Paden–Kahan subproblems are a set of solved geometric problems which occur frequently in inverse kinematics of common robotic manipulators. Although the set of problems is not exhaustive, it may be used to simplify inverse kinematic analysis for many industrial robots. Beyond the three classical subproblems several others have been proposed.
Simplification strategies For a structure equation defined by the product of exponentials method, Paden–Kahan subproblems may be used to simplify and solve the inverse kinematics problem. Notably, the matrix exponentials are non-commutative. Generally, subproblems are applied to solve for particular points in the inverse kinematics problem (e.g., the intersection of joint axes) in order to solve for joint angles.
Eliminating revolute joints Simplification is accomplished by the principle that a rotation has no effect on a point lying on its axis. For example, if the point p {\textstyle p} is on the axis of a revolute twist ξ {\textstyle \xi } , its position is unaffected by the actuation of the twist. To wit: e ξ ^ θ p = p {\displaystyle e^{{\widehat {\xi }}\theta }p=p}
Thus, for a structure equation e ξ ^ 1 θ 1 e ξ ^ 2 θ 2 e ξ ^ 3 θ 3 = g {\displaystyle e^{{\widehat {\xi }}_{1}\theta _{1}}e^{{\widehat {\xi }}_{2}\theta _{2}}e^{{\widehat {\xi }}_{3}\theta _{3}}=g} where ξ 1 {\textstyle \xi _{1}} , ξ 2 {\textstyle \xi _{2}} and ξ 3 {\textstyle \xi _{3}} are all zero-pitch twists, applying both sides of the equation to a point p {\textstyle p} which is on the axis of ξ 3 {\textstyle \xi _{3}} (but not on the axes of ξ 1 {\textstyle \xi _{1}} or ξ 2 {\textstyle \xi _{2}} ) yields e ξ ^ 1 θ 1 e ξ ^ 2 θ 2 e ξ ^ 3 θ 3 p = g p {\displaystyle e^{{\widehat {\xi }}_{1}\theta _{1}}e^{{\widehat {\xi }}_{2}\theta _{2}}e^{{\widehat {\xi }}_{3}\theta _{3}}p=gp} By the cancellation of ξ 3 {\textstyle \xi _{3}} , this yields e ξ ^ 1 θ 1 e ξ ^ 2 θ 2 p = g p {\displaystyle e^{{\widehat {\xi }}_{1}\theta _{1}}e^{{\widehat {\xi }}_{2}\theta _{2}}p=gp} which, if ξ 1 {\textstyle \xi _{1}} and ξ 2 {\textstyle \xi _{2}} intersect, may be solved by Subproblem 2.
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