Robot.ikine_LM
- Robot.ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
Levenberg-Marquadt Numerical Inverse Kinematics Solver
A method which provides functionality to perform numerical inverse kinematics (IK) using the Levemberg-Marquadt method.
See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.
- Parameters:
end (
Union
[str
,Link
,Gripper
,None
]) – the link considered as the end-effectorstart (
Union
[str
,Link
,Gripper
,None
]) – the link considered as the base frame, defaults to the robots’s base frameq0 (
Union
[ndarray
,List
[float
],Tuple
[float
],Set
[float
],None
]) – The initial joint coordinate vectorilimit (
int
) – How many iterations are allowed within a search before a new search is startedslimit (
int
) – How many searches are allowed before being deemed unsuccessfultol (
float
) – Maximum allowed residual error Emask (
Union
[ndarray
,List
[float
],Tuple
[float
],Set
[float
],None
]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priorityjoint_limits (
bool
) – Reject solutions with joint limit violationsseed (
Optional
[int
]) – A seed for the private RNG used to generate random joint coordinate vectorsk (
float
) – Sets the gain value for the damping matrix Wn in the next iteration. See synopsismethod (
Literal
['chan'
,'wampler'
,'sugihara'
]) – One of “chan”, “sugihara” or “wampler”. Defines which method is used to calculate the damping matrix Wn in thestep
methodkq (
float
) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solutionkm (
float
) – The gain for maximisation. Setting to 0.0 will remove this completely from the solutionps (
float
) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limitpi (
Union
[ndarray
,float
]) – The influence angle/distance (in radians or metres) in null space motion becomes active
Synopsis
The operation is defined by the choice of the
method
kwarg.The step is deined as
\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( \mat{A}_k \right)^{-1} \bf{g}_k \\ % \mat{A}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} + \mat{W}_n\end{split}\]where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is non-singular and positive definite. The performance of the LM method largely depends on the choice of \(\mat{W}_n\).
Chan’s Method
Chan proposed
\[\mat{W}_n = λ E_k \mat{1}_n\]where λ is a constant which reportedly does not have much influence on performance. Use the kwarg
k
to adjust the weighting term λ.Sugihara’s Method
Sugihara proposed
\[\mat{W}_n = E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), and \(l\) is the length of a typical link within the manipulator. We provide the variable
k
as a kwarg to adjust the value of \(w_n\).Wampler’s Method
Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the
k
kwarg.Examples
The following example makes a
panda
robot object, makes a goal poseTep
, and then solves for the joint coordinates which result in the poseTep
using theikine_LM
method.>>> import roboticstoolbox as rtb >>> panda = rtb.models.Panda() >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) >>> panda.ikine_LM(Tep) IKSolution(q=array([ 0.3059, -0.3126, -0.2739, -2.199 , -0.0959, 1.9959, 0.8611]), success=True, iterations=8, searches=1, residual=4.99812874879148e-07, reason='Success')
Notes
The value for the
k
kwarg will depend on themethod
chosen and the arm you are using. Use the following as a rough guidechan, k = 1.0 - 0.01
,wampler, k = 0.01 - 0.0001
, andsugihara, k = 0.1 - 0.0001
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).
See also
Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the
Robot
class