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Multibody modeling and nonlinear control of the pantograph/catenary system

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Abstract

In this paper, a closed-chain multibody model of a pantograph/catenary system is developed and used for the optimal design of a nonlinear controller based on an open-loop control architecture. The goal of the nonlinear controller is the reduction of the contact force arising from the pantograph/catenary interaction and, at the same time, the suppression of the mechanical vibrations of the pantograph mechanism. The analytical formulation employed in this paper for describing the nonlinear dynamics of the pantograph/catenary multibody system considers a Lagrangian approach and is based on a redundant set of generalized coordinates. The contact forces generated by the pantograph/catenary interaction are modeled in this work employing an elastic force element collocated between the pantograph pan-head and a moving support. The external support follows a prescribed motion law that simulates the periodic deployment of the catenary system. On the other hand, in this investigation, the algebraic constraints arising from the closed-loop topology of the pantograph multibody system are enforced employing a method based on the Udwadia–Kalaba equations recently developed in the field of analytical dynamics. Furthermore, the problem of the determination of an effective feedforward controller for reducing the pantograph/catenary contact force is formulated in this work as a nonlinear optimal control problem. For this purpose, the solution of the control optimization problem is carried out by using an adjoint-based computational procedure. Numerical simulations demonstrate the effectiveness of the nonlinear controller obtained in this investigation for the pantograph/catenary multibody system.

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Author contributions

This research paper was principally developed by Carmine Maria Pappalardo. A great support was provided by Marco Claudio De Simone. The detailed review carried out by Domenico Guida considerably improved the quality of the work.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, Salerno, Italy

    Carmine M. Pappalardo, Marco C. De Simone & Domenico Guida

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  1. Carmine M. Pappalardo
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  3. Domenico Guida
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Appendices

Appendix A

In this appendix, the mathematical descriptions of the geometric configurations of the components that form the pantograph multibody model are reported.

1.1 A.1 Crank geometric configuration

$$\begin{aligned} {{\mathbf{A}}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\cos ({\theta _2}(t))}&{}\quad { - \sin ({\theta _2}(t))}&{}\quad 0\\ {\sin ({\theta _2}(t))}&{}\quad {\cos ({\theta _2}(t))}&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}} \right] \end{aligned}$$
(94)
$$\begin{aligned} {{\mathbf{R}}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + {L_2}\cos ({\theta _2}(t))}\\ {2{L_1}\sin ({\theta _1}(t)) + {L_2}\sin ({\theta _2}(t))}\\ 0 \end{array}} \right] \end{aligned}$$
(95)
$$\begin{aligned} {{\mathbf{r}}_2}({P_2},t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + \left( {{L_2} + {{{\bar{x}}}_2}({P_2})} \right) \cos ({\theta _2}(t))}\\ {2{L_1}\sin ({\theta _1}(t)) + \left( {{L_2} + {{{\bar{x}}}_2}({P_2})} \right) \sin ({\theta _2}(t))}\\ 0 \end{array}} \right] \end{aligned}$$
(96)
$$\begin{aligned} {{\varvec{{\bar{\omega }} }}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ {{{{\dot{\theta }} }_2}(t)} \end{array}} \right] \end{aligned}$$
(97)
$$\begin{aligned} {{\mathbf{J}}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - {L_2}\sin ({\theta _2}(t))}&{}\quad 0&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {{L_2}\cos ({\theta _2}(t))}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(98)
$$\begin{aligned} {{\mathbf{L}}_2}({P_2},t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - \left( {{L_2} + {{{\bar{x}}}_2}({P_2})} \right) \sin ({\theta _2}(t))}&{}\quad 0&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {\left( {{L_2} + {{{\bar{x}}}_2}({P_2})} \right) \cos ({\theta _2}(t))}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(99)
$$\begin{aligned} {{{{\bar{\varvec{\Omega }}} }}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0 \end{array}} \right] . \end{aligned}$$
(100)

