# Nonholonomic Systems

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Nonholonomic Systems

mechanical systems that have imposed on them nonholonomic constraints (kinematic constraints that do not reduce to geometric constraints) in addition to purely geometric constraints. (SeeHOLONOMIC SYSTEMS.) A sphere rolling on a rough plane without slipping is an example of a nonholonomic system. In this case, the constraint imposed is a constraint not only on the position of the center of the sphere (geometric constraint) but also on the velocity of the point of contact between the sphere and the plane; this velocity must be zero at any moment of time, which is a kinematic constraint that does not reduce to a geometric constraint.

Mathematically, nonholonomic constraints are directly expressed by nonintegrable equations of the form

f (xi, i,zi, ẋi, ẏi, żi, , t) = 0

where xi, yi, and zi are the coordinates of the points of the mechanical system; and xi, yi, and zi, are the components of the velocities of the points, equal to the derivatives of the coordinates with respect to time t.

The motion of nonholonomic systems is studied using special equations, such as the Chaplygin or Appell equations, or equations that can be obtained from differential variational principles of mechanics.

### REFERENCE

Dobronravov, V. V. Osnovy mekhaniki negolonomnykh sistem. Moscow,

S. M. TARG

References in periodicals archive ?
A Framework for the Stabilization of General Nonholonomic Systems With an Application to the Plate-Ball Mechanism.
It should be noted that although the theorem of Sussman [18] or Goodwine and Burdick [19] provides sufficient conditions for the controllability of nonholonomic systems, they cannot be applied in this study owing to the nonnegative restriction of the thruster forces.
However, the novelty of the new approach lies in its ability to design continuous nonlinear control laws to translate locally rigid formations of nonholonomic systems tagged with dynamic constraints.
In the past few years, nonholonomic systems especially their extended version ([1, 2]) have received considerable attention.
The authors of [4, 5] studied the course of the development and the status of nonholonomic systems as well as their principles of dynamics and control methods.
On the other hand, since the significant development that has witnessed the field of fuzzy modeling and control and more particularly model-based control, a new theory about fuzzy switched systems has emerged as an answer to more complicated real systems analysis and synthesis requirements such as multiple nonlinear systems, switched nonlinear systems, and second-order nonholonomic systems [3-10].
On the other hand, the inverted system can be underactuated and nonholonomic systems. Yue et al.
Nijmeijer, "A recursive technique for tracking control of nonholonomic systems in chained form," IEEE Transactions on Automatic Control, vol.
Wheeled mobile robots (WMRs) are typical nonholonomic systems and they have attracted the attention of many researchers as they do not satisfy Brockett's necessary condition [1].
Nonholonomic systems, which can model many classes of mechanical systems such as mobile robots and wheeled vehicles, have attracted intensive attention over the past decades.
Besides they have proved to be useful in Mechanics [2, 4, 7, 16, 24], in the theory of nonholonomic systems [3, 9, 18] in control theory [6], in field theory [16], in quantum and classical gravity [22, 23].
His research applies to both theory and applications in a wide range of problems, including nonholonomic systems, space and mobile robots, haptic interfaces and robots for telesurgery and remote diagnostics, control of structural vibration, and control of rotors supported by magnetic bearings.

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