Denavit-Hartenberg transformation

Example of a kinematic chain based on a robot; with coordinate systems and DH parameters

The Denavit-Hartenberg transformation (DH transformation) from 1955 is a mathematical process that uses homogeneous matrices and the so-called Denavit-Hartenberg convention (DH convention) to transfer spatial coordinate systems (OKS) within kinematic Describes chains . Above all, it facilitates the calculation of the direct kinematics (forward kinematics) and is now considered the standard method , especially in the field of robotics .

DH convention

The following requirements are necessary:

1. the axis lies along the axis of movement of the nth joint ${\displaystyle z_{n-1}}$
2. the -axis is the cross product of -axis and -axis (= x ). ${\displaystyle x_{n-1}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle z_{n}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle z_{n}}$
3. the coordinate system is supplemented by the -axis so that it results in a right-handed system . ${\displaystyle y_{n}}$

For the first joint, the axis is taken from the second joint. ${\displaystyle x}$

DH transformation

The actual DH transformation from the object coordinate system (OKS) to the OKS consists of the following individual transformations being carried out one after the other: ${\displaystyle T_{n-1}}$ ${\displaystyle T_{n}}$

• a rotation (joint angle) around the -axis so that the -axis is parallel to the -axis ${\displaystyle \theta _{n}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle x_{n-1}}$ ${\displaystyle x_{n}}$

${\displaystyle \operatorname {Rot} (z_{n-1},\theta _{n})={\begin{pmatrix}\cos \theta _{n}&-\sin \theta _{n}&0&0\\\sin \theta _{n}&\cos \theta _{n}&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}}$

• a translation (joint distance) along the - axis to the point where and intersect ${\displaystyle d_{n}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle x_{n}}$

${\displaystyle \operatorname {Trans} (z_{n-1},d_{n})={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&d_{n}\\0&0&0&1\\\end{pmatrix}}}$

• a translation (arm element length) along the axis in order to bring the origins of the coordinate systems into congruence ${\displaystyle a_{n}}$ ${\displaystyle x_{n}}$

${\displaystyle \operatorname {Trans} (x_{n},a_{n})={\begin{pmatrix}1&0&0&a_{n}\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}}$

• a rotation (twisting) around the -axis in order to convert the -axis into the -axis ${\displaystyle \alpha _{n}}$ ${\displaystyle x_{n}}$ ${\displaystyle z_{n-1}}$ ${\displaystyle z_{n}}$

${\displaystyle \operatorname {Rot} (x_{n},\alpha _{n})={\begin{pmatrix}1&0&0&0\\0&\cos \alpha _{n}&-\sin \alpha _{n}&0\\0&\sin \alpha _{n}&\cos \alpha _{n}&0\\0&0&0&1\\\end{pmatrix}}}$

In matrix notation, the overall transformation then reads (to be interpreted from left to right):

${\displaystyle {\begin{aligned}{}^{n-1}T_{n}=&\operatorname {Rot} (z_{n-1},\theta _{n})\cdot \operatorname {Trans} (z_{n-1},d_{n})\cdot \operatorname {Trans} (x_{n},a_{n})\cdot \operatorname {Rot} (x_{n},\alpha _{n})\\&\\=&{\begin{pmatrix}\cos \theta _{n}&-\sin \theta _{n}\cos \alpha _{n}&\sin \theta _{n}\sin \alpha _{n}&a_{n}\cos \theta _{n}\\\sin \theta _{n}&\cos \theta _{n}\cos \alpha _{n}&-\cos \theta _{n}\sin \alpha _{n}&a_{n}\sin \theta _{n}\\0&\sin \alpha _{n}&\cos \alpha _{n}&d_{n}\\0&0&0&1\\\end{pmatrix}}\,.\end{aligned}}}$

