Constrained mechanical systems modeling and control: A free-floating space manipulator case as a multi-constrained system

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Highlights

  • Multi-constrained multibody systems.

  • Control oriented dynamics modeling framework for constrained systems.

  • Control strategy for tracking predefined tasks.

  • Free-floating space vehicles control.

Abstract

The paper presents a development of a modeling framework for constrained multibody systems, for which constraints may arise either as kinematic, task-based, design, control or origin from conservation laws. The framework is control oriented since its dynamic modules yield nonlinear system dynamics in reduced state forms, i.e., ready to employ control algorithms. It encompasses systems subjected to any equality first order constraints and in this regard it unifies modeling for control applications. The framework properties support numerical computation and simulation studies for nonlinear system models due to the dynamic modeling unification and the reduction procedure built into it. The framework application is illustrated by an example of dynamics modeling and designing a tracking controller for a free-floating space manipulator, which is underactuated, subjected to conservation laws and task-based constraints.

Introduction

Modeling constrained systems, specifically nonholonomic, is strongly oriented to further model applications. This is due to properties of these system models that may manifest either at the level of their modeling or model applications. In mechanics, there are usually identified two types of constraints that are both material, i.e. the constraints imposed on system configurations or on kinematics, which specify the non-slipping roll condition. Kinematic control models for such systems are obtained based upon the nonholonomic constraint equations to which control inputs selected out of first order derivatives of state variables are added. This kinematic control model is nonlinear and due to the Brockett condition it requires a nonlinear approach in a controller design  [1]. A dynamic control model, in turn, is usually obtained using the Lagrange approach so the reduction procedure has to be applied to get the reduced state dynamics for a controller design. If modeling is oriented for off-line simulation or motion analysis, then the form of a dynamic model may not be a concern.

In control, there are more constraint sources. Usually, four constraint sources are listed. They are kinematic constraints, conservation laws, design and control constraints, which often come from the underactuation, and task-based constraints like a trajectory to follow  [2]. Also, specific properties of control oriented dynamic models are required due to on-line computation with a control law in a closed-loop. The constraints as listed above have to be handled at the modeling, control design or implementation levels.

Modeling for control applications of systems with kinematic constraints is well established and it is usually based upon the Lagrange approach  [3], [4]. A control framework that incorporates a system dynamics is usually developed based on two-level control architecture. The lower control level operates within a kinematic model to stabilize a system motion to a desired trajectory. The upper control level uses a dynamic model and stabilizes feedback obtained on the lower control level  [3].

Systems like space manipulators and vehicles are frequently classified as nonholonomic robotic systems together with hopping robots, wheeled mobile robots and multi-fingered robotic hands. This classification suggests that dynamic equations of motion of a space manipulator should have a similar form and be obtained using the same methods as for other nonholonomic systems. However, nonholonomic properties of space manipulators do not result from explicit kinematic constraints but occur due to the symmetry in system dynamics. Also, a space manipulator becomes underactuated when its base is free to rotate.

Underactuated systems are usually considered as a separate class of nonholonomic control systems, as in  [5], where a derivation of their control dynamics is based upon the Lagrange approach. There, the reduction procedure has to be applied to eliminate the Lagrange multipliers in order to obtain control dynamics. More details on modeling and control of underactuated systems can be found in  [6] and references there.

The motivation for this work is three folded. Firstly, for many constrained systems classified as nonholonomic, modeling methods are specialized  [4], [7], [8], [9], [10], [11]. The well known example is the dynamically equivalent manipulator (DEM) and virtual manipulator concepts that are used to derive dynamic models of space manipulators  [9], [10]. DEM is widely used for workspace analysis, trajectory planning, simulation and control of the space manipulators, see e.g.  [12]. This specialization in modeling is clearly visible for space systems, which are described as being nonholonomic but their dynamics is generated using the Lagrange–d’Alembert equations and the free-floating constraint, and other task-based constraints are added at the level of a controller design. A reduction procedure is suggested to develop their dynamic models in  [8], in the same way as for wheeled mobile robots. In  [11] Lagrange–d’Alembert equations for holonomic systems are applied to obtain dynamic models of free-floating space vehicles. However, they are still referred to as nonholonomic and it may cause some confusion. The same approach to modeling space vehicles, i.e. using Lagrange equations, and treated it as a routine part of a control design process can be also found in [13], [14], [15], [16], [17]. Control oriented modeling of other nonholonomic systems subjected to task-based constraints is discussed in  [18]. In the paper we advocate a different modeling approach, which yields a system constrained dynamics in a reduced state form, i.e. it is free of the constraint reaction forces and thus reduced in dimension since the state vector q is the only unknown.

Secondly, a control law designed either for tracking or stabilization, has to reflect a system behavior in the task space rather than in the joint space, and cope with the underactuation that manifests for free-floating space vehicles, i.e. its base is not actuated. In this regard, a modeling framework should account for underactuation, explicitly include switching between the spaces, and be able to incorporate various constraints on a system. The modeling method proposed herein satisfies these requirements.

