# Robot.ikine_QP

Robot.ikine_QP(Tep, end=None, start=None, 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)

Quadratic Programming Numerical Inverse Kinematics Solver

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

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

Parameters:
Raises:

ImportError – If the package qpsolvers is not installed

Synopsis

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 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.

  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/robot/IK.py", line 1331, in __init__
"the package qpsolvers is required for this class. \nInstall using 'pip"
ImportError: the package qpsolvers is required for this class.
Install using 'pip install qpsolvers'


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_GN

Implements the IK_GN class as a method within the ETS class

ikine_NR

Implements the IK_NR class as a method within the ETS class

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class