Mathematical modelling and nonlinear control of a rear-wheel-drive vehicle by using the Newton-Euler equations. II. (English) Zbl 1535.70108
Summary: This is Part 2 of a two part series of works dealing with the mathematical modelling and nonlinear control of a rear-wheel-drive vehicle by using the Newton-Euler equations and the d’Alembert principle. The vehicle considered in Part 1 [the author, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 26, No. 6, 381–446 (2019; Zbl 1472.70053)] is subject to a set of holonomic and nonholonomic velocity constraints that are not independent. Thus, in Part 2, the Newton-Euler equations for constrained multibody systems are extended for the case where the velocity constraints may not be independent. The results indicate that in the case of independent velocity constraints the Lagrange multipliers are unique while in the case of dependent velocity constraints the Lagrange multipliers are not unique. In addition, two state space forms of the Newton-Euler equations for constrained multibody systems are derived and denoted by the abbreviations NE1 and NE2. State space form NE1 requires the derivation of the kinematic model of the system while state space form NE2 does not require the kinematic model. The vectors of Lagrange multipliers associated with state space forms NE1 and NE2 are denoted by \(\lambda_{\mathrm{NE}1}\) and \(\lambda_{\mathrm{NE}2}\), respectively. A method is proposed to practically compute \(\lambda_{\mathrm{NE}1}\), \(\lambda_{\mathrm{NE}2}\), by using the Moore-Penrose generalized inverse. The proposed method yields vectors of Lagrange multipliers \(\lambda_{\mathrm{NE}1}\), \(\lambda_{\mathrm{NE}2}\), that have minimum Euclidean norm and is applicable to cases of independent and not independent velocity constraints. The obtained expressions for \(\lambda_{\mathrm{NE}1}\), \(\lambda_{\mathrm{NE}2}\), do not have the same form and contain the Moore-Penrose generalized inverse of a different matrix term. By employing additional derivations it is shown that \(\lambda_{\mathrm{NE}1}=\lambda_{\mathrm{NE}2}\). Thus, Part 1 applies the results of Part 2 and employs state space form NE1 in order to derive the kinematic and dynamic models of the vehicle by using all the velocity constraints in their original form.
MSC:
| 70Q05 | Control of mechanical systems |
| 70E55 | Dynamics of multibody systems |
| 70F20 | Holonomic systems related to the dynamics of a system of particles |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
Keywords:
d’Alembert principle; generalized constraint force; nonholonomic/holonomic velocity constraint; reduced dynamic model; Lagrange multiplier; Moore-Penrose generalized inverseCitations:
Zbl 1472.70053References:
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