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Mathematical formulas from the sciences
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
G gravitational constant≈6.67×10-11Nm2kg2 $ \left( { \approx 6.67 \times 10^{ - 11} \frac{{{\text{N}}\,{\text{m}}^{ 2} }}{{{\text{kg}}^{2} }}} \right) $
Relative equilibria, stability and bifurcations in Hamiltonian galactic tidal models
Published in Dynamical Systems, 2021
J. L. Zapata, F. Crespo, S. Ferrer
Following previous works [1, 2], we translate this problem into the Hamiltonian formalism. More precisely, moving to a set of galactic-rate rotating coordinates and , the Hamiltonian function associated to objects coming from the Oort cloud brings in the effects of galactic tides and is given as a perturbed Keplerian system where the zero order is the Kepler system with gravity parameter , being G and M the gravitational constant and the Sun mass, respectively. The perturbing potential is made of the Coriolis term , and the galactic tidal symmetric and asymmetric terms, see [1] In what follows, we will assume the following notation: Thus, the small parameter will no longer be given explicitly and our analysis is carried out for generic values of α, β and γ. However, for the study of the stability of relative equilibria given in Section 4, we use explicit values with astrodynamical meaning. Precisely, we consider obtained from [1] as with .
A fault tolerant tracking control for a quadrotor UAV subject to simultaneous actuator faults and exogenous disturbances
Published in International Journal of Control, 2020
A model of the quadrotor UAV with the configuration depicted in Figure 1 can be represented by the following equations (Merheb et al., 2015): where U1, U2, U3 and U4 are the control inputs that produce thrust and torques in the rotors. Kftx, Kfty and Kftz are the translational drag coefficients, and Kfax, Kfay and Kfaz are the aerodynamic friction coefficients. Ix, Iy and Iz are the moments of inertia of the quadrotor with respect to the corresponding axes. m is the mass of the quadrotor and g is the gravitational constant.
Smartphone localization inside a moving car for prevention of distracted driving
Published in Vehicle System Dynamics, 2020
Gregory Johnson, Rajesh Rajamani
The lateral acceleration is treated as a force input at the centre of gravity (CG) in the moment balance equation taken about the roll centre. This couples the roll dynamics to the lateral dynamics through the lateral acceleration term: where is the moment of inertia taken at the roll centre, is the distance from the roll centre to the CG, is the vehicle mass and is the gravitational constant. More detailed simulation models for roll dynamics can be found in other papers from this journal, such as [10–12]. However, the model presented here is adequate for the roll dynamics in order to evaluate in simulations how the lateral motion and acceleration of the car cause roll and how these signals influence the accelerometer and gyroscope signals read by the sensors on the phone. Extensive experimental evaluations of the developed algorithm using measurements on real car maneuvers will later be presented to serve as a more reliable indicator of the developed algorithm’s effective performance.