1.2 A.2 Lower arm geometric configuration

$$\begin{aligned} {{\mathbf{A}}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\cos ({\theta _3}(t))}&{}\quad { - \sin ({\theta _3}(t))}&{}\quad 0\\ {\sin ({\theta _3}(t))}&{}\quad {\cos ({\theta _3}(t))}&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}} \right] \end{aligned}$$
(101)
$$\begin{aligned} {{\mathbf{R}}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + 2{L_2}\cos ({\theta _2}(t)) + {L_3}\cos ({\theta _3}(t))}\\ {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t)) + {L_3}\sin ({\theta _3}(t))}\\ 0 \end{array}} \right] \end{aligned}$$
(102)
$$\begin{aligned} {{\mathbf{r}}_3}({P_3},t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + 2{L_2}\cos ({\theta _2}(t)) + \left( {{L_3} + {{{\bar{x}}}_3}({P_3})} \right) \cos ({\theta _3}(t))}\\ {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t)) + \left( {{L_3} + {{{\bar{x}}}_3}({P_3})} \right) \sin ({\theta _3}(t))}\\ 0 \end{array}} \right] \end{aligned}$$
(103)
$$\begin{aligned} {{\varvec{{\bar{\omega }} }}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ {{{{\dot{\theta }} }_3}(t)} \end{array}} \right] \end{aligned}$$
(104)
$$\begin{aligned} {{\mathbf{J}}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - 2{L_2}\sin ({\theta _2}(t))}&{}\quad { - {L_3}\sin ({\theta _3}(t))}&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {2{L_2}\cos ({\theta _2}(t))}&{}\quad {{L_3}\cos ({\theta _3}(t))}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(105)
$$\begin{aligned} {{\mathbf{L}}_3}({P_3},t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - 2{L_2}\sin ({\theta _2}(t))}&{}\quad { - \left( {{L_3} + {{{\bar{x}}}_3}({P_3})} \right) \sin ({\theta _3}(t))}&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {2{L_2}\cos ({\theta _2}(t))}&{}\quad {\left( {{L_3} + {{{\bar{x}}}_3}({P_3})} \right) \cos ({\theta _3}(t))}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(106)
$$\begin{aligned} {{{{\bar{\varvec{\Omega }}} }}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1&{}\quad 0 \end{array}} \right] . \end{aligned}$$
(107)

1.3 A.3 Upper arm geometric configuration

$$\begin{aligned} {{\mathbf{A}}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\cos ({\theta _2}(t) - \beta )}&{}\quad { - \sin ({\theta _2}(t) - \beta )}&{}\quad 0\\ {\sin ({\theta _2}(t) - \beta )}&{}\quad {\cos ({\theta _2}(t) - \beta )}&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}} \right] \end{aligned}$$
(108)
$$\begin{aligned} {{\mathbf{R}}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + 2{L_2}\cos ({\theta _2}(t)) + {L_4}\cos ({\theta _2}(t) - \beta )}\\ {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t)) + {L_4}\sin ({\theta _2}(t) - \beta )}\\ 0 \end{array}} \right] \end{aligned}$$
(109)
$$\begin{aligned} {{\mathbf{r}}_4}({P_4},t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + 2{L_2}\cos ({\theta _2}(t)) + \left( {{L_4} + {{{\bar{x}}}_4}({P_4})} \right) \cos ({\theta _2}(t) - \beta )}\\ {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t)) + \left( {{L_4} + {{{\bar{x}}}_4}({P_4})} \right) \sin ({\theta _2}(t) - \beta )}\\ 0 \end{array}} \right] \end{aligned}$$
(110)
$$\begin{aligned} {{\varvec{{\bar{\omega }} }}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ {{{{\dot{\theta }} }_2}(t)} \end{array}} \right] \end{aligned}$$
(111)
$$\begin{aligned} {{\mathbf{J}}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - 2{L_2}\sin ({\theta _2}(t)) - {L_4}\sin ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {2{L_2}\cos ({\theta _2}(t)) + {L_4}\cos ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(112)
$$\begin{aligned} {{\mathbf{L}}_4}({P_4},t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - 2{L_2}\sin ({\theta _2}(t)) - \left( {{L_4} + {{{\bar{x}}}_4}({P_4})} \right) \sin ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {2{L_2}\cos ({\theta _2}(t)) + \left( {{L_4} + {{{\bar{x}}}_4}({P_4})} \right) \cos ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(113)
$$\begin{aligned} {{{{\bar{\varvec{\Omega }}} }}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0 \end{array}} \right] . \end{aligned}$$
(114)

