# Robot.ikine_GN

Robot.ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)

Gauss-Newton Numerical Inverse Kinematics Solver

A method which provides functionality to perform numerical inverse kinematics (IK) using the Gauss-Newton method.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Note

When using this method with redundant robots (>6 DoF), pinv must be set to True

Parameters:

Synopsis

Each iteration uses the Gauss-Newton optimisation method

$\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} \right)^{-1} \bf{g}_k \\ \bf{g}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \vec{e}_k\end{split}$

where $$\mat{J} = {^0\mat{J}}$$ is the base-frame manipulator Jacobian. If $$\mat{J}(\vec{q}_k)$$ is non-singular, and $$\mat{W}_e = \mat{1}_n$$, then the above provides the pseudoinverse solution. However, if $$\mat{J}(\vec{q}_k)$$ is singular, the above can not be computed and the GN solution is infeasible.

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ikine_GN method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_GN(Tep)
IKSolution(q=array([-0.792 ,  1.2674, -0.2142, -2.8253,  0.1563,  1.0283, -2.2696]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')


Notes

When using the this method, the initial joint coordinates $$q_0$$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

• J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

• J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

ikine_LM

Implements the IK_LM class as a method within the ETS class

ikine_NR

Implements the IK_NR class as a method within the ETS class

ikine_QP

Implements the IK_QP class as a method within the ETS class

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class