page_title: Paden–Kahan subproblems
section_title: Solution
hierarchical_section_title: Paden–Kahan subproblems / Subproblem 2: Rotation about two subsequent axes / Solution
caption_attribution_description: English: An illustration of Paden-Kahan subproblem 2, showing a case with 2 intersecting circles and therefore two solutions (c, c2) with both solutions highlighted. Created in Geogebra.
context_page_description: 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.
context_section_description: Provided that the two axes are not parallel (i.e., ), let  be a point such that 
   In other words,  represents the point to which  is rotated around one axis before it is rotated around the other axis to be coincident with . Each individual rotation is equivalent to Subproblem 1, but it’s necessary to identify one or more valid solutions for  in order to solve for the rotations.
  
  Let  be the point of intersection of the two axes:
  Define the vectors ,  and . Therefore,
  This implies that , , and . Since ,  and  are linearly independent,  can be written as
  
  The values of the coefficients may be solved thus:
  , and
  The subproblem yields two solutions in the event that the circles intersect at two points; one solution if the circles are tangential; and no solution if the circles fail to intersect.
caption: An illustration of Paden–Kahan subproblem 2, showing a case with two intersecting circles and therefore two solutions. Both solutions (c, c2) are highlighted.
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