Underactuated System

In this paper, hybrid control synthesis is proposed for a class of 2-degrees-of-freedom (DOF) underactuated mechanical systems with Coulomb friction in the joints. The control objective is to regulate both actuated and unactuated joints to desired positions. The proposed synthesis combines second-order sliding-mode (SOSM) control methods and a pure technical solution of imposing a relatively stronger dry friction on the unactuated joint as compared to that of the actuated joint. This design feature allows us to decouple the hybrid synthesis procedure into two steps. At the first step, a SOSM control algorithm is utilized to drive the unactuated link to the desired position in finite time. Once this task is achieved, the control input is switched to another SOSM algorithm which drives the actuated link to its desired position. As the amplitude of the controller used at the second step is smaller than the Coulomb friction level in the unactuated link, but it is greater than that in the actuated link, the unactuated link remains at rest, whereas the actuated link is regulated to the endpoint of interest. Performance issues of the developed synthesis, including robustness features of the closed-loop system, are illustrated in an experimental study made for a laboratory horizontal 2-DOF pendulum.

A systematic approach for designing analytical dynamics and servo
control of constrained mechanical systems is proposed. Fundamental
equation of constrained mechanical systems is first obtained
according to Udwadia-Kalaba approach which is applicable to
holonomic and nonholonomic constrained systems no matter whether
they satisfy the D'Alember's principle. The performance
specifications are modeled as servo constraints.
Constraint-following servo control is used to realize the servo
constraints. For this inverse dynamics control problem, the
determination of control inputs is based on the Moore-Penrose
generalized inverse to complete the specified motion. Second-order
constraints are used in the dynamics and servo control. Constraint
violation suppression methods can be adopted to eliminate constraint
drift in the numerical simulation. Furthermore, this proposed
approach is  applicable to not only fully actuated but also
underactuated and redundantly actuated mechanical systems. Two-mass spring system and coordinated robot system are presented as examples for illustration.

Citation: Xiaoli Liu, Shengchao Zhen, Kang Huang, Han Zhao, Ye-Hwa Chen, Ke Shao. A systematic approach for designing analytical
dynamics and servo control of constrained mechanical systems. IEEE/CAA Journal of Automatica Sinica, 2015,  2(4): 382-393

In this paper, the dynamic model of a wheeled inverted pendulum (e.g., Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view. First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations. This model is compared with the previous models derived for similar systems. The strong accessibility condition is checked and the maximum relative degree of the system is found. Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated. The resulting equations are then used to design two novel controllers. The first one is a two-level velocity controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch (pendulum angle from the vertical) within a specified range. The second controller is also a two-level controller which stabilizes the vehicle's position to the desired point, while again keeping the pitch bounded between specified limits. Simulation results are provided to show the efficacy of the controllers using realistic data.

This paper investigates the sliding mode control of the Ball on a Beam system. A static and a dynamic sliding mode controllers are designed using a simplified model of the system; the simplified model renders the system feedback linearizable. Then, a static and a dynamic sliding mode controllers are designed using the complete model of the Ball on a Beam system. Simulation results indicate that the proposed controllers work well. The four proposed controllers are implemented using an experimental setup. Implementation results indicate that the proposed control schemes work well. As expected, it is found that the proposed two controllers which are designed using the complete model of the system gave better performances than the ones designed using the simplified model of the system. In addition, the experimental results indicate the two dynamic controllers greatly reduce the chattering usually associated with sliding mode controllers.

In this paper, the dynamic model of a wheeled inverted pendulum (e.g. Segway [9], Quasimoro [8], Joe [6]) is analyzed from a controllability and feedback linearizability point of view. First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic noslip constraints into considerations. This model is compared with the previous models derived for similar systems. The strong accessibility condition is checked and the maximum relative degree of the system is found. Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated. The resulting equations are then used to design a two-level controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch within a specified range. Simulation results are provided to show the efficacy of the controller.

