Stabilization of underactuated four-link gymnast robot using torque-coupled method

In this paper, we use an underactuated four-link gymnast robot (UFGR) with a passive first joint to model a gymnast on the bar, and consider the stabilization of the UFGR at the straight-up position. First, we introduce a coupled relationship between control torques for the UFGR. It decouples some state variables of the UFGR from others and changes the nonlinear UFGR system into a cascade-connected system. And then, we use an energy-based method to design a stabilizing controller for the zero dynamics of the cascade-connected system. And the triangle lemma guarantees the control objective of the UFGR to be achieved. The torque-coupled method transforms the stabilization of an underactuated four-link manipulator into that of an underactuated two-link acrobot. This makes the structure of the control system simple. Moreover, our proposed control strategy is easy to extend to the stabilization control of other multi-input nonlinear underactuated systems. Simulation results using the characteristics parameters of a gymnast demonstrate the validity of the proposed method.     

A controller for 2-DOF underactuated mechanical systems with discontinuous friction

We propose a controller for a class of 2-DOF underactuated mechanical systems with discontinuous friction in the unactuated joint. The control objective is the regulation of the unactuated variable while the position and speed of the actuated joint remain bounded. The unactuated joint is considered as a mechanical system with discontinuous friction but continuous, artificial control input given by a term depending on the actuated positions and velocities. The proposed controller guarantees the convergence of the position error of the unactuated joint to zero, and it is robust with respect to some uncertainty in the discontinuous friction coefficients. We illustrate the technique with its application to two systems.     

Stabilization of underactuated planar acrobot based on motion-state constraints

An underactuated planar acrobot (UPA) is a two-link manipulator with a passive first joint. The control objective for a UPA in a horizontal plane is to move it from an initial position and stabilize it at a target position. Since a UPA is not small-time locally controllable, motion control is a big challenge. This paper presents a new control strategy based on constraints on the motion state. First, constraints on the angles and angular velocities of the two links are devised by integrating the dynamic equation of the UPA. Next, the state constraints are analyzed to discover the motion characteristics. Then, the optimal target position is obtained by solving a motion optimization problem that incorporates those characteristics. Finally, a control strategy based on the Lyapunov function method is devised to achieve the control objective. A numerical example demonstrates the validity of the control strategy.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Reachable area of an underactuated space manipulator subjected to simple spinning

Manipulators with free links are called underactuated manipulators, and in these systems, the number of actuators is less than the degree of freedom of the manipulator. Motivations of the study on the underactuated manipulators are, for example, to construct back-up strategies in case of actuator failure in extreme missions such as space robots and to realize cost reduction by employing fewer actuators. In this paper, we investigate the reachable area of the tip of a two-link underactuated manipulator whose first link (active link) is equipped through an actuator to a base rotated with a constant angular velocity. We produce pitchfork bifurcations in the free link (second link) by motion with a high-frequency component of the active link and actuate perturbations of the pitchfork bifurcations by changing the configuration of the active link with respect to the direction of the centrifugal force. We investigate the reachable area of the tip of the manipulator under the control by analytical approach and experimentally discuss the validity of the theoretical prediction.     

Integrated mechanical and control design of underactuated multibody systems

Multibody systems are called underactuated if they have less control inputs than degrees of freedom, e.g. due to passive joints or body flexibility. For trajectory tracking of underactuated multibody systems often advanced modern nonlinear control techniques are necessary. The analysis of underactuated multibody systems might show that they possess internal dynamics. Feedback linearization is only possible if the internal dynamics remain bounded, i.e. the system is minimum phase. Also feed-forward control design for minimum phase systems is much easier to realize than for non-minimum phase systems. However, often the initial design of an underactuated multibody system is non-minimum phase. Therefore, in this paper a procedure for integrated mechanical and control design is proposed such that minimum phase underactuated multibody systems are obtained. Thereby an optimization-based design process is used, whereby the geometric dimensions and mass distribution of the multibody systems are altered.     

