polynomials of the second degree in respect of p; a ,b, c with relevant indexes--constant coefficients (any of them can be equal to zero); [F.sub.i](t)--the generalized outside force acting along the [x.sub.i]--th generalized coordinate; [H.sub.i] (t) is the component of the
generalized force acting along the [x.sub.i]--th generalized coordinate obtained on differentiation of the functions T, [PI], [PHI] in the cases of their direct dependence on time t (for example, when the equations (3) and (4) include nonstationary members [[gamma].sub.j] and [[??].sub.j]); [x.sub.1], ..., [x.sub.n] are principal generalized coordinates (each of them corresponds to one of the equations (1)), and; [x.sub.n+1], ..., [x.sub.m] are redundant generalized coordinates when they are used.
Considering the negative velocity difference effect, Helbing and Benno Tilch proposed a
generalized force model (GFM) [11].
With the above velocity and acceleration, the
generalized force F and torque N on each link at the center of mass can be obtained from the end effector to the root as follows:
where T is the total kinetic energy, [PI]--the potential energy, [Q.sub.i]--the i-th
generalized force, [h.sub.j]--the equation of the j-th constraint, [[lambda].sub.j]--the Lagrange multiplier corresponding to the j-th constraint.
These actions generate the first
generalized force, which is determined by the principle of virtual work as,
and Q is the
generalized force vector projected in the joint subspace and can be calculated as follows:
Generally for this purpose relation of the external torque vector with
generalized force through the Jacobian matrix of manipulator is used.
Q = the
generalized force applied to the ith generalized coordinate
Because the
generalized force vector [F.sub.R] is a nonlinear function vector of [e.sub.R] and [lambda], the right side of Equation (56) should be simplified before each iteration.