controllability and stability analysis of planar snake robot locomotion this paper contributes to the understanding of snake robot locomotion by employing nonlinear system analysis tools for investigating fundamental properties of snake robot dynamics the paper has five contributions 1 a partially feedback linearized model of a planar snake robot influenced by viscous ground friction is developed 2 a stabilizability analysis is presented proving that any asymptotically stabilizing control law for a planar snake robot to an equilibrium point must be time varying 3 a controllability analysis is presented proving that planar snake robots are not controllable when the viscous ground friction is isotropic but that a snake robot becomes strongly accessible when the viscous ground friction is anisotropic the analysis also shows that the snake robot does not satisfy sufficient conditions for small time local controllability stlc 4 an analysis of snake locomotion is presented that easily explains how anisotropic viscous ground friction enables snake robots to locomote forward on a planar surface the explanation is based on a simple mapping from link velocities normal to the direction of motion into propulsive forces in the direction of motion 5 a controller for straight line path following control of snake robots is proposed and a poincare map is investigated to prove that the resulting state variables of the snake robot except for the position in the forward direction trace out an exponentially stable periodic orbit

speed observation and position feedback stabilization of partially linearizable mechanical systems the problems of speed observation and position feedback stabilization of mechanical systems are addressed in this paper our interest is centered on systems that can be rendered linear in the velocities via a partial change of coordinates it is shown that the class is fully characterized by the solvability of a set of partial differential equations pdes and strictly contains the class studied in the existing literature on linearization for speed observation or control a reduced order globally exponentially stable observer constructed using the immersion and invariance methodology is proposed the design requires the solution of another set of pdes which are shown to be solvable in several practical examples it is also proven that the full order observer with dynamic scaling recently proposed by karagiannis and astolfi obviates the need to solve the latter pdes finally it is shown that the observer can be used in conjunction with an asymptotically stabilizing full state feedback interconnection and damping assignment passivity based controller preserving asymptotic stability

feedback linearization and stabilization of second order non holonomic chained systems this paper presents a theoretical framework for non regular feedback linearization and stabilization of second order non holonomic chained systems by giving a new criterion for the problem of non smooth non regular feedback linearization it is proved that second order chained systems are non regular static state feedback linearizable a discontinuous control law is obtained based on linear system theory and the inversion technique the design mechanism is generalized to higher order non holonomic chained systems simulation studies are carried out to show the effectiveness of the approach

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

on brockett s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback the necessity of brockett s condition for stabilizability of nonlinear systems by smooth feedback is shown by an argument based on properties of a degree for set valued maps to persist when the class of controls is enlarged to include discontinuous feedback

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

controllability along a trajectory a variational approach this paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point this is accomplished by defining high order variations that can be continuously summed the peculiar property of these variations is that they may be generated by thin conditions as relations in the lie algebra associated with a control system this property leads to a variational interpretation of sussmann type sufficient conditions of local controllability neutralization of obstructions and it allows the use of different weights for neutralization

exponential stabilization of an underactuated autonomous surface vessel this paper studies the problem of controlling the planar position and orientation of an autonomous surface vessel using two independent thrusters it is first shown that although the system is not asymptotically stabilizable to a given configuration using a time invariant continuous feedback it is strongly accessible and small time locally controllable at any equilibrium time invariant discontinuous feedback laws are then constructed to asymptotically stabilize the system to the desired configuration with exponential convergence rates a simulation example is included to demonstrate the results

a geometric approach to feedback stabilization of nonlinear systems with drift the paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems the class of systems of interest includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback the approach is independent of the selection of a lyapunov type function but requires the solution of a nonlinear programming satisficing problem stated in terms of the logarithmic coordinates of flows as opposed to other approaches point to point steering is not required to achieve asymptotic stability instead the flow of the controlled system is required to intersect periodically a certain reachable set in the space of the logarithmic coordinates

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

robust point stabilization of underactuated mechanical systems via the extended chained form this paper addresses point stabilization for the extended chained form ecf a control system that may be used to model a number of mechanical underactuated systems a control law is proposed based on well known hybrid open loop feedback techniques which exponentially stabilizes the origin of a dynamic extension of the ecf and ensures a degree of robustness to additive disturbance terms that may represent for instance model uncertainties numerical simulations are included to illustrate the performance of the presented stabilizers

swing up control based on virtual composite links for n link underactuated robot with passive first joint this paper concerns the swing up control of an n link revolute robot moving in the vertical plane with the first joint being passive and the others being active the goal of this study is to design and analyze a swing up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point with all links in the upright position to achieve this challenging objective while preventing the robot from becoming stuck at an undesired closed loop equilibrium point we first address the problem of how to iteratively devise a series of virtual composite links to be used for designing a coordinate transformation on the angles of all the active joints second we devise an energy based swing up controller that uses a new lyapunov function based on that transformation third we analyze the global motion of the robot under the controller and establish conditions on the control parameters that ensure attainment of the swing up control objective specifically we determine the relationship between the closed loop equilibrium points and a control parameter finally we verify the theoretical results by means of simulations on a 4 link model of a gymnast on the high bar this study not only unifies some previous results for acrobots and three link robots with a passive first joint but also provides insight into the energy and passivity based control of underactuated multiple degree of freedom systems

