Figure 2 - uploaded by Silvère Bonnabel
Content may be subject to copyright.
Contrarily to the EKF, the gain computation does not directly depend on the last estimate of the state in the case of the IEKF.
Source publication
In this paper, we consider the problem of tracking a reference trajectory for
a simplified car model based on unicycle kinematics, whose position only is
measured, and where the control input and the measurements are corrupted by
independent Gaussian noises. To tackle this problem we devise a novel
observer-controller: the invariant Linear Quadrati...
Context in source publication
Similar publications
Robots operating autonomously in household environments must be capable of interacting with articulated objects on a daily basis. They should be able to infer each object's underlying kinematic linkages purely by observing its motion during manipulation. This work proposes a framework that enables robots to learn the articulation in objects from us...
Serial 3R orthogonal manipulators have been studied recently and it has been proved that they can exhibit good performances in term of workspace size and kinematic properties. The aim of this work is to analyze their dynamic performances, and compare them with anthropomorphic manipulators, which are very popular in industry.
Static and dynamic anal...
This paper proposes a kinematics path-following algorithm that drives a unicycle-type robot to a desired path, respecting velocity saturation constraint on the actuators. The solution, based on an extension of a globally convergent non-singular path following solution, is formally shown. Simple conditions for the selection of the gains are provided...
Citations
... Additionally, there is substantial evidence that the use of the Lie group structure in the manifold space SE(3) plays a significant role in probabilistic robotics [20]. In the literature [21], the concept of an Invariant Extended Kalman Filter (INEKF) based on Lie groups has been proposed to address the inconsistency issues in the EKF series. Compared to the probability density defined on groups, as mentioned in [22], the exponential mapping method employed by INEKF is more efficient and stable. ...
Accurate localization of robots in unstructured environments poses challenges due to low localization accuracy and local trajectory oscillation caused by complex feature points when using Euclidean-based filtering methods. In this study, we propose a novel 3D robot localization method named LIF-M that leverages a manifold-based approach in conjunction with an unscented Kalman filter (UKF-M). Additionally, a relocalization algorithm is designed to ensure localization stability. The proposed method addresses the limitations of conventional Euclidean-based filtering methods by incorporating manifold-based techniques, providing a more comprehensive representation of the complex geometric features. We introduce the manifold concept, where the relevant definition is defined and utilized within the LIF-M framework. By combining left and right invariants, we effectively reduce noise uncertainty, resulting in improved localization accuracy. Moreover, we employ sigma points as a matrix representation of the state points’ space in order to seamlessly transition between the matrix space and the vector representation of the tangent space. Experimental tests and error calculations were conducted to evaluate the performance of various algorithm frameworks, and the results demonstrated the importance of the manifold-based approach for accurate attitude estimation. Compared to the standard UKF, the manifold space equips LIF-M with better robustness and stability in unstructured environments.
... Matrix Lie groups can be used to compactly and accurately represent vehicle attitude, pose, or extended pose, which in turn can be leveraged in state estimation and control problems [4]- [9]. In particular, an invariant linear quadratic Gaussian controller defined on SE(2) is used to control a simplified car in [6], and invariant linear quadratic regulator (ILQR) using an error defined on SE 2 (3) is used to control a quadrotor in [7]. The use of an invariant error definition, instead of a traditional multiplicative error definition [10], along with a particular type of process model, results in Jacobians that are state-independent. ...
... The dynamics used for control design do not exactly fit the invariant framework. Nevertheless, the invariant approach to control is followed due to the straightforward linearization process and the reduced dependence of the Jacobians on attitude [6], [7]. ...
... Consider the guidance and control structure shown in Fig. 1. By using the control inputs in (6), the dynamics of the tandem rotor helicopter are differentially flat [19]. A set of flat outputs, equal to the number of inputs, exists such that all of the system states and control inputs can be represented by the flat outputs and their derivatives [21]. ...
This letter considers model predictive control of a tandem-rotor helicopter. The error is formulated using the matrix Lie group $SE_2(3)$. A reference trajectory to a target is calculated using a quartic guidance law, leveraging the differentially flat properties of the system, and refined using a finite-horizon linear quadratic regulator. The nonlinear system is linearized about the reference trajectory enabling the formulation of a quadratic program with control input, attitude keep-in zone, and attitude error constraints. A non-uniformly spaced prediction horizon is leveraged to capture the multi-timescale dynamics while keeping the problem size tractable. Monte-Carlo simulations demonstrate robustness of the proposed control structure to initial conditions, model uncertainty, and environmental disturbances.
