This work was developed in the context of our MSc dissertations: A Collaborative Work Cell to Improve Ergonomics and Productivity by João Cunha, and Human-Like Motion Generation through Waypoints for Collaborative Robots in Industry 4.0 by João Pereira, in which we got to work with the collaborative robotic arm UR10 e-series.
Before producing our kinematics solution, we conducted a comprehensive literature review on the UR robots’ kinematics (which can be consulted in the "Articles" folder of this repository) and realised there was a lack of thorough and detailed analysis of their kinematics. Additionally, we found the Universal Robots’ documentation confusing and unclear.
Thus, the main goal of this work is to provide an explicit and transparent guide into the UR robots kinematics (using by reference the UR10 e-series) by expanding on the literature we found and describing every part of our analysis.
We present a forward kinematic solution based on the Modified Denavit-Hartenberg convention and an inverse kinematic solution based on a geometric analysis. The code is available in C++ and MATLAB, both with integration with CoppeliaSim.
The solution was built using Microsoft Visual Studio 2019, and C++ 20 standard.
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Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
In the context of this solution it is used to simplify the creation of 2D arrays, and matrix operations.
To use Eigen:
- Download the source code from their website;
- Extract the contents of the .rar to a directory of your choice (usually directly to your C:/ drive);
- On your project properties:
- C/C++ -> Additional Include Directories -> your Eigen directory e.g. C:\Eigen;
- now you can include
#include <Eigen\Dense>.
This isn't necessary if you only want to use the forward and inverse kinematics solutions. To run the CoppeliaSim integration, you have to install CoppeliaSim. To use their C++ remote API:
- Include the following files in your project directory: extApi.h, extApi.c, extApiPlatform.h, and extApiPlatform.c; which are located in CoppeliaSim's installation directory, under programming/remoteApi;
- Define
NON_MATLAB_PARSINGandMAX_EXT_API_CONNECTIONS=255(and optionallyDO_NOT_USE_SHARED_MEMORY) as a preprocessor definition; - Include additional directories remoteApi and include;
- If you aren't interested in using the secure version of their calls (like fopen_s), you need to place a definition of
_CRT_SECURE_NO_DEPRECATEbefore your included header files.
Benchmarking consists of:
- getting compute times of forward and inverse kinematics functions;
- getting the error of the solutions.
The solutions were tested for a set of 100000 randomly generated target tip poses (the scripts were run in a Ryzen 5 3600 CPU at 4.28GHz).
MATLAB's compute times are 10x slower than with C++, so the C++ solution is obviously recommended for use.
| MATLAB | C++ | |
|---|---|---|
| Forward Kinematics | 1.832571E-05 | 1.39434e-06 |
| Inverse Kinematics | 1.612797E-04 | 1.07403E-05 |
To compute the average error the following flowchart was followed:
| x | y | z | alpha | beta | gamma | |
|---|---|---|---|---|---|---|
| average error | 7.45E-14 | 1.86E-14 | 7.45E-14 | 4.14E-06 | 4.14E-06 | 4.14E-06 |
Units in: x, y, z metres, alpha, beta, gamma radians.