Kinematics and dynamics of planar mechanisms reinterpreted in rigid-body’s configuration space

Abstract

Mechanisms with lower mobility can be studied by using tools that are directly deduced from those of spatial kinematics as screw theory. Nevertheless, ad-hoc tools that fully exploit the peculiarities of the displacement subgroups these mechanisms move in are usually more efficient both in showing mechanisms’ features and when used to conceive numerical algorithms. Planar displacements constitute a three-dimensional subgroup with many peculiarities that allow the use of simplified tools (e.g., complex numbers) for studying planar mechanisms. Here, the systematic use of three-dimensional vector spaces to represent link poses and velocities in planar motion and planar system of forces is investigated. The result is a novel coherent set of tools that make it possible to geometrically describe kinematics and dynamics of planar mechanisms in the three-dimensional configuration space of links’ planar poses. The effectiveness of this novel approach is shown through a case study.

Appendices

Appendix 1

The following vector equations can be written (see Fig. 5)

$$ \left( {{\mathbf{I}}_{71}^{(1)} {-}{\mathbf{A}}} \right) = {\text{w}}_{1} ( {{\mathbf{B}}{-}{\mathbf{A}}}) $$

(42a)

$$ \left( {{\mathbf{I}}_{71}^{(1)} {-}{\mathbf{A}}} \right) = {\text{w}}_{2} ( {{\mathbf{M}}{-}{\mathbf{L}}}) + ( {{\mathbf{L}}{-}{\mathbf{A}}}) $$

(42b)

$$ \left( {{\mathbf{I}}_{71}^{(2)} {-}{\mathbf{A}}} \right) = {\text{w}}_{3} ( {{\mathbf{B}}{-}{\mathbf{A}}}) $$

(42c)

$$ \left( {{\mathbf{I}}_{71}^{(2)} {-}{\mathbf{A}}} \right) = {\text{w}}_{4} ( {{\mathbf{E}}{-}{\mathbf{D}}}) + ( {{\mathbf{D}}{-}{\mathbf{A}}}) $$

(42d)

where the scalar coefficients w_j, for j = 1,…,4, are unknowns to be determined; whereas, the explicit expressions of (B–A), (L–A) and (D–A) are given by Eqs. (25) and those of (M–L) and (E–D) by Eqs. (26).

The linear elimination of (I ⁽¹⁾₇₁ –A) and (I ⁽²⁾₇₁ –A) from Eqs. (42) yields the following two vector equations

$$ {\text{w}}_{1} ( {{\mathbf{B}}{-}{\mathbf{A}}}) - {\text{w}}_{2} ( {{\mathbf{M}}{-}{\mathbf{L}}}) = ( {{\mathbf{L}}{-}{\mathbf{A}}}) $$

(43a)

$$ {\text{w}}_{3} ( {{\mathbf{B}}{-}{\mathbf{A}}}) - {\text{w}}_{4} ( {{\mathbf{E}}{-}{\mathbf{D}}}) = ( {{\mathbf{D}}{-}{\mathbf{A}}}) $$

(43b)

which are two linear systems of two scalar equations in two unknowns whose solution is

$$ {\text{w}}_{1} = \frac{{({\mathbf{L}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{M}}{-}{\mathbf{L}})]}}{{({\mathbf{B}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{M}}{-}{\mathbf{L}})]}} $$

(44a)

$$ {\text{w}}_{2} = - \frac{{({\mathbf{L}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{B}}{-}{\mathbf{A}})]}}{{({\mathbf{M}}{-}{\mathbf{L}}) \cdot [{\mathbf{k}} \times ({\mathbf{B}}{-}{\mathbf{A}})]}} $$

(44b)

$$ {\text{w}}_{3} = \frac{{({\mathbf{D}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{E}}{-}{\mathbf{D}})]}}{{({\mathbf{B}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{E}}{-}{\mathbf{D}})]}} $$

(44c)

$$ {\text{w}}_{4} = - \frac{{({\mathbf{D}}{-}{\mathbf{A}}) \cdot [{\mathbf{k}} \times ({\mathbf{B}}{-}{\mathbf{A}})]}}{{({\mathbf{E}}{-}{\mathbf{D}}) \cdot [{\mathbf{k}} \times ({\mathbf{B}}{-}{\mathbf{A}})]}} $$

(44d)

Formulas (44) provide the explicit expressions of the scalar coefficients w_j, for j = 1,…,4, as a function of the mechanism configuration. Eventually, the introduction of such expressions into Eqs. (42) yields the explicit expressions of (I ⁽¹⁾₇₁ –A) and (I ⁽²⁾₇₁ –A) as a function of the mechanism configuration.