... One of the methods used to solve optimization problems with uncertain parameters is the linear quadratic gaussian (LQG) [12]. There are several reports in varying fields regarding the advantages of using this control method: for example, power system [13], [14], mechanical control problem [15], [16], power flow [17], unmanned aerial vehicle [18], simplified car [19], acquisition process [20], slung transportation [21], vehicle automation [22], [23], buck converter controlling [24], robotic & electronic systems [25], [26], and optics system [27]. ...
This study formulates a dynamical system for the control of a single product inventory system in accordance with the random value of demand and the percentage of damaged product during the delivery process. The formulated model has the form of a linear state-space system comprising of two disturbances, which represents the random value of demand and the percentage of the damaged product during delivery. The optimal value of the product amount ordered to the supplier is properly calculated by using the linear quadratic gaussian (LQG) method. The controller is used by the manager to make inventory level decisions under the uncertainty of demand and damaged items during the product delivery process. The result showed that the optimal product order for each review time was achieved, and the inventory level was used to obtain the right set point properly. Moreover, based on comparison with other research results, the proposed model was well performed.
... Reserved to systems defined on Lie groups, it has been mainly driven by applications to localization and guidance, where it appears as a slight modification of the multiplicative EKF (MEKF), widely known and used in the world of aeronautics. It has been proved to possess theoretical local convergence properties the EKF lacks in [8], to be an improvement over the EKF in practice (see e.g., [5, 4, 17, 31] and more recently [8] where the EKF is outperformed), and has been successfully implemented in industrial applications to navigation (see the patent [7]). In the present paper, we slightly generalize the IEKF framework, to make it capable to handle very general observations (such as range and bearing or bearing only observations), and we show how the derived IEKF-SLAM, a simple variant of the EKF-SLAM, allows remedying the inconsistency of EKF-SLAM stemming from the non-observability of the orientation and origin of the global frame. ...
... Extensive Monte-Carlo simulations have illustrated the consistency of the new method and the striking improvement over EKF, UKF, OC-EKF, and more remarkably over the ideal EKF also, which is the -impossible to implement -variant of the EKF where the system is linearized about the true trajectory. Note that, the IEKF approach may prove relevant beyond SLAM to some other problems in robotics as well, such as autonomous navigation (see [4] ), and in combination with controllers, notably for motion planning purposes, see [17]. In [9] the IEKF has proved to possess global asymptotic convergence properties on a simple localization problem of a wheeled robot, which is a strong property. ...
In this paper we address the inconsistency of the EKF-based SLAM algorithm
that stems from non-observability of the origin and orientation of the global
reference frame. We prove on the non-linear two-dimensional problem with point
landmarks observed that this type of inconsistency is remedied using the
Invariant EKF, a recently introduced variant ot the EKF meant to account for
the symmetries of the state space. Extensive Monte-Carlo runs illustrate the
theoretical results.
This letter considers model predictive control of a tandem-rotor helicopter. The error is formulated using the matrix Lie group
$SE_{2}{(}3{)}$
. A reference trajectory to a target is calculated using a quartic guidance law, leveraging the differentially flat properties of the system, and refined using a finite-horizon linear quadratic regulator. The nonlinear system is linearized about the reference trajectory enabling the formulation of a quadratic program with control input, attitude keep-in zone, and attitude error constraints. A non-uniformly spaced prediction horizon is leveraged to capture the multi-timescale dynamics while keeping the problem size tractable. Monte-Carlo simulations demonstrate robustness of the proposed control structure to initial conditions, model uncertainty, and environmental disturbances.
This letter considers optimal control of a quadrotor unmanned aerial vehicles (UAV) using the discrete-time, finite-horizon, linear quadratic regulator (LQR). The state of a quadrotor UAV is represented as an element of the matrix Lie group of double direct isometries,
$SE_2(3)$
. The nonlinear system is linearized using a left-invariant error about a reference trajectory, leading to an optimal gain sequence that can be calculated offline. The reference trajectory is calculated using the differentially flat properties of the quadrotor. Monte-Carlo simulations demonstrate robustness of the proposed control scheme to parametric uncertainty, state-estimation error, and initial error. Additionally, when compared to an LQR controller that uses a conventional error definition, the proposed controller demonstrates better performance when initial errors are large.