In this paper, we address the trajectory tracking issue of underactuated dynamic systems using a special examplea pendulum-driven cart-pole system. We first develop the dynamic model of the pendulum-driven cartpole system. To drive the cart in one specified direction, we propose a six-step motion strategy and an optimal algorithm to select the six-step strategy profile. To track the desired profile, we first investigate two control algorithms: openloop control (OLC) and closed-loop control (CLC). Later on, based on the learning of the OLC and the CLC, we propose a new control algorithm called simple switch control (SSC) to overcome the issues caused by the real experiment. Robustness of the proposed three algorithms is investigated by using a variable parameter on the system. Finally, extensive simulation studies are conducted to demonstrate the proposed algorithms.

This report presents the modeling and controller design of three degree of freedom Helicopter/ Twin rotor control system. The system is multi-input multi-output (MIMO) having highly nonlinear dynamics and static input nonlinearities. The dynamics of the system have been derived using Euler-Lagrange approach. The system has two inputs and three outputs .The present model is derived with the purpose of accurate simulation of the Twin rotor system (Helicopter) when it is moving upward. The task is control the azimuth and elevation angle and the height. The system is linearized and an optimal controller has been designed to control the azimuth angle, elevation angle as well as the height. At the end we have presented the open loop and closed loop responses of the system. A set of model parameters is also given at the end.

This article covers research work about unmanned helicopters that is being developed by the Portuguese SME IntRoSys, S.A. in the context of a generic project targeting the development of a Multi-Robot system for landmine detection. As helicopters are complex multivariable nonlinear underactuated systems with strong coupling in some control loops, they are an interesting subject of research for the control and robotic research communities. Therefore, a wide range of techniques have been proposed in the literature to control such vehicles, ranging from classic control to soft-computing methods. The main goal for this article is to present a survey about existing low level control architectures for unmanned helicopters and to discuss which control design criteria (design requirements) are relevant when they are operating in the context of Humanitarian Demining.

A theoretical framework is established for the dynamics and control of underactuated systems, defined as systems which have fewer inputs than degrees of freedom. Control system formulation of underactuated systems is addressed and the class of second-order nonholonomic systems is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results.

Sloshing of liquid in a tank is important in several areas including launch vehicles carrying liquid fuel in space application, ships, and liquid cargo carriages. Hence modeling and characterization of nonlinear slosh dynamics is critical for study of dynamics of these systems. Additionally control of sloshing liquid offers a challenging problem of control of underactuated systems. To study slosh dynamics, develop useful identification schemes, and design and verify slosh control algorithms, a new 2DOF actuation slosh rig is reported in this paper considering the fact that most of the times the liquid tanks are subjected to linear as well as pitching excitation. The paper discusses mechatronic design and several advantages offered by the new design. Furthermore, a mathematical model of the rig is developed using Lagrange formulation assuming two-pendulum model for slosh. Slosh parameter identification with the rig is demonstrated in pitching and linear excitation cases. Nonlinear parameter identification schemes developed using simplified version of rig model are used for the purpose. Further results on compensation of slosh and rotary slosh phenomena are presented. Thus the proposed rig is ideal tool for study, identification, and control of slosh phenomena.

This paper deals with a control approach dedicated to stable limit cycle generation for underactuated mechanical systems. The proposed approach is based on partial nonlinear feedback linearization and dynamic control for optimal periodic reference trajectories tracking. The computation of the reference trajectories is performed in order to optimize the behavior of the whole dynamics of the system and especially its zero dynamics at the end of each cycle. Simulation results as well as experiments show the performance and the efficiency of the proposed control scheme.

Underactuated manipulators have to be controlled by following the restriction that the number of actuators is less than the number of generalized coordinates, because they have some passive joints. In this article, we propose a logic-based switching mechanism using a fuzzyenergy region. Boundary curves to separate an error energy region into several energy subregions are constructed by fuzzy logic. Therefore, fuzzy related parameters are optimized by a genetic algorithm (GA). This method is applied to an underactuated system with a drift term, such as a 2-DOF planar manipulator that has only one active joint. The effectiveness of the method is illustrated with some simulations.

