On Euler-Lagrange’s Equations: A New Approach

ABSTRACT

A new formalism is proposed to study the dynamics of mechanical systems composed of N connected rigid bodies, by introducing the concept of 6N-dimensional composed vectors. The approach is based on previous works by the authors where a complete formalism was developed by means of differential geometry, linear algebra, and dynamical systems usual concepts. This new formalism is a method for the description of mechanical systems as a whole and not as each separate part. Euler-Lagrange’s Equations are easily obtained by means of this formalism.

Keywords:
composed vectors; connected rigid bodies; dynamics

RESUMO

Um novo formalismo é proposto para estudar a dinâmica de sistemas mecânicos compostos por N corpos rígidos conectados, introduzindo o conceito de vetores compostos 6N-dimensionais. A abordagem é baseada em trabalhos anteriores dos autores, nos quais um formalismo completo foi desenvolvido por meio dos conceitos usuais de geometria diferencial, álgebra linear e sistemas dinâmicos. Esse novo formalismo é um método para a descrição de sistemas mecânicos como um todo e não como cada parte separada. As equações de Euler-Lagrange são facilmente obtidas por meio desse formalismo.

Palavras-chave:
vetores compostos; corpos rígidos conectados; dinâmica

1 INTRODUCTION

Works by Cortizo and Giacaglia ², Kottke ⁸, and Giacaglia and Kottke ⁵ have described in details the formalism, which is the theoretical background of this work, by the introduction of virtual linear velocities and angular velocities and force-torque composed vectors representing kinematical and dynamical quantities of all links involved.

In the present work we show a compact and straightforward method to obtain Euler-Lagrange’s Equations for such a mechanical system by simply writing down the Newton-Euler dynamical equations condensed into composed vectors and then multiplying such equations by a properly chosen composed kinematical vector.

The links are considered to be connected by rotational joints and acted on by arbitrary forces and torques, including torques resulting from frictional forces arising in the joints. The final differential equations are transformed by simple scalar products and it is shown that the resulting differential equations are completely equivalent to the differential equations obtained from the Lagrangian of the system. The method is applicable to connected rigid multi body systems in the presence of holonomic and nonholonomic constraints, a possibility available by the methodology used in the present work ⁶. Figure 1 shows an example of a simple system composed by two connected joints, and torque and forces applied.

Figure 1:
System with two constrained plane links.

Figure 2: gives the geometry of this system, showing the position and orientation of each torque and forces applied.

Figure 2:
Links geometry and applied forces and torques.

The first of Figure 2 is the link OA and the second is the link AB. We have set the lengths to be 2L ₂ and 2L ₁, for simplicity of the geometry and mechanics involved.

Classical work by Roberson ¹¹, Wittenburg ¹⁵, and Shabana ¹⁴ on multi-bodies systems use classical approaches to this matter, treating each body as a separate entity of these systems. Schiehlen ¹² presents a substantial collection of the most efficient algorithms for calculating rigid-body dynamics, but as we understand no composed vectors, suggested here, are used. This is also observed in revised work of this author Schiehlen ¹³.

A more elaborate algorithm is given by Featherstone ³ where the author informs that rigid body dynamics algorithms presented are the subject of computational rigid-body dynamics through the medium of spatial 6D vector notation.

In the present article we actually use an algorithm with 6N-dimensional vectors, where N is the number of connected bodies involved.

2 DYNAMICS OF THE SYSTEM

We consider a mechanical system constituted by N rigid links connected by rotational joints. The position and orientation of each link is represented by the coordinates of its centre of mass and by a set of three Euler angles each. The number of degrees of freedom of the system, using a set of generalised coordinates, for instance the Denawitt-Hartenberg parameters ⁷, is represented by $n<N$generalised coordinates$q=(q^1,q^2,...,q^n)$, associated to $\dot{q}=({\dot{q}}^1,...,{\dot{q}}^n)$generalised velocities and $\ddot{q}=({\ddot{q}}^1,...,{\ddot{q}}^n)$generalised accelerations associated to configuration q. The linear velocities of the centre mass of each link and their angular velocities, represented by vectors

$$\begin{gathered} {\overrightarrow{\upsilon }}_i(q,\dot{q})=\sum _{\alpha =1}^n{q}^{\alpha }{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right) \\ {\overrightarrow{\omega }}_i(q,\dot{q})=\sum _{\alpha =1}^n{q}^{\alpha }{\overrightarrow{\omega }}_{\alpha i}\left(q\right) \end{gathered}$$ (2.1)

