Alessandro Melis, Tarek Bouazza, Hassan Alnahhal et al.

Attitude estimation methods typically rely on full vector measurements from inertial sensors such as accelerometers and magnetometers. This paper shows that reliable estimation can also be achieved using only scalar measurements, which naturally arise either as components of vector readings or as independent constraints from other sensing modalities. We propose nonlinear deterministic observers on $\mathbf{SO}(3)$ that incorporate gyroscope bias compensation and guarantee uniform local exponential stability under suitable observability conditions. A key feature of the framework is its robustness to partial sensing: accurate estimation is maintained even when only a subset of vector components is available. Experimental validation on the BROAD dataset confirms consistent performance across progressively reduced measurement configurations, with estimation errors remaining small even under severe information loss. To the best of our knowledge, this is the first work to establish fundamental observability results showing that two scalar measurements under suitable excitation suffice for attitude estimation, and that three are enough in the static case. These results position scalar-measurement-based observers as a practical and reliable alternative to conventional vector-based approaches.

Soulaimane Berkane, Andrea Bisoffi, Dimos V. Dimarogonas

For a vehicle moving in an $n$-dimensional Euclidean space, we present a construction of a hybrid feedback that guarantees both global asymptotic stabilization of a reference position and avoidance of an obstacle corresponding to a bounded spherical region. The proposed hybrid control algorithm switches between two modes of operation: stabilization (motion-to-goal) and avoidance (boundary-following). The geometric construction of the flow and jump sets of the hybrid controller, exploiting a hysteresis region, guarantees robust switching (chattering-free) between the stabilization and avoidance modes. Simulation results illustrate the performance of the proposed hybrid control approach for a 3-dimensional scenario.

Mouaad Boughellaba, Soulaimane Berkane, Abdelhamid Tayebi

This paper addresses the distributed attitude synchronization problem for a network of rigid-body systems on the special orthogonal group SO(3). Each agent measures, in its body frame, its own angular velocity and a set of vectors whose corresponding directions in the inertial frame are unknown. Under an undirected, connected, and acyclic interaction graph topology, we develop four distributed synchronization schemes relying solely on local vector measurements, without the need for attitude estimation and attitude exchange between agents. Specifically, two leaderless schemes are proposed at the kinematic and dynamic levels to achieve synchronization to a common unknown orientation. In addition, two leader-follower schemes are proposed to align all agents with a prescribed constant orientation defined by reference vector measurements available only to a designated leader. All control laws are formulated directly on SO(3), preserving the geometric structure of the attitude dynamics. A rigorous stability analysis is provided showing that the closed-loop systems achieve almost global asymptotic stability, which is the strongest stability property one can achieve on SO(3) with smooth controllers. %Compared with existing vector-measurement-based approaches that provide only local stability or convergence results, the proposed methods significantly strengthen the theoretical guarantees while maintaining a fully distributed architecture. Numerical simulations are provided to illustrate the effectiveness and performance of the proposed distributed control schemes.

Alessandro Melis, Tarek Bouazza, Soulaimane Berkane et al.

This paper addresses the problem of estimating the relative pose (position and orientation) and velocity of a vehicle with respect to a moving target, where both are equipped with Inertial Measurement Units (IMUs), assuming the availability of relative position or bearing measurements. The body-target relative dynamics are formulated on $\mathbf{SE}_2(3)$ and recast into a linear time-varying (LTV) model in the ambient space $\mathbb{R}^{15}$, on which a deterministic Riccati observer is designed. We analyze the uniform observability (UO) conditions required to guarantee global exponential convergence of the estimation error in the ambient space for both measurement cases. In the case of relative position measurements, UO requires only a persistence-of-excitation condition on the target acceleration, whereas for bearing measurements, additional conditions are required. Building on this, a nonlinear complementary filter on $\mathbf{SO}(3)$ is designed to provide a smooth estimate of the orientation component of the state with almost global asymptotic stability. Finally, simulation results are provided to validate the proposed solution.

