Stereographic SEW Angle

Implementation of stereographic SEW (shoulder-elbow-wrist) angle for 7-DOF robot arms as well as inverse kinematics solutions for a number of 7-DOF arms from "Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators". The IK procedures also work with the conventional SEW angle.

Abstract: Seven degree-of-freedom (DOF) robot arms have one redundant DOF which does not change the motion of the end effector. The redundant DOF offers greater manipulability of the arm configuration to avoid obstacles and singularities, but it must be parameterized to fully specify the joint angles for a given end effector pose. For 7-DOF revolute (7R) manipulators, we introduce a new concept of generalized shoulder-elbow-wrist (SEW) angle, a generalization of the conventional SEW angle but with an arbitrary choice of the reference direction function. The SEW angle is widely used and easy for human operators to visualize as a rotation of the elbow about the shoulder-wrist line. Since other redundancy parameterizations including the conventional SEW angle encounter an algorithmic singularity along a line in the workspace, we introduce a special choice of the reference direction function called the stereographic SEW angle which has a singularity only along a half-line, which can be placed out of reach. We prove that such a singularity is unavoidable for any parameterization. We also include expressions for the SEW angle Jacobian along with singularity analysis. Finally, we provide efficient and singularity-robust inverse kinematics solutions for most known 7R manipulators using the general SEW angle and the subproblem decomposition method. These solutions are often closed-form but may sometimes involve a 1D or 2D search in the general case. Search-based solutions may be converted to finding zeros of a high-order polynomial. Inverse kinematics solutions, examples, and evaluations are available in a publicly accessible repository.

For examples using the polynomial method, see the subproblem-polynomial repo.

For 6-DOF IK solutions see ik-geo. Code in this repo depends on that code.

This repo also depends on matlab-diagrams for a few visualizations.

Interactive Demos

Conventional SEW Angle

Stereographic SEW Angle

Multimedia Extensions

Comparing conventional and stereographic SEW angles

Sawyer inverse kinematics solutions using 1D search

Folder Breakdown

+SEW_IK: Inverse kinematics solutions for different robot types

+SEW_IK_setups: Test classes for IK solutions

+robot_kin: Kinematic parameters for robot examples

IK_helpers: Functions to help with robot IK (and forward kinematics)

SEW_comparison_demo: Demonstration of stereographic vs conventional SEW angle

correctness_tests: Tests to verify the correctness IK solutions

robot_examples: Demonstration of IK for specific robots

unit_tests: Testing for smaller helper functions

verify_eqns: Verify equations used in paper

Contributing

If you have any questions, improvements you'd like to make, or even ideas or requests for improvements, please start a GitHub issue or send an email.

Tutorial: KUKA LBR iiwa 14 R820

Here we will demonstrate finding all IK solutions for the KUKA LBR iiwa 14 R820 using MATLAB. This robot falls into the R-2R-3R kinematic family and therefore has closed-form IK solutions.

You can follow along by running robot_examples/kuka.m.

Download and add to path

Other than MATLAB, you will need to download the following GitHub repos onto your local machine:

1. stereo-sew (This repo, for 7-DOF IK code and SEW code)
2. ik-geo (For subproblem solution code)

Then, add the code the the MATLAB path. Right click on each folder in MATLAB and click Add To Path > Selected Folders and Subfolders

Define robot kinematics parameters

IK-Geo uses the product of exponentials convention to define the robot kinematic parameters. (If you already have the Denavit-Hartenberg parameters, you can use dh_to_kin.m to convert.)

Kinematic parameters for the KUKA LBR iiwa 14 R820 are already defined in +robot_kin/kuka.m. The columns of kin.Pare $p_{i, i+1}$ and $p_{7T}$, the columns of kin.H are $h_{i}$, and kin.joint_type is all zeros because all joints are revolute.

kin = robot_kin.kuka();

Define desired end effector pose and SEW angle

The 6-DOF end effector pose is defined in this base by $R_{07}$ and $R_{0T}$. The remaining 1 DOF is parameterized by the SEW angle. In this case we use the stereographic SEW angle $\pi/6$ with $e_t$ pointing in the negative $z$ direction and $e_r$ pointing in the positive $y$ direction.

R_07 = rot([1;0;0], pi/4);
p_0T = [0.2; 0.3; 0.4];
SEW = sew_stereo([0;0;-1], [0;1;0]);
psi = pi/6;

Perform inverse kinematics

We call the IK_2R_2R_3R() function using the kinematic parameters and desired 7-DOF pose. Each column of Q is a vector q of joint angles. Each element of is_LS_vec is 0 if the solution is exact and 1 if the solution is an approximate continuous solution. (This is 1 for IK branches where the desired pose is on the boundary or outside of the workspace.)

[Q, is_LS_vec] = SEW_IK.IK_2R_2R_3R(R_07, p_0T, SEW, psi, kin)

In this case, all 8 solutions are exact.

Double check solutions using forward kinematics

If we take each q and use forward kinematics to find the resultant end effector pose and SEW angle, we see all IK solutions are correct. All errors are zero to a tolerance of less than 1.0e-15.

i = 1;
[R_07_t, p_0T_t, P_SEW] = fwdkin_inter(kin, Q(:,i), [1, 3, 5]);
psi_t = SEW.fwd_kin(P_SEW(:,1), P_SEW(:,2), P_SEW(:,3));

R_07_t - R_07
p_0T_t - p_0T
wrapToPi(psi_t - psi)