Vectors ${\overrightarrow{\nu }}_{\alpha i}$ and ${\overrightarrow{\omega }}_{\alpha i}$ defined in Eq. (2.1) are the linear and angular velocities of link $i(i=1,...,n)$ in the state where all generalised velocities are zero except the one (α) which is unitary. The absolute value of vector ${\overrightarrow{\nu }}_{\alpha i}$ is the distance 𝓁_i between point C _i and a reference joint J _i . Vectors ${\overrightarrow{\omega }}_{\alpha i}$ are dimensionless. The linear and angular accelerations are given by

$\frac{d{\overrightarrow{\upsilon }}_i}{dt}\left(q,\dot{q},\ddot{q}\right)=\frac{d}{dt}\left[\sum _{\alpha =1}^n{\dot{q}}^{\alpha }{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right)\right]=\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right)+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\frac{\partial {\overrightarrow{\upsilon }}_{\beta i}}{\partial q^{\gamma }\left(q\right)}$ (2.2)

$\frac{d{\overrightarrow{\omega }}_i}{dt}\left(q,\dot{q},\ddot{q}\right)=\frac{d}{dt}\left[\sum _{\alpha =1}^n{\dot{q}}^{\alpha }{\overrightarrow{\omega }}_{\alpha i}\left(q\right)\right]=\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }{\overrightarrow{\omega }}_{\alpha i}\left(q\right)+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\frac{\partial {\overrightarrow{\omega }}_{\beta i}}{\partial q^{\gamma }\left(q\right)}$ (2.3)

The distributions of actions on each link is the pair $\left({\overrightarrow{F}}_i,{\overrightarrow{N}}_i\right)$ and corresponds to all actions (forces and torques) acting on link i, using as pole their centres of mass $C_i(i=1,...,n)$. Euler equations for any particular link i are given by

$$\begin{gathered} {\overrightarrow{N}}_i\left(q,\dot{q},\ddot{q}\right)=I_i\left(\frac{d{\overrightarrow{\omega }}_i}{dt}\right)+{\overrightarrow{\omega }}_i\times I_i\left({\overrightarrow{\omega }}_i\right)= \\ =I_i\left[\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }{\overrightarrow{\omega }}_{\alpha i}\left(q\right)+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\frac{\partial {\overrightarrow{\omega }}_{\beta i}}{\partial q^{\gamma }}\left(q\right)\right]+ \\ +\left[\sum _{\beta =1}^n{\dot{q}}^{\beta }{\overrightarrow{\omega }}_{\beta i}\left(q\right)\right]\times I_i\left[\sum _{\gamma =1}^n{\dot{q}}^{\gamma }{\overrightarrow{\omega }}_{\gamma i}\left(q\right)\right]=\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }\left\{I_i\left[{\overrightarrow{\omega }}_{\alpha i}\left(q\right)\right]\right\}+ \\ +\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\left\{\left[\frac{\partial \left(I_i{\overrightarrow{\omega }}_{\beta i}\right)}{\partial q^{\gamma }}\left(q\right)\right]+\left[{\overrightarrow{\omega }}_{\beta i}\left(q\right)\times I_i\left[{\overrightarrow{\omega }}_{\gamma i}\left(q\right)\right]\right]\right\} \end{gathered}$$ (2.4)

Newton’s Equation for the force acting on the centre of mass C _i of link i is given by

$$\begin{gathered} {\overrightarrow{F}}_i\left(q,\dot{q},\ddot{q}\right)=m_i\left(\frac{d{\overrightarrow{\upsilon }}_i}{dt}\right)+m_i\left[\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right)+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\frac{\partial {\overrightarrow{\upsilon }}_{\beta i}}{\partial q^{\gamma }}\left(q\right)\right] \\ =\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }\left[m_i{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right)\right]+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\left[m_i\frac{\partial {\overrightarrow{\upsilon }}_{\beta i}}{\partial q^{\gamma }}\left(q\right)\right] \end{gathered}$$ (2.5)

The results of the above evaluations corresponding to the 3n set $(q,\dot{q},\ddot{q})$ is represented by ${\overrightarrow{F}}_i(q,\dot{q},\ddot{q})$ and ${\overrightarrow{N}}_i(q,\dot{q},\ddot{q})$ for $i=1,...,n$.