Alessandro Melis, Soulaimane Berkane, Tarek Hamel

Attitude estimation using scalar measurements, corresponding to partial vectorial observations, arises naturally when inertial vectors are not fully observed but only measured along specific body-frame vectors. Such measurements arise in problems involving incomplete vector measurements or attitude constraints derived from heterogeneous sensor information. Building on the classical complementary filter on SO(3), we propose an observer with a modified innovation term tailored to this scalar-output structure. The main result shows that almost-global asymptotic stability is recovered, under suitable persistence of excitation conditions, when at least three inertial vectors are measured along a common body-frame vector, which is consistent with the three-dimensional structure of SO(3). For two-scalar configurations - corresponding either to one inertial vector measured along two body-frame vectors, or to two inertial vectors measured along a common body-frame vector - we further derive sufficient conditions guaranteeing convergence within a reduced basin of attraction. Different examples and numerical results demonstrate the effectiveness of the proposed scalar-based complementary filter for attitude estimation in challenging scenarios involving reduced sensing and/or novel sensing modalities.

Soulaimane Berkane

This tutorial presents a control-oriented introduction to aided inertial navigation systems using a Lie-group formulation centered on the extended Special Euclidean group SE_2(3). The focus is on developing a clear and implementation-oriented geometric framework for fusing inertial measurements with aiding information, while making the role of invariance and symmetry explicit. Recent extensions, including higher-order state representations, synchronous observer designs, and equivariant filtering methods, are discussed as natural continuations of the same underlying principles. The goal is to provide readers with a coherent system-theoretic perspective that supports both understanding and practical use of modern aided inertial navigation methods.

Mayur Sawant, Soulaimane Berkane, Ilia Polusin et al.

We develop an autonomous navigation algorithm for a robot operating in two-dimensional environments cluttered with obstacles having arbitrary convex shapes. The proposed navigation approach relies on a hybrid feedback to guarantee global asymptotic stabilization of the robot towards a predefined target location while ensuring the forward invariance of the obstacle-free workspace. The main idea consists in designing an appropriate switching strategy between the move-to-target mode and the obstacle-avoidance mode based on the proximity of the robot with respect to the nearest obstacle. The proposed hybrid controller generates continuous velocity input trajectories when the robot is initialized away from the boundaries of the unsafe regions. Finally, we provide an algorithmic procedure for the sensor-based implementation of the proposed hybrid controller and validate its effectiveness through some simulation results.

Miaomiao Wang, Soulaimane Berkane, Abdelhamid Tayebi

This paper deals with the simultaneous estimation of the attitude, position and linear velocity for vision-aided inertial navigation systems. We propose a nonlinear observer on $SO(3)\times \mathbb{R}^{15}$ relying on body-frame acceleration, angular velocity and (stereo or monocular) bearing measurements of some landmarks that are constant and known in the inertial frame. Unlike the existing local Kalman-type observers, our proposed nonlinear observer guarantees almost global asymptotic stability and local exponential stability. A detailed uniform observability analysis has been conducted and sufficient conditions are derived. Moreover, a hybrid version of the proposed observer is provided to handle the intermittent nature of the measurements in practical applications. Simulation and experimental results are provided to illustrate the effectiveness of the proposed state observer.

Soulaimane Berkane, Andrea Bisoffi, Dimos V. Dimarogonas

In this paper we present a hybrid feedback approach to solve the navigation problem of a point mass in the n-dimensional space containing an arbitrary number of ellipsoidal shape obstacles. The proposed hybrid control algorithm guarantees both global asymptotic stabilization to a reference and avoidance of the obstacles. The intuitive idea of the proposed hybrid feedback is to switch between two modes of control: stabilization and avoidance. The geometric construction of the flow and jump sets of the proposed hybrid controller, exploiting hysteresis regions, guarantees Zeno-free switching between the stabilization and the avoidance modes. Simulation results illustrate the performance of the proposed hybrid control approach for 2-dimensional and 3-dimensional scenarios.