Numerical Techniques
Many six- and five-DOF kinematic structures lack the necessary architectural simplicity for closed-form inverse kinematic solutions. Solving such manipulators requires the use of numerical iterative techniques. For six-DOF robots, equation (2.14) can be expressed as a system of six nonlinear equations in the six joint variables of the form
fl(e1, e2, e3, e4, e5, e6) = Px f2(91' e2' e3' G4' e5' e6) = Py f3(81' e2, e3, e4, 5, 6) = Pz f4(el, e2, 83, e4, e5, e6) = a f5(e1, e2, e3, e4, S5, e6) = E f6(811 e2, e3, e4, e5, e6) =
where px, Py, and pz are the coordinates of the origin of the end-effector frame and a, e, and are either the Euler angles or the roll-pitch-yaw angles derived- from the orientation matrix R of the end-effector frame (Paul 1981).
The six-dimensional equation is then solved by use of a direct or modified Newton-Raphson ot- similar technique. Multidimensional iterative techniques for solving the inverse kinematics problem of manipulators of arbitrary architecture are described by Angeles (1985-, -1986), Goldenberg- Benhabib', & Fentorn (1985)."- Goldenberg:and Lawrence (1985). The computational efficiency of these
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