Inverse Manipulator Kinematics (1)Read Chapter 4 and Appendix CI. Introduction• Forward kinematics- Given joint angles →compute the position of the gripper (foot) with respect to the base (the body)• Inverse kinematics- An opposite problem to the forward kinematics: given the position of the gripper (foot) calculate the joint angles –much harder FixedFunction of θi(di), i = 1, …nGivenqiII. Use OSU Hexapod as an examplen = 3 < 6• Specify the position of the foot with 3 DOF•• Focus on the position only•• is the position of the foot in the body coordinate frame 1 0001Position of the foot in the foot coordinate frameSteps for solving the I.K. problem (1)• Step (a): Isolate - the manipulator (leg) transformation matrix1= 0001)10001000100011= 3.2416.691.44111= 112.82.512.82.512.80 0 013.2416.691.441Steps for solving the I.K. problem (1)• Step (b): Solve for , i= 1, 2, 3– How many equations in Step (a)? - 3– How to solve the equations?– Some special techniques have to be used• Inverse Kinematics Procedure– For simple manipulators:– Algebraic approach which gives a closed-form solution; not an iterative numeric approach– Good example in the text (Section 4.7, pp. 117-121) which uses PUMA 560 robot• Principle #1: Separate out the dependence on the joint angles between the right and left sides of the equation • Principle #2: If two equations involved 2 joint angles which are about consecutive parallel axes, then square and add the equations to solve for the anglesqii=0, ±90; di=0Principle #1 details: Separate out the dependence on the joint angles between the right and left sides of the equation 1= 3.2416.691.4410000000010011= 0001012.82.5012.80 0 013.2416.691.441• From the above you can solve for :1 1.44• Other two equationsUse Principle #2 to solve for and . 2.5 3.24 16.69+12.83.24
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