Remark 13.1.1.
This assumption is pivotal to Lagrange formulation, and it has important consequences in practice.
Section 13.1 Partial Kinematic Screws
In the Lagrangian formalism, the configuration of a system \(\Sigma\) of one or more rigid bodies is assumed to be defined by \((n+1)\) independent coordinates 1 \((q_1, q_2, \ldots , q_n, t)\) so that the position of any point of \(\Sigma\) relative to a referential \(\cE\) (of origin \(O\)) can be viewed as a vector function \(\br_{OP} (q_1, q_2, \ldots , q_n, t)\) of these \((n+1)\) variables. Hence the velocity of \(P\) relative to \(\cE\) can be evaluated as follows:
\begin{equation}
\vel_{P/\cE} = \sum_{i=1}^n \frac{\partial \br_{OP}}{\partial q_i} \, \dq_i
+ \frac{\partial \br_{OP}}{\partial t}\tag{13.1.1}
\end{equation}
Equation (13.1.1) shows that the velocity of any point \(P\) of the system can be defined by the values of \((2n+1)\) variables \((\bq, \dbq, t)\text{,}\) denoting by \(\dbq\) the set of variables \((\dq_1, \ldots,\dq_n)\text{.}\) The Lagrangian formulation hinges on the following assumption
\begin{equation}
(\bq, \dot{\bq}, t) \text{ form a set of independent variables}. \tag{13.1.2}
\end{equation}
Remark 13.1.2.
In equation (13.1.1), the partial derivatives \({\partial \br_{OP}}/{\partial q_i}\) and \({\partial \br_{OP}}/{\partial t}\) depend on the choice of referential \(\cE\) and should be denoted \(({\partial \br_{OP}}/{\partial q_i})_\cE\) and \(({\partial \br_{OP}}/{\partial t})_\cE\text{,}\) respectively. In what follows, we omit this dependence for simplicity.
One key consequence is the following identity:
\begin{equation}
\frac{\partial\vel_{P/\cE}}{\partial \dq_i} = \frac{\partial \br_{OP}}{\partial q_i} \tag{13.1.3}
\end{equation}
for \(i=1,\ldots, n\text{,}\) leading to the following result.
Theorem 13.1.3. Partial Kinematic Screw.
Given a rigid body \(\cB\) parametrized by \((q_1, q_2, \cdots q_n)\) satisfying assumption (13.1.2), the vector fields
\begin{equation*}
P\in \cB \mapsto \frac{\partial \br_{OP}}{\partial q_i} \qquad (i=1,\ldots,n)
\end{equation*}
and
\begin{equation*}
P\in \cB \mapsto \frac{\partial \br_{OP}}{\partial t}
\end{equation*}
define screws denoted as \(\{\cV_{\cB/\cE}^{q_i}\}\) and \(\{\cV_{\cB/\cE}^{t}\}\text{.}\) The kinematic screw of body \(\cB\) can then be expressed as
\begin{equation}
\{\cV_{\cB/\cE} \}= \dq_1 \{\cV_{\cB/\cE}^{q_1}\} + \cdots +\dq_n \{\cV_{\cB/\cE}^{q_n}\} + \{\cV_{\cB/\cE}^{t}\}\tag{13.1.4}
\end{equation}
\(\{\cV_{\cB/\cE}^{q_i}\}\) is called the partial kinematic screw of body \(\cB\) with respect to coordinate \(q_i\text{.}\)
Proof.
For any two points \(P\) and \(Q\) attached to body \(\cB\text{,}\) we take the partial derivative of both sides of equation \(\vel_Q = \vel_P + \bom_\cB \times \br_{PQ}\) w.r.t.~variable \(\dq_i\) (fixing index \(i=1,\ldots,n\)) to find
\begin{equation*}
\frac{\partial \vel_Q}{\partial \dq_i} = \frac{\partial \vel_P}{\partial \dq_i}
+ \frac{\partial \bom_\cB}{\partial \dq_i}\times \br_{PQ}
\end{equation*}
since vector \(\br_{PQ}\) does not depend upon \(\dq_i\text{.}\) This shows that the field \(P\in \cB \mapsto \frac{\partial\vel_P}{\partial \dq_i} =
\frac{\partial \br_{OP}}{\partial q_i}\) is a screw of resultant \(\frac{\partial \bom_\cB}{\partial \dq_i}\text{.}\) Equation (13.1.1) implies that the field \(P\in \cB \mapsto \frac{\partial \br_{OP}}{\partial t}\) also defines a screw. Equation (13.1.4) is then equivalent to ((13.1.1).
The following examples illustrate the calculation of the partial kinematic screws \(\{\cV_{\cB/\cE}^{q_i}\}\) in accordance with assumption (13.1.2).
Example 13.1.4.
Consider the planar motion of plate
1 in referential 0. Its lower edge \(A\) remains in contact with a horizontal support \((O,\bx_0,\bz_0)\) while remaining in contact with a vertical step of height \(h\text{.}\) The configuration of the plate is defined by the coordinates \(\bq = (x, \theta)\) as shown in Figure 13.1.5.
