Kinematics. Overview. Forward Kinematics. Example: 2-Link Structure. Forward Kinematics. Forward Kinematics

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1 Overvew Knemacs Tomas Funkouser Prnceon Unversy C0S 46, Sprng 004 Knemacs Consders only moon Deermned by posons, veloces, acceleraons Dynamcs Consders underlyng forces Compue moon from nal condons and pyscs Example: -Lnk Srucure Two lnks conneced by roaonal jons Forward Knemacs Anmaor specfes jon angles: Θ and Θ Compuer fnds posons of end-effecor: X Θ End-Effecor Θ Θ Θ X ( l cos Θ + l cos( Θ + Θ), l sn Θ + l sn( Θ + Θ )) Forward Knemacs Jon moons can be specfed by splne curves Forward Knemacs Jon moons can be specfed by nal condons and veloces Θ Θ Θ Θ Θ Θ (0) 60 Θ (0) 50 Θ dθ. d dθ 0. d

2 Example: -Lnk Srucure Wa f anmaor knows poson of end-effecor Anmaor specfes end-effecor posons: X Compuer fnds jon angles: Θ and Θ : Θ Θ End-Effecor Θ Θ Θ cos ( l sn( Θ) x + ( l + l cos( Θ)) y Θ ( l sn( Θ )) y + ( l + l cos( Θ )) x x + x l l l l End-effecor posons can be specfed by splne curves Θ Θ x Problem for more complex srucures Sysem of equaons s usually under-defned Mulple soluons Θ Θ y Θ Tree unknowns: Θ, Θ, Θ Two equaons: x, y Soluon for more complex srucures: Fnd bes soluon (e.g., mnmze energy n moon) Non-lnear opmzaon Θ Θ Θ Ballboy Fujo, Mllron, Ngan, & Sanock Prnceon Unversy

3 Summary of Knemacs Forward knemacs Specfy condons (jon angles) Compue posons of end-effecors Inverse knemacs Goal-dreced moon Specfy goal posons of end effecors Compue condons requred o aceve goals Overvew Knemacs Consders only moon Deermned by posons, veloces, acceleraons Dynamcs Consders underlyng forces Compue moon from nal condons and pyscs Inverse knemacs provdes easer specfcaon for many anmaon asks, bu s compuaonally more dffcul Dynamcs Smulaon of pyscs nsures realsm of moon Anmaor specfes consrans: Wa e caracer s pyscal srucure s» e.g., arculaed fgure Wa e caracer as o do» e.g., jump from ere o ere wn me Wa oer pyscal srucures are presen» e.g., floor o pus off and land How e moon sould be performed» e.g., mnmze energy Lasseer `87 Compuer fnds e bes pyscal moon sasfyng consrans Example: parcle w je propulson x() s poson of parcle a me f() s force of je propulson a me Parcle s equaon of moon s: mx' ' f mg 0 Suppose we wan o move from a o b wn 0 o w mnmum je fuel: Mnmze 0 f ( ) d subjec o x( 0 )a and x( )b Dscreze me seps: x x x' x+ x + x x' ' x + x + x m x' ' f mg f Mnmze subjec o x 0 a and x b 0

4 Solve w erave opmzaon meods Advanages: Free anmaor from avng o specfy deals of pyscally realsc moon w splne curves Easy o vary moons due o new parameers and/or new consrans Callenges: Specfyng consrans and objecve funcons Avodng local mnma durng opmzaon Adapng moon: Adapng moon: Orgnal Jump Heaver Base Hurdle Adapng moon: Edng moon: Sk Jump L e al. `99 4

5 Morpng moon: Advanages: Free anmaor from avng o specfy deals of pyscally realsc moon w splne curves Easy o vary moons due o new parameers and/or new consrans Callenges: Specfyng consrans and objecve funcons Avodng local mnma durng opmzaon Glecer `98 Dynamcs Oer pyscal smulaons: Rgd bodes Sof bodes Clo Lquds Gases ec. Ho Gases (Foser & Meaxas `97) Clo (Baraff & Wkn `98) Summary Knemacs Forward knemacs» Anmaor specfes jons (ard)» Compue end-effecors (easy - assn 4!) Inverse knemacs» Anmaor specfes end-effecors (easer)» Solve for jons (arder) Dynamcs Space-me consrans» Anmaor specfes srucures & consrans (eases)» Solve for moon (ardes) Also oer pyscal smulaons 5