1.4 A.4 Pan-head geometric configuration

$$\begin{aligned} {{\mathbf{u}}_5}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ {x(t)}\\ 0 \end{array}} \right] \end{aligned}$$
(115)
$$\begin{aligned} {{\mathbf{R}}_5}(t)= & {} \left[ {\begin{array}{*{20}{l}} {2{L_1}\cos ({\theta _1}(t)) + 2{L_2}\cos ({\theta _2}(t)) + 2{L_4}\cos ({\theta _2}(t) - \beta )}\\ {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t)) + 2{L_4}\sin ({\theta _2}(t) - \beta ) + x(t)}\\ 0 \end{array}} \right] \end{aligned}$$
(116)
$$\begin{aligned} {{\mathbf{J}}_5}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}\sin ({\theta _1}(t))}&{}\quad { - 2{L_2}\sin ({\theta _2}(t)) - {L_4}\sin ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 0\\ {2{L_1}\cos ({\theta _1}(t))}&{}\quad {2{L_2}\cos ({\theta _2}(t)) + {L_4}\cos ({\theta _2}(t) - \beta )}&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] . \end{aligned}$$
(117)

Appendix B

In this appendix, the mathematical descriptions of the inertia terms and of the generalized forces of the components that form the pantograph multibody model are reported.

1.1 B.1 Crank inertia terms and generalized forces

$$\begin{aligned} {{\mathbf{M}}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} {4{m_2}L_1^2}&{}\quad {2{L_1}{L_2}{m_2}\cos ({\theta _1}(t) - {\theta _2}(t))}&{}\quad 0&{}\quad 0\\ {2{L_1}{L_2}{m_2}\cos ({\theta _1}(t) - {\theta _2}(t))}&{}\quad {\frac{4}{3}{m_2}L_2^2}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(118)
$$\begin{aligned} {{\mathbf{C}}_2}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}{\begin{array}{*{20}{l}} {2{L_1}{L_2}{m_2}{{{\dot{\theta }} }_2}(t)}\\ {\cdot \sin ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad 0&{}\quad 0\\ {\begin{array}{*{20}{l}} { - 2{L_1}{L_2}{m_2}{{{\dot{\theta }} }_1}(t)}\\ {\cdot \sin ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(119)
$$\begin{aligned} {{\mathbf{Q}}_{{\mathrm{v}},2}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{L_1}{L_2}{m_2}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _2^2(t)}\\ {2{L_1}{L_2}{m_2}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _1^2(t)}\\ 0\\ 0 \end{array}} \right] \end{aligned}$$
(120)
$$\begin{aligned} {{\mathbf{Q}}_{g,2}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{m_2}g{L_1}\cos ({\theta _1}(t))}\\ { - {m_2}g{L_2}\cos ({\theta _2}(t))}\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$
(121)

1.2 B.2 Lower arm inertia terms and generalized forces

$$\begin{aligned} {{\mathbf{M}}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} {4{m_3}L_1^2}&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_3}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {2{L_1}{L_3}{m_3}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _3}(t))} \end{array}}&{}\quad 0\\ {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_3}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad {4{m_3}L_2^2}&{}\quad {\begin{array}{*{20}{l}} {2{L_2}{L_3}{m_3}}\\ {\quad \cdot \cos ({\theta _2}(t) - {\theta _3}(t))} \end{array}}&{}\quad 0\\ {\begin{array}{*{20}{l}} {2{L_1}{L_3}{m_3}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _3}(t))} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {2{L_2}{L_3}{m_3}}\\ {\quad \cdot \cos ({\theta _2}(t) - {\theta _3}(t))} \end{array}}&{}\quad {\frac{4}{3}{m_3}L_3^2}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(122)
$$\begin{aligned} {{\mathbf{C}}_3}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_3}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {2{L_1}{L_3}{m_3}{{{\dot{\theta }} }_3}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _3}(t))} \end{array}}&{}\quad 0\\ {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_3}{{{\dot{\theta }} }_1}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t))} \end{array}}&{}\quad 0&{}\quad {\begin{array}{*{20}{l}} {2{L_2}{L_3}{m_3}{{{\dot{\theta }} }_3}(t)}\\ {\quad \cdot \sin ({\theta _2}(t) - {\theta _3}(t))} \end{array}}&{}\quad 0\\ {\begin{array}{*{20}{l}} { - 2{L_1}{L_3}{m_3}{{{\dot{\theta }} }_1}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _3}(t))} \end{array}}&{}\quad {\begin{array}{*{20}{l}} { - 2{L_2}{L_3}{m_3}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _2}(t) - {\theta _3}(t))} \end{array}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(123)
$$\begin{aligned} {{\mathbf{Q}}_{{\mathrm{v}},3}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_3}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _2^2(t) - 2{L_1}{L_3}{m_3}\sin ({\theta _1}(t) - {\theta _3}(t)){\dot{\theta }} _3^2(t)}\\ {4{L_1}{L_2}{m_3}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _1^2(t) - 2{L_2}{L_3}{m_3}\sin ({\theta _2}(t) - {\theta _3}(t)){\dot{\theta }} _3^2(t)}\\ {2{L_1}{L_3}{m_3}\sin ({\theta _1}(t) - {\theta _3}(t)){\dot{\theta }} _1^2(t) + 2{L_2}{L_3}{m_3}\sin ({\theta _2}(t) - {\theta _3}(t)){\dot{\theta }} _2^2(t)}\\ 0 \end{array}} \right] \end{aligned}$$
(124)
$$\begin{aligned} {{\mathbf{Q}}_{g,3}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{m_3}g{L_1}\cos ({\theta _1}(t))}\\ { - 2{m_3}g{L_2}\cos ({\theta _2}(t))}\\ { - {m_3}g{L_3}\cos ({\theta _3}(t))}\\ 0 \end{array}} \right] . \end{aligned}$$
(125)

