Lagrange equations in the presence of holonomic constraints
are of the form :
The literature takes into account the type of mobility of mobile manipulators with four possible configurations: (1) Type (h, h) where both platform and manipulator contain holonomic constraints, (2) Type (h, nh) where the platform is holonomic and manipulator is nonholonomic, (3) Type (nh, h) where the platform is nonholonomic but the manipulator is holonomic, and (4) Type (nh, nh) where both the platform and the manipulator contain nonholonomic constraints.
The challenges associated with motion planning and control of the Type (nh, h) are amplified by the intimate coupling of the nonholonomic and holonomic constraints arising from the amalgamation of an articulated robotic arm and a wheeled platform.
These position constraints are known as the holonomic constraints of the 2MM system .
where [C.sup.h.sub.q] is the Jacobian matrix of the holonomic constraints and [C.sup.nh.sub.[??]] = C(q, t) is the Jacobian matrix of the nonholonomic constraints.
where the Q vector includes all external and elastic forces, Qc are the generalized reaction forces due to holonomic constraints, and [Q.sub.t] are the generalized forces due to the nonholonomic constraints.
The holonomic constraints can be considered as listed in Table 4; the total number of holonomic constraints is 58; therefore, the system has 5 degrees of freedom.
The number of Lagrange multipliers associated with holonomic constraints is 58 multipliers; among them, there are 9 trivial multipliers.
Within this framework, we defined an extremum problem, either with holonomic constraints
or nonholonomic constraints of equality type, if the Pfaff system (5.1) is completely integrable or not, respectively.
This paper is concerned with extending the HHT-[alpha] method to sys tems of overdetermined differential-algebraic equations (ODAEs) with index 3 constraints and their underlying index 2 constraints, e.g., to systems in mechanics having holonomic constraints. An extension of the HHT-[alpha] method to index 2 DAEs, e.g., to systems in mechanics with nonholonomic constraints, is briefly discussed as well.
In mechanics (3.1d) represents holonomic constraints, [lambda] represents Lagrange multipliers, and r (t, y, [lambda]) = -[M.sup.-1][g.sup.T.sub.y] (t, y) [lambda] where M is the mass matrix and -[g.sup.T.sub.y] (t, y) [lambda] represents reaction forces coming from the constraints .
Egeland, "Control of vehicles with second holonomic constraints
: underactuated vehicles," in Proceedings of the European Control Conference, pp.