The composed vector represented by $\overrightarrow{v}\overrightarrow{w}$ is given by the

$\overrightarrow{\nu }\overrightarrow{\omega }=\left({\overrightarrow{\nu }}_1,{\overrightarrow{\omega }}_1,{\overrightarrow{\nu }}_2,{\overrightarrow{\omega }}_2,...,{\overrightarrow{\nu }}_n,{\overrightarrow{\omega }}_n\right)$ (2.6)

and from the above expressions for the linear and angular velocities it is found that

$\overrightarrow{\nu }\overrightarrow{\omega }=\sum _{\alpha =1}^n{\dot{q}}^{\alpha }\overrightarrow{\nu }{\overrightarrow{\omega }}_{\alpha }$ (2.7)

where

$\overrightarrow{\nu }{\overrightarrow{\omega }}_{\alpha }=\left({\overrightarrow{\nu }}_{\alpha 1},{\overrightarrow{\omega }}_{\alpha 1},{\overrightarrow{\nu }}_{\alpha 2},{\overrightarrow{\omega }}_{\alpha 2},...,{\overrightarrow{\nu }}_{\alpha n},{\overrightarrow{\omega }}_{\alpha n}\right)$ (2.8)

with components defined as above. The 6n-dimensional vector $\overrightarrow{v}\overrightarrow{w}$ is the vector of virtual velocities associated to the state $(q,\dot{q})$ of the system.

The pair force-torque is represented by $\left({\overrightarrow{F}}_i,{\overrightarrow{N}}_i\right)$,$(q,\dot{q},\ddot{q})$ where the components vectors are given by ${\overrightarrow{F}}_i(q,\dot{q},\ddot{q})$ and ${\overrightarrow{N}}_i(q,\dot{q},\ddot{q})$ with $(i=1,...,n)$ and corresponding to the 3n-dimensional set $(q,\dot{q},\ddot{q})$.

Newton-Euler equations give the resulting forces ${\overrightarrow{F}}_i(q,\dot{q},\ddot{q})$ and torques ${\overrightarrow{N}}_i(q,\dot{q},\ddot{q})$ acting on link $i(i=1,...,N)$ corresponding to the 3n-dimensional set $(q,\dot{q},\ddot{q})$.

These 2n three dimensional vectors can be composed in only one 6n-dimensional vector $\overrightarrow{F}\overrightarrow{N}(q,\dot{q},\ddot{q})$, given by

$\overrightarrow{F}\overrightarrow{N}(q,\dot{q},\ddot{q})=\left({\overrightarrow{F}}_1(q,\dot{q},\ddot{q}),{\overrightarrow{N}}_1(q,\dot{q},\ddot{q}),...,{\overrightarrow{F}}_n(q,\dot{q},\ddot{q}),{\overrightarrow{N}}_n(q,\dot{q},\ddot{q})\right)$ (2.9)

The dynamical equations for the mechanical system considered are obtained as follows. The expressions derived above (Eqs. (2.4) and Eqs. (2.5) ) for ${\overrightarrow{F}}_i(q,\dot{q},\ddot{q})$ and ${\overrightarrow{N}}_i(q,\dot{q},\ddot{q})$ can be condensed in a unique composed 6n-dimensional vector

$\overrightarrow{F}\overrightarrow{N}(q,\dot{q},\ddot{q})=\sum _{\alpha =1}^n{\ddot{q}}^{\alpha }\overrightarrow{P}{\overrightarrow{L}}_{\alpha }\left(q\right)+\sum _{\beta =1}^n\sum _{\gamma =1}^n{\dot{q}}^{\beta }{\dot{q}}^{\gamma }\overrightarrow{X}{\overrightarrow{Y}}_{\beta \gamma }\left(q\right)$ (2.10)

Where

$\overrightarrow{P}{\overrightarrow{L}}_{\alpha }=\left({\overrightarrow{P}}_{\alpha 1},{\overrightarrow{L}}_{\alpha 1},{\overrightarrow{P}}_{\alpha 2},{\overrightarrow{L}}_{\alpha 2},...,{\overrightarrow{P}}_{\alpha n},{\overrightarrow{L}}_{\alpha n}\right)$ (2.11)

and

$\overrightarrow{X}{\overrightarrow{Y}}_{\beta \gamma }=\left({\overrightarrow{X}}_{\beta \gamma 1},{\overrightarrow{Y}}_{\beta \gamma 1},{\overrightarrow{X}}_{\beta \gamma 2},{\overrightarrow{Y}}_{\beta \gamma 2},...,{\overrightarrow{X}}_{\beta \gamma n},{\overrightarrow{Y}}_{\beta \gamma n}\right)\left(\alpha ,\beta ,\gamma =1,...,n\right)$ (2.12)

are given by

$$\begin{gathered} {\overrightarrow{P}}_{\alpha i}\left(q\right)=m_i{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right) \\ {\overrightarrow{L}}_{\alpha i}\left(q\right)=I_i\left[{\overrightarrow{\omega }}_{\alpha i}\left(q\right)\right] \end{gathered}$$ (2.13)

