ETS.ikine_QP

ETS.ikine_QP(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]

Quadratic programming numerical inverse kinematics solver

Parameters:

Tep (ndarray | SE3) – the desired end-effector pose
q0 (Union[ndarray, List[float], Tuple[float, ...], None]) – the initial joint coordinate vector
ilimit (int) – maximum iterations allowed per search
slimit (int) – maximum search attempts before failure
tol (float) – maximum allowed residual error E
mask (Union[ndarray, List[float], Tuple[float, ...], None]) – a 6-vector weighting Cartesian DoF error priority
joint_limits (bool) – reject solutions with joint limit violations
seed (int | None) – seed for the RNG used to generate random joint configurations
kj – gain for joint velocity norm minimisation
ks – gain adjusting the cost of slack (intentional error)
kq (float) – gain for joint limit avoidance (0.0 disables)
km (float) – gain for manipulability maximisation (0.0 disables)
ps (float) – minimum joint approach distance to limit (radians or metres)
pi (ndarray | float) – null-space influence distance (radians or metres)

Returns:

IK solution

Raises:

ImportError – if the package qpsolvers is not installed

A method that provides functionality to perform numerical inverse kinematics (IK) using a quadratic programming approach.

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

Each iteration uses the following approach

\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]

where the QP is defined as

\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ \text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ \mathcal{A} \vec{x} &\leq \mathcal{B}, \\ \vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]

with

\[\begin{split}\vec{x} &= \begin{pmatrix} \dvec{q} \\ \vec{\delta} \end{pmatrix} \in \mathbb{R}^{(n+6)} \\ \mathcal{Q} &= \begin{pmatrix} \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} \end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ \mathcal{J} &= \begin{pmatrix} \mat{J}(\vec{q}) & \mat{1}_{6} \end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ \mathcal{C} &= \begin{pmatrix} \mat{J}_m \\ \bf{0}_{6 \times 1} \end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ \mathcal{A} &= \begin{pmatrix} \mat{1}_{n \times n + 6} \\ \end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ \mathcal{B} &= \eta \begin{pmatrix} \frac{\rho_0 - \rho_s} {\rho_i - \rho_s} \\ \vdots \\ \frac{\rho_n - \rho_s} {\rho_i - \rho_s} \end{pmatrix} \in \mathbb{R}^{n} \\ \vec{x}^{-, +} &= \begin{pmatrix} \dvec{q}^{-, +} \\ \vec{\delta}^{-, +} \end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, \(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the cost of the norm of the slack vector in the optimiser, \(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and \(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

Examples

The following example gets the ets of 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_QP method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda().ets()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_QP(Tep)
!! ImportError: the package qpsolvers is required for this class. 
Install using 'pip install qpsolvers' [ERR robot/ETS.py:110:roboticstoolbox.robot.ETS.ETS.ikine_QP (stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.rst)]

Notes

When using 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).