Nonholonomic constraints

One might confound constraints given on velocity level with nonholonomic constraints. Though we will not treat nonholonomic constraints in this book we will give a definition to show the difference  

Definition 1.3.3 A constraint G p)p = 0 is called holonomic, iff there exists a function g p) with g p) = G p). Otherwise the constraint is called nonholonomic. 

in order to be a holonomic constraint every row of G p) must be a gradient of a function. A necessary condition for a vector to be a gradient is that its derivative is a symmetric matrix, e.g. [OR70]. Thus, the z-th constraint is nonholonomic, iff 

Nonholonomic constraints occur in mechanics for example when pure rolling of two surfaces is modeled, see [RS88]. In practice pure rolling is an idealization, if creepage effects can be neglected. 

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The differential form dXj = akjdqk + aktdt = 0 defines a general linear constraint condition. For an integrable or holonomic constraint, expressed by Xj( qk, t) = 0, the coefficients are partial derivatives of X such that akj = and cikt — Nonintegrable or nonholonomic constraints are defined by the differ-... 

The equations of motion, generalized to include holonomic or nonholonomic constraints with Lagrange multipliers, are... 

The present treatment covers mainly general holonomic constraints, with only a brief mention of nonholonomic constraints in the context of the application of Gauss s principle of least constraint in MD simulations. A holonomic constraint, a = 0, is defined as an algebraic equation connecting the coordinates of the particles (and in general also the time, but not here). To completely understand the various constraint methods used in modem simulations, it is appropriate to start the discussion from first principles. 

Although this chapter is concerned primarily with holonomic constraints, we comment on the role of nonholonomic constraints in the present context. Because the analytical dynamics theory of a system of particles subject to holonomic constraints is well established, the issues in (holonomic) constraint dynamics are mainly algorithmic in nature, as seen in this chapter. The situation is more complex for molecular dynamics with nonholonomic constraints, where theoretical difficulties exist. " ... 

R. Kutteh, preprint (1997). Analytical Dynamics with General Nonholonomic Constraints. 

Notice that the no-slip condition imposed a nonholonomic constraint described by (5.3), that it means that the mobile robot can only move in the direction normal to the axis of the driving wheels. 

In the EEM method. Gauss s principle of least constraint is invoked to derive the equations of motion of the system of particles with holonomic constraints. However, it is well known that when holonomic constraints are involved, the equations of motion can be derived from either D Alembert s principle, Hamilton s principle, or by means of a third approach. Application of Gauss s principle in this case offers no advantage over these other approaches. Gauss s principle is also exploited in the EEM method to enforce a nonholonomic temperature constraint in constant-temperature MD simulations. Again, the same equations of motion can be obtained by alternative means. ... 

Mobile robots are nonholonomic systems due to the constraints imposed on their kinematics. The equations describing the constraints cannot be integrated symbolically to obtain explicit relationships between robot positions in local and global coordinate s frames. Hence, control problems that involve them have attracted attention in the control community in the last years. 

The second need puts nonholonomic contraints to the parameters e and b. These constraints are determined by noting that for... 

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