Vectors ${\overrightarrow{P}}_{\alpha i}$ and ${\overrightarrow{L}}_{\alpha i}$ above defined represent the linear and angular momenta of link i in the state of the system where only the α velocity is different from zero and unitary, all other being zero.

We also have that

${\overrightarrow{X}}_{\beta \gamma i}\left(q\right)=m_i\frac{\partial {\overrightarrow{\upsilon }}_{\beta i}}{\partial q^{\gamma }}\left(q\right){\overrightarrow{Y}}_{\beta \gamma i}\left(q\right)=I_i\left[\frac{\partial {\overrightarrow{\omega }}_{\beta i}}{\partial q^{\gamma }}\left(q\right)\right]+\left[{\overrightarrow{\omega }}_{\beta i}\left(q\right)\right]\times I_i\left[{\overrightarrow{\omega }}_{\gamma i}\left(q\right)\right]$ (2.14)

for $i=1,...,n$.

It should be noted that the scalar product of two composed vectors must satisfy the physical meaning of such product. As an example, consider the power associated to the composed vector

$\overrightarrow{F}\overrightarrow{N}\left(q,\dot{q},\ddot{q}\right)=\left({\overrightarrow{F}}_1\left(q,\dot{q},\ddot{q}\right),{\overrightarrow{N}}_1\left(q,\dot{q},\ddot{q}\right),...,{\overrightarrow{F}}_n\left(q,\dot{q},\ddot{q}\right),{\overrightarrow{N}}_n\left(q,\dot{q},\ddot{q}\right)\right)$ (2.15)

The scalar product of this force-torque action by the linear-angular velocity composed vector $\overrightarrow{v}\overrightarrow{\omega }=({\overrightarrow{v}}_1,{\overrightarrow{\omega }}_1,{\overrightarrow{v}}_2,{\overrightarrow{\omega }}_2,...,{\overrightarrow{v}}_n,{\overrightarrow{\omega }}_n)$ is given by

$\overrightarrow{F}\overrightarrow{N}\left(q,\dot{q},\ddot{q}\right)\circ \overrightarrow{\upsilon \omega }\left(q,\dot{q}\right)=\sum _{i=1}^n\left[{\overrightarrow{F}}_i\circ {\overrightarrow{\nu }}_i+{\overrightarrow{N}}_i\circ {\overrightarrow{\omega }}_i\right]$ (2.16)

The differential equations of motion of the composed system derived by simple vector algebra give an equivalent result as that given by the explicit form of Euler-Lagrange’s Equations ^{4), (10}. In fact, consider the composed vectors, for $\alpha =1,2,...,n$,

$\overrightarrow{\nu }{\overrightarrow{\omega }}_{\alpha }=\left({\overrightarrow{\nu }}_{\alpha 1},{\overrightarrow{\omega }}_{\alpha 1},...,{\overrightarrow{\nu }}_{\alpha n},{\overrightarrow{\omega }}_{\alpha n}\right)$ (2.17)

where vectors ${\overrightarrow{v}}_{\alpha i},{\overrightarrow{w}}_{\alpha i}$ have been defined by Eq. (2.1),

$\left\{\begin{array}{l}{\overrightarrow{\upsilon }}_i\left(q,\dot{q}\right)=\sum _{\alpha =1}^n{\dot{q}}^{\alpha }{\overrightarrow{\upsilon }}_{\alpha i}\left(q\right)\\ {\overrightarrow{\omega }}_i\left(q,\dot{q}\right)=\sum _{\alpha =1}^n{\dot{q}}^{\alpha }{\overrightarrow{\omega }}_{\alpha i}\left(q\right)\end{array}\right.$ (2.18)

