The quasi-constrained dynamics is composed of motions not only onto the constraint space but also onto the unconstraint or the "freedom" space, which may occur when the system escapes or slips away from its constraint manifold during high-speed motion. Traditionally, slippage phenomenon, which has been an issue in the automobile industry, is usually ignored because of its high frequency and strong nonlinear features. Conventional Frobenius theorem is focused on the holonomic dynamics, which are integrable on the freedom space. On the other hand, a complementary Frobenius theorem (CFT) is proposed to release conventional constraints from "hard" to "soft' models. In this paper, we derive a geometric formulation instead of its algebraic counterpart for acatastatically nonholonomic systems in the viewpoint of topology. Besides, we propose a mixed fuzzy controller (MFC) for the nonholonomic system with escaping motions, which includes a traditional controller for the hard subsystem and a non-tradtional controller with fuzzy rules for another soft subsystem on the constraint manifold. The closedloop stability of'the nonholonomic system with an MFC scheme will be proved under admissible condtions. Finally, th proposed algorithm is applied to a wheeled vehicle with an anti-lock braking system (ABS) under the assumption of Coulomb's viscous friction. Computer simulation is used to jus* the results.
B a s i c concepts about t h e creep behavior of nonholon o m i c constrained (NC) s y s t e m s w e r e discussed in [5]. Two f u n d a m e n t a l t y p e s o f creep k i n e m a t i c s w e r e proposed a n d t h e hybrid o n e s could be m a d e . In this w o r k , w e e x t e n d to t h e creep d y n a m i c s of typical NC syst e m s such a s t h e disk, sleigh a n d wheel. First, certain reductive m o d e l s o f constrained m o t i o n s u c h a s ideal, relative, pair, a n d general-pair m o d e l s are explored and related. Secondly, by t h e i n v a r i a n t m a n i f o l d m e t h o d of singular perturbation, f u n d a m e n t a l rotational a n d traverse creep d y n a m i c s a n d a hybrid o n e are calculated in a n a p p r o x i m a t e w a y . T h e y c a n be close to t h e c o mplete s y s t e m a s possible. T h e r e f o r e , t h e bridge between reductzve a n d i n v a r i a n t a n a l y s e s c a n be m a d e by t h e m o d e l of general-pair creep a n d c a n help us u n d e r s t a n d physical i m p l i c a t i o n s behind t h e a p p r o x i m a t e solutions. It i s p r o v e n t h a t advanced vehicle techniques, s u c ha s t h e anti-lock braking s y s t e m a n d a special tracking control s y s t e m , c a n be realized b y t h e proposed quasi constrained creeps.
M o s t studies about nonholonomic s y s t e m s were developed u n d e r t h e absolute relation of constraint equations. S u c h ideally constrained s y s t e m s are k n o w n somewhat relaxed in real world, which are claimed as complementarily constrained problems [lz]. S y s t e ms with viscous f r i c t i o n s a n d / o r jlexibilities are typical cases at which t h e constraints are pseudo-violated rather t h a n violated. T h e s e nonideal kinematics are treated i n t e r m s of nonholonomic creep behaviors, classified b y t w o f u n d a m e n t a l types : rotation creep an,d traverse creep. Creep c o e f i c i e n t s are defined and proposed t o illustrate t h e kinematical relations for such complementarily constrained s y s t e m s . It can be s h o w n that m o s t nonholonomic vehicles can be described b y both creep types, while t h e trailers attached t o a car falls i n t o the traverse type only. Therefore, we conclude the existing algorithms for nonholonomic m o t i o n planning and control should be modified as intrinsic creeps take place zn nature. T h e creep n o t i o n can also be applied t o the s y m m e t r y s y s t e m s of conservation of angular m o m e n t u m .
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