1.3 B.3 Upper arm inertia terms and generalized forces

$$\begin{aligned} {{\mathbf{M}}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} {4{m_4}L_1^2}&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t)) }\\ { + 2{L_1}{L_4}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad 0\\ {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t)) }\\ { + 2{L_1}{L_4}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {\frac{4}{3}{m_4}L_4^2 + 4{m_4}L_2^2 }\\ { + 4{m_4}{L_2}{L_4}\cos (\beta )} \end{array}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(126)
$$\begin{aligned} {{\mathbf{C}}_4}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_4}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t)) }\\ {+2{L_1}{L_4}{m_4}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad 0\\ {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_4}{{{\dot{\theta }} }_1}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t)) }\\ { - 2{L_1}{L_4}{m_4}{{{\dot{\theta }} }_1}(t)}\\ {\sin ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \end{aligned}$$
(127)
$$\begin{aligned} {{\mathbf{Q}}_{{\mathrm{v}},4}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_4}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _2^2(t) }\\ { - 2{L_1}{L_4}{m_4}\sin ({\theta _1}(t) - {\theta _2}(t) + \beta ){\dot{\theta }} _2^2(t)} \end{array}}\\ {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_4}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _1^2(t) }\\ { + 2{L_1}{L_4}{m_4}\sin ({\theta _1}(t) - {\theta _2}(t) + \beta ){\dot{\theta }} _1^2(t)} \end{array}}\\ 0\\ 0 \end{array}} \right] \end{aligned}$$
(128)
$$\begin{aligned} {{\mathbf{Q}}_{g,4}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{m_4}g{L_1}\cos ({\theta _1}(t))}\\ { - 2{m_4}g{L_2}\cos ({\theta _2}(t)) - {m_4}g{L_4}\cos (\beta - {\theta _2}(t))}\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$
(129)