Controllability of an underactuated spacecraft with one thruster under disturbance

For an underactuated spacecraft using only one thruster,the attitude controllability with respect to the orbit frame is studied in the presence of periodical oscillation disturbance,which provides a preconditional guide on designing control law for underactuated attitude control system.Firstly,attitude dynamic model was established for an underactuated spacecraft,and attitude motion was described using the special orthogonal group(SO(3)).Secondly,Liouville theorem was used to confirm that the flow generated by the drift vector of the underactuated attitude control system is volume-preserving.Furthermore,according to Poincare’’s recurrence theorem,we draw conclusions that this drift field is weakly positively poisson stable(WPPS).Thirdly,the sufficient and necessary condition of controllability was obtained on the basis of lie algebra rank condition(LARC).Finally,the controllable conditions were analyzed and simulated in different cases of inertia matrix with the installed position of thruster.     

Low-complexity robust tracking of high-order nonlinear systems with application to underactuated mechanical dynamics

This paper presents a low-complexity design approach with predefined transient and steady-state tracking performance for global practical tracking of uncertain high-order nonlinear systems. It is assumed that all nonlinearities and their bounding functions are unknown and the reference signal is time varying. A simple output tracking scheme consisting of nonlinearly transformed errors and positive design parameters is presented in the presence of virtual and actual control variables with high powers where the error transformation technique using time-varying performance functions is employed. Contrary to the existing results using known nonlinear bounding functions of model nonlinearities, the proposed tracking scheme can be implemented without using nonlinear bounding functions (i.e., the feedback domination design), any adaptive and function approximation techniques for estimating unknown nonlinearities. It is shown that the tracking performance of the proposed control system is ensured within preassigned bounds, regardless of high-power virtual and actual control variables. The motion tracking problem of an underactuated unstable mechanical system with unknown model parameters and nonlinearities is considered as a practical application, and simulation results are provided to show the effectiveness of the proposed theoretical result.     

Controlling the inverted pendulum by means of a nested saturation function

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.     

Uniform ultimate boundedness for underactuated mechanical systems as mismatched uncertainty disappeared

Nonlinear Dynamics - We propose to design control for uncertain underactuated mechanical systems. The underactuated mechanical system is to follow prescribed holonomic or nonholonomic constraints....     

Global stabilization of 2-DOF underactuated mechanical systems—an equivalent-input-disturbance approach

This paper presents a new method of globally stabilizing a non-linear underactuated mechanical system with two degrees of freedom (DOF). It is based on the idea of equivalent input disturbance (EID), and designing the controller requires only the state variables of position, not velocity. The design procedure has two steps: (1) Use a global homeomorphic coordinate transformation to convert the original system into a new non-linear system. This changes the problem of stabilizing the original system into one of stabilizing the new system. (2) Divide the new system into linear and non-linear parts and take the non-linear part to be an artificial disturbance, thereby enabling use of the EID approach to globally asymptotically stabilize the new system at the origin. The new method was tested through numerical simulations on three well-known 2-DOF underactuated mechanical systems (TORA, beam ball, inertia wheel pendulum). The results demonstrate its validity and its superiority over others.     

Nonlinear stabilizing control for a class of underactuated mechanical systems with multi degree of freedoms

Nonlinear Dynamics - This paper investigates the generation of some novel bursting patterns in active control oscillator with multiple time delays. We present the bursting patterns, including...     

Effect of the deformation in the inertia forces on the inverse dynamics of planar flexible mechanical systems

The inverse dynamics problem for articulated structural systems such as robotic manipulators is the problem of the determination of the joint actuator forces and motor torques such that the system components follow specified motion trajectories. In many of the previous investigations, the open loop control law was established using an inverse dynamics procedure in which the centrifugal and Coriolis inertia forces are linearized such that these forces in the flexible model are the same as those in the rigid body model. In some other investigations, the effect of the nonlinear centrifugal and Coriolis forces is neglected in the analysis and control system design of articulated structural systems. It is the objective of this investigation to study the effect of the linearization of the centrifugal and Coriolis forces on the nonlinear dynamics of constrained flexible mechanical systems. The virtual work of the inertia forces is used to define the complete nonlinear centrifugal and Coriolis force model. This nonlinear model that depends on the rate of the finite rotation and the elastic deformation of the deformable bodies is used to obtain the solution of the inverse dynamics problem, thus defining the joint torques that produce the desired motion trajectories. The effect of the linearization of the mass matrix as well as the centrifugal and Coriolis forces on the obtained feedforward control law is examined numerically. The results presented in this investigation are obtained using a slider crank mechanism with a flexible connecting rod.     