Consider Eq. (2.11) and the scalar product

$$\begin{gathered} \overrightarrow{P}{\overrightarrow{L}}_{\alpha }\left(q\right)\circ \overrightarrow{\nu }{\overrightarrow{\omega }}_{\gamma }\left(q\right)=\sum _{i=1}^n\left({\overrightarrow{P}}_{\alpha i}\circ {\overrightarrow{\nu }}_{\gamma i}+{\overrightarrow{L}}_{\alpha i}\circ {\overrightarrow{\omega }}_{\gamma i}\right)= \\ =\sum _{i=1}^n\left[m_i\left({\overrightarrow{\nu }}_{\alpha i}\circ {\overrightarrow{\nu }}_{\gamma i}\right)+I_i\left({\overrightarrow{\omega }}_{\alpha i}\right)\circ {\overrightarrow{\omega }}_{\gamma i}\right]=a_{\gamma \alpha }\left(q\right) \end{gathered}$$ (2.19)

The coefficients a _γα (q) are the coefficients of the quadratic terms ${\dot{q}}^{\gamma }{\dot{q}}^{\alpha }$ in the total kinetic energy of the system, that is

$T=\frac{1}{2}\sum _{\alpha }\sum _{\beta }{a}_{\alpha \beta }{\dot{q}}^{\alpha }{\dot{q}}^{\beta }$ (2.20)

The first part of Eq. (2.19) on the right hand side is the linear kinetic energy (except for a factor $\frac{{\dot{q}}^{\alpha }{\dot{q}}^{\gamma }}{2}$ of link i because the velocity of the centre of mass C _i of this link associated to the generalised coordinate (angle) q ^α is precisely ${\overrightarrow{\nu }}_{\alpha i}$ except for a factor ${\dot{q}}^{\alpha }$. The second part is the rotational kinetic energy of this same link except for the same factor $\frac{{\dot{q}}^{\alpha }{\dot{q}}^{\gamma }}{2}$.

The same scalar product is now applied to the second term on the right-hand side of Eq. (2.11), giving

$\overrightarrow{X}{\overrightarrow{Y}}_{\alpha \beta }\circ \overrightarrow{\nu }{\overrightarrow{\omega }}_{\gamma }\left(q\right)=\sum _{i=1}^n{m}_i\frac{\partial {\overrightarrow{\nu }}_{\alpha i}}{\partial q^{\beta }}\circ {\overrightarrow{\nu }}_{\gamma i}+\sum _{i=1}^n\left[\frac{\partial I_i\left({\overrightarrow{\omega }}_{\alpha i}\right)}{\partial q^{\beta }}+{\overrightarrow{\omega }}_{\alpha i}\times I_i\left({\overrightarrow{\omega }}_{\beta i}\right)\right]\circ {\overrightarrow{\omega }}_{\beta i}={\alpha }_{\beta \alpha }^{\gamma }\left(q\right)$ (2.21)

The coefficients $a_{\beta \alpha }^{\gamma }\left(q\right)$ correspond to the Christoffel Brackets $\left[\begin{array}{c}\gamma \\ \beta \alpha \end{array}\right]$ resulting from the quadratic part of Euler-Lagrange’s Equations when the constraints are time independent and considering unitary linear and angular velocities. Taking into account the definition of the coefficients a _γα (q) given by Eq. (2.21) and comparing Eq. (2.22) with the classical definition ^{1), (4}

$\left[\begin{array}{c}i\\ \beta \alpha \end{array}\right]=\frac{\partial a_{i\alpha }}{\partial q^{\beta }}+\frac{\partial a_{i\beta }}{\partial q^{\alpha }}-\frac{\partial a_{\alpha \beta }}{\partial q^i}$ (2.22)

it is seen that there is an exact correspondence between the two definitions.

The same scalar product applied to the force-torque composed vector given by Eq. (2.16), gives

$\overrightarrow{F}\overrightarrow{N}\left(q,\dot{q},\ddot{q}\right)\circ \overrightarrow{\nu }{\overrightarrow{\omega }}_{\gamma }\left(q\right)=\sum _{i=1}^n{\overrightarrow{F}}_i\circ {\overrightarrow{\nu }}_{\gamma i}+\sum _{i=1}^n{\overrightarrow{N}}_i\circ {\overrightarrow{\omega }}_{\gamma i}=Q_{\gamma }$ (2.23)

Quantities Q _γ are the generalised forces and give a direct physical interpretation of these quantities. From Eq. (2.23) the dimension of the generalised forces associated to an angular generalised coordinate q ^α shows that it corresponds to the torque of the applied force with respect to joint J _i .

