Robot.ik_GN

Robot.ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)

Fast numerical inverse kinematics by Gauss-Newton optimization

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose or pose trajectory
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | None) – initial joint configuration (default to random valid joint configuration contrained by the joint limits of the robot)
ilimit (int) – maximum number of iterations per search
slimit (int) – maximum number of search attempts
tol (float) – final error tolerance
mask (ndarray | None) – a mask vector which weights the end-effector error priority. Corresponds to translation in X, Y and Z and rotation about X, Y and Z respectively
joint_limits (bool) – constrain the solution to being within the joint limits of the robot (reject solution with invalid joint configurations and perfrom another search up to the slimit)
pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse
pinv_damping (float) – Damping factor for the psuedo-inverse

Return type:

tuple[ndarray, int, int, int, float]

Returns:

tuple (q, success, iterations, searches, residual)

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding to the robot end-effector pose Tep which is an SE3 or ndarray object. This method can be used for robots with any number of degrees of freedom. This is a fast solver implemented in C++.

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

The return value sol is a tuple with elements:

If success == 0 the q values will be valid numbers, but the solution will be in error. The amount of error is indicated by the residual.

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.ik_GN(Tep)
(array([-1.0952, -0.5422,  0.9183, -2.177 ,  0.472 ,  1.8969,  0.4155]), 1, 333, 20, 1.4070724355676836e-10)

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.

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).