1.4 B.4 Pan-head inertia terms and generalized forces

$$\begin{aligned} {{\mathbf{M}}_5}(t)= & {} \left[ {\begin{array}{*{20}{l}} {4{m_5}L_1^2}&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_5}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t)) }\\ { + 4{L_1}{L_4}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad {\begin{array}{*{20}{l}} {2{L_1}{m_5}}\\ {\quad \cdot \cos ({\theta _1}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_5}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t)) }\\ { + 4{L_1}{L_4}{m_4}}\\ {\quad \cdot \cos ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {4{m_5}L_2^2 + 4{m_5}L_4^2 }\\ { + 8{m_5}{L_2}{L_4}\cos (\beta )} \end{array}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {\begin{array}{*{20}{l}} {2{m_5}{L_2}}\\ {\quad \cdot \cos ({\theta _2}(t)) }\\ { + 2{m_5}{L_4}}\\ {\quad \cdot \cos (\beta - {\theta _2}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {2{L_1}{m_5}}\\ {\quad \cdot \cos ({\theta _1}(t))} \end{array}}&{}\quad {\begin{array}{*{20}{l}} {2{m_5}{L_2}}\\ {\quad \cdot \cos ({\theta _2}(t)) }\\ { + 2{m_5}{L_4}}\\ {\quad \cdot \cos (\beta - {\theta _2}(t))} \end{array}}&\quad 0&\quad {{m_5}} \end{array}} \right] \end{aligned}$$
(130)
$$\begin{aligned} {{\mathbf{C}}_5}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0&{}\quad {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_5}{{{\dot{\theta }} }_2}(t)}\\ {\cdot \sin ({\theta _1}(t) - {\theta _2}(t)) }\\ { + 4{L_1}{L_4}{m_5}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad 0\\ {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_5}{{{\dot{\theta }} }_1}(t)}\\ {\cdot \sin ({\theta _1}(t) - {\theta _2}(t)) }\\ { - 4{L_1}{L_4}{m_5}{{{\dot{\theta }} }_1}(t)}\\ {\quad \quad \cdot \sin ({\theta _1}(t) - {\theta _2}(t) + \beta )} \end{array}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ { - 2{L_1}{m_5}\sin ({\theta _1}(t)){{{\dot{\theta }} }_1}(t)}&{}\quad {\begin{array}{*{20}{l}} {2{L_4}{m_5}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin (\beta - {\theta _2}(t)) }\\ { - 2{L_2}{m_5}{{{\dot{\theta }} }_2}(t)}\\ {\quad \cdot \sin ({\theta _2}(t))} \end{array}}&\quad 0&\quad 0 \end{array}} \right] \end{aligned}$$
(131)
$$\begin{aligned} {{\mathbf{Q}}_{{\mathrm{v}},5}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} { - 4{L_1}{L_2}{m_5}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _2^2(t) }\\ { - 4{L_1}{L_4}{m_5}\sin ({\theta _1}(t) - {\theta _2}(t) + \beta ){\dot{\theta }} _2^2(t)} \end{array}}\\ {\begin{array}{*{20}{l}} {4{L_1}{L_2}{m_5}\sin ({\theta _1}(t) - {\theta _2}(t)){\dot{\theta }} _1^2(t) }\\ { + 4{L_1}{L_4}{m_5}\sin ({\theta _1}(t) - {\theta _2}(t) + \beta ){\dot{\theta }} _1^2(t)} \end{array}}\\ 0\\ {\begin{array}{*{20}{l}} {2{L_1}{m_5}\sin ({\theta _1}(t)){\dot{\theta }} _1^2(t) }\\ { - 2{L_4}{m_5}\sin (\beta - {\theta _2}(t)){\dot{\theta }} _2^2(t) }\\ { + 2{L_2}{m_5}\sin ({\theta _2}(t)){\dot{\theta }} _2^2(t)} \end{array}} \end{array}} \right] \end{aligned}$$
(132)
$$\begin{aligned} {{\mathbf{Q}}_{g,5}}(t)= & {} \left[ {\begin{array}{*{20}{l}} { - 2{m_5}g{L_1}\cos ({\theta _1}(t))}\\ { - 2{m_5}g{L_2}\cos ({\theta _2}(t)) - 2{m_5}g{L_4}\cos (\beta - {\theta _2}(t))}\\ 0\\ { - {m_5}g} \end{array}} \right] \end{aligned}$$
(133)

1.5 B.5 Pneumatic actuator generalized force

$$\begin{aligned} {{\mathbf{Q}}_{k,1}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\left( {{p_1} - {k_1}\left( {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t))} \right) } \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { - {k_1}\left( {\left( {{L_3} + {L_E}} \right) \sin ({\theta _3}(t)) - {H_E}} \right) 2{L_1}\cos ({\theta _1}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {\left( {{p_1} - {k_1}\left( {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t))} \right) } \right) 2{L_2}\cos ({\theta _2}(t)) }\\ { - {k_1}\left( {\left( {{L_3} + {L_E}} \right) \sin ({\theta _3}(t)) - {H_E}} \right) 2{L_2}\cos ({\theta _2}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {\left( {{p_1} - {k_1}\left( {2{L_1}\sin ({\theta _1}(t)) + 2{L_2}\sin ({\theta _2}(t))} \right) } \right) \left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)) }\\ { - {k_1}\left( {\left( {{L_3} + {L_E}} \right) \sin ({\theta _3}(t)) - {H_E}} \right) \left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t))} \end{array}}\\ 0 \end{array}} \right] \end{aligned}$$
(134)
$$\begin{aligned} {{\mathbf{Q}}_{r,1}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} { - {r_1}\left( {2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t)} \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { - {r_1}\left( {2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { - {r_1}\left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)){{\dot{\theta }}_3}(t)2{L_1}\cos ({\theta _1}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} { - {r_1}\left( {2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t)} \right) 2{L_2}\cos ({\theta _2}(t)) }\\ { - {r_1}\left( {2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) 2{L_2}\cos ({\theta _2}(t)) }\\ { - {r_1}\left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)){{\dot{\theta }}_3}(t)2{L_2}\cos ({\theta _2}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} { - {r_1}\left( {2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t)} \right) \left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)) }\\ { - {r_1}\left( {2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) \left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)) }\\ { - {r_1}\left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t)){{\dot{\theta }}_3}(t)\left( {{L_3} + {L_E}} \right) \cos ({\theta _3}(t))} \end{array}}\\ 0 \end{array}} \right] . \end{aligned}$$
(135)