With the above definitions, the equations of motion for the generalised coordinates assume the known form

$\sum _{\alpha }{a}_{\gamma \alpha }{\ddot{q}}^{\alpha }+\sum _{\alpha }\sum _{\beta }\left[\begin{array}{c}\gamma \\ \beta \alpha \end{array}\right]{\dot{q}}^{\alpha }{\dot{q}}^{\beta }=Q_{\gamma }$ (2.24)

From Eq. (2.24) we obtain the explicit form of Euler-Lagrange’s Equations for a system with time independent constraints given by

${\ddot{q}}^s+\sum _{\alpha }\sum _{\beta }\left(\begin{array}{c}s\\ \beta \alpha \end{array}\right){\dot{q}}^{\alpha }{\dot{q}}^{\beta }=Q^s$ (2.25)

where we have used the usual Christoffel Parentheses ⁹⁾ and the generalised forces Q ^s corresponding to the action on the generalised coordinate q ^s .

The above development shows how the use of composed vectors leads to Euler-Lagrange’s Equations of motion of a system composed of connected rigid bodies. The computation of all quantities involved in Eq. (2.19) is straightforward from the computation of forces and torques acting on the system.

3 APPLICATION: TWO-LINK ARTICULATED ARM

A simple example will be used in order to follow the mathematical formalism without complicating the physical system. We consider the simple two-link system shown in Figure 1 and representing two articulated links on a fixed base.

The rotational joint J ₁ is articulated to a fixed support. The rotational joint J ₂ is connecting links OA and AB whose lengths are L ₁ and 2L ₂. The system moves on a fixed vertical plane. The generalised coordinates are the angles ${\theta }_1=q^1\text{e}{\theta }_2=q^2$, that links OA and AB make with a fixed horizontal line, so that its time derivatives are the absolute angular velocities of each link. This is done instead of defining the angle $\theta ={\theta }_2-{\theta }_1$ between links OB and AO, as it is done in usual robots’ kinematics. For the purpose of this example there is no point in adopting this notation. The positions of points C ₁ and C ₂ are given by

$C_1-O=L_1(c_1{\overrightarrow{e}}_1+s_1{\overrightarrow{e}}_2)$ (3.1)

$C_2-O=2L_1(c_1{\overrightarrow{e}}_1+s_1\hat{j})+L_2(c_2{\overrightarrow{e}}_1+s_2\hat{j})$ (3.2)

where $\left({\overrightarrow{e}}_1,{\overrightarrow{e}}_2\right)$ is an inertial base and $c_1=\cos q^1,s_1=\sin q^1,c_2=\cos q^2\text{and}{s}_2=\sin q^2$.

The linear velocities of these points are

$$\begin{gathered} {\overrightarrow{v}}_1={\dot{q}}^1{\overrightarrow{v}}_{11}+{\dot{q}}^2{\overrightarrow{v}}_{21} \\ {\overrightarrow{v}}_{11}=L_1(-s_1{\overrightarrow{e}}_1+c_1{\overrightarrow{e}}_2) \\ {\overrightarrow{v}}_{21}=\overrightarrow{0} \\ {\overrightarrow{v}}_2={\dot{q}}^1{\overrightarrow{v}}_{21}+{\dot{q}}^2{\overrightarrow{v}}_{22} \\ {\overrightarrow{v}}_{12}=2L_1(-s_1{\overrightarrow{e}}_1+c_1{\overrightarrow{e}}_2) \\ {\overrightarrow{v}}_{22}=L_2(-s_2{\overrightarrow{e}}_1+c_2{\overrightarrow{e}}_2) \end{gathered}$$ (3.3)

The angular velocities of each link are

$$\begin{gathered} {\overrightarrow{\omega }}_1={\dot{q}}^1{\overrightarrow{\omega }}_{11} \\ {\overrightarrow{\omega }}_2={\dot{q}}^2{\overrightarrow{\omega }}_{22} \end{gathered}$$ (3.4)

where

${\overrightarrow{\omega }}_{11}={\overrightarrow{\omega }}_{22}={\overrightarrow{e}}_3$ (3.5)

since angles have been chosen to be with respect to an inertial line.