time varying exponential stabilization of a rigid spacecraft with two control torques rigid body models with two controls cannot be locally asymptotically stabilized by continuous feedbacks which are functions of the state only this impossibility no longer holds when the feedback is also a function of time and time varying asymptotically stabilizing feedbacks have already been proposed however due to the smoothness of the feedbacks the convergence rate is only polynomial in this paper exponential convergence is obtained by considering time varying feedbacks which are only continuous

sliding mode control of underactuated multibody systems and its application to shape change control in this article we introduce an approach based on sliding mode control to design full state feedback controllers for stabilisation of underactuated non linear multibody systems we define first order sliding surfaces as a linear combination of actuated and unactuated coordinate tracking errors lyapunov stability analysis guarantees that all system trajectories reach and remain on the sliding surfaces however stability of the sliding surfaces depends on the equilibrium manifold if the system has isolated equilibrium points it is linearly controllable and asymptotic stability can be guaranteed under certain conditions otherwise the control system fails brockett s necessary condition for existence of a smooth stabilising feedback in the latter case if the total momentum is conserved the closed loop control system will be marginally stable consequently a procedure is proposed to achieve an asymptotically stable discontinuous control law through sliding surface redefinition and shape changes it i

rapidly exponentially stabilizing control lyapunov functions and hybrid zero dynamics this paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models systems with impulse effects through control lyapunov functions the periodic orbit is assumed to lie in a c1 submanifold z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics the associated restriction dynamics are termed the hybrid zero dynamics the orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics prior results on the stabilization of such periodic orbits with respect to the full order dynamics of the system with impulse effects have relied on input output linearization of the dynamics transverse to the zero dynamics manifold the principal result of this paper demonstrates that a variant of control lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface z can be used to achieve exponential stability of the periodic orbit in the full order dynamics thereby significantly extending the class of stabilizing controllers the main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control lyapunov functions

on the stabilization in finite time of locally controllable systems by means of continuous time varying feedback law it is proven that if for any positive time t there exists an open loop control u a t depending continuously on the initial data a vanishing for a 0 and steering a small neighborhood of 0 into 0 in time t then the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds

class preserving automorphisms of finite groups abstract we construct a family f of frobenius groups having abelian sylow subgroups and non inner class preserving automorphisms we show that any a group that is a finite solvable group with abelian sylow subgroups has a sub quotient belonging to f provided it has a non inner class preserving automorphism as a consequence we obtain that for metabelian a groups or a groups with elementary abelian sylow subgroups class preserving automorphisms are necessarily inner automorphisms the same is true for a finite group whose sylow subgroups of odd order are all cyclic and whose sylow 2 subgroups are either cyclic dihedral or generalized quaternion some applications are given

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

dynamics and control of a class of underactuated mechanical systems this paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems defined as systems with fewer inputs than degrees of freedom control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified controllability and stabilizability results are derived for this class of underactuated systems examples are included to illustrate the results these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots motivated by the problem of controlling walking in a biped with series compliant actuation this paper develops two main theorems relating to the stabilization of periodic orbits in systems with impulse effects the first main result shows that when a periodic orbit of a system with impulse effects lies within a hybrid invariant manifold there exist local coordinate transforms under which the jacobian linearization of the poincare return map has a block upper triangular structure one diagonal block is the linearization of the system as restricted to the hybrid invariant manifold also called the hybrid zero dynamics the other is the product of two sensitivity matrices related to the transverse dynamics one pertaining to the impact map and the other pertaining to the closed loop vector field when either of these sensitivity matrices is sufficiently close to zero the stability of the return map is determined solely by the stability of the hybrid zero dynamics the second main result of the paper details the construction of a hybrid invariant manifold such as that required by the first main theorem forward invariance follows from the methods of byrnes and isidori and impact invariance is achieved by a novel construction of impact updated control parameters in addition to providing impact invariance the construction allows entries of the impact sensitivity matrix of the transverse dynamics to be made arbitrarily small a simulation example is provided where stable walking is achieved in a 5 link biped with series compliant actuation

configuration controllability of simple mechanical control systems in this paper we present a definition of configuration controllability for mechanical systems whose lagrangian is kinetic energy with respect to a riemannian metric minus potential energy a computable test for this new version of controllability is derived this condition involves an object which we call the symmetric product of particular interest is a definition of equilibrium controllability for which we are able to derive computable sufficient conditions examples illustrate the theory

local equilibrium controllability of multibody systems controlled via shape change we study local equilibrium controllability of shape controlled multibody systems the multibody systems are defined on a trivial principal fiber bundle by a lagrangian that characterizes the base body motion and shape dynamics a potential dependent on an advected parameter e g uniform gravitational potential is considered this potential breaks base body symmetries but a symmetry subgroup is assumed to exist symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter based on these properties sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed these results are applied to two classes of shape controlled multibody systems in a uniform gravitational field multibody attitude systems and neutrally buoyant multibody underwater vehicles