Thirdly, separate calculations are usually performed to obtain a final control oriented dynamic model for space vehicles. For example, in  [15], [16] a space of allowable kinematic motions for a space vehicle has to be determined separately. This is due to the task-based constraints that are not merged into its dynamics. As a consequence, a preplanned trajectory for a space vehicle may lead to motions not allowed for it. The modeling method, which we propose merges all constraints, which can be presented by geometric or differential equations, into the constrained dynamics or a reference dynamics that enable planning motions that respect the constraints.

The paper presents a development of a modeling framework for constrained multibody systems, which unifies the approach to modeling constraint systems. The constraints on systems may be kinematic, design, control, task-based and origin from the conservation law. The underlying modeling method yields dynamic models in a reduced state form so they are ready for control applications. It means that preprocessing work for decoupling constraint reaction forces is eliminated at the level of a model generation. This makes the framework analytically and numerically effective, saves effort in the generation of constraint reactions free control dynamic models, and finally reduces numerical simulation effort.

The paper is organized as follows. In Section  2 constraint sources on mechanical systems are discussed from points of view of mechanics and control. Also, the underlying approach to modeling constrained systems is presented. Section  3 presents a control oriented dynamic modeling framework for constrained multibody systems. A specific example of a multi-constrained system, i.e. a free-floating space manipulator is detailed in Section  4. Its dynamic model as well as a tracking controller design is obtained based upon the framework. The paper closes with conclusions and the list of references.

Section snippets

Constrained dynamics for control applications

The sources of constraints put upon control systems are kinematic, conservation laws, task-based (let us limit them to configuration and first order constraints, i.e. specified by first order differential equations with respect to coordinates), design and control including underactuation. The first two constraint types are material and they are well known in classical mechanics and control  [7]. The other types are non-material, what means that a designer or a control engineer put them on a

A control oriented modeling framework

A control oriented dynamic modeling framework for constrained multibody systems is presented in Fig. 1. It is model-based and provides feedback loops with control inputs. It encompasses systems with material and non-material constraints up to first order, as specified in Section  2.

It offers the simplification of an automated process of the derivation of constrained dynamics, the reduction of the computation effort and verification of singularities. This is due to the underlying modeling method

Example—a free-floating space manipulator control

The selected example of a space manipulator illustrates advantages of the application of the modeling framework. The manipulator model is presented in Fig. 2. It is an underactuated system model with its base that is free to rotate. It is constrained by a task constraint, i.e. its end-effector is to move along a pre-specified trajectory, and by the conservation of its angular momentum. Usually underactuated systems are treated separately as special cases of nonholonomic systems  [5], [6]. Also,

Conclusions

The control oriented dynamic modeling framework for constrained multibody systems is presented in the paper. It can serve any multibody system subjected to holonomic or first order nonholonomic constraints, which may origin from modeling or control requirements. The framework application supports simplification of an automated process of a derivation of a constrained dynamics, helps reduction of computation effort and enables the verification of preplanned motion execution possibilities by

Acknowledgment

This work was supported by Grant No. UMO-2011/01/B/ST10/06966 from the National Science Centre, Poland.

Elżbieta M. Jarzębowska is currently with the Institute of Aeronautics and Applied Mechanics at the Power and Aeronautical Engineering Department, Warsaw University of Technology, Warsaw, Poland. She received the B.S., M.S., and Ph.D., D.Sc. degrees in Mechanical engineering, Control and mechanics of constrained systems, from the Warsaw University of Technology.

Her fields of research expertise and teaching include dynamics modeling and analysis of multibody systems, nonlinear control of

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Elżbieta M. Jarzębowska is currently with the Institute of Aeronautics and Applied Mechanics at the Power and Aeronautical Engineering Department, Warsaw University of Technology, Warsaw, Poland. She received the B.S., M.S., and Ph.D., D.Sc. degrees in Mechanical engineering, Control and mechanics of constrained systems, from the Warsaw University of Technology.

Her fields of research expertise and teaching include dynamics modeling and analysis of multibody systems, nonlinear control of multibody systems including nonholonomic, underactuated, UAV, and wheeled robotic systems, and geometric control theory.

She was involved in research projects for Automotive Research Centre and Engineering Research Centre for Reconfigurable Machining Systems at the University of Michigan, Ann Arbor, MI. Also, she gained valuable experience when worked for Ford Motor Company Research Laboratories, Dearborn, MI.

She is a member of ASME, IEEE, GAMM, IFToMM Technical Committee of Mechatronics, and InternationalSAR.

Her hobbies are psychology, swimming, yachting, and travels.

K. Pietrak is currently a Ph.D. candidate at the Institute of Power Engineering, Power and Aeronautical Engineering Department, Warsaw University of Technology, Warsaw, Poland. He received his M.S. degree in mechanical engineering from the Warsaw University of Technology. His field of research includes modeling of thermal systems, inverse problem theory and numerical methods. His hobbies are history, psychology and music.

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