Correction to: Adaptive RBFNN finite-time control of normal forms for underactuated mechanical systems

Nonlinear Dynamics - The list of authors in the original publication was incomplete. The complete list of authors is shown here, including the third author, Faiçal Mnif.     

Steady-state analysis of multibody systems with reference to vehicle dynamics

This paper presents a systematic methodology and formulation for determining the steady-state response of multibody systems. The equations of motion for a general multibody system are described in terms of a set of relative joint accelerations. Then, the differential equations of motion are converted to a set of algebraic equations for the steady-state response. These equations are derived based upon a set of conditions that must exist for the steady state. The application of this formulation in determining the steady-state response of a vehicle moving in a circular path is shown. The multibody model of the vehicle for two- or four-wheel steering is presented. The results of the steady-state simulation are compared with those obtained from a transient dynamic analysis.     

Dynamics of flexible mechanical systems with contact-impact and plastic deformations

A computer based formulation for the analysis of mechanical systems is investigated as a feasible method to predict the impact response of complex structural systems. A general methodology for the dynamic analysis of rigid-flexible multibody systems using a number of redundant Cartesian coordinates and the method of the Lagrange multipliers is presented. The component mode synthesis is then used to reduce the number of flexible degrees of freedom. In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. This concept is used by associating revolute nonlinear actuators with constitutive relations corresponding to the collapse behavior of the structural components. The contact of the system components is described using a continuous force model based on the Hertz contact law with hysteresis damping. The effect and importance of structural damping schemes in flexible bodies are also addressed here. Finally, the validity of this methodology is assessed by comparing the results of the proposed models with those obtained in different experimental tests where: a beam collides transversally with a rigid block; a torque box impacts a rigid barrier.     

Control structure with disturbance identification for Lagrangian systems

A new control structure for a class of uncertain Lagrangian systems that solves the regulation and tracking control problems is proposed. This structure has good robustness properties, similar to sliding-mode-type controllers; and is free from chattering. The control structure is based on a discontinuous, robust observer, which displays a second-order sliding mode yielding an exponential convergence to the state of the plant in spite of the existence of non-vanishing disturbances and parameter uncertainties. At the same time, with the aid of a low-pass filter, this observer is employed to estimate the perturbations affecting the plant. This disturbance estimation is used to compensate the perturbations acting on the plant, improving the controller robustness. The performance of the proposed control structure is illustrated numerically and experimentally.     

Some aspects of control systems as dynamical systems

Linear control semigroupsL=Gl(d,R) are associated with semilinear control systems of the form whereA:R ^m gl(d,R) is continuous in some open set containingU. The semigroupL then corresponds to the solutions with piecewise constant controls, i.e., L acts in a natural way onR ^d {0}, on the sphereS ^d–1, and on the projective spaceP ^d–1. Under the assumption that the group generated byL in Gl(d,R) acts transitively onP ^d–1, we analyze the control structure of the action ofL onP ^d–1: We characterize the sets inP ^d–1, where the system is controllable (the control sets) using perturbation theory of eigenvalues and (generalized) eigenspaces of the matrices g L For nonlinear control systems on finitedimensional manifoldsM, we study the linearization on the tangent bundleTM and the projective bundleP M via the theory of Morse decompositions, to obtain a characterization of the chain-recurrent components of the control flow onU×PM. These components correspond uniquely to the chain control sets onP M, and they induce a subbundle decomposition ofU×TM. These results are used to characterize the chain control sets ofL acting onP ^d–1 and to compare the control sets and chain control sets.Research supported in part by NSF Grant DMS 8813976 and DFG Grant Co 124/6-1.     

On the Helmholtz-Boltzmann thermodynamics of mechanical systems

In an 1884 paper, Boltzmann showed that for a one-dimensional mechanical system with a convex potential energy that depends on a parameter V, it is possible to define a temperature T, pressure p, and entropy S that satisfy the Gibbs relation TdS = de + p dV, where . In the paper we review the extension of the Boltzmann construction to general natural mechanical systems endowed with a fibration over the (possibly multidimensional) space of macroscopic parameters. Moreover, for certain discrete mechanical systems with non-convex potential energies, which are used as models for phase transitions in solids, we compare the thermodynamic pressure p = p(e,V) introduced above with the quasi-static, macroscopic, stress-strain relation.Received: 20 February 2002, Accepted: 19 May 2003PACS: 5.20.y, 5.45.a, 5.70.ce Correspondence to: F. Cardin