# Nonholonomic Systems

## 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 ?
Mobayen, "Finite-time tracking control of chained-form nonholonomic systems with external disturbances based on recursive terminal sliding mode method," Nonlinear Dynamics, vol.
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.
In [19], the underactuated nonholonomic systems with chained form were investigated by output feedback control law; the designed controller rendered the state variables to zero within finite time.
In [30], a class of nonholonomic systems in chained form which can model mobile robots and wheeled vehicles was studied, the finite time state feedback controller was addressed; however, the method requires the sway velocity satisfying the first-order nonholonomic constraints (the sway velocity must be zero); it cannot be applied to the control of UAVs.
Zhang, "Finite-time stabilization of uncertain nonholonomic systems in feedforward-like form by output feedback," ISA Transactions, vol.
In holonomic systems, the control input degrees are equal to total degrees of freedom, whereas, nonholonomic systems have less controllable degrees of freedom as compared to total degrees of freedom and have restricted mobility due to the presence of nonholonomic constraints.
A detailed survey of stabilization of nonholonomic systems can be found in [3] and a survey of underactuated mechanical systems is given in [4].
Astolfi, "Discontinuous control of nonholonomic systems," Systems and Control Letters, vol.
Lee, "Adaptive stabilization of uncertain nonholonomic systems by state and output feedback," Automatica, vol.
Liu, "Adaptive stabilization of a class of high-order uncertain nonholonomic systems with unknown control coefficients," International Journal of Adaptive Control and Signal Processing, vol.
Zhang, "Saturated finite-time stabilization of uncertain nonholonomic systems in feedforward-like form and its application," Nonlinear Dynamics, vol.

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