Efficient formulation of the Gibbs–Appell equations for constrained multibody systems

Abstract

In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell formulation. Without constructing the Gibbs function, the proposed method provides a minimal set of first-order dynamic equations applicable for multibody constrained systems. Transforming the Newton–Euler equations from physical coordinates to quasivelocity spaces eliminate constraint reaction forces from motion equations. In this study, we develop a general procedure to select effective quasivelocities, which indicate that the proposed quasivelocities satisfy constraints, eliminate Lagrange multipliers, and reduce the number of dynamic equations to degrees of freedom. Besides, we test the validity and efficiency of the proposed approach using three constrained dynamical systems as illustrative examples. Finally, we compare the simulation results with Udwadia–Kalaba, augmented Lagrangian, and other conventional methods.

Appendices

Appendix A

Selecting quasivelocities based on the general proposed procedure in this research and using Eq. (35), we have

$$ {\left \{ \dot{{\gamma }_{\mathrm{1}}}\right \} }_{r\mathrm{\times 1}} = {\left [a\right ]}_{r\times m}{\left \{ \dot{q} \right \} }_{m\mathrm{\times 1}} + {\left \{ b\right \} }_{r \mathrm{\times 1}} = {\left \{ 0\right \} }_{r\mathrm{\times 1}}, $$

$$ {\left \{ \dot{{\gamma }_{\mathrm{2}}}\right \} }_{p\mathrm{\times 1}} = {\left [{\left [0\right ]}_{p\times r}\ { \left [y\right ]}_{p\times p}\right ]}_{p\times n}{ \left \{ \dot{q}\right \} }_{n\mathrm{\times 1}}, $$

(68)

where $\left [y\right ]$ is an arbitrary square matrix $\in {\mathbb{R}}^{p}\times {\mathbb{R}}^{p}$, and

$$ {\left \{ \dot{\gamma }\right \} }_{n\mathrm{\times 1}} = \left \{ \textstyle\begin{array}{c} \dot{{\gamma }_{\mathrm{1}}} \\ \dot{{\gamma }_{\mathrm{2}}} \end{array}\displaystyle \right \} = \left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}_{r\times r} & {\left [a_{ \mathrm{2}}\right ]}_{r\times p} \\ {\left [0\right ]}_{p\times r} & {\left [y\right ]}_{p \times p} \end{array}\displaystyle \right ]{\left \{ \dot{q}\right \} }_{n\mathrm{\times 1}} + \left \{ \textstyle\begin{array}{c} {\left \{ b\right \} }_{r\mathrm{\times 1}} \\ {\left \{ z\right \} }_{p\mathrm{\times 1}} \end{array}\displaystyle \right \} . $$

(69)

Based on Eq. (12), we have

$$ \left [w\right ] = \left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}_{r\times r} & {\left [a_{ \mathrm{2}}\right ]}_{r\times p} \\ {\left [0\right ]}_{p\times r} & {\left [y\right ]}_{p \times p} \end{array}\displaystyle \right ]. $$

(70)

Differentiation of Eq. (69) with respect to time yields

$$ \left \{ \ddot{q}\right \} = \left [u\right ]{\left [w\right ]}^{ -\mathrm{1}}\left \{ \ddot{\gamma }\right \} + \left \{ \overline{v}(q,\dot{q},\dot{\gamma },t)\right \} . $$

(71)

By partitioning a matrix into four blocks, it can be inverted clockwise as

$$ {\left [ \textstyle\begin{array}{c@{\quad }c} \mathrm{A} & \mathrm{B} \\ \mathrm{C} & \mathrm{D} \end{array}\displaystyle \right ]}^{-\mathrm{1}} = \left [ \textstyle\begin{array}{c@{\quad }c} {\mathrm{A}}^{-\mathrm{1}} + {\mathrm{A}}^{- \mathrm{1}}\mathrm{B}{\left (\mathrm{D-C}{\mathrm{A}}^{- \mathrm{1}}\mathrm{B}\right )}^{-\mathrm{1}}\mathrm{C}{ \mathrm{A}}^{-\mathrm{1}} & -{\mathrm{A}}^{ -\mathrm{1}}\mathrm{B}{\left (\mathrm{D-C}{\mathrm{A}}^{ -\mathrm{1}}\mathrm{B}\right )}^{-\mathrm{1}} \\ -{\left (\mathrm{D-C}{\mathrm{A}}^{-\mathrm{1}} \mathrm{B}\right )}^{-\mathrm{1}}\mathrm{C}{\mathrm{A}}^{ -\mathrm{1}} & {\left (\mathrm{D-C}{\mathrm{A}}^{- \mathrm{1}}\mathrm{B}\right )}^{-\mathrm{1}} \end{array}\displaystyle \right ]\ , $$

(72)

where A, B, C, and D have arbitrary sizes. (A and D should be squared so that they can be inverted.) Furthermore, A and $\left (\mathrm{D-C}{ \mathrm{A}}^{-\mathrm{1}}\mathrm{B}\right )$ must be nonsingular, and

$$\begin{aligned} {\left [w\right ]}^{-\mathrm{1}} &= {\left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}_{r\times r} & {\left [a_{ \mathrm{2}}\right ]}_{r\times p} \\ {\left [0\right ]}_{p\times r} & {\left [y\right ]}_{p \times p} \end{array}\displaystyle \right ]}^{-\mathrm{1}} \\ &= \left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} + { \left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}}\left [a_{ \mathrm{2}}\right ]{\left [y\right ]}^{-\mathrm{1}}\left [0 \right ]{\left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} & -{\left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} \left [a_{\mathrm{2}}\right ]{\left [y\right ]}^{-\mathrm{1}} \\ -{\left [y\right ]}^{-\mathrm{1}}\left [0\right ]{ \left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} & {\left [y \right ]}^{-\mathrm{1}} \end{array}\displaystyle \right ]. \end{aligned}$$