An interesting problem in robotics is to minimize the required time to force a manipulator to travel between two specific points (positioning time). In this paper a concurrent structure-control redesign approach is proposed in order to find the minimum positioning time of an underactuated robot manipulator, by considering a synergetic combination between the structural parameters and a bang-bang control law. The problem consists in finding the structural parameters of the system and the switching intervals of the bang-bang control that simultaneously minimize the positioning time, subject to the input constraint and the structural parameter constraint. The concurrent structure-control redesign approach is stated as a dynamic optimization problem (DOP). The projected gradient method is used to solve the DOP. The effectiveness of the proposed concurrent redesign approach is shown via simulation and experimental results on an underactuated system called the Pendubot.

An interesting problem in robotics is to minimize the required time to force a manipulator to travel between two specific points (positioning time). In this paper a concurrent structure-control redesign approach is proposed in order to find the minimum positioning time of an underactuated robot manipulator, by considering a synergetic combination between the structural parameters and a bang–bang control law. The problem consists in finding the structural parameters of the system and the switching intervals of the bang–bang control that simultaneously minimize the positioning time, subject to the input constraint and the structural parameter constraint. The concurrent structure-control redesign approach is stated as a dynamic optimization problem (DOP). The projected gradient method is used to solve the DOP.The effectiveness of the proposed concurrent redesign approach is shown via simulation and experimental results on an underactuated system called the Pendubot.

The dynamics and kinematics of manipulators that have fewer actuators than degrees of freedom are studied. These underactuated manipulators arise in a number of important applications such as free-flying space robots, hyperredundant manipulators, manipulators with structural flexibility, etc. In the analysis such underactuated manipulators are decomposed into component active and passive arms. This decomposition allows techniques previously developed for regular (fully actuated) manipulators to be applied to underactuated systems. Spatial operator identities are used to develop closed-form expressions for the generalized accelerations for the system. These expressions form the basis for a recursive O(N) dynamics algorithm. The structure of this algorithm is a hybrid of known forward and inverse dynamics algorithms for regular manipulators. Expressions and computational algorithms are also developed for the generalized and disturbance Jacobians for underactuated manipulators. The application of the results in the paper to space manipulators is also described. NOMENCLATURE Coordinate free spatial notation is used throughout this_ paper (see [5], for additional details). The notation 1 denotes the cross-product matrix associated with the threedimensional vector 1, while z* denotes the transpose of a matrix z. In the stacked notation used in this paper, indices are used to identify quantities pertinent to a specific link. Thus for instance, V denotes the vector of the spatial velocities for all the links, and V ( k ) denotes the spatial velocity vector for the kth link. Some key quantities used in this paper are defined number of links in the manipulator inboard (body) frame for the kth link outboard frame on the ( k + 1)th link number of degrees of freedom for the kth hinge = freedom for the manipulator E R2'('), the vector of generalized coordinates for the kth hinge E the kth hinge n r ( k ) , the total number of degree of k = l the vector of generalized velocities for . The authors are with the Jet Propulsion Laboratory, California Institute of IEEE Log Number 9212604. Technology 4800 Oak Grove Drive, Pasadena, CA 91 109. E !R3, the vector from the kth to the j t h body frame =(o A I f ( k , j ) I ) the spatial transformation operator between the j t h and the kth hinges E !R6x'(k), the joint map matrix for the kth hinge the mass of the kth link E R3, the vector from o k to the center of mass of the kth link E R3x31 the inertia matrix for the kth link referred to ok inertia of the kth link referred to 0 1 , = ( : [ : {) E R6, the spatial velocity of the kth link referred to ok, with w ( k ) and w(k) denoting the angular and linear velocity components respectively -V ( k + l)] E R6, the Coriolis acceleration for the kth link referred to C)k

1. THE MATCHING CONDITION This note presents an application of the method developed by Auckly, et al. (2000), to stabiliza-tion of a ball and beam system. The results are fully described in (Andreev, et al., (2000), Auckly, Kapitanski (2000), and Auckly, et al. (2000)). An ...

A recent approach to the control of underactuated systems is to look for control laws which will induce some specified structure on the closed loop system. In this paper, we describe one matching condition and an approach for finding all control laws that fit the condition. After an analysis of the resulting control laws for linear systems, we present the results from an experiment on a nonlinear ball and beam system.

