IK_GN - Gauss-Newton Numerical IK

class roboticstoolbox.robot.IK.IK_GN(name='IK Solver', 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)[source]

Bases: IKSolver

Gauss-Newton Numerical Inverse Kinematics Solver

A class which provides functionality to perform numerical inverse kinematics (IK) using the Gauss-Newton method. See step method for mathematical description.


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

  • name (str) – The name of the IK algorithm

  • ilimit (int) – How many iterations are allowed within a search before a new search is started

  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • tol (float) – Maximum allowed residual error E

  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority

  • joint_limits (bool) – Reject solutions with joint limit violations

  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate vectors

  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of the normal inverse

  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solution

  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely from the solution

  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limit

  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion becomes active


The following example gets the ets of a panda robot object, instantiates the IK_GN solver class using default parameters, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the solve method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda().ets()
>>> solver = rtb.IK_GN(pinv=True)
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> solver.solve(panda, Tep)
IKSolution(q=array([ 1.2886,  1.4862, -2.1934, -2.0771,  1.244 ,  1.1468, -0.096 ]), success=True, iterations=53, searches=4, residual=9.796561164911834e-10, reason='Success')


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. When the the problem is solvable, it converges very quickly.

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.


  • 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).

See also


An abstract super class for numerical IK solvers


Implements IKSolver using the Newton-Raphson method


Implements IKSolver using the Levemberg-Marquadt method


Implements IKSolver using a quadratic programming approach

Changed in version 1.0.3: Added the Gauss-Newton IK solver class


step(ets, Tep, q)

Performs a single iteration of the Gauss-Newton optimisation method

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

Private Methods

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits