Moment (physics) | EPFL Graph Search

In physics, a moment is a mathematical expression involving the product of a distance and physical quantity. Moments are usually defined with respect to a fixed reference point and refer to physical quantities located some distance from the reference point. In this way, the moment accounts for the quantity's location or arrangement. For example, the moment of force, often called torque, is the product of a force on an object and the distance from the reference point to the object. In principle, any physical quantity can be multiplied by a distance to produce a moment. Commonly used quantities include forces, masses, and electric charge distributions. Elaboration In its most basic form, a moment is the product of the distance to a point, raised to a power, and a physical quantity (such as force or electrical charge) at that point: : \mu_n = r^n,Q, where Q is the physical quantity such as a force applied at a point, or a point charge, or a poi

Push recovery with stepping strategy based on time-projection control

Salman Faraji, Auke Ijspeert, Seyed Hamed Razavi

In this paper, we present a simple control framework for on-line push recovery with dynamic stepping properties. Due to relatively heavy legs in our robot, we need to take swing dynamics into account and thus use a linear model called 3LP which is composed of three pendulums to simulate swing and torso dynamics. Based on 3LP equations, we formulate discrete LQR controllers and use a particular time-projection method to adjust the next footstep location on-line during the motion continuously. This adjustment, which is found based on both pelvis and swing foot tracking errors, naturally takes the swing dynamics into account. Suggested adjustments are added to the Cartesian 3LP gaits and converted to joint-space trajectories through inverse kinematics. Fixed and adaptive foot lift strategies also ensure enough ground clearance in perturbed walking conditions. The proposed structure is robust, yet uses very simple state estimation and basic position tracking. We rely on the physical series elastic actuators to absorb impacts while introducing simple laws to compensate their tracking bias. Extensive experiments demonstrate the functionality of different control blocks and prove the effectiveness of time-projection in extreme push recovery scenarios. We also show self-produced and emergent walking gaits when the robot is subject to continuous dragging forces. These gaits feature dynamic walking robustness due to relatively soft springs in the ankles and avoiding any Zero Moment Point (ZMP) control in our proposed architecture.

2018

Complete network characterisation of stratified planar circuits using the method of moments technique: an integrated approach to lumped and wave ports

Laleh Golestanirad, Michael Mattes, Juan Ramon Mosig

This study gives a full treatment of the network characterisation of multi-port planar circuits embedded in shielded stratified media with an integrated approach to lumped ports and wave ports. The details of the mathematical formulation for the retrieval of scattering parameters are given in three cases: networks possessing only lumped ports, networks possessing only wave ports and networks with a combination of both types. A detailed description of the computer algorithm is presented to help the reader easily reproduce our results. The implementation of combined ports in the framework of planar multi-layered structures is particularly useful in analysing systems that have waveguide connections and planar circuitries in their structure, such as cavity-backed antennas and their associated feeding systems, dielectric resonator antennas or compound microwave filters. In such structures one can divide the system into the waveguide structure and the planar circuitries and analyse each part separately with an appropriate and efficient numerical technique. Network characterisations will be consequently communicated at the port interfaces. This strategy enhances the total performance of the computational scheme to a great extent, while not affecting the accuracy of the full-wave analysis.

Institution of Engineering and Technology2011

Competing Jahn-Teller distortions and ferrimagnetic ordering in the geometrically frustrated system Ni1-xCuxCr2O4

Philip Pattison

Competing Jahn-Teller distortions combined with geometrical frustration give rise to a rich phase diagram as a function of x(Cu) and temperature in the spinel system Ni1-xCuxCr2O4. The Jahn-Teller distortion of the end members acts in opposite ways, with an elongation of the NiO4 tetrahedra resulting in a structural transition at T-S1 = 317K in NiCr2O4, but a flattening in the CuO4 tetrahedra at T-S1 = 846K in CuCr2O4. In both cases the symmetry is lowered from cubic (Fd (3) over barm) to tetragonal (I4(1)/amd) on cooling. In order to follow the influence of Jahn-Teller active Ni2+ and Cu2+ ions on the structural and magnetic properties of chromium spinels, we have investigated a series of samples of Ni1-xCuxCr2O4 by x-ray and neutron powder diffraction. In the critical range 0.10 < x(Cu) < 0.20, strong orthorhombic distortions were observed, where competing Jahn-Teller activities between the Cu2+ and Ni2+ ions result in distortions along both the a and c axes. For Ni0.85Cu0.15Cr2O4, the orthorhombic structure (Fddd) is stabilized up to T-S2 = 368(2) K, close to the first structural phase transition at T-S1 = 374(2) K. A ferrimagnetic spin alignment of the Ni/Cu and chromium atoms sets in at much lower temperature TC = 95K in this compound. The end members NiCr2O4 and CuCr2O4 undergo this ferrimagnetic transition at TC = 74 and 135 K, respectively. These transitions are accompanied by the structural change to the orthorhombic symmetry which relieves the frustration. NiCr2O4 and Ni0.85Cu0.15Cr2O4 undergo a second magnetic transition at TM2 = 24 and 67K due to a superimposed antiferromagnetic ordering of the Cr moments resulting in a noncollinear magnetic structure. In the system Ni1-xCuxCr2O4, the magnetic transitions TC and TM2 merge with increasing copper content up to x(Cu) similar to 0.5. For the Ni-rich chromites, geometrical frustration causes a strong reduction of the chromium moments, where magnetic long-range order coexists with a disordered spin-liquid-like or a reentrant-spin-glass-like state. This paper provides insight into the interplay between the Jahn-Teller effect, geometrical frustration, and long-range magnetic order in these complex systems.

Amer Physical Soc2015

Competing Jahn-Teller distortions and ferrimagnetic ordering in the geometrically frustrated system Ni1-xCuxCr2O4

Philip Pattison

Amer Physical Soc2015