scholarly journals Tracking control for underactuated non-minimum phase multibody systems

2021 ◽  
Author(s):  
Thomas Berger ◽  
Svenja Drücker ◽  
Lukas Lanza ◽  
Timo Reis ◽  
Robert Seifried
Keyword(s):  
Tracking Control ◽  
Multibody Systems ◽  
Control Design ◽  
Differential Algebraic Equations ◽  
Minimum Phase ◽  
Algebraic Equations ◽  
Internal Dynamics ◽  
Kinematic Loops ◽  
Servo Constraints ◽  
Kinematic Loop

AbstractWe consider tracking control for multibody systems which are modeled using holonomic and non-holonomic constraints. Furthermore, the systems may be underactuated and contain kinematic loops and are thus described by a set of differential-algebraic equations that cannot be reformulated as ordinary differential equations in general. We propose a control strategy which combines a feedforward controller based on the servo-constraints approach with a feedback controller based on a recent funnel control design. As an important tool for both approaches, we present a new procedure to derive the internal dynamics of a multibody system. Furthermore, we present a feasible set of coordinates for the internal dynamics avoiding the effort involved with the computation of the Byrnes–Isidori form. The control design is demonstrated by a simulation for a nonlinear non-minimum phase multi-input, multi-output robotic manipulator with kinematic loop.

Servo-Constraints for Control of Flexible Multibody Systems With Contact

2013 ◽  
Author(s):  
Robert Seifried ◽  
Markus Burkhardt
Keyword(s):  
Numerical Solution ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Inverse Model ◽  
Flexible Multibody Systems ◽  
Algebraic Equations ◽  
Internal Dynamics ◽  
End Effector ◽  
Flexible Multibody ◽  
Servo Constraints

This paper presents inversion based feedforward control design for flexible multibody systems with kinematic loops and end-effector contact. The inverse model provides for a given desired output trajectories, e.g. end-effector point and contact force, the required control inputs for exact output reproduction. A very appealing and efficient model inversion approach for such multibody systems is the use of so-called servo-constraints. These can be seen as an extension of classical mechanical constraints and yield a set of differential-algebraic equations. This allows an efficient numerical solution without burdensome symbolic manipulations. In addition, the use of servo-constraints allows the straight-forward treatment of flexible multibody systems with various topologies. The arising set of differential-algebraic equations describes the inverse model. The inverse model might be purely algebraic or include a dynamical part, which is called internal dynamics in nonlinear control theory. For its numerical solution it is advisable to transform the set of differential-algebraic equations to its underlying set of ordinary differential equations. The solution method for this internal dynamics depends then on its stability. For systems with unstable internal dynamics, as considered in this paper, a solution can be computed from a boundary-value problem. The efficiency of this approach is demonstrated for a flexible multibody system with a kinematic loop and a closed end-effector contact.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Control Constraint Realization for Multibody Systems

2009 ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic under-actuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that Differential-Algebraic Equations (DAE) need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

The Dynamic Analysis of Multibody Systems with Uncertain Parameters Using Interval Method

2012 ◽  
Vol 152-154 ◽  
pp. 1555-1561 ◽  
Author(s):  
Jing Lai Wu ◽  
Yun Qing Zhang
Keyword(s):  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Uncertain Parameters ◽  
Algebraic Equations ◽  
Scanning Method ◽  
Inclusion Function ◽  
Bounded Interval ◽  
Interval Variables ◽  
Multibody Dynamic ◽  
Computational Aspects

The theoretical and computational aspects of interval methodology based on Chebyshev polynomials for modeling multibody dynamic systems in the presence of parametric uncertainties are proposed, where the uncertain parameters are modeled by uncertain-but-bounded interval variables which only need the bounds of uncertain parameters, not necessarily knowing the probabilistic distribution. The Chebyshev inclusion function which employs the truncated Chevbyshev series expansion to approximate the original function is proposed. Based on Chebyshev inclusion function, the algorithm for solving the nonlinear equations with interval parameters is proposed. Combining the HHT-I3 method, this algorithm is used to calculate the multibody systems dynamic response which is governed by differential algebraic equations (DAEs). A numerical example that is a slider-crank with uncertain parameters is presented, which shows that the novel methodology can control the overestimation effectively and is computationally faster than the scanning method.

