Verification of the higher order kinematic analyses equations

• Published in: European journal of mechanics, A, Solids Vol. 61; pp. 198 - 215
• Main Authors: López-Custodio, P.C., Rico, J.M., Cervantes-Sánchez, J.J., Pérez-Soto, G.I., Díez-Martínez, C.R.
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Masson SAS 01.01.2017; Elsevier BV
• Subjects: Chain mobility; Chains; Derivatives; Higher order analysis; Jacobi matrix method; Jacobian matrix; Kinematics; Lie product; Mathematical analysis; Matrix algebra; Matrix methods; Screw theory; Velocity; Lie product; Jacobian matrix; Screw theory; Higher order analysis
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2016.09.010

In this contribution the higher order analyses equations for a kinematic chain are deduced by successive derivation of the velocity analysis equation. In order to simplify the process, a generic expression formed by the Jacobian matrix times a general vector of appropriate dimension is defined. Deriving this expression and forming similar generic expressions, it is possible to obtain first, second and third derivatives of the original expression. From these derivatives, the second -acceleration-, third -jerk- and fourth -jounce- order analyses equations are easily obtained. These equations are compared with those previously obtained by Lerbet and Duffy and his coworkers, and they are found to be in complete agreement. Finally, these expressions are employed to determine the local mobility of three single-loop kinematic chains in singular positions in which most of the techniques for the mobility determination fail. The first example is a kinematotropic linkage. In the second example, the expressions are employed to prove that a seemingly movable linkage is, in reality, a structure; an important detail in this example is that the expressions for the fourth order analysis are required. Finally, a reconfigurable linkage is dealt with, in the third example. •A different way to come up with the higher-order analyses equations derived by Duffy, Rico and Gallardo is shown.•The method allows comparing the equations derived by Duffy, Rico and Gallardo with those derived by Lerbet, showing that they are equivalent.•A more compact notation for the higher order kinematic analyses is developed.•The method allows to show that the equations derived by Duffy, Rico and Gallardo are equivalent to those derived by Lerbet.•The equations obtained can be applied to both open and closed kinematic chains.