Rotation Matrices Three interpretations of rotational matrices Representing the coordinates of a point in two different frames

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1 From Lat Cla Numerial Integration Stabilit v. hoie of te ie Firt orer v. higher orer metho obot Kinemati obot onfiguration Configuration ae Joint oorinate v. workae oorinate Poition Kinemati otation Tranlation otation Matrie Three interretation of rotational matrie ereenting the oorinate of a oint in two ifferent frame 3 otational Tranformation Three interretation of rotational matrie (ont ) Orientation of a tranforme oorinate frame with reet to a fie frame Now aume i a fie oint on the rigi objet with fie oorinate frame O The oint an be rereente in the frame O ( ) again b the rojetion onto the bae frame w v u w v u w v u w v u w v u w v u w v u w v u 4 otating a vetor Three interretation of rotational matrie (ont ) otating a vetor about an ai in a fie frame i.e. in the ame frame E: rotate v about b / v o in in o / / v v

2 Proertie of rotation matrie Summar: Column (row) of are mutuall orthogonal Eah olumn (row) of i a unit vetor T et The et of all n n matrie that have thee roertie are alle the Seial Orthogonal grou of orer n SOn 5 Sine An Comoition of otation Matrie SO SO Then w/ reet to the urrent frame E: three frame O O O BUT EMBE: In general member of SO(3) o not ommute 6 Aoiativit 3 Allow u to ombine rotation: eall: 3D rotation General 3D rotation: SO3 Seial ae Bai rotation matrie Comoition of rotation E: rereent rotation about the urrent -ai b followe b about the urrent -ai o in o in o o o in in in o in o in o in o in in o o in in o o in in o o in in o 7 8

3 3 9 Parameteriing rotation There are three arameter that nee to be eifie to reate arbitrar rigi bo rotation We will eribe three uh arameteriation:. Euler angle. oll Pith Yaw angle 3. Ai/Angle Parameteriing rotation Euler angle otation b about the -ai followe b about the urrent -ai then about the urrent -ai ZYZ Parameteriing rotation oll Pith Yaw angle Three oneutive rotation about the fie rinial ae: Yaw ( ) ith ( ) roll ( ) XYZ Parameteriing rotation Ai/Angle rereentation An rotation matri in SO(3) an be rereente a a ingle rotation about a uitable ai through a et angle For eamle aume that we have a unit vetor: Given we want to erive k : Intermeiate te: rojet the -ai onto k: Where the rotation i given b: k k k k k k

4 4 3 igi motion igi motion i a ombination of rotation an tranlation Define b a rotation matri () an a ilaement vetor () the grou of all rigi motion () i known a the Seial Euliean grou SE(3) Conier three frame O O an O an orreoning rotation matrie an Let be the vetor from the origin o to o from o to o For a oint attahe to o we an rereent thi vetor in frame o an o : 3 3 SO n SO SE Homogeneou tranform We an rereent rigi motion (rotation an tranlation) a matri multiliation Define: Now the oint an be rereente in frame O : Where the P an P are: H H P H P H P P 5 Homogeneou tranform The matri multiliation H i known a a homogeneou tranform an we note that Invere tranform: 3 H SE H T T 6 Homogeneou tranform Bai tranform: Three ure tranlation three ure rotation b a b a Tran Tran Tran ot ot ot

5 Homogeneou ereentation Homogeneou Tranformation ereentation of oint & vetor Proertie. Sum & ifferene of vetor are vetor. Sum of a vetor an a oint i a oint 3. Differene between two oint i a vetor 4. Sum of two oint == meaningle A C B 7 8 Veloit Kinemati Poition kinemati Maniulator: elate the en-effetor oition & orientation to joint variable Wheele obot: elate the robot oition & orientation to wheel angle obot Veloit Kinemati Forwar Kinemati: Invere Kinemati: Veloit kinemati Maniulator: elate the en-effetor linear an angular veloitie w/ joint veloitie Wh Not? Wheele obot: elate the robot linear & angular veloitie w/ wheel veloitie Derive follow a bottom-u aroah Mut onier ontraint imoe b eah wheel Wheel tie together via geometr of robot hai Not Straightforwar 9 5

6 eall the Mobile obot Configuration Sae Configuration Seifiation of oition for all oint on the robot Kinemati Degree of Freeom The minimum number of arameter to efine a onfiguration Configuration Sae Set of all oible onfiguration of the robot In 3-D a rigi link ha 6 DOF Contraint remove DOF evolute & Primati joint imoe 5 ontraint Two Kin of Contraint Inequalit Contraint No two objet an ou the ame ae Contraint (ont.) The evolute Joint Holonomi Contraint: Poition i limite to a ubet of the onfiguration ae C Free Configuration (oe): Holonomi Contraint Inequalitie Contraine Configuration 3 4 6

7 Dimenion of the Configuration Sae Configuration Sae: Set of all oible onfiguration of the robot Dimenion of the Configuration: Number of DOF Number of Holonomi Contraint A Different Te of Contraint Non-holonomi ontraint: Veloit (NOT oition) i limite to a ubet of the onfiguration ae The robot an reah anwhere in the onfiguration ae BUT It i uner-atuate thu veloit i ontraine. 5 6 The Iealie Knife-Ege Contraint The Iealie Knife-Ege Contraint Configuration (oe): Single Point of Contat at C Veloit ontraine to be along the knife ege Single ontat oint at C Veloit ontraine to be along the knife ege Lateral Veloit = Veloit at oint C Contraine veloit: Poition i NOT ontraine! 7 8 7

8 Holonomi v. Non-holonomi: How to Tell The intantaneou veloit at the ontat oint i: Non-Holonomi Contraint The robot an reah everwhere in the onfiguration ae BUT it i uner-atuate an thu the veloit i ontraine. Wheel olling in the Plane The intantaneou veloit of the wheel enter i Teting for Integrabilit of Contraint Given a ontraint in the form of The tet: Whih i integrable: Let Thi how the ontraint i HOLONOMIC! then if ontraint an be integrate 9 3 Mobile obot Drive Differential Steering Co-aial wheel Ineenentl riven Two-imenional Non-holonomiall ontraine Differential Steering: Forwar Kinemati Given the robot geometr an wheel ee what i the robot veloit? Let: r wheel raiu obot Poition l ale length right wheel ee left wheel ee We want 3 3 8

9 Differential Steering: Forwar Kinemati (ont.) Given the robot geometr an wheel ee what i the robot veloit? Let: r wheel raiu Goal l ale length right wheel ee left wheel ee Differential Steering Intantaneou Center of Curvature ICC Forwar Veloit Angular Veloit Worl Coorinate Differential Steering: Invere Kinemati

10 Differential Steering Benefit Simle ontrution Zero minimum turning raiu Drawbak Small error in wheel ee tranlate to large oition error equire two rive motor Wheel-firt i namiall untable Trile: Forwar Kinemati Steerable owere front wheel Free-inning rear wheel r front wheel raiu wheelbae front wheel ee Forwar Veloit Angular Veloit Trile: Intantaneou Center of Curvature Trile: Invere Kinemati What wheel ee an angle are neear to roue a eire robot veloit? 39 4

11 Trile Benefit Doe not require aurate ee mathing Drawbak Non-ero minimum turning raiu More omliate owertrain ICC Akerman Steering 4 4