Integrated Design and Minimum Sensitivity Design of Controlled Multibody Systems

2005 ◽  
Author(s):  
Jason Carrigan ◽  
Atul Kelkar ◽  
Prakash Krishnaswami
Keyword(s):  
Design Process ◽  
Multibody Systems ◽  
System Analysis ◽  
Mechanical Design ◽  
Integrated Design ◽  
Solution Process ◽  
Differential Algebraic Equations ◽  
Element Formulation ◽  
Algebraic Equations ◽  
System Variables

The design methods that have been traditionally used for controlled mechanical systems suffer from three major drawbacks. First, the design process is generally sequential, with the mechanical design being done first and frozen before the control system design is done. Secondly, the design is usually tuned to improve performance only without worrying about the sensitivity of the system’s performance to small variations in the system parameters. Third, there is a lack of systematic guidance for traversing the design space and arriving at a high quality design. In this work, we propose a design approach that addresses all three of these concerns. This approach first extends the constrained multi-element formulation for multibody systems to include a generic controller model. This gives the basic capability to simulate controlled multibody systems in a general way by numerically solving a set of differential-algebraic equations (DAE’s). A direct differentiation technique is then applied to the unified mathematical model to obtain a set of DAE’s in the sensitivities of the system variables. This is then used to compute the sensitivity of any performance function of interest. The system analysis and sensitivity analysis are then treated as inputs to a suitable nonlinear programming problem (NLP). The NLP serves as a vehicle to unify mechanical system and control criteria in the design process, and to incorporate sensitivity considerations along with performance considerations. The NLP also provides the means for automating the solution process through the use of optimization algorithms. Two representative example, including an industrial problem, are solved using this method. The results clearly show that the methodology is feasible and leads to a vast improvement in the quality of the final design, whatever the design considerations may be.

Elimination of Redundant Cut Joint Constraints for Multibody System Models

2003 ◽  
Vol 126 (3) ◽  
pp. 488-494 ◽  
Author(s):  
A. Mu¨ller
Keyword(s):  
Multibody Systems ◽  
Computational Effort ◽  
Basis Matrix ◽  
Constraint Equations ◽  
Kinematic Loops ◽  
Joint Constraints ◽  
Rigid Multibody Systems ◽  
Kinematic Loop ◽  
Rigid Multibody

The problem of dependent cut joint constraints for kinematic loops in rigid multibody systems is addressed. The constraints are reduced taking into account the subalgebra generated by the screw system of the kinematic loop. The elimination of dependent constraint equations is based on constructing a basis matrix of the screw algebra generated by loop’s screw system. This matrix is configuration independent and thus always valid. The determination of the sufficient constraints is achieved with a SVD or QR decomposition of this matrix. Unlike all other proposed approaches the presented method is singularity consistent because it is not the Jacobian which is decomposed, but instead a basis matrix for the loop algebra. Since this basis is obtained after a finite number of cross products the computational effort is negligible. Furthermore, because the elimination process is only necessary once in advance of the integration/simulation process, it proved valuable even if it does not remove all dependent constraints, as for paradoxical mechanisms.

A Novel Approach to Real-Time Flexible Multibody Simulation: Sub-System Global Modal Parameterization

2011 ◽  
Author(s):  
Frank Naets ◽  
Gert H. K. Heirman ◽  
Wim Desmet
Keyword(s):  
Model Reduction ◽  
Real Time ◽  
Relative Motion ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Flexible Multibody Systems ◽  
System Level ◽  
Algebraic Equations ◽  
Global Modal Parameterization ◽  
Flexible Multibody

This paper introduces a novel model reduction technique, namely Sub-System Global Modal Parameterization (SS-GMP), for real-time simulation of flexible multibody systems. In the past, other system-level model reduction techniques have been proposed for this purpose, but these were limited in applicability due to the large storage requirements for systems with many rigid degrees-of-freedom (DOFs). However, in the SS-GMP approach, the motion of a mechanism is split up into a global motion and a relative motion of the (sub-)system. The relative motion is then reduced according to the Global Modal Parameterization, which is a model reduction procedure suitable for closed chain flexible multibody systems. In combination with suitable explicit solvers, the SS-GMP approach enables (hard) real-time simulations due to the strong reduction in the number of DOFs and the conversion of a system of differential-algebraic equations into a system of ordinary differential equations. The proposed approach is validated numerically with a quarter-car model. This fully flexible mechanism is simulated faster than real-time on a regular PC with the SS-GMP approach while providing accurate results.

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