Inverse Manipulator Kinematics (2)Read Chapter 4 and Appendix CPrinciple #2 details: If two equations involved 2 joint angles which are about consecutive parallel axes, then square and add the equations to solve for the angles• Recall that using Principle #1, you obtain the following two equation 2.5 3.24 16.69+12.83.24 16.69-12.8Constant: let it be (1)(2)Square (1) and (2) and add the two together3.2416.69+12.8+2(3.24)(12.8)(+2(16.69)(12.8)(1712.882.94 427.26(3)Using (3) one can solve for 3.How to solve for 2?Approach for 2• Separate 2(move it to the left) 002.52.50 10 00 00 11= 0000112.8000 0 013.2416.691.4411= 3.2416.691.441From the first two equations, you can obtain the following′2.5 3.24 16.69+12.82.53.24 16.69Approach for 2 ‐continued• From the above two equations you can obtain the following:′ ′ • Now you have three sets of equations:• For 1• For 3• For 2 (see the top) 1.4482.94 427.26= Solve for review trigonometric functions 1.44• Convert to cylindrical coordinates: 0, Ø ,Ø; ØwhereØθ Øθ1.44 θØ) .• Is the solution in the second or third quadrant? – it usually generates two solutions: θ.. Solve for 82.94 427.26= • Similar to 1: also two solutions: θ.... 435.2 79.02Solve for ′ ′ • In general, if we have two equations as the above, we have only one solution:θ θ θ θ θ , • So how many solutions we could have: 2 × 1 × 2 = 4• Are those solutions physically possible?OSU Hexapod3311Front View• Two solutions for 3- Knee up - natural- Knee down (green) – not physically possible• Two solutions for 1- Outside the body - natural- Inside the body – not physically
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