The method of cut-joint kinematic constraint: velocity propagations

Abstract

Based on a recent formulation for parametric design of mechanical systems using kinematic analysis, we reported an approach to automated design by converting mechanical part geometry into a mechanism by breaking the part into sub-parts attached by higher pair joints. In order to evaluate the performance of a mechanism where a design change has been introduced into one of its links, we extend the formulation in this paper to address the propagation of this change and its effect on the velocity of other members. This analysis will limit itself to mechanisms. Velocity propagations due to modifications in the design of a mechanical part are analytically addressed using a proposed working coordinates formulation based on the cut-joint kinematic constraint method (presented earlier). Velocity vectors in state-vector notation are derived and treated as variables such that propagations of working coordinates of a link are obtained with respect to position and orientation. The underlying theory is presented and several planar and spatial mechanisms are treated.

Appendix

A pair of bodies i and j shown in Fig. 14 are connected by a joint. The global reference frame is xyz, the body reference frame is x'y'z', and the joint reference frame is $x^{{\prime \prime }}_{i} y^{{\prime \prime }}_{i} z^{{\prime \prime }}_{i}$.

Fig. 14

Kinematic notation for vectors

The position and orientation of body i is uniquely determined by vector r _i and Euler parameter P _i where ${\mathbf{P}}_{i} = [\begin{array}{*{20}c} {{e_{o} }} & {{{\mathbf{e}}^{T} }} \\ \end{array} ]^{T}$and where e _o and ${\mathbf{e}}^{T} = [\begin{array}{*{20}c} {{e_{1} }} & {{e_{2} }} & {{e_{3} }} \\ \end{array} ]^{T}$ are minimal representation Euler parameters specifying the orientation of link i. The transformation matrix A _i , written as

$${\mathbf{A}}_{i} = (2e_{{\user2{0}}} ^{2} - 1){\mathbf{I}} + 2({\mathbf{ee}}^{{\user2{T}}} + e_{{\user2{0}}} \widetilde{{\mathbf{e}}})$$

(28)

where I is the identity matrix, is an orthogonal transformation matrix which transforms a vector in the body i reference frame to the global reference frame.

The position of body j can be defined with respect to body i (Fig. 4) by

$${\mathbf{r}}_{j} = {\mathbf{r}}_{i} + {\mathbf{A}}_{i} {\mathbf{s}}^{\prime }_{{ij}} + {\mathbf{A}}_{i} {\mathbf{C}}_{{ij}} {\mathbf{d}}^{{\prime \prime }}_{{ij}} ({\mathbf{q}}_{j} )$$

(29)

where q _j are the generalized coordinates. The angular velocity of body j, denoted by ω _j , can be expressed as $\omega _{j} = \omega _{i} + {\mathbf{H}}_{j} {\mathop {\mathbf{q}}\limits^\cdot }_{j}$, where the transformation matrix H _j has columns that represent the axes of rotation for each component of the relative velocity vector ${\mathop {\mathbf{q}}\limits^\cdot }_{j}$ of body j with respect to body i . Differentiating Eq. 29 and combining with the equation for ω _j as an augmented matrix yields

$${\left[ {\begin{array}{*{20}c} {{{\mathop {\mathbf{r}}\limits^\cdot }_{j} + \widetilde{{\mathbf{r}}}_{j} \omega _{j} }} \\ {{\omega _{j} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{{\mathop {\mathbf{r}}\limits^\cdot }_{i} + \widetilde{{\mathbf{r}}}_{i} \omega _{i} }} \\ {{\omega _{i} }} \\ \end{array} } \right]} + {\left[ {\begin{array}{*{20}c} {{\frac{{\partial {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} }} + \widetilde{{\mathbf{r}}}_{j} {\mathbf{H}}_{j} }} \\ {{{\mathbf{H}}_{j} }} \\ \end{array} } \right]}{\mathop {\mathbf{q}}\limits^\cdot }_{j} $$

(30)

where the tilde operator (the symbol ∼) is used to denote the skew-symmetric matrix generated by the associated vector. Equation 30 written in state-vector notation is

$$\widehat{{\mathbf{Y}}}_{j} = \widehat{{\mathbf{Y}}}_{i} + {\mathbf{B}}_{j} {\mathop {\mathbf{q}}\limits^\cdot }_{j} $$