(73)

Substituting Eq. (70) into Eq. (72), we have

$$ {\left [\Omega \right ]}_{(m\times m )} = \left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} & -{ \left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}}\left [a_{ \mathrm{2}}\right ]\left [\beta \right ] \\ \left [0\right ] & \left [\beta \right ] \end{array}\displaystyle \right ]\ ,\ \left [\beta \right ] = {\left [y\right ]}^{ -\mathrm{1}}. $$

(74)

Multiplying the constraint matrix $\left [\mathrm{a}\right ]\$and $\left [\Omega \right ]$ yields

$$ \left [\mathrm{a}\right ]\left [\Omega \right ] = \left [ \textstyle\begin{array}{c@{\quad }c} \left [a_{\mathrm{1}}\right ] & \left [a_{\mathrm{2}}\right ] \end{array}\displaystyle \right ]\left [ \textstyle\begin{array}{c@{\quad }c} {\left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}} & -{ \left [a_{\mathrm{1}}\right ]}^{-\mathrm{1}}\left [a_{ \mathrm{2}}\right ]\left [\beta \right ] \\ \left [0\right ] & \left [\beta \right ] \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad }c} \left [I\right ] & \left [0\right ] \end{array}\displaystyle \right ]. $$

(75)

Finally, Eq. (27) is proved as follows:

$$ {\left [\Omega \right ]}^{T}{\left [\mathrm{a}\right ]}^{T} \left \{ \lambda \right \} = \left [ \textstyle\begin{array}{c} {[I]}_{r\times r} \\ {\mathrm{[0]}}_{p\times r} \end{array}\displaystyle \right ]\left \{ \lambda \right \} . $$

(76)

Appendix B

In this annex, we prove that $\ \frac{\partial {\tilde{\boldsymbol{a}}}_{G}}{\partial \ddot{\boldsymbol{q}}} = \frac{\partial {\tilde{\boldsymbol{v}}}_{G}}{\partial \dot{\boldsymbol{q}}}$. This is an important equation to reduce calculations, where ${\tilde{v}}_{G}\left (\dot{\boldsymbol{q}},\boldsymbol{q},t\right ) \ \text{and}\ {\tilde{a}}_{G}\left (\ddot{\boldsymbol{q}}, \dot{\boldsymbol{q}},\boldsymbol{q},t\right )$ denote the velocity and acceleration vectors of the mass center, respectively, and are defined as follows:

$$ \left \{ {\tilde{a}}_{G}\right \} = {[a_{Gx},a_{Gy},a_{Gz},{ \dot{\omega }}_{x},{\dot{\omega }}_{y},{\dot{\omega }}_{z}]}^{T} \ , \ \left \{ {\tilde{v}}_{G}\right \} = {[v_{Gx},v_{Gy},v_{Gz},{ \omega }_{x},{\omega }_{y},{\omega }_{z}]}^{T}. $$

(77)

To simplify the formulation, the velocity vector can be written in following matrix notation:

$$ {\tilde{v}}_{G}\left (\dot{\boldsymbol{q}},\boldsymbol{q},t\right ) = \boldsymbol{\eta }\ \dot{\boldsymbol{q}} + \boldsymbol{\kappa }\ , $$

(78)

where $\boldsymbol{\eta }(\boldsymbol{q},t)\in { \mathbb{R}}^{\mathrm{6}}\times {\mathbb{R}}^{n}$ and $\boldsymbol{\kappa }(\boldsymbol{q},t) \in {\mathbb{R}}^{\mathrm{6}}$ are functions of generalized coordinates, vector $\boldsymbol{q}$, and time. In addition, the generalized acceleration vector can be expressed in terms of kinematic characteristics by taking the time derivative of Eq. (78) leading to the equation

$$ \left \{ {\tilde{a}}_{G}\right \} = \frac{d}{dt}{\tilde{v}}_{G} \left (\dot{\boldsymbol{q}},\boldsymbol{q},t\right ) = \boldsymbol{\eta }\ddot{\boldsymbol{q}} + \dot{\boldsymbol{\eta }}(\dot{\boldsymbol{q}},\boldsymbol{q},t )\dot{\boldsymbol{q}} + \dot{\boldsymbol{\kappa }} \ (\dot{\boldsymbol{q}},\boldsymbol{q},t). $$

(79)

Using definitions (78) and (79), we conclude that the $\mathit{jacoobian}\left (\left \{ {\tilde{a}}_{G}\right \} ; \left \{ \ddot{q}\right \} \right )$ is equal to $\operatorname{jacoobian}\left (\left \{ {\tilde{v}}_{G}\right \} ;\left \{ \dot{q}\right \} \right )$. In fact, we have

$$ \frac{\partial \left \{ {\tilde{a}}_{G}\right \} }{\partial \left \{ \ddot{q}\right \} } = \frac{\partial \left \{ {\tilde{v}}_{G}\right \} }{\partial \left \{ \dot{q}\right \} } = \boldsymbol{\eta }. $$

(80)

Substituting Eq. (80) into Eq. (25) gives the alternative form of $\left \{ U^{\mathrm{*}}\right \}$ as

$$ {\left \{ U^{\mathrm{*}}\right \} }_{(n\mathrm{\times 1)}} = \sum ^{N}_{i=1}{ \frac{\partial { \left \{ {\tilde{v}}_{G}\right \} }_{i}}{\partial \left \{ \dot{q}\right \} } \ {\left \{ \mathrm{U}\right \} }_{i}}. $$

(81)