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.

This work is focused on the motion control of an underactuated brachiation robot with 3 links. We present the modeling of the dynamics of the robot and introduce the application of the model-based predictive control (MPC) using a linearized model of the system. The robot has 3 revolute joints but only one of them is actuated, i.e., the robot has less control inputs than degrees of freedom. The investigation of the MPC to control such a system is due to the fact that MPC is able to deal with constraints (state or input limitations) in a direct way, obtaining an optimal control law. Also, it is investigated the use of a linearized model for prediction of system state. Simulation results are shown for complete arm switching, as well as some directions for future developments

A theoretical framework is established for the dynamics and control of underactuated systems, defined as systems which have fewer inputs than degrees of freedom. Control system formulation of underactuated systems is addressed and the class of second-order nonholonomic systems is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results. Consider first a dynamic system with configuration manifold Q. Let (4, q ) = (ql,. . . , qn, q', . . . , 4") denote local coordinates on M = TQ. We refer to q , q , and q

A recent approach to the control of underactuated systems is to look for control laws which will induce some speci3ed structure on the closed loop system. In this paper, we describe one matching condition and an approach for 3nding all control laws that 3t the condition. After an analysis of the resulting control laws for linear systems, we present the

We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely charactarized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order PDEs for the second set of variables (g, V). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.

In this paper we introduce a new method to design control laws for non-linear underactuated systems. Our method will often work in situations where standard control techniques fail. Even when standard techniques apply, we believe that our approach will achieve better performance. This is because we produce an infinite dimensional family of

A recent approach to the control of underactuated systems is to look for control laws which will induce some specified structure on the closed loop system. In this paper, we describe one matching condition and an approach for finding all control laws that fit the condition. After an analysis of the resulting control laws for linear systems, we present the results from an experiment on a nonlinear ball and beam system.

We propose a hybrid controller for a class of 2-DOF underactuated mechanical systems with discontinuous friction in the unactuated joint. The control objective is the regulation of both unactuated and actuated variables. The design of the control law is divided in two parts. First we design a discontinuous control for the unactuated joint, then we propose another discontinuous control for the actuated joint. The proposed controller guarantees the convergence of the position error to zero, and it is robust with respect to some uncertainty in the discontinuous friction coefficients. We illustrate the technique with its application to a physical system.

An observer-based controller is proposed for the walking of a biped without feet, i.e. an underactuated system in single support phase. The originality is both: first, the observer is based on second-order sliding mode approach and is original in biped robot context. Secondly, an existing "simplified" Poincaré's sections-based analysis of the stability of the walking is adapted to nonlinear system with not fully available state variables.

In this paper a "Robust Fixed Point Transformation (RFPT)" based adaptive control of an underactuated physical system is stabilized by adaptively tuning only one parameter of a single fuzzy membership function. This approach serves as an alternative to Lyapunov's "direct" method that suffers from mathematical difficulties when a Lyapunov function candidate has to be found for the control of a dynamically singular or badly conditioned system as an underactuated cart plus double pendulum. In our case the reaction forces of the directly driven two rotary axles are used for controlling the linear motion of the cart within possible physical limits. As a modification of the common TORA (Translational Oscillations with an Eccentric Rotational Proof Mass Actuator) that has only one counterweight, the present solution applies two counterweights: one of them is actively used and the other one is kept in a "safe" position while the system is far from the dynamic singularity. When the singularity is approached the reserved axle takes the active role and the previously used weight is moved back to the nonsingular region. This session results in oscillatory motion that precisely has to be implemented by the controller. Instead developing complete and generally useful system model this approach extracts information on the recent behavior of the controlled system only in the given control situation by using only three adaptive control parameters. Two of them can be kept fixed but the third one may need fine tuning for stable control. Now a formerly proposed intricate tuning strategy is replaced by simple fuzzy tuning. It is illustrated by simulations that the controller can precisely track the prescribed trajectory even in the presence of considerable modeling errors and badly conditioned dynamics.