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In (1) and (2), m is the total mass of the projectile; (u, v, w) are the velocity components, respectively; (X, Y, Z) are the resultant force components of the aerodynamic force, the canard-guided control force, and the resultant force of the Magnus force and gravity, respectively; ([p.sub.f], [p.sub.a], q, r) are the rotation speed components, where [p.sub.f] is the rotation speed of the forebody and [p.sub.a] is the rotation speed of the afterbody; ([L.sub.f], [L.sub.a], M, N) are the moment components of the afterbody, where [L.sub.f] indicates the rolling moment of the forebody and [L.sub.a] indicates the rolling moment subjected contributed by the afterbody; while I is the moment of inertia matrix.
where [mathematical expression not reproducible] denote the joint displacements, velocities, and acceleration values, respectively, [tau] [member of] [R.sup.nx1] stands for the vector of generalized forces/torques applied, M(q) [member of] [R.sup.nxn] is the symmetric positive-definite inertia matrix, C(q, [??]) [member of] [R.sup.nxn] is the matrix of centripetal and Coriolis torques, g(q) [member of] [R.sup.n] is the vector of gravitational torques, and, finally, f([??]) [member of] [R.sup.n] is the friction torque vector.
J = diag{[J.sub.x], [J.sub.y], [J.sub.z]}: the inertia matrix of the quadrotor.
This mathematical description is the most accurate of the three listed models because of inertia matrix using with a centre of mass position vectors from the CAD model, which are with transformation matrices all necessary parameters to need to know.
where D(q), H(q,q), and G(q) are the inertia matrix, centrifugal and Coriolis force vector and gravity vector of the leg, respectively.
where [] = diag ([J.sub.d*1],..., [J.sub.d*1]) and [I.sub.d] = diag ([*1],..., [*n]) are diagonal inertia matrices of independent and dependent states, M is the inertia matrix and T is the torque matrix.
The automatic segmentation of the bone of interest from both acquired volumes is the starting point for a good estimate of the location and orientation parameters, which is here performed through the computation of the inertia matrix. The subsequent steps of registration are performed on the basis of principal inertial axes of each volume.
where index i indicates master 1 or master 2, [q.sub.m](t) [member of] [R.sup.nx1] is the joint position of the master; [[??].sub.m](t) is the joint velocity; [M.sub.m]([q.sub.m]) [member of] [R.sup.n x n] is the inertia matrix; [C.sub.m]([q.sub.m], [[??].sub.m]) is the matrix representing centripetal and Coriolis torques; [g.sub.m]([q.sub.m]) is the gravitational torque; [f.sub.h] is the torque caused by the human operator force; and [[tau].sub.m] is the torque applied to the master by the controller.
On the other hand, we assume the symmetric positive-definite inertia matrix M(q) with uncertainty [DELTA]M(q).
where m is the payload mass, [sup.P]I is the inertia matrix with respect to frame {P}, [F.sub.3x1] = [[[[tau].sub.1],[[tau].sub.2],[[tau].sub.3]].sup.T] and [M.sub.3x1] = [[[[tau].sub.4], [[tau].sub.5], [[tau].sub.6]].sup.T].
Featherstone and Orin [12] and Featherstone [13] presented an efficient approach for utilizing spatial algebra to model multibody systems and efficiently factor the inertia matrix for rigid body systems.