Appendix 2

The time derivative of Eqs. (27a) and (27b) yields

$$ {\text{n}}_{11} {\dot{\uptheta }}_{ 2 1} + {\text{n}}_{12}\upomega_{71} + {\text{n}}_{13} {\dot{\text{q}}}_{ 1} = 0 $$

(45a)

$$ {\text{n}}_{21} {\dot{\uptheta }}_{ 2 1} + {\text{n}}_{22}\upomega_{71} + {\text{n}}_{23} {\dot{\text{q}}}_{ 2} = 0 $$

(45b)

where

$$ \begin{aligned} {\text{n}}_{11} &= {\text{ab}}\left[ \sin (\uptheta_{71} +\upgamma)\cos\uptheta_{21} \right. \\ &\left. \quad- \cos(\uptheta_{71} +\upgamma) \sin \uptheta_{21} \right] \\ & \quad {-}\,{\text{a}}^{2} \left( \cos\uptheta_{21} \sin q_{1} \right. \\ & \left.\quad - \sin \uptheta_{21} \cos q_{1}\right) + {\text{ca}}\,\cos\uptheta_{21} \\ \end{aligned}$$

(46a)

$$ \begin{aligned} {\text{n}}_{12} & = {\text{ab}}\left[\cos(\uptheta_{71} +\upgamma)\left( { \sin \uptheta_{21}- \sin q_{1} } \right) \right.\\ &\left. \quad - \sin (\uptheta_{71} +\upgamma)\left({\cos\uptheta_{21} -\cos q_{1}} \right)\right] \\&\quad + {\text{cb}}\,\cos(\uptheta_{71} + {\upgamma }) \\\end{aligned} $$

(46b)

$$ \begin{aligned} {\text{n}}_{13} & = {\text{ab}}\left[ \sin q_{1} \cos(\uptheta_{71} +\upgamma) \right. \\ &\left.\quad {-}\,\cos q_{1} \sin (\uptheta_{71} +\upgamma)\right] \\ & \quad {-}\,{\text{a}}^{2} ( \sin \uptheta_{21} \cos q_{1} -\cos\uptheta_{21} \sin q_{1} )\\ &\quad {-}\,{\text{ca}}\,\cos q_{1} \\ \end{aligned} $$

(46c)

$$ \begin{aligned} {\text{n}}_{21} & = {\text{ab}}( \sin \uptheta_{71} \cos\uptheta_{21} - \cos\uptheta_{71} \sin \uptheta_{21} ) \\ & \quad {-}\,{\text{a}}^{2} (\cos\uptheta_{21} \sin q_{2} - \sin \uptheta_{21} \cos q_{2} ) \\ &\quad+ {\text{ha}}\, \sin \uptheta_{21} \\ \end{aligned} $$

(46d)

$$ \begin{aligned} {\text{n}}_{22} & = {\text{ab}}\left[\cos\uptheta_{71} ( \sin \uptheta_{21} - \sin q_{2} )\right. \\ & \left.\quad -\, \sin \uptheta_{71} (\cos\uptheta_{21} - \cos q_{2} )\right] + {\text{hb}}\, \sin \uptheta_{71} \\ \end{aligned} $$

(46e)

$$ \begin{aligned} {\text{n}}_{23} & = {\text{ab}}(\cos\uptheta_{71} \sin q_{2} - \sin \uptheta_{71} \cos q_{2} ) \\ & \quad{-}\,{\text{a}}^{2} ( \sin \uptheta_{21} \cos q_{2} - \cos\uptheta_{21} \sin q_{2} )\\ &\quad {-}\,{\text{ha}}\, \sin q_{2} \\ \end{aligned} $$

(46f)

The subtraction of the product of Eq. (45a) by n₂₁ from the product of Eq. (45b) by n₁₁ yields

$$ \left( {{\text{n}}_{22} {\text{n}}_{11} - {\text{n}}_{21} {\text{n}}_{12} } \right)\upomega_{71} = {\text{n}}_{21} {\text{n}}_{13} {\dot{\text{q}}}_{ 1} - {\text{n}}_{23} {\text{n}}_{11} {\dot{\text{q}}}_{ 2} $$

(47)

whose comparison with Eq. (28b) gives, for g₁ and g₂, the following explicit expressions as a function of the mechanism configuration

$$ {\text{g}}_{1} = \frac{{{\text{n}}_{21} {\text{n}}_{13} }}{{{\text{n}}_{22} {\text{n}}_{11} - {\text{n}}_{21} {\text{n}}_{12} }} $$

(48a)

$$ {\text{g}}_{2} = \frac{{{\text{n}}_{23} {\text{n}}_{11} }}{{{\text{n}}_{21} {\text{n}}_{12} - {\text{n}}_{22} {\text{n}}_{11} }} $$

(48b)