For the circular cross section, the local projection of the gravity vector
along the tangent is given by Equation (10).
Section 3 is devoted to the estimation of the gravity vector
. The attitude determination algorithm is detailed in Section 4.
Introduce the following notation: [alpha] is the inclination angle of the rotation axis relative to the vertical and [beta] is the angle between the plane OXZ and the plane containing the z-axis and the gravity vector
where [m.sub.t] is the mass of the top, G is the center of gravity of the top, [[theta].sub.t] is the inclination angle, g is acceleration due to gravity, [r.sub.t] is distance from fulcrum point to the center of gravity of the top, [[??].sub.t] is angular momentum vector for the rotation of the top around its own axis, [[??].sub.t] is gravity position vector, [[??].sub.g] is gravity vector
. Angular momentum vector [[??].sub.g] is generated in a direction shown in Figure 1 according to the right-hand rule.
Inclination or tilt angle sensing can be determined by the gravity vector
and its projection on the axes.
The Deflection of the Vertical (DOV)as one basic research topic in geodesy and geophysics can characterize thedirection of gravity vector
and provide one tight local tie among space geodetic techniques (Shen et al., 2015).
If measurements from sensor array (accelerometer, gyroscope and magnetometer) are available, then gravity vector
g = [[[g.sub.x], [g.sub.y], [g.sub.z]].sup.T] can be estimated using orientation represented in quaternion (8)
where [mathematical expression not reproducible] denote the joint position (generalized coordinates), velocity, and acceleration vectors, respectively; M(q) [member of] [R.sup.4x4] is the inertia matrix; V(q, [??]) [member of] [R.sup.4x1] is the centripetal-Coriolis vector; G(q) [member of] [R.sup.4x1] is the gravity vector
; and r(t) [member of] [R.sup.4x1] is the control vector of generalized exogenous torques/forces supplied by the joint actuators.
Matrix w is raw data from the accelerometer collected at 6 stationary positions (when device lays on each side of the device and normalized gravity vector
is zero in two axis).
Also, a technique to calculate gravity vector
from the quaternion is incorporated into preprocessing module as MEMS triaxis accelerometer senses gravity plus translational acceleration of wand.