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.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

Ride Dynamics of a Flexible Multibody Model of an Urban Bus

2007 ◽  
Author(s):  
G. Georgiou ◽  
A. Badarlis ◽  
S. Natsiavas
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Accurate Information ◽  
Algebraic Equations ◽  
Multibody Model ◽  
Road Excitation ◽  
Urban Bus ◽  
Stiffness And Damping

Dynamic response of a large order mechanical model of an urban bus is investigated. The emphasis is first put on developing a quite complete model, which can be utilized in order to extract sufficiently reliable and accurate information related to its dynamics in a fast way. Since some of the components of the bus undergo large rigid body rotation, in addition to motion resulting from their deformability, a multibody dynamics framework is adopted. This implies that the resulting equations of motion appear in the form of a strongly nonlinear set of differential-algebraic equations, which are difficult to handle even numerically. In fact, the modeling becomes more involved because all the significant nonlinearities appearing in the interconnections of the structural components and especially in the front and rear suspension subsystems of the bus are taken into account. In order to alleviate some of these complexities, the number of degrees of freedom of each component, associated with its deformability, is reduced drastically by applying an appropriate coordinate condensation methodology. Finally, this model is employed and numerical results are obtained for motions resulting from typical road excitation. In particular, selected response quantities related to ride comfort are examined for characteristic combinations of the bus suspension stiffness and damping parameters.

scholarly journals Speed Oscillations of a Vehicle Rolling on a Wavy Road

2021 ◽  
Vol 11 (21) ◽  
pp. 10431
Author(s):  
Walter V. Wedig
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Travel Speed ◽  
Differential Algebraic Equations ◽  
Polar Coordinates ◽  
Algebraic Equations ◽  
Traffic Lights ◽  
Analytical Approximations ◽  
Vertical Vibrations ◽  
Nonlinear Equations Of Motion

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential–algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

Control Constraint of Underactuated Aerospace Systems

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Alessandro Fumagalli
Keyword(s):  
Feedforward Control ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Point Of View ◽  
Underactuated Systems ◽  
Theoretical Point ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Multibody Problems

This paper discusses the problem of control constraint realization applied to the design of maneuvers of complex underactuated systems modeled as multibody problems. Applications of interest in the area of aerospace engineering are presented and discussed. The tangent realization of the control constraint is discussed from a theoretical point of view and is used to determine feedforward control of realistic underactuated systems. The effectiveness of the computed feedforward input is subsequently verified by applying it to more detailed models of the problems, in the presence of disturbances and uncertainties in combination with feedback control. The problems are solved using a free general-purpose multibody software that writes the constrained dynamics of multifield problems formulated as differential-algebraic equations. The equations are integrated using unconditionally stable algorithms with tunable dissipation. The essential extension to the multibody code consisted of the addition of the capability to write arbitrary constraint equations and apply the corresponding reaction multipliers to arbitrary equations of motion. The modeling capabilities of the formulation could be exploited without any undue restriction on the modeling requirements.

Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems

2007 ◽  
Vol 3 (1) ◽  
Author(s):  
Olivier A. Bauchau ◽  
André Laulusa
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constraint Violation ◽  
New Techniques ◽  
Index Reduction ◽  
Reduction Techniques ◽  
Constraint Enforcement ◽  
The Relationship

A hallmark of multibody dynamics is that most formulations involve a number of constraints. Typically, when redundant generalized coordinates are used, equations of motion are simpler to derive but constraint equations are present. Approaches to dealing with high index differential algebraic equations, based on index reduction techniques, are reviewed and discussed. Constraint violation stabilization techniques that have been developed to control constraint drift are also reviewed. These techniques are used in conjunction with algorithms that do not exactly enforce the constraints. Control theory forms the basis for a number of these methods. Penalty based techniques have also been developed, but the augmented Lagrangian formulation presents a more solid theoretical foundation. In contrast to constraint violation stabilization techniques, constraint violation elimination techniques enforce exact satisfaction of the constraints, at least to machine accuracy. Finally, as the finite element method has gained popularity for the solution of multibody systems, new techniques for the enforcement of constraints have been developed in that framework. The goal of this paper is to review the features of these methods, assess their accuracy and efficiency, underline the relationship among the methods, and recommend approaches that seem to perform better than others.

Control Constraint Realization Applied to Underactuated Aerospace Systems

2011 ◽  
Author(s):  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Alessandro Fumagalli
Keyword(s):  
Position Control ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Point Of View ◽  
Theoretical Point ◽  
Algebraic Equations ◽  
Control Constraint

This paper discusses the problem of control constraint realization applied to the design of maneuvers of complex under-actuated systems modeled as multibody problems. Applications of interest in the area of aerospace engineering are presented and discussed. The tangent realization of the control constraint is discussed from a theoretical point of view and used to determine feedforward control of realistic under-actuated systems. The effectiveness of the computed feedforward input is subsequently verified by applying it to more detailed models of the problems, in presence of disturbances and uncertainties in combination with feedback control. The proposed applications consist in the position control of a complex closed chain mechanism representative of a robotic system, the control of a simplified model of a canard and a conventional air vehicle in the vertical plane, and the angular velocity control of a wind-turbine. In the aeromechanics examples, the tangent realization of the control relies on the availability of the Jacobian matrix of an aeroelastic model. All problems are solved using a free general-purpose multibody software that writes the constrained dynamics of multi-field problems in form of Differential-Algebraic Equations (DAE). The equations are integrated using A/L-stable algorithms. The essential extension to the multibody code consisted in the addition of the capability to write arbitrary constraint equations and apply the corresponding reaction multipliers to arbitrary equations of motion. This allowed to exploit the modeling capabilities of the formulation without any undue restriction on the modeling requirements.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sa¨nger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Secondly, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sänger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

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