The accelerations of these points are also easily found

$$\begin{gathered} \frac{d{\overrightarrow{v}}_1}{dt}={\ddot{q}}^1{\overline{v}}_{11}+{\dot{q}}^1{\dot{q}}^1{\overrightarrow{a}}_{11} \\ {\overrightarrow{a}}_{11}=-L_1\left(c_1{\overrightarrow{e}}_1+s_1{\overrightarrow{e}}_2\right) \\ \frac{d{\overrightarrow{v}}_2}{dt}={\ddot{q}}^1{\overline{v}}_{12}+{\ddot{q}}^2{\overrightarrow{v}}_{22}+{\dot{q}}^1{\dot{q}}^1{\overrightarrow{a}}_{12}+{\dot{q}}^2{\dot{q}}^2{\overrightarrow{a}}_{22} \\ {\overrightarrow{a}}_{12}=-2L_1\left(c_1{\overrightarrow{e}}_1+s_1{\overrightarrow{e}}_2\right) \\ {\overrightarrow{a}}_{22}=-L_2\left(c_2{\overrightarrow{e}}_1+s_2{\overrightarrow{e}}_2\right) \\ {\dot{\overrightarrow{\omega }}}_1={\ddot{q}}^1{\overrightarrow{e}}_3 \\ {\dot{\overrightarrow{\omega }}}_2={\ddot{q}}^2{\overrightarrow{e}}_3 \end{gathered}$$ (3.6)

For the example of Figure 1 it is found that

$$\begin{gathered} {\overrightarrow{F}}_1\left(q,\dot{q},\ddot{q}\right)=-m_{1}g{\hat{e}}_2 \\ {\overrightarrow{F}}_2\left(q,\dot{q},\ddot{q}\right)=-m_{2}g{\hat{e}}_2 \\ {\overrightarrow{N}}_1\left(q,\dot{q},\ddot{q}\right)=N_1{\overrightarrow{e}}_3 \\ {\overrightarrow{N}}_2\left(q,\dot{q},\ddot{q}\right)=N_2{\overrightarrow{e}}_3 \end{gathered}$$ (3.7)

In the equation

$\overrightarrow{FN}\left(q,\dot{q},\ddot{q}\right)=\sum _{\alpha =1}^2{\ddot{q}}^{\alpha }{\overrightarrow{PL}}_{\alpha }\left(q\right)+\sum _{\beta =1}^2\sum _{\gamma =1}^2{\dot{q}}^{\beta }{\dot{q}}^{\gamma }{\overrightarrow{XY}}_{\beta \gamma }\left(q\right)$ (3.8)

we find that

$$\begin{gathered} {\overrightarrow{PL}}_1=\left[m_1{\overline{v}}_{11}\left(q^1\right),I_1{\overrightarrow{e}}_3,m_2{\overline{v}}_{12}\left(q^1\right),\overrightarrow{0}\right] \\ {\overrightarrow{PL}}_2=\left[m_2{\overline{v}}_{21}\left(q^1\right),\overrightarrow{0},m_2{\overline{v}}_{22}\left(q^1\right),I_2{\overrightarrow{e}}_3\right] \\ {\overrightarrow{XY}}_{11}\left(q\right)=\left[m_1{\overrightarrow{a}}_{11}\left(q^1\right),\overrightarrow{0},m_2{\overrightarrow{a}}_{12}\left(q^1\right),\overrightarrow{0}\right] \\ {\overrightarrow{XY}}_{22}\left(q\right)=\left[\overrightarrow{0},\overrightarrow{0},m_2{\overrightarrow{a}}_{22}\left(q^2\right),\overrightarrow{0}\right] \end{gathered}$$ (3.9)

and all other composed vectors being zero.

The differential equation of motion is

$\overrightarrow{FN}\left({\overrightarrow{F}}_1,{\overrightarrow{N}}_1,{\overrightarrow{F}}_2,{\overrightarrow{N}}_2\right)={\ddot{q}}^1{\overrightarrow{PL}}_1+{\ddot{q}}^2{\overrightarrow{PL}}_2+{\dot{q}}^1{\dot{q}}^1{\overrightarrow{XY}}_{11}+{\dot{q}}^2{\dot{q}}^2{\overrightarrow{XY}}_{22}$ (3.10)

with the above definitions for the coefficients of this equation.