(31)

where B _j is the velocity transformation matrix, defined as ${\mathbf{B}}_{j} = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} }} + \widetilde{{\mathbf{r}}}_{j} {\mathbf{H}}_{j} }} \\ {{{\mathbf{H}}_{j} }} \\ \end{array} } \right]}$, $\widehat{{\mathbf{Y}}}_{j}$ is the state-vector velocity of body j, defined as $\widehat{{\mathbf{Y}}}_{j} = {\left[ {\begin{array}{*{20}c} {{{\mathop {\mathbf{r}}\limits^\cdot }_{j} + \widetilde{{\mathbf{r}}}_{j} \omega _{j} }} \\ {{\omega _{j} }} \\ \end{array} } \right]}$, and the Cartesian velocity vector ${\mathbf{Y}}_{j} = {\left[ {\begin{array}{*{20}c} {{{\mathop {\mathbf{r}}\limits^\cdot }_{j} }} & {{\omega _{j} }} \\ \end{array} } \right]}^{T}$ can be recovered from

$${\mathbf{Y}}_{j} = {\mathbf{T}}_{j} \widehat{{\mathbf{Y}}}_{j} $$

(32)

where the transformation matrix ${\mathbf{T}}_{j} = {\left[ {\begin{array}{*{20}c} {{\mathbf{I}}} & {{ - \widehat{{\mathbf{r}}}_{j} }} \\ {{\mathbf{0}}} & {{\mathbf{I}}} \\ \end{array} } \right]}$.

1.1 Variational form of the position and velocity state-vectors

We define the term working coordinates (denoted by wc) as the position and orientation of the link that is permitted to undergo design changes denoted by r _wc and P _wc, respectively. Changes in ${\mathbf{s}}^{\prime }_{{ij}}$ and P _ij affect the kinematic relations between bodies and influence the dynamic characteristics of the underlying mechanism. The goal of the derivation in this section is to obtain the variational position and velocity state vectors. To derive the general design propagation equations, the variations of Eq. 29 are determined as

$$\delta {\mathbf{r}}_{j} = \delta {\mathbf{r}}_{i} + \delta \widetilde{{\mathbf{\pi }}}_{i} {\left( {{\mathbf{r}}_{j} - {\mathbf{r}}_{i} } \right)} + \frac{{\partial {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} }}\delta {\mathbf{q}}_{j} + {\mathbf{A}}_{i} \delta {\mathbf{s}}^{\prime }_{{ij}} - 2\widetilde{{\mathbf{d}}}_{{ij}} {\mathbf{A}}_{i} {\mathbf{E}}_{{ij}} \delta {\mathbf{P}}_{{ij}} $$

(33)

and the virtual rotation vector as

$$\delta {\mathbf{\pi }}_{j} = \delta {\mathbf{\pi }}_{i} + {\mathbf{H}}_{j} ({\mathbf{A}}_{i} ,{\mathbf{q}}_{j} )\delta {\mathbf{q}}_{j} + 2{\mathbf{A}}_{i} {\mathbf{E}}_{{ij}} \delta {\mathbf{P}}_{{ij}} $$

(34)

where the global vector δπ, is the virtual rotation of the transformation matrix A from the body reference frame x'y'z' to the global reference frame xyz and $\delta \widetilde{{\mathbf{\pi }}}$ is its skew symmetric matrix defined as $\delta \widetilde{{\mathbf{\pi }}} = \delta {\mathbf{AA}}^{T}$ and E _ij is the Euler parameter semi-rotation matrix such that

$${\mathbf{E}}_{{{\user2{ij}}}} = [ - {\mathbf{e}},\widetilde{{\mathbf{e}}} + e_{{\user2{0}}} {\mathbf{I}}]$$

(35)

In state-vector notation, Eqs. 33 and 34 are written as

$$\delta \widehat{{\mathbf{z}}}_{j} = \delta \widehat{{\mathbf{z}}}_{i} + {\mathbf{B}}_{j} \delta {\mathbf{q}}_{j} + {\mathbf{M}}_{i} \delta {\mathbf{s}}^{\prime }_{{ij}} + 2{\mathbf{N}}_{j} {\mathbf{A}}_{i} {\mathbf{E}}_{{ij}} \delta {\mathbf{P}}_{{ij}} $$

(36)

where $\delta \widehat{{\mathbf{z}}}_{j} = {\left[ {\begin{array}{*{20}c} {{\delta {\mathbf{r}}_{j} + \widetilde{{\mathbf{r}}}_{j} \delta {\mathbf{\pi }}_{j} }} \\ {{\delta {\mathbf{\pi }}_{j} }} \\ \end{array} } \right]}$ ${\mathbf{M}}_{i} = {\left[ {\begin{array}{*{20}c} {{{\mathbf{A}}_{i} }} \\ {{\mathbf{0}}} \\ \end{array} } \right]}$ and ${\mathbf{N}}_{j} = {\left[ {\begin{array}{*{20}c} {{\widetilde{{\mathbf{r}}}_{j} - \widetilde{{\mathbf{d}}}_{{ij}} }} \\ {{\mathbf{I}}} \\ \end{array} } \right]}$. To transform to Cartesian coordinates, the relationship between the variational position state-vector $\delta \widehat{{\mathbf{z}}}_{j}$ and variational position vector δz _j is $\delta {\mathbf{z}}_{j} = {\mathbf{T}}_{j} \delta \widehat{{\mathbf{z}}}_{j}$ where T _j is defined in Eq. 32. Taking the variation of Eq. 5, the variational velocity state-vector is obtained as

$$\delta \widehat{{\mathbf{Y}}}_{j} = \delta \widehat{{\mathbf{Y}}}_{i} + {\mathbf{B}}_{j} \delta {\mathop {\mathbf{q}}\limits^\cdot }_{j} + \delta {\mathbf{B}}_{j} {\mathop {\mathbf{q}}\limits^\cdot }_{j} $$

(37)

where

$$\delta \widehat{{\mathbf{Y}}}_{j} = {\left[ {\begin{array}{*{20}c} {{\delta {\mathop {\mathbf{r}}\limits^\cdot }_{j} + \widetilde{{\mathbf{r}}}_{j} \delta \omega _{j} + \delta \widetilde{{\mathbf{r}}}_{j} \omega _{j} }} \\ {{\delta \omega _{j} }} \\ \end{array} } \right]}$$

(38)

$$\delta {\mathbf{B}}_{j} = {\left[ {\begin{array}{*{20}c} {{\widetilde{{\mathbf{r}}}_{j} \delta {\mathbf{H}}_{j} + \delta \widetilde{{\mathbf{r}}}_{j} {\mathbf{H}}_{j} + \delta \widetilde{\pi }_{i} \frac{{\partial {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} }} + {\left( {2{\mathbf{A}}_{i} {\mathbf{E}}_{{ij}} \delta {\mathbf{P}}_{{ij}} } \right)}\frac{{\partial {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} }} + \frac{{\partial ^{2} {\mathbf{d}}_{{ij}} }}{{\partial {\mathbf{q}}_{j} ^{2} }}\delta {\mathbf{q}}_{j} }} \\ {{\delta {\mathbf{H}}_{j} }} \\ \end{array} } \right]}$$

(39)

The relationship between the variational velocity state-vector and the variational velocity vector is

$$\delta {\mathbf{Y}}_{j} = {\mathbf{T}}_{j} \delta \widehat{{\mathbf{Y}}}_{j} - {\mathbf{Q}}_{j} $$

(40)

where $\delta {\mathbf{Y}}_{j} = [\begin{array}{*{20}c} {{\delta {\mathop {\mathbf{r}}\limits^\cdot }_{j} ^{T} }} & {{\delta \omega _{j} }} \\ \end{array} ]^{T}$ and ${\mathbf{Q}}_{j} = - \delta {\mathbf{T}}_{j} \widehat{{\mathbf{Y}}}_{j} = [\begin{array}{*{20}c} {{\delta \widetilde{{\mathbf{r}}}_{j} \omega _{j} }} & {{\text{0}}} \\ \end{array} ]^{T}$ The above formulation will be used to obtain the kinematic sensitivity of propagated changes in design variables.