b - NTNU

3.1 Newton-Euler Equations of Motion
about CG
Coordinate-free vector: A vector v b/n , velocity of {b} with respect to {n}, is defined by its
magnitude and direction but without reference to a coordinate frame.
i
Coordinate vector: A vector v b/n
decomposed in the inertial reference frame is denoted by v b/n
Newton-Euler Formulation
Newton's Second Law relates mass m, acceleration v g/i and force f g according to:
mv g/i fg
Isaac Newton (1642-1726)
where the subscript g denotes the center of gravity (CG).
Euler's First and Second Axioms
Euler suggested to express Newton's Second Law in terms of conservation
of both linear momentum p g and angular momentum h according to:
g
i
d
dt p g Leonhard Euler (1707-1783)
fg
p g mv g/i #
f g and m g are forces/moments about CG
i
d
is the angular velocity of frame b relative frame i
dt h g m g h g I g b/i # b/i
I g is the inertia dyadic about the body's CG
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Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)