We now consider the composed vectors

$$\begin{gathered} {\overrightarrow{v\omega }}_1=\left[{\overline{v}}_{11}\left(q^1\right),{\overrightarrow{e}}_3,{\overline{v}}_{12}\left(q^1\right),\overrightarrow{0}\right] \\ {\overrightarrow{v\omega }}_2=\left[{\overline{v}}_{21}\left(q^1\right),\overrightarrow{0},{\overline{v}}_{22}\left(q^1\right),{\overrightarrow{e}}_3\right] \end{gathered}$$ (3.11)

It is easily found that

$$\begin{gathered} \overrightarrow{FN}\circ {\overrightarrow{v\omega }}_1=-\left(m_1+2m_2\right)g{l}_1{c}_1+N_1=Q_1 \\ \overrightarrow{FN}\circ {\overrightarrow{v\omega }}_2=-m_{2}g{l}_2{c}_2+N_2=Q_2 \end{gathered}$$ (3.12)

It is also easily found that

$$\begin{gathered} {\overrightarrow{PL}}_1\circ {\overrightarrow{v\omega }}_1=\left(m_1+4m_2\right)l_1^2+I_1=a_{11}\left(q\right) \\ {\overrightarrow{PL}}_1\circ {\overrightarrow{v\omega }}_2=2l_1{l}_2{m}_2{c}_{12} \\ {c}_{12}=\cos \left(q^1-q^2\right)=a_{21} \\ {\overrightarrow{PL}}_2\circ {\overrightarrow{v\omega }}_1=2m_2{l}_1{l}_2{c}_{12}=a_{12} \\ {\overrightarrow{PL}}_2\circ {\overrightarrow{v\omega }}_2=m_2{l}_2^2+I_2=a_{22} \end{gathered}$$ (3.13)

From the definition of the Christoffel Brackets, in this particular example it is found that the only non-zero bracket is

$$\begin{gathered} a_{22}^1=2m_2{l}_1{l}_2{s}_{12} \\ {s}_{12}=\sin \left(q^1-q^2\right) \end{gathered}$$ (3.14)

The differential equations are given by

$\left[\left(m_1+4m_2\right)l_1^2+I_1\right]{\ddot{q}}^1+\left[2m_2{l}_1{l}_2{c}_{1}2\right]{\ddot{q}}^2+\left[2m_2{l}_1{l}_2{s}_{12}\right]{\dot{q}}^2{\dot{q}}^2=\left(m_1+2m_2\right)g{l}_1{c}_1+N_1$ (3.15)

$\left[2l_1{l}_2{m}_2{c}_{12}\right]{\ddot{q}}^1+\left[m_2{l}_2^2+I_2\right]{\ddot{q}}^2=-m_{2}g{l}_2{c}_2+N_2$ (3.16)

Collecting constant coefficients

$A_1{\ddot{q}}^1+B_1\cos \left(q^1-q^2\right){\ddot{q}}^2+C_1\sin \left(q^1-q^2\right){\dot{q}}^2{\dot{q}}^2=D_1\cos q^1{N}_1$ (3.17)

$A_2\cos \left(q^1-q^2\right){\ddot{q}}^1+B_2{\ddot{q}}^2=C_2+D_2\cos q^2+N_2$ (3.18)

In order to have explicit equations for ${\ddot{q}}^1$ and for ${\ddot{q}}^2$ it is a simple matter of solving the above system for these two quantities. In this simple example, expressing ${\ddot{q}}^1$ in terms of all other terms in Eq. (3.18) and substituting into Eq. (3.17) we obtain a differential equation containing only the first and second derivatives of q ² and the first derivative of q ¹ with coefficients functions of both variables q ¹ and q ². Obviously, this is the well-known problem of a double pendulum with the additional complication of the presence of torques applied to the two arms of the pendulum.

4 CONCLUSION

It has been shown that the use of composed linear-angular velocities and force-torque vectors, a new method of representing kinematical and dynamical quantities, is a very efficient and compact form to derive the dynamical equations of multi-body, as compared with classical approaches.

ACKNOWLEDGEMENTS

Dr. Giorgio Eugenio Oscare Giacaglia thanks his ”Productivity Scholarship in Technological Development and Innovative Extension”, supported by the Brazilian National Council for Scientific and Technological Development (CNPq), grant number 314689/2018-4.

Dr. Wendell de Queiro’z Lamas thanks his ”Productivity Scholarship in Research”, supported by the Brazilian National Council for Scientific and Technological Development (CNPq), grant number 300992/2018-1.

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Publication Dates

• Publication in this collection
  03 Aug 2020
• Date of issue
  May-Aug 2020

History

• Received
  20 Oct 2019
• Accepted
  26 Apr 2020