雅可比矩阵逆解

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详细介绍了雅可比矩阵逆解的思路与方法,英文版,值得研究
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 nvent artificial constraints e EXamples 2DOF arm See s&s textbook 2.11 ff Analytical Inverse Kinematics of a 2 DOF Arm y x=l cos 0, +L2 cos(0, +02) e y=4 sin, +l2sin(0, +02) X o Inverse Kinematics 2 =√x2+y 2=12+12-2l1cosy + r= arccos 21l tane→1= arctan y Sin 2- arctan x-l, cos 8, Iterative Solutions of Inverse Kinematics Resolved motion rate control x=J(0) 0=J(0)X Properties Only holds for high sampling rates or low Cartesian velocities a local solution"that may be globally"inappropriate o Problems with singular postures o 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 Essential in resolved motion Rate ethods: The Jacobian Jacobian of direct kinematics Analytical Jacobian 6 dx df(e) J(0) 000 o In general, the Jacobian(for Cartesian positions and orientations ) has the following form(geometrical Jacobian) H1 f() 卩1 -1 for a prismatic」ont where ail(p-pi-\for a revolute joint p i is the vector from the origin of the world coordinate system to the origin of the i-th link coordinate system, p is the vector from the origin to the endeffector end, and z is the i-th joint axis(p. 72 S&s) The Jacobian Transpose Method △O=J()△x perating Principle Project difference vector Dx on those dimensions g which can reduce it the most Advantag es Simple computation(numerically robust No matrix inversions ● Disadvantages Needs many iterations until convergence in certain configurations(e. g Jacobian has very small coefficients) Unpredictable joint configurations Non conservative Jacobian Transpose Derivation Minimize cost function F=5(xarget -x)( larger -x) (xm-f()(xm-(6) with respect to aby gradient descent T dF △0=-0 df(e) target CJ(6) CJ(6)△x Jacobian Transpose Geometric Intuition Target 3 0 The pseudo Inverse Method △0=J(6(J(6)()△x=J△x perating Principle Shortest path in q-space ° Advantages Computationally fast (second order method) ● Disadvantages: Matrix inversion necessary(numerical problems) Unpredictable joint configurations Non conservative

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