Numerical Integration of Underactuated Mechanical Systems Subjected to Mixed Holonomic and Servo Constraints

Underactuated mechanical systems have fewer control inputs than degrees of freedom. The specified in time outputs, equal in number to the number of inputs, lead to servo-constraints on the system. The servo-constraint problem is then a specific inverse simulation problem in which an input control strategy (feedforward control) that forces an underactuated system to complete the partly specified motion is determined. Since mechanical systems may be “underactuated” in several ways, and the control forces may be arbitrarily oriented with respect to the servo-constraint manifold, this is, in general, a challenging task. The use of servo-constraints in the inverse dynamics analysis of underactuated systems is discussed here with an emphasis on diverse possible ways of the constraint realization. A formulation of the servo-constraint problem in configuration coordinates is compared with a setting in which the actuated coordinates are replaced with the outputs. The governing equations can then be set either as ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). The existence and nonexistence of an explicit solution to the servo-constraint problem is further discussed, related to so-called flat systems (with no internal dynamics) and nonflat systems (with internal dynamics). In case of nonflat systems, of paramount importance is stability of the internal dynamics. Simple case studies are reported to illustrate the discussion and formulations.

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Open-Loop Control of Underactuated Mechanical Systems Using Servo-Constraints: Analysis and Some Examples

Applications of Differential-Algebraic Equations: Examples and Benchmarks - Differential-Algebraic Equations Forum ◽

10.1007/11221_2018_4 ◽

2018 ◽

pp. 81-122

Author(s):

Svenja Otto ◽

Robert Seifried

Keyword(s):

Mechanical Systems ◽

Open Loop ◽

Open Loop Control ◽

Underactuated Mechanical Systems ◽

Loop Control ◽

Servo Constraints

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Stabilization of Internal Dynamics of Underactuated Systems by Periodic Servo-Constraints

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455417400041 ◽

2017 ◽

Vol 17 (05) ◽

pp. 1740004 ◽

Cited By ~ 4

Author(s):

László Bencsik ◽

László L. Kovács ◽

Ambrus Zelei

Keyword(s):

Stability Analysis ◽

Dynamic Behavior ◽

Multibody Systems ◽

Degrees Of Freedom ◽

Mechanical Systems ◽

Underactuated Mechanical Systems ◽

Algorithmic Approach ◽

Servo Constraints ◽

And Control ◽

Dependent Coordinates

The model-based motion control of underactuated, multiple degree-of-freedom, complex multibody systems is in focus. Underactuated mechanical systems possess less number of independent control inputs than degrees-of-freedom. The main difficulty in their control is caused by the dynamics of the uncontrolled part of the system. The complexity of multibody systems makes the dynamical and control formulation difficult. The direct application of traditional control techniques available in the literature can lead to unstable dynamic behavior in many cases. In order to avoid instability, these general methods are usually adapted for specific problems in an intuitive way. Here, we present a direct, more algorithmic approach, and propose the use of periodic servo-constraints to overcome stability problems and enhance the dynamic behavior. An exact, stability analysis-based method is also proposed for tuning the control parameters. A stability analysis procedure is developed which is directly applicable for investigating the dynamics of mechanical systems described by dependent coordinates and mathematically formulated as a set of algebraic differential equations.

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Observer-based robust integral of the sign of the error control of class I of underactuated mechanical systems: Theory and real-time experiments

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211031396 ◽

2021 ◽

pp. 014233122110313

Author(s):

Afef Hfaiedh ◽

Ahmed Chemori ◽

Afef Abdelkrim

Keyword(s):

Real Time ◽

Error Control ◽

Degrees Of Freedom ◽

Sliding Mode ◽

Mechanical Systems ◽

Class I ◽

Control Input ◽

Underactuated Mechanical Systems ◽

Control Scheme ◽

And Performance

In this paper, the control problem of a class I of underactuated mechanical systems (UMSs) is addressed. The considered class includes nonlinear UMSs with two degrees of freedom and one control input. Firstly, we propose the design of a robust integral of the sign of the error (RISE) control law, adequate for this special class. Based on a change of coordinates, the dynamics is transformed into a strict-feedback (SF) form. A Lyapunov-based technique is then employed to prove the asymptotic stability of the resulting closed-loop system. Numerical simulation results show the robustness and performance of the original RISE toward parametric uncertainties and disturbance rejection. A comparative study with a conventional sliding mode control reveals a significant robustness improvement with the proposed original RISE controller. However, in real-time experiments, the amplification of the measurement noise is a major problem. It has an impact on the behaviour of the motor and reduces the performance of the system. To deal with this issue, we propose to estimate the velocity using the robust Levant differentiator instead of the numerical derivative. Real-time experiments were performed on the testbed of the inertia wheel inverted pendulum to demonstrate the relevance of the proposed observer-based RISE control scheme. The obtained real-time experimental results and the obtained evaluation indices show clearly a better performance of the proposed observer-based RISE approach compared to the sliding mode and the original RISE controllers.

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