The inverse of this matrix

${\displaystyle {\begin{aligned}{}^{n-1}T_{n}^{-1}=&\operatorname {Rot} (x_{n},-\alpha _{n})\cdot \operatorname {Trans} (x_{n},-a_{n})\cdot \operatorname {Trans} (z_{n-1},-d_{n})\cdot \operatorname {Rot} (z_{n-1},-\theta _{n})\\&\\=&{\begin{pmatrix}\cos -\theta _{n}&-\sin -\theta _{n}&0&-a_{n}\\\sin -\theta _{n}\cos -\alpha _{n}&\cos -\theta _{n}\cos -\alpha _{n}&-\sin -\alpha _{n}&-\sin(-\alpha _{n})(-d_{n})\\\sin -\alpha _{n}\sin -\theta _{n}&\sin -\alpha _{n}\cos -\theta _{n}&\cos -\alpha _{n}&\cos(-\alpha _{n})(-d_{n})\\0&0&0&1\\\end{pmatrix}}\\&\\=&{\begin{pmatrix}\cos \theta _{n}&\sin \theta _{n}&0&-a_{n}\\-\sin \theta _{n}\cos \alpha _{n}&\cos \theta _{n}\cos \alpha _{n}&\sin \alpha _{n}&-d_{n}\sin \alpha _{n}\\\sin \alpha _{n}\sin \theta _{n}&-\cos \theta _{n}\sin \alpha _{n}&\cos \alpha _{n}&-d_{n}\cos \alpha _{n}\\0&0&0&1\\\end{pmatrix}}\end{aligned}}}$

describes the transformation of a point from the OKS to the OKS . Correspondingly, the original matrix can also be interpreted as a transformation of a point from the OKS to the OKS if the position vector of the point is multiplied from the right of the matrix. It should be noted that the matrix multiplication is generally not commutative and thus the calculation sequence of the overall transformation is not interchangeable. ${\displaystyle T_{n}}$ ${\displaystyle T_{n-1}}$ ${\displaystyle {}^{n-1}T_{n}}$ ${\displaystyle T_{n-1}}$ ${\displaystyle T_{n}}$

The parameters and are also called Denavit-Hartenberg parameters . ${\displaystyle \theta _{n},d_{n},a_{n}}$ ${\displaystyle \alpha _{n}}$

With open kinematic chains , and are variable sizes during the movement of the robot, depending on its special geometry and dimensions. In the case of a rotary joint, it is variant and constant, in the case of a sliding joint, the opposite is true. and on the other hand, both rotational and sliding joints are invariant quantities and only need to be determined once for each individual arm element for the subsequent calculation of the direct kinematics . ${\displaystyle \theta _{n}}$ ${\displaystyle d_{n}}$ ${\displaystyle \theta _{n}}$ ${\displaystyle d_{n}}$ ${\displaystyle \alpha _{n}}$ ${\displaystyle a_{n}}$

literature

• Hans-Jürgen Siegert, Siegfried Bocionek: Robotics, programming of intelligent robots. Springer Verlag 1996, ISBN 3-540-60665-3 .
• Wolfgang Weber: Industrial robots, methods of control and regulation. Carl Hanser Verlag, Munich Vienna, 2009, ISBN 978-3-446-41031-2 .
• Jorge Angeles: Fundamentals of Robotic Mechanical Systems. Springer Verlag, New York, 1997, ISBN 0-387-94540-7 .
• Friedrich Pfeiffer, Eduard Reithmeier: Robot Dynamics. Teubner Verlag, Stuttgart, 1987, ISBN 3-519-02077-7 .
• Miomir Vukobratvic: Introduction to Robotics. Springer Verlag, Berlin, 1989, ISBN 0-387-17452-4 .
• John J. Craig: Introduction to Robotics, Mechanics and Control. Pearson Prentice Hall, NJ 07458, 2005, ISBN 0-201-54361-3 .

Web links

Commons : Denavit-Hartenberg-Transformation  - collection of pictures, videos and audio files

• A visualization for determining the Denavit-Hartenberg parameters is available in English at:
  • Denavit-Hartenberg Reference Frame Layout on YouTube
  • 1280x720 MPEG-4 (MP4; 49.8 MB),
  • 640x360 MPEG-4 (MP4; 19.2 MB)
• 3D visualization for determining the Denavit-Hartenberg parameters (German): Denavit-Hartenberg parameters 3D video tutorial for a KUKA industrial robot on YouTube
• Denavit-Hartenberg Parameters (English)