1.6 B.6 Pan-head suspension generalized force

$$\begin{aligned} {{\mathbf{Q}}_{k,2}}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ 0\\ { - {k_2}x(t)} \end{array}} \right] \end{aligned}$$
(136)
$$\begin{aligned} {{\mathbf{Q}}_{r,2}}(t)= & {} \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ 0\\ { - {r_2}\dot{x}(t)} \end{array}} \right] . \end{aligned}$$
(137)

1.7 B.7 Pantograph/catenary generalized contact force

$$\begin{aligned} {{\mathbf{Q}}_{k,3}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {{k_3}\left( {{H_0} + s(t)} \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { + {k_3}\left( { - 2{L_1}\sin ({\theta _1}(t)) + 2{L_4}\sin (\beta - {\theta _2}(t))} \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { + {k_3}\left( { - 2{L_2}\sin ({\theta _2}(t)) - x(t)} \right) 2{L_1}\cos ({\theta _1}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {{k_3}\left( {{H_0} + s(t) - 2{L_1}\sin ({\theta _1}(t))} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) }\\ { + {k_3}\left( {2{L_4}\sin (\beta - {\theta _2}(t))} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) }\\ { + {k_3}\left( { - 2{L_2}\sin ({\theta _2}(t)) - x(t)} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) } \end{array}}\\ 0\\ {\begin{array}{*{20}{l}} {{k_3}\left( {{H_0} + s(t) - 2{L_1}\sin ({\theta _1}(t)) + 2{L_4}\sin (\beta - {\theta _2}(t))} \right) }\\ { + {k_3}\left( { - 2{L_2}\sin ({\theta _2}(t)) - x(t)} \right) } \end{array}} \end{array}} \right] \end{aligned}$$
(138)
$$\begin{aligned} {{\mathbf{Q}}_{r,3}}(t)= & {} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {{r_3}\left( {\dot{s}(t) - 2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t) - 2{L_4}\cos (\beta - {\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) 2{L_1}\cos ({\theta _1}(t)) }\\ { + {r_3}\left( { - 2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t) - \dot{x}(t)} \right) 2{L_1}\cos ({\theta _1}(t))} \end{array}}\\ {\begin{array}{*{20}{l}} {{r_3}\left( {\dot{s}(t) - 2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t)} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) }\\ { + {r_3}\left( { - 2{L_4}\cos (\beta - {\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) }\\ { + {r_3}\left( { - 2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t) - \dot{x}(t)} \right) 2\left( {{L_4}\cos (\beta - {\theta _2}(t)) + {L_2}\cos ({\theta _2}(t))} \right) } \end{array}}\\ 0\\ {\begin{array}{*{20}{l}} {{r_3}\left( {\dot{s}(t) - 2{L_1}\cos ({\theta _1}(t)){{{\dot{\theta }} }_1}(t) - 2{L_4}\cos (\beta - {\theta _2}(t)){{{\dot{\theta }} }_2}(t)} \right) }\\ { + {r_3}\left( { - 2{L_2}\cos ({\theta _2}(t)){{{\dot{\theta }} }_2}(t) - \dot{x}(t)} \right) } \end{array}} \end{array}} \right] . \end{aligned}$$
(139)

1.8 B.8 Control actuator generalized force

$$\begin{aligned} {{\mathbf{Q}}_u}(t) = \left[ {\begin{array}{*{20}{l}} 0\\ 0\\ 0\\ {u(t)} \end{array}} \right] . \end{aligned}$$
(140)

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Pappalardo, C.M., De Simone, M.C. & Guida, D. Multibody modeling and nonlinear control of the pantograph/catenary system. Arch Appl Mech 89, 1589–1626 (2019). https://doi.org/10.1007/s00419-019-01530-3

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