3.4 Quotient manifolds 3.4.1 Theory of quotient manifolds 3.4.2 Functions on quotient manifolds 3.4.3 The real projective space RP n−1 3.4.4 The Grassmann manifold Grass(p, n) 3 3.5 Tangent vectors and differential maps CONTENTS ix 5.3 Riemannian connection 5.3.1 Symmetric connections 5.3.2 Definition of the Riemannian connection 5.3.3 Riemannian connection on Riemannian submanifolds 5.3.4 Riemannian connection on quotient manifolds 5.4 Geodesics, exponential mapping, and parallel translation 5.5 Riemannian Hessian operator 5.6 Second covariant derivative* 5.7 Notes and references 6. Newton's Method 6.1 Newton's method on manifolds 6.2 Riemannian Newton method for real-valued functions 6.3 Local convergence 6.3.1 Calculus approach to local convergence analysis 6.4 Rayleigh quotient algorithms 6.4.1 Rayleigh quotient on the sphere 6.4.2 Rayleigh quotient on the Grassmann manifold 6.4.3 Generalized eigenvalue problem 6.4.4 The nonsymmetric eigenvalue problem 6.4.5 Newton with subspace acceleration: Jacobi-Davidson 6.5 Analysis of Rayleigh quotient algorithms 6.5.1 Convergence analysis 6.5.2 Numerical implementation 6.6 Notes and references 7. Trust-Region Methods 7.1 Models 7.1.1 Models in R n 7.1.2 Models in general Euclidean spaces 7.1.3 Models on Riemannian manifolds 7.

16International audienceThis paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO(3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on S(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle

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Abstract. We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications -computing an invariant subspace of a matrix and the mean of subspaces -are worked out. (2000): 65J05, 53C05, 14M15. Mathematics Subject Classifications