Rigid body dynamics using Clifford algebra

In this paper the dynamics of rigid bodies is recast into a Clifford algebra formalism. Specifically, the algebra C(0, 6, 2), is used and it is shown how velocities, momenta and inertias can be represented by elements of this algebra. The equations of motion for a rigid body are simply derived by differentiating the momentum of the body. © 2008 Birkḧauser Verlag Basel/Switzerland. The final publication is available at Springer via http://dx.doi.org/10.1007/s00006-008-0144-

Robust Pose Control of Robot Manipulators Using Conformal Geometric Algebra

A controller, based on sliding mode control, is proposed for the n-link robotic manipulator pose tracking problem. The point pair (a geometric entity expressed in geometric algebra) is used to represent position and orientation of the end-effector of a manipulator. This permits us to express the direct and differential kinematics of the endeffector of the manipulator in a simple and compact way. For the control, a sliding mode controller is designed with the following properties: robustness against perturbations and parameter variations, finite time convergence, and easy implementation. Finally, the application, of the proposed controller in a 6 DOF robotic manipulator is presented via simulation.Consejo Nacional de Ciencia y Tecnologí

Robust Tracking of Bio-Inspired References for a Biped Robot Using Geometric Algebra and Sliding Mode Control

Controlling walking biped robots is a challenging problem due to its complex and uncertain dynamics. In order to tackle this, we propose a sliding mode controller based on a dynamic model which was obtained using the conformal geometric algebra approach (CGA). The CGA framework permits us to use lines, points, and other geometric entities, to obtain the Lagrange equations of the system. The references for the joints of the robot were bio-inspired in the kinematics of a walking human body. The first and second derivatives of the reference signal were obtained through an exact robust differentiator based on high order sliding mode. The performance of the proposed control schemes are illustrated through simulation.ITESO, A.C

Robust Tracking of Bio-Inspired References for a Biped Robot Using Geometric Algebra and Sliding Modes

Controlling walking biped robots is a challenging problem due to its complex and uncertain dynamics. In order to tackle this, we propose a sliding mode controller based on a dynamic model which was obtained using the conformal geometric algebra approach (CGA). The CGA framework permits us to use lines, points, and other geometric entities, to obtain the Lagrange equations of the system. The references for the joints of the robot were bio-inspired in the kinematics of a walking human body. The first and second derivatives of the reference signal were obtained through an exact robust differentiator based on high order sliding modes. The performance of the proposed control scheme is illustrated through simulation.CINVESTA

Visual Servoing and Robust Object Manipulation Using Symmetries and Conformal Geometric Algebra

Object tracking and manipulation is an important process for many applications in robotics and computer vision. A novel 3D pose estimation of objects using reflectionally symmetry formulated in Conformal Geometric Algebra (CGA) is proposed in this work. The synthesis of the kinematics model for robots and a sliding mode controller using the CGA approach is described. Real time implementation results are presented for the pose estimation of object using a stereo vision system.ITESO, A.C.CINVESTA

Multi-mode Flight Sliding Mode Control System for a Quadrotor

There is a wide range of applications for unmanned aerial vehicles that requires the capability of having several and robust flight controllers available. This paper presents the main framework of a multimode flight control system for a quadrotor based on the super twisting control algorithm. The design stages for the four flight control modes encompassing manual, altitude, GPS fixed and autonomous mode are presented. The stability proof for each flight mode is carried out by means of Lyapunov functions while the stability analysis for the complete system, when a transition from one mode to another occurs, is demonstrated using the switching nonlinear systems theory. The performance of the proposed framework is demonstrated in a simulation study taking into account external disturbances.ITESO, A.C.CINVESTAV-IP

Model-based visual self-localization using geometry and graphs

Abstrac

Geometric Algebra Model of Distributed Representations

Formalism based on GA is an alternative to distributed representation models developed so far --- Smolensky's tensor product, Holographic Reduced Representations (HRR) and Binary Spatter Code (BSC). Convolutions are replaced by geometric products, interpretable in terms of geometry which seems to be the most natural language for visualization of higher concepts. This paper recalls the main ideas behind the GA model and investigates recognition test results using both inner product and a clipped version of matrix representation. The influence of accidental blade equality on recognition is also studied. Finally, the efficiency of the GA model is compared to that of previously developed models.Comment: 30 pages, 19 figure

The Absolute Line Quadric and Camera Autocalibration

We introduce a geometrical object providing the same information as the absolute conic: the absolute line quadric (ALQ). After the introduction of the necessary exterior algebra and Grassmannian geometry tools, we analyze the Grassmannian of lines of P^3 from both the projective and Euclidean points of view. The exterior algebra setting allows then to introduce the ALQ as a quadric arising very naturally from the dual absolute quadric. We fully characterize the ALQ and provide clean relationships to solve the inverse problem, i.e., recovering the Euclidean structure of space from the ALQ. Finally we show how the ALQ turns out to be particularly suitable to address the Euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters, providing new linear and non-linear algorithms for this problem. We also provide experimental results showing the good performance of the techniques