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CS545_Lecture_9

# CS545_Lecture_9 - CS545-Contents IX Inverse Kinematics...

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CS545—Contents IX Inverse Kinematics Analytical Methods Iterative (Differential) Methods Geometric and Analytical Jacobian Jacobian Transpose Method Pseudo-Inverse Pseudo-Inverse with Optimization Extended Jacobian Method Reading Assignment for Next Class See http://www-clmc.usc.edu/~cs545

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The Inverse Kinematics Problem Direct Kinematics Inverse Kinematics Possible Problems of Inverse Kinematics Multiple solutions Infinitely many solutions No solutions No closed-form (analytical solution) x = f θ ( ) θ = f 1 x ( )
Analytical (Algebraic) Solutions Analytically invert the direct kinematics equations and enumerate all solution branches Note: this only works if the number of constraints is the same as the number of degrees-of-freedom of the robot What if not? Iterative solutions Invent artificial constraints Examples 2DOF arm See S&S textbook 2.11 ff

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Analytical Inverse Kinematics of a 2 DOF Arm Inverse Kinematics: l 1 l 2 x y x = l 1 cos θ 1 + l 2 cos θ 1 + θ 2 ( ) y = l 1 sin θ 1 + l 2 sin θ 1 + θ 2 ( ) l = x 2 + y 2 l 2 2 = l 1 2 + l 2 2 l 1 l cos γ γ = arccos l 2 + l 1 2 l 2 2 2 l 1 l y x = tan ε θ 1 = arctan y x γ θ 2 = arctan y l 1 sin θ x l 1 cos θ 1 θ 1 γ
Iterative Solutions of Inverse Kinematics Resolved Motion Rate Control Properties Only holds for high sampling rates or low Cartesian velocities “a local solution” that may be “globally” inappropriate Problems with singular postures Can be used in two ways: As an instantaneous solutions of “which way to take “ As an “batch” iteration method to find the correct configuration at a target x = J θ ( ) θ θ = J θ ( ) # x

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