Find the corresponding partial kinematic screws \(\{\cV_{1/0}^{x}\}\) and \(\{\cV_{1/0}^{\theta}\}\text{.}\)
Solution.
The velocity \(\vel_A = \dx \bx_0\) of \(A\) and the angular velocity \(\bom_{1/0}=\dte \bz_0\) give the expression of kinematic screw \(\{\cV_{1/0}\}\) of body
1
\begin{equation*}
\{\cV_{1/0}\} = \begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0 \end{Bmatrix}_A =
\dte \; \begin{Bmatrix} \bz_0 \\\\ \bze \end{Bmatrix}_A
+
\dx \; \begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_A
\end{equation*}
This gives the expressions of partial kinematic screws \(\{\cV_{1/0}^{x}\}\) and \(\{\cV_{1/0}^{\theta}\}\text{:}\)
\begin{equation*}
\{\cV_{1/0}^{x}\} =\begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_A , \qquad
\{\cV_{1/0}^{\theta}\} = \begin{Bmatrix} \bz_0 \\\\ \bze \end{Bmatrix}_A
\end{equation*}
Note that there exists a holonomic constraint between the coordinates \((x,\theta)\text{:}\)
\begin{equation*}
x + h \cot\theta = \text{const.}
\end{equation*}
which guarantees contact at \(Q\) and leads to the velocity \(\vel_{Q\in 1/0}= \dx \cos\theta \bx_1\text{.}\) This constraint must be ignored since it violates the requirement that \((x,\te, \dx, \dte)\) be independent variables. The slip velocity at \(Q\) compatible with this requirement is given by \(\vel_{Q\in 1/0}= \dx\bx_0 +\dte \frac{h}{\sin\theta} \by_1\) according to the expression of \(\{\cV_{1/0}\}\text{.}\)
Example 13.1.6.
A sphere
1 of center \(G\) and radius \(r\) rolls without slipping on a horizontal plane \((O,\bx_0, \by_0)\) of a referential 0. Its configuration is defined by the Cartesian coordinates \((x,y)\) of center \(G\) and the Euler angles \((\psi, \theta, \phi )\) as defined in Figure 13.1.7.
Find the corresponding partial kinematic screws \(\{\cV_{1/0}^{q_i}\}\) for \(q= x,y, \psi, \theta, \phi\text{.}\)
Solution.
The velocity \(\vel_G = \dx \bx_0+ \dy \by_0\) of \(G\) and the angular velocity \(\bom_{1/0}= \dpsi \bz_0 + \dte \bu + \dphi \bz_1\) give the expression of kinematic screw \(\{\cV_{1/0}\}\)
\begin{align*}
\{\cV_{1/0}\} \amp =
\begin{Bmatrix}\dpsi \bz_0 + \dte \bu + \dphi \bz_1 \\ \dx \bx_0 + \dy \by_0\end{Bmatrix}_G\\
\amp =
\dx \begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_G
+
\dy \begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_G
+
\dpsi \begin{Bmatrix} \bz_0 \\\\ \bze \end{Bmatrix}_G
+
\dte \begin{Bmatrix} \bu \\\\ \bze \end{Bmatrix}_G
+
\dphi \begin{Bmatrix} \bz_1 \\\\ \bx_0 \end{Bmatrix}_G
\end{align*}
This gives the expressions of the partial kinematic screws \(\{\cV_{1/0}^{q}\}\) for \(q=x,y,\psi,\te,\phi\text{:}\)
\begin{equation*}
\{\cV_{1/0}^{x}\} =\begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_G , \quad
\{\cV_{1/0}^{y}\} =\begin{Bmatrix} \bze \\\\ \by_0 \end{Bmatrix}_G
\end{equation*}
\begin{equation*}
\{\cV_{1/0}^{\psi}\} = \begin{Bmatrix} \bz_0 \\\\ \bze \end{Bmatrix}_G,\quad
\{\cV_{1/0}^{\theta}\} = \begin{Bmatrix} \bu \\\\ \bze \end{Bmatrix}_G,\quad
\{\cV_{1/0}^{\phi}\} = \begin{Bmatrix} \bz_1 \\\\ \bze \end{Bmatrix}_G
\end{equation*}
Note that the no-slip condition \(\vel_{I\in 1/0} =\bze\) at contact point \(I\) must not be taken into account in accordance with the requirement that the variables
\begin{equation*}
(x,y,\psi,\te,\phi,\dx,\dy,\dpsi,\dte,\dphi)
\end{equation*}
are assumed to be independent. Then slip velocity \(\vel_{I\in 1/0}\) takes the expression
\begin{equation*}
\vel_{I\in 1/0} =\dx \bx_0 + \dy \by_0 + r (\dte \bv + \dphi \sin\te \bu) \neq \bze
\end{equation*}
These examples demonstrate that it is always possible to guarantee assumption (13.1.2) even in the case of holonomic or non-holonomic constraint equations between the variables \((\bq, \dbq, t)\text{.}\)
\(\danger\)Care must then be taken to consistently ignore all constraints which may exist between the variables \((\bq, \dbq, t)\text{.}\) This last